Gyrotropic Magnetic Effect and the Magnetic Moment on the Fermi Surface

# Gyrotropic Magnetic Effect and the Magnetic Moment on the Fermi Surface

Shudan Zhong Department of Physics, University of California, Berkeley, California 94720, USA    Joel E. Moore Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA    Ivo Souza Centro de Física de Materiales, Universidad del País Vasco, 20018 San Sebastían, Spain Ikerbasque Foundation, 48013 Bilbao, Spain
July 20, 2019
###### Abstract

The current density induced in a clean metal by a slowly-varying magnetic field is formulated as the low-frequency limit of natural optical activity, or natural gyrotropy. Working with a multiband Pauli Hamiltonian, we obtain from the Kubo formula a simple expression for in terms of the intrinsic magnetic moment (orbital plus spin) of the Bloch electrons on the Fermi surface. An alternate semiclassical derivation provides an intuitive picture of the effect, and takes into account the influence of scattering processes in dirty metals. This “gyrotropic magnetic effect” is fundamentally different from the chiral magnetic effect driven by the chiral anomaly and governed by the Berry curvature on the Fermi surface, and the two effects are compared for a minimal model of a Weyl semimetal. Like the Berry curvature, the intrinsic magnetic moment should be regarded as a basic ingredient in the Fermi-liquid description of transport in broken-symmetry metals.

Introduction.— When a solid is placed in a static magnetic field the nature of the electronic ground state can change, leading to striking transport effects. A prime example is the integer quantum Hall effect in a quasi-two-dimensional metal in a strong perpendicular field Thouless et al. (1982). Novel magnetotransport effects have also been predicted to occur in 3D topological (Weyl) metals, such as an anomalous longitudinal magnetoresistence Nielsen and Ninomiya (1983); Son and Spivak (2013), and the chiral magnetic effect (CME), where an electric pulse induces a transient current  Son and Yamamoto (2012); both are related to the chiral anomaly that was originally discussed for Weyl fermions in particle physics Adler (1969); Bell and Jackiw (1969). In all these phenomena the role of the static  field is to modify the equilibrium state, but an  field is still required to put the electrons out of equilibrium and drive the current (since , the vector potential is time dependent even for a static  field).

Recently, the intriguing proposal was made that a pure  field could drive a dissipationless current in certain Weyl semimetals where isolated band touchings [the “Weyl points” (WPs)] of opposite chirality are at different energies Zyuzin et al. (2012). The existence of such an effect was later questioned Vazifeh and Franz (2013), and the initial interpretation as an equilibrium current was discounted. (Indeed, that would a violate a “no-go theorem” attributed to Bloch that forbids macroscopic current in a bulk system in equilibrium Yamamoto (2015).) Subsequent theoretical work suggests that the proposed effect can still occur in transport, as the current response to a  field oscillating at low frequencies Chen et al. (2013); Goswami and Tewari (); Chang and M.-F.Yang (2015); Chang and Yang (2015).

At present the effect is still widely regarded as being related to the chiral anomaly Chen et al. (2013) (or, more generally, to the Berry curvature of the Bloch bands Goswami and Tewari (); Chang and M.-F.Yang (2015); Chang and Yang (2015); Goswami et al. (2015)), and is broadly characterized as a type of CME. We show in this Letter that the experimental implications and microscopic origin of this effect are both very different from the CME (as defined in Ref. Son and Yamamoto (2012), consistent with the particle-physics literature Kharzeev (2014)). Experimentally, the effect is realized as the low-frequency limit of natural gyrotropy 111The term natural gyrotropy refers to the time-reversal-even part of the optical response of a medium at linear order in the wave vector of light Landau and Lifshitz (1984); Agranovich and Ginzburg (1984). The reactive part gives rise to natural optical rotation, and the dissipative part to natural circular dichroism. Furthermore, polar crystals display natural gyrotropy effects unrelated to optical rotation Agranovich and Ginzburg (1984). Gyrotropic effects that are time-reversal-odd and zeroth order in the wave vector of light (e.g., Faraday rotation and magnetic circular dichroism Landau and Lifshitz (1984)) are not considered in this work. in clean metals (see also Ref. Goswami et al., 2015), and we will call it the “gyrotropic magnetic effect” (GME). Both and  optical fields drive the gyrotropic current, but at frequencies well below the threshold for interband absorption () their separate contributions can be identified. In nonpolar metals, the induced gyrotropic current can be inferred from optical rotation measurements. The GME is predicted to occur not only in certain Weyl semimetals, but in any optically active metal; it is necessary that the structure lacks an inversion center, and it is sufficient that the structure is either chiral Landau and Lifshitz (1984); Flack (2003); Newnham (2005) or polar Agranovich and Ginzburg (1984).

Existing expressions for the natural gyrotropy current in metals involve the Berry curvature of all the occupied states (and velocities of empty bands) Goswami and Tewari (); Chang and M.-F.Yang (2015); Chang and Yang (2015); Goswami et al. (2015), at odds with the notion that transport currents are carried by states near the Fermi level . Integrals over all occupied states involving the Berry curvature also appear in calculations of a part of the low-frequency optical activity Orenstein and Moore (2013); Hosur and Qi (2015); Zhong et al. (2015), and of the anomalous Hall effect (AHE); in the case of the AHE, a Fermi surface (FS) reformulation exists Haldane (2004). We find that the GME is not governed by the chiral anomaly or the Berry curvature, but by the intrinsic magnetic moment of the Bloch states on the FS. Our analysis also takes into account the finite relaxation time in real materials, which is shown to weaken the effect at the lowest frequencies. The magnitude of the GME is estimated for the predicted chiral Weyl semimetal SrSi Huang et al. ().

CME versus GME.— Both effects can be discussed by positing a linear relation between and :

 ji=αijBj. (1)

Suppose we use linear response to evaluate for a clean metal, describing the  field in terms of a vector potential that depends on both and . The result will depend on the order in which the and limits are taken Chen et al. (2013); Goswami and Tewari (); Chang and M.-F.Yang (2015), much as the compressibility and conductivity are different limits of electrical response. The CME tensor can be obtained from Eq. (1) in the equilibrium or static limit of the magnetic field (setting before sending ), with an additional step needed to describe the -field pulse. The GME tensor is extracted directly from Eq. (1) in the transport or uniform limit (sending before ) that describes conductivities in experiment. (Here, “” means , but note that because the clean limit is assumed; effects caused by finite relaxation times in dirty samples will be discussed later.) Only is a material property, since the details of the -field pulse producing nonequilibrium are missing from . Below we derive microscopic expressions for both.

Chiral magnetic effect.— The tensor calculated in the static limit is isotropic, , with

 αstat=−e2ℏ∑n∫[dk]f0kn(vkn⋅Ωkn)=0, (2)

where , the integral is over the Brillouin zone, is the equilibrium occupation factor, is the band velocity, is the Berry curvature, and is the electron charge. Equation (2) was derived in Ref. Zhou et al. (2013) using the semiclassical formalism Xiao et al. (2010), and we obtain the same result from linear response 222See Supplemental Material at http://cmt.berkeley.edu/suppl/zhong-arxiv15-suppl.pdf, which includes Refs. [29-43], for (i) the derivation of Eqs. (2) and (8) from linear response, (ii) the derivation from Eq. (9) of the non-FS formula for given in Refs. Goswami and Tewari (); Chang and M.-F.Yang (2015), (iii) the derivation of Eq. (18) from Eq. (17) combined with phenomenological relations, (iv) an analysis of the gyrotropic response in polar metals, and of the gyrotropic response induced in Weyl semimetals by the chiral anomaly, (v) the derivation of a reciprocity relation for the natural gyrotropy of a metal with a smooth interface, and (vi) the identification of a traceless Berry-curvature piece in the full GME response tensor of Eq. (9).. The fact that vanishes (see below) is in accord with Bloch’s theorem Yamamoto (2015).

To turn the above “quasiresponse” into , let us recast Eq. (2) as a FS integral. Integrating by parts produces two terms. The one containing picks up monopole contributions from the occupied WPs, and vanishes because each WP appears twice with opposite signs Gosálbez-Martínez et al. (2015). In the remaining term we write , with the FS normal at , and introduce the Chern number of the th Fermi sheet in band  Haldane (2004); Gosálbez-Martínez et al. (2015). After assigning different chemical potentials to different sheets to account for the effect of the -field pulse, Eq. (2) becomes , leading to the current density  Son and Yamamoto (2012); Yamamoto (2015). In equilibrium , and using we find , as per Eq. (2).

For a Weyl semimetal with two Fermi pockets with and placed at slightly different chemical potentials and  333With our sign convention for the Berry curvature, a right-handed WP acts as a source in the lower band and as a sink in the upper band Xiao et al. (2010). An enclosing pocket, either electronlike or holelike, has Chern number . [Fig. 1(a)], a current develops:

 j=(e2/h2)B(μR−μL). (3)

Gyrotropic magnetic effect.— Symmetry considerations already suggest a link between the GME and natural gyrotropy. Both and are odd under time reversal , and is odd under spatial inversion , while is  even, and so according to Eq. (1) the GME is  even and  odd, the same as natural gyrotropy Note1 ().

To make the connection precise, consider the current density induced by a monochromatic electromagnetic field at first order in :

 ji(ω,q)=Πijl(ω)Aj(ω,q)ql. (4)

The -even part of the response tensor is antisymmetric () under . It has nine independent components, and can be repackaged as a rank-2 tensor using Hornreich and Shtrikman (1968); Malashevich and Souza (2010)

 ΠAijl = iεilpαGMEjp−iεjlpαGMEip (5a) αGMEij = 14iεjlp(ΠAlpi−2ΠAilp). (5b)

At nonabsorbing frequencies is real and is purely imaginary, but otherwise both are complex.

From now on we assume , so that only intraband absorption can occur. In this regime satisfies

 jBi = −iωPBi=αGMEijBj (6a) MEi = −(i/ω)αGMEjiEj, (6b)

where and , and and are oscillating moments induced by and respectively. The natural gyrotropy current is . In the long-wavelength limit Eq. (6a) describes a transport current induced by a time-varying in an optically active metal (the direct GME), and Eq. (6b) describes a macroscopic magnetization induced by ; this inverse GME has been previously discussed for polar Edelstein (2011) and chiral Yoda et al. (2015) metals.

To derive Eq. (6), consider a finite sample of size . Using Eq. (20) of Ref. Malashevich and Souza (2010) for we find 444To recover the bulk result from Eq. (7), the limit should be taken faster than the limit, consistent with the order of limits discussed earlier for transport.

 αGMEij=(ω/2i)(χemij−χme% ji)+(E.Q. terms)\,. (7)

“E.Q.” denotes electric quadrupole terms that keep origin independent at higher frequencies Buckingham and Dunn (1971); Malashevich and Souza (2010), but do not contribute to or when , as they are higher order in than the first term. The low-frequency gyrotropic response is controlled by the magnetoelectric susceptibilities and . The dynamic polarization can be decomposed into -even and -odd parts and  555This decomposition is obtained by invoking the Onsager relation  Melrose and McPhedran (1991)., and Eq. (6a) corresponds to the former. Similarly, Eq. (6b) gives the -even part of the magnetization induced by . (The -odd part of the magnetoelectric susceptibilities describes the linear magnetoelectric effect in insulators such as CrO.)

In brief, the GME is the low-frequency limit of natural gyrotropy in -broken metals, in much the same way that the AHE is the transport limit of Faraday rotation in -broken metals. While the intrinsic AHE is governed by the geometric Berry curvature Xiao et al. (2010); Haldane (2004) and becomes quantized by topology in Chern insulators, the GME is controlled by a nongeometric quantity, the intrinsic magnetic moment of the Bloch states on the FS 666Here, the term geometric refers to the intrinsic geometry of the Bloch-state fiber bundle. The orbital moment of Bloch electrons can be considered geometric in a different sense: it is the imaginary part of a complex tensor whose real part gives the inverse effective mass tensor, i.e., the curvature of band dispersions Gao et al. (2015)..

To establish this result let us return to periodic crystals and derive a bulk formula for at . From the Kubo linear response in the uniform limit, we obtain Note2 ()

 ΠAijl = e2ωτ1−iωτ∑n∫[dk]∂f∂ϵkn[−gs2meεiplvkn,jSkn,p (8) + vkn,iℏIm⟨∂jukn|Hk−ϵkn|∂lukn⟩−(i↔j)].

[The calculation was carried out for a clean metal where formally and  Allen (2006). Alternately one could retain a finite to give a phenomenological relaxation time in dirty metals, and indeed the semiclassical relaxation-time calculation to be presented shortly gives the same Drude-like dependence on as Eq. (8).] is the expectation value of the spin of a Bloch state, is the spin  factor of the electron, and is the electron mass. Inserting Eq. (8) into Eq. (5b) gives

 αGMEij=iωτe1−iωτ∑n∫[dk](∂f/∂ϵkn)vkn,imkn,j, (9)

where is the magnetic moment of a Bloch electron, whose orbital part is Xiao et al. (2010)

 morbkn=e2ℏIm⟨∂kukn|×(Hk−ϵkn)|∂kukn⟩. (10)

At zero temperature, we can replace in Eq. (9) with to obtain the FS formula

A nonzero requires broken symmetry, but the GME can only occur if is broken: with symmetry present and , leading to . Without spin-orbit coupling, only the orbital moment contributes.

Equations (6) and (11) are our main results. The GME is fully controlled by the bulk FS and vanishes trivially for insulators, contrary to the AHE where the FS formulation misses possible quantized contributions Haldane (2004).

According to Eq. (11), the reactive response is suppressed by scattering when . It increases with , and levels off for (satisfying this condition without violating requires sufficiently clean samples). The opposite is true for the dissipative response , which drops to zero at and becomes strongest at . In this lowest-frequency limit , and Eqs. (6b) and (9) for the induced magnetization reduce to the expression in Ref. Yoda et al. (2015). Thus, in the dc limit only a dissipative inverse GME occurs in dirty metals.

Semiclassical picture.— Our discussion of the GME assumed from the outset . Since this is the regime where the semiclassical description of transport in metals holds Ashcroft and Mermin (1976), it is instructive to rederive Eqs. (6) and (11) by solving the Boltzmann equation. This provides an intuitive picture of the GME and its modification by scattering processes. The key ingredient beyond previous semiclassical approaches Orenstein and Moore (2013); Hosur and Qi (2015); Zhong et al. (2015) is the correction to the band energy and the band velocity (as opposed to the Berry-curvature anomalous velocity) in the presence of a magnetic field Chang and M.-F.Yang (2015); Xiao et al. (2010): , where .

In a static  field, the conduction electrons reach a new equilibrium state with as the distribution function Chang and M.-F.Yang (2015), and the current vanishes according to Eq. (2). Under oscillating fields the electrons are in an excited state with a distribution function which we find by solving the Boltzmann equation in the relaxation-time approximation,

 ∂tgkn+˙r∂gkn∂r+˙k∂gkn∂k=−[gkn−f0kn(B)]/τ, (12)

where is the relaxation time to return to the instantaneous equilibrium state described by (for a slow spatial variation of ). Using the semiclassical equations Xiao et al. (2010), the distribution function to linear order in and is , with

 f1kn=∂f/∂ϵkn1−qω⋅vkn+iωτ[mkn⋅B+(ie/ω)E⋅vkn], (13)

which at reduces to the result in Ref. Chang and M.-F.Yang, 2015.

As the current associated with vanishes, the current induced by an oscillating  field is obtained by multiplying the first term in Eq. (13) with the unperturbed band velocity. The result in the long-wavelength limit is

 jB=iωτe1−iωτ∑n∫[dk](∂f/∂ϵkn)vkn(mkn⋅B), (14)

in agreement with Eqs. (6a) and (9). Conversely, inserting the second term of Eq. (13) in the bulk expression for  Xiao et al. (2010) leads to Eqs. (6b) and (9) for the magnetization induced by an oscillating  field.

GME in two-band models.— Consider a situation where only two bands are close to , and couplings to more distant bands can be neglected when evaluating the orbital moment on the FS (for simplicity, we focus here on the orbital contribution). The Hamiltonian written in the basis of the identity matrix and the three Pauli matrices is , with eigenvalues , where and . Equation (10) becomes

 morbkt,i=−eℏεijl12d2kdk⋅(∂jdk×∂ldk). (15)

For orientation we study a minimal model for a Weyl semimetal where the FS consists of two pockets surrounding isotropic WPs of opposite chirality. We allow the WPs to be at different energies (this requires breaking both and ), but is assumed close to both [Fig. 1(b)]. Near each WP the Hamiltonian is , where labels the WP, and are its energy and chirality (positive means right-handed), is measured from the WP, and is the Fermi velocity. From Eq. (15), for , and only the trace piece survives is Eq. (11); in the clean limit each pocket contributes

 ¯¯¯¯αGMEν=∓13e2h2χνℏvFkF=13e2h2χν(ϵν−ϵF), (16)

where the minus (plus) sign in the middle expression corresponds to (). Summing over and using  Nielsen and Ninomiya (1981) gives . For a minimal model , and the GME current is

 jB=(e2/3h2)(ϵR−ϵL)B. (17)

Equation (17) looks deceptively similar to Eq. (3) for the CME current. The prefactor is different, but the key difference is in the meaning of the various quantities, and in their respective roles. To stress this point, in both equations we have placed the “force” that drives the current at the end, after the equilibrium parameter that enables the effect. The GME current is driven by the oscillating  field, while and are band structure parameters, with reflecting the degree of structural symmetry breaking that allows the effect to occur. Equation (3) is “universal” because of the topological nature of the FS integral involved, while Eq. (17) is for spherical pockets surrounding isotropic Weyl nodes. For generic two-band models the traceless part of is generally nonzero 777For any number of bands, the traceless part of includes Note2 () the Berry-curvature piece found previously Zhong et al. (2015), and the full tensor satisfies Note2 () the microscopic constraint from time-reversal symmetry previously shown for that traceless piece Zhong et al. (2015)., and the non-FS expression of Refs. Goswami and Tewari (); Chang and M.-F.Yang (2015) for the orbital contribution to the trace can be recovered from Eq. (9Note2 ().

We emphasize that breaking is not required for the GME. If  is present (and broken), the minimum number of WPs is four, not two Young et al. (2012). In the class of -symmetric Weyl materials so far discovered, relates WPs of the same chirality and energy. Mirror symmetries connect WPs of opposite chirality so that , as expected since these symmetries tend to exclude optical rotation Flack (2003); Newnham (2005). Fortuitously, the predicted Weyl material SrSi has misaligned WPs of opposite chirality due to broken mirror symmetry Huang et al. (). Its rotatory power  can be estimated from the energy splitting between WPs. Neglecting anisotropy effects and spin contributions that were not included in Eq. (17), each WP pair contributes Note2 ()

 ρ=(2α/3hc)(ϵL−ϵR), (18)

with the fine-structure constant and the speed of light. The calculated splitting  eV Huang et al. () gives  rad/mm per node pair, about the same as  rad/mm for quartz at  Newnham (2005). This should be measurable in a frequency range from the infrared (above which the semiclassical assumptions break down) down to , which depends on crystal quality. When the rotatory power vanishes in equilibrium, but a nonequilibrium gyrotropic effect can still occur due to the chiral anomaly Hosur and Qi (2015); Note2 (). In polar metals, the tensor acquires am antisymmetric part (equivalent to a polar vector ) that does not contribute to optical rotation, but which leads to a transverse GME of the form  Note2 ().

In summary, we have elucidated the physical origin of currents induced by low-frequency magnetic fields in metals in terms of the magnetic moment on the FS, and discussed the experimental implications. Unlike the CME Parameswaran et al. (2014) or the photoinduced AHE Mak et al. (2014), no detailed model of nonequilibrium is required to quantify the GME, and efficient ab initio methods already exist to compute the needed orbital moments Lopez et al. (2012).

We thank Q. Niu, J. Orenstein, D. Pesin, and D. Vanderbilt for useful comments, and also thank D. Vanderbilt for calling our attention to Ref. Yoda et al. (2015) and suggesting a possible connection with the present work. We acknowledge support from Grant No. NSF DMR-1507141 (S. Z.), from the DOE LBL Quantum Materials Program and Simons Foundation (J. E. M.), and from Grants No. MAT2012-33720 from the Spanish Ministerio de Economía y Competitividad and No. CIG-303602 from the European Commission (I. S.).

Note added.— Along with the present paper, the role of orbital moments in the natural gyrotropy of metals was also recognized in Ref. Ma and Pesin (2015).

## References

• Thouless et al. (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale,  and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
• Nielsen and Ninomiya (1983) H. B. Nielsen and M. Ninomiya, Phys. Lett. 130B, 389 (1983).
• Son and Spivak (2013) D. T. Son and B. Z. Spivak, Phys. Rev. B 88, 104412 (2013).
• Son and Yamamoto (2012) D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109, 181602 (2012).
• Adler (1969) S. Adler, Phys. Rev. 177, 2426 (1969).
• Bell and Jackiw (1969) J. S. Bell and R. Jackiw, Nuovo Cim. A 60, 47 (1969).
• Zyuzin et al. (2012) A. A. Zyuzin, S. Wu,  and A. A. Burkov, Phys. Rev. B 85, 165110 (2012).
• Vazifeh and Franz (2013) M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 027201 (2013).
• Yamamoto (2015) N. Yamamoto, Phys. Rev. D 92, 085011 (2015).
• Chen et al. (2013) Y. Chen, S. Wu,  and A. A. Burkov, Phys. Rev. B 88, 125105 (2013).
• (11) P. Goswami and S. Tewari, arXiv:1311.1506 .
• Chang and M.-F.Yang (2015) M.-C. Chang and M.-F.Yang, Phys. Rev. B 91, 115203 (2015).
• Chang and Yang (2015) M.-C. Chang and M.-F. Yang, Phys. Rev. B 92, 205201 (2015).
• Goswami et al. (2015) P. Goswami, G. Sharma,  and S. Tewari, Phys. Rev. B 92, 161110(R) (2015).
• Kharzeev (2014) D. E. Kharzeev, Prog. Part. Nucl. Phys. 75, 133 (2014).
• (16) The term natural gyrotropy refers to the time-reversal-even part of the optical response of a medium at linear order in the wave vector of light Landau and Lifshitz (1984); Agranovich and Ginzburg (1984). The reactive part gives rise to natural optical rotation, and the dissipative part to natural circular dichroism. Furthermore, polar crystals display natural gyrotropy effects unrelated to optical rotation Agranovich and Ginzburg (1984). Gyrotropic effects that are time-reversal-odd and zeroth order in the wave vector of light (e.g., Faraday rotation and magnetic circular dichroism Landau and Lifshitz (1984)) are not considered in this work.
• Landau and Lifshitz (1984) L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, Oxford, 1984).
• Agranovich and Ginzburg (1984) V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, 2nd ed. (Springer, Berlin, 1984).
• Flack (2003) H. D. Flack, Helv. Chim. Acta 86, 905 (2003).
• Newnham (2005) R. E. Newnham, Properties of Materials (Oxford University Press, Oxford, 2005).
• Orenstein and Moore (2013) J. Orenstein and J. E. Moore, Phys. Rev. B 87, 165110 (2013).
• Hosur and Qi (2015) P. Hosur and X.-L. Qi, Phys. Rev. B 91, 081106 (2015).
• Zhong et al. (2015) S. Zhong, J. Orenstein,  and J. E. Moore, Phys. Rev. Lett. 115, 117403 (2015).
• Haldane (2004) F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004).
• (25) S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, M. Neupane, H. Zheng, D. Sanchez, A. Bansil, G. Bian, H. Lin,  and M. Z. Hasan, arXiv:1503.05868 .
• Zhou et al. (2013) J.-H. Zhou, J. Hua, Q. Niu,  and J.-R. Shi, Chin. Phys. Lett. 30, 027101 (2013).
• Xiao et al. (2010) D. Xiao, M.-C. Chang,  and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
• (28) See Supplemental Material at http://cmt.berkeley.edu/suppl/zhong-arxiv15-suppl.pdf, which includes Refs. [29-43], for (i) the derivation of Eqs. (2) and (8) from linear response, (ii) the derivation from Eq. (9) of the non-FS formula for given in Refs. Goswami and Tewari (); Chang and M.-F.Yang (2015), (iii) the derivation of Eq. (18) from Eq. (17) combined with phenomenological relations, (iv) an analysis of the gyrotropic response in polar metals, and of the gyrotropic response induced in Weyl semimetals by the chiral anomaly, (v) the derivation of a reciprocity relation for the natural gyrotropy of a metal with a smooth interface, and (vi) the identification of a traceless Berry-curvature piece in the full GME response tensor of Eq. (9).
• Blount (1962) E. I. Blount, Solid State Phys. 13, 305 (1962).
• Yafet (1963) Y. Yafet, Solid State Phys. 14, 1 (1963).
• Wooten (1972) F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).
• Dressel and Grüner (2002) M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press, Cambridge, UK, 2002).
• Harrison (1980) W. A. Harrison, Solid State Theory (Dover, New York, 1980).
• Yao et al. (2008) W. Yao, D. Xiao,  and Q. Niu, Phys. Rev. B 77, 235406 (2008).
• Natori (1975) K. Natori, J. Phys. Soc. Jpn. 39, 1013 (1975).
• Hosur et al. (2013) P. Hosur, A. Kapitulnik, S. A. Kivelson, J. Orenstein,  and S. Raghu, Phys. Rev. B 87, 115116 (2013).
• Barron (2004) L. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, Cambridge, UK, 2004).
• Halperin (1992) B. Halperin, in The Physics and Chemistry of Oxide Superconductors, Springer-Verlag Proceedings of Physics Vol 60, edited by Y. Iye and H. Yasuoka (Springer-Verlag, Berlin, 1992) p. 439.
• Fried (2014) A. D. Fried, Phys. Rev. B 90, 121112 (2014).
• Agranovich and Yudson (1973) V. M. Agranovich and V. I. Yudson, Opt. Commun. 9, 58 (1973).
• Bokut and Serkyukov (1974) V. V. Bokut and A. N. Serkyukov, Zh. Prikl. Spectrosk. 20, 677 (1974) [J. Appl. Spectrosc. (USSR) 20, 513 (1974)].
• Vinogradov (2002) A. P. Vinogradov, Phys. Usp. 45, 331 (2002).
• Armitage (2014) N. P. Armitage, Phys. Rev. B 90, 035135 (2014).
• Gosálbez-Martínez et al. (2015) D. Gosálbez-Martínez, I. Souza,  and D. Vanderbilt, Phys. Rev. B 92, 085138 (2015).
• (45) With our sign convention for the Berry curvature, a right-handed WP acts as a source in the lower band and as a sink in the upper band Xiao et al. (2010). An enclosing pocket, either electronlike or holelike, has Chern number .
• Hornreich and Shtrikman (1968) R. M. Hornreich and S. Shtrikman, Phys. Rev. 171, 1065 (1968).
• Malashevich and Souza (2010) A. Malashevich and I. Souza, Phys. Rev. B 82, 245118 (2010).
• Edelstein (2011) V. M. Edelstein, Phys. Rev. B 83, 113109 (2011).
• Yoda et al. (2015) T. Yoda, T. Yokoyama,  and S. Murakami, Sci. Rep. 5, 12024 (2015).
• (50) To recover the bulk result from Eq. (7), the limit should be taken faster than the limit, consistent with the order of limits discussed earlier for transport.
• Buckingham and Dunn (1971) A. D. Buckingham and M. D. Dunn, J. Chem. Soc. A , 1988 (1971).
• (52) This decomposition is obtained by invoking the Onsager relation  Melrose and McPhedran (1991).
• Melrose and McPhedran (1991) D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge University Press, Cambridge, UK, 1991).
• (54) Here, the term geometric refers to the intrinsic geometry of the Bloch-state fiber bundle. The orbital moment of Bloch electrons can be considered geometric in a different sense: it is the imaginary part of a complex tensor whose real part gives the inverse effective mass tensor, i.e., the curvature of band dispersions Gao et al. (2015).
• Gao et al. (2015) Y. Gao, S. A. Yang,  and Q. Niu, Phys. Rev. B 91, 214405 (2015).
• Allen (2006) P. B. Allen, in Conceptual Foundations of Materials Properties: A Standard Model for Calculation of Ground- and Excited-State Properties, Contemporary Concepts of Condensed Matter Science, Vol. 1, edited by S. G. Louie and M. L. Cohen (Elsevier, New York, 2006) p. 165.
• Ashcroft and Mermin (1976) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks-Cole, Belmont, MA, 1976).
• Nielsen and Ninomiya (1981) H. Nielsen and M. Ninomiya, Nucl. Phys. B185, 20 (1981).
• (59) For any number of bands, the traceless part of includes Note2 () the Berry-curvature piece found previously Zhong et al. (2015), and the full tensor satisfies Note2 () the microscopic constraint from time-reversal symmetry previously shown for that traceless piece Zhong et al. (2015).
• Young et al. (2012) S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele,  and A. M. Rappe, Phys. Rev. Lett. 108, 140405 (2012).
• Parameswaran et al. (2014) S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin,  and A. Vishwanath, Phys. Rev. X 4, 031035 (2014).
• Mak et al. (2014) K. F. Mak, K. L. McGill, J. Park,  and P. L. McEuen, Science 344, 1489 (2014).
• Lopez et al. (2012) M. G. Lopez, D. Vanderbilt, T. Thonhauser,  and I. Souza, Phys. Rev. B 85, 014435 (2012).
• Ma and Pesin (2015) J. Ma and D. A. Pesin, Phys. Rev. B 92, 235205 (2015).
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