Entanglement entropy on a fuzzy sphere with a UV cutoff
Hong Zhe Chen Joanna L. Karczmarek
hep-th
















    1 21.1 
        
         0.5in
        
        
        Entanglement entropy on a fuzzy sphere with a UV cutoff
         0.5in
        Hong Zhe Chen and Joanna L. Karczmarek   
         0.3in
        
        Department of Physics and Astronomy
        
        University of British Columbia,
        Vancouver, Canada V6T 1Z1
         0.5in
        
        We introduce a UV cutoff into free scalar field theory on the 
        noncommutative (fuzzy) two-sphere.  Due to the IR-UV connection,
        varying the UV cutoff allows us to control the effective
        nonlocality scale of the theory.
        In the resulting fuzzy geometry, we establish which degrees
        of freedom lie within a specific geometric subregion 
        and compute the associated vacuum entanglement entropy.
        Entanglement entropy for regions smaller than the 
        effective nonlocality scale is extensive, while
        entanglement entropy for regions larger than 
        the effective nonlocality scale follows the area law.
        This reproduces features previously obtained in the 
        strong coupling regime through holography.  
        We also show that mutual information is unaffected
        by the UV cutoff.
        
        
        § INTRODUCTION
        
        
        As a tool used to investigate phenomena from
        the nature of quantum criticality to the holographic origin
        of space time, entanglement entropy in quantum field theories
        has received wide interest.
        One of the strengths of  geometric entanglement entropy (entanglement entropy associated
        with a particular region of space) is that its definition is
        independent of the details of the field theory, such as its degrees of freedom, and of the quantum state under consideration.
        A reflection of this universality is the simple holographic interpretation
        of geometric entanglement entropy as an area of a minimal surface <cit.>.
        
        Since entanglement entropy is universal, it is a good tool to study 
        unconventional quantum field theories, such as those lacking in
        locality.  Nonlocal theories have been argued to be
        relevant to the study of holographic duals of flat space <cit.>,
        while noncommutative theories in particular have been argued to be related to
        black hole horizons and information scrambling 
        <cit.>.
        Unsurprisingly, entanglement entropy in nonlocal theories can have drastically different
        properties than it does in local theories.  For example,
        the now very-well established `area law'—in the vacuum state,
        the leading term of entanglement entropy grows (at most) proportionately
        to the area of the boundary of the region of interest—
        can fail if locality is not present.
        Studying entanglement entropy in nonlocal theories can help us understand which
        properties of entanglement entropy are dependent on locality and how.  Conversely,
        detailed behaviour of entanglement entropy in nonlocal theories can illuminate
        features of nonlocal theories such as the presence or absence of a UV-IR connection.
        
        
        Vacuum entanglement entropy in noncommutative field theories has been studied both
        holographically 
        <cit.> 
        and directly <cit.>. 
        Holographically, for regions whose size is below a certain critical
        length scale L_trans, the entanglement entropy grows with the volume of the region,
        while the area law is restored for regions whose 
        dimensions are larger than the critical length scale L_trans.  
        The critical length scale L_trans is proportional to the UV cutoff,
        revealing information about the
        UV-IR connection.  It is not immediately clear whether the holographic
        approach is truly measuring geometric entanglement entropy in the
        field theory, as existence of well defined
        geometric regions in the nonlocal boundary theory corresponding to 
        regions of AdS boundary is not immediately obvious.
        To remove these doubts, entanglement entropy was studied for 
        a free field on a fuzzy (noncommutative) sphere directly, using a finite N
        matrix model <cit.>.
        However, owing to an inevitable
        relationship between the noncommutativity parameter 
        √(θ), 
        the circumference of the sphere R and the UV cutoff λ_UV:
        
        2 R^2 = N θ   and   λ_UV = R/N ,
        
        the expected critical length scale 
        L_trans∼θ / λ_UV 
        is larger than the radius of the sphere, making
        it impossible to examine regions whose dimensions larger 
        than L_trans.
        
        In this paper, we give a prescription for lowering the UV cutoff
        in a scalar field theory on a fuzzy sphere, 
        increasing the associated wavelength 
        from λ_UV = R/N to λ̃_UV = R/n,
        and defining geometric regions in a theory with a UV cutoff.   
        In this simple theory, we are able to recover the phase transition 
        previously seen holographically, as described above.
        As can be seen in Figure <ref>, imposing a
        lower UV cutoff has no effect on entanglement entropy associated 
        with a spherical cap of angular size θ when θ is small.
        For larger regions, entanglement
        entropy becomes proportional to the length of the boundary of the
        region, 2π R sinθ.    As expected,
        this transition from extensive entanglement entropy to area law
        happens at an angular size θ_trans which increases
        proportionately to 1/λ̃_UV∼ n (Figure <ref>).
        
        The rest of this paper is organized as follows.  
        We work with a free scalar field theory on a noncommutative—or fuzzy—sphere
        with R=1 throughout the paper.
        In section
        <ref> we describe our approach to computing entanglement
        entropy associated with a spherical cap region in the presence of
        a UV cutoff.  In section <ref> we present detailed
        behaviour of the entanglement entropy as we vary θ, n, N and the
        mass of the scalar field.  In section <ref>, we
        discuss  mutual information.
        
        
        
        
        
        
        § METHODOLOGY
        
        
        
        
        The Hamiltonian for a real scalar field theory on a fuzzy sphere
        of radius one is given by
        
        H = 4 π/N ( (ϕ̇)^2  - ∑_i=1,2,3 [L_i,ϕ]^2  + μ^2  ϕ^2  )
         ,
        
        where ϕ is an N× N Hermitian matrix representing the
        scalar field, whose mass is μ. 
        L_i generate an irreducible
        representation of the SU(2) algebra:
        
        [L_i, L_j] = ∑_k iϵ_ijk L_k ,   ∑_i=1,2,3 L_i^2 = (N^2-1)/4 .
        
        The Hamiltonian (<ref>) represents N^2 coupled harmonic oscillators
        of equal masses.  We work in a basis where L_3 is diagonal.
        In <cit.> it was noticed that Hamiltonian (<ref>) is a 
        sum of 2N+1 independent (mutually non-interacting) sectors.
        This decomposition can be viewed as a consequence of rotational
        symmetry.  To implement it, we need to first `complexify' the 
        scalar field ϕ.  The Hermitian matrix ϕ represents the configuration
        space of our system, ℝ^N^2.  First, let's repackage
        the information contained in ϕ to make its nature as a real
        vector space easier to see, by defining a real matrix
        ϕ̃=  (ϕ + ϕ^T) + 1/2i(ϕ - ϕ^T).
        Matrices ϕ and ϕ̃ are in one-to-one correspondence
        with each other, with ϕ = (ϕ̃+ ϕ̃^T) +
        i/2(ϕ̃- ϕ̃^T), and ϕ̃ is
        a real matrix iff ϕ is Hermitian, thus each entry in matrix ϕ̃
        is an independent real field.
         The Hamiltonian (<ref>) can
        be rewritten in terms of ϕ̃,
        
        H = 4 π/N ( ϕ̇̃̇^Tϕ̇̃̇ - ∑_i=1,2,3 [L_i,ϕ̃^T][L_i,ϕ̃]  + μ^2  ϕ̃^Tϕ̃ ) , 
        
        while the U(N) invariant metric (from which the measure
        derives) on configuration space is given by
        
        ds^2 =  dϕ^2  =  dϕ̃^T dϕ̃ .
        
        It is now easy to see that the Hamiltonian (<ref>)
        contains 2N+1 independent sectors.
        In (<ref>), consider ϕ̃
        to be a complex matrix and replace transpose with Hermitian
        conjugate.  On real matrices ϕ̃, the Hamiltonian
        is unchanged.  Now, consider the action of e^iα L_3
        on the complexified ϕ̃: ϕ̃→
        e^iα L_3ϕ̃e^-iα L_3. It is easy to see
        that the Hamiltonian is unchanged under this action.  
        Since under this action each term ϕ̃_ab
        in the complexified ϕ̃ acquires a phase e^i(a-b)α,
        we learn that the Hamiltonian contains terms with 
        (ϕ̃^†)_dcϕ̃_ab=
        ϕ̃_cd^⋆ϕ̃_ab only if (a-b)-(c-d)
        is zero.  Reducing ϕ̃ back to a real matrix,
        we see that the Hamiltonian does indeed contain 2N+1 independent
        sectors V^(m), each consisting of terms ϕ̃_ab with
        a fixed a-b=m, for m from -N to N.  We are able to compute entanglement
        entropy in each of those sectors separately, greatly speeding
        up the numerical analysis.
        
        In each sector V^(m), the Laplacian 
         operator 
         Δ : ϕ̃→∑_i=1,2,3 [L_i,[L_i,ϕ̃]]
        has the non-degenerate eigenvalues
        j(j+1) with corresponding eigenvectors v^(m)_j
        for j from |m| to N-1.
        To impose a UV cutoff, we focus on a subspace of V^(m) spanned
        by the  n-|m| eigenvectors v^(m)_j
        for j from |m| to n-1.
        Denote such a subspace with V^(m)_n and let
         O^(m)_n: V^(m)→ V^(m)_n
        be a linear operator such that
        O^(m)_n(v^(m)_j) = v^(m)_j for j∈{|m|, …, n-1}
        and O^(m)_n(v^(m)_j) = 0 for j∈{n, …, N-1}.
        Under the natural inner product, the eigenvectors of the 
        Laplacian are orthogonal, so we also have the transpose
        of O^(m)_n, (O^(m)_n)^T: V^(m)_n → V^(m)
        with  (O^(m)_n)^T(v^(m)_j) = v^(m)_j for j∈{|m|, …, n-1}.
        (We use a transpose rather than 
        a Hermitian conjugate as V^(m) and V^(m)_n are real vector spaces.)
        
        In <cit.> it was argued that the degrees of 
        freedom associated with a spherical cap centered on the north pole
        are represented by matrix entries ϕ_ab 
        (or ϕ̃_ab) with a+b < N (1-cos(θ)).
        For maximal control over
        selection of degrees of freedom, we define an operator Z,
        which `measures' the position of any degree of freedom along the
        L_3 (or z) axis.  This operator acts on any matrix entry
        ϕ_ab by multiplying it by N+1-(a+b).  When Z is restricted to
        a single sector V^(m), we denote the corresponding
        operator Z^(m).  
        Thus, we can easily identify, given θ, which degrees of freedom in
        each sector V^(m) are inside or outside the spherical cap.
        We denote with P_θ^(m) the operator acting on
        a sector V^(m) which projects onto the degrees
        of freedom within a given cap (with (P^(m)_θ)^2=
        P^(m)_θ).  
        
        The UV cutoff is associated with a projection operator
        (O^(m)_n)^T (O^(m)_n) which does not commute with
        P_θ^(m) or Z^(m).
        This is not surprising, and implies that imposing a cutoff `blurs' the edges
        of the spherical cap region.
        To consider a spherical cap in a theory with a UV
        cutoff, we can replace P^(m)_θ with
        P^(m)_θ,n:=O_n^(m) P_θ^(m)  
        (O_n^(m))^T, which acts on V^(m)_n.
        P^(m)_θ,n is not a projection operator itself.
        In a UV-restricted theory,
        we would identify degrees of freedom inside
        the spherical cap with an eigenspace of P^(m)_θ,n whose
        eigenvalues are greater than half.
        This is sensible as the eigenvalues of P^(m)_θ,n are between 0 and 1 by
        construction, with most of them close to either 0 or 1 and only
        a small subset falling in-between.  
        
        A better (it turns out) method to assign the degrees of freedom uses
        the operator Z^(m)_n:=O_n^(m) Z^(m)  (O_n^(m))^T.
        The inside of the spherical cap is associated with eigenvectors
        of Z^(m)_n whose eigenvalues are greater than z=(N-)cosθ.
        While one might think naively that methods based on operators
        P^(m)_θ,n and Z^(m)_n are
        interchangeble (at least in the large N limit), this turns
        out to not be the case.  While for entanglement entropy, the two
        methods do produce similar (though not identical) results, 
        significant artifacts are introduced into mutual information if the operator
        P^(m)_θ,n is used to assign degrees of freedom belonging to a 
        region (see section <ref>). 
        This highlights the delicate nature of delineating geometric
        regions in a fuzzy geometry.
        
        All the computations presented in the paper were
        obtained using Z_n^(m).
        In each UV-controlled sector V^(m)_n, we have two 
        self-adjoint operators, the Laplacian Δ and Z^(m)_n.  If we write
        the Hamiltonian in the basis in which Z^(m)_n is diagonal,
        we have a system of n-|m| coupled harmonic oscillators.
        We can use well-established procedures <cit.>
         for computing the entanglement
        entropy of a selected number of those oscillators (using the above
        criteria for which  degrees of freedom fall inside/outside
        a given spherical cap).
        
        
        
        
        § ENTANGLEMENT ENTROPY
        
        
        Leading order entanglement entropy of a spherical
        cap region on a commutative sphere is 
        proportional to n sinθ, where n is 
        a discretization parameter such that the total
        number of degrees of freedom is proportional to n^2<cit.>.
        We can think of the discretized commutative sphere
        as a noncommutative sphere with  N taken to infinity while
        n is held fixed.  We introduce a notation, S(N,n;θ)
        for the entanglement entropy of a spherical cap in a 
        noncommutative sphere; S(∞,n;θ) = const· nsinθ
        is then the local (commutative) result.
        
        In Figure <ref>, entanglement entropy 
        in noncommutative field theory
        is plotted against cap 
        size for a selection of UV cutoff parameters n. 
        For each value of n<N, a sine curve is plotted at an 
        amplitude matching the entanglement entropy at θ=π/2. 
        It is clear that imposing a UV cutoff has no effect on entanglement
        entropy for small cap sizes, and that entanglement entropy becomes
        proportional to the length of the cap's boundary once the cap
        is large enough at a given UV cutoff.  This is further highlighted in
        Figure <ref>, where the entanglement entropy is plotted
        as a function of the cap's area (top) and as a function of the length of the
        cap's boundary (bottom).  At any given UV cutoff, as θ grows, 
        there is a transition from extensive behaviour characteristic of 
        a highly non-local theory to area-law behaviour characteristic of
        a local theory.
        
        In Figures  <ref> and <ref>, 
        we highlight the point at which this transition happens with colored squares.
        Since this is a smooth cross-over rather than a sharp phase transition, the
        exact position of the cross-over is not well defined; we choose to
        use the intersection between sine curves (which represent behaviour expected 
        from a local theory) and the interpolated curve for n=N (which represents 
        nonlocal behaviour).  We have computed this cross-over point as a function
        of the cutoff parameter n; its roughly linear behaviour with n can be seen in Figure
        <ref> (bottom).
        
        This linear behaviour of the transition point between non-local and local
        behaviour in entanglement entropy is a clear demonstration of the UV-IR
        connection on the fuzzy sphere.  The UV-IR connection in noncommutative
         theories <cit.> implies that for a theory with a noncommutativity
         scale √(θ) defined by 
        [x_1,x_2]=iθ,
        the nonlocality scale is θ/λ_UV,
        where λ_UV is the UV cutoff wavelength.  On the fuzzy
        sphere with radius R, we have θ = R^2/J 
        (where N=2J+1),
        and the UV cutoff is λ̃_UV = R/n, implying that
        the transition should happen at θ_trans = const· n/N,
        consistent with the behaviour shown in Figure <ref>.
        
        
        
        
        
        
        To examine the behaviour of entanglement entropy in more detail, 
        we begin with dependence on the quantization parameter N.
        As was already observed in <cit.>, entanglement entropy
        S for cap with a fixed angular size grows linearly with N.  
        This is peculiar in a nonlocal theory with 
        N^2 degrees of freedom: we could naively expect entanglement entropy to
        be proportional to N^2.  However, the theory's nonlocality is not
        complete, allowing it to reproduce the behaviour of the local theory
        when expected.
        In Figure <ref>, we plot S(N,n;θ)/N as a function of θ,
        to show that it converges to a function of n/N and θ.
        
        Further, at a fixed N, the entanglement entropy of a hemisphere,
        S(N,n,π/2), is proportional to the cutoff parameter n,
        as can be seen in Figure <ref> (top). 
        Together, Figures <ref> and <ref> establish that
        the entanglement entropy of any spherical cap with θ>θ_trans
        is proportional to the UV cutoff parameter n.  
        
        Entanglement entropy for small regions 
        is smaller than the local theory would predict.
        Further, for small regions, the entanglement entropy is independent
        of the imposed UV cutoff (though it is still proportional to N).
        This is a sign that only low-momentum modes participate in entanglement
        entropy when θ < θ_trans.  
        
        
        
        In summary, in the large N limit, we can write
        
        S(N,n;θ) = 
        
        N s_extensive(θ) 
              for θ<θ_trans(n/N) 
        
        C n sin(θ) 
              for θ>θ_trans(n/N) 
         ,
        
        where C is a constant,  
        θ_trans is a function of n/N only,
        and s_extensive(θ) ∼θ^2 for
        small θ.
        The above result includes behaviour of a commutative theory,
        S(∞,n;θ)=C nsin(θ), since for n/N=0, 
        θ_trans=0.
        
        
        
        
        
        
        
        
        
        
        Finally, we explore mass dependence of the entanglement
        entropy.  In Figure <ref>, we show that entanglement
        entropy with a UV cutoff has similar behaviour to that without
        an imposed UV cutoff (which was discussed in <cit.>),
        being insensitive to changes in mass as long as μ < n
        and tending towards zero for μ >n.
        In Figure <ref>, we plot S against θ for 
        low and high masses. In the low mass case, the entanglement entropy
        is nearly unchanged from μ=1.  In the high mass case, as the
        mass term dominates the kinetic energy terms, entanglement decreases,
        still however retaining some of the qualitative features of the intermediate
        mass case.
        Figure <ref> further compares 
        the μ=1 and μ=10^-3 cases.  We see that the main
        effect of reducing the mass is a nearly-constant shift in
        the entanglement entropy that is the result of an appearance
        of a very-light mode at small mass.  
        
        
        
        
         
        
        § MUTUAL INFORMATION
        
        Another quantity that behaves in an interesting way in
        strongly coupled theories <cit.> is mutual information.
        Mutual information is a useful quantity to study because it is
        UV finite and—perhaps because of that— is the same
        on the fuzzy sphere as it is on the ordinary sphere 
        <cit.>.  Given this, we compute
        mutual information to validate our choice of method for
        the assignment of degrees of freedom.
        
        
        
        
        Mutual information is defined for a pair of regions.  Taking
        those to be two polar caps  C_1 and C_2 centered at opposite 
        poles of the sphere, mutual information is given by
        
            I = S(C_1)+S(C_2)-S(C_1 ∪ C_2) .
        
        To be able to compute it, we need to be able to assign degrees of freedom
        (subspaces of V^(m)_n)
        to caps C_1 and C_2 as well as their union C_1 ∪ C_2.
        One condition we must impose is that the linear subspace
        associated with C_1 must be orthogonal to the linear
        subspace associated with C_2; otherwise, we would
        have a situation where two functions, despite of having support
        on just one of two disjoint regions, have a non-zero overlap.
        This can be easily and naturaly achieved with the operator
        Z_n^(m) defined in section <ref>.  Given
        angular sizes θ_1 and θ_2 
        of the two spherical caps C_1 and C_2 (with
        θ_2 measured from the south pole), we have
        z_1=(N-)cosθ_1 and   z_2=-(N-)cosθ_2.
        Degrees of freedom within C_1 are given by eigenvectors of
        Z_n^(m) with eigenvalues greater than z_1
        and  degrees of freedom within C_2 are given by eigenvectors of
        Z_n^(m) with eigenvalues less than z_2.  Because
        these eigenvectors are mutually orthogonal, C_1 ∪ C_2
        can be simply associated with the direct sum of these
        two subspaces.
        
        
        
        
        For simplicity, we compute mutual information I(N,n;θ)
        when the two polar caps C_1 and C_2 
        have the same angular size θ.  
        Following our notation for entanglement entropy,
        we denote mutual information on a commutative sphere with
        I(∞,n;θ).
        In <cit.> it was shown that
        I(n,n;θ) is independent of n and that I(n,n;θ)=I(∞,n;θ). 
        Given this, we would expect that I(N,n;θ) would be independent
        of n over the entire range of N∈[n,∞).  This is
        indeed what is seen in Figure <ref>:
        mutual information seems unaffected by the cutoff, except for
        some artifacts having to do with discretization of the angle θ,
        which should go away at large N.  
        This fact is further evidence that our prescription for
        identifying degrees of freedom associated with spherical cap
        regions works.  Mutual information turns out to be a nontrivial
        test for the validity of degree of freedom assignment.
        In the reminder of the paper, we discuss different
        approaches to this assignment and explain why they appear to be inadequate.
        
        
        
        
        In section <ref> we defined a nearly-projection operator
        P^(m)_θ,n:=O^(m)_n P^(m)_θ  (O^(m)_n)^T.
        We now define[Omitting indices at this point
        to reduce clutter.]P_1:=O^(m)_n P^(m)_north,θ_1  
        (O^(m)_n)^T
        and P_2:=O^(m)_n P^(m)_south,θ_2  (O^(m)_n)^T
        and try to associate eigenvectors of P_1 and P_2
        with eigenvalues greater than half with degrees of freedom of the
        corresponding spherical cap.
        Unfortunately,
        while the associated subspaces are linearly
        independent vector spaces,[This is easy 
        to show.  Let v be any normalized linear combination of eigenvectors
        of P_1 with an eigenvalue greater than half.  Then,
        v^T P_1 v >.  Since Id_V_m-P_θ_1-P_θ_2 
        is obviously a positive semidefinite operator,
        Id_V_m^n-P_1-P_2 is also a positive
        semidefinite operator.  Therefore, v^T P_2 v < and v
        cannot be a linear combination of eigenvectors
        of P_2 with an eigenvalue greater than half.]
        they are not guaranteed to be orthogonal.
        In addition to re-assigning degrees of freedom slightly
        to fix this, we also need to make sure that the vector
        space associated with C_1 ∪ C_2 is the direct
        sum of vector spaces associated with C_1 and C_2.
        This requirement introduces an ambiguity, because there
        are degrees of freedom that do not appear to correspond to C_1
        or C_2, but which could reasonably be thought of as belonging to C_1∪ C_2.[
        Consider that there might exist a normalized vector v in V_n^(m)
        such that v^T P_1 v < and v^T P_2 v <, but
        v^T (P_1 +  P_2) v >.]
        We can choose to associate  a direct
        sum of vector spaces associated with C_1 and C_2
        as the vector space associated with C_1 ∪ C_2.
        We call this the `exclusive' method as it appears to
        include the smallest possible number of degrees of freedom with C_1 ∪ C_2.
        An alternative is an `inclusive' method, where we
        first fix the vector space associated with C_1 ∪ C_2
        as the span of eigenvectors of 
        P_1 + P_2 with eigenvalues greater than half, and then
        decide how to break this vector space into a sum of two pieces 
        associated with C_1 and C_2 respectively.  
        This second method includes more degrees of freedom in C_1 ∪ C_2
        than the first.  
        In either of the two cases, it is necessary to decide how to break
        a vector space into two pieces associated with either spherical
        cap.  We do this by examining the eigenvectors of P_1-P_2
        (restricted to the space associated with C_1 ∪ C_2). Those
        associated with positive eigenvalues we assign to cap C_1 and
        those associated with negative eigenvalues we assign to cap C_2.
        Notice that if there is no UV cutoff imposed (n=N), P_1, P_2
        and P_1+P_2 are projection operators, none
        of the above ambiguities arise and both methods reduce to
        that using Z^(m)_n.
        
        
        Not only are the assignment methods based on P_1 and P_2
        much more complicated than the one based on Z^(m)_n,
        using them yields different results for mutual information.
        Artifacts appear in mutual information I(N,n;θ) for
        θ larger than the transition θ_trans.
        Those artifacts are UV finite in the exclusive assignment method 
        and grow like N in the inclusive assignment method.
        It would be interesting to understand further why out of
        several
        supperficially similar methods for assigning degrees of
        freedom, only one leads to sensible mutual information.
        
        
        
        
        
        
        
        
        § ACKNOWLEDGMENTS
        
        
        This work was completed with support from the
        Natural Sciences and Engineering Council
        of Canada (NSERC), grant SAPIN-2016-00032.
        
        
        JHEP
