Hall viscosity and geometric response in the Chern-Simons matrix model of the Laughlin states
Matthew F. Lapa Taylor L. Hughes





















    [email address: ]lapa2@illinois.eduDepartment of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, 61801-3080Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, 61801-3080
        
        We study geometric aspects of the Laughlin fractional quantum Hall (FQH) states using a description of these states in terms of
        a matrix quantum mechanics model known as the Chern-Simons matrix model (CSMM). This model was proposed by 
        Polychronakos as a regularization of the noncommutative Chern-Simons theory description of the Laughlin states proposed 
        earlier by Susskind. Both models can be understood as describing the electrons in a FQH state as forming a noncommutative 
        fluid, i.e., a fluid occupying a noncommutative space. Here we revisit the CSMM in light of recent work on geometric response in 
        the FQH effect, with the goal of determining whether the CSMM captures this aspect of the physics of the Laughlin states. 
        For this model we compute the Hall viscosity, Hall conductance in a non-uniform electric field, and the Hall viscosity in the 
        presence of anisotropy (or intrinsic geometry). 
        Our calculations show that the CSMM captures the guiding center contribution to the known
        values of these quantities in the Laughlin states, but lacks the Landau orbit contribution. The interesting correlations in a  
        Laughlin state are contained entirely in the guiding center part of the state/wave function, and so we conclude that the CSMM 
        accurately describes the most important aspects of the physics of the Laughlin FQH states, including the Hall viscosity and
        other geometric properties of these states which are of current interest.
        
        Hall viscosity and geometric response in the Chern-Simons matrix model of the Laughlin states
            Taylor L. Hughes
            December 30, 2023
        =============================================================================================
        
        
        
        
        § INTRODUCTION
        
        
        
        In the past few years there has been tremendous progress in the understanding of the geometric properties of 
        quantum Hall states. The role of geometry in the quantum Hall effect first came to prominence in early work on 
        Hall viscosity <cit.> 
        (sometimes called odd viscosity), and there has been much work on Hall viscosity since 
        then <cit.>. 
        Recent work on geometry in the quantum Hall effect has gone in two separate directions. First, there is the study of 
        the properties of quantum Hall states on curved spatial manifolds 
        (Riemann surfaces) <cit.>. Second, there is the
        study of intrinsic geometry and anisotropy in quantum Hall states on 
        flat space <cit.>.
        In the past year a very interesting
        new theory of quantum Hall states has been proposed, known as the bi-metric theory, and this theory promises
        to unify the two separate directions of research on geometry in the quantum Hall effect <cit.>.
        
        In a separate line of development, Susskind proposed in 2001 that Laughlin fractional quantum Hall (FQH) states
        could be described by noncommutative Chern-Simons (NCCS) theory <cit.>. 
        This is a deformation of ordinary Chern-Simons theory in which the theory is formulated on a noncommutative analog of 
        the flat space ℝ^2 consisting of “coordinates" x̂^1 and x̂^2 obeying a nontrivial 
        commutation relation 
        
        	[x̂^1,x̂^2]=iθ .
        
        Here θ is a real parameter with dimensions of length squared that characterizes the degree of 
        noncommutativity of the theory. The original motivation for this proposal was Susskind's observation that the gauge
        symmetry of NCCS theory provides a discretization of the symmetry under area-preserving diffeomorphisms that is present
        in a description of a FQH state as a charged fluid in a magnetic field. In particular, this discretization was argued to capture
        the “granularity" of a fluid constructed from electrons, and in the NCCS theory description each electron is associated with a
        fundamental area of size 2π|θ|. In addition, in the NCCS theory a quantization rule <cit.> enforces 
        θ= ℓ_B^2 m , m∈ℤ ,
        
        where ℓ_B is the magnetic length, and so one finds (for m>0) that the fluid described by the NCCS theory has a number 
        density ρ_0=1/2πℓ_B^2 m, exactly as in the ν=1/m Laughlin state.
        
        Susskind's original proposal has been followed up by many 
        authors <cit.>. 
        Of all of these subsequent works, the
        work of Polychronakos is particularly important for this article. In Ref. P1, Polychronakos proposed a
        regularization of the NCCS theory which is known as the Chern-Simons matrix model (CSMM).
        This is a particular regularization of the NCCS theory which can be viewed as a quantum mechanics model in which the
        degrees of freedom are N× N matrices
        (we discuss the model in detail and make this statement precise below). The quantum ground state
        of the CSMM having θ=ℓ_B^2 m (m>0) is known to describe a uniform droplet of “noncommutative fluid" with 
        a density and area matching that of the ν=1/m Laughlin state. Polychronakos has also demonstrated that
        excitations in this model carry the appropriate fractional charge of the quasihole excitations in the Laughlin state.
        
        Despite the successes in describing the basic properties of the Laughlin FQH states using NCCS theory and the CSMM, 
        there have not been any attempts to study geometric properties of the Laughlin states in the context of these
        noncommutative models. Therefore, our goal in this article is to answer the following question: does the CSMM 
        accurately describe the geometric properties of the Laughlin states? 
        
        The particular geometric properties that we are
        concerned with are the Hall viscosity, the Hall conductance in a non-uniform electric field, and the
        Hall viscosity in the presence of anisotropy (or intrinsic geometry). 
        We compute all of these quantities in the CSMM and we find that the
        results in the CSMM contain only the guiding center contribution to the known values for these quantities in the
        Laughlin states. For example, the full Hall viscosity coefficient
        for the Laughlin ν=1/m state is given by <cit.>η_tot= ħρ_0 m/4 ,
         
        while for the CSMM with θ=ℓ_B^2 m we find[In the literature the quantity m-1/2 is referred
        to either as the anisospin (Refs. GGB,gromov-son) or minus the guiding center spin
        (Refs. haldane2009,haldane2011,park-haldane) of the ν=1/m Laughlin state.]
         (after regularization) 
        η_CSMM,reg= 1/2ħρ_0 (m-1/2) ,
        
        which is exactly the (regularized) guiding center Hall viscosity of the ν=1/m 
        Laughlin state <cit.>. The need for regularization of the
        guiding center part of the Hall viscosity has been discussed in Refs. haldane2009,haldane2011,park-haldane. 
        In this paper we also give a fluid
        interpretation of this regularization in the context of the CSMM.
        
        Based on our calculations we conclude quite generally that the CSMM and NCCS theory descriptions of the Laughlin 
        FQH states capture the guiding center contribution to the geometric properties of these states, but lack the Landau
        orbit contribution. We argue that this is not surprising since in the fluid interpretation of the CSMM and
        NCCS theories, the cyclotron frequency ω_c is sent to infinity by sending the mass of the particles in the fluid
        to zero. This is analogous to a projection into a Landau level (which freezes out the Landau orbit degrees of freedom), and so it 
        makes sense that only the guiding center contribution remains.
        The Landau orbit contribution is often considered to be less important since the interesting correlations in a Laughlin
        state are contained entirely in the guiding center part of the state/wave function. Therefore we find that the CSMM
        description is able to capture the most important contributions to the geometric properties of the Laughlin 
        states. We expect that our results will rekindle interest in noncommutative
        models of the FQH effect, as these models clearly have a role to play in the investigation of geometric properties of 
        FQH states.
        
        This paper is organized as follows. In Sec. <ref> we review the notion of Hall viscosity. 
        In Secs. <ref> and <ref> we give a comprehensive review of the NCCS theory and CSMM, the
        fluid interpretation of these models, and their relation to the Laughlin states. In Sec. <ref> we
        compute the Hall viscosity in the CSMM. In Sec. <ref> we compute the Hall conductance of 
        the CSMM in a non-uniform electric field. In Sec. <ref> we give a fluid interpretation of the regularization of the
        guiding center part of the Hall viscosity in which one subtracts the extensive contribution to this quantity. Finally, in 
        Sec. <ref> we present a modified version of the CSMM incorporating anisotropy, 
        and we compute the Hall viscosity for the modified model. Sec. <ref> presents our 
        conclusions. The paper also includes two appendices. 
        In Appendix <ref> we review the form of the quantum generators of the 
        action of the group U(N) on the fields of the CSMM, as this information is necessary for the quantization of this model which 
        we review in Sec. <ref>. In Appendix <ref> we present the details of the calculation of the Hall
        viscosity of the CSMM (which is presented in Sec. <ref> of the main text), 
        which involves a Kubo formula approach inspired by Ref. bradlyn2012.
        
        
        
        § REVIEW OF HALL VISCOSITY
        
        
        In this section we review the concept of Hall viscosity following the derivation and point of view in 
        Ref. park-haldane. We also emphasize, again following Ref. park-haldane, 
        the separation of the Hall viscosity tensor into two parts: the Landau orbit contribution and the guiding center
        contribution. Finally, we review the form of both parts of the Hall viscosity tensor for typical FQH trial states including the 
        Laughlin states. The example of the Laughlin states is of particular interest for the rest of the paper when we 
        compare to the results obtained in the CSMM, which has been argued to describe the physics of the Laughlin states.
        
        
        
         §.§ Hall viscosity calculation
        
        
        The Hall viscosity can be computed by studying the response of a FQH state to time-dependent area-preserving 
        deformations (APDs). Before we review the calculation of the Hall viscosity, we briefly recall 
        the setup of the quantum Hall problem. We consider N electrons on the plane, each with a charge 
        -e<0, in the presence of a constant background magnetic field of strength B >0 and pointing in the positive z direction.
        Let r_j be the position 
        coordinates of the N electrons, where j=1,…,N, is a particle label. We write r^a_j with a=1,2, for the 
        two components of the vector r_j (i.e., a=1,2, labels the two directions of space).
        In this situation the electron coordinate operators r^a_j break up into two parts as
        
        	r^a_j = R^a_j + R̃^a_j ,
        
        where R^a_j are known as the guiding center coordinates, and R̃^a_j are the Landau orbit 
        coordinates. These coordinates obey the commutation relations 
        [R^a_j, R^b_k]    =    iℓ_B^2 ^abδ_jk
        [R̃^a_j, R̃^b_k]    =    -iℓ_B^2 ^abδ_jk
        [R^a_j, R̃^b_k]    =    0 ,
        
        where ℓ_B^2= ħ/eB is the square of the magnetic length ℓ_B.
        
        The Hall viscosity is defined as the response of the system (more precisely, the ground state) to time-dependent
        APDs of the electron coordinates. These APDs are generated by Hermitian operators Λ^ab
        which are a linear combination of guiding center and Landau orbit parts,
        Λ^ab= Λ^ab-Λ^ab .
        
        The operators Λ^ab generate APDs of the guiding center coordinates and have the form
        Λ^ab= 1/4ℓ_B^2∑_j=1^N {R^a_j,R^b_j} ,
        
        where {·,·} denotes an anti-commutator, while Λ^ab generates APDs of the Landau orbit 
        coordinates, and Λ^ab is defined like Λ^ab but with
        the guiding center coordinates R^a_j replaced by the Landau orbit coordinates R^a_j.
        One can show that these generators obey the Lie algebras
        
            [Λ^ab,Λ^cd]    = i/2(^bcΛ^ad + ^bdΛ^ac + ^acΛ^bd+^adΛ^bc) 
            [Λ^ab,Λ^cd]    =  -i/2(^bcΛ^ad + ^bdΛ^ac + ^acΛ^bd+^adΛ^bc)  .
        
        In addition, it is clear that [Λ^ab,Λ^cd]=0.
        The generators Λ^ab (and also Λ^ab) can be expressed in terms of the generators of the Lie 
        algebra of the group SU(1,1), and we will
        use this fact later[Physicists can read about the group SU(1,1) in Ref. perelomov, for example].
        
        Finite (as opposed to infinitesimal) APDs of the electron coordinates are implemented by conjugation by 
        the unitary operators[Here, 
        and in the rest of the article, we use a summation convention in which we sum over all indices which are repeated once as
        a subscript and once as a superscript. All other summations will be indicated explicitly.]
        	U()= e^i_abΛ^ab ,
         
        where _ab is a constant, symmetric tensor with unit determinant (thus, the APDs are spatially uniform since _ab
        does not depend on the spatial coordinates). For example, acting on the electron
        coordinates gives
        
        	U()r^a_j U()^= r^a_j + ^ab_bcr^c_j + … ,
        
        where the ellipses denote higher order terms in _ab.  
        
        The APDs that we have been considering so far are closely related to strains in continuum mechanics. Suppose
        the vector r is the location of a point in a solid or fluid before a deformation, and r'(r) is the location of 
        that same point after the deformation. Then for small deformations the strain tensoru_ab is defined in terms of 
        the displacement vector u(r)= r'(r)- r as
        
        	u_ab= 1/2( u_a/ r^b +  u_b/ r^a) ,
        
        where u^a are the components of u(r) and u_a= δ_abu^b.
        If we consider small APDs in the quantum Hall problem (i.e., we work to linear order in _ab), then we find a strain
        tensor 
        
        	u_ab = 1/2( δ_ac^cd_db + (a↔ b) ) .
        
        In particular, we find for the trace ∑_a=1^2 u_aa= 0, which means that the APDs are indeed area-preserving (the
        trace of the strain tensor determines the change in the area of a small element of the fluid or solid at the
        location r). The strain
        tensor is also spatially uniform since _ab does not depend on the spatial coordinates r^a. Therefore, the
        APDs that we have been considering can be understood as a special case of a strain in continuum mechanics, namely, a
        spatially uniform and area-preserving strain. In what follows we sometimes use the terms APD and strain interchangeably
        although, strictly speaking, the former is a special case of the latter.
        
        
        Consider a FQH system described by a Hamiltonian H_0. 
        Under a time-independent APD parametrized by _ab the Hamiltonian is transformed to 
        
        	H()= U()H_0 U()^ .
        
        We can define the generalized force associated with this APD as
        
        	F^ab= - H()/_ab|_=0 = -i[Λ^ab,H_0] .
        
        If we subject the system to a time-dependent APD _ab(t), then we can compute the expectation
        value of F^ab in the time-dependent ground state |ψ(t) in an expansion in time derivatives of 
        _ab(t). In fact, as was argued in Ref. bradlyn2012, one should actually compute the expectation value of  
        U((t))F^abU((t))^ instead of F^ab. We discuss this point in more detail in the context of our
        Kubo formula calculation of the Hall viscosity for the CSMM in Appendix <ref>, but 
        just mention here that this replacement corresponds to expressing the generalized force in terms of the coordinates of the
        deformed system. 
        
        We now compute the expectation value of U((t))F^abU((t))^ in an expansion
        in time derivatives of _ab(t) as 
        
            ψ(t)| U((t))F^abU((t))^ |ψ(t)   = 
            ψ_0| F^ab|ψ_0   + Γ^abcd_cd(t) + … ,
        
        where |ψ_0 denotes the initial state of the system, the overdot on _cd(t) denotes a time derivative,
         and Γ^abcd is a four index tensor which is 
        independent of the parameters _ab(t) (in principle there could also be an elastic term which is proportional
        to _ab(t), but this term is not present for a fluid state). Park and Haldane then define the full Hall viscosity 
        tensor η^abcd_tot (with all indices raised) as 
        η^abcd_tot= Γ^abcd/A ,
        
        where A denotes the area of the quantum Hall droplet (recall that we are working on the infinite plane, so we must
        assume that the quantum Hall droplet occupies a finite area A). The 
        intuition behind this definition is that η^abcd_tot encodes the linear response of the “generalized stress"
        U((t))F^abU((t))^/A to the “rate of strain" encoded by the tensor _cd(t).
        We also note here that for a droplet of quantum Hall fluid the area A of the droplet can be expressed as 
        A= 2πℓ_B^2 N_ϕ, where N_ϕ is the number of fundamental flux quanta Φ_0= h/e piercing the 
        droplet.
        
        Using adiabatic perturbation theory, Park and Haldane showed that 
        
            η^abcd_tot   = iħ/Aψ_0|[Λ^ab,Λ^cd]|ψ_0
               =  iħ/Aψ_0|[Λ^ab,Λ^cd]|ψ_0+iħ/Aψ_0|[Λ^ab,Λ^cd]|ψ_0
               ≡η^abcd_H + η^abcd_H .
        
        Thus, the full Hall viscosity tensor breaks up into two parts: the guiding center Hall viscosity tensor 
        η^abcd_H, and the Landau orbit Hall viscosity tensor η^abcd_H.
        
        The expression for the full Hall viscosity tensor can be simplified further by using the algebra of APD generators 
        from Eq. (<ref>) to find
        η^abcd_tot = 1/2( ^acη^bd_tot + ^adη^bc_tot + (a↔ b)) ,
        
        where the symmetric two-index tensor η^ab_tot also breaks up into guiding center and Landau orbit parts as
        η^ab_tot= η^ab_H + η^ab_H
        
        with
        η^ab_H    =    -ħ/Aψ_0|Λ^ab|ψ_0
        η^ab_H    =   ħ/Aψ_0|Λ^ab|ψ_0 .
        
        In what follows we also refer to these two-index tensors as “Hall viscosity tensors".
        Ref. park-haldane emphasized that the guiding center contribution η^ab_H to η^ab_tot 
        has a physical interpretation in 
        terms of the intrinsic electric dipole moment along the edge of a FQH state, and in fact must be proportional to the
        symmetric tensor which determines this dipole moment in order to balance the force 
        on a FQH edge in an inhomogeneous electric field (see also Ref. wiegmann2012 for a complementary discussion of 
        this boundary dipole moment from a different point of view). We now review the form of the two parts of the 
        Hall viscosity tensor for typical FQH trial states including the Laughlin states.  
        
        
        
        
         §.§ Values in quantum Hall trial states
        
        
        In this section we consider the form of the guiding center and Landau orbit Hall viscosity tensors η^ab_H and
        η^ab_H for typical FQH trial states including the Laughlin states. In the operator, or 
        Heisenberg, approach (as 
        opposed to the Schrodinger approach using wave functions) a state vector for a trial FQH state is constructed using ladder operators b_j and b^_j defined in terms of the 
        guiding center coordinates as
        
        	b_j= 1/ℓ_B √(2)(R^1_j + iR^2_j) ,
        
        and also ladder operators a_j and a^_j defined in terms of the Landau orbit coordinates as
        
        	a_j= 1/ℓ_B√(2)(R^1_j - iR^2_j) .
        
        We define |0_a and |0_b to be the Fock vacuum states annihilated by the a_j and b_j operators, 
        respectively. In terms of these, a typical FQH trial state in the n^th Landau level has 
        the form
        
        	|ψ_0= [∏_j=1^N (a^_j)^n/√(n!)] F(b^_1,…,b^_N)|0_a ⊗ |0_b ,
        
        where F(b^_1,…,b^_N) is a homogeneous polynomial of N variables, and which is either symmetric (for 
        bosons) or antisymmetric (for fermions) under exchange of any two variables. We use Deg[F] to denote the total 
        degree of the polynomial function F. Then if we scale all arguments of F by a numerical factor λ, we have
        
        	 F(λ b^_1,…,λ b^_N)= λ^Deg[F] F(b^_1,…,b^_N) .
        
        Let 𝒩_b= ∑_j=1^N b^_j b_j be the total number operator for the N guiding center ladder 
        operators. Then the homogeneity property of F implies that |ψ_0 is an eigenvalue of 𝒩_b with 
        eigenvalue Deg[F].
        
        To compute η^ab_H for these trial FQH states we use a connection between the
        APD generators and the generators of the group SU(1,1) (see, for example, Ref. perelomov). 
        Define the operators
        
        	K_0    =   1/2∑_j=1^N(b^_j b_j + 1/2) 
        
        	K_+   =   1/2∑_j=1^N (b^_j)^2 
        
        	K_-   =   1/2∑_j=1^N (b_j)^2  .
        
        These operators obey the commutation relations of the Lie algebra of the group SU(1,1),
        
        	[K_0,K_±]    =   ± K_±
        [K_-,K_+]    =    2K_0 .
        
        The Fock space of the oscillators b_j forms a (reducible) 
        representation of this algebra, and the generators Λ^ab can be expressed in terms of the SU(1,1) generators
        as
        Λ^11   =    K_0 + 1/2K_+ + 1/2K_-
        Λ^22   =    K_0 - 1/2K_+ - 1/2K_-
        and
        Λ^12=Λ^21= -i/2(K_--K_+) .
        
        
        It is clear that the state |ψ_0 is an eigenstate of K_0 with eigenvalue
        1/2(Deg[F]+N/2). It then follows that the expectation values ψ_0 |K_±| ψ_0
        are equal to zero as K_±|ψ_0 is orthogonal to |ψ_0. Then, for the trial state
        parametrized by the function F, we have
        ψ_0|Λ^ab|ψ_0= 1/2[Deg[F]+N/2]δ^ab .
        
        A similar computation shows that for a trial state in the n^th Landau level we have
        ψ_0|Λ^ab|ψ_0 = 1/2(nN+N/2)δ^ab ,
        
        which follows since the product ∏_j=1^N (a^_j)^n/√(n!) is a homogeneous polynomial in the
        a^_j of total degree n N.
        
        For the case of the ν= 1/m Laughlin state (m a positive integer) we have
        
        	F(b^_1,…,b^_N)= ∏_j<k(b^_j - b^_k)^m ,
        
        and so
        Deg[F]= 1/2m N (N-1) .
        
        If we consider this Laughlin state in the lowest Landau level (n=0) then we find that
        
            ψ_0|Λ^ab|ψ_0   = 1/2[1/2m N^2 + ( 1-m/2)N]δ^ab
            ψ_0|Λ^ab|ψ_0   = N/4δ^ab ,
        
        and so
        η^ab_H= -ħ/A1/2[1/2m N^2 + ( 1-m/2)N]δ^ab ,
        
        while 
        η^ab_H= ħ/4N/Aδ^ab .
        
        Both of these tensors are proportional to the identity matrix (in this rotation-invariant case), and it is convenient to denote the 
        constants of proportionality by
        η_H= -ħ/A1/2[1/2m N^2 + ( 1-m/2)N]
        
        and 
        η_H=  ħ/4N/A
        so that we can simply write η^ab_H= η_H δ^ab and similarly for η^ab_H.
        
        For a Laughlin FQH droplet with ν=1/m, and consisting of a large number N of particles, we have 
        A≈ 2πℓ_B^2 m N.
        Then, in its current form, the coefficient η_H in the guiding center Hall viscosity tensor is the sum of an extensive 
        (order N) term and an intensive (order 1) term. Since A itself is proportional to N, the extensive term in η_H 
        comes from the superextensive (order N^2) term in ψ_0|Λ^ab|ψ_0. 
        This term is associated with a uniform rotational 
        motion (in fact, it is just the orbital angular momentum) of the FQH fluid, and
        so it has been argued that one should subtract this term when defining the guiding center Hall 
        viscosity <cit.>. If we make this subtraction then we end up with the regularized quantities
        ψ_0|Λ^ab|ψ_0_reg   =   1/2[( 1-m/2)N]δ^ab
        η_H,reg   =    -ħ/2( 1-m/2)ρ_0  ,
        
        where ρ_0= 1/2πℓ_B^2 m= N/A is the density of the ν=1/m Laughlin FQH state at large 
        N. We discuss the physical interpretation of this regularization scheme in the context of the CSMM
        in Sec. <ref>.
        
        The Landau orbit contribution η_H does not require regularization as it only consists of an intensive term. In terms
        of the density ρ_0 of the Laughlin state this coefficient has the form
        η̃_H= ħρ_0/4 .
        
        Then the full Hall viscosity coefficient for the ν=1/m Laughlin state (in the lowest Landau level and 
        after regularization of the guiding center part) is
        η_tot=η̃_H+η_H,reg= ħρ_0 m/4 ,
        
        as originally found by Read <cit.>. It is interesting to observe that since ρ_0=1/2πℓ_B^2 m, the
        full Hall viscosity coefficient η_tot actually does not depend on the filling fraction of the Laughlin state (i.e., it 
        does not depend on m).
        
        The coefficient 1-m/2 appearing in η_H,reg is what Haldane has termed the “guiding center spin"
        of a FQH state. This
        coefficient has been denoted as “s" in Ref. haldane2011 and “s" in Ref. park-haldane.
        It is also equal to minus the “anisospin" defined in Refs. GGB,gromov-son, and denoted there by 
        ς. We choose to adopt the notation of  Refs. GGB,gromov-son and so we write
        η_H,reg = ħ/2ςρ_0  
        with ς= m-1/2. We see that unlike the full Hall viscosity coefficient η_tot, the
        guiding center contribution to the Hall viscosity has a clear dependence on m.
        It follows that different Laughlin states cannot be distinguished by their full Hall viscosity η_tot, but
        they can be distinguished by their guiding center Hall viscosity η_H,reg which, moreover, has been argued to be
        connected to the physical property of intrinsic electric dipole moment at the edge of the FQH state <cit.>.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        § NONCOMMUTATIVE CHERN-SIMONS THEORY
        
        
        In this section we review Susskind's noncommutative Chern-Simons (NCCS) theory description of the Laughlin FQH 
        states <cit.>. This will pave the way for the discussion of the Chern-Simons matrix model in the next section,
        as the Chern-Simons matrix model can be thought of as a particular regularization of the NCCS theory. To prepare the reader 
        for this discussion in this section we first make a few remarks about the two different formulations (“operator" vs. 
        “star product" formulations) of noncommutative field theory. We then present the NCCS theory in both formulations. Finally,
        we discuss the NCCS theory in the limit of weak noncommutativity, and its connection with the dynamics of a fluid of 
        charged particles in a magnetic field. From this connection one sees that the full NCCS theory should be understood as
        describing a fluid of charged particles in a magnetic field on a noncommutative space. 
        Our discussion of noncommutative field
        theory closely follows that in Refs. DN,harvey,szabo. For the fluid picture of the NCCS theory we follow
        Refs. susskind,P2. Readers who are already familiar with noncommutative field theory and the NCCS theory
        may want to skip this section.
        
        
        
         §.§ Two formulations of noncommutative field theory
        
        
        Consider a classical field theory in 2+1 dimensions in which the two-dimensional space is taken to be ℝ^2, and
        let x= (x^1,x^2) denote the spatial coordinates. We denote a general field in this theory as
        Φ(t,x). In such a field theory the fields Φ(t,x) at a fixed time t are elements of the ordinary algebra of 
        functions on ℝ^2 (the commutative algebra generated by pointwise addition and multiplication of functions of
        x). The noncommutative deformation of the this theory that we consider consists of replacing
        the ordinary space ℝ^2 with a “noncommutative plane" whose two spatial coordinates do not commute with
        each other. The time direction will always be commutative in this article, i.e., we consider
        theories in two noncommutative spatial dimensions and one commutative (or ordinary) time direction. 
        
        In the noncommutative deformation of the classical field theory, the fields (again at a fixed time t) instead take values in the
        algebra ℝ^2_θ which consists of all complex linear combinations of products of position variables
        x̂^a, a=1,2, satisfying the commutation relation
        
        	[x̂^1,x̂^2]= iθ . 
        Here θ is a constant real number with dimensions of length squared; it controls the “strength" of the 
        noncommutativity of this theory. The algebra ℝ^2_θ comes equipped with a conjugation operator
        “" (which one can think of as Hermitian conjugation), and the operators x̂^a are assumed to be invariant
        under this operation[For any complex number c and any y∈ℝ^2_θ we have
        (c y)^= c y^, where c is the complex conjugate of c.]. We see that the algebra 
        ℝ^2_θ is nothing but the universal enveloping 
        algebra of the Heisenberg algebra specified by x̂^a and the commutation relation of Eq. (<ref>). 
        The operators x̂^a are sometimes said to be coordinates on a “noncommutative plane".  In the noncommutative 
        theory the notion of a point no longer makes sense, and the smallest area that one can resolve is of order θ. 
        
        In the noncommutative field theory, the notion of integration over space is replaced with a trace in a representation of
        the Heisenberg algebra of the noncommutative coordinates x̂^a. Usually this representation is taken to be the
        Fock representation in which the ladder operators
        â   =   1/√(2θ)(x̂^1 + ix̂^2) 
        â^   =   1/√(2θ)(x̂^1 - ix̂^2)
        
        act on a Fock space ℋ_F generated by the action of the raising operator â^ 
        on a vacuum state |0 which is annihilated by the lowering operator â. The action functional for the
        noncommutative field theory then takes the form
        
        	S = ∫ dt Tr_ℋ_F{ (⋯) }
        where (⋯) denotes a Lagrangian written in terms of fields Φ̂(t) which are
        operators on the space ℋ_F, and whose matrix elements are functions of time. 
        
        It is natural to call the formulation of noncommutative field theory that we have just described the “operator formulation." 
        We now describe an alternative formulation, which one might call the “star-product formulation," which may be more familiar 
        to some readers. In this formulation one instead works with fields Φ(t,x) which are ordinary functions of the 
        coordinates x on ℝ^2, but replaces the ordinary product  of functions with the 
        Groenewold-Moyal star product“⋆", which is defined as follows. For any two functions 
        f(x) and g(x) of x we have
        
        	f(x)⋆ g(x)    =    e^i/2θ^ab/ y^a/ z^af(y)g(z)|_y=z=x
           =    f(x)g(x)+ i/2θ^ab_a f(x)_b g(x) + … ,   
        and where in the last line the ellipses denote terms of order θ^2 and higher. For two functions f(x) and 
        g(x) which vanish at spatial infinity we have the important property that
        ∫ d^2x f(x) ⋆ g(x)= ∫ d^2x f(x)  g(x) ,
        
        which follows after integration by parts on the higher derivative terms in the star product. There is no analogous result for 
        integrals of star products of three or more functions.
        
        These two formulations of noncommutative field theory are related by the Wigner-Weyl mapping of functions and
        operators. This mapping is as follows. Let f(x) be an ordinary function on ℝ^2 and let
        f̃(k) = ∫ d^2x f(x) e^-i k_a x^a
        be its Fourier transform. Then we can define a Weyl-ordered operator f̂ by taking the inverse Fourier
        transform but replacing x^a with x̂^a in the exponential,
        f̂= ∫d^2k/(2π)^2 f̃(k) e^i k_a x̂^a . 
        One can check that this mapping satisfies the following properties which will be needed later: 
        f̂ĝ   =   f⋆ g
        Tr_ℋ_F{f̂}   =   1/2πθ∫ d^2 x f(x) .	
        
        To check the second property one can express the trace over ℋ_F using a basis {|x^1} of eigenstates
        of x̂^1 as 
        Tr_ℋ_F{f̂}= ∫_-∞^∞dx^1  x^1|f̂| x^1
        and then plug in the expression Eq. (<ref>) for f̂.
        
        The Chern-Simons matrix model that we study below is a particular regularization of the NCCS theory in its
        operator formulation. Therefore, for our purposes we generally find that the operator formulation of the
        NCCS theory is more convenient. However, the star product formulation is still useful for the study of the behavior of the 
        theory near the commutative limit θ→ 0, and so we will have occasion to use both formulations of the NCCS theory
        in what follows.
        
        
        
         §.§ NCCS theory in the operator formulation
        
        
        We now review the operator formulation of the NCCS theory. In the operator formulation, the
        NCCS theory consists of three fields
        X̂^a(t), a=1,2, and Â_0(t). All fields should be thought of as operators on the Fock space 
        ℋ_F whose matrix elements are functions of time. In addition, all fields are Hermitian (i.e., all fields are
        invariant under the “" operation on the algebra ℝ^2_θ). 
        We also consider the theory on a time interval of length
        T and assume periodic boundary conditions in time so that X̂^a(0)=X̂^a(T), and likewise for Â_0(t). 
        In addition to the noncommutativity parameter θ, the theory includes various coupling constants including e>0, an 
        electric charge, and B>0, a constant magnetic field. We discuss the physical interpretation of this theory as 
        representing a charged fluid in a magnetic field later in this section (and we will see that the charge of the particles which 
        make up this fluid is actually q=-e <0). 
        
        The action for the NCCS theory in the operator formulation takes the form
        
        	S_NCCS= -eB/2∫_0^T dt Tr_ℋ_F{_abX̂^a D_0X̂^b + 2 θÂ_0 } , 
        where we introduced a covariant derivative
        
        	D_0 X̂^b= Ẋ̂̇^b + i[X̂^b,Â_0] .
        
        and where the dot denotes a time derivative. The field Â_0 functions as a Lagrange multiplier and its equation of
        motion yields the constraint
        
        	[X̂^1,X̂^2]=iθ . 
        This constraint can only be satisfied by operators X̂^a on an infinite-dimensional space. This
        is due to the fact that if the variables X̂^a were finite-dimensional matrices, then the trace of the left-hand side of the
        equation is zero while the trace of the right-hand side would be proportional to the size of the matrices.
        The CSMM discussed in the next section is a modification of the NCCS theory which features a modified 
        constraint that can be satisfied by operators (matrices) on a finite-dimensional space.
        
        
        If we ignore the term containing 2θÂ_0 for a moment, then 
        one can check that the action is invariant under the gauge transformation
        X̂^a    →   V̂X̂^a V̂^
        Â_0    →   V̂Â_0 V̂^ + i V̂V̇̂̇^,
        
        where V̂(t) is an arbitrary time-dependent unitary operator on the Fock space ℋ_F. In particular,
        this follows from the fact that, under this transformation, the covariant derivative transforms as 
        D_0 X̂^b →V̂ D_0 X̂^b V̂^. To understand these gauge transformations in the presence of 
        the term 2θÂ_0, we need to constrain the allowed V̂'s that we consider <cit.>. To motivate 
        this restriction we now briefly discuss some aspects of the geometry of the noncommutative plane.
        
        Consider the occupation number basis {|n}_n∈ℕ of the Fock space ℋ_F 
        (|n∝ (â^)^n|0). The
        radius squared operator R̂^2= δ_abx̂^ax̂^b is diagonal in this basis and we have 
        R̂^2|n= 2θ(n+1/2)|n. Thus, the occupation number n can be
        identified with the distance squared from the origin in the noncommutative plane. We now restrict our attention to gauge 
        transformations defined by unitary operators V̂(t) which act as the identity 
        on states |n with n sufficiently large, say n>N_0. The actual value of N_0 is not important for the argument.
        This is the noncommutative analogue of requiring gauge transformations in a commutative gauge theory on
        the space ℝ^2 to tend to the identity at spatial infinity. 
        
        With this restriction on possible gauge transformations, the unitary operator V̂(t) defines a map from the 
        periodic time interval [0,T) to U(N_0), the group
        of unitary matrices of size N_0. Large gauge transformations are those V̂(t) which correspond to a nontrivial element
        of the homotopy group π_1(U(N_0))=ℤ. The full NCCS action is not invariant under these large gauge
        transformations because of the presence of the 2θÂ_0 term. 
        In Ref. PN, Polychronakos and Nair have 
        shown that requiring the exponential e^iS_CSMM/ħ to be invariant under these large
        gauge transformations enforces a 
        quantization rule on θ which states that
        
        	e Bθ= ħ m, m∈ℤ ,
        
        or
        θ= ℓ_B^2 m , m∈ℤ , 
        where ℓ_B^2= ħ/eB is the square of the magnetic length defined earlier. This quantization rule is the
        noncommutative analogue of the level quantization which obtains in ordinary (say SU(N)) Chern-Simons theory on a 
        commutative space.
        
        
        
         §.§ NCCS theory in the star product formulation
        
        
        We now discuss the NCCS theory in the star product formulation. In this form the theory looks very similar to the
        ordinary Chern-Simons theory (i.e., Chern-Simons theory on the commutative space ℝ^2). We proceed by deriving the star
        product formulation of the NCCS theory from the operator formulation by using the Wigner-Weyl mapping discussed earlier
        in this section. To do this we need to know how spatial derivatives are represented in the operator formulation of the
        theory. Derivative operators _a in the operator formulation of noncommutative field theory are defined by
        _1= ix̂^2/θ , _2=-ix̂^1/θ 
        and one can check that
        
        	[_a,x̂^b]=δ_a^b ,
        
        just as one has for ordinary derivatives of functions on ℝ^2. In addition, in the Wigner-Weyl mapping one has
        
        	[_a,f̂]= _a f ,
        
        so under this mapping the ordinary derivative of a function f(x) with respect to x^a is mapped to the commutator
        of _a with f̂ (i.e., the adjoint action of _a on f̂).
        
        The first step towards deriving the star product formulation of NCCS theory is to make a change of variables in the operator
        formulation by defining two new fields Â_a, a=1,2, which are related to the fields X̂^a by
        X̂^a= x̂^a + θ^abÂ_b .
        
        Under a gauge transformation the new fields transform as[This is derived by requiring the gauge transformation
        of x̂^a+θ^abÂ_b to coincide with the gauge transformation of X̂^a from 
        Eq. (<ref>).]Â_a→V̂Â_aV̂^ + iV̂[_a,V̂^] .
        
        This transformation resembles the transformation of an ordinary non-Abelian gauge field. In addition, in the
        new variables, the NCCS constraint of Eq. (<ref>) becomes
        F̂_12= 0 ,
        
        where we defined the field strength for noncommutative gauge theory as
        F̂_ab= [_a,Â_b]-[_b,Â_a]- i[Â_a,Â_b] . 
        
        Thus, the constraint in NCCS theory is an exact noncommutative analogue of the constraint enforced by
        the temporal component of the gauge field in ordinary Chern-Simons theory on a commutative space.
        
        After tedious algebra (including many uses of the cyclic property of the trace) 
        one can show that after performing this transformation the NCCS action takes the form
        
            S_NCCS = -eBθ^2/2   ∫_0^T dt Tr_ℋ_F{^abÂ_aÂ̇_b - ^abÂ_0 [_a,Â_b] 
            
             + ^abÂ_b   [_a,Â_0] + 2i/3^μνÂ_μÂ_νÂ_} ,
        
        where the Greek indices μ,ν, run over the range 0,1,2. There is one subtle point in the derivation of
        this equation which involves a term which is a total time derivative. Specifically, after the transformation from the 
        X̂^a variables to the Â_a variables one finds a term
        
        	-eB/2∫_0^T dt Tr_ℋ_F{ -θx̂^a Â̇_a} .
        
        Since x̂^a has no time dependence this term is a total derivative, and it evaluates to zero since we assumed periodic
        boundary conditions on all fields in the time direction. 
        
        Finally, we apply the Wigner-Weyl mapping to write the NCCS action in the star product formulation as
        
            S_NCCS= eBθ/4π∫_0^T dt ∫ d^2x    ^μν( A_μ⋆_νA_
            
            -   2i/3A_μ⋆ A_ν⋆ A_) .
        
        The quantization condition on θ (Eq. (<ref>)) then implies that the coefficient of the action is 
        eBθ/4π= ħ m/4π  .
        
        Then, in units where ħ=1, we find the NCCS action at level m∈ℤ. If we take ℓ_B^2→ 0, 
        which also sends θ→ 0, then we recover the ordinary U(1) Chern-Simons theory at level m (again
        with ħ=1 for now),
        
        	S_CS= m/4π∫_0^T dt ∫ d^2x ^μνA_μ_νA_ .
        
        For completeness we note here that in the star product formulation the noncommutative field strength is
        
        	F_μν= _μA_ν - _νA_μ - i(A_μ⋆ A_ν - A_ν⋆ A_μ) ,
        
        and the equation of motion of the NCCS theory is equivalent to F_μν=0, just like in ordinary Chern-Simons
        theory. 
        
        
        
         §.§ Fluid interpretation of the NCCS theory at small θ
        
        
        We now discuss the behavior of the NCCS theory in the limit of weak noncommutativity in which θ is assumed to 
        be small. Note that since θ has units, and since there is no other length scale in the problem to compare θ
        to, it is more accurate to say that in this section we study a truncation of the NCCS theory at first order
        in θ. In the star product formulation of the theory this truncation simply amounts to neglecting terms of order
        θ^2 and higher in the star product of functions.
        In this limit we will see that the NCCS theory has an 
        interpretation as describing a fluid of charged particles in 
        a constant magnetic field B, as was discussed by Susskind <cit.> (see also Refs. jackiw2002,P2). 
        
        To consider the NCCS theory in the regime of small θ we start by using the cyclic property of the trace
        to write the action in the form 
        
            S_NCCS= -eB/2∫_0^T dt    Tr_ℋ_F{_abX̂^aẊ̂̇^b 
            
            +    2Â_0 (θ + i[X̂^1,X̂^2])} .
        
        We then use the Wigner-Weyl mapping, and keep only the terms up to order θ in the star product, to find that in the
        limit of small θ
            S_NCCS→ -eB/21/2πθ   ∫_0^T dt ∫ d^2x ( _abX^aẊ^b 
            
            +    2θ A_0 (1-^ab_a X^1 _b X^2)) .
        
        
        
        
        
        Susskind observed that in this limit the NCCS theory describes the dynamics of a charged fluid at constant density
        ρ_0=1/2πθ in a 
        constant magnetic field B, and in the limit where the cyclotron frequency is sent to infinity. In fact, in Susskind's original 
        derivation he starts with the fluid description and then observes that it coincides with the small θ limit of the 
        NCCS theory. 
        
        We now briefly remind the reader of this connection between the NCCS theory and fluid dynamics.
        
        The starting point is the Lagrange description[The relation between noncommutative gauge theory
        and the Lagrange description of a fluid is discussed in detail in Ref. jackiw2002.] 
        of a fluid of charged particles moving on the plane ℝ^2 in a 
        background electromagnetic field. In the Lagrange
        description of a fluid one keeps track of the motion of the individual particles in the fluid, and measures their current position
        with respect to some reference configuration. In this description we use coordinates x to describe the reference
        configuration of the fluid and coordinates X^a(t,x), a=1,2, to describe the configuration of the fluid at a later time
        t. Without loss of generality, we may assume that X^a(0,x)=x^a. Thus, X^a(t,x) is the position, at time
        t, of the fluid particle which was at position x^a at t=0. We also assign a constant density ρ_0 to the fluid in 
        the reference configuration.
        
        The action for a Lagrange fluid made up of particles of mass M and charge q in the presence of a background 
        electromagnetic field takes the form
        
            S = ∫_0^T dt∫ d^2x ρ_0 (1/2M δ_abẊ^a   Ẋ^b + q 𝒜_a(t,X)Ẋ^a  
             
              -q    (t,X)  ) ,
        
        where 𝒜_a(t,X) and (t,X) are the vector and scalar potentials, respectively,
        for the external electromagnetic
        field. Intuitively, this action is just the sum over all particles in the fluid of the 
        ordinary action for a massive charged particle in a background electromagnetic field. However, the discrete sum over particle 
        labels has been replaced with an integration over the reference coordinates x weighted with the density ρ_0 in 
        the reference configuration. The reference coordinates x can therefore be considered as a set of continuous particle
        labels.
        
        To see the connection of the fluid model to the NCCS theory we first
        place the system in a uniform background magnetic field with strength B>0. 
        This can be accomplished by setting (t,X)=0 and
        𝒜_a(t,X)= -B/2_abX^b ,
        
        where we have chosen the symmetric gauge for the vector potential. Next, we set the mass of the particles to zero, 
        M=0. This corresponds to taking the cyclotron frequency ω_c=e B/M to infinity, which is similar to a 
        projection into the lowest Landau level (since ħω_c is the energy gap between Landau levels). Finally, we
        take the charge of the particles to be q=-e with e>0. Then at this point the action reads as
        
        	S= -e B/2ρ_0 ∫_0^T dt∫ d^2x _abX^aẊ^b .
        
        Note that ρ_0 can be pulled out of the integral since we assumed it was constant.
        We also mention here that
        our conventions for the direction of the magnetic field and the charge of the particles in the fluid exactly matches our 
        conventions for the setup of the quantum Hall problem from Sec. <ref>.
        
        
        
        
        
        
        The next step is
        to incorporate a Lagrange multiplier which enforces the constraint that the fluid remains at the constant density
        ρ_0 at all times. The density ρ(t,X) of the fluid at time t is related to the initial density ρ_0 by the 
        Jacobian ^ab_a X^1 _b X^2 of the map from the reference coordinates
        to the fluid coordinates X at time t as
        ρ(t,X) ^ab_a X^1 _b X^2= ρ_0 ,
        
        where we remind the reader that _a is a shorthand for / x^a, i.e., a derivative with respect to the
        reference coordinates x^a. Then the constraint that ρ(t,X)=ρ_0 for all t can be written as
        ^ab_a X^1 _b X^2=1.
        
        
        
        
        We denote the Lagrange multiplier enforcing this constraint by A_0(t,x), and write the action with the constraint
        included in the form 
        
            S= -e B/2ρ_0 ∫_0^T dt   ∫ d^2x (_abX^aẊ^b  
            
            +    2θ A_0 ( 1 - ^ab_a X^1 _b X^2)) ,
        
        where we have introduced a parameter θ with units of (length)^2. With this choice, the Lagrange multiplier field 
        A_0 has units of (time)^-1. 
        
        We can now see that the small θ limit of the NCCS action from Eq. (<ref>) is exactly the
        action for a fluid of particles with charge q=-e at the constant density ρ_0=1/2πθ in a constant 
        background magnetic
        field B in the limit in which the cyclotron frequency is taken to infinity. This limit is analogous to the projection into the
        lowest Landau level, and it is the physical reason why this fluid theory (and the NCCS theory) is expected to describe
        FQH physics in the lowest Landau level <cit.>. In the full NCCS theory we should then interpret the fields
        X̂^a(t) as describing the positions of particles in a fluid on a noncommutative space, as discussed by 
        Susskind <cit.> (see also Ref. P2 for a review of the physics of 
        such noncommutative fluids).  
        
        
        
        
        § THE CHERN-SIMONS MATRIX MODEL
        
        
        In this section we discuss the Chern-Simons matrix model (CSMM), which was introduced
        by Polychronakos in Ref. P1. This model can be thought of as a particular regularization of the 
        operator formulation of the NCCS theory, in which the fields X̂^a(t) (which were
        operators on the infinite-dimensional Fock space ℋ_F) are now finite N× N matrices X^a(t)
        instead. Note that we do not use a hatted notation for the finite size matrix variables of the NCCS theory. 
        The parameter N serves as a regulator which should be taken to infinity to recover the NCCS theory discussed
        in the previous section. The fluid interpretation of the NCCS theory carries over to the CSMM, so we still interpret the matrix
        variables X^a(t) as representing the coordinates of particles in a fluid on a noncommutative space, only now the fluid turns
        out to occupy a finite area of this space. In other words, the CSMM is a model of a finite droplet droplet of 
        noncommutative fluid.
        
        Since the CSMM can be difficult to understand, we begin this section by making a few
        remarks about our notation and conventions, and then discuss some subtleties of this model. 
        We then review the quantization of this model following Refs. P1,HVR. Finally, we review (following
        the original discussion in Ref. P1) the calculation 
        of the area A and density ρ_0 of the droplet of noncommutative fluid represented by the ground state of the CSMM.
        We will then be able to identify the CSMM having θ=ℓ_B^2 m as describing the ν=1/m Laughlin state
        by comparing the results for ρ_0 and A to the known answers for a droplet of FQH fluid in the ν=1/m
        Laughlin state in the limit of a large number of particles N. 
        
        
        
         §.§ Some remarks on notation
        
        
        The CSMM, and especially the quantization of this model, can be quite tricky due to two 
        separate noncommutative structures which appear. First, at the classical level the degrees of freedom in this model are 
        Hermitian N× N matrix variables X^1, X^2,
        A_0, as well as a complex vector Ψ of length N. All of these variables are functions of time. Since some of the 
        variables are matrix variables, ordinary (i.e., classical) matrix multiplication of these variables is 
        not commutative. Next, upon quantization of the model, the matrix elements of X^1, X^2, and A_0 
        (and also the components of Ψ) become
        operators on a separate Hilbert space, which is unrelated to the vector space on which the classical matrix variables 
        act. Thus, in the quantized matrix model there are two sources of noncommutativity. The
        first source is the fact that we are dealing with matrix variables from the start, and the second source comes from the
        fact that the matrix elements of the original matrix variables are now operators on a second Hilbert space, and so multiplication
        of individual matrix elements does not commute either, but for a different reason. 
        
        In an attempt to present this model in as clear a manner as possible, we will adhere to the following notational 
        conventions. First, we use [·,·]_M to denote a matrix commutator of classical matrices, and use 
        [·,·] (with no subscript) to denote the commutator of quantum operators. 
        We also reserve the symbol  to denote Hermitian conjugation of quantum operators. In all manipulations
        with classical matrix variables, we instead use an overline to denote complex conjugation of a matrix and a superscript `T' to 
        denote a transpose. So if A is an N× N matrix variable, then A^T is its transpose conjugate, i.e., if 
        A has matrix elements A_jk, then the matrix elements of A^T are (A^T)_jk=A_kj (and
        Hermitian matrices satisfy the relation A^T=A). As we 
        mentioned before, in the quantum theory the matrix elements A_jk are promoted to operators on a Hilbert
        space. We denote the Hermitian conjugate (with respect to the inner product on this Hilbert space) of the operator A_jk
        by A^_jk. Note that for a generic matrix variable A it is entirely possible that the operator A^_jk
        is not the same as the operator (A^T)_jk. In what follows we also make every effort to avoid using `i' as an index,
        and instead try to reserve it for the symbol meaning √(-1), and occasionally for the differential geometry operation 
        i_v of interior multiplication by a vector field v.
        
        
        
         §.§ Description of the model
        
        
        In this subsection we describe the CSMM of the Laughlin quantum Hall states <cit.>. 
        The degrees of 
        freedom in this model are two N× N matrices X^a(t), a=1,2, an N× N matrix A_0(t), and a complex
        vector Ψ(t) of length N. All degrees of freedom depend on time. The matrices X^a(t) and A_0(t) are all Hermitian
        and so they have real eigenvalues. The variables X^a are to be interpreted as 
        coordinates in the Lagrange description of a fluid on the noncommutative plane, in accordance with the physical ideas of 
        Susskind and Polychronakos <cit.>(and as we reviewed at the end of Sec. <ref>). 
        The number N will later be identified with
        the number of electrons in a Landau level. The action for the CSMM takes the form
        
            S_CSMM   = -eB/2∫_0^T dt Tr{_abX^a D_0 X^b + 2θ A_0 
            
            +    δ_abX^a X^b } + ∫_0^T Ψ^T(iΨ̇+ A_0 Ψ) ,
        
        where
        
        	D_0 X^b    :=   Ẋ^b + i[X^b,A_0]_M 
           =   Ẋ^b- i[A_0,X^b]_M
        
        is a covariant derivative. Here we view Ψ as a column vector and Ψ^T denotes the row vector whose
        elements are the complex conjugates of the elements of Ψ. 
        In addition, e and B are the same charge and constant magnetic field from Sec. <ref>,  is a 
        frequency (the term with  is a quadratic potential for the noncommutative coordinates X^a), and θ is a 
        parameter with units of length squared. We assume periodic boundary conditions on all the fields in the time direction,
        for example X^a(0)=X^a(T), so that the time direction is a circle of circumference T. Note that the action as written here 
        differs slightly in the details (signs, etc.) from Ref. P1, but is consistent with our interpretation of this model
        and the NCCS theory as describing a noncommutative fluid of particles with charge -e<0.
        
        At this point we would like to emphasize that the frequency ω appearing in the parabolic potential
        term of the CSMM has no relation to the
        cyclotron frequency ω_c=e B/M in the quantum Hall problem. Indeed, as we discussed in
        Sec. <ref>, the NCCS theory (and therefore the CSMM as well) describes a charged fluid in a magnetic field
        in the limit in which the mass M of the particles making up the fluid has been sent to zero. This sends the cyclotron
        frequency ω_c to infinity. Therefore, the CSMM contains no information related to the cyclotron frequency or
        the energy of a Landau level.
        
        We now discuss the gauge symmetry in the CSMM.
        If we ignore the term with 2θ A_0 for a moment, then we can see that the rest of the action is invariant under a
        U(N) gauge transformation
        
        	X^a    →    VX^a V^T 
        
        	A_0    →    VA_0 V^T + i V V̇^T 
        Ψ   →    VΨ ,
        
        where V(t) is an arbitrary time-dependent U(N) matrix. The presence of the term 2θ A_0 in the action means that
        the action is not invariant under large U(N) gauge transformations which are maps from [0,T) → U(N)
        which correspond to a nontrivial element in the homotopy group π_1(U(N))=ℤ. Since we would like 
        e^iS_CSMM/ħ to be invariant under any gauge transformation, these large gauge transformations enforce a 
        quantization rule on θ (the argument is identical to the argument for the full NCCS theory from Sec. <ref>) 
        which states that
        
        	e Bθ= ħ m, m∈ℤ ,
        
        or
        θ= ℓ_B^2 m , m∈ℤ . 
        
        
        The gauge field A_0 can be interpreted as a matrix Lagrange multiplier. If we look at the equation of motion resulting from a 
        variation of A_0, then we find that A_0 enforces the constraint
        
        	ieB[X^1,X^2]_M+eBθ𝕀 - ΨΨ^T= 0 . 
        This constraint should be compared with Eq. (<ref>) for the NCCS theory. In the NCCS case the
        contribution from the vector Ψ is absent and the constraint can only be realized by infinite-dimensional matrices (i.e.,
        operators on ℋ_F).
        It is the presence of the vector Ψ which allows this constraint to be 
        realized by finite-dimensional matrices, and this is why the CSMM can be thought of as a regularization of the NCCS
        theory. We refer the reader to Ref. P1 for the detailed analysis of the constraint in the classical solution of 
        the CSMM (which is also closely related to the Calogero model of interacting particles in one spatial dimension).
        In this paper our main focus is on the solution of the model in the quantum case. 
        
        We now make a few remarks and set up some notation relating to the transformation properties of the fields under the
        action of the group U(N). The field Ψ transforms in the fundamental representation of U(N). We indicate this
        by writing the components of Ψ with an upper Latin index, Ψ^j, j=1,…,N. Under a U(N) transformation we 
        have
        Ψ^j →V^j_k Ψ^k ,
        
        where V^j_k are the matrix elements of a unitary matrix V in U(N). Next, the transpose conjugate Ψ^T
        transforms in the anti-fundamental representation of U(N), Ψ^T→Ψ^TV^T. We indicate this
        by writing the components of Ψ^T with a lower index, Ψ_j, j=1,…,N (and recall that the components 
        of Ψ^T are just the complex conjugates of the components of Ψ). In components we have
        Ψ_j →Ψ_k (V^T)^k_j .
        
        Finally, the matrix variables X^a transform in the adjoint representation of U(N), X^a→ V X^a V^T. Thus, the
        index structure of X^a is such that it has one upper and one lower index, (X^a)^j_k, j,k=1,…,N. The component
        form of the U(N) transformation is then
        (X^a)^j_k →V^j_ℓ(X^a)^ℓ_m(V^T)^m_k .
        
        These conventions will be extremely useful later when we try to write down quantum states that respect the constraint
        of the CSMM. 
        
        We already mentioned that the matrix variables X^a are Hermitian matrices. Thus, their matrix elements (X^a)^j_k
        are generically complex numbers. For the quantization of this system it will be more convenient to parametrize X^a
        in terms of scalar variables which are manifestly real. Then, when we quantize the theory, these real variables will be
        promoted to Hermitian operators on the quantum Hilbert space. Our choice of parametrization is as follows. First, let 
        T^A, A=1,…,N^2-1, be the N× N generators, in the fundamental representation, of the Lie algebra of SU(N). 
        The matrices T^A are all Hermitian and traceless, and can be normalized to obey the relations
        Tr{T^A T^B}   =   δ^AB
        [T^A,T^B]_M   =    i ∑_C=1^N^2-1 f^ABC T^C ,
        
        where f^ABC are the structure constants for SU(N). These structure constants have a very important property which
        is that they are antisymmetric under exchange of any two indices A,B, or C (typically one only expects
        antisymmetry under A ↔ B). We will take advantage of this property later on. Using the generators
        T^A, we can parametrize X^a (for a=1,2) as
        
        	X^a(t)= x^a_0(t)𝕀/√(N) + ∑_A=1^N^2-1 x^a_A(t) T^A , 
        where x^a_0(t) and x^a_A(t), A=1,…,N^2-1, are N^2 real scalar variables. In the quantum theory these 
        variables will be promoted to Hermitian operators. The factor of √(N) on the identity matrix term was chosen for 
        convenience. 
        
        The Poisson brackets for this system can be obtained from the corresponding symplectic form, which can in turn be read off
        from the action (which is first order in time derivatives). The full symplectic form on the phase space for this system
        is
        Ω= Ω_X+Ω_Ψ
        with 
        Ω_X= -e B∑_A=0^N^2-1 dx^1_A∧ dx^2_A
        
        
        
        and
        Ω_Ψ= -i dΨ^j∧ dΨ_j .
        
        Our conventions for Poisson brackets are as follows. To any function f on phase space we associate a vector field
        v_f defined as the solution to the equation df= -i_v_fΩ. Then the Poisson bracket of any two functions
        f and g is given by {f,g}= i_v_fi_v_gΩ. Using this convention we obtain the classical
        Poisson brackets (with A,B=0,…,N^2-1 now)[The reader should beware that the symbol B is now being used
        for two purposes. It is the strength of the magnetic field felt by the noncommutative fluid described by the CSMM and NCCS 
        theory, and it is also (along with the capital Latin letters A,C,…) an index on the SU(N) generators 
        T^A and the variables x_A. It should be clear from the context whether B represents the magnetic field strength or an
        index.]{ x^1_A ,x^2_B}   =   1/eBδ_AB
        
        	{Ψ^j, Ψ_k}   =    -iδ^j_k .
        
        Upon quantization, in which we replace Poisson brackets with commutators as { f,g}→ -i/ħ[f,g], we find
        the commutation relations in the quantum CSMM to be
        
        	[x^1_A ,x^2_B]    =    iℓ_B^2δ_AB
        
        	[ Ψ^j, Ψ_k ]    =   ħδ^j_k ,
        
        where ℓ_B^2=ħ/eB is the magnetic length.
        
        Finally, when the gauge field A_0 is set to zero, the Hamiltonian for this system is given by
        
        	H_CSMM= eB/2Tr{δ_abX^a X^b} .
        
        All of the energy in the system is associated with the harmonic trap, and the only energy scale is associated with frequency
         of the harmonic trap. 
        
        We now review the quantization of this model.
        
        
        
         §.§ Quantization of the CSMM
        
        
        We now discuss the quantization of the CSMM. Instead of trying to solve the constraint before quantization, we follow
        previous approaches to this model and first quantize, then impose the constraint on quantum states, i.e., physical states should
        be annihilated by the constraint operator. As we discussed above, upon quantization the matrix elements of X^1
        and X^2 and the components of Ψ obey the quantum commutation relations from Eq. (<ref>).
        
        
        
        
        
        In what follows we instead work with the oscillator variables
        
        	b^j= 1/√(ħ)Ψ^j  ,
        
        with b^_j= 1/√(ħ)Ψ_j, and
        
        	a_A = 1/ℓ_B√(2)( x^1_A + i x^2_A) ,
        
        with a^_A = 1/ℓ_B√(2)( x^1_A - i x^2_A). These variables obey the commutation relations
        [a_A, a^_B]    =   δ_AB
        [ b^j, b^_k]   =   δ^j_k .
        
        
        The Hamiltonian for this system has the form
        
        	H_CSMM   =   eB/2δ_ab(X^a)^j_k (X^b)^k_j 
           =   eB/2∑_A=0^N^2-1δ_ab x^a_A x^b_A .
        
        In terms of the oscillator variables a_A and a^_A this becomes
        
        	H_CSMM= ħN^2/2 + ħ∑_A=0^N^2-1a^_A a_A .
        
        Note that the first term represents the zero point energy of N^2 harmonic oscillators.
        
        Next we turn to an analysis of the constraint. Classically, and in terms of the variables x^a_A, the constraint
        from Eq. (<ref>) takes the form
        
        	-eB ∑_A,B,C=1^N^2-1x^1_A x^2_B f^ABC T^C + eB θ𝕀 - ΨΨ^T= 0 .
        
        To interpret the constraint in the quantum theory we study its j,k matrix element
        
        	-eB ∑_A,B,C=1^N^2-1x^1_A x^2_B f^ABC(T^C)^j_k + eB θδ^j_k - Ψ^j Ψ_k = 0 .
        
        In terms of the oscillator variables one can show that this matrix element of the constraint takes the form 
        
            iħ/2∑_A,B,C=1^N^2-1(a^_A a_B + a_B a^_A)f^ABC   (T^C)^j_k  
            
            +     eB θδ^j_k - ħ b^j b^_k = 0 .
        
        Note that in deriving this expression we needed to use the antisymmetry of the structure constants f^ABC under 
        exchange of its indices. Finally, we use the commutation relations of the oscillator variables to rewrite this as
        
        	iħ∑_A,B,C=1^N^2-1a^_A a_B f^ABC(T^C)^j_k +  (eB θ-ħ) δ^j_k - ħ b^_k b^j = 0 ,
        
        where we used the fact that ∑_A,B=1^N^2-1δ_AB f^ABC=0. Note the shift in the coefficient of the
        δ^j_k term which resulted from this manipulation[In Ref. P1 Polychronakos instead
        performs normal-ordering of the constraint by making the replacementb^j b^_k → b^_k b^j. There is 
        then no shift of the coefficient of the δ^j_k term. This difference between normal-ordering the constraint vs. treating
        it as is completely accounts for the fact that Polychronakos found that the CSMM with θ=ℓ_B^2 m describes 
        the ν=1/m+1 Laughlin state, while we will find that it describes the ν=1/m Laughlin state (if
        we treated the constraint like Polychronakos then this would result in a trivial replacement of m→ m+1 in all results in this 
        article). Our treatment of the constraint is also identical to the treatment in Ref. tong2016, 
        which discusses new Chern-Simons matrix models which can describe non-Abelian FQH states (our m is equal to their
        k+1 for their model with p=1).]. Finally,
        we define G^j_k to be the j,k matrix element of the constraint, but divided by a factor of ħ for convenience,
        G^j_k= i∑_A,B,C=1^N^2-1a^_A a_B f^ABC(T^C)^j_k +  ( θ/ℓ_B^2-1) δ^j_k - b^_k b^j  . 
        
        In the quantum theory physical states |ψ will be those states which satisfy 
        G^j_k|ψ= 0 , ∀ j,k .
        
        To understand the form of the physical states |ψ we now analyze the constraint. First set j=k and sum over
        all j. Then the constraint implies that
        
        	b^_j b^j |ψ= N( θ/ℓ_B^2-1)|ψ .
        
        Now we already know that θ is quantized as an integer, θ=ℓ_B^2 m, m∈ℤ. If we take m>0,
        then this equation reads as
        
        	b^_j b^j |ψ= N(m-1)|ψ . 
        Thus, we find that the total number of b^j quanta in physical states must be equal to N(m-1). 
        
        Next, we consider the off-diagonal components of the constraint. For this it is convenient to instead consider 
        
        	G^A := G^j_k (T^A)^k_j ,
        
        which is the trace of the product of the constraint matrix (with elements G^j_k) and a generator T^A of SU(N).
        We find that these operators take the form
        
        	G^A= -i (𝒪_X(T^A) + 𝒪_Ψ(T^A) ) ,
        
        where 𝒪_X(T^A) and 𝒪_Ψ(T^A) are the quantum operators which generate the
        action of the SU(N) generator T^A on the X^a and Ψ variables, respectively. We define these operators and
        demonstrate their properties in Appendix <ref>. Thus, the set of constraints
        
        	G^A|ψ = 0 , A=1,…, N^2-1 
        simply expresses the fact that physical states must be singlets under the total SU(N) action, as originally noted by 
        Polychronakos <cit.>. 
        
        To summarize, we find that the constraint in the CSMM breaks up into two separate parts. The first is associated with the
        U(1) part of the total U(N) action and states that physical states |ψ obey Eq. (<ref>). 
        The second part is associated with the SU(N) part of U(N) and states that physical states should be 
        singlets under the SU(N) action. Now that we understand the constraint, we can write down a basis of physical
        states satisfying this constraint. To this end we introduce the matrix-valued operator[Perhaps a more precise notation for 
        this operator would be 
        A^= a^_0⊗𝕀/√(N) + ∑_B=1^N^2-1 a^_B⊗ T^B, which expresses
        the fact that A^ acts on the tensor product ℋ_Q⊗ℋ_N of an infinite-dimensional Hilbert 
        space ℋ_Q which arises upon quantization of the model, and an N-dimensional vector space ℋ_N 
        on which the classical matrix variables X^a act.]
        	A^= a^_0 𝕀/√(N) + ∑_B=1^N^2-1 a^_B T^B
        
        with matrix elements
        (A^)^j_k= a^_0 1/√(N)δ^j_k + ∑_B=1^N^2-1 a^_B (T^B)^j_k .
        
        Then, as was shown by Hellerman and Van Raamsdonk in Ref. HVR, 
        one possible basis for all physical states is given by states of the form
        
        	|{c_1,…,c_N}= Tr[(A^)^N]^c_N⋯Tr[A^]^c_1|ψ_0
        where each c_j ∈ℕ for j=1,…,N, and 
        
        	|ψ_0= [^j_1⋯ j_Nb^_j_1(b^A^)_j_2⋯(b^(A^)^N-1)_j_N]^(m-1)|0 .
        Note that all U(N) indices j,k, etc. are contracted in these expressions, and so every operator present is a
        singlet under the SU(N) action. The overall power of m-1 in |ψ_0 is required to satisfy the U(1) part of the
        constraint coming from Eq. (<ref>).
        
        Since the Hamiltonian of the CSMM just counts the total number of a_A quanta in a state, we find that
        |ψ_0 is the unique ground state of the CSMM, and that it has an energy
        
        	E_0    =   ħ[N^2/2 + 1/2(m-1)N(N-1)] 
           =   ħ[1/2mN^2 +(1-m/2)N] .
        
        The excited states |{c_1,…,c_N} then have an energy
        
        	E({c_1,…,c_N})= E_0 + ħ∑_j=1^N c_j j .
        
        It follows that the partition function of the CSMM at an inverse temperature β is 
        just
        
        	Z= Tr_Q[e^-β H_CSMM]= q^1/2mN^2 +(1-m/2)N∏_j=1^N1/1-q^j ,
        
        where Tr_Q[·] denotes a trace over the quantum Hilbert space (consisting of physical states obeying the
        constraint of the CSMM), and where we defined q= e^-βħ. As N→∞ the product
        ∏_j=1^N1/1-q^j becomes the partition function for the oscillator modes of a single chiral boson,
        which we know is the edge theory of a Laughlin fractional quantum Hall state. 
        
        
        
        
        
         §.§ Density of the droplet
        
        
        Here we review the calculation of the density of the FQH droplet described by the CSMM in the large N limit. We will
        see from this calculation that the CSMM with θ=ℓ_B^2 m corresponds to the  
        Laughlin state at filling fraction ν=1/m. We do not find ν=1/m+1 as we
        treated the constraint of Eq. (<ref>)as is instead of normal-ordering it as in Polychronakos'
        original paper <cit.>. 
        
        We compute the density of the droplet following the reasoning outlined by Polychronakos <cit.>. 
        The key is to examine the eigenvalue of the operator 
        Tr{δ_ab X^a X^b }= ∑_A=0^N^2-1δ_abx^a_A x^b_A
         
        in the ground state |ψ_0 of the CSMM (the trace here is a matrix trace). Since this operator is proportional
        to H_CSMM we have Tr{δ_ab X^a X^b }|ψ_0= R^2|ψ_0 where
        the eigenvalue R^2 is given by
        
        	R^2= 2ℓ_B^2(mN(N-1)/2 +N/2) . 
        We interpret this eigenvalue as a sum
        of contributions from N different particles at different radial positions by writing it as
        
        	R^2= ∑_j=1^NR^2_j ,
        
        where 
        
        	R^2_j= 2ℓ_B^2(m (j-1) + 1/2) .
        
        Indeed, the R^2_j can be thought of as the eigenvalues of the classical matrix δ_abX^a X^b, since the
        operator R^2 is equal to the trace of this matrix.
        Thus, we think of the ground state of the droplet as containing N particles at definite radial positions R_j but with
        complete uncertainty in their angular position. 
        
        
        In addition, since R^2_j is linear in j, the area π (R^2_j- R^2_j-1)=2πℓ_B^2 m of the annulus 
        between 
        consecutive particles is independent of j. This implies that the particles are distributed uniformly, i.e., the density is a 
        constant within the droplet.
        
        The size of the droplet is given by
        the largest value of R^2_j, which is
        
        	R^2_N = 2ℓ_B^2(m(N-1)+1/2)≈ 2ℓ_B^2 m N
        
        for large N. Then at large N we compute the density as being that of N particles evenly spread out over a disk
        of radius R^2_N≈ 2ℓ_B^2 m N, and so 
        ρ_0 = N/π R^2_N≈1/2πℓ_B^2 m ,
        
        which is exactly the density of the Laughlin state with filling fraction ν=1/m (in the limit of a large number N
        of electrons).
        
        
        
        § HALL VISCOSITY OF THE CSMM
        
        
        
        We now compute the Hall viscosity in the CSMM following the calculation of Park and Haldane <cit.> (which
        we reviewed in Sec. <ref>). We find that the Hall viscosity tensor contains only a single contribution,
        and that this contribution is equal to the guiding center Hall viscosity of the Laughlin state. In other words, the CSMM
        lacks the Landau orbit contribution to the Hall viscosity, but does contain the (physically important) guiding center contribution.
        
        To compute the Hall viscosity in this system we recall that in the fluid interpretation of the NCCS theory
        and the CSMM (which we reviewed at the end of Sec. <ref>), 
        the variables X^a represent a noncommutative analogue of fluid coordinates in 
        a Lagrange description of a fluid <cit.>. In the case of the CSMM, this is a finite droplet of noncommutative 
        fluid. Thus, to compute the Hall viscosity we first need to identify the quantum operators Λ^ab which 
        generate APDs (or strains) of the noncommutative fluid coordinates X^a. 
        Since we expand the noncommutative coordinates in terms of
        the scalar variables x^a_A, A=0,…,N^2-1, we can instead search for operators which implement APDs 
        of these variables. These operators will then automatically implement the correct transformations of the
        X^a coordinates, as the operators do not act on the matrix indices of the X^a variables. 
        
        Since the commutation relations of the variables x^a_A are identical to the commutation relations of the 
        guiding center coordinates in the quantum Hall problem, we immediately see that the desired operators are given by
        Λ^ab=1/4ℓ_B^2∑_A=0^N^2-1{ x^a_A, x^b_A} .
        	
        
        These operators obey the same algebra as in Eq. (<ref>). It follows that the unitary operators which 
        implement the APDs are U()= e^i_abΛ^ab, with _ab a constant
        symmetric matrix. To first order in _ab we have (for all A=0,…,N^2-1)
        
        	U() x^a_A U()^= x^a_A + ^ab_bcx^c_A+⋯
        which implies (for all j,k)
        
        	U() (X^a)^j_k U()^= (X^a)^j_k + ^ab_bc(X^c)^j_k+⋯
        
        It is important to note that the APD generators Λ^ab act only on the physical position indices a of the
        variables X^a. There is no action at all on the U(N) indices j,k of the matrix elements (X^a)^j_k. Thus,
        the generators Λ^ab act identically on all matrix elements of X^a, and so they are indeed the correct 
        quantum generators of APDs of the noncommutative fluid coordinates X^a (which we recall are actually 
        N× N Hermitian matrices in the classical theory).
        
        Now we want to compute the Hall viscosity in the ground state |ψ_0 of the CSMM. We compute
        this using a Kubo formula approach similar to that of Ref. bradlyn2012. 
        We present the Kubo formula calculation of the Hall viscosity in Appendix <ref>. Our result is that
        the Hall viscosity tensor in this model takes the form (A is the area of the droplet)
        η^abcd_CSMM= iħ/Aψ_0| [Λ^ab,Λ^cd]|ψ_0 .
         
        We note that the tensor η^abcd_CSMM contains only a single contribution, as opposed to the 
        two separate terms (guiding center and Landau orbit contributions) appearing in the discussion of the Hall viscosity tensor from 
        Sec. <ref>.
        Note that in deriving this result it was crucial
        that the CSMM has a unique ground state and a finite energy gap set by the frequency  of the harmonic trap. 
        
        Due to the commutation relations of the generators Λ^ab (which are the same as 
        Eq. (<ref>)), the four index tensor η^abcd_CSMM
        can again be expressed in terms of a symmetric two-index tensor
        η^ab_CSMM = -ħ/Aψ_0|Λ^ab|ψ_0 .
        
        Therefore, to compute the Hall viscosity tensor of the CSMM, we just need to compute the expectation values 
        ψ_0|Λ^ab|ψ_0. To compute these we first note that the CSMM Hamiltonian can be
        written as
        
        	H_CSMM= ħδ_abΛ^ab= ħ (Λ^11+Λ^22) .
        
        From this we can already deduce that
        ψ_0|δ_abΛ^ab|ψ_0= E_0/ħ= 1/2mN^2 +(1-m/2)N .  
        We can go further and compute the individual expectation values of Λ^11 and Λ^22 
        by deriving a Virial theorem for the CSMM. To derive this theorem consider the operator
        
        	Q= ∑_A=0^N^2-1 x^1_A x^2_A .
        
        A short computation shows that
        
        	[Q,δ_abΛ^ab]= 2iℓ_B^2(-Λ^11+Λ^22) .
        
        If we take the expectation value of this equation in the state |ψ_0 (or any eigenstate of 
        δ_abΛ^ab), then we find that
        ψ_0|Λ^11|ψ_0= ψ_0|Λ^22|ψ_0 .
        
        Combining this result with Eq. (<ref>) gives the result that
        ψ_0|Λ^11|ψ_0= ψ_0|Λ^22|ψ_0= 1/2[1/2mN^2 +(1-m/2)N ] .
        
        
        Finally, it remains to compute the expectation value of the off-diagonal generator 
        Λ^12=Λ^21. 
        In terms of the oscillator variables a_A and a^_A this operator takes the form
        Λ^12= 1/4i∑_A=0^N^2-1( a_A a_A - a^_A a^_A ) .
        
        Now all eigenstates of H_CSMM are eigenstates of the total number operator for the a_A oscillators. Since
        Λ^12 clearly does not commute with the total number operator, we immediately conclude that 
        the expectation value of Λ^12 in any eigenstate of H_CSMM is zero. 
        
        Therefore our final result for the expectation value of the APD generators 
        Λ^ab in the CSMM ground state is
        ψ_0|Λ^ab|ψ_0=  1/2[1/2mN^2 +(1-m/2)N ]δ^ab .
        
        This means that we can write η^ab_CSMM = η_CSMMδ^ab, where
        the coefficient η_CSMM of Hall viscosity in this model is equal to 
        η_CSMM = -ħ/A1/2[1/2m N^2 + ( 1-m/2)N] .
        
        Now since A= π R_N^2≈ 2πℓ_B^2 m N for the CSMM at large N, this exactly matches the result (before 
        regularization) for the guiding center Hall viscosityη_H of the  ν=1/m 
        Laughlin state. The Landau orbit contribution η_H is absent in the CSMM. 
        Finally, as was the case for the ordinary Laughlin state, 
        this result can be regularized by subtracting off 
        the extensive term in η_CSMM 
        (or the superextensive term in ψ_0|Λ^ab|ψ_0).
        We discuss a fluid interpretation of this regularization of the Hall viscosity later in Sec. <ref>.
        
        
        
        
        § HALL CONDUCTANCE OF THE CSMM IN A NON-UNIFORM ELECTRIC FIELD
        
        
        In this section we study the Hall conductance of the CSMM when it is subjected to a non-uniform electric field. Our motivation
        for studying this setup is the well-known result of Hoyos and Son which shows that in a quantum Hall state 
        the Hall conductance σ_H(k) at finite wave vector k has a universal contribution at order k^2 
        (k^2= δ^abk_a k_b) which is related to the Hall viscosity <cit.> (see also Ref. bradlyn2012
        for a Kubo formula approach to this relation). 
        We find a similar contribution in the CSMM, but depending only on the guiding center Hall viscosity as opposed to the
        full Hall viscosity. Again, this is not surprising as we only expect the CSMM to describe the dynamics
        of the guiding center degrees of freedom in a FQH state.
        
        In this section we first review the result of Ref. hoyos-son on the Hall conductance at finite wave vector. We
        then warm up by calculating the Hall conductance of the CSMM subjected to a uniform electric field. The reason for
        this is that there are several subtle points associated with the computation of the Hall conductance in the CSMM that we
        want to explain clearly. Finally, we compute the Hall conductance of the CSMM in a 
        non-uniform electric field, where we find a result which resembles the result of Hoyos and Son <cit.>, but 
        with the full Hall viscosity replaced by the guiding center Hall viscosity. We note here
        that the Hall conductance of the NCCS theory in a uniform electric
        field was computed previously in Refs. hansson2001,hansson2003 at the classical level by solving the 
        equations of motion for the NCCS theory in a uniform electric field. We therefore emphasize that our treatment in this section 
        deals directly with the quantized CSMM theory as opposed to the classical NCCS theory. 
        
        
        
         §.§ The result of Hoyos and Son
        
        
        We start by reviewing the result of Ref. hoyos-son. Consider a quantum Hall system in a non-uniform
        electric field E= (E(x),0) pointing in the x^1 direction, and where the
        spatial dependence is only on the x^1 coordinate, so that _2 E(x)=0. 
        The Hall conductance σ_H(k) at finite wave vector is defined by the relation
        
        	j^2(k)= σ_H(k) E(k) ,
        
        where j^2(k) is the Fourier transform of the charge current in the x^2 direction, and E(k) is the
        Fourier transform of E(x). The result of  Ref. hoyos-son is that (recall that E(x) is a function 
        of only x^1)
        σ_H(k)/σ_H(0)= 1 + C_2 (k_1 ℓ_B)^2+⋯ ,
        
        where the Hall conductance at zero wave vector is simply (ν is the filling fraction)
        σ_H(0)= νe^2/h .
        
        The coefficient C_2 is given by
        
        	C_2=η_tot/ħρ_0 - 2π/νℓ_B^2/ħω_cB^2ℰ”(B) , 
        
        where η_tot denotes the full Hall viscosity of the quantum Hall state (as opposed to just the guiding center
        part), ℰ(B) is the energy density of the quantum Hall state viewed as a function of the external field
        B, and ℰ”(B) denotes the second derivative of ℰ(B) with respect to B. In addition,
        ρ_0 denotes the number density of the quantum Hall state, and ω_c=eB/M is the cyclotron frequency,
        where M is the mass of the particles making up the quantum Hall state. As an example, for a 
        quantum Hall state consisting of N electrons in the lowest Landau level and occupying an area A, we have
        ℰ(B)= ħω_c/2N/A= ħω_c/2ρ_0, and for a Laughlin 
        ν=1/m state this gives ℰ(B)= ħω_c/4πℓ_B^2 m. 
        
        In the context of the CSMM, the quantity that we actually compute is the current at the location of the center of mass
        of the droplet (we explain the reason for this in the next subsection). 
        Therefore we need to Fourier transform the result of Hoyos and Son back to real space in order to compare
        with our calculation in the CSMM later in this section. In real space we find that
        
        	j^2(x)= νe^2/h( E(x) - C_2 ℓ_B^2 ^2_1E(x) + …) .
        
        In particular, at the origin x=0 (where the center of mass of a uniform droplet would be located) we have
        
        	j^2(x=0)= νe^2/h( E^(0) - C_2 ℓ_B^2 E^(2) + …) , 
        where E^(0) and E^(2) are the coefficients in the Taylor series expansion of E(x) about the
        origin,
        
        	E(x)= E^(0) + E^(1) x^1 + 1/2! E^(2) (x^1)^2 + … ,
        
        and where we again remind the reader that we assumed that E(x) has no x^2 dependence. 
        
        
        
         §.§ Uniform electric field
        
        
        We now compute the Hall conductance of the CSMM in a uniform electric field. Our reason for treating this simple
        case first is to highlight a few subtleties in the calculation of the Hall conductance of the CSMM. The first subtlety is 
        associated with the fact that one cannot resolve individual points in space in the CSMM, since the spatial coordinates
        are actually the noncommuting matrices X^1 and X^2. However, in the CSMM one can still define a notion of the
        center of mass coordinate of the FQH droplet, and the expectation value of this center of mass coordinate can be
        computed in any state |ψ of the quantized CSMM. We define the center of mass coordinates X^a_COM
        as
        
        	X^a_COM= 1/NTr{X^a} = x^a_0/√(N) ,
        
        where in the second equality we evaluated the trace and found that X^a_COM is proportional to the variable
        x^a_0 introduced in Eq. (<ref>) of Sec. <ref>.
        To motivate this definition we simply note that if the X^a were diagonal matrices, then their diagonal elements could 
        be interpreted as the positions of N particles, and then 1/NTr{X^a} would agree with the usual
        definition of the center of mass coordinate of N particles (assuming all particles have equal masses).
        
        Our strategy to compute the Hall conductance in the CSMM is to compute the drift velocity v_drift of the
        center of mass coordinate when the system is placed in an electric field E. We can then use the fact that the
        CSMM describes a droplet of particles with charge -e and density ρ_0=1/2πℓ_B^2 m (computed in 
        Sec. <ref>) to 
        compute the charge current j_COM at the center of mass as
        j_COM = -eρ_0 v_drift .
        
        The result can then be compared with the result of Hoyos and Son for the current at the origin (location of the center of mass)
        as expressed in Eq. (<ref>).
        
        
        
        Next, we need to discuss the issue of how to couple the CSMM to an external electric field. This can be done using
        the fluid interpretation of this theory from Sec. <ref>. First, recall from Sec. <ref> that
        an ordinary charged fluid on commutative flat space ℝ^2 can be coupled to a background electromagnetic
        field by including vector and scalar potentials 𝒜_a(t,X) and (t,X), respectively, in the action for 
        the Lagrange description of this fluid, Eq. (<ref>). In our case we are only interested in
        adding a scalar potential (t,X) for the external electric field. Using the fluid interpretation 
        we can incorporate this potential into the NCCS theory by adding a term to the NCCS action of the form
        
        	S_EM= e∫_0^T dt Tr_ℋ_F{(X̂,t)} ,
        
        where the operator (X̂,t) is the operator representing the scalar potential for the
        external electromagnetic field (and recall that the charge of the particles is q=-e).
        
        In defining the operator (X̂,t) we encounter an ordering ambiguity. For example
        if the scalar potential for the electric field configuration on a commutative space is
        (t,X)= X^1X^2, then we could define (X̂,t)= X̂^1X̂^2, 
        (X̂,t)= X̂^2X̂^1, or the symmetric Weyl ordering 
        (X̂,t)= 1/2(X̂^1X̂^2+X̂^2X̂^1), for example. 
        We choose
        to use Weyl ordering since this is consistent with our use of Weyl ordering to go between star product
        and operator formulations of noncommutative field theory (recall Eq. (<ref>)), however, in
        the examples of this section we do not actually encounter this ordering ambiguity. Weyl-ordering for the
        external field was also adopted by the authors of Ref. hansson2001, 
        who also considered the NCCS theory in the presence of external fields. 
        
        Finally, to couple the CSMM to the
        external electromagnetic field we use the same action S_EM as above but replace the operators X̂^a
        on the infinite-dimensional space ℋ_F with the finite N× N matrix variables of the CSMM. From this
        action we can then read off the new Hamiltonian for the CSMM coupled to the external electric field.
        
        
        There is one more subtlety with the calculation of the Hall conductance of the CSMM that we need to address before
        we can proceed. The issue is that the parabolic potential in the CSMM competes with the applied electric field to determine
        the long time behavior of the CSMM in the presence of the electric field. This is best illustrated for the case of the 
        CSMM in a constant electric field E^(0) pointing in the x^1 direction. The Hamiltonian describing this system is
        
        	H'    =    H_CSMM + e E^(0)Tr{X^1}
           =    H_CSMM + e E^(0)N X^1_COM , 
        and where the trace is a classical matrix trace. To derive this Hamiltonian we used the
        fluid interpretation of the CSMM theory and incorporated a scalar potential (t,X)= -E^(0)X^1 to describe
        the coupling to a constant electric field in the x^1 direction. 
        This Hamiltonian can be immediately diagonalized by noting that
        
        	H' = T(R)H_CSMM T(R)^ - e N (E^(0))^2 /2 B ,
        
        where T(R) is a unitary translation operator[We have
        [X^a_COM,X^b_COM]= iℓ_B^2/N^ab and 
        T(R)X^a_COM T(R)^= X^a_COM+ R^a.] (similar to a magnetic 
        translation) of the form
        
        	T(R)= exp{-i _ab N X^a_COM R^b/ℓ_B^2} ,
        
        and where in this case
        R= (E^(0)/B,0) .
        
        The ground state of this Hamiltonian is |ψ_0'= T(R)|ψ_0 and represents a stationary state
        with ψ_0'|X^1_COM|ψ_0' = -E^(0)/B and 
        ψ_0'|X^2_COM|ψ_0'=0, which corresponds to the equilibrium position in the total potential
        
        	V= eBN/2δ_abX^a_COMX^b_COM + e E^(0)N X^1_COM
        felt by the center of mass. 
        
        We see that if we simply diagonalize the Hamiltonian H' for the CSMM in the 
        presence of the external field, we find
        no time dependence and, in the ground state, the center of mass of the droplet just sits at its equilibrium position 
        (-E^(0)/B,0) under the influence of the combined forces of the parabolic potential and the 
        applied electric field. 
        
        To compute the Hall conductance of this model we instead need to consider a non-equilibrium situation in which we start
        with the system in the ground state |ψ_0 of the unperturbed CSMM (which we will now assume is properly 
        normalized) and then suddenly turn on the 
        electric field. We then study the time evolution of the center of mass coordinate at small times 
        t≪1/, where 1/ is the time scale set by the parabolic potential. Therefore
        we consider the “quantum quench" problem in which the state of the system at time t is given by
        
        	|ψ(t)= e^-iH't/ħ|ψ_0 ,
        
        where |ψ_0 is the ground state of the unperturbed CSMM Hamiltonian H_CSMM, and H' is the perturbed
        CSMM Hamiltonian including the applied electric field. We then compute
        
            ψ(t)|X^a_COM|ψ(t)    = ψ_0|X^a_COM|ψ_0
            
             + it/ħ   ψ_0|[H',X^a_COM] |ψ_0 + …
        
        and identify the drift velocity v_drift of the center of mass with the term linear in t in this expansion,
        
        	v^a_drift= i/ħψ_0|[H',X^a_COM] |ψ_0 . 
        
        We now consider the case of a uniform electric field E^(0) pointing in the x^1 direction so that H' takes the form 
        shown in Eq. (<ref>). In this case the drift velocity evaluates to 
        v_drift= (0,-E^(0)/B) .
        
        Then the non-zero part of the charge current at the center of mass of the droplet, at times t≪1/, is
        
        	j^2_COM   =    eρ_0E^(0)/B
           =   νe^2/h E^(0) ,
        
        with ν=1/m, and where we used ρ_0=1/2πℓ_B^2 m. Therefore we find that the
        Hall conductance of the CSMM with θ=ℓ_B^2 m is given by
        σ_H= 1/me^2/h ,
        
        exactly as in the ν=1/m Laughlin state.
        
        For the case of a uniform electric field we can actually go further and compute the full time dependence of the center of mass
        coordinate. We find that
        ψ(t)|X^1_COM|ψ(t)   =   E^(0)/B( -1+cos( t) ) 
        ψ(t)|X^2_COM|ψ(t)   =    -E^(0)/Bsin( t) .
        
        We see that the center of mass moves in a large circle about its equilibrium position (-E^(0)/B,0), but
        that at early times t≪1/ the droplet drifts in the x^2 direction with velocity vector
        v_drift= (0,-E^(0)/B).
        
        
        
        
        
        
         §.§ Non-uniform electric field
        
        
        We now compute the Hall conductance of the CSMM in a non-uniform electric field. We consider an electric field which
        points in the x^1 direction, and which depends only on the x^1 coordinate. Since we are interested in contributions
        to the current which depend on the second derivative of the electric field, it is sufficient to consider an electric field which
        depends at most quadratically on the x^1 coordinate. Thus, for an ordinary classical charged fluid
        described by the action of Eq. (<ref>), we would add a scalar potential of the form
        (t,X)= -E^(0)X^1 - 1/2E^(1)(X^1)^2- 1/3!E^(2)(X^1)^3 ,
        
        which corresponds, after computing minus the spatial gradient, to an electric field E=(E(X),0) with
        
        	E(X)= E^(0) + E^(1)X^1 + 1/2E^(2)(X^1)^2 .
        
        The coefficients E^(j), j=0,1,2 in this expression (which are all fixed real numbers) can be understood as the 
        coefficients in the Taylor expansion of E(X) about the origin. 
        
        This form of the scalar potential for the ordinary classical fluid, combined with the considerations from earlier in this section
        on how to couple the CSMM to external fields, leads to a Hamiltonian 
        
        	H'= H_CSMM + H_1
         
        with
        
        	H_1= eTr{ E^(0)X^1 + 1/2E^(1)(X^1)^2+ 1/3!E^(2)(X^1)^3 } ,
        
        where the trace denotes a matrix trace. This Hamiltonian then describes the CSMM in the presence of a non-uniform 
        electric field in the x^1 direction. 
        To compute the Hall conductance we again consider a time-dependent problem where the state at 
        time t is given by |ψ(t)= e^-iH't/ħ|ψ_0 with |ψ_0 the ground state of
        H_CSMM. The drift velocity is again given by Eq. (<ref>) and since 
        ψ_0|[H_CSMM,X^a_COM]|ψ_0=0 (since |ψ_0 is an eigenstate of H_CSMM), 
        this reduces to
        
        	v^a_drift= i/ħψ_0|[H_1,X^a_COM] |ψ_0 . 
        
        It remains to actually compute the matrix element ψ_0|[H_1,X^a_COM] |ψ_0. 
        
        To compute this matrix element we first note that we already know the answer for the term in H_1 proportional to 
        E^(0) from the previous subsection. 
        Next, we can immediately see that the term proportional to E^(1) will vanish since the commutator
        of Tr{(X^1)^2} with X^a_COM is linear in the center of mass coordinate and we know
        that ψ_0|X^a_COM |ψ_0=0 in the unperturbed ground state of the CSMM. To handle the term 
        proportional to E^(2) we use Eq. (<ref>) to find that
        
            Tr{(X^1)^3} =    (x^1_0)^3/√(N)+3/√(N)x^1_0∑_A=1^N^2-1x^1_A x^1_A 
             +   ∑_A,B,C=1^N^2-1x^1_A x^1_B x^1_C Tr{T^A T^B T^C} .
        
        Then we have [Tr{(X^1)^3},X^1_COM]=0 and
        
        	[Tr{(X^1)^3},X^2_COM]= 3 iℓ_B^2/N∑_A=0^N^2-1x^1_A x^1_A  .
        
        We find that v^1_drift=0, while 
        
        	v^2_drift   =    -E^(0)/B +i/ħ(eE^(2)/3!)3 iℓ_B^2/Nψ_0|∑_A=0^N^2-1x^1_A x^1_A|ψ_0
           =    -E^(0)/B - e E^(2)ℓ_B^4/ħ Nψ_0|Λ^11|ψ_0
           =    -E^(0)/B + E^(2)ℓ_B^2/Bη_CSMM/ħρ_0 ,
        
        where we used the fact that ψ_0|Λ^11|ψ_0= -A/ħη_CSMM 
        and ρ_0=N/A.
        If we now compute j^2_COM= -eρ_0 v^2_drift then we find that
        
        	j^2_COM   =   νe^2/h( E^(0) - E^(2)ℓ_B^2 η_CSMM/ħρ_0) 
           =   νe^2/h( E^(0) - E^(2)ℓ_B^2 η_H/ħρ_0) ,
        
        where the second line follows from the fact that η_CSMM=η_H, where 
        η_H was the guiding center Hall viscosity for the Laughlin state.
        Finally, we should regularize this expression to obtain a finite answer for the current in the N→∞ limit. This 
        just amounts to the replacement η_H →η_H,reg in the final expression (we discuss the physical
        interpretation of this regularization in Sec. <ref>). Therefore our final expression for the center of mass
        current in a non-uniform electric field is
        
        	j^2_COM= νe^2/h( E^(0) - E^(2)ℓ_B^2 η_H,reg/ħρ_0) . 
        Eq. (<ref>) is the main result of this section. 
        
        It is interesting to compare Eq. (<ref>) with the result of Hoyos and Son, Eq. (<ref>),
        where the coefficient C_2 was given in Eq. (<ref>). We see that the CSMM result contains a contribution
        like the first term in C_2, but with the total Hall viscosity η_tot replaced with the guiding center Hall viscosity
        η_H,reg. As we remarked earlier, this makes sense because we only expect the CSMM to describe the dynamics
        of the guiding center degrees of freedom in the quantum Hall problem. We also find that the CSMM result does not contain
        any contribution resembling the second term in C_2 which is proportional to ℰ”(B). This is also not 
        surprising since the CSMM itself does not contain any information about the energy associated with electrons filling
        a Landau level. 
        Indeed, we can see from the fluid interpretation of the NCCS theory from Sec. <ref>
        that the NCCS theory (and therefore the CSMM theory which is a regularization of it), is obtained by sending the
        energy scale ħω_c to infinity. Therefore we find that the CSMM accurately captures the
        guiding center contribution to the response of a FQH state to a non-uniform electric field.
        
        
        
        § N→∞ LIMIT, REGULARIZATION OF THE HALL VISCOSITY, AND FLUID INTERPRETATION
        
        
        In Ref. park-haldane Park and Haldane argued that one should regularize the guiding center Hall viscosity
        by subtracting the extensive term in 
        η_H= -ħ/A1/2[1/2m N^2 + ( 1-m/2)N], which amounts
        to subtracting the term 1/2m N^2 from  
        1/2m N^2 + ( 1-m/2)N  .
        
        In the quantum Hall problem this regularization (or something similar to it) is necessary to obtain a finite value for 
        the guiding center Hall viscosity in the thermodynamic limit N→∞. 
        
        
        
        
        
        In this section we give an interpretation of this regularization scheme
        in the context of the fluid interpretation (reviewed in the last subsection of Sec. <ref>) of the NCCS theory and 
        CSMM. Our starting point is to note that the expectation value ψ_0|Λ^ab|ψ_0
        in the CSMM is actually proportional to the total angular momentum of the state |ψ_0. 
        The fact that the Hall viscosity is related to angular momentum has
        been discussed extensively in Ref. read-rezayi, so this is not a new observation. However, this observation
        will allow us to understand the origin of the superextensive term 1/2m N^2 in
        ψ_0|Λ^ab|ψ_0, and to explain why it should be subtracted when computing the Hall viscosity
        of the CSMM. 
        
        We start by deriving an expression for the angular momentum in the CSMM theory. To do this we use the fluid interpretation of 
        the NCCS theory and CSMM from the last part of Sec. <ref>. Our derivation of the expression for the angular
        momentum consists of several steps. First, we derive an expression for the angular momentum of a classical fluid 
        of charged particles on a commutative space ℝ^2 and in the presence of a constant background magnetic
        field. Next, we take the limit in which the mass of the particles making up the fluid goes to zero. 
        We then perform the noncommutative 
        deformation of the expression for the angular momentum
        to obtain an expression for the angular momentum in NCCS theory. Finally, the 
        expression for the angular momentum in NCCS theory can also be used for the CSMM, after we replace the infinite-dimensional
        operator variables in the NCCS theory with the N× N matrix variables of the CSMM.
        
        We start with the action for a fluid of particles of mass M, charge q=-e, and constant (initial) density ρ_0 in a 
        constant magnetic field B (see the discussion in the last subsection of Sec. <ref>), 
        
        	S = ∫_0^T dt∫ d^2x ρ_0 (1/2M δ_abẊ^aẊ^b - eB/2_ab X^aẊ^b ) , 
        
        where we remind the reader that for the classical fluid the fields X^a(t,x) are ordinary functions of time t and
        spatial coordinates x∈ℝ^2. For now we omit the Lagrange multiplier field A_0(t,x) which keeps
        the density fixed to ρ_0 at all times, as this term plays no role in the definition of the angular momentum of the
        theory. The momentum variables P_a(t,x) canonically conjugate to X^a(t,x) are obtained by differentiating 
        the Lagrangian[We define the Lagrangian ℒ by S=∫ dt ∫ d^2x ρ_0 ℒ.] with 
        respect to Ẋ^a, and we have
        
        	P_1    =    MẊ^1 +eB/2X^2 
        
        	P_2    =    MẊ^2 -eB/2X^1  .
        
        
        The expression for the angular momentum of this fluid is then 
        
        	L_z    =   ∫ d^2x ρ_0 ( X^1 P_2 - X^2 P_1) 
           =   ∫ d^2x ρ_0 { M_abX^aẊ^b - eB/2δ_abX^a X^b} ,
        
        and the limit M→ 0 gives
        
        	L_z =  -∫ d^2x ρ_0 eB/2δ_abX^a X^b . 
        Next, we set ρ_0=1/2πθ as is appropriate for the fluid interpretation of NCCS theory, and 
        we perform the noncommutative deformation of this expression (see Sec. <ref>) by replacing 
        1/2πθ∫ d^2x (⋯) →Tr_ℋ_F{⋯} and
        X^a(t,x)→X̂^a(t). This gives an expression for the angular momentum in NCCS theory,
        
        	L_z= -e B/2Tr_ℋ_F{δ_abX̂^aX̂^b} .
        
        Finally, we obtain an expression for the angular momentum of the CSMM by replacing the operators
        X̂^a(t) with the N× N matrix variables X^a(t) of the CSMM, and by replacing the trace over
        the infinite-dimensional space ℋ_F by the trace for N× N matrices, 
        
        	L_z,CSMM= -eB/2Tr{δ_abX^a X^b} .
        
        
        We now compute the angular momentum in the quantum ground state |ψ_0 of the CSMM. We first use the
        expansion of Eq. (<ref>) to write L_z,CSMM as 
        
        	L_z,CSMM   =    -e B/2∑_A=0^N^2-1δ_abx^a_A x^b_A 
           =    -ħδ_abΛ^ab ,
        
        where Λ^ab are the strain generators for the CSMM introduced in Sec. <ref>.
        We see that our derivation of the angular momentum for the CSMM theory makes sense since 
        δ_abΛ^ab
        is exactly the operator which generates rotations of the noncommutative coordinates X^a in the CSMM.
        
        For the ground state of the CSMM we have L_z,CSMM|ψ_0 = L_0|ψ_0 with
        
        	L_0= -ħ[1/2m N^2 + ( 1-m/2)N ] ,
        
        and our previous results for ψ_0|Λ^ab|ψ_0 and η_CSMM 
        can be rewritten in the form
        ψ_0|Λ^ab|ψ_0   =    -1/2ħL_0δ^ab
        η_CSMM   =   1/2L_0/A .
        
        Thus, we see that the Hall viscosity coefficient η_CSMM (before regularization) is equal to
        one half the angular momentum density L_0/A in the ground state of the CSMM (compare with the
        angular momentum interpretation of the Hall viscosity from Ref. read-rezayi). Finally, we also note
        that L_0 is exactly the guiding center part of the angular momentum of the Laughlin ν=1/m state.
        In the lowest Landau level the Landau orbit contribution to the angular momentum is simply ħN/2, which
        leads to the total angular momentum of the Laughlin state 
        L_z,ν=1/m= ħ[-1/2mN^2+mN/2].
        
        We now give a fluid interpretation of the superextensive (order N^2) term in L_0, which is equal to 
        -1/2ħ m N^2. This can be 
        rewritten in terms of the density ρ_0= 1/2πℓ_B^2 m and radius 
        R^2_N ≈ 2ℓ_B^2 m N of the droplet described by the CSMM as
        
        	-π/4e Bρ_0 R^4_N .	
        
        This is exactly the angular momentum of a droplet of radius R_N of the classical fluid described by
        the small θ limit of the NCCS action in the presence of an additional parabolic potential, as we now describe. 
        
        Recall that in the small θ limit the NCCS theory is described by the fluid action of Eq. (<ref>).
        Let us add to this action a parabolic potential term which is the commutative analogue of the 
        potential term in the CSMM action, 
        
        	S_para= -eB/2ρ_0 ∫_0^T dt∫ d^2x δ_abX^a X^b ,
        
        where ρ_0=1/2πθ. The equations of motion which result from Eq. (<ref>)
        plus S_para are Ẋ^1= X^2 and Ẋ^2= - X^1, as well as the constant density
        constraint enforced by A_0. For the initial condition X^a(0,x)=x^a the solution to these equations
        can be expressed concisely as 
        
        	X^1(t,x)+i X^2(t,x) = (x^1 + i x^2) e^-i t .
        
        Finally, using Eq. <ref> for the angular momentum we find that a droplet of radius ℛ
        has angular momentum
        
        	L_orb   =    -∫_|x|≤ℛ d^2x ρ_0 eB/2δ_abX^a X^b 
           =    -π/4e Bρ_0 ℛ^4,
        
        where “orb" stands for “orbital"  since this angular momentum is associated with an overall rotation of the
        fluid. 
        
        We see that the superextensive term in L_0 is exactly the orbital angular momentum of a classical fluid on a commutative
        space in a magnetic field undergoing uniform rotational motion. Based on this observation, 
        and using the anisospinς=m-1/2 defined earlier, the full angular momentum in 
        the ground state of the CSMM can be written as
        
        	L_0= L_orb + ħς N .
        
        Now that we have identified the orbital contribution to the total angular momentum the remaining extensive term, 
        which has a coefficient ς, can be interpreted as a spin angular momentum for the N particles in the
        fluid, in keeping with the interpretations of Hall viscosity of 
        Refs. haldane2009,haldane2011,park-haldane,read2009,read-rezayi.
        
        
        Now that we understand the connection between the expectation value ψ_0|Λ^ab|ψ_0 
        and the
        total angular momentum L_0 of the state |ψ_0, we can give a fluid interpretation of the regularization scheme
        for the guiding center Hall viscosity proposed in Ref. park-haldane. Specifically, the regularization scheme
        of  Ref. park-haldane corresponds to subtracting the orbital contribution to L_0,
        η_CSMM,reg= 1/2(L_0-L_orb/A)= 1/2ħςρ_0 .
        
        This can be justified by noting that the classical charged fluid in a constant magnetic field and on ordinary commutative
        space does not exhibit a Hall viscosity[This can be seen directly by writing down the equations of motion for this
        classical fluid in the Euler description (i.e., in terms of mass density and velocity fields), 
        and then noting that no viscosity term is present. The Euler equations for a charged
        fluid in a magnetic field and a general external potential appear, for example, in Eqns. (46) and (47) 
        of Ref. guralnik2001.], 
        and so the Hall viscosity in the fluid described by the CSMM must only be due to the 
        remaining terms in L_0 which do not have an interpretation in terms of the classical fluid on a commutative space. 
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        § HALL VISCOSITY IN THE PRESENCE OF ANISOTROPY
        
        
        In this section we introduce a simple modification of the CSMM which incorporates a constant unimodular 
        metric g_ab (i.e., a constant metric with determinant equal to 1). This metric parametrizes an anisotropy or
        intrinsic geometry of a FQH state,  as discussed in the works of Haldane and 
        collaborators <cit.>. As emphasized by 
        Haldane <cit.>, introducing a unimodular metric g_ab into the guiding center part of a
        FQH state enables one to see the clear separation
        of the full Hall viscosity tensor η^abcd_tot
        into Landau orbit and guiding center contributions. When such a metric is used in the construction of the
        guiding center part of a FQH state, the guiding center Hall viscosity tensor η^ab_H is modified to be proportional to 
        g^ab (the inverse metric
        of g_ab with g^abg_bc=δ^a_c) instead of δ^ab. In this section we show that for our modified
        CSMM, the two-index Hall viscosity tensor η^ab_CSMM is also modified to be proportional to 
        g^ab. This confirms that our modification of the CSMM does indeed correspond to incorporating a nontrivial metric
        g_ab into the definition of the guiding center part of a FQH state. We also note here that the introduction of a second 
        metric (in addition to the the metric of space) into the quantum Hall problem is exactly the starting point for the construction of 
        the bi-metric theory of FQH states of Refs. GGB,gromov-son.
        
        The action for our modified CSMM takes the form
        
            S_CSMM   = -eB/2∫_0^T dt Tr{_abX^a D_0 X^b + 2θ A_0 
               +  g_abX^a X^b }  + ∫_0^T Ψ^T(iΨ̇+ A_0 Ψ) .
        
        Note that the only change is the replacement of δ_ab with g_ab in the quadratic potential term. This is the
        only part of the action which could conceivably depend on a metric, since the time derivative term already uses the epsilon 
        symbol _ab to contract indices. 
        To quantize this system we make a change to a new set of variables X̃^a 
        which diagonalize the potential term but, crucially, obey the same commutation relations as the original variables. In other
        words, the symplectic form on the phase space of this model takes the same form in the new variables
        as in the old ones. Therefore the Poisson brackets and quantum commutation relations of the new variables will be identical
        to those for the old variables.
        
        To describe this change of variables we decompose the metric and inverse metric in terms of coframes e^a_a and
        frames E^a_a as
        
        	g_ab   =    e^a_a δ_ab e^b_b 
        
        	g^ab   =    E^a_aδ^ab E^b_b .
        
        Note that we use new indices a,b=1,2 for the internal indices of the frames and coframes. The frames and 
        coframes satisfy the relations E^a_ae^a_b = δ^a_b and 
        E^a_ae^b_a = δ^b_a, which just express the fact that the matrices e and E (with
        entries e^a_a and E^a_a, respectively) are inverses of each other. In addition, it is possible to choose 
        det(e)=det(E)=1. This can be seen as follows. First, note that the relation between g_ab and 
        e^a_a can be expressed in matrix form as g= e^T e, where g is the matrix with entries g_ab. 
        This implies that det(e)^2=det(g)=1, so that det(e)=± 1. However, the
        parametrization of g in terms of e is invariant under the transformation e → S e for any matrix S∈ O(2), i.e., any 
        S such that S^T S=𝕀. Then if for some reason we found a decomposition of g with det(e)=-1, we
        can always switch to a new parametrization with det(e)=1 by replacing e with Se for any S∈ O(2) with
        det(S)=-1. Then, since E=e^-1 as matrices, we also guarantee that det(E)=1.
        
        Using the frames and coframes we introduce new matrix variables X^a as
        X^a   =    e^a_a X^a 
        
        	X^a    =    E^a_aX^a .
        
        In terms of these variables we have
        
        	g_abX^a X^b= δ_abX^aX^b 
        and, crucially, 
        _abX^a D_0 X^b    =   _abE^a_aE^b_bX^aD_0 X^b
           =   det(E)_abX^aD_0 X^b
           =   _abX^aD_0 X^b .
        
        We can then carry out the quantization of this modified CSMM using the X̃^a variables in exactly the
        same way that we quantized the original CSMM in Sec. <ref>. For example we would start by 
        expanding the X̃^a in terms of a new set of real scalar variables x^a_A (A=0,…,N^2-1)
        exactly as in Eq. (<ref>). This procedure results in a new ground state
        |ψ_0 for the modified CSMM depending on the unimodular metric g_ab.
        
        We can now calculate the Hall viscosity in this modified CSMM. The setup for this calculation is the
        same as in Sec. <ref> and, in particular, we still apply an APD (or strain) to 
        the physical position variables X^a and 
        not the new variables X^a. The final expression for the two-index Hall viscosity tensor 
        η^ab_CSMM is now proportional to the expectation value of the strain generators 
        Λ^ab in the ground state |ψ_0 of the modified CSMM,
        η^ab_CSMM = -ħ/Aψ_0|Λ^ab|ψ_0 .
        
        The expectation value ψ_0|Λ^ab|ψ_0 is easily computed by writing 
        Λ^ab= E^a_aE^b_bΛ^ab, where
        Λ^ab= 1/4ℓ_B^2∑_A=0^N^2-1{x^a_A,x^b_A}
        are the strain generators for the new variables, and by noting that
        ψ_0|Λ^ab|ψ_0= 1/2[1/2m N^2 + ( 1-m/2)N]δ^ab ,
        
        which follows since all quantities here are in terms of the new “tilde" variables. Then the original expectation value of interest
        evaluates to 
        ψ_0|Λ^ab|ψ_0   =   1/2[1/2m N^2 + ( 1-m/2)N]δ^ab E^a_aE^b_b
           =   1/2[1/2m N^2 + ( 1-m/2)N]g^ab .
        
        After regularization, which consists of subtracting off the order N^2 term in this expectation value, the
        Hall viscosity tensor for the modified CSMM takes the form
        η^ab_CSMM,reg= -ħ/A1/2( 1-m/2)N g^ab = η_CSMM,reg g^ab ,
        
        where η_CSMM,reg= 1/2ħςρ_0 as before, and where we defined 
        ρ_0=N/A. We find that the Hall viscosity tensor for the modified CSMM is exactly the guiding center
        part of the Hall viscosity tensor of the Laughlin state with nontrivial guiding
        center metric g_ab <cit.>.
        
        We close this section by calculating the area A and the shape of the droplet of fluid described by the ground 
        state |ψ_0 of the modified CSMM. To do this we follow the method from the end of Sec. <ref>
        and consider the eigenvalue of Tr{ g_abX^a X^b} when acting on the state |ψ_0. We again
        find that Tr{ g_abX^a X^b}|ψ_0= R^2|ψ_0 with the same eigenvalue
        R^2 from Eq. (<ref>), and we can again interpret R^2 as a sum of contributions from 
        N particles, R^2= ∑_j=1^NR^2_j with R^2_j= 2ℓ_B^2(m (j-1) + 1/2). However,
        the interpretation of the shape of the droplet is different now since g_abX^a X^b is a general quadratic form
        of the noncommutative position coordinates. In the simple case where g_ab=δ_ab, we argued that the
        droplet was circular, with the j^th particle located somewhere on a circle of radius R_j. In this case we will
        argue that the droplet has the shape of an ellipse, with the particular geometry of the ellipse determined by the
        eigenvectors and eigenvalues of the metric g_ab considered as a matrix, and where the j^th particle is now
        located somewhere on an ellipse whose size is determined by R_j.
        
        To facilitate this analysis we use a convenient parametrization <cit.> of the unimodular metric g_ab in 
        terms of a single complex parameter γ∈ℂ, |γ|<1, and write
        
        	g= 1/1-|γ|^2[ (1+γ)(1+γ)    i (γ-γ);    i (γ-γ) (1-γ)(1-γ) ] .
        
        If we also write γ= tanh(/2)e^iβ for real >0 and a real phase β, then we find that
        the matrix g has the decomposition 
        
        	g= S D S^T
        with
        
        	S= [  cos(β/2)  sin(β/2); -sin(β/2)  cos(β/2) ]
        and
        
        	D=[  e^   0;   0 e^- ] .
        
        Here e^± are the eigenvalues of g and the columns of the matrix S are the normalized eigenvectors of g.
        In component form we can also write
        
        	g_ab= S^a_a D_abS^b_b ,
        
        where for S^a_a, a indexes the rows of the matrix S and a indexes the columns. 
        
        
        
        We now introduce new noncommutative coordinates (i.e., matrices) Y^a defined as
        
        	Y^a= S^a_a X^a ,
        
        and in terms of these we have
        
        	g_abX^a X^b    =    D_abY^aY^b
           =    e^ (Y^1)^2 + e^-(Y^2)^2 .
        
        We now see that in the modified CSMM with metric g_ab, we can interpret the j^th particle
        as residing on an ellipse with the lengths of the minor and major axes of that ellipse given by
        r_1,j= e^-/2R_j and r_2,j= e^/2R_j[Recall that the equation 
        a^2 x^2 + b^2 y^2= R^2 describes an ellipse in the (x,y) plane with the lengths of the two axes of the ellipse given
        by R/a and R/b.]. 
        Furthermore, this ellipse has its minor and major axes lined up with
        the axes of the Y^a coordinate system, which is rotated from the X^a coordinate system by an angle of
        β/2 as shown in Fig. <ref>. The area of the ellipse where the j^th particle is located is
        π r_1,j r_2,j = π R^2_j, and since R^2_j is linear in j, we again find that
        the particle density is constant inside the droplet. 
        Finally, the area of the droplet is equal to the area of the ellipse for particle N which 
        is A = π R^2_N ≈ 2πℓ_B^2 m N, just as in the ordinary CSMM. 
        
        We conclude that the modified CSMM
        incorporating the unimodular metric g_ab describes an elliptical droplet of fluid with the same area A and constant
        density ρ_0 as the ordinary CSMM, and where the details of the shape of the ellipse are determined by the eigenvalues
        and eigenvectors of the metric g_ab. In addition, since the density ρ_0 is the same as for the original
        CSMM, we find that the coefficient 
        η_CSMM,reg= 1/2ħςρ_0 of Hall viscosity for the
        CSMM with g_ab≠δ_ab is numerically equal to the coefficient for the case where g_ab=δ_ab. The
        only difference between these two cases is the structure of the Hall viscosity tensor, since for g_ab≠δ_ab the
        two index tensor η^ab_CSMM,reg is proportional to g^ab instead of δ^ab.
        	
        
        
        § CONCLUSION
        
        
        In this paper we investigated the geometric properties of the Laughlin FQH states within the CSMM description of 
        these states which, roughly speaking, models these states as a charged fluid in a magnetic field and propagating on a 
        noncommutative space. 
        We focused our attention on the specific properties of Hall viscosity, Hall conductance in a non-uniform electric field,
        and the Hall viscosity in the presence of anisotropy. We found that the answers for these quantities calculated from
        the CSMM description contain only the guiding center contribution to the known answers for these quantities in the 
        Laughlin states. 
        
        These results lead us to the general conclusion that the CSMM description of the Laughlin FQH states 
        accurately captures the guiding center contribution to the geometric properties of these states, but lacks the 
        Landau orbit contribution. As we remarked in the Introduction, the Landau orbit contribution is often considered to be a
        trivial contribution since the interesting correlations in the Laughlin state are contained in the guiding center part of its
        wave function/state vector. Therefore we find that the CSMM description captures the most important contribution, namely the 
        guiding center contribution, to the physics of the Laughlin FQH states. However, any attempt to completely describe
        the Laughlin states using the CSMM or NCCS theory must also include some auxiliary degrees of freedom which account for
        the missing Landau orbit contributions to the geometric properties of these states.
        
        There are several possible directions for future work in this area. One direction would be to continue to develop the
        fluid interpretation of the CSMM. One goal of this work would be to find an appropriate definition of a density 
        operator ρ(x) which is a function of a commutative two-dimensional coordinate x∈ℝ^2
        and which is defined on length scales much larger than the scale set by θ in the noncommutative theory. One 
        could then check whether this density operator satisfies the Girvin-Macdonald-Plaztman algebra, and also attempt to
        compute the static structure factor and compare to the known answer for the Laughlin states <cit.>. 
         Another goal of this work would be to connect the CSMM description of the Laughlin states with 
        a different fluid description of these states, which is Wiegmann's vortex fluid description <cit.>.
        In this description the Laughlin FQH state with N electrons is modeled as a rotating incompressible fluid containing N point 
        vortices each carrying a quantized circulation Γ which depends on the filling fraction of the Laughlin state.
        On this topic we note that Bettelheim has recently introduced a method for defining density and velocity fields in the
        CSMM which are functions of a commutative coordinate x in Ref. bettelheim2013, and it
        would be interesting to develop his approach further and to use it to connect with Wiegmann's vortex fluid description. 
        We also note that the problem of defining density operators in NCCS theory and the CSMM has been considered before in 
        Refs. hansson2001,hansson2003.
        
        A second direction for future work would be to investigate the Hall viscosity and other geometric response properties
         in matrix models which
        describe other more complicated FQH states. For example, a matrix model for the Jain states <cit.> 
        has been proposed in 
        Ref. cappelli-jain. More recently, the authors of Ref. tong2016 proposed a class of 
        matrix models for the Blok-Wen series of non-Abelian FQH states <cit.>. 
        It would also be interesting to search for new matrix models which can describe
        other FQH states of interest.
        
        
        
        
        
        
        
        
        
        
        
        
        
        We acknowledge useful discussions with E. Fradkin, R. Leigh, B. Bradlyn, A. Gromov, M. Stone, T. Zhou, P. Di Francesco, 
        R. Kedem, P. Wiegmann, and V. Pasquier. 
        M.F.L. wishes to acknowledge the hospitality of the Simons Center for Geometry and Physics during the 2017
         workshop “Strongly correlated topological phases of matter", 
        and the hospitality of the Institut d'Etudes Scientifiques in Cargèse
        during the 2017 “Exact methods in low-dimensional physics" summer school, where parts of this research were conducted.
        M.F.L. and T.L.H. acknowledge support from the US National
        Science Foundation under grant DMR 1351895-CAR. We also gratefully acknowledge the support of the Institute
        for Condensed Matter Theory at the University of Illinois at Urbana-Champaign.  
        
        
        
        
        § QUANTUM GENERATORS OF THE U(N) ACTION
        
        
        In this Appendix we consider the form of the quantum generators of the U(N) transformations of the matrix
        model variables X^a and Ψ. We use this result in Sec. <ref> to show that the
        constraint of Eq. (<ref>) simply forces physical states in the CSMM to be singlets under the SU(N) action, and 
        to carry a certain total charge under the U(1) action. This information is sufficient to write down a basis of physical
        states (states respecting the constraint) for the model following Ref. HVR.
        
        We start with the generators for the U(N) transformation of the complex vector variable Ψ. Under a U(N)
        transformation by a matrix V we have Ψ→ VΨ or in components
        Ψ^j →V^j_k Ψ^k .
        
        We are interested in the infinitesimal form of this transformation, so we take V= e^iT for a Hermitian matrix T (the
        Lie algebra of the group U(N) consists of the N× N Hermitian matrices). Then to first order in 
        T we have Ψ→Ψ + i TΨ. In components, the first order change in Ψ^j generated by T is
        δ_TΨ^j = i T^j_kΨ^k .
        
        
        We now look for a quantum operator 𝒪_Ψ(T) such that 
        
        	[𝒪_Ψ(T),Ψ^j] =  i T^j_kΨ^k ,
        
        i.e., the quantum commutator of 𝒪_Ψ(T) with Ψ^j implements the infinitesimal U(N) action generated
        by T (this is what we mean when we say that a quantum operator generates the U(N) action).
        The correct operator is (in terms of b^j instead of Ψ^j)
        𝒪_Ψ(T)= -i b^_j T^j_k b^k .
        
        Thus, 𝒪_Ψ(T) is the quantum operator which generates the U(N) transformation V=e^iT acting 
        on Ψ. One can also check that the operators 𝒪_Ψ(T) obey the Lie algebra of U(N). To check
        this it is sufficient to check that the map T↦𝒪_Ψ(T) is a Lie algebra homomorphism, i.e., that
        
        	[𝒪_Ψ(T_1),𝒪_Ψ(T_2)]= 𝒪_Ψ(-i[T_1,T_2]_M) ,
        
        and it is straightforward to verify that this relation holds for our generators 𝒪_Ψ(T).
        
        Next we consider the matrix variables X^a. Under a U(N) transformation we have
        X^a→ V X^a V^T. Writing V=e^iT as before, we find that to first order in T we have 
        X^a→ X^a + i[T,X^a]_M. Note that for T= 𝕀, i.e., for U(1) transformations, the
        matrix variables X^a are invariant. Therefore we can restrict our attention to SU(N) transformations for the
        X^a variables. We then choose T to be one of the generators T^A of SU(N), and examine the infinitesimal
        action of V= e^iT^A on the scalar variables x^a_0 and x^a_A, A=1,…,N^2-1, which appear in the expansion 
        of X^a from Eq. (<ref>). We have
        δ_T^A X^a    =    i[T^A,X^a]_M 
           =    i∑_B=1^N^2-1x^a_B [T^A,T^B]_M 
           =    -∑_B,C=1^N^2-1 x^a_B f^ABC T^C .
        
        From this we read off that δ_T^A x^a_0= 0 (reflecting the invariance under U(1) transformations), and
        δ_T^A x^a_B= -∑_C=1^N^2-1 x^a_C f^ACB , B=1,…,N^2-1 .
        
        
        We now look for a quantum operator 𝒪_X(T^A) which generates this action on the variables x^a_A 
        (A=1,…,N^2-1), i.e., an operator which commutes with x^a_0 and satisfies
        
        	[𝒪_X(T^A), x^a_B]= -∑_C=1^N^2-1 x^a_C f^ACB
        for B=1,…,N^2-1. One can check that the correct operator is (in terms of the oscillator variables a_A)
        𝒪_X(T^A)= ∑_B,C=1^N^2-1 f^ACB a^_B a_C .
        
        
        This completes the construction of the quantum generators of the U(N) action on the X^a and Ψ variables
        in the CSMM. This is all the information which is needed to analyze the j≠ k elements of the 
        CSMM constraint G^j_k from Eq. (<ref>).
        
        
        
        
        
        
        
        § KUBO FORMULA APPROACH TO HALL VISCOSITY IN THE CSMM
        
        
        In this Appendix we use a Kubo formula approach inspired by Ref. bradlyn2012 to compute the 
        Hall viscosity in the ground state of the CSMM. For this computation we subject the CSMM to a 
        time-dependent APD (or strain) parametrized by _ab(t) such that
        the dynamics of the system is described by the time-dependent Hamiltonian
        
        	H((t))= U((t))H_CSMMU((t))^ .
        
        Here the operator U((t)) is the APD generator for the CSMM which we derive in Sec. <ref> 
        of the main text. We also assume that at the time t_0 we have _ab(t_0)= 0 so that 
        |ψ(t_0)= |ψ_0, which
        is the ground state of the CSMM from Eq. (<ref>). As we discussed in Sec. <ref>, 
        the generalized force associated with the APD parametrized by the coefficients _ab is
        
        	F^ab= - H()/_ab|_=0 = -i[Λ^ab,H_CSMM] .
        
        To calculate the Hall viscosity we need to compute the expectation value of the generalized force
        F^ab in the state |ψ(t) of the system, where |ψ(t) is the solution to the time-dependent 
        Schrodinger equation
        
        	H((t))|ψ(t)= i ħ/ t|ψ(t) .
        
        We now discuss the details of this computation. 
        
        First, to set up this problem in a form which is amenable to perturbation theory and the Kubo formula, we make a 
        time-dependent change of states by writing
        
        	|ψ(t)= U((t))|ϕ(t) .
        
        The state |ϕ(t) is then the solution to a time-dependent Schrodinger equation with a new Hamiltonian H'(t) given by
        
        	H'(t)= H_CSMM + V(t) 
        
        with
        
        	V(t)    =    -iħ U((t))^ U((t))/ t
           ≈   ħ_ab(t)/ tΛ^ab + … ,
        
        where in the second line we expanded the perturbation V(t) to first order in _ab(t). The new Hamiltonian 
        H'(t) is now expressed as a time-independent term H_CSMM plus a time-dependent perturbation 
        V(t), and is therefore in a form[The change of basis from |ψ(t) to |ϕ(t) is equivalent to the 
        change from the “x" to the “X" variables in Ref. bradlyn2012. 
        We thank Barry Bradlyn for helpful discussions on this point.] 
        which is amenable to an application of standard linear response theory.
        
        To compute the Hall viscosity we naively want to compute the expectation value of F^ab in the state
        |ψ(t). However, in Ref. bradlyn2012 the authors argued that one should instead compute the
        expectation value of U((t))F^abU((t))^, which is equivalent to expressing the generalized force F^ab in 
        terms of the strained coordinates U((t)) x^a_A U((t))^ instead of the original coordinates x^a_A of the
        CSMM (in the language of Ref. bradlyn2012, we express the generalized force in terms of the “X"
        variables as opposed to the unstrained “x" variables). The reason for this is as follows. We view the
        APD parametrized by _ab(t) as an active transformation (i.e., we physically deform the fluid/CSMM), and so in the
        computation of the response to this APD we should use the generalized force expressed in terms of the coordinates of the 
        deformed system. Now we have
        ψ(t) | U((t))F^abU((t))^|ψ(t) = ϕ(t) |F^ab|ϕ(t) ,
        
        and so it remains to compute the expectation value ϕ(t) |F^ab|ϕ(t).
        
        In interaction picture perturbation theory in the strength of the potential V(t), the expectation value of any 
        time-independent operator A in the state |ϕ(t) is given by the standard Kubo formula as 
        
            ϕ(t)|A|ϕ(t)-   ϕ(t_0)|A|ϕ(t_0) = 
            
             -   i/ħ∫_t_0^t dt' ϕ(t_0)|[A_I(t),V_I(t')]|ϕ(t_0) + … ,
        
        where A_I(t)= e^i H_CSMM(t-t_0)/ħAe^-i H_CSMM(t-t_0)/ħ is in the interaction picture 
        defined by evolution with H_CSMM, and likewise for 
        V_I(t')= e^i H_CSMM(t'-t_0)/ħV(t')e^-i H_CSMM(t'-t_0)/ħ. Note also that for any
        time-independent A we have A_I(t_0)= A, and we also have |ϕ(t_0)=|ψ(t_0)= |ψ_0.
        
        For the application to the calculation of the Hall viscosity we set A= F^ab and keep only the term in V(t) 
        which is linear in the parameters _ab(t). This yields the expression
         F^ab_t -  F^ab_t_0 = -i∫_t_0^t dt'  [F^ab_I(t),Λ^cd_I(t')]_t_0_cd(t')/ t' ,
        
        where we used the shorthand notation F^ab_t ≡ϕ(t)|F^ab|ϕ(t), etc. 
        Next, since [F^ab_I(t),Λ^cd_I(t')]_t_0=  [F^ab_I(t-t'+t_0),Λ^cd_I(t_0)]_t_0, 
        this can be rewritten as
         F^ab_t -  F^ab_t_0 = -∫_-∞^∞ dt' 𝒳^abcd(t-t')_cd(t')/ t' , 
        where we defined the response function
        𝒳^abcd(t)= lim_→ 0^+ i Θ(t)  [F^ab_I(t+t_0),Λ^cd_I(t_0)]_t_0e^- t ,
        
        and where we also sent t_0→ -∞ in Eq. (<ref>). Note that in Eq. (<ref>) the Heaviside function 
        Θ(t-t') allows us to extend the upper limit of the integral over t' to +∞, while the presence of the factor
        e^-(t-t') allows us to send t_0→ -∞. 
        
        Next we perform a Fourier transform[Our convention for Fourier transforms is
        f(ω)= ∫_-∞^∞ dt f(t) e^iω t, 
        f(t)= ∫_-∞^∞dω/2π f(ω) e^-iω t.] and consider the 
        frequency-dependent response function
        
            𝒳^abcd(ω)    = ∫_-∞^∞dt 𝒳^abcd(t) e^iω t
               = lim_→ 0^+ i ∫_0^∞ dt e^iω_+t [F^ab_I(t+t_0),Λ^cd_I(t_0)]_t_0 ,
        
        where ω_+= ω+i. Now we note that 
        
        	F^ab_I(t+t_0)=  -i[Λ^ab_I(t+t_0),H_CSMM]= ħd Λ^ab_I(t+t_0)/dt ,
        
        where we used the equation of motion for Λ^ab_I(t+t_0) in the interaction picture. Then an integration by 
        parts with respect to t in the expression for 𝒳^abcd(ω) yields a “strain-strain" form of the
        response function 𝒳^abcd(ω) analogous to Eq. (3.5) of Ref. bradlyn2012,
        
            𝒳^abcd(ω)   = -iħ [Λ^ab(t_0),Λ^cd(t_0)]_t_0
               + lim_→ 0^+ħω_+∫_0^∞ dt e^iω_+t [Λ^ab(t+t_0),Λ^cd(t_0)]_t_0 .
        
        In the case where the unperturbed Hamiltonian has a unique ground state and a finite energy gap one finds that
        lim_ω→ 0𝒳^abcd(ω)    =    -iħ [Λ^ab(t_0),Λ^cd(t_0)]_t_0
           =    -iħψ_0| [Λ^ab,Λ^cd]|ψ_0 ,
        
        i.e., the first term in Eq. (<ref>) gives the full response at ω=0 <cit.>. These
        assumptions (unique ground state and finite energy gap) hold for the CSMM for any finite value of ω̃, and
        so this formula for the response at ω=0 can be applied to the CSMM[One
        should not confuse ω, the frequency appearing in the Fourier transform of the response function, with ω,
        which sets the strength of the parabolic potential in the CSMM.].
        We note that this form
        of the response at ω=0 is what one obtains from a Hall viscosity calculation using adiabatic perturbation 
        theory <cit.>.
        
        Finally, we can complete the calculation of F^ab_t ≡ϕ(t)|F^ab|ϕ(t) to lowest order
        in time derivatives of _cd(t). First, after a Fourier transformation (taking t_0→ -∞ in order to do the
        integration over t') we can write
         F^ab_t -  F^ab_t_0= ∫_-∞^∞dω/2π iω𝒳^abcd(ω)_cd(ω) e^-iω t .
        
        Next, we expand 𝒳^abcd(ω) about ω=0 as
        
            𝒳^abcd(ω) = -iħψ_0| [Λ^ab,Λ^cd]|ψ_0 + …
        
        and invert the Fourier transformation to find
         F^ab_t -  F^ab_t_0= iħψ_0| [Λ^ab,Λ^cd]|ψ_0_cd(t)/ t + …
        For a system with an area A (A= 2πℓ_B^2 m N for the CSMM with θ=ℓ_B^2 m) we then find that the Hall 
        viscosity tensor is given by
        η^abcd_CSMM= iħ/Aψ_0| [Λ^ab,Λ^cd]|ψ_0 ,
         
        and this tensor encodes the linear response of the “generalized stress" F^ab/A to the “rate of strain" given by
        _cd(t)/ t.
        
        
        
        
        
        
        
        
        
        
        
        
        52
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