Magnetoelectric Polarizability: a Microscopic Perspective

PHYSICAL REVIEW RESEARCH 2, 033126 (2020)

Magnetoelectric polarizability: A microscopic perspective

Perry T. Mahon * and J. E. Sipe† Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7

(Received 9 March 2020; revised 17 June 2020; accepted 24 June 2020; published 23 July 2020)

We extend a field theoretic approach for the investigation of the electronic charge-current density response of crystalline systems to arbitrary electromagnetic fields. The approach leads to the introduction of microscopic polarization and magnetization fields, as well as free charge and current densities, the dynamics of which are described by a lattice gauge theory. The spatial averages of such quantities constitute the fields of macroscopic electrodynamics. We implement this formalism to study the modifications of the orbital electronic properties of a class of insulators due to uniform dc electric and magnetic fields, at zero temperature. To first order in the electric and magnetic fields, the free charge and current densities vanish; thus the linear effect of such fields is captured by the first-order modifications of the microscopic polarization and magnetization fields. Associated with the dipole moment of the microscopic polarization (magnetization) field is a macroscopic polarization (magnetization), for which we extract various tensors relating it to the electric and magnetic fields. We focus on the orbital magnetoelectric polarizability (OMP) tensor, and find the accepted expression as derived from the “modern theory of polarization and magnetization.” Since our results are based on the spatial averages of microscopic polarization and magnetization fields, we can identify the distinct contributions to the OMP tensor from the perspective of this microscopic theory, and we establish the general framework in which extensions to finite frequency can be made.

DOI: 10.1103/PhysRevResearch.2.033126

I. INTRODUCTION the definition of macroscopic quantities, as it did for Lorentz, links to a microscopic picture follow from the definitions. In Interest in describing the response of insulators to external the “modern theory,” the macroscopic polarization was found electromagnetic fields dates back to the earliest studies of to be related to the dipole moment of a Wannier function and electricity and magnetism. In pioneering work near the start of its associated nucleus [3]. The ambiguity of which nucleus to the twentieth century, Lorentz [1] based his definition of the associate with a given Wannier function—the “closest,” or one macroscopic polarization and magnetization fields on a phys- some number of lattice spacings away?—leads to a “quantum ical picture of molecules with electric and magnetic moments of ambiguity” in the macroscopic polarization itself. Such [2], and from that perspective addressed the response of the ambiguities are inherent to the “modern theory,” and can macroscopic quantities to the electromagnetic field. generally be related to the behavior and description of charges Near the end of the twentieth century a new approach, and currents at the surface of a finite sample [9]. called the “modern theory of polarization and magnetization,” We have recently argued [10] that it is useful to expand was introduced [3–6]. Largely focused on the static response upon the approach of Lorentz by introducing microscopic of crystalline materials to uniform fields, the microscopic polarization and magnetization fields in bulk crystals, and underpinning was now electronic Bloch eigenfunctions [7], defining the corresponding macroscopic fields as their spatial or alternately the spatially localized Wannier functions that averages; in general, microscopic “free” charges and currents, could be constructed from them. However, a macroscopic and their spatial averages, are also introduced, and the result- perspective was taken to define the polarization and magne- ing description takes the form of a generalized lattice gauge tization. For example, if one imagined a slow variation in theory. The strategy employed is an extension of that used to material parameters leading to a macroscopic current density introduce microscopic polarization and magnetization fields J, the polarization (or at least its change) could be defined for atoms and molecules [11,12], which itself is an exten- through J = dP/dt [8]. Thus, instead of a microscopic picture sion of Lorentz’ characterization of molecules by a series of the underlying position and motion of charges leading to of multipole moments. Such microscopic polarization and magnetization fields allow for the visualization of electronic dynamics, in the sense that perturbative modifications to these *[email protected] microscopic fields arising due to the electromagnetic field can †[email protected] be found and exhibited if one has the Wannier functions in hand; existing schemes, which are primarily ab initio based Published by the American Physical Society under the terms of the [13,14], can be used to construct such Wannier functions. In Creative Commons Attribution 4.0 International license. Further the usual “long-wavelength limit” of optics, where the electric distribution of this work must maintain attribution to the author(s) field is varying in time but its variation in space is neglected, and the published article’s title, journal citation, and DOI. we recover the standard results for crystalline solids. In other

2643-1564/2020/2(3)/033126(21) 033126-1 Published by the American Physical Society PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020) instances where comparison with the “modern theory” is With this, our goal is to formulate the relation of these site possible, such as the modification of the polarization due quantities to the electric and magnetic fields evaluated at that to a static or uniform electric field, or similar modifications site. The OMP tensor is extracted upon taking these electric in systems expected to exhibit the quantum anomalous Hall and magnetic fields to be uniform in space and independent effect, we also find agreement [10]. of time. Our results are in complete agreement with those of The approach we implement has the advantage that it can the “modern theory,” as would be expected, but in the process be employed to describe the effects of spatially varying, time- we achieve some insight into the microscopic origin of the dependent electromagnetic fields.1 Exploring its characteri- distinct contributions to (1). In particular, we can compare our zation of this generalized optical response, in the linear and results for the OMP tensor with what would be expected in nonlinear regimes, is our main program. There is, however, an a “molecular crystal limit,” a model in which at each lattice interesting overlap of our program with that of the “modern site there is a molecule with orbitals that share no common theory,” and that is in calculating the static modification of the support with the orbitals of molecules at other lattice sites. polarization due to a uniform magnetic field, and the static And with the development of the formalism presented here we modification of the magnetization due to a uniform electric position ourselves to extend this approach to describe material field. This phenomenon is termed the magnetoelectric effect, response at finite frequency. and for a class of insulators2 in the “frozen-ion” approxima- After some preliminary discussion to begin Sec. II,we tion, where spin contributions are also neglected, both of these extend the formalism [10] where necessary in order to cal- modifications are described by the orbital magnetoelectric culate the modification of a site quantity due to arbitrary polarizability (OMP) tensor [9,15–19], electromagnetic fields. This is a very general development, and only in the later sections do we restrict ourselves to the ∂ i ∂ l il P M α = = . (1) limit of uniform and static electric and magnetic fields. The ∂Bl E = 0 ∂E i E = 0 B = 0 B = 0 calculations are made in Sec. III; in Sec. IV, we show that the accepted expression for the OMP tensor is reproduced. Unlike its generalization to finite frequency, the OMP tensor We also calculate the OMP tensor in the molecular crystal is nonvanishing only when both spatial-inversion and time- limit by two approaches. The first is a direct molecular physics 3 reversal symmetry are broken in the unperturbed system, calculation, and the second is by taking the appropriate limit and it is composed of two distinct terms: the Chern-Simons of our general expressions; they agree, as they should. We also and the cross-gap contributions. The former is isotropic and discuss the nature of both the Chern-Simons and cross-gap entirely a property of the subspace spanned by the cell- contributions from the perspective of this microscopic theory. periodic functions associated with the originally occupied In Sec. V, we conclude. energy eigenfunctions, while the latter involves both occupied and excited eigenvectors, and the corresponding energies, II. PERTURBATIVE MODIFICATIONS OF THE of the unperturbed system. The Chern-Simons contribution SINGLE-PARTICLE DENSITY MATRIX has generated particular interest in the literature because of its topological features; there is a discrete ambiguity in its The electronic response of a crystalline insulator is a value, which can be used to identify Z2 topological insulators consequence of the evolution of the fermionic electron field [9,20,21]. As well, while the analytic structure of the cross- operator, ψˆ (x, t). We assume that, in the Heisenberg picture, gap contribution is of the form one would expect to find from the dynamics of this object is governed by the usual treatment of linear response using a Kubo formal- ∂ψˆ , ism, the Chern-Simons contribution takes a rather unexpected (x t) ih¯ = H0 x, p (x, t) + eφ(x, t) ψˆ (x, t), (2) form. ∂t mc The expression for the OMP tensor found via the “modern where e =−|e| is the electron charge, theory” is well established [17,18]. In this paper, we present a calculation of the OMP tensor within our framework of , = − e , , pmc(x t) p(x) A(x t) identifying microscopic polarization and magnetization fields. c

It is a special case of our general approach, in which by and H0(x, p(x)) is the differential operator that governs the using a set of orthogonal functions that are well-localized dynamics of the electron field in the unperturbed infinite spatially one can associate a portion of a total quantity with crystal. The presence of a classical electromagnetic field, the point about which each of these functions is localized; described by its vector and scalar potentials, A(x, t) and a total quantity can be decomposed into “site” contributions. φ(x, t), has been included through the usual minimal coupling prescription. In writing (2) the independent particle approximation is made, neglecting any interactions apart from those 1While in past work [10] and in this paper we treat the electromag- described by the coupling of the electron field operators to netic field classically, quantum mechanical effects can, in principle, the electromagnetic field, the associated electric and magnetic be taken into account. fields of which are taken to be the macroscopic Maxwell 2 This includes both ordinary and Z2 topological insulators. We fields; local field corrections are thus neglected. The Maxwell discuss this further below. fields are assumed to be nonvanishing only at times greater 3More precisely, the OMP tensor vanishes modulo a discrete am- than an initial time at which the system is taken to be in its un- biguity when time-reversal or inversion symmetry are present in the perturbed zero temperature ground state, and the expectation unperturbed system. values of pairs of (Heisenberg) field operators ψˆ (x, t) and their

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6 adjoints in the unperturbed ground state are used to construct This class includes ordinary insulators and Z2 topological the minimal coupling Green functions [10]. insulators [24], but excludes, for example, Chern insulators Implementing the frozen-ion approximation, we take [26]. Generally a filling factor fn is associated with each |ψnk 2 that is either 0 or 1, and here we introduce an analogous filling p(x) |α H x, p(x) = + V (x), (3) factor fα associated with each R that is also either 0 or 1; 0 2m the latter can be inferred directly from the occupancy of the where V (x) is the spatially periodic lattice potential that char- energy eigenvectors used in the construction of a particular = acterizes the crystal structure and satisfies V (x) = V (x + R) ELWF. Thus, for the class of insulators we consider, Unα (k) = for all Bravais lattice vectors R, and 0 only if fn fα [13]. The ELWFs (5) can generally be expressed as = h¯ ∇ − e . p(x) Astatic(x) (4) dk i c x|αR= eik · (x−R)x|αk, uc π 3 In (4) we have allowed for the presence of an “internal,” static, BZ (2 ) cell-periodic magnetic field described by the vector potential where another set of cell-periodic functions, {x|αk},have A (x), where A (x) = A (x + R). The inclusion of been introduced and are formed via static static static such an “internal” field respects the discrete translational sym- |αk≡ U α (k)|nk. metry of the crystal, but generically leads to a Hamiltonian (3) n with broken time-reversal symmetry, which will be important n in what follows. In future publications we plan to include both Here, x|αk is in general not the cell-periodic part of an the spin-orbit and Coulomb interactions that we neglect here. energy eigenfunction. The sets of vectors {|αk} and {|nk} In spatially periodic systems, a set of exponentially lo- are related by a (in general) multiband gauge transformation calized Wannier functions (ELWFs), {WαR(x) ≡x|αR}, can characterized by U (k)[9]. Although we use Roman and Greek generally be constructed [22–26]via subscripts in the notation Unα (k), as we will primarily be considering a transformation from the cell-periodic part of uc −ik · R energy eigenfunctions to cell-periodic functions that are not |αR= dke U α (k)|ψ , (5) 3 n nk associated with energy eigenfunctions, this need not necessar- (2π ) BZ n ily hold. Indeed, the simplest type of gauge transformation, where uc is the unit cell volume, and although of course it would not generally lead to |αk associated with ELWFs, is one that involves the |nk associated 1 ik · x ψnk(x) ≡x|ψnk= e unk(x)(6)with each band individually; such a transformation is achieved (2π )3 −iλ (k) by taking Unα (k)tobeoftheformδnαe n , where, for a are eigenfunctions of (3) that are normalized over the infinite given n and k, λn(k) ∈ R. Rather than a special limit of the ψ |ψ =δ δ − crystal such that mk nk nm (k k ). A periodic gauge general multiband transformation, this could be considered choice is made such that the energy eigenvectors |ψnk and as simply a new choice of Bloch eigenvectors; under Bloch’s the unitary operator U (k) are periodic over the first Brillouin theorem, energy eigenvectors are uniquely defined only within 4 zone. Associated with each Bloch eigenfunction ψnk(x)is a k-dependent phase [23], even at k points where there is no an energy Enk and a cell-periodic function unk(x) ≡x|nk degeneracy. Here, however, it is considered as one type of satisfying the orthogonality relation (mk|nk) = δnm; we adopt Unα (k), associated with a gauge transformation of the U(1) the notation type. That is, we consider the vectors {|nk} fixed at the start, and use the term “gauge dependent” for quantities that depend | ≡ 1 ∗ (g h) g (x)h(x)dx (7) generally on the Unα (k) and their derivatives, including Unα (k) uc uc of the U(1) type. for functions g(x) ≡x|g and h(x) ≡x|h that are periodic The ELWFs are an important element of our approach, over a unit cell, where the integration is over any unit cell. because we use them to introduce “site” quantities, and define Also, we restrict our study to three-dimensional systems.5 the macroscopic polarization and magnetization in terms of Here and below n is a band index, α is a type index, andh ¯k their moments, as we discuss in detail below. Of course, not denotes a crystal-momentum within the first Brillouin zone. all gauge transformations of an initial set {|nk} will lead to In this paper we initiate our considerations with the zero ELWFs via (5), as indicated by the example given above. temperature ground state of an insulator, and consider the Nonetheless, the various tensors that describe the modification class of insulators for which the sets of occupied and unoc- of the electronic quantities due to the Maxwell fields, includ- cupied energy eigenfunctions each map to a set of ELWFs. ing αil , must be such that the resulting charge and current densities in the bulk are not only invariant with respect to the

4 More precisely, |ψnk+G=|ψnk and U (k + G) = U (k)forany reciprocal lattice vector G. For more details, see, e.g., Vanderbilt [9]. 6By “ordinary insulator” we mean crystalline insulators supporting 5The derived expressions can later be applied to lower dimen- Bloch energy eigenvectors for which there exists no topological sional systems by conﬁning the Bloch and Wannier functions to obstruction to choosing a smooth gauge that can respect some the appropriate subspace of R3. However, in systems with spatial underlying symmetry of the system. For instance, there exists no dimension less than three, the Chern-Simons contribution vanishes. obstruction to choosing a time-reversal or inversion symmetric gauge Thus three-dimensional systems are of primary interest here. for a system with the same discrete symmetry.

033126-3 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020) choices of Unα (k) that lead to ELWFs, but in fact to all choices expression for (15) follows from the form of the integral over of Unα (k); that is, the charge and current densities in the bulk the unit cell and the use of (10). Indeed, a more general form must be gauge invariant. This is plausible because it would be of (14), (15) can be derived involving the matrix elements of possible—although we would argue much less convenient and pa(x) in the basis of the cell-periodic functions x|αk using less interesting physically—to calculate those charge and cur- the same strategy.8 rent densities directly from the minimal coupling Hamiltonian In previous work [10], we considered the calculation of without ever introducing Wannier functions. And we shall see the expectation values of the electronic charge and current that this gauge invariance does indeed hold. density operators for a crystalline insulator perturbed by an A useful identity [23]is electromagnetic field. Noting that the lesser, equal time single- particle Green function can be used to find such quantities, ∗ a uc ik · R a W (x)x Wα (x)dx = dke ξ˜ (k), (8) we employed a set of spatially localized, “adjusted Wannier βR 0 π 3 βα (2 ) BZ functions” as a basis in which to expand the electron field where operator and its adjoint in an effort to associate portions of the full electronic Green function with individual lattice sites. ξ˜ a (k) ≡ i(βk|∂ αk) (9) βα a Upon identifying such “site Green functions,” and thereby is the non-Abelian Berry connection associated with the set identifying “site charge and current densities,” we defined {|αk}; here and below, superscript indices indicate Carte- microscopic polarization and magnetization fields associated sian components, repeated Cartesian components are summed with each lattice site using the same functions that are used a over, and we adopt the shorthand ∂a ≡ ∂/∂k . The object (9) in atomic and molecular physics to relate the microscopic is related to the non-Abelian Berry connection associated with polarization and magnetization fields of atoms and molecules the set {|nk}, to the microscopic charge and current densities9; we call these functions “relators.” The full microscopic polarization ξ a ≡ |∂ , mn(k) i(mk ank) (10) and magnetization fields are given by summing the respective via site contributions. We then insisted that these microscopic polarization and magnetization fields, together with the elec- ξ˜ a † = ξ a + Wa , Umβ (k) βα(k)Uαn(k) mn(k) mn(k) (11) tronic charge and current density expectation values, satisfy αβ the expressions arising in classical macroscopic electrody- Here we have defined the Hermitian matrix Wa [9], populated namics relating such quantities. This led to the identification by elements of microscopic “free” electronic charge and current densities, which take predictable forms. At zero temperature the Wa ≡ ∂ † , mn(k) i aUmα (k) Uαn(k) (12) first-order modifications of both the free charge and current α densities due to the Maxwell fields vanish for the class of which, for the class of insulators we consider, is nonzero insulators considered here, even for electromagnetic fields in the x-ray regime [10]. As a consequence, the first-order only if fm = fn. Under the aforementioned periodic gauge choice, all objects appearing in (11) are periodic over the perturbative modifications to the expectation values of the first Brillouin zone. In what follows, the k dependence of the electronic charge and current density operators resulting from preceding objects is usually kept implicit. the electromagnetic field can be found directly from the A consequence of the Hamiltonian (3) and the resulting corresponding first-order modifications to the microscopic dynamics (2) of the electron field operator and its adjoint polarization and magnetization fields. is that the differential operators associated with the spatial A quantity central to the calculation of both the micro- components of the conserved current take the form scopic polarization and magnetization fields is the single- particle density matrix, ηα β (t). Thus, a starting point in de- e R ; R a , = a , , = a , , Jmc x p(x); t J x pmc(x t) pmc(x t) (13) scribing the effect of an electromagnetic field to a crystalline m insulator is identifying how the Maxwell fields affect this in the usual fashion,7 where Ja(x, p(x)) are the analogous object. The single-particle density matrix evolves according differential operators arising for the unperturbed system. As to [10] a result, another useful identity is ∂ηα β (t) μ ν R ; R R1; R2 = η , ih¯ Fα β (t) μR1;νR2 (t) (16) ∗ ∂ R ; R ψ a ψ = a δ − , t μν nk (x)p (x) nk(x)dx pnn(k) (k k ) (14) R1R2 where where the matrix elements are μ ν R1; R2 i(R ,R1,R ; t) F (t) = δνβδ e H¯α μ (t) m im αR ;βR R2R R ; R1 a = δ ∂ + − ξ a . pnn(k) n n aEnk (En k Enk ) nn(k) (15) h¯ h¯ i(R,R ,R; t) − δμαδ 2 ¯ R1R e HνR2;βR (t) This can be shown by breaking the integral in (14)into 0 − e R , t δνβδμαδ δ . the sum of integrals over unit cells; the sum over Bravais R ( ) R2R R1R (17) lattice vectors yields the Dirac delta function in (14), and the

8Rodrigo A. Muniz, J. L. Cheng, and J. E. Sipe (unpublished). 7See, e.g., Peskin and Schroeder [32]. 9For a review and references to original work see Ref. [12].

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0 , ,..., , that The definitions of (R1 R2 RN ; t), R (R t), and i(R,R ,R; t) H¯νR ;μR (t) are as given earlier [10], and are provided in 1 ¯ 2 1 e HαR ;μR1 (t) the Appendices. The first of these quantities is related to i(R ,R ,R ,R ; t) 0 = a 1 ¯ , − , δαμδ the magnetic flux through the surface generated by connecting e HαR ;μR1 (Ra t) e R (Ra t) R R1 , ,..., the points (R1 R2 RN ) with straight lines, when the usual (18) choice of straight-line paths for the relators is adopted. The and second is related to the electric field along the path connecting , , i (R R1 R ; t) ¯ points (R , R ). The third quantity can be understood as a gen- e HμR1;βR (t) eralized “hopping” matrix element. Each of terms appearing i(R,R ,R ,R; t) 0 = e 1 b H¯μ β (R , t) − e (R , t)δμβ δ , in (17) is gauge invariant in the electromagnetic sense, and R1; R b R b R R1 η consequently so too is αR ;βR (t). In Appendix A,weshow (19) for any lattice sites Ra and Rb. We have defined

h¯ ∂(R , x, R ; t) h¯ ∂(R , x, R ; t) ∗ i(R1,x,Ra; t) a 1 a 2 i(Ra,x,R2; t) H¯μR ;νR (Ra, t) ≡ χμ (x, t)e HR (x, t) + + e χνR (x, t)dx 1 2 R1 a 2 ∂t 2 ∂t 2

∗ ih¯ ∂χν (x, t) ∂χμ (x, t) i(R1,x,Ra; t) ∗ R2 R1 i(Ra,x,R2; t) − e χμ (x, t) − χνR (x, t) e dx (20) 2 R1 ∂t ∂t 2

and To prepare for later perturbative analysis we ex- H , ≡ , , − 0 , , pand all quantities in powers of the electromagnetic Ra (x t) H0 x p(x Ra; t) e R (x t) (21) a field, where (0) (1) e η = η + η +··· , p(x, R ; t) ≡ p(x) − (x, t), (22) αR ;βR (t) αR;βR αR;βR (t) a c Ra as before [10]. Here, R (x, t) is related to the Maxwell (0) (1) a ¯μ ν , = + ¯ , +··· , H R1; R2 (Ra t) HμR ;νR HμR ;νR (Ra t) magnetic field along the path connecting points (Ra, x), and is 1 2 1 2 defined in Appendix C. The functions in the set {χαR(x, t)} are etc., where the superscript (0) denotes the contribution to the generally not orthogonal, but they are related to the mutually total quantity that is independent of the electromagnetic field, orthogonal “adjusted Wannier functions” introduced earlier the superscript (1) denotes the contribution that is first-order [10], and they depend only on the Maxwell magnetic field in the electric and magnetic fields, and so on. Using (16), and not the vector potential used to describe it. In the limit and equating terms appearing with the same powers of the η(0) of a weak magnetic field, a perturbative expansion can be Maxwell fields, the zeroth-order term αR;βR is found to constructed for each χαR(x, t)[10], the lowest order terms of satisfy the same equation of motion as the unperturbed single- which are particle density matrix, and so i χ , = − (0) αR(x t) WαR(x) Wγ R1 (x) η = fαδαβδ , (25) 2 αR ;βR R R γ R1 while from (20) it is found that ∗ × Wγ (y)(R1, y, R; t)WαR(y)dy +··· R1 (0) ∗ = , ν . HμR ;νR WμR (x)H0 x p(x) W R2 (x)dx (26) (23) 1 2 1

Choosing Ra = Rb, one can then re-express (17)as Next, implementing the usual Fourier series analysis via μ ν R1; R2 i(R ,Ra,R1,R ; t) − ω F (t) = δνβδ e H¯α μ (R , t) i t αR ;βR R2R R ; R1 a g(t) ≡ e g(ω), (27) , , , ω − δ δ i (R R2 Ra R ; t) , μα R R e H¯νR ;βR (Ra t) 1 2 the first-order modification to the single-particle density ma- ∂ , , (R Ra R ; t) trix due to the electromagnetic field can be identified, via (16), − h¯ δνβδμαδ δ . (24) ∂t R2R R1R as

(1) α |ψ ψ |μ ,ω ν |ψ ψ |β R mk mk R1 Hμ ν (Ra ) R2 nk nk R η(1) ω =− R1; R2 α β ( ) fnm dkdk + R ; R − − ω + BZ Emk Enk h¯( i0 ) μνR1R2 mn i ∗ + fβα W (x) (R , x, R ; ω) + (R , x, R ; ω) Wβ (x)dx (28) 2 αR a a R

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(Appendix B), where fnm ≡ fn − fm and fβα ≡ fβ − fα; recall that for the class of insulators of interest here there are well defined filling factors associated with the orbital type indices, but in general this is not so. In identifying (28), we have also introduced (1) ∗ (1) H (R ,ω) ≡ Wμ (x)H (x,ω)Wν (x)dx. (29) μR1;νR2 a R1 Ra R2 Often in optics one is interested in the effects of macroscopic Maxwell fields that vary little over electron correlation lengths, which for the class of insulators we consider are on the order of the lattice constant. In Appendix C, we show that if this approximation is made, and if |y − x| and |z − x| are on the order of lattice constants, then 1 a(x,ω) albBl (y,ω)(xb − yb), (30) y 2 1 0(x,ω) (xl − yl )E l (y,ω) + (x j − y j )(xl − yl )F jl (y,ω), (31) y 2 e (z, x, y; ω) − labBl (y,ω)(za − ya )(xb − yb), (32) 2¯hc where E(y, t) is the Maxwell electric field, B(y, t) is the Maxwell magnetic field,

1 ∂E j (y, t) ∂E l (y, t) F jl (y, t) ≡ + , (33) 2 ∂yl ∂y j and abd is the Levi-Civita symbol. Implementing the approximate expressions (30), (31), as well as (3) and (27), the first-order modification to (21) is found to be (1) l l l e j j l l jl e lab l b b a H (x,ω) −e x − R E (Ra,ω) − x − R x − R F (Ra,ω) + B (Ra,ω) x − R p (x). (34) Ra a 2 a a 2mc a In the above, we have used straight-line paths in the relators [10], and in what follows, this choice is always made. Equations (28)–(34) make clear the motivation for introducing the arbitrary lattice site Ra; it serves as a reference site for the electromagnetic field. This will prove useful when considering the response of a quantity associated with site R to a spatially varying electromagnetic field; choosing Ra = R, the modification to that site quantity is related to the Maxwell field evaluated at that site. However, in this paper we restrict ourselves to uniform electric and magnetic fields in the dc limit. Thus F jl (x, t) vanishes and η(1) ω = ω = αR;βR ( ) 0 only if 0. We retain only the nonvanishing, first-order perturbative modifications arising from the electric η(E ) η(B) and magnetic fields, which we denote by αR;βR and αR;βR , respectively, such that η(1) ω = = η(E ) + η(B) . αR;βR ( 0) αR;βR αR;βR (35) Implementing (28), the first-order perturbative modification to the single-particle density matrix due to a uniform dc electric field is found to be ik · (R−R ) † ξ l (E ) l dk e Uαm mnUnβ η = e E f (36) αR ;βR uc nm (2π )3 E − E mn BZ mk nk

(Appendix B), where E ≡ E(Ra,ω = 0), for any Ra, is now the uniform dc electric field. The result (36) is consistent with previous work [10], where this expression was derived via a different method. Notably (36) is written as a single Brillouin zone integral, unlike (28). This feature is expected upon comparison to the usual perturbative treatment, in which, when spatial variation of the electric field is neglected, a k-conserving interaction term arises that results in perturbative modifications being given by single Brillouin zone integrals [27]. While here we assume the macroscopic Maxwell electric field is uniform, more generally this reduction to a single k integral holds as a consequence of the approximation that the electric field is to vary little over electron correlation lengths, allowing a Taylor series expansion about each lattice site. In the limit of uniform fields of interest here, the Ra dependence of (36) vanishes. There are two distinct contributions to the first-order perturbative modification due to a uniform dc magnetic field, (B) e uc lab l dk ik · (R−R ) † ab η = B f e Uα B (k)U β αR ;βR 4¯hc nm (2π )3 m mn n mn BZ e uc lab l dk ik · (R−R ) † † b + B f e (∂ Uα )U β − Uα (∂ U β ) ξ , (37) 4¯hc nm (2π )3 a m n m a n mn mn BZ where we have defined E − E E − E ∂ (E + E ) Bab (k) ≡ i sk nk ξ a ξ b + sk mk ξ a ξ b − 2 a mk nk ξ b , (38) mn E − E ms sn E − E ms sn E − E mn s mk nk mk nk mk nk

and B ≡ B(Ra,ω = 0), for any Ra, is the uniform dc magnetic not simply the individual contributions of the terms of (28). field. We mention that the two terms appearing in (37)are Moreover, in the first-order perturbative modifications of the

033126-6 MAGNETOELECTRIC POLARIZABILITY: A MICROSCOPIC … PHYSICAL REVIEW RESEARCH 2, 033126 (2020) quantities considered below, the first term of (37)givesrise where above and below the sums range over all lattice vectors to gauge invariant contributions, while the final term will give and orbital types, and rise to gauge dependent contributions.10 e i(R,x,R ) ∗ ρβ α (x, R) = (δ + δ )e χ (x)χα (x). R ; R 2 RR RR βR R III. MODIFICATION OF P AND M DUE TO (41) UNIFORM DC E AND B FIELDS Meanwhile, there are two contributions to each site magneti- By implementing the perturbative modifications of the zation field, single-particle density matrix (35), we now calculate the first-order modifications of the electric and magnetic dipole mR(x) = m¯ R(x) + m˜ R(x). (42) moments—found from the microscopic polarization and mag- The first of these, m¯ (x), corresponds to the “local” or netization fields, respectively—due to uniform dc electric R “atomic-like” contribution to each site magnetization field, and magnetic fields. In this section we restrict earlier [10], and is related to the electronic current density associated with more general expressions to this limit. Thus a function pre- that site, j (x), via the relator αib(x; y, R) (see Appendix C).11 viously written in terms of frequency components g(ω)(27), R It is given by will simply be given by the single nonvanishing component g ≡ g(ω = 0). Furthermore, as previously mentioned, to first- i 1 ib b m¯ (x) ≡ α (x; y, R) j (y, R)dy ηα β , order in the electromagnetic field, the free charge and current R c βR ;αR R ; R densities vanish at zero temperature for the class of insulators αβRR we consider [10], as would be expected physically, and so (43) those quantities do not appear here. where

, A. Summary of formalism jβR ;αR (x R) 1 i(R,x,R ) ∗ Introducing a set of “adjusted Wannier functions” that is = δ χ , , χ 4 RR e βR (x) J x p(x R) αR (x) an orthonormal basis of the electronic Hilbert space allows 1 ∗ ∗ i(R,x,R ) + δ , , χ χ for the lesser, equal time single-particle Green function to 4 RR J x p(x R) βR (x) e αR (x) be exactly decomposed as a sum of site contributions [10]. 1 ∗ i(R ,x,R ) ∗ + δ , , χ χ Consequently, so too can be the microscopic polarization and 4 RR J x p(x R) e βR (x) αR (x) magnetization fields, such that 1 ∗ i(R,x,R ) + δ χ , , χ . 4 RR βR (x) J x p(x R) e αR (x) (44) = , = , p(x) pR(x) m(x) mR(x) (39) It is not obvious that there is another contribution to each site R R magnetization field, as each atomic-like contribution is found where the sums range over all Bravais lattice vectors R.In from a site electronic current density, and these collectively the limit of uniform dc electric and magnetic fields treated compose the total current. However, in extended systems perturbatively, the discrete translational symmetry that was where the charge-current density associated with each site present in the unperturbed Hamiltonian (3) is now lost [cf. need not be conserved, there is in general an additional term, (2)], but the polarization and magnetization fields associated m˜ R(x)[10]. Adopting the terminology of the “modern theory,” with each lattice site remain physically equivalent; that is it corresponds to the “itinerant” contribution to each site magnetization field, and is given by p (x) = p (x − R ), m + (x) = m (x − R ), R+R R R R R i 1 ib b for any R and R . This is a result of the fact that it is the electric m˜ (x) ≡ α (x; y, R) j˜ (y, R)dy ηα β . R c βR ;αR R ; R and magnetic fields, not the vector and scalar potentials, that αβRR enter in the expressions that follow. Thus, in its perturbative (45) modifications, the system retains its periodic nature in the limit of uniform dc Maxwell fields. Here

Each site polarization ﬁeld, pR(x), is related to the elec- ˜ , = 1 δ + δ ˜ , jβR ;αR (x R) 2 ( RR RR ) jβR ;αR (x) (46) tronic charge density associated with that site, ρR(x), via the “relator” si(x; y, R) (see Appendix C), as shown previously with [10]. It is given by b b αR ;βR j˜ (x) =− s (x; z, R ) (z)dz βR ;αR 3 R3 i ≡ i , ρ , η , pR(x) s (x; y R) βR ;αR (y R)dy αR ;βR R3 αβR R 1 b αR ;βR − s (x; R2, R1)ς , (47) (40) 2 R2R1 R1R2

10As previously discussed, in this paper, we move the gauge free- dom of the energy eigenvectors, and thus the gauge dependence of 11The relator αib(x; y, R) is not to be confused with the OMP tensor ξ a αil the connections mn, into the Unα matrices. introduced in Eq. (1).

033126-7 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020) where Expanding (51) in powers of the electromagnetic field, we αR;βR e find the zeroth-order term to be ς = δRR δRR H¯βR ;αR − δRR δRR H¯βR ;αR R2R1 ih¯ 2 1 2 1 2 1 1 2 μi(0) = i − i ρ(0) , η(0) (48) R (y R ) βR;αR (y R)dy αR;βR αβRR and ∗ i αR;βR = e fα Wα (x)x Wα0(x)dx = ∇ · , 0 R (x) jβR ;αR (x R3) α 3 1 αR ;βR dk + ρν μ , . i i R2; R1 (x R3)FμR ;νR (49) = e f ξ + W , (55) 1 2 uc n 3 nn nn ih¯ μν (2π ) R1R2 n BZ Typically it is the modifications of the electric and mag- wherewehaveused(8)–(11). This corresponds to the un- netic dipole moments due to electromagnetic perturbations perturbed ground state site electric dipole moment, and upon that one is interested in studying; these correspond to the implementing (54), the usual expression [3]forP(0) is re- spatial integrals of the microscopic polarization and magne- produced. It is well known that the unperturbed macroscopic tization fields [10], respectively, polarization is unique modulo a “quantum of ambiguity.” This ambiguity originates from the gauge dependence of (55), and μi ≡ i ,νi ≡ i . it has been shown that the gauge dependent term of (55) R pR(x)dx R mR(x)dx (50) contributes only to this “quantum” [3]. Importantly, it is only Wi From (40), we find the diagonal elements of the matrix that appear in (55), and as a result, even a U(1) gauge transformation can give rise to this “quantum of ambiguity.” This is discussed further μi = i − i ρ , η R (y R ) βR ;αR (y R)dy αR ;βR (51) in Sec. IV. αβR R Turning to the site magnetic dipole moment, we expand (52) and (53) in powers of the electromagnetic field. The and νR = ν¯R + ν˜R, where, from (43), zeroth-order terms are found to be iab iab νi = a − a b , η ¯R (y R ) jβR;αR (y R)dy αR ;βR νi(0) = a − a b(0) , 2c ¯ fα (y R ) jα α (y R)dy αβ R c R ; R R R α 2 R (52) e = iab ∗ a b fα Wα0(x)x p (x)Wα0(x)dx (56) is the atomic-like contribution to the site magnetic dipole 2mc α moment and, from (45), and iab i a a b iab ν˜ = (y − R ) j˜ (y, R)dy ηα β i(0) a a b(0) R βR ;αR R ; R ν˜ = fα (y − R ) j˜ (y, R)dy 2c R αR ;αR αβR R 2c αR (53) e iab a (0) ∗ b = fαR Im Hα γ Wγ (x)x Wα0(x)dx , is the itinerant contribution to the site magnetic dipole mo- 1 0; R1 R1 2¯hc αγ ment. The macroscopic polarization and magnetization can be R1 found from their respective site dipole moments introduced (57) above, and are taken to be which, together, form the unperturbed ground state site mag- μ ν netic dipole moment, ν(0) = ν¯ (0) + ν˜ (0). Separately (56) and ≡ R , ≡ R . R R R P M (54) (57) are “multiband gauge dependent;” we adopt this phrase to uc uc describe quantities that are gauge dependent only if there are In the limit of uniform dc Maxwell fields, both the site electric degenerate Bloch energy eigenvectors. Here (56) and (57)are and magnetic dipole moments are independent of R, and as a in fact only gauge dependent if there are degenerate occupied consequence, the macroscopic polarization and magnetization energy eigenvectors, and in the limit of isolated valence bands fields are uniform. both (56) and (57) become gauge invariant individually. This is in agreement with past results [5]. Nonetheless, even if B. Unperturbed expressions the occupied bands are not isolated the sum of (56), (57)is generally a gauge invariant quantity and thus there is no ambi- We begin by confirming that our microscopic treatment guity in the value of the unperturbed ground state macroscopic yields the standard expressions for the unperturbed ground magnetization. Implementing (54), the usual expression [4,5] state macroscopic polarization and magnetization, more usu- for M(0) is reproduced. ally constructed from macroscopic arguments [28]. While (51)–(53) have been defined to include only valence and conduction electron contributions, the contributions from ion C. First-order perturbative modifications cores can be identified as well (see Mahon et al. [10]). We now turn to the first-order modifications of the Carte- μ ν However, we focus only on the former contributions here. sian components of R and R and thus, through (54), to

033126-8 MAGNETOELECTRIC POLARIZABILITY: A MICROSCOPIC … PHYSICAL REVIEW RESEARCH 2, 033126 (2020)

μ ν the components of P and M, due to an electromagnetic field. the moments R and R on those fields, even though in this Generally, the site quantities we consider are of the form contribution (62) the electron populations remain as they were before the system was perturbed. There is a familiar analog ≡ η , R βR ;αR (R) αR ;βR (58) to this in the response of an atom to the Maxwell magnetic αβRR field. Considering a single electron, the initial operator for the magnetic dipole moment νatom = (e/2mc)X P, where where R indicates one of the components of μ or νR, R here X and P are the position and momentum operators of ηα β is the single-particle density matrix, and β α (R), R ; R R ; R ν → / − / which we call a “site quantity matrix element,” is of the form the electron, becomes atom (e 2mc)X (P eA(X) c) when the magnetic field is nonvanishing. For a uniform mag- 1 = δ + δ = / βR ;αR (R) 2 ( RR RR ) βR ;αR ; (59) netic field, we can take A(X) (B X) 2, giving μ ν 2 see (41), (51)for R, and (44), (46), (52), (53)for R. e e ν = X P − X2B − (X · B)X . (63) atom 2mc 4mc2 1. Dynamical and compositional modifications The second term gives a contribution when the expectation The first-order modification to (58) due to an electromag- value is taken, even in the ground state (say a 1s orbital), netic field thus has two types of contributions, and gives the diamagnetic response of the atom. The contribu- (1) = (1;I) + (1;II), tions from the compositional modification to (60) are of this R R R (60) form. Nonetheless, in the extended systems we consider it is the first arising from the combination of the unperturbed important to note that during the perturbative analysis many expression for βR ;αR (R) and the first-order modification of lattice sites and all orbital types may be involved; for instance, η αR ;βR due to the Maxwell fields, observe that (46)–(49) would be used in constructing (53). We also distinguish between the first-order modifications (1;I) 1 (0) (1) (0) (1) ≡ η + η , (61) arising from the uniform dc Maxwell electric and magnetic R 2 αR;βR βR ;αR βR ;αR αR;βR αβR fields individually, such that and the second from the combination of the first-order mod- (1) = (E ) + (B), R R R (64) ification of βR;αR (R) due to the Maxwell fields and the unperturbed expression for ηαR;βR ,(25), where each of these modifications is composed of a dynamical and a compositional term, (1;II) (1) (1) ≡ fα (R) = fα . (62) R αR ;αR αR;αR (E ) = (E;I) + (E;II),(B) = (B;I) + (B;II). αR α R R R R R R

We refer to the first contribution (61) as “dynamical” 2. Induced polarization because it arises from modifications of the single-particle density matrix, which captures the electronic transition amplitude a. Modification due to the electric field. We begin by between various lattice sites and orbital types, due to the considering modifications due to the electric field. From (41), χ electromagnetic field. Notably in this type of modification of and the fact that αR(x) only depends on the magnetic field , , = a site quantity associated with R, that lattice vector always and (R x R) 0, it is clear that appears as at least one of the lattice vector indices identifying (E ) ρ (y, R) = 0, (65) the relevant single-particle density matrix elements. This is αR ;αR expected physically; a site quantity associated with R is μ(E ) and so there is no compositional modification to R .The affected by electrons moving between different orbital types first-order modification is entirely dynamical, as described at that lattice site, and by electrons moving from the region above, and given by “nearest” R to regions “nearest” other R . Contributions to η η = (61) arising from βR ;αR and αR;βR , with R R,area i(E ) i i (0) (E ) μ = (y − R )ρ (y, R)dy η consequence of the extended nature of the system, in which R βR ;αR αR ;βR αβRR electrons are not confined to regions of space; such contributions vanish in the limit that the crystalline solid is considered dk ξ l ξ i = e2 E l f mn nm . (66) simply as a periodic array of “isolated molecules,” which we uc nm 3 BZ (2π ) Emk − Enk call the “molecular crystal limit” (see Sec. IV B). Conversely, mn contributions to (61) arising from ηαR;βR take the form of Implementing (54), we find the usual result from perturbation single-site modifications. theory [29]. This modification is gauge invariant, in that the The second contribution (62) to the first-order modifica- final line of (66) is independent of the unitary transformation tion of a site quantity associated with R depends on the 12 μi(E ) l Unα (k). Also, the modification of R due to E is the same first-order modification of the site quantity matrix element as that of μl(E ) due to E i. (1) R associated only with lattice site R, αR;αR, and with orbital types α that are originally occupied. It is not associated with any change in the single-particle density matrix, but rather with the dependence of the associated site quantity matrix 12As previously discussed, in this paper, we move the gauge free- elements on the electromagnetic field itself; thus we call it dom of the energy eigenvectors, and thus the gauge dependence of ξ a a “compositional” modification. It leads to a dependence of the connections mn, into the Unα matrices.

033126-9 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020) b. Modification due the magnetic field. The first-order e2 dk + uc iabE l f Re iξ l Wb ξ a modification due to the magnetic field has nonvanishing dy- 2¯hc nm (2π )3 mn ns sm mns BZ namical and compositional modifications. Considering first the compositional modification, we find E − E + sk nk ξ l Wa ξ b . Re i mn ns sm (70) Emk − Enk i(B;II) i i (B) μ = fα (y − R )ρ (y, R)dy R αR ;αR b. Modification of the itinerant contribution due to the elec- αR tric field. This involves the first-order modification of (53) 2 e uc lab l dk i a due to the electric field, and has nonvanishing modifications of = B fα Re ξ˜ ∂ ξ˜ . (67) π 3 αγ b γα both compositional and dynamical origin. The compositional 2¯hc αγ BZ (2 ) modification is 2 We later simplify this term; notably it is gauge dependent. Re- i(E;II) e uc iab l dk l a ν˜ = E fα Re ξ˜ ∂ ξ˜ , (71) calling the modification of the single-particle density matrix R 3 αγ b γα 2¯hc BZ (2π ) due to the magnetic field (37), we find αγ and the dynamical modification is μi(B;I) R νi(E;I) ˜R = i − i ρ(0) , η(B) (y R ) β α (y R)dy α β 2 ∂ + R ; R R ; R e uc iab l dk b(Emk Enk ) a l αβ = E f ξ ξ R R 4¯hc nm (2π )3 E − E nm mn mn BZ mk nk e2 dk ∂ (E + E ) = uc labBl f b mk nk ξ a ξ i − nm π 3 − mn nm Esk Enk l a b 2¯hc BZ (2 ) Emk Enk − 2 Re iξ W ξ . (72) mn E − E mn ns sm s mk nk E − E + sk nk Re iξ a ξ b ξ i E − E ms sn nm Both (70) and (71) are gauge dependent in general, while s mk nk (72) is multiband gauge dependent. Very generally, there is a e2 dk simplification that occurs when (70)–(72) are summed to form + uc labBl f Re iξ i Wa ξ b , (68) 2¯hc nm (2π )3 ns sm mn the total site magnetic dipole moment: the term appearing in mns BZ the final line of (72) cancels with the term appearing in the which is also gauge dependent. The two separate terms ap- final line of (70), and as a result the gauge dependent terms ν(E ) pearing in the final equality of (68) originate individually from appearing in the total R do not explicitly depend on the the two terms of (37). In going from the first to the final energies Enk. Wi = equality, we have implemented (8), (11), and used mn 0 only if fm = fn, which holds for the class of insulators IV. MICROSCOPIC ORIGIN OF CONTRIBUTIONS considered here. Similar arguments are used in the following TO THE OMP TENSOR subsection when finding (70), (72). In Sec. IV, we explicitly A. Constructing the OMP tensor combine (67) and (68), and show that the usual OMP tensor [18] is reproduced. The OMP tensor, which describes the first-order modification of the macroscopic polarization due to a uniform dc magnetic field, is defined through 3. Induced magnetization i(B) = αil l In this work, we only consider modifications of the site P B magnetic dipole moment arising due to the electric field. and, from (54), (67), and (68), is found to be We defer to a later study the modification of the magnetic 2 dipole moment due to the magnetic field, as considered in this e dk ∂b(Emk + Enk ) αil = lab f ξ a ξ i framework. 2¯hc nm (2π )3 E − E nm mn mn BZ mk nk a. Modification of the atomic-like contribution due to the E − E electric field. This involves the first-order modification of + sk mk Re iξ a ξ b ξ i (52) due to the electric field. The compositional modification E − E ns sm mn s mk nk vanishes, as from (44) it is clear that e2 dk + lab ξ i Wb ξ a b(E ) , = , fnm Re i mn ns sm jα α (y R) 0 (69) 2¯hc (2π )3 R ; R mns BZ 2 following the argument leading to (65), and so this modifica- e lab dk i a + fα Re ξ˜ ∂ ξ˜ , (73) tion is entirely dynamical. Using (36), we find π 3 αγ b γα 2¯hc αγ BZ (2 ) e2 dk ∂ (E + E ) ν¯ i(E ) = uc iabE l f b mk nk ξ a ξ l after some manipulation of the band indices. If we now R 4¯hc nm (2π )3 E − E nm mn mn BZ mk nk define an analogous tensor describing the first-order modification of the macroscopic magnetization due to a uniform dc E − E + 2 sk mk Re iξ a ξ b ξ l electric field, we find that, upon combining (70), (71), and E − E ns sm mn αil s mk nk (72), this modification is described by the same tensor

033126-10 MAGNETOELECTRIC POLARIZABILITY: A MICROSCOPIC … PHYSICAL REVIEW RESEARCH 2, 033126 (2020) introduced above, but with the order of the indices switched, rewrite the sums to be over occupied states (the set {|vk}) and such that unoccupied states (the set {|ck}). We first consider the term in (73) involving the ratio of energy differences, which can be Mi(E ) = αliE l . traced back to (70), and adopt the shorthand |n≡|nk;we This is the usual result in theh ¯ω Egap limit, which is find effectively the condition we have initially assumed. E − E In the previous section we made a concerted effort to lab f sk mk Re iξ a ξ b ξ i nm E − E ns sm mn identify dynamical and compositional contributions to the var- mns mk nk ious first-order modifications; distinguishing between these αil Evk − Evk proves useful here. Because the tensor describes the first- = 2lab − Re (∂ v|v )(∂ v |c)(c|∂ v) i l l E − E b a i order modification of both P due to B , and M due to cvv vk ck i E , we can focus on the terminology associated with only E − E the magnetization. In what follows, we illustrate how the + ck c k Re (∂ v|c)(c|∂ c)(c|∂ v) E − E b a i three types of nonvanishing terms—the dynamical atomic- ccv vk ck like, the compositional itinerant, and the dynamical itinerant lab modifications—combine to give an OMP tensor having the + 2 Re (∂iv|c)(c|∂av )(v |∂bv) usual form cvv αil = αil + αil , G CS (74) + Re (∂iv|c)(∂ac|∂bv) . (75) αil αil where G contains only cross-gap contributions and CS,the cv Chern-Simons contribution, is a property of the subspace spanned by the originally occupied |nk. We find that the The first set of ...’s are identified as cross-gap contributions αil αil ... cross-gap contribution, G, originates from a combination of and will be included in G. The second set of ’s, together αil the dynamical atomic-like (70) and dynamical itinerant (72) with the penultimate and final terms of (73), form CS;the αil terms, whereas CS has contributions from the dynamical penultimate term of (73) originates from the gauge dependent atomic-like term as well, but also from the compositional term arising in the dynamical atomic-like modification (70) itinerant term (71). that does not explicitly depend on energy, and the final term of In order to compare with past results [17,18], we will re- (73) originates from the compositional itinerant modification express (73) in terms the cell-periodic functions x|nk and (71). We find

e2 dk αil = lab 2 Re (∂ v|c)(c|∂ v)(v|∂ v) + Re (∂ v|c)(∂ c|∂ v) CS 2¯hc (2π )3 i a b i a b BZ cvv cv 2 2 e lab dk i b a e lab dk i a + f Re iξ W ξ + fα Re ξ˜αγ ∂ ξ˜γα 2¯hc nm (2π )3 mn ns sm 2¯hc (2π )3 b mns BZ αγ BZ 2 il e abc dk a c 2i a b c a c 2i a b c =−δ ξ ∂bξ − ξ ξ ξ + ∂bW W − W W W (76) π 3 vv v v vv v v1 v1v vv v v vv v v1 v1v 2¯hc BZ (2 ) 3 3 vv vv v1 vv vv v1 (Appendix D), which is the usual Chern-Simons contribution to the OMP tensor [15–18], and is multiband gauge dependent in the sense introduced after Eq. (57). Due to this gauge dependence, this contribution to the OMP tensor is multivalued but, like P(0), has been shown to be unique modulo a quantum of ambiguity [9]. Furthermore, this contribution is isotropic, and the corresponding quantity vanishes in systems of spatial dimension less than three. αil The remaining terms compose G, in accordance with (74). These terms originate from the dynamical atomic-like modification (70), and the dynamical itinerant modification (72). We find e2 dk ∂ (E + E ) αil = lab − b ck vk Re (∂ v|c)(c|∂ v) G hc¯ (2π )3 E − E a i BZ cv vk ck E − E E − E − vk v k Re (∂ v|v)(∂ v|c)(c|∂ v) + ck c k Re (∂ v|c)(c|∂ c)(c|∂ v) , (77) E − E b a i E − E b a i cvv vk ck ccv vk ck which is in agreement with the usual expression for the It has been pointed out that the net gauge dependence of cross-gap contribution [17,18]. Importantly this contribution the OMP tensor, through the Chern-Simons contribution, has is found to be gauge invariant; this is a result of the fact that no effect on the induced charge-current density in the bulk the multiband gauge dependent terms appearing in (70) and [16,18]. We present a slightly different formulation of that (72) canceled one another. argument here, but starting with the same assumption used

033126-11 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020) in earlier arguments: While the derivation we have presented which can be easily confirmed using (81), under the condition here, as well as that in the approach of the “modern theory,” that at t =−∞the system is unperturbed. holds strictly only for uniform dc fields, we can expect that In the past, the origin of the ambiguity arising in the OMP if the macroscopic Maxwell electric and magnetic fields that tensor has, like that in the unperturbed macroscopic polariza- we consider are varying very slowly in both space and time, tion, been understood in the context of finite sized systems αil αil αil the same tensors CS and G can be used to good approxima- [9,30,31]. At a surface where cannot be treated as uniform tion.13 For such slowly varying fields, the first-order response the argument in the above paragraph breaks down, and the to the electromagnetic field in the bulk of a medium can be surface current that will arise shows an ambiguity that reflects expressed as the well-known gauge dependence of αCS [15,16]. However, it (1) , = (1) , + α , , seems there should be an equivalent bulk interpretation. That P (x t) P (x t) CSB(x t) certainly holds for the unperturbed macroscopic polarization; (78) M(1)(x, t) = M(1)(x, t) + α E(x, t), in a calculation where the energy eigenvectors are chosen and CS fixed at the start, a “quantum of ambiguity” arises from the αil = δil α where we have taken CS CS, and at the level of linear gauge dependent term of (55). This term has been shown [3] response, we have to depend only on the phase of the determinant of U (k), and as such, is not qualitatively different whether the bands are i(1) , = χ il l , + αil l , , P (x t) E E (x t) GB (x t) isolated or not. Indeed, this ambiguity can be understood at the Mi(1)(x, t) = χ il Bl (x, t) + αliE l (x, t), level of a U(1) gauge transformation, which in the simplest B G −ik · R of cases takes Unα (k)offormδnαe ; this corresponds χ il χ il where E and B are the usual linear electric and magnetic to changing the site with which each Wannier function is susceptibilities. Inside the bulk material, we then immediately associated and, in turn, this changes P(0) by a discrete amount, find that the induced macroscopic charge and current densities proportional to R. However, such an interpretation cannot be that result, used to understand the ambiguity associated with the OMP tensor; the gauge dependent term of (76) vanishes in the limit ρ(1)(x, t) =−∇ · P(1)(x, t), of isolated bands, and thus must interpreted on the more ∂P(1)(x, t) (79) general grounds of a multiband gauge transformation. Yet, J(1)(x, t) = + c∇ M(1)(x, t), ∂t at some level these ambiguities appear to be related, since the terms giving rise to them are constructed from the same can also be written as object, the Wa matrix; (55) contains only diagonal matrix ρ(1)(x, t) =−∇ · P(1)(x, t), elements, while (76) contains off-diagonal matrix elements as well. Perhaps it is from this perspective that a bulk in- ∂P(1)(x, t) J(1)(x, t) = + c∇ M(1)(x, t). (80) terpretation of the discrete ambiguity associated (76) can be ∂t formulated. That is, the contributions from the Chern-Simons coefficient, αCS, completely cancel each other. In deriving (80) from (79) B. Limiting cases we have used Faraday’s law and Gauss’ law for magnetism, In this section, we explore the magnetoelectric effect in ∂B(x, t) the limit of isolated molecules. First we construct the relevant c∇ E(x, t) + = 0, ∇ · B(x, t) = 0, (81) ∂t tensor for a single molecule, and then use that to construct the OMP tensor of a crystal in the “molecular crystal limit,” which which of course must be assumed to hold for E(x, t) and we take to be a model where there is a molecule at each lattice B(x, t), no matter how slowly they are varying in space and site with orbitals that have no common support with orbitals of time. Interestingly, the relation between {P(1)(x, t), M(1)(x, t)} molecules associated with other lattice sites. Finally, we show and {P (1)(x, t), M (1)(x, t)} can be understood as another kind that the molecular crystal limit so obtained is in agreement of “gauge dependence.” Very generally, such sets of fields lead with the appropriate limit of our general expressions (76) and to the same induced charge-current densities when (77). P(1)(x, t) = P(1)(x, t) + ∇ a(x, t), (82) 1. A single molecule 1 ∂a(x, t) M(1) x, t = M(1) x, t − + ∇b x, t , ( ) ( ) ∂ ( ) As pointed out earlier (Appendix D of Mahon el al. [10]), c t the response of a molecule to an electromagnetic field can be for a general vector field a(x, t) and a general scalar field treated via the same approach we have used here for a crystal, b(x, t); here the sets of fields {P(1)(x, t), M(1)(x, t)} and by constructing microscopic polarization and magnetization {P (1)(x, t), M (1)(x, t)} of (78) are related by fields from the electronic Green function, with the expectation t values of the electric and magnetic dipole moments following b(x, t) = 0, a(x, t) = cαCS E(x, t )dt , (83) from the single-particle density matrix (51)–(53). However, −∞ for a molecule (or an atom), it is also possible to follow a more common strategy in molecular physics [11,12], where microscopic polarization and magnetization operators are introduced, leading to operators associated with the electric 13See, e.g., Zhong et al. [33]. and magnetic dipole moments. We present that approach

033126-12 MAGNETOELECTRIC POLARIZABILITY: A MICROSCOPIC … PHYSICAL REVIEW RESEARCH 2, 033126 (2020) here (Appendix C of Mahon et al. [10]) to better make the νˆP + νˆD(t)aregivenby connection between this calculation and molecular physics. We take the initially unperturbed system to be described by μˆ = e ψˆ †(x)xψˆ (x)dx, (3,4), where we include an Astatic(x)—which of course need e not be periodic, since we are considering a localized system— νˆ = ψˆ †(x) x p(x) ψˆ (x)dx, to guarantee the breaking of time-reversal symmetry, and P 2mc | |→∞ consider a V (x) that vanishes as x and that does not e2 satisfy inversion symmetry about any point. The latter condi- νˆ (t) =− ψˆ †(x) (x · x)B(t) − x · B(t) x ψˆ (x)dx. D 2 tion could arise in a molecule because of a noncentrosymmet- 4mc ric configuration of the nuclei, or even in an atomic system Note that the last expression is the field theoretic analog of 0 because of an imposed dc electric field that is considered part (63). Taking Hˆ |G=EG|G, we then calculate the first-order of the unperturbed Hamiltonian. We consider a “special point” perturbative modifications due to static fields E and B.The R = 0, which for a molecule could be taken to be, say, the diamagnetic term in (85) will make no contribution to first center of mass of the ion cores and for an atom could be taken order, and if we denote the excited states by |H (with energies as the position of the ion core. In the frozen-ion approximation EH ), then by standard perturbation theory the modification of the contribution of the ions to the multipole moments of a the expectation value of the electric dipole moment operator μ(B) molecule will not affect the perturbative response calculation due to the magnetic field, ˆ atom, and the modification of the we make, so we neglect them. We introduce a “special point” expectation value of the magnetic dipole moment operator due ν(E ) electron field operator (see Appendix C of Mahon et al. [10]) to the electric field, ˆ atom,aregivenby ψˆ , sp(x t), μi(B) = αil l , νi(E ) = αli l , ˆ atom ˇ B ˆ atom ˇ E (86) ψˆ , = −i(x,0; t)ψˆ , , sp(x t) e (x t) where G|μˆ i|HH|νˆ l |G where αˇ il = 2Re P . (87) E − E e H H G (x, 0; t) ≡ sa(w; x, 0)Aa(w, t)dw, hc¯ We expand the field operator ψˆ (x) in terms of orbitals Wv (x) and A(x, t) is again the vector potential describing the electro- that are occupied in the ground state and orbitals Wc(x) that magnetic field. Then the Hamiltonian operator governing the are not, evolution of ψˆsp(x, t)is ψˆ (x) = cˆvWv (x) + cˆcWc(x), (88) v c ˆ = ψˆ † , , , − 0 , Hsp(t) sp(x t) H0 x p(x 0; t) e 0(x t) wherec ˆv andc ˆc are fermionic annihilation operators generat- ing single-particle eigenfunctions of H0(x, p(x)) with energies × ψˆsp(x, t)dx, (84) Ev and Ec, respectively, with | = , †| = . , 0 , cˆc G 0 cˆv G 0 where p(x 0; t) is given by (22) and 0(x t)by(C3); we have Then the expression (87) becomes ∂ψˆsp(x, t) ih¯ = ψˆ (x, t), Hˆ (t) . (μi ) (νl ) ∂t sp sp αˇ il = 2Re vc P cv , (89) E − E c,v c v We begin the evolution at a time t0 before the electromagnetic field is nonzero, taking as the (Heisenberg) ket the ground where state |G; at this time, we have ψˆ (x, t ) = ψˆ (x, t ) ≡ ψˆ (x). sp 0 0 (μi ) = exi , The dynamics can equivalently be described in a Schrödinger cv cv picture where the field operator is fixed at ψˆ (x) and the ket with | evolves from G at t0 according to a Hamiltonian operator i ≡ ∗ i xcv Wc (x)x Wv (x)dx (90) Hˆ (t) having the form of (84), but with ψˆsp(x, t) replaced ψˆ by (x). Using the approximate expressions (30), (31)for and (x, t) and 0(x, t), and neglecting the variation of the 0 0 e electric and magnetic fields over the atom or molecule, we νl = lab W ∗(x)xapb(x)W (x)dx ˆ P cv 2mc c v can write H(t)as e 0 1 lab a ∗ b Hˆ (t) = Hˆ − μˆ · E(t) − νˆ · B(t) − νˆ (t) · B(t), (85) = x W (x)p (x)Wv (x)dx P 2 D 2mc cn n n ≡ , ≡ , where E(t) E(0 t) and B(t) B(0 t), ie = lab − a b . (En Ev )xcnxnv ˆ 0 † 2¯hc H = ψˆ (x)H0 x, p(x) ψˆ (x)dx, n Here we have inserted a complete set of states {|n}, with and the operator for the electric dipole moment μˆ , and op- single-particle energies {En}, into the first expression, where erators for the paramagnetic (νˆP) and diamagnetic (νˆD(t)) the label n ranges over all v and c, and in going to the third line contributions to the magnetic dipole moment operator νˆ(t) = we have used the commutation relation of x and H0(x, p(x)) to

033126-13 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020)

b b 2. The molecular crystal limit write the matrix element of p (x) in terms of xnv in the usual way. From (89), we then have We can now construct the “molecular crystal limit,” in which we consider a periodic array of molecules where the e2 E − E αˇ il = lab n v Re ixi xa xb . (91) orbitals associated with a molecule at a given lattice site share hc¯ E − E vc cn nv vcn c v no common support with those of molecules associated with other lattice sites; again, we take the electric and magnetic As in a solid, this vanishes unless both time-reversal sym- fields that modify the electronic properties of the molecules to metry and inversion symmetry are broken: For if there is be the macroscopic Maxwell fields. Since the first-order modi- time-reversal symmetry the orbitals Wn(x) can be chosen to be fications to the electric and magnetic moments associated with real and the quantity inside the brackets is purely imaginary, each molecule are given by (86), using the expressions (54) i while if there is inversion symmetry the matrix elements xvc for P and M, together with the defining equation (1)forthe themselves vanish. orbital magnetoelectric polarizability tensor, from (92)–(94), Splitting the sum over n in (91) into a sum over filled states we have v and a sum over empty states c, we can write αil = αil + αil , ˚ ˚ G ˚ CS (95) αˇ il = αˇ il + αˇ il , (92) G CS where simply where αˇ il αˇ il α˚ il = G , α˚ il = CS , (96) 2 − G CS il e lab Ec Ec i a b uc uc α = ix x x ˇ G − Re vc cc c v hc¯ Ec Ev αil αil vcc with ˇ G and ˇ CS given by (93); here the circle accent identifies 2 − that the molecular crystal limit has been taken. e lab Ev Ev i a b + Re ix x x (93) Alternately, rather than building up the molecular crystal hc¯ E − E vc cv v v vvc c v limit by assembling a collection of molecules, we can imagine starting with a full band-structure calculation and taking the and limit where the Wannier functions associated with each lattice 2 site have no common support with the Wannier functions il e lab i a b αˇ = Re i x x x . associated with a different lattice site. We also take the ELWFs CS hc¯ vc cc c v vcc to be eigenfunctions of the unperturbed Hamiltonian, (3), which requires taking E → E . These conditions lead to In the second of these expressions, we sequentially put nk n simplifications in the general expressions (76) and (77), and when they are employed the result should reproduce the = − , = − , molecular crystal limit (95) and (96). A first simplification c n v c n v is that, since the bands are flat, ∂a(Emk + Enk ) → 0. Further, where both n and n range over all states; noting that the sums taking the orbitals introduced in (88) to be the ELWFs Wv0(x) over n and n give no net contribution, we have and Wc0(x), the flat bands can be identified by taking Unα (k) = δnα in (5) relating |αR with |ψnk. Hence the Hermitian e2 matrices Wa (12) vanish, and we need not distinguish between αil = lab i a b , ˇ CS Re i xvv xvv xvv the connections (9) and (10); the lack of common support for hc¯ vv v orbitals associated with different lattice sites, together with ξ a where the quantity in brackets is real. In three dimensions at (8), also implies that cv, etc., are independent of k, and from least two of i, l, a, b must be identical; if i = l the expression (8), (9), and (10), we take is found to vanish, and in general we can write ξ a = |∂ → a , cv (k) i(ck avk) xcv 2 il il ie cab c a b wherewehaveused(90) and αˇ = δ x x x . (94) CS 3¯hc vv v v v v vvv dk = 1. uc π 3 In the presence of uniform fields E and B, and in the BZ (2 ) frozen-ion approximation, each magnetoelectric tensor com- Applying these molecular crystal conditions to the general ex- ponentα ˇ il of a molecule can then be written (92)asthe pressions (76) and (77), we indeed recover (96), as expected. αil sum of a Chern-Simons-like term (94) and a cross-gap-like Note that in this limit there is no gauge dependence in ˚ CS, αil αil term (93). As in a solid, the Chern-Simons-like contribution since the bands are isolated, and as well both ˚ CS and ˚ G arise depends only on the occupied orbitals, and describes an solely from dynamical contributions. For while a calculation isotropic modification, regardless of how complicated might of the magnetic susceptibility would involve a compositional be the structure of the molecule. Note also that in a situation contribution due to the modification of the diamagnetic term, where all relevant initially occupied (unoccupied) orbitals even in the molecular crystal limit [see the discussion around are degenerate, Ec = Ec (Ev = Ev ), the cross-gap-like term (63)], the contributions to the magnetoelectric effect in that vanishes, in line with the vanishing of the cross-gap term in a limit are purely dynamical; were the calculation for a single solid when all relevant initially occupied (unoccupied) bands molecule here done in terms of the Green function strategy are degenerate [18]. used for a crystal, the modification would result from changes

033126-14 MAGNETOELECTRIC POLARIZABILITY: A MICROSCOPIC … PHYSICAL REVIEW RESEARCH 2, 033126 (2020) in ηα0;β0. Of course, in the molecular crystal limit all contribu- as a reference site for the electric and magnetic fields when tions are what we have called atomic-like rather than itinerant. calculating modifications to site quantities due to those fields. This strategy will be useful in future work, where we plan to take into account the spatial variations of the electric and C. Microscopic origin of αil and αil G CS magnetic fields. However, in the calculation reported here we While the qualitative features of the two contributions have restricted ourselves to the limit of uniform dc electric and to the OMP tensor have been discussed earlier [18], the magnetic fields. We began by reproducing the usual electric microscopic nature of the approach implemented here can susceptibility [29], and then found the orbital magnetoelectric provide further insight into the character of the cross-gap and polarizability tensor. Generally, the OMP tensor is written as Chern-Simons contributions. a combination of the isotropic, Chern-Simons contribution Since both of these contributions are nonvanishing in the and the cross-gap contribution; this microscopic theory repro- molecular crystal limit, neither can simply be understood as duces the usual result [15–18]. entirely a consequence of the delocalized nature of Bloch elec- In the course of the perturbative analysis it became evident trons. For the Chern-Simons contribution, this is in agreement that there are generally two distinct types of modifications with earlier work [18] where a particular model for a molecule that contribute to the tensors that relate site quantities to the at a lattice site was constructed that exhibits a Chern-Simons- Maxwell fields. The first type arose from modifications to like modification; our expression (94) for the Chern-Simons- the single-particle density matrix due to the electromagnetic like modification of an arbitrary molecule generalizes that field, and were termed “dynamical.” The other type arose result. However, neither contribution can be understood as a from modifications to the diagonal elements of site quantity purely “localized moleculelike contribution” either, because matrices, and were termed “compositional.” The electric sus- αil αil the full expressions for CS (76) and G (77) contain terms that ceptibility was found to arise from only a dynamical modifi- vanish in the molecular crystal limit; this is again in agreement cation. In our analysis of the magnetoelectric effect, we found with earlier arguments [18]. three terms with distinct microscopic origins combine to form When moving from the molecular crystal limit of the the OMP tensor: a dynamical modification to the atomic-like Chern-Simons and cross-gap tensors, where the only con- contribution, and dynamical and compositional modifications tributions are atomic-like, to the full crystal expressions, to the itinerant contribution. We found the Chern-Simons both acquire itinerant contributions. In addition, while the contribution to arise from a combination of parts of the cross-gap tensor is purely dynamical in nature, both in the atomic-like dynamical modification, and the itinerant compo- molecular crystal limit and more generally, the Chern-Simons sitional modification. The cross-gap contribution was found tensor acquires a compositional modification when moving to arise from the remainder of the atomic-like dynamical from the molecular crystal limit to the general expression modification, and the itinerant dynamical modification. We for a crystal. This suggests that perhaps it is through this have also compared our expressions with those that arise in compositional modification that a bulk interpretation of the the molecular crystal limit, a model in which a periodic array discrete ambiguity associated with the Chern-Simons tensor of isolated molecules is considered. can be constructed. We plan to explore this conjecture in a As is well known, the linearly induced macroscopic bulk future publication. charge and current densities that result from uniform dc electric and magnetic fields are gauge invariant, but at a surface where the bulk arguments underlying this result are not valid, V. CONCLUSION gauge dependent currents emerge. The ambiguity associated We have implemented a previously developed [10]mi- with such surface currents can be studied in detail via the croscopic theory of polarization and magnetization to study microscopic theory that underpins the approach taken here; modifications of the orbital electronic properties of a class of we plan to consider this in future work. We also intend to insulators due to uniform dc electric and magnetic fields, at extend the calculations presented in this paper to take into zero temperature. To first-order in the Maxwell fields, the free account the frequency dependence of response tensors in gen- charge and current densities vanish [10]; in future work, we eral. This will lead to a description of frequency-dependent plan to extend this type of investigation to metallic systems magnetoelectric effects, such as optical activity, for which the for which these quantities would be relevant. Thus the pertur- breaking of both spatial-inversion symmetry and time-reversal bative modifications of both the electronic charge and current symmetry in the unperturbed material system is not required. density expectation values due to the Maxwell fields can be A microscopic understanding of the mechanisms giving rise found directly from the corresponding modifications of the to such effects is now accessible via this formalism. microscopic polarization and magnetization fields. Associated with the dipole moment of the site microscopic polarization (magnetization) field is a macroscopic polarization (magneti- ACKNOWLEDGMENTS zation), for which we extract various tensors relating it to the Maxwell fields. We thank Ivo Souza, Rodrigo A. Muniz, Sylvia Swiecicki, A quantity central in any calculation implementing this mi- and Julen Ibanez-Azpiroz for insightful discussions. This croscopic formalism is the single-particle density matrix. We work was supported by the Natural Sciences and Engineering began by re-expressing the equation governing its dynamics Research Council of Canada (NSERC). P.T.M. acknowledges to include an arbitrary lattice site, Ra, which is to be used a PGS-D scholarship from NSERC.

033126-15 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020)

APPENDIX A: INTRODUCING ARBITRARY LATTICE SITES Here we work out an expression for , , i (R R R ;t) = A + B + C + D e H¯αR;ηR (t) (A1)

(see Eq. (35) of [10] for the deﬁnition of H¯αR;ηR (t)), where 1 i(R,R,R;t) ∗ i(R,x,R;t) A = e χα (x, t)e H (x, p(x, R ; t))χη (x, t)dx, 2 R 0 R 1 i(R,R,R;t) ∗ i(R,x,R;t) B = e H (x, p(x, R; t))χα (x, t) χη (x, t)e dx, 2 0 R R e i(R,R,R;t) i(R,x,R;t) ∗ 0 0 C =− e e χα (x, t) (x, t) + (x, t) χη (x, t)dx, 2 R R R R ∗ ∂χ , ∂χ , ih¯ i(R,R,R;t) i(R,x,R;t) ∗ ηR (x t) αR(x t) D =− e e χα (x, t) − χη (x, t) dx. 2 R ∂t ∂t R Looking at the ﬁrst of these terms, we note that −i(Ra,x,R ;t) i(Ra,x,R ;t) p(x, R ; t) = e p(x, Ra; t)e , so 1 i(R,R ,R ;t) ∗ i(R,x,R ;t) −i(Ra,x,R ;t) i(Ra,x,R ;t) A = e χα (x, t)e e H (x, p(x, R ; t))e χη (x, t)dx. 2 R 0 a R Now (R, R , R ; t) + (R, x, R ; t) − (Ra, x, R ; t) = (R, x, Ra, R , R ; t) = (R, x, Ra; t) + (R, Ra, R , R ; t) and so 1 i(R,Ra,R ,R ;t) ∗ i(R,x,Ra;t) i(Ra,x,R ;t) A = e χα (x, t)e H (x, p(x, R ; t))e χη (x, t)dx. 2 R 0 a R Similarly, since

−i(Ra,x,R;t) i(Ra,x,R;t) p(x, R; t) = e p(x, Ra; t)e , we have 1 i(R,R,R;t) −i(R ,x,R;t) i(R ,x,R;t) ∗ i(R,x,R;t) B = e e a H (x, p(x, R ; t))e a χα (x, t) χη (x, t)e dx 2 0 a R R 1 i(R,R,R;t) i(R ,x,R;t) ∗ i(R,x,R;t) i(R ,x,R;t) = e H (x, p(x, R ; t))e a χα (x, t) χη (x, t)e e a dx. 2 0 a R R Now (R, R , R ; t) + (R, x, R ; t) + (Ra, x, R; t) = (R, Ra, x, R , R ; t) = (Ra, x, R ; t) + (R, Ra, R , R ; t), so 1 i(R,R ,R,R;t) i(R ,x,R;t) ∗ i(R ,x,R;t) B = e a H (x, p(x, R ; t))e a χα (x, t) χη (x, t)e a dx 2 0 a R R 1 i(R,Ra,R ,R ;t) ∗ i(R,x,Ra;t) ∗ i(Ra,x,R ;t) = e H (x, p(x, R ; t)) e χα (x, t) e χη (x, t)dx. 2 0 a R R And then 1 i(R,Ra,R ,R ;t) ∗ i(R,x,Ra;t) i(Ra,x,R ;t) A + B = e χα (x, t)e H (x, p(x, R ; t))e χη (x, t)dx 2 R 0 a R 1 i(R,Ra,R ,R ;t) ∗ i(R,x,Ra;t) ∗ i(Ra,x,R ;t) + e H (x, p(x, R ; t)) e χα (x, t) e χη (x, t)dx 2 0 a R R , , , ∗ , , , , = i (R Ra R R ;t) χ , i (R x Ra;t) , , i (Ra x R ;t)χ , . e αR(x t)e H0(x p(x Ra; t))e ηR (x t)dx

033126-16 MAGNETOELECTRIC POLARIZABILITY: A MICROSCOPIC … PHYSICAL REVIEW RESEARCH 2, 033126 (2020)

The last form is not “explicitly Hermitian,” but it will be convenient. Next, since (R, Ra, R , R ; t) + (R, x, Ra; t) + (Ra, x, R ; t) = (R, R , R ; t) + (R, x, R ; t), (A2) we can write , , , ∗ , , , , A + B + C = i (R Ra R R ;t) χ , i (R x Ra;t)H , i (Ra x R ;t)χ , e αR(x t)e Ra (x t)e ηR (x t)dx i(R,R,R;t) i(R,x,R;t) ∗ 0 e 0 e 0 + e e χα (x, t) e (x, t) − (x, t) − (x, t) χη (x, t)dx, R Ra 2 R 2 R R wherewehaveused(21). Now 0 e 0 e 0 e 0 0 e 0 0 e (x, t) − (x, t) − (x, t) = (x, t) − (x, t) + (x, t) − (x, t) Ra 2 R 2 R 2 Ra R 2 Ra R e e = 0 (x, t) + 0(R, t) + 0 (x, t) + 0(R, t) 2 Ra x 2 Ra x e 0 0 0 e 0 0 0 = (x, t) + (R , t) + (Ra, t) + (x, t) + (R, t) + (Ra, t) 2 Ra x R 2 Ra x R e 0 0 − (R , t) + (R , t) 2 R a R a ∂ , , ∂ , , h¯ (Ra x R ; t) h¯ (Ra x R; t) e 0 0 = + − (R , t) + (R , t) , 2 ∂t 2 ∂t 2 R a R a and recalling (A2), we have h¯ ∂(R , x, R ; t) h¯ ∂(R , x, R; t) i(R,Ra,R ,R ;t) ∗ i(R,x,Ra;t) a a i(Ra,x,R ;t) A + B + C = e χα (x, t)e H (x, t) + + e R Ra 2 ∂t 2 ∂t e i(R,R,R;t) 0 0 i(R,x,R;t) ∗ × χη (x, t)dx − e (R , t) + (R , t) e χα (x, t)χη (x, t)dx. R 2 R a R a R R As the function χα (x, t) satisfies the modified orthogonality relation [10] R , , ∗ i (R x R ;t)χ , χ , = δ δ , e αR(x t) ηR (x t)dx αη RR we find h¯ ∂(R , x, R ; t) h¯ ∂(R , x, R; t) i(R,Ra,R ,R ;t) ∗ i(R,x,Ra;t) a a A + B + C = e χα (x, t)e H (x, t) + + R Ra 2 ∂t 2 ∂t , , × i (Ra x R ;t)χ , − 0 , δ δ , e ηR (x t)dx e R(Ra t) αη RR

, and so from (A1)wehave(18), where we have deﬁned H¯αR;ηR (Ra t)asin(20). We also require , , , , , , , i (R R R ;t) ¯ = i (R R R;t) ¯ = i (R Ra R R;t) ¯ , − 0 , δ δ e HηR ;βR (t) e HηR ;βR (t) e HηR ;βR (Ra t) e R (Ra t) ηβ R R , , , = i (R R Ra R ;t) ¯ , − 0 , δ δ , e HηR ;βR (Ra t) e R (Ra t) ηβ R R wherewehaveused(20). Since the lattice site Ra is arbitrary, we can as well write (19) for any lattice site Rb.

(0) (0) APPENDIX B: PERTURBATIVE MODIFICATIONS where H is given by (26). Thus η (t) evolves as the αR ;λR3 αR ;βR OF THE SINGLE-PARTICLE DENSITY MATRIX unperturbed single-particle density matrix, and consequently Beginning with the equation of motion for the single- (0) η (t) = fαδαβδ , particle density matrix αR ;βR R R as expected. ∂ηα β (t) R ; R μR1;νR2 From (B1) it is found that the first-order modification to the ih¯ = F (t)ημ ν (t), (B1) ∂t αR ;βR R1; R2 single-particle density matrix due to the electromagnetic field μνR R 1 2 evolves according to then expanding all quantities in powers of the electromagnetic (1) ∂η (t) field, and then matching powers, at zeroth order, we find αR ;βR = (0) η(1) − η(1) (0) ih¯ Hα λ λ β (t) α λ Hλ β ∂t R ; R3 R3; R R ; R3 R3; R λR ∂η(0) 3 α β (t) μ μ R ; R = (0) η(0) − η(0) (0) , + R1; R1 (1) , ih¯ Hα λ λ β (t) α λ Hλ β fμFα β (t) (B2) ∂t R ; R3 R3; R R ; R3 R3; R R ; R λR3 μR1

033126-17 PERRY T. MAHON AND J. E. SIPE PHYSICAL REVIEW RESEARCH 2, 033126 (2020) the ﬁnal term of which is found to be for which, from (B2), we ﬁnd

μR1;μR1 (1) (1) μ = βα ¯ , . (1) f Fα β (t) f Hα β (Ra t) ∂η (t) R ; R R ; R mk;nk (1) μR ih¯ = (E − E )η (t)+ f ψ |μR 1 ∂ mk nk mk;nk nm mk 1 t μν It is useful to deﬁne the intermediate quantity R1R2 (1) × H¯μ ν (Ra, t)νR2|ψ . η = ψ |μ η ν |ψ , R1; R2 nk mk;nk (t) mk R1 μR1;νR2 (t) R2 nk (B3)

μνR1R2 Then, implementing the usual Fourier analysis via (27), we find (1) ψ |μ ,ω ν |ψ mk R1 H¯μ ν (Ra ) R2 nk η(1) ω =− R1; R2 , ( ) fnm + mk;nk − − ω + Emk Enk h¯( i0 ) μνR1R2 where 0+ entering in the denominator describes the “turning on” of the electromagnetic field at t > −∞. Finally, using (20) and the inverse of (B3), we find (1) α |ψ ψ |μ ,ω ν |ψ ψ |β R mk mk R1 Hμ ν (Ra ) R2 nk nk R η(1) ω =− R1; R2 α β ( ) fnm dkdk + R ; R E − E − h¯(ω + i0 ) μνR R mn BZ mk nk 1 2 i ∗ + fβα W (x) (R , x, R ; ω) + (R , x, R ; ω) Wβ (x)dx, (B4) 2 αR a a R where H (1) (R ,ω) is given by (29). We now apply this result to the case of uniform dc fields and find the first-order μR1;νR2 a modification to the single-particle density matrix due to a electric field, Eq. (36). The second term of (B4) vanishes trivially, and for the first we make use of (29) and (34) to find (E ) l ∗ l l H (R ,ω) =−eE Wμ (x) x − R Wν (x)dx, (B5) μR1;νR2 a R1 a R2 where E ≡ E(Ra,ω = 0) for any Ra, in this limit. Then, recalling (5), we find 2 i(k · R−k · R ) † e Uα (k)Unβ (k ) − · · ∗ η(E ) = l uc m ik R1 ik R2 l † α β eE fnm dkdk e e Umμ(k) WμR −R (x)x Wν0(x)dx Uνn(k ) R ; R π 6 − 1 2 (2 ) BZ Emk Enk mn μνR1R2 3 i(k · R−k · R ) † e Uα (k)Unβ (k ) − − · − · = l uc m ξ˜ l † i(k k1 ) R1 i(k k1 ) R2 eE fnm dkdk dk1 Umμ(k) μν (k1)Uνn(k ) e e π 9 − (2 ) BZ Emk Enk μν mn R1 R2 ik · (R−R ) † l e U (k)ξ (k)U β (k) = eE l uc f dk αm mn n , (2π )3 nm E − E mn BZ mk nk where we have used the identity straight-line path; see Ref. [10], where we find 1 uc − · ei(k k ) R = δ(k − k ), (B6) si(w; x, y) = (xi − yi )δ(w − y − u(x − y))du, (2π )3 R 0 as well as (8) and (11), in going to the final expression. Notice 1 αlj(w; x, y) = lmj (xm − ym )δ(w − y − u(x − y))udu, that the approximation of an electric field that varies little on 0 the scale of the lattice constant gives rise to a simplified form (C1) (E ) of (29), which in turn allows η to be written as a single αR ;βR for our relators, and we will consider the quantities η(B) Brillouin zone integral. To derive the expression for αR;βR j , = αlj w , l w, w, a similar procedure is followed; however, one must also use y (x t) ( ; x y)B ( t)d (C2) (15). In this case, the approximation that the magnetic field (B) varies little on the scale of the lattice constant allows η αR ;βR 0(x, t) = si(w; x, y)E i(w, t)dw, (C3) to be written as a single Brillouin zone integral. y and APPENDIX C: NEARLY UNIFORM hc¯ ELECTROMAGNETIC FIELDS (x, z, y; t) = si(w; x, z)Ai(w, t)dw e We now work out some of the general expressions for our relators and quantities dependent on them in the limit of nearly + si(w; y, x)Ai(w, t)dw uniform electromagnetic fields. By this we mean that we keep the electric field and its first derivatives at the “expansion + i w , i w, w. point,” but only the magnetic field at that point. We use a s ( ; z y)A ( t)d (C4)

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The “expansion point” here is y. We ﬁrst consider while j , l , αlj w , w. i j j y (x t) B (y t) ( ; x y)d s (w; x, z)(w − y )dw Now = si(w; x, z)(z j − y j )dw + si(w; x, z)(w j − z j )dw αlj(w; x, y)dw = i − i j − j + 1 i − i j − j 1 1 (x z )(z y ) (x z )(x z ) = lmj (xm − ym )udu = lmj(xm − ym ), 2 2 0 = (xi − yi ) − (zi − yi ) (z j − y j ) so , 1 , − . 1 i i i i j j j j y(x t) 2 B(y t) (x y) + (x − y ) − (z − y ) (x − y ) − (z − y ) 2

Next, we consider = 1 i − i j − j − 1 i − i j − j 0 , i , i w , w (x y )(z y ) (z y )(x y ) y (x t) E (y t) s ( ; x y)d 2 2 i ∂E (y, t) 1 i i j j 1 i i j j + si(w; x, y)(wk − yk )dw. + (x − y )(x − y ) − (z − y )(z − y ), ∂yk 2 2 where in the third line we have used (C5), (C6); similarly The terms we need are 1 i w , w = i − i = i − i 1 s ( ; x y)d (x y )du x y (C5) si(w; y, x)(w j − y j )dw =− (xi − yi )(x j − y j ), 0 2 and 1 and ﬁnally si(w; x, y)(wk − yk )dw = (xi − yi )(xk − yk )udu 1 0 si(w; z, y)(w j − y j )dw = (zi − yi )(z j − y j ). 1 2 = (xi − yi )(xk − yk )(C6) 2 So, in all so we have 1 ∂E i(y, t) j − j i w , + i w , + i w , w 0(x, t) (xi − yi )E i(y, t) + (xi − yi )(xk − yk ) (w R ) s ( ; x z) s ( ; y x) s ( ; z y) d y 2 ∂yk

1 1 i i j j 1 i i j j = (xi − yi )E i(y, t) + (xi − yi )(xk − yk ) = (x − y )(z − y ) − (z − y )(x − y ), 2 2 2

∂ i , ∂ k , × 1 E (y t) + 1 E (y t) and 2 ∂yk 2 ∂yi hc¯ , , 1 (x z y; t) = (xi − yi )E i(y, t) + (xi − yi )(xk − yk )F ik (y, t), e 2 1 ∂Ai(y, t) wherewehaveused(33). Finally, we look at (x, z, y; t). This (xi − yi )(z j − y j ) − (zi − yi )(x j − y j ) 2 ∂y j can be done “by hand” when the magnetic ﬁeld is uniform, but in what follows we work it out formally. 1 ∂Ai(y, t) ∂A j (y, t) = − (xi − yi )(z j − y j ), ∂ j ∂ i hc¯ 2 y y (x, z, y; t) e and since Ai(y, t) si(w; x, z) + si(w; y, x) + si(w; z, y) dw ∂A j (y, t) ∂Ai(y, t) − = kijBk (y, t), ∂ i ∂ j y y ∂Ai(y, t) + (w j − y j ) si(w; x, z) + si(w; y, x) we have ∂y j hc¯ 1 + i w , w +··· (x, z, y; t) − B(y, t) · (x − y) (z − y) . s ( ; z y) d e 2 From (C5), we have We collect all of these approximate expressions in (31,30,32). j , 0 , The expansions of y (x t) and y (x t) derived here can also si(w; x, z) + si(w; y, x) + si(w; z, y) dw = 0, be derived using a formal expansion of the relators (C1) about u = 0.

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APPENDIX D: CONSTRUCTING THE CHERN-SIMONS + lab ξ i + Wi ξ a + Wa Wb δ − δ Wb fn ps ps mn mn i sm np i sm np CONTRIBUTION TO THE OMP TENSOR nmps Here we outline the steps in going from the first to the = lab ξ i + Wi ξ b ξ a − Wb Wa fn nm nm i ms sn i ms sn second equality of Eq. (76). The final term of the first line nms of (76) can be re-expressed, using (11), as proportional to the + lab ξ i + Wi Wb ξ a + Wa Brillouin zone integral of i fn ns ns sm mn mn nms − ξ i + Wi ξ a + Wa Wb , sm sm mn mn ns lab ξ˜ i ∂ ξ˜ a fα αγ b γα iab∂ ξ a = iab ξ b ξ a where we have used the identities b mn i s ms sn αγ iab∂ Wa =−iab Wb Wa and b mn i s ms sn; in the following we lab i i † a a † Wa = = = f ξ + W U γ ∂ U ξ + W U α U will also often use mn 0 only if fm fn. We now con- n ps ps s b γ m mn mn n αp W αγ nmps sider the contributions to (76) at each order in .Wefirst consider the terms quadratic in W; their contribution to (76) = lab ξ i + Wi ∂ ξ a + Wa fn nm nm b mn mn is proportional to the Brillouin zone integral of nm lab −ξ i Wb Wa + Wi Wb ξ a + ξ i Wb Wa − Wi ξ a Wb − ξ i Wa Wb Re ifn nm ms sn ns sm mn ns sm mn sm mn ns sm mn ns nms = lab ξ a Wi Wb + ξ b Wa Wi + ξ i Wb Wa Re ifn mn ns sm mn ns sm sm mn ns nms = δil lab ξ a Wl Wb + ξ b Wa Wl + ξ l Wb Wa Re ifn mn ns sm mn ns sm sm mn ns nms

= δil cab ξ a Wc Wb , Re ifn nm ms sn nms where in going from the second to third line we have used the fact that, in three-dimensions, at least two of i, l, a, b must be identical; if i = l the expression is found to vanish. This sort of argument is often used in what follows. The contribution to (76) that is linear in W [notice the penultimate term of (76) also contributes here] is proportional to the Brillouin zone integral of lab Wi ξ b ξ a + ξ i Wb ξ a − ξ i ξ a Wb − lab ξ i Wb ξ a Re ifn nm ms sn ns sm mn sm mn ns fnmRe i nm ms sn nms mns = lab Wi ξ b ξ a − ξ i ξ a Wb + lab ξ i Wb ξ a Re ifn nm ms sn sm mn ns fmRe i nm ms sn nms mns = δil lab ξ b ξ a Wl + ξ l ξ b Wa + ξ a ξ l Wb Re ifn ms sn nm sm mn ns sm mn ns nms

= δil cab ξ b ξ a Wc . Re ifn nm ms sn nms Now the combined contribution of the linear and quadratic in W terms is proportional to δil abc ξ b ξ a Wc + ξ a Wc Wb = δil abc ∂ ξ a Wc + ξ a Wc Wb fn dkRe i nm ms sn i nm ms sn fn dkRe b ns sn i nm ms sn nms BZ ns BZ m = δil abc − ξ a ∂ Wc + ξ a Wc Wb fn dkRe ns b sn i nm ms sn ns BZ m = 0, where we have used an integration by parts on the initially linear in W term. This “miraculous cancellation” is also presented in Appendix C of Vanderbilt [9]. Thus, the contribution cubic in W is the only gauge dependent term that has not vanished, or been canceled. Its contribution to (76) is proportional to the Brillouin zone integral of lab −Wi Wb Wa + Wi Wb Wa − Wi Wa Wb Re ifn nm ms sn ns mn sm sm mn ns nms

lab i a b =− Re iW W W vv v v1 v1v vv v1

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cab il c a b =−δ Re iW W W 3 vv v v1 v1v vvv 1 il abc a c 2i a b c =−δ Re ∂ W W − W W W . (D1) b vv1 v1v vv v v1 v1v 3 vv1 vv1v The contribution of this term to (76) is proportional to the Brillouin zone integral of the well-known term arising from the gauge-transformation of the Chern-Simons 3-form (see, e.g., Eq. C.19 of Vanderbilt [9]). The contribution independent of W is proportional [notice the ﬁrst term of (76) contributes here] to the Brillouin zone integral of lab ξ i ξ b ξ a + lab ∂ | |∂ |∂ + ∂ | ∂ |∂ Re ifn nm ms sn 2 Re ( iv c)(c av )(v bv) Re ( iv c)( ac bv) nms cv cvv lab lab = Re (∂iv|m)(∂bm|∂av) − (∂iv|c)(∂bc|∂av) + 2 Re (∂iv|c)(c|∂av )(v |∂bv) vm cvv lab lab = Re (∂iv|v )(∂bv |∂av) + 2 Re (∂iv|c)(c|∂av )(v |∂bv) , (D2) vv cvv which is equivalent to Eqs. (A11a)+(A11b) of Essin et al. [18]. Then, in all, (76) results from the combination of (D1)+(D2).

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