Semi-classical wave packet dynamics in non-uniform electric fields
Matthew F. Lapa Taylor L. Hughes












    [email address: ]mlapa@uchicago.eduKadanoff Center for Theoretical Physics, University of Chicago, Illinois 60637, USA[email address: ]hughest@illinois.eduDepartment of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, 61801-3080, USA
        
        We study the semi-classical theory of wave packet dynamics in crystalline solids extended to include the effects of a non-uniform 
        electric field. In particular, we derive a correction to the semi-classical equations of motion (EOMs) for the dynamics of the wave packet
        center that depends on the gradient of the electric field and on the quantum metric 
        (also called the Fubini-Study, Bures, or Bloch metric) on the
        Brillouin zone. We show that the physical origin of this term is a contribution to the total energy of the
        wave packet that depends on its electric quadrupole moment and on the electric field gradient. 
        We also derive an equation relating the electric quadrupole moment of a sharply-peaked wave packet to the quantum 
        metric evaluated at the wave packet center in reciprocal space. Finally, we explore the physical consequences of this
        correction to the semi-classical EOMs. We show that in a metal with broken time-reversal and inversion
        symmetry, an electric field gradient can generate a longitudinal current which 
        is linear in the electric field gradient, and which depends on the quantum metric at the Fermi surface. 
        We then give two examples of concrete lattice models in which this effect occurs. Our results show that non-uniform 
        electric fields can be used to probe the quantum geometry of the electronic bands in metals and open the door to further 
        studies of the effects of non-uniform electric fields in solids.  
        
        Semi-classical wave packet dynamics in non-uniform electric fields
            Taylor L. Hughes
            December 2017 (Updated Feb 2018 and Dec 2018)
        ==================================================================
        
        
        
        Electron wave packets in a solid placed in an applied electric field experience an anomalous contribution to 
        their velocity which has its origin in the Berry curvature of the electronic 
        bands. This anomalous velocity is responsible for the quantized Hall conductivity of Chern insulators, and the 
        intrinsic contribution to the anomalous Hall effect in metals, among other things, 
        and these effects can all be understood within
        a framework based on the semi-classical equations of motion (EOMs) for electron wave packets in 
        solids <cit.> (see also
        the review <cit.>). Combining the semi-classical EOMs with a Boltzmann equation approach to transport
        is particularly useful in the search for novel physical consequences of band geometry and 
        topology <cit.>.
        
        One issue which is not addressed by the current semi-classical framework is the effect of a
        non-uniform electric field on wave packet motion. However, it is known that non-uniform electric fields 
        probe some of the most subtle and interesting effects in condensed matter systems, for example the
        Hall viscosity in quantum Hall systems <cit.> and electrical multipole moments in 
        insulators <cit.>. 
         In addition, it is likely that non-uniform electric fields can 
        have significant effects in metals where the partially filled conduction band can respond quickly to an applied field.
        These expectations motivate a systematic study of the semi-classical EOMs in an expansion in 
        spatial derivatives of the external electric field. 
        
        In this article we initiate this study by considering the semi-classical dynamics of electron wave packets in the presence
        of a constant electric field gradient. We derive a correction to the usual 
        semi-classical EOM for the time derivative of the wave packet center in real space. This correction 
        depends on the gradient of the electric field, and on the quantum metric <cit.> on the Brillouin zone (BZ)  for the electronic band
        whose states are used in the construction of the wave packet. The quantum metric (also called the 
        Fubini-Study metric, Bures metric, Bloch metric, etc.) has previously been studied in 
        the context of band theory and the semi-classical EOMs in 
        Refs. marzari-vanderbilt,matsuura2010momentum,Neupert,kolodrubetz2013classifying,legner2013relating,
        ma2014quantum,gao2014field,gao2015geometrical,srivastava2015signatures,piechon2016geometric,freimuth2017geometrical,
        liang2017wave,ozawa2018steady,iskin2018quantum,ozawa2018extracting,bleu2018effective,iskin2018exposing.
        
        
        The correction to the semi-classical EOMs that we derive depends on the derivative
        of the quantum metric. As a consequence, we show that this correction does not affect transport in insulators. 
        On the other hand, we show one direct effect of this correction on transport in metals where it can lead
        to a longitudinal current proportional to the electric field gradient and to the quantum metric at
        the Fermi surface. Thus, our result shows that the quantum geometry of bands in metals can be probed by a 
        transport experiment using a non-uniform electric field. We now turn to an explanation of our results.
        
        
        
        Setup: We study the dynamics of electrons of charge Q=-e in a crystal and in the presence of a 
        time-independent electric field E(x). 
        Let x̂^μ, p̂_ν, μ,ν=1,…,D, denote the position
        and momentum operators for a single electron in D spatial dimensions, with 
        [x̂^μ,p̂_ν]=iħδ^μ_ν. The single particle Hamiltonian is
        Ĥ=Ĥ_0 + Q(x̂) ,
        
        where Ĥ_0 is a Hamiltonian for an electron in a periodic potential V(x̂), for example the 
        standard non-relativistic Hamiltonian 
        Ĥ_0= 1/2mδ^μνp̂_μp̂_ν + V(x̂)
        for particles of mass m. In fact, our only requirement for Ĥ_0 is that it be subject to Bloch's theorem. 
        The second term in Ĥ captures the coupling to the electric field 
        E(x), which is determined by the potential (x) as 
        E_μ(x)= -(x)/ x^μ. 
        
        Bloch's theorem implies that Ĥ_0 has a basis of eigenstates (“Bloch waves") |ψ_n,q which are labeled 
        by a band index n and a wavevector q (in the first BZ) and obey 
        Ĥ_0|ψ_n,q= ℰ_n(q)|ψ_n,q, where 
        ℰ_n(q) are the energy eigenvalues. In addition, we can write
        |ψ_n,q= e^i q_μx̂^μ|u_n,q where the function
        u_n,q(x):= x|u_n,q has the periodicity of the crystal lattice. 
        Note that the Bloch states are time-independent since we have made the
        simplifying assumption that our Hamiltonian is time-independent.
        We normalize the Bloch states so that ψ_n,q|ψ_n',q'= δ_n,n'δ^(D)(q-q'),
        which implies that u_n,q| u_n',q = δ_n,n'. Here, the
        the inner product of the |u_n,q is defined as integration over the real space unit cell times a factor of 
        (2π)^D/v_c, where v_c is the volume of the real space unit cell <cit.>.
        We also introduce a crystal momentum operator q̂_μ which is diagonal
        in the basis of Bloch states and satisfies q̂_μ|ψ_n,q= q_μ|ψ_n,q.
        
        We are interested in the leading corrections to the semi-classical EOMs due to a non-zero electric field gradient,
        and so we choose a potential of the form 
        (x)= -E^(0)_μx^μ-1/2E^(0)_μνx^μx^ν,
        where E^(0)_μ and E^(0)_μν (with E^(0)_μν=E^(0)_νμ) are 
        two sets of constant parameters. The components of the electric field are then 
        E_μ(x)= E^(0)_μ + E^(0)_μνx^ν.
        We see that E^(0)_μ specify the uniform part of the electric field, while E^(0)_μν specify the 
        electric field gradient.
        
        Wave packets and their first moments: We study the time-evolution (using the full Hamiltonian Ĥ) 
        of a wave packet |Ψ(t) constructed from the Bloch states 
        |ψ_n,q. We assume that the wave packet is constructed from states within a single band,
        and so we drop the band index n from the notation. We define this wave packet state as 
        |Ψ(t)= ∫ d^Dq a(q,t)|ψ_q,
        where a(q,t) is a complex amplitude which must satisfy the normalization condition 
        ∫ d^Dq |a(q,t)|^2=1 (q integrals run
        over the first BZ). By plugging into the Schrodinger equation 
        iħd/dt|Ψ(t) = Ĥ|Ψ(t), one can show that a(q,t) satisfies 
        
            iħȧ(q,t)= a(q,t)ℰ(q) + Q ∫ d^Dq' a(q',t) ψ_q|(x̂)|ψ_q' ,
        
        where the dot denotes a time derivative. 
        
        The semi-classical EOMs for wave packet dynamics in solids can be derived by studying the dynamics
        of the first momentsX^μ(t) and K_μ(t) of the wave packet in position and reciprocal space, 
        respectively. These are defined by X^μ(t) = Ψ(t)|x̂^μ|Ψ(t) and
        K_μ(t) = Ψ(t)|q̂_μ|Ψ(t)= ∫ d^Dq q_μ |a(q,t)|^2.
        We derive the semi-classical EOMs for X^μ(t) and K_μ(t) by first computing the 
        exact expressions for Ẋ^μ(t) and K̇_μ(t), and then truncating these expressions using the
        assumption that the wave packet is sharply-peaked about the locations X^μ(t) and K_μ(t) in position and 
        reciprocal space. To derive the equations for Ẋ^μ(t) and K̇_μ(t) we simply 
        differentiate the expressions for X^μ(t) and K_μ(t) with respect to time, and then we substitute in 
        for ȧ(q,t) and ȧ(q,t) using Eq. (<ref>) and its complex conjugate.
        
        After a tedious but straightforward calculation, we find that the equation for 
        Ẋ^μ(t) takes the form
        
            Ẋ^μ(t) =1/ħℰ(q̂)/ q_μ_t - 1/ħQE^(0)_νΩ^μν(q̂)_t  -1/2ħQ E^(0)_ν{x̂^, Ω^μν(q̂)}_t - 1/2ħQ E^(0)_ν g^ν(q̂)/ q_μ_t ,
        
        where ·_t denotes an expectation value in the state |Ψ(t), and
        {·,·} denotes an anti-commutator (third term on the right-hand side). In this equation Ω^μν(q)
        is the Berry curvature, which is expressed in terms of the Berry connection 
        𝒜^μ(q)= i u_q|  u_q/ q_μ as 
        Ω^μν(q) = 𝒜^ν(q)/ q_μ-𝒜^μ(q)/ q_ν. The quantity g^μν(q) is the quantum metric on the BZ, and is defined as
        
            g^μν(q)=1/2(  u_q/ q_μ|  u_q/ q_ν   -  u_q/ q_μ|u_q u_q|  u_q/ q_ν
               + (μ↔ν) ) .
        
        Both Ω^μν(q) and g^μν(q) are invariant under a gauge transformation 
        |u_q→ e^-i f(q)|u_q for any function f(q). The equation for 
        K̇_μ(t) is much simpler, and it takes the form
        K̇_μ(t)= 1/ħQ E_μ(X(t)) , 
        where E_μ(X(t))= E^(0)_μ + E^(0)_μνX^ν(t) is the electric field at the location of the
        first moment X^μ(t).
        
        To derive these equations, it is necessary to use explicit expressions for the matrix elements 
        ψ_q|x̂^μ|ψ_q' and ψ_q|x̂^μx̂^ν|ψ_q'
        of the position operator in the Bloch states. We record these expressions in Eqs. (3) and (4) of the Supplemental Material <cit.>.
        In the derivation we also used several integrations by parts in integrals over the BZ, and we neglected boundary 
        terms. If the amplitudes a(q,t) or the Berry connection 𝒜^μ(q) are not single-valued, then there could be 
        some interesting, subtle additions to these modified EOMs. 
        In the Supplemental Material we show that by a suitable choice of gauge for the
        Bloch states |ψ_q, we can make a(q,t) single-valued for all t. In that case the only possible source of
        boundary corrections is the Berry connection. Here we assume that no boundary corrections arise, and we
        leave a detailed discussion of any alternatives to future work.
        
        To obtain the semi-classical EOMs for 
        X^μ(t) and K_μ(t) we make the substitutions
        x̂^μ→ X^μ(t) and q̂_μ→ K_μ(t) in all
        expectation values in Eq. (<ref>) and Eq. (<ref>). Our result, which is one of the main results of 
        this article, is that the semi-classical EOMs take the form
        
            Ẋ^μ   = 1/ħℰ(K)/ K_μ -Ω^μν(K)K̇_ν-1/2ħQE^(0)_ν g^ν(K)/ K_μ
            K̇_μ   = 1/ħQ E_μ(X) ,
        
        where we also used the second equation to rewrite part of the Ẋ^μ(t) equation in terms of 
        K̇_μ(t). The main difference compared to the usual semi-classical EOMs is the term
        -1/2ħQ E^(0)_ν g^ν(K)/ K_μ.
        This new term depends on the gradient of the electric field, since it depends on E^(0)_ν but not
        E^(0)_μ, and it also probes the geometry of the band 
        structure since it involves the quantum metric g^ν(K). 
        
        Interpretation: We now show that the new term in 
        (<ref>) arises from an electric field-induced correction to the energy of the wave packet. 
        In the absence of an electric field we have
        Ψ(t)|Ĥ_0|Ψ(t) = ∫ d^Dq |a(q,t)|^2ℰ(q) ≈ℰ(K), 
        where K(t) is the wave packet center. In the presence of the electric field, we show in the 
        Supplemental Material that the wave packet energy takes the form
        
            Ψ(t)|Ĥ|Ψ(t)   ≈ℰ(K) - Q E^(0)_μX^μ
               - 1/2Q E^(0)_μν( X^μX^ν + g^μν(K) ) 
               ≡ℰ_eff(X,K) .
        
        As a result, the corrected semi-classical EOM for Ẋ^μ(t) can be rewritten as
        Ẋ^μ = 1/ħℰ_eff(X,K)/ K_μ -Ω^μν(K)K̇_ν .
         
        We can also rewrite the equation for K̇_μ(t) as 
        K̇_μ = -1/ħℰ_eff(X,K)/ X^μ.
        
        In this form, the correction to the Ẋ^μ(t) and K̇_μ(t) equations closely 
        resembles a similar correction which occurs for electrons 
        in a magnetic field. In that case the correction to the wave packet energy arises from the magnetic moment of the 
        wave packet <cit.>. In the present case of a non-uniform electric field, the corrections to the energy depend on 
        the dipole moment X^μ(t) of the wave packet (the term proportional to E^(0)_μ), and on the
        quadrupole moment of the wave packet (the term proportional to E^(0)_μν). Indeed, in the 
        Supplemental Material we show that for a wave packet |Ψ(t) sharply peaked at position K in reciprocal space, the 
        quadrupole moment is given by
        Ψ(t)|x̂^μx̂^ν|Ψ(t)≈ X^μX^ν + g^μν(K) . 
        The correction due to the dipole moment is already present in the case of a uniform electric field, and it does not alter the
        semi-classical EOMs. On the other hand, the correction proportional to the quadrupole moment is only present in a non-uniform 
        field, and it does alter the semi-classical EOMs. We also note that to find
        the g^μν(K) term in ℰ_eff(X,K), we need to expand
        QΨ(t)|(x̂)|Ψ(t) to second order about the wave packet center in real space, and so this
        term cannot be found using the first order expansion of Ref. sundaram-niu.
        
        Physical consequences: We now discuss physical consequences of the new term in Eq. (<ref>) for
        transport in solids. Within the semi-classical approach, the current density j^μ(r) at position r in the
        material is given by j^μ(r)= Q∫ d^DXd^DK/(2π)^D f(X,K,t)Ẋ^μδ^(D)(X-r), where f(X,K,t) is the non-equilibrium distribution function which specifies the occupation, at
        time t, of the volume element d^DXd^DK/(2π)^D at position (X,K) in phase space.
        The full distribution function can be obtained by solving the Boltzmann equation. In the relaxation time approximation, 
        with relaxation time τ, f(X,K,t) takes the form of a power series in τ, 
        f(X,K,t)= f_0(K) + O(τ), where f_0(K) is the equilibrium distribution function specifying
        the occupied states in reciprocal space at temperature T <cit.>. 
        In what follows, we will be interested in the currents which come from 
        this zeroth order contribution, which captures the intrinsic part of the linear response of the system to the applied electric field. 
        The zeroth order contribution to the current is then j^μ_0(r)= Q∫d^DK/(2π)^D f_0(K)
        Ẋ^μ|_X=r. In D=2, for example, j^μ_0(r) contains the intrinsic contribution 
        to the anomalous Hall effect. Using Eq. (<ref>), we find that 
        j^μ_0(r) contains the additional term
        
        j^μ_geom.(r)= -Q^2/2ħ∫d^DK/(2π)^D f_0(K) E^(0)_ν g^ν(K)/ K_μ , 
        which involves the electric field gradient and the quantum metric. We will refer to j^μ_geom. as the 
        geometric current.
        
        The geometric current is easiest to understand in the case of a metal in D=1 dimension 
        (so μ,ν=1 in all equations). Recall that we considered wave packets constructed from states in a 
        single band. We assume a partial filling of this band such that the Fermi level ℰ_F crosses the band at the set of wave 
        numbers {k_I,+,k_I,-}_I∈{1,…,n_F} for some integer n_F (so 2n_F is the total number of Fermi points). Our 
        notation means that ℰ(K_1)/ K_1 is positive at a + Fermi point and negative at a - Fermi point 
        (we assume that ℰ_F is chosen so that there is no Fermi point where ℰ(K_1)/ K_1 
        vanishes). At temperature T=0 the distribution function f_0(K_1) is equal to 1 if ℰ(K_1)≤ℰ_F
        and zero otherwise. After an integration by parts, and using
        f_0(K_1)/ K_1= ∑_I=1^n_F[ δ(K_1-K_I,-) - δ(K_1-K_I,+)], 
        we find that (h=2πħ)
        
        	j^1_geom. = -1/2Q^2/hE^(0)_11∑_I=1^n_F[ g^11(K_I,+)- g^11(K_I,-)] , 
        which is non-zero if the sum does not equal zero. 
        
        
        
        Next, we consider a similar example for a metal in D=2. To illustrate the nontrivial response
        we compute j^1_geom. as an example. We again consider
        a single band and we assume the Fermi surface consists of a single 
        closed contour 𝒞. For simplicity, we assume further that the parts of 𝒞 to the left and right of the
        K_2 axis can be specified by single-valued functions h_L(K_2), h_R(K_2), such that K_1=h_L(K_2) defines
        the part of 𝒞 to the left of the K_2 axis, and K_1=h_R(K_2) defines the part to the right (note that
        for a generic 𝒞 the functions h_R/L(K_2) would not be single-valued). Let
        K_2,+ >0 and K_2,-<0 be the two points where 𝒞 intersects the K_2 axis. This situation
        is illustrated in Fig. <ref>. At T=0 the distribution function for this metal is f_0(K)=1 for
        K inside 𝒞, and zero otherwise.
        We then have f_0(K)/ K_1= δ(K_1-h_L(K_2))-δ(K_1-h_R(K_2)), and so 
        
            j^1_geom.   = -Q^2/2 hE^(0)_ν/2π( ∫_K_2,-^K_2,+ dK_2  g^ν(K)|_K_1=h_R(K_2)
               - ∫_K_2,-^K_2,+ dK_2 g^ν(K)|_K_1=h_L(K_2)) .
        
        Since j^1_geom. involves an integral of g^μν(K) only on 𝒞, we see that this current is a Fermi 
        surface property, like the intrinsic contribution to the anomalous Hall effect <cit.>, and it vanishes in insulators (which have
        a full band, f_0(K)=1∀ K). 
        
        
        Symmetry analysis: In systems with time-reversal symmetry we have
        ℰ(K)=ℰ(-K) and g^μν(K)= g^μν(-K), and identical
        conditions hold in the case of inversion symmetry. These conditions imply that j^μ_geom.=0.
        To prove this we use these conditions to first replace g^μν(K) in Eq. (<ref>) with
        g^μν(-K). Next, we use the fact that f_0(K)=f_0(-K) if f_0(K) is a function of 
        ℰ(K) only (as it would be in thermal equilibrium or at T=0). Finally, we 
        change integration variables from K to -K to find that time-reversal
        or inversion symmetry imply that j^μ_geom.= - j^μ_geom., and so 
        j^μ_geom.=0. Therefore we must break these symmetries to obtain j^μ_geom.≠ 0.
        
        Examples in D=1 and D=2: We now discuss two examples of lattice models of metals in D=1 and D=2
        which yield a non-zero geometric current. We present the detailed results for the geometric current in these models in 
        the Supplemental Material. In D=1 we consider the two-band model with Bloch Hamiltonian 
        
        	H_1D(k)= Asin(k)𝕀 + sin(k)σ^x + (m+1-cos(k))σ^z , 
        where 𝕀 is the 2× 2 identity matrix and σ^x,y,z are the Pauli matrices. In D=2 we consider
        the two-band model with Bloch Hamiltonian 
        
            H_2D(k)   = Asin(k_1)𝕀 + sin(k_1)σ^x + sin(k_2)σ^y 
               + (m+2-cos(k_1)-cos(k_2))σ^z  .
        
        In both cases we choose the parameters m and A so that there is an energy gap between the two bands of the model. We
        then fill the lower band and partially fill the upper band to a Fermi energy
        ℰ_F to obtain a model of a metal. In the Supplemental Material we show that under these conditions, and when the parameter
        A≠ 0, both of these models display a nontrivial geometric current in the presence of a non-uniform electric field in the X^1 
        direction (i.e., E^(0)_11≠ 0). In both cases the condition A≠ 0 is required to break inversion and/or time-reversal 
        symmetry, which then allows for a non-zero j^μ_geom. according to our previous discussion.
        
        Discussion: A natural question to ask is how one can distinguish the geometric current of Eq. (<ref>)
        from a more typical longitudinal current of the Drude form. The Drude contribution has the form
        
        	j^1_Drude(r^1)= τQ^2/hE_1(r^1)∑_I=1^n_F[v(k_I,+)-v(k_I,-) ] ,
        
        where τ is the relaxation time and v(k)=ℰ(k)/ k. To distinguish this from Eq. (<ref>) 
        we choose an electric field which is a pure gradient around an origin,
        E_1(r^1)= E^(0)_11r^1. 
        We then compute the average of the current over a 
        spatial region centered at that origin r^1∈[-L/2,L/2]. 
        We find that ∫_-L/2^L/2 dr^1 j^1_Drude(r^1)=0, while 
        ∫_-L/2^L/2dr^1  j^1_geom.(r^1)= -L Q^2 E^(0)_11/2h∑_I=1^n_F[ g^11(K_I,+)- g^11(K_I,-)] . 
        Thus, a spatial average of the current about the origin can distinguish between these two kinds of responses when the electric
        field is a pure gradient (“pure" refers to the fact that E_1(0)=0 and E_1(r^1) is linear near r^1=0). 
        
        Eq. (<ref>) shows that information about the quantum metric at the Fermi points can be extracted
        from a transport experiment using an electric field which is a pure gradient. By averaging the current over a spatial region
        which is symmetric about the origin, any Drude contribution to the current will be canceled. Then, since
        L (the length of the spatial region), Q, E_11^(0), and h are known to the experimenter, the
        signed sum ∑_I=1^n_F[ g^11(K_I,+)- g^11(K_I,-)] can be extracted from this transport data.
        
        A second natural question concerns the conditions under which the electric field gradient term is expected to 
        significantly alter the semi-classical dynamics. After all, if the electric field varies slowly over the width of the wave packet, then
        it should be reasonable to neglect the gradient term. To understand the relevant scales we use Eq. (<ref>), 
        which implies that the squared spread Ψ(t)|x̂^μx̂^ν|Ψ(t) - X^μX^ν of a wave
        packet sharply peaked at K in reciprocal space is equal to g^μν(K). 
        For simplicity, consider the case of D=1. Then the width of the wave packet is
        √(g^11(K)) and so the change of the electric field over the width of the wave packet is 
        Δ E_1≈ E^(0)_11√(g^11(K)). If Δ E_1 ≪ E^(0)_1 
        (the uniform part of the electric field), then we can
        neglect the gradient term. On the other hand, we must include this gradient term if Δ E_1 is comparable to or
        larger than E^(0)_1.
        
        
        Conclusion: In this article we extended the semi-classical theory of electron wave packet
        motion in solids to incorporate the effects of a non-uniform electric field. In particular, we 
        systematically calculated corrections to the semi-classical EOMs in an expansion in derivatives of the electric field, and 
        we obtained the correction proportional to the first derivative of the electric field.
        Our main result, shown in Eqs. (<ref>), is a correction to the semi-classical EOM for the wave packet 
        center in real space which depends on the electric field gradient, and on the quantum metric 
        g^μν(q) on the BZ. 
        We then gave a physical interpretation of this new term as arising from the energy associated with the electric quadrupole 
        moment of the wave packet in the presence of the non-uniform electric field. 
        We also showed that this correction to the semi-classical EOMs does not affect transport in insulators, 
        but does lead to a nontrivial transport signature in metals with 
        broken time-reversal and inversion symmetry. Specifically, we showed that in such metals an electric field gradient
        can generate a longitudinal current which is proportional to the electric field gradient and to the quantum 
        metric at the Fermi surface. Since the current depends only on the quantum metric at the Fermi surface, 
        we expect that it will be robust to the inclusion of interaction or disorder effects, as in the case of the
        anomalous Hall effect in metals <cit.>.
        
        
        We envision at least two possible directions for future work. The first would be to understand the corrections to the
        semi-classical EOMs (<ref>) which involve higher derivatives of the electric field. 
        The correction proportional to the second derivative
        would be particularly interesting as it should allow for a derivation of an analog of the formula of Hoyos and Son <cit.>, 
        which relates the finite wave vector Hall conductivity
        of a quantum Hall system to the Hall viscosity, but in the context of Chern insulators (where there is no magnetic
        field) instead of Landau levels. 
        A second direction would be to derive semi-classical EOMs for the higher moments of the
        wave packet, for example the second moments X^μν(t):=Ψ(t)|x̂^μx̂^ν|Ψ(t) 
        and K_μν(t):=Ψ(t)|q̂_μq̂_ν|Ψ(t) in position and reciprocal space, respectively.
        In particular, it would be interesting to understand how these second moments respond to non-uniform electric fields. 
        We leave these topics for future work.
        
        Note added: After this work was completed we became aware of Ref. gao-xiao, which obtained many
        of the same results as part of a study of nonreciprocal directional dichroism in crystalline solids.
        
        Acknowledgments: We thank A. Alexandradinata for a useful conversation. M.F.L. acknowledges the support of the Kadanoff 
        Center for Theoretical Physics at 
        the University of Chicago. T.L.H thanks the U.S. National Science Foundation under grant DMR 1351895-CAR for support.
        
        
        
        
        
        
        
        
        
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        @bib@innerbibempty
        
         [Karplus and Luttinger(1954)]KL
          authorauthorR. Karplus and authorJ. M. Luttinger, @noop journaljournalPhys. Rev. volume95, pages1154
          (year1954)NoStop
         [Kohn and Luttinger(1957)]Kohn-Luttinger
          authorauthorW. Kohn and authorJ. M. Luttinger, @noop journaljournalPhys. Rev. volume108, pages590
          (year1957)NoStop
         [Luttinger(1958)]luttinger1958
          authorauthorJ. M. Luttinger, @noop journaljournalPhys. Rev. volume112, pages739
          (year1958)NoStop
         [Adams and Blount(1959)]adams-blount
          authorauthorE. Adams and authorE. Blount,@noop journaljournalJ. Phys. Chem.
          Solids volume10, pages286 (year1959)NoStop
         [Chang and Niu(1995)]chang1995berry
          authorauthorM.-C. Chang and authorQ. Niu,@noop journaljournalPhys. Rev. Lett.volume75, pages1348 (year1995)NoStop
         [Chang and Niu(1996)]chang-niu
          authorauthorM.-C. Chang and authorQ. Niu,@noop journaljournalPhys. Rev. Bvolume53, pages7010 (year1996)NoStop
         [Sundaram and Niu(1999)]sundaram-niu
          authorauthorG. Sundaram and authorQ. Niu,@noop journaljournalPhys. Rev. Bvolume59, pages14915 (year1999)NoStop
         [Haldane(2004)]haldane2004
          authorauthorF. D. M.Haldane, @noop journaljournalPhys. Rev. Lett. volume93, pages206602 (year2004)NoStop
         [Xiao et al.(2010)Xiao,
          Chang, and Niu]niu-review
          authorauthorD. Xiao, authorM.-C. Chang,   and authorQ. Niu, @noop
          journaljournalRev. Mod. Phys. volume82, pages1959 (year2010)NoStop
         [Moore and Orenstein(2010)]Moore2010
          authorauthorJ. E. Moore and authorJ. Orenstein, @noop journaljournalPhys. Rev. Lett. volume105, pages026805 (year2010)NoStop
         [Orenstein and Moore(2013)]Moore2013
          authorauthorJ. Orenstein and authorJ. E. Moore, @noop journaljournalPhys.
          Rev. B volume87, pages165110
          (year2013)NoStop
         [Zhong et al.(2015)Zhong,
          Orenstein, and Moore]Moore2015
          authorauthorS. Zhong, authorJ. Orenstein,   and authorJ. E. Moore,@noop journaljournalPhys. Rev. Lett.volume115, pages117403 (year2015)NoStop
         [Sodemann and Fu(2015)]Fu2015
          authorauthorI. Sodemann and authorL. Fu,@noop journaljournalPhys. Rev. Lett.volume115, pages216806 (year2015)NoStop
         [Hoyos and Son(2012)]hoyos-son
          authorauthorC. Hoyos and authorD. T. Son, @noop journaljournalPhys.
          Rev. Lett. volume108, pages066805
          (year2012)NoStop
         [Bradlyn et al.(2012)Bradlyn, Goldstein, and Read]bradlyn2012
          authorauthorB. Bradlyn, authorM. Goldstein,
           and authorN. Read,@noop journaljournalPhys. Rev. Bvolume86, pages245309 (year2012)NoStop
         [Benalcazar et al.(2017)Benalcazar, Bernevig, and Hughes]benalcazar2017
          authorauthorW. A. Benalcazar, authorB. A. Bernevig,  and authorT. L. Hughes, @noop journaljournalScience volume357, pages61
          (year2017)NoStop
         [Wheeler et al.(2018)Wheeler, Wagner, and Hughes]wheeler2018
          authorauthorW. A. Wheeler, authorL. K. Wagner,
           and authorT. L. Hughes,@noop journaljournalarXiv
          preprint:1812.06990  (year2018)NoStop
         [Kang et al.(2018)Kang,
          Shiozaki, and Cho]cho2018
          authorauthorB. Kang, authorK. Shiozaki,   and authorG. Y. Cho,@noop journaljournalarXiv preprint:
          1812.06999  (year2018)NoStop
         [Provost and Vallee(1980)]provost-vallee
          authorauthorJ. P. Provost and authorG. Vallee, @noop journaljournalCommun. Math. Phys. volume76, pages289 (year1980)NoStop
         [Marzari and Vanderbilt(1997)]marzari-vanderbilt
          authorauthorN. Marzari and authorD. Vanderbilt, @noop journaljournalPhys. Rev. B volume56, pages12847
          (year1997)NoStop
         [Matsuura and Ryu(2010)]matsuura2010momentum
          authorauthorS. Matsuura and authorS. Ryu,@noop journaljournalPhys. Rev. Bvolume82, pages245113 (year2010)NoStop
         [Neupert et al.(2013)Neupert, Chamon, and Mudry]Neupert
          authorauthorT. Neupert, authorC. Chamon,   and authorC. Mudry,@noop journaljournalPhys. Rev. Bvolume87, pages245103 (year2013)NoStop
         [Kolodrubetz et al.(2013)Kolodrubetz, Gritsev, and Polkovnikov]kolodrubetz2013classifying
          authorauthorM. Kolodrubetz, authorV. Gritsev,  and authorA. Polkovnikov, @noop journaljournalPhys. Rev. B volume88, pages064304
          (year2013)NoStop
         [Legner and Neupert(2013)]legner2013relating
          authorauthorM. Legner and authorT. Neupert, @noop journaljournalPhys.
          Rev. B volume88, pages115114
          (year2013)NoStop
         [Ma(2014)]ma2014quantum
          authorauthorY.-Q. Ma, @noop journaljournalPhys. Rev.
          E volume90, pages042133 (year2014)NoStop
         [Gao et al.(2014)Gao,
          Yang, and Niu]gao2014field
          authorauthorY. Gao, authorS. A. Yang,   and authorQ. Niu, @noop
          journaljournalPhys. Rev. Lett. volume112, pages166601 (year2014)NoStop
         [Gao et al.(2015)Gao,
          Yang, and Niu]gao2015geometrical
          authorauthorY. Gao, authorS. A. Yang,   and authorQ. Niu, @noop
          journaljournalPhys. Rev. B volume91, pages214405 (year2015)NoStop
         [Srivastava and Imamoğlu(2015)]srivastava2015signatures
          authorauthorA. Srivastava and authorA. Imamoğlu, @noop journaljournalPhys. Rev. Lett. volume115, pages166802 (year2015)NoStop
         [Piéchon et al.(2016)Piéchon, Raoux, Fuchs, andMontambaux]piechon2016geometric
          authorauthorF. Piéchon, authorA. Raoux,
          authorJ.-N. Fuchs,  andauthorG. Montambaux,@noop journaljournalPhys. Rev. Bvolume94, pages134423 (year2016)NoStop
         [Freimuth et al.(2017)Freimuth, Blügel, and Mokrousov]freimuth2017geometrical
          authorauthorF. Freimuth, authorS. Blügel,  and authorY. Mokrousov, @noop journaljournalPhys. Rev. B volume95, pages184428
          (year2017)NoStop
         [Liang et al.(2017)Liang,
          Peotta, Harju, and Törmä]liang2017wave
          authorauthorL. Liang, authorS. Peotta,
          authorA. Harju,  and authorP. Törmä, @noop journaljournalPhys. Rev. B volume96, pages064511 (year2017)NoStop
         [Ozawa(2018)]ozawa2018steady
          authorauthorT. Ozawa, @noop journaljournalPhys.
          Rev. B volume97, pages041108
          (year2018)NoStop
         [Iskin(2018a)]iskin2018quantum
          authorauthorM. Iskin, @noop journaljournalPhys.
          Rev. A volume97, pages033625
          (year2018a)NoStop
         [Ozawa and Goldman(2018)]ozawa2018extracting
          authorauthorT. Ozawa and authorN. Goldman, @noop journaljournalPhys.
          Rev. B volume97, pages201117
          (year2018)NoStop
         [Bleu et al.(2018)Bleu,
          Malpuech, Gao, and Solnyshkov]bleu2018effective
          authorauthorO. Bleu, authorG. Malpuech,
          authorY. Gao,  and authorD. D. Solnyshkov, @noop journaljournalPhys. Rev. Lett. volume121, pages020401 (year2018)NoStop
         [Iskin(2018b)]iskin2018exposing
          authorauthorM. Iskin, @noop journaljournalPhys.
          Rev. A volume97, pages063625
          (year2018b)NoStop
         [Blount(1962)]blount
          authorauthorE. I. Blount, in @noop booktitleSolid state
          physics, Vol. volume13 (publisherElsevier,year1962) pp. pages305–373NoStop
         [Lapa and Hughes()]supp
          authorauthorM. F. Lapa and authorT. L. Hughes, @noop howpublishedSupplemental material
          for “Semi-classical wave packet dynamics in non-uniform electric
          fields”NoStop
         [Gao and Xiao(2018)]gao-xiao
          authorauthorY. Gao and authorD. Xiao,@noop journaljournalarXiv:1810.02728 (year2018)NoStop
         [King-Smith and Vanderbilt(1993)]king-smith
          authorauthorR. King-Smith and authorD. Vanderbilt, @noop journaljournalPhys. Rev. B volume47, pages1651
          (year1993)NoStop
        
        
        § SUPPLEMENTAL MATERIAL FOR “SEMI-CLASSICAL WAVE PACKET DYNAMICS IN NON-UNIFORM ELECTRIC FIELDS”
        
        
         §.§ Quadrupole moment of the wave packet state
        
        
        Here we present an approximate calculation of the quadrupole moment 
        Ψ(t)|x̂^μx̂^ν|Ψ(t)
        of the wave packet state |Ψ(t) introduced in the main text. Our calculation uses the assumption that the wave
        packet is sharply peaked in reciprocal space. The quadrupole moment enters the calculation of 
        the wave packet energy Ψ(t)|Ĥ|Ψ(t), since we have
        
            Ψ(t)|Ĥ|Ψ(t)= Ψ(t)|Ĥ_0|Ψ(t)+ QΨ(t)|(x̂)|Ψ(t) ,
        
        and for our choice of potential the second term here takes the form
        Ψ(t)|(x̂)|Ψ(t)   =    -E^(0)_μΨ(t)|x̂^μ|Ψ(t)
           -   1/2E^(0)_μνΨ(t)|x̂^μx̂^ν|Ψ(t) .
        
        By definition of the wave packet center, we have Ψ(t)|x̂^μ|Ψ(t)= X^μ in the
        first term here. The second term involves the quadrupole moment of the wave packet. We now turn to the calculation of the
        quadrupole moment. 
        
        Our calculation of the quadrupole moment uses two important formulas for the matrix elements of 
        x̂^μ and x̂^μx̂^ν in the Bloch states |ψ_q which diagonalize 
        Ĥ_0. These formulas are:
        ψ_q|x̂^μ|ψ_q'= i/ q_μδ^(D)(q-q') + δ^(D)(q-q')𝒜^μ(q) 
        and
        
            ψ_q   |x̂^μx̂^ν|ψ_q' = ^2/ q_μ q_ν'δ^(D)(q-q') - i 𝒜^ν(q)/ q_μ'δ^(D)(q-q') 
               -  i 𝒜^μ(q)/ q_ν'δ^(D)(q-q') -δ^(D)(q-q') u_q| ^2 u_q/ q_μ q_ν .
        
        
        To proceed, we first use Eq. (<ref>) to find the exact expression for the quadrupole moment,
        Ψ(t)|x̂^μx̂^ν|Ψ(t)= ∫ d^Dq {a(q,t)/ q_μ a(q,t)/ q_ν + ia(q,t) a(q,t)/ q_μ𝒜^ν(q) 
        
        	+ ia(q,t) a(q,t)/ q_ν𝒜^μ(q)- |a(q,t)|^2  u_q| ^2 u_q/ q_μ q_ν} . 
        
        We are interested in the approximate evaluation of this expression in the case that the wave packet is sharply peaked 
        in reciprocal space. To do this, we follow the method of Sundaram and Niu <cit.>
        and first simplify the exact expression for X^μ under the 
        assumption of a sharply peaked wave packet. Using (<ref>) we find the exact formula
        
        	X^μ= ∫ d^Dq { i a(q,t) a(q,t)/ q_μ + |a(q,t)|^2 𝒜^μ(q)} .
        
        Next, we multiply and divide the first term by a(q,t) and use the fact that 
        |a(q,t)|^2≈δ^(D)(q-K) (because the wave packet is sharply peaked in reciprocal space) 
        to find the approximate expression
        
        	X^μ≈[i 1/a(q,t) a(q,t)/ q_μ]_q=K + 𝒜^μ(K) . 
        We now use this to simplify the formula for Ψ(t)|x̂^μx̂^ν|Ψ(t). 
        
        We start by integrating by parts in the second term in Eq. (<ref>) to obtain
        Ψ(t)|x̂^μx̂^ν|Ψ(t)= ∫ d^Dq {a(q,t)/ q_μ a(q,t)/ q_ν - ia(q,t)/ q_μ a(q,t)𝒜^ν(q) - i |a(q,t)|^2𝒜^ν(q)/ q_μ
        
        	+ ia(q,t) a(q,t)/ q_ν𝒜^μ(q)- |a(q,t)|^2  u_q| ^2 u_q/ q_μ q_ν} . 
        
        This can be rewritten exactly as
        Ψ(t)|x̂^μx̂^ν|Ψ(t)   =   ∫ d^Dq {(a(q,t)/ q_μ + i𝒜^μ(q)a(q,t))( a(q,t)/ q_ν - i𝒜^ν(q)a(q,t)) - |a(q,t)|^2 𝒜^μ(q)𝒜^ν(q) 
           -     i |a(q,t)|^2𝒜^ν(q)/ q_μ - |a(q,t)|^2  u_q| ^2 u_q/ q_μ q_ν} .
        
        Next, we can write
         u_q| ^2 u_q/ q_μ q_ν   =   / q_μ(  u_q|  u_q/ q_ν) -  u_q/ q_μ|  u_q/ q_ν
           =    -i 𝒜^ν(q)/ q_μ -  u_q/ q_μ|  u_q/ q_ν .
        
        Plugging this back into our expression for the quadrupole moment gives
        Ψ(t)|x̂^μx̂^ν|Ψ(t)   =   ∫ d^Dq {(a(q,t)/ q_μ + i𝒜^μ(q)a(q,t))( a(q,t)/ q_ν - i𝒜^ν(q)a(q,t)) - |a(q,t)|^2 𝒜^μ(q)𝒜^ν(q) 
           +    |a(q,t)|^2  u_q/ q_μ|  u_q/ q_ν} .
        
        We now multiply and divide the first term here by |a(q,t)|^2, use 
        |a(q,t)|^2≈δ^(D)(q-K), and use Eq. (<ref>) and its complex conjugate to find that
        Ψ(t)|x̂^μx̂^ν|Ψ(t)≈ X^μX^ν- 𝒜^μ(K)𝒜^ν(K) +  u_K/ K_μ|  u_K/ K_ν .
        
        Now we can write
        𝒜^μ(K)𝒜^ν(K)    =     -  u_K|  u_K/ K_μ u_K|  u_K/ K_ν
           =    u_K/ K_μ| u_K u_K|  u_K/ K_ν ,
        
        and so we find that
        Ψ(t)|x̂^μx̂^ν|Ψ(t)≈ X^μX^ν +  u_K/ K_μ|  u_K/ K_ν -  u_K/ K_μ| u_K u_K|  u_K/ K_ν .
        
        We have nearly arrived at the final answer. The last ingredient is to return to the original expression 
        Eq. (<ref>) and note that since
        ∫ d^Dq a(q,t)/ q_μ a(q,t)/ q_ν= ∫ d^Dq a(q,t)/ q_ν a(q,t)/ q_μ ,
        
        (to see it, integrate by parts twice to exchange the derivatives) we could have replaced the original integral expression for 
        Ψ(t)|x̂^μx̂^ν|Ψ(t) with a 
        symmetrized version of it, 
        Ψ(t)|x̂^μx̂^ν|Ψ(t)= 1/2[Eq. (<ref>)+(μ↔ν)]. Going through analogous manipulations for this symmetrized expression then yields our final
        expression for the quadrupole moment of |Ψ(t),
        Ψ(t)|x̂^μx̂^ν|Ψ(t)≈ X^μX^ν + g^μν(K) .
        
        
        We see that this expression involves the quantum metric evaluated at the wave packet center in reciprocal space. 
        In addition, we find that the connected part of the quadrupole moment is given by
        Ψ(t)|x̂^μx̂^ν|Ψ(t) - Ψ(t)|x̂^μ|Ψ(t)Ψ(t)|x̂^ν|Ψ(t)≈  g^μν(K) .
        
        We learn from this expression that if we assume that the wave packet is sharply peaked at position K
        in reciprocal space, then the spread of the wave packet in real space — which is measured by the connected part of the
        quadrupole moment — is given by the quantum metric at the location K.
        
        If we use our result for the quadrupole moment of the wave packet, then we find that the total wave packet energy is given
        by the approximate expression  
        
            Ψ(t)|Ĥ|Ψ(t)   ≈ℰ(K) - Q E^(0)_μX^μ
               - 1/2Q E^(0)_μν( X^μX^ν + g^μν(K) ) .
        
        This is the quantity that we called ℰ_eff(X,K) in the main text. 
        
        
        
        
        
         §.§ Issues related to possible boundary terms
        
        
        In the main text we noted that in our derivation of Eqs. 3 and 5 (of the main text) we integrated by parts in integrals over the
        Brillouin zone (BZ) and we neglected boundary terms. In this section we prove that in a suitable choice of gauge for the Bloch states 
        |ψ_q, namely the periodic gauge, the wave packet amplitude a(q,t) can be chosen to be a periodic function 
        on the BZ. This means that in this gauge the only possible boundary terms which could appear in the derivation of Eqs. 3 and 5 are
        terms associated with the Berry connection 𝒜^μ(q). It is well-known that for certain band
        structures the Berry connection for a given band may not be periodic on the BZ, and this might lead to nontrivial boundary terms which
        would appear as corrections to Eqs. 3 and 5 of the main text. We leave an exploration of such boundary terms for future work. 
        
        The periodic gauge for the Bloch states |ψ_q is defined by the condition
        
        	|ψ_q+G=|ψ_q
        for all reciprocal lattice vectors G (see, for example, Ref. king-smith). Recall also that the
        wave packet state that we consider in the main text has the form
        
        	|Ψ(t)= ∫ d^Dq a(q,t)|ψ_q .
        
        We now prove the following. Suppose that we adopt the periodic gauge for the Bloch states |ψ_q, and
        we also choose the wave packet amplitude a(q,0) at time t=0 to obey the periodicity condition
        
        	a(q+G,0)=a(q,0) 
        
        for all reciprocal lattice vectors G. Then at all later times t>0 the amplitude a(q,t) remains periodic,
        i.e., we have
        
        	a(q+G,t)=a(q,t) 
        
        for all reciprocal lattice vectors G and for all t>0. 
        
        To prove this we first recall the equation of motion for a(q,t),
        
            iħȧ(q,t)= a(q,t)ℰ(q) + Q ∫ d^Dq' a(q',t) ψ_q|(x̂)|ψ_q' .
        
        The equation of motion for a(q+G,t) is then
        
            iħȧ(q+G,t)   = a(q+G,t)ℰ(q+G) 
            
            +    Q ∫ d^Dq' a(q',t) ψ_q+G|(x̂)|ψ_q' .
        
        Since G is a reciprocal lattice vector we always have
        ℰ(q+G)=ℰ(q)
        
        since ℰ(q), the energy of the Bloch state |ψ_q, is periodic in reciprocal space. In
        addition, in the periodic gauge we also have Eq. (<ref>). This means that in the 
        periodic gauge the equation of motion for a(q+G,t) takes the form
        
            iħȧ(q+G,t)   = a(q+G,t)ℰ(q) 
            
            +    Q ∫ d^Dq' a(q',t) ψ_q|(x̂)|ψ_q' .
        
        Now let us define the quantity
        
        	b(q,t):= a(q+G,t)-a(q,t) .
        
        By subtracting the equations of motion for a(q+G,t) and a(q,t), we find that
        b(q,t) evolves in time according to the simple equation
        
        	iħḃ(q,t)= b(q,t)ℰ(q)
        
        with solution
        
        	b(q,t)= e^-iℰ(q)t/ħb(q,0) .
        
        It follows that if we choose b(q,0)= a(q+G,0)-a(q,0)=0, then we have
        b(q,t)= a(q+G,t)-a(q,t)=0 for all later times t, and this completes the proof.
        
        
        
         §.§ Details of the example lattice models displaying a nontrivial geometric current
        
        
        In this section we present the details of the calculation of the geometric current in the two example lattice models that we mentioned in the
        main text. 
        
        
        
          §.§.§ An example in D=1
        
        
        We start with an example of a lattice model in D=1 which yields a non-zero j^1_geom.. 
        We consider the two-band model with Bloch Hamiltonian (we write k≡ k_1 since we are in D=1)
        
        	H(k)= Asin(k)𝕀 + sin(k)σ^x + (m+1-cos(k))σ^z , 
        where 𝕀 is the 2× 2 identity matrix and σ^x,y,z are the Pauli matrices. We assume that
        the parameters m and A are chosen so that there is an energy gap between the two bands of the model. To obtain
        a model of a metal we then completely fill the lower band and partially fill the upper band to a Fermi energy
        ℰ_F (we work at zero temperature). For A=0 this model
        has inversion and time-reversal symmetry, which are both given by H(-k)=σ^z H(k)σ^z since H(k) is real. Thus,
        according to the symmetry analysis in the main text, we need A≠ 0 to obtain a non-zero geometric current. The
        band energies for this model are given by ℰ_±(k)= ±λ(k) + Asin(k) with
        λ(k)= √(sin^2(k)+(m+1-cos(k))^2), and the quantum metric for the upper band is given by
        
        	g_+(k)= [(1+m)cos(k)-1]^2/4λ(k)^4 .
        
        In particular, g_+(k) is independent of A as the identity matrix term in H(k) does not change the
        form of the eigenvectors of H(k). 
        
        We now show that this model yields a non-zero j^1_geom.. 
        Let I_1D(ℰ_F) denote the integral
        
        	I_1D(ℰ_F)= ∫ dk f_0(k)  g_+(k)/ k . 
        This integral is a function of the Fermi energy ℰ_F, since f_0(k), the equilibrium distribution function at zero 
        temperature, is completely determined by ℰ_F. From Eq. 11 of the main text, the geometric current in the wire in 
        the presence of an electric field gradient will be proportional to I_1D(ℰ_F). For the parameter
        values m=1, A=0.1, we plot the function I_1D(ℰ_F) in Fig. <ref>c for a range of values of 
        ℰ_F which span the entire “+” band of the model. We find that I_1D(ℰ_F)≠ 0 in general, and 
        so this model has j^1_geom.≠ 0 for generic values of ℰ_F. For reference, the band 
        energies ℰ_±(k) and the quantum metric g_+(k) for the parameter values m=1, A=0.1 are plotted
        in Fig. <ref>a and Fig. <ref>b.
        
        
        
          §.§.§ An example in D=2
         
        
        We now briefly discuss a model in D=2 which yields a non-zero 
        j^μ_geom.. This model has a Bloch Hamiltonian given by
        
            H(k)   = Asin(k_1)𝕀 + sin(k_1)σ^x + sin(k_2)σ^y 
               + (m+2-cos(k_1)-cos(k_2))σ^z  .
        
        For A=0 this model can describe a Chern insulator on the square lattice when the lower band is full and 
        -4<m<-2 or -2<m<0. Here we consider the model for m>0, and we fill the lower band completely and only partially
        fill the upper band to obtain a model of a metal. For m>0 the upper band has a minimum at k=(0,0) and
        maxima at the BZ corners k=(±π,±π). The quantum metric for this model is independent of A
        and is given in Eq. 62 of Ref. matsuura2010momentum. For A=0 this model has inversion symmetry
        given by H(-k)=σ^z H(k)σ^z, and so we need A≠ 0 to have j^μ_geom.≠ 0. To illustrate 
        the nontrivial response we consider a non-uniform electric
        field in the X^1 direction only, so we only turn on E^(0)_11≠ 0. As a result, we only need to study the
        term in j^1_geom. which contains g^11(k). In this case, 
        j^1_geom. will be proportional to the integral
        
        	I_2D(ℰ_F)= ∫ d^2k f_0(k)  g^11(k)/ k_1 , 
        	
        where f_0(k) again denotes the zero temperature distribution function determined by
        ℰ_F. We plot I_2D(ℰ_F) in Fig. <ref>d for a range of ℰ_F values which stretches
        from the bottom to the top of the upper band in this model. The plot again shows that I_2D(ℰ_F)≠ 0 in
        general, which implies a non-zero j^1_geom. for generic values of ℰ_F.
