The Berry Phase and Luttinger’s Anomalous Velocity in Metals

New Topological Ingredients in the Fermi-Liquid Theory of Metals

F. D. M. Haldane, Princeton University.

See: F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004) (cond-mat/0408417)

Talk presented at the IRG1 workshop “Strongly Correlated Electronic Materials”, Princeton Center for Complex Materials, January 28-29, 2005

work supported by NSF MRSEC DMR02-13706

© F. D. M. Haldane 2005haldane@princeton.edu v 0.01

Hall effect in isotropic (cubic) metals:

ρxy = R0Bz

Hall effect in ferromagnetic metals with B parallel to a magnetization in the z-direction, and isotropyin the x-y plane:

ρxy = ρxys + R0Bz

The anomalous extra term is constant when Hz is large enough to eliminate domain structures.

What non-Lorentz force is providing the sideways deflection of the current? Is it intrinsic, or due to scattering of electrons by impurities or local non-uniformities in the magnetization?

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Dissipationless Anomalous Hall Current in the Ferromagnetic Spinel CuCr2Se4−xBrx.∗

Wei-Li Lee1, Satoshi Watauchi2†, V. L. Miller2, R. J. Cava2,3, and N. P. Ong1,3‡1Department of Physics, 2Department of Chemistry,

3Princeton Materials Institute, Princeton University, New Jersey 08544, U.S.A.(Dated: May 26, 2004)

In a ferromagnet, an applied electric field E invariably produces an anomalous Hall current JHthat flows perpendicular to the plane defined by E and M (the magnetization). For decades, thequestion whether JH is dissipationless (independent of the scattering rate), has been keenly debatedwithout experimental resolution. In the ferromagnetic spinel CuCr2Se4−xBrx, the resistivity ρ (atlow temperature) may be increased 1000 fold by varying x(Br), without degrading the M. Weshow that JH/E (normalized per carrier, at 5 K) remains unchanged throughout. In addition toresolving the controversy experimentally, our finding has strong bearing on the generation and studyof spin-Hall currents in bulk samples.

A major unsettled question in the study of electrontransport in a ferromagnet is whether the anomalousHall current is dissipationless. In non-magnetic metals,the familiar Hall current arises when electrons moving incrossed electric (E) and magnetic (H) fields are deflectedby the Lorentz force. However, in a ferromagnet subjectto E alone, a large, spontaneous (anomalous) Hall cur-rent JH appears transverse to E (in practice, a weak Hserves to align the magnetic domains) (1,2). Questionsregarding the origin of JH , and whether it is dissipation-less, have been keenly debated for decades. They haveemerged anew because of fresh theoretical insights andstrong interest in spin currents for spin-based electron-ics. Here we report measurements in the ferromagnetCuCr2Se4−xBrx which establish that, despite a 100-foldincrease in the scattering rate from impurities, JH (percarrier) remains constant, implying that it is indeed dis-sipationless.

In 1954, Karplus and Luttinger (KL)(3,4) proposed apurely quantum-mechanical origin for JH . An electronin the conduction band of a crystal lattice spends partof its time in nearby bands because of admixing causedby the (intracell) position operator X. In the process, itacquires a spin-dependent ‘anomalous velocity’ (5). KLpredicted that the Hall current is dissipationless: JH re-mains constant even as the longitudinal current (J||E)is degraded by scattering from added impurities. A con-ventional mechanism was later proposed (6) whereby theanomalous Hall effect (AHE) is caused instead by asym-metric scattering of electrons by impurities (skew scat-tering). Several authors (7,8,9) investigated the theoret-ical ramifications of these competing models. The roleof impurities in the anomalous-velocity theory was clari-fied by Berger’s side-jump model (7). A careful account-ing of various contributions (including side-jump) to the

∗Science 303, 1647 (2004).†Permanent address of S. W. : Center for Crystal Science andTechnology, University of Yamanashi, 7 Miyamae, Kofu, Ya-manashi 400-8511, Japan‡To whom correspondence should be addressed E-mail:npo@princeton.edu

AHE in a semiconductor has been given by Nozières andLewiner (NL) who derive X = λk × S, with λ the en-hanced spin-orbit parameter, k the carrier wavevectorand S its spin (9). In the dc limit, NL obtain the AHEcurrent

JH = 2ne2λE× S, (1)

where n is the carrier density and e the charge. As noted,JH is linear in S but independent of the electron lifetimeτ .

In modern terms, the anomalous velocity term of KLis related to the Berry phase (10), and has been applied(11) to explain the AHE in Mn-doped GaAs (12). Theclose connection of the AHE to the Berry phase has alsobeen explored in novel ferromagnets in which frustrationleads to spin chirality (13,14,15). In the field of spin-tronics, several schemes have been proposed to producea fully polarized spin current in thin-film structures (16),and in bulk p-doped GaAs (17). The AHE is intimatelyrelated to these schemes, and our experimental resultshave bearing on the spin-current problem.

In an AHE experiment (1), the observed Hall resistiv-ity is comprised of two terms,

ρxy = R0B + ρ′xy, (2)

with B the induction field, R0 the ordinary Hall coef-ficient, and ρ′xy the anomalous Hall resistivity. A di-rect test of the dissipationless nature of JH is to checkwhether the anomalous Hall conductivity σ′H (defined asρ′xy/ρ

2) changes as impurities are added to increase 1/τ(and ρ) (3,7). A dissipationless AHE current implies thatρ′xy ∼ ρ

α, with α = 2. By contrast, in the skew scatteringmodel, α = 1.

Tests based on measurements at high temperatures(77-300 K) yield exponents in the range αexp = 1.4-2.0(18,19). However, it has been argued (20) that, at highT , both models in fact predict α = 2, a view supportedby detailed calculations (21). To be meaningful, the testmust be performed in the impurity-scattering regime overa wide range of ρ. Unfortunately, in most ferromagnets,ρ′xy becomes too small to measure accurately at low T .Results on α in the impurity-scattering regime are verylimited.

2

The copper-chromium selenide spinel CuCr2Se4, ametallic ferromagnet with a Curie temperature TC ∼430 K, is particularly well-suited for testing the AHE.Substituting Se with Br in CuCr2Se4−xBrx decreases thehole density nh (22). However, because the coupling be-tween local moments on Cr is primarily from 90o su-perexchange along the Cr-Se-Cr bonds (23), this doesnot destroy the magnetization. We have grown crystalsof CuCr2Se4−xBrx by chemical vapor transport [detailsgiven in Supporting Online Materials (SOM) (24)]. In-creasing x from 0 to 1 in our crystals decreases nh by afactor of ∼30 (Fig. 1A), while TC decreases from 430 Kto 230 K. The saturated magnetization Ms at 5 K corre-sponds to a Cr moment that actually increases from ∼2.6to 3 µB (Bohr magneton) (Fig. 1B).

FIG. 1: (A) The hole density nh (solid circles) inCuCr2Se4−xBrx vs. x determined from R0 at 400 K (one holeper formula unit corresponds to nh = 7.2 × 10

21cm−3). TheCurie temperature TC is shown as open circles. (B) Curves ofthe magnetization M vs. H at 5 K in 3 samples (x values indi-cated). The saturation value Ms = 3.52, 3.72, 3.95 (10

5A/m)for x = 0, 0.5, 1.0, respectively. (C) The resistivity ρ vs. T in10 samples with Br content x indicated (a, b indicate differentsamples with the same x). Values of nh in all samples fall inthe metallic regime (for x = 1, nh = 1.9 × 10

20 cm−3).

As shown in Fig. 1C, all samples except the ones withx = 1.0 lie outside the localization regime. In the ‘metal-lic’ regime, the low-T resistivity increases by a factor of

∼270, as x increases from 0 to 0.85, and is predominantlydue to a 70-fold decrease in τ . The hole density nh de-creases by only a factor of 4. In the localization regime(x = 1.0), strong disorder causes ρ to rise gradually withdecreasing T . We emphasize, however, that these sam-ples are not semiconductors (ρ is not thermally activated,and nh = 1.9 × 1020 cm−3 is degenerate).

The field dependence of the total Hall resistivity (Eq.2) is shown for x = 0.25 (Fig. 2A) and 1.0 (B). See SOM(24) for measurement details. The steep increase in |ρxy|in weak H reflects the rotation of domains into alignmentwith H. Above the saturation field Hs, when ρ′xy is con-stant, the small ordinary Hall term R0B is visible as alinear background (24). As in standard practice, we usedR0 measured above TC to find the nh plotted in Fig. 1A.

FIG. 2: Curves of the observed Hall resistivity ρxy = R0B +Rsµ0M vs. H (at temperatures indicated) in CuCr2Se4−xBrxwith x = 0.25 (Panel A) and x = 1.0 (B). In (A), the anoma-lous Hall coefficient Rs changes sign below 250 K, becomesnegative, and saturates to a constant value below 50 K. How-ever, in (B), Rs is always positive and rises to large values atlow T (note difference in scale).

By convention, the T dependence of the AHE signal isrepresented by the anomalous Hall coefficient Rs(T ) de-fined by ρ′xy = Rsµ0M (µ0 is the vacuum permeability).

T=5K

2

The copper-chromium selenide spinel CuCr2Se4, ametallic ferromagnet with a Curie temperature TC ∼430 K, is particularly well-suited for testing the AHE.Substituting Se with Br in CuCr2Se4−xBrx decreases thehole density nh (22). However, because the coupling be-tween local moments on Cr is primarily from 90o su-perexchange along the Cr-Se-Cr bonds (23), this doesnot destroy the magnetization. We have grown crystalsof CuCr2Se4−xBrx by chemical vapor transport [detailsgiven in Supporting Online Materials (SOM) (24)]. In-creasing x from 0 to 1 in our crystals decreases nh by afactor of ∼30 (Fig. 1A), while TC decreases from 430 Kto 230 K. The saturated magnetization Ms at 5 K corre-sponds to a Cr moment that actually increases from ∼2.6to 3 µB (Bohr magneton) (Fig. 1B).

FIG. 1: (A) The hole density nh (solid circles) inCuCr2Se4−xBrx vs. x determined from R0 at 400 K (one holeper formula unit corresponds to nh = 7.2 × 10

21cm−3). TheCurie temperature TC is shown as open circles. (B) Curves ofthe magnetization M vs. H at 5 K in 3 samples (x values indi-cated). The saturation value Ms = 3.52, 3.72, 3.95 (10

5A/m)for x = 0, 0.5, 1.0, respectively. (C) The resistivity ρ vs. T in10 samples with Br content x indicated (a, b indicate differentsamples with the same x). Values of nh in all samples fall inthe metallic regime (for x = 1, nh = 1.9 × 10

20 cm−3).

As shown in Fig. 1C, all samples except the ones withx = 1.0 lie outside the localization regime. In the ‘metal-lic’ regime, the low-T resistivity increases by a factor of

∼270, as x increases from 0 to 0.85, and is predominantlydue to a 70-fold decrease in τ . The hole density nh de-creases by only a factor of 4. In the localization regime(x = 1.0), strong disorder causes ρ to rise gradually withdecreasing T . We emphasize, however, that these sam-ples are not semiconductors (ρ is not thermally activated,and nh = 1.9 × 1020 cm−3 is degenerate).

The field dependence of the total Hall resistivity (Eq.2) is shown for x = 0.25 (Fig. 2A) and 1.0 (B). See SOM(24) for measurement details. The steep increase in |ρxy|in weak H reflects the rotation of domains into alignmentwith H. Above the saturation field Hs, when ρ′xy is con-stant, the small ordinary Hall term R0B is visible as alinear background (24). As in standard practice, we usedR0 measured above TC to find the nh plotted in Fig. 1A.

FIG. 2: Curves of the observed Hall resistivity ρxy = R0B +Rsµ0M vs. H (at temperatures indicated) in CuCr2Se4−xBrxwith x = 0.25 (Panel A) and x = 1.0 (B). In (A), the anoma-lous Hall coefficient Rs changes sign below 250 K, becomesnegative, and saturates to a constant value below 50 K. How-ever, in (B), Rs is always positive and rises to large values atlow T (note difference in scale).

By convention, the T dependence of the AHE signal isrepresented by the anomalous Hall coefficient Rs(T ) de-fined by ρ′xy = Rsµ0M (µ0 is the vacuum permeability).

example of a very large AHE

• Karplus and Luttinger (1954): proposed an intrinsic bandstructure explanation, involving Bloch states, spin-orbit coupling and the imbalance between majority and minority spin carriers.

• A key ingredient of KL is an extra “anomalous velocity” of the electrons in addition to the usual group velocity.

• More recently, the KL “anomalous velocity” was reinterpreted in modern language as a “Berry phase” effect.

• In fact, while the KL formula looks like a band-structure effect, I have now found it is a new fundamental Fermi liquid theory feature (possibly combined with a quantum Hall effect.)

This has revealed that the Fermi liquid theory of metals has some additional fundamental ingredients which Landau and coworkers were too early to spot!

The DC conductivity tensor can be divided into a symmetric Ohmic (dissipative) part and an antisymmetric non-dissipative Hall part:

σab = σabOhm + σabHall

In the limit T →0, there are a number of exact statements that can be made about the DC Hall conductivity of a translationally-invariant system.

For non-interacting Bloch electrons, the Kubo formula gives an intrinsic Hall conductivity (in both 2D and 3D)

σabHall =e2

!1

VD

∑nk

Fabn (k)Θ(εF − εn(k))

This is given in terms of the total Berry curvature of occupied states with band index n and Bloch vector k.

If the Fermi energy is in a gap, so every band is either empty or full, this is a topological invariant:(integer quantized Hall effect)

σxy =e2

!12π

ν ν = an integer(2D)

σab =e2

!1

(2π)2#abcKc K = a reciprocal vector G (3D)

Implication: If in 2D, ν is NOT an integer, the non-integer part MUST BE A FERMI SURFACE PROPERTY!

In 3D, any part of K modulo a reciprocal vector also must be a Fermi surface property!

In 3D G = νG0, where G0 indexes a family of lattice planes with a 2D QHE.

TKNN formula

2D zero-field Quantized Hall Effect

• 2D quantized Hall effect: σxy = νe2/h. In the absence of interactions between the particles, ν must be an integer. There are no current-carrying states at the Fermi level in the interior of a QHE system (all such states are localized on its edge).

• The 2D integer QHE does NOT require Landau levels, and can occur if time-reversal symmetry is broken even if there is no net magnetic flux through the unit cell of a periodic system. (This was first demonstrated in an explicit “graphene” model shown at the right.).

• Electronic states are “simple” Bloch states! (real first-neighbor hopping t1, complex second-neighbor hopping t2e

iφ, alternating onsite potential M.)

FDMH, Phys. Rev. Lett. 61, 2015 (1988).

Fabn (k) = ∇akAbn(k) −∇bkAan(k)

Aan(k) = −i∫

UCdV u∗n(k,x)∇akun(k,x)

eiφ(Γ) = exp i∮

dAn

Berry curvature is analog of a magnetic field in Bloch space. “Vector potential” = Berry connection,curl of “Vector potential” = Berry curvature.

The Berry connection here derives from the k-dependence of the periodic part of the Bloch wavefunction

The Berry phase factor is the analog of a Bohm-Aharonov factor for a closed path in k-space

• The Berry curvature acts in k-space like a magnetic flux density acts in real space.

• Covariant notation ka, ra is used here to emphasize the duality between k-space and r-space, and expose metric dependence or independence (a ∈{x,y,z }).

Semiclassical dynamics of Bloch electrons

write magnetic flux density as an antisymmetric tensor

Motion of the center of a wavepacket of band-n electrons centered at k in reciprocal space and r in real space: (Sundaram and Niu 1999)

Fab(r) = !abcBc(r)

Note the “anomalous velocity” term! (in addition to the group velocity)

Karplus and Luttinger 1954

!dkadt

= eEa(r) + eFabdrb

dt

!dra

dt= ∇akεn(k) + !Fabn (k)

dkbdt

k, t

k+δk, t+δt

x

x

Current flow as a Bloch wavepacket is accelerated

• If the Bloch vector k (and thus the periodic factor in the Bloch state) is changing with time, the current is the sum of a group-velocity term (motion of the envelope of the wave packet of Bloch states) and an “anomalous” term (motion of the k-dependent charge distribution inside the unit cell)

• If both inversion and time-reversal symmetry are present, the charge distribution in the unit cell remains inversion symmetric as k changes, and the anomalous velocity term vanishes.

“anomalous” flow

regular flow

2D case: “Bohm-Aharonov in k-space”

• The Berry phase for moving a quasiparticle around the Fermi surface is only defined modulo 2π:

• Only the non-quantized part of the Hall conductivity is defined by the Fermi surface!

σxy =e2

h

(ΦBerryF

2π

)σxy =

e2

!1

(2π)2

∫d2k∇k ×A(k)n(k)

σxy =e2

!1

(2π)2

∮FS

dkF A(k)

• even the quantized part of Hall conductance is determined at the Fermi energy (in edge states necessarily present when there are fully-occupied bands with non-trivial topology)

• All transport occurs AT the Fermi level, not in “states deep below the Fermi energy”. (transport is NOT diamagnetism!)

Application of formula to composite fermion Fermi

surface at ν = 1/m

Flux enclosed by path of displaced electron

around vortex:

Berry phase for moving composite quasiparticlearound Fermi-surface

σxyAHE =e2

h

1m

independent of Fermi surface shape!

a composite fermion is modeledas an electron laterally displaced from the center of the m-vortex that is bound to it.

2π"2

× 1m

non-quantized part of 3D case can also be expressed as a Fermi surface integral

• There is a separate contribution to the Hall conductivity from each distinct Fermi surface manifold.

• Intersections with the Brillouin-zone boundary need to be taken into account.

“Anomalous Hall vector”:K =

∑α

Kα(modulo G)

integral of Fermi vectorweighted by Berry curvature on FS

Berry phase aroundFS intersection with

BZ boundary

This is ambiguous up to a reciprocal vector, which is a non-FLT quantized Hall edge-state contribution

SΓ2Γ1

Γ1

Γ2

Γ3

Kα =12π

(∫Sα

d2SFFkF +r∑

i=1

Gi

∮Γiα

dA

)

• The Fermi surface formulas for the non-quantized parts of the Hall conductivity are purely “geometrical” (referencing both k-space and Hilbert space geometry)

• Such expressions are so elegant that they “must” be more general than free-electron band theory results!

• This is true: they are like the Luttinger Fermi surface volume result, and can be derived in the interacting system using Ward identities.

Is there also a “Spin Hall effect?”• These formulas don’t seem to allow further

generalization to give a “spin Hall effect” as a separate fundamental phenomenon. But...

• Each distinct Fermi surface manifold has its own dissipationless Hall current, and these in general will flow in different directions.

• Different manifolds can support slightly different chemical potentials in the non-equilibrium steady state. This will be higher at the surface to which that manifold’s AHE is flowing towards. This applies even if the total AHE cancels because of unbroken time reversal symmetry.

• suggests “spin” in “spin-Hall” is really just Fermi-surface manifold label. This is adiabatically conserved, but relaxes via non-adiabatic processes. This generalizes the separate conservation laws of 1D “Fermi points”

The new FLT ingredient: quasiparticle wavefunctions

• Quasiparticle states on the Fermi surface (only) have Bloch wavefunctions which are naturally defined by the single-particle Green’s function of the interacting system, without reference to any fictional non-interacting precursor.

• Here s is a 2-component parameterization of the Fermi surface, and the 3-component unit vector Ω is a coherent-state parameterization of 2-fold “spin degeneracy” if time-reversal and inversion symmetry are unbroken. The periodic factor uσ(r;s,Ω) is the source of the new FLT ingredients.

Ψqpσ (r; s,Ω) = eikF (s)·ruσ(r; s,Ω)

Local Bloch-state basis:

• R are lattice translations and G are the reciprocal lattice vectors:

• • k is a periodic Bloch vector in the Brillouin

zone, x is a periodic position relative to the origin in the unit cell:

• Ψ†σ(k,x) creates an electron in this state.

U(G) = exp i(iG · r); T (R) = exp(ip · R/!)

T (R)|k,x,σ〉 = eik·R|k,x,σ〉U(G)|k,x,σ〉 = eiG·x|k,x,σ〉

Sz|k,x,σ〉 = σ|k,x,σ〉

|k + G,x + R,σ〉 = |k,x,σ〉

The single-particle Green’s function:

• Δσσ'(E,k,x,x') is positive-definite Hermitian for E ≠0, vanishes when E = 0;

• !-1σσ'(0,k,x,x') is Hermitian when T=0 (THE key FLT property! (Pauli principle)).

• The defining relation for both the Fermi surface and the quasiparticle wavefunction is the Hermitian zero-mode eigenproblem: (in the free-electron case, this is just Hu = EFu)

Gσσ′(ω,k,x,x′) =∫

dt

2πie−iωt〈Tt{Ψσ(k,x, 0),Ψ†σ(k,x, t)}〉T

Ψ†σ(k,x, t) = eiHtΨ†σ(k,x)e

−iHt, H = H − µN

limT→0

∑σ′

∫UC

d3x′ G−1σσ′(0,kF (s),x,x′)uσ′(x′, s,Ω) = 0

limT→0

Σσσ′(ω,k,x,x′) = Σ0σσ′(k,x,x′) +

∫ ∞−∞

dE∆σσ′(E,k,x,x′)!ω − E(1 + i"+)

*In this talk, the issue of the Kohn-Luttinger BCS instability will be “swept under the rug”.

G−1σσ′(ω,k,x,x′) = !ω11σσ′(x,x′) − Σσσ′(ω,k,x,x′)

All properties of the FLT fixed point except the Landau function derive from the one-electron Green’s function:• k-space geometry: Fermi vector kF(s), and outward normal nF(s).

• energetics: Fermi speed vF(s), and Landau function f(s,Ω,s',Ω') = f0(s,s') + f

ij(s,s')ΩiΩ'j, intrinsic magnetic moment µai(s)Ωi .

• NEW ... Hilbert space geometry: Berry gauge potential Aμ (s,Ω) and “quantum distance” dQ(s,s').

• Non fixed-point properties: quasiparticle residue Z(s) and asymptotic low-temperature inelastic scattering path length l(s,T).

Γ(E, s) = π−1〈u(s,Ω)|∆(E,kF (s))|u(s,Ω)〉

Z(s) = 〈u(s,Ω)|∂ωG−1(0,kF (s))|u(s,Ω)〉

vF (s)nF (s) = −〈u(s,Ω)|∇kG−1(0,kF (s))|u(s,Ω)〉

µia(s)Ωi = 〈u(s,Ω)|µea|u(s,Ω)〉

!(s, T ) = !vF (s)/Γ(πkBT, s)

The Landau ground state energy density functional

simplified notation for Fermi surface area sum (Sα means primitive region e.g., in BZ)

Change in electron density

Change in energy density

Landau function coupling different Fermi surface points

δk‖F (s) ≡ n̂F (s) · δkF (s)δkF component normal to original Fermi surface

• The Landau functional gives the change in energy density if the Fermi surface is changed from kF to kF + δkF .

δn[δkF ] =∑

s

δk‖F (s)

δh[δkF ] = 12∑

s

!vF (s)(δk‖F (s)

)2+ 12

∑s,s′

f(s, s′)δk‖F (s)δk‖F (s

′)

Fermi speed

∑s

≡∑α

∫Sα

d2SF(2π)3

d2SF = (n̂F (s) · ∂µkF (s) × ∂νkF (s)) dsµ ∧ dsν

Bures-Uhlmann “Quantum distance” betweentwo (pure) states in Hilbert space.

• symmetric, satisfies the triangle inequality.• d=0 for equivalent states (independent of

phase), d=1 for orthogonal states. generally, 0 ≤ d ≤ 1.

• Generates a Riemann (quadratic) metric.

dQ(ΨA,ΨB) = (1 − |〈ΨA|ΨB〉|)1/2

dQ(ΨA,ΨB)) =

(1 −

(2

∫d2Ω4π

|〈ΨA(Ω)|ΨB(Ω′)〉|2)1/2)1/2

non-degenerate (orthogonal or unitary) case:

doubly-degenerate (symplectic) case:

Berry connection (gauge fields on Fermi surface)

• without spin-splitting, there are three distinct Fermi surface gauge fields:

•

−i∑

σ

∫UC

d3xu∗σ(x, s,Ω)∂

∂sµuσ(x, s,Ω) = A0µ(s) + Aiµ(s)Ωi

−i∑

σ

∫UC

d3xu∗σ(x, s,Ω)∂

∂Ωiuσ(x, s,Ω) = SAi(s,Ω)

SAi(s,Ω) : standard Berry spin conection (S = 12 )(∂/∂Ωi)Aj(s,Ω) = "ijkΩk

A0µ(s),µ = 1, 2: a Z(2) Berry gauge field(∂µA0ν − ∂νA0ν = 0.)

Aiµ(s): a SO(3) non-Abelian Berry gauge field:Its non-Abelian Berry curvature field is:F iµν(s) = ∂µAiν − ∂νAiµ + S"ijkAjµAkν

Effect of spin-orbit coupling when the Fermi surface is doubly-degenerate.

• The Z(2) gauge field detects Dirac point singularities where two Fermi surfaces touch: These only occur in the absence of spin orbit coupling.

• The SO(3) gauge field is only non-trivial when spin-orbit coupling is present, and there is a non-vanishing non-Abelian SO(3) Berry curvature field.

There are clearly very interesting possibilities when the Fermi surface has spin degeneracy, but I will now specialize to the simpler case where time-reversal symmetry is broken, the Fermi surface is non-degenerate, and there is a non-zero magnetization. In this case there is a U(1) Berry gauge field on the Fermi surface. (This also happens if inversion symmetry is broken)

An exact formula for the T=0 DC Hall conductivity:

• While the Kubo formula gives the conductivity tensor as a current-current correlation function, a Ward-Takahashi identity allows the ω→0, T→0 limit of the (volume-averaged) antisymmetric (Hall) part of the conductivity tensor to be expressed completely in terms of the single-electron propagator!

• The formula is a simple generalization and rearrangement of a 2+1D QED3 formula obtained by Ishikawa and Matsuyama (Z. Phys C 33, 41 (1986), Nucl. Phys. B 280, 523 (1987)), and later used in their analysis of possible finite-size corrections to the 2D QHE.

{cki, c†kj} = δkk′δij(PBC, discretized k)

Ka = limη→0+

!abc2π

∫BZ

d3k

∫ ∞−∞

dω

2πeiωηTr

(∂G

∂ω∇bkG−1G∇ckG−1

)

Gij(k,ω) = −i∫ ∞−∞

dt e−iωt〈0|Tt{cki(t), c†kj(0)}|0〉exact (interacting) T=0 propagator

limω,T→0

σabH (ω, T ) =e2

!#abc

(2π)2Kc

antisymmetric part of conductivity tensor

agrees with Kubo for free electrons, but is quite generally EXACT at T=0 for interacting Bloch electrons with local current conservation (gauge invariance).

• Simple manipulations now recover the result unchanged from the free-electron case.

• After 43 years, the famous Luttinger (1961) theorem relating the non-quantized part of the electron density to the Fermi surface volume now has a “partner”.

Ka = limη→0+

!abc2π

∫BZ

d3k

∫ ∞−∞

dω

2πeiωηTr

((∇bk

∂

∂ω(lnG))(G∇ckG−1)

)

Other Fermi surface formulas.

• Tsuji’s formula for the (small B) dissipative Hall conductivity (controlled by the local inelastic scattering path length at each FS point) can be written as a true FS integral without Tsuji’s restriction to cubic materials

dkFa = kab(s)dnbF nF is outward normal vectorkab(s) is the FS radius of curvature field:

In a suitable coordinate basis:naF n

bF = diag(0,0,1).

kab(s) = diag(k1(s),k2(s),0) principal radiiK(s) = (k1k2)−1 (Gaussian curvature)

σabH =e2

!∑

s

"2(s)#abcK(s)kcd(s)eBd

!σabOhmic =

e2

!∑

s

"(s)naF (s)nbF (s)

*can be related to Ong’s 2D formula!

For the Future:

• General reformulation of FLT for arbitrary Fermi surface geometry and topology. Bosonization revisited? Use differential geometry of manifolds

• non-Abelian SO(3) Berry effects on spin-degenerate Fermi surface?

• role of “quantum distance” ? (approach weak localization by adding disorder to FLT, not interactions to disordered free electrons?)