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

###### 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 ^{1}^{1}1The
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 :

(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

(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 ^{2}^{2}2See 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
^{3}^{3}3With 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:

(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 :

(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)

(5a) | |||||

(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

(6a) | |||||

(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 ^{4}^{4}4To 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.

(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
^{5}^{5}5This 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 ^{6}^{6}6Here, 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 ()

(8) | |||||

[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

(9) |

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

(10) |

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

(11) |

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,

(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

(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

(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

(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

(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

(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 ^{7}^{7}7For 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. (9) Note2 ().

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 ()

(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 m 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).

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