Anomalous Hall Effect, Dissipationless Currents,

and Quasiparticle Berry Phases:

a New Topological Ingredient 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 November 17, 2004 at the Yukawa International Seminar 2004 (YKIS 2004), Yukawa Institute for Fundamental Physics, Yukawa Memorial Hall, Kyoto University, Kyoto, Japan,

November 17-19 2004

© F. D. M. Haldane 2004haldane@princeton.edu v 1.01

Unfinished business in the Fermi-liquid theory of metals

• 1957-62: development of Landau’s Fermi-liquid theory for 3He, and its extension to metals. Thought to have been conceptually completed. Metals are thought of as “almost like 3He”, except for Coulombic screening effects, and a non-spherical Fermi surface (from Bloch states instead of plane-wave states).

• 1978-84: development of quantum-geometric ideas: Bures distance (1978), the quantum Riemann metric induced by parametric quantum states (Provost and Vallee, 1980), the Berry phase (1984). No impact on Fermi-liquid theory.

• 2004: a surprise! It becomes apparent that the Fermi surface of metals has hitherto-overlooked “quantum geometric” properties (absent in 3He) which endow the Fermi-liquid theory of metals with a potentially-rich topological structure. . ...

• The first new physical result - a 50-year-old controversy is resolved: the “anomalous Hall effect” in ferromagnetic metals is revealed to be a fundamental topological Fermi liquid property. More to come?

• Q: Does Fermi liquid theory describe a valid ground state of interacting electrons? Doesn’t it ignore non-perturbative effects such as the BCS instability (always marginally relevant in some channel, at low enough temperatures, according to Kohn and Luttinger)? Does it really make sense at T=0?

• A: If both time-reversal AND spatial inversion symmetry are broken, the Fermi liquid fixed point is certainly stable against pairing in a finite range of interaction strengths. It appears that a true quantum phase transition could occur at T=0 between such a Fermi liquid and a superconductor.

• This suggests that the Fermi liquid is a true T=0 phase, even though it is in principle (marginally) unstable if the Fermi surface has inversion symmetry in k-space.

Is Fermi-liquid theory a valid description of a T=0 phase of matter?

• A generic non-singular Fermi surface consists of one or more differentiable,

oriented non-intersecting 2-manifolds kF(s), s ∈Sα embedded in Euclidean reciprocal k-space, and periodically repeated on a Bravais lattice of reciprocal lattice vectors G. A primitive region (e.g., the Brillouin zone) must be chosen (a different choice may be made on each manifold).

• A positive differentiable “Fermi speed” function vF(s) is defined at each distinct

Fermi surface point, and a “Landau f-function” f(s,s’) couples every distinct pair of points.

• A wavefunction renormalization factor 0 <Z(s) ≤1is defined at each Fermi surface point. Note that this characterizes the 1-quasiparticle weight in the bare electronic form factor, and is not a fundamental FLT fixed-point property.

• All these quantities except f(s,s’) can be determined from the (exact) inverse single-particle propagator at zero temperature.

Traditional elements of Fermi-liquid theory, as applied to metals:

It turns out that this list is incomplete! Two further fundamental ingredients of the FLT of metals are needed, and can be related to the exact inverse single-particle propagator: a “Riemannian quantum metric tensor” and a “Berry gauge connection” are also defined at each point on the Fermi surface!

Structure of the single-electron propagator (Green’s function)

Σij(k,ω;T ) = limη→0+

1π

∫ ∞

−∞dE

∆ij(k, E;T )ω − E(1 + iη)

G−1ij (k,ω;T ) = (ω + µ(T ))δij − H(1)

ij (k) − Σij(k,ω;T )

limE,T→0

∆ij(k, E;T ) = 0

c†kicreates an electron in a state with Bloch vector k and “internal” state i (i ∈spin + position in the unit cell).

Propagator

Inverse propagator

Self-energy matrix

Positive Hermitianself-energy matrix spectral function

Fundamental Fermi-liquid property:(consequence of Pauli principle)

Hqpij (k) = (Hqp

ji (k))∗ = − limω,T→0

G−1ij (k,ω;T )

Defines a Hermitian “single quasiparticle Hamiltonian”

∆ij(k, E;T ) = ∆ji(k, E;T )∗ ≥ 0

Gij(k,ω;T ) = −i

∫ ∞

−∞dt e−iωt〈Ttcki(t), c†kj(0)〉T

“imaginary part of self energy vanishes at the Fermi energy at T=0”

• The Fermi surface is defined by the set of Bloch vectors at which the “single quasiparticle Hamiltonian” has a zero eigenvalue.

• The “zero-mode” wavefunctions are the periodic parts of the quasiparticle Bloch states:

• They are differentiable on generic non-singular Fermi surfaces, and define the until-now-overlooked “quantum geometry” of the Fermi surface

• Their expectation value of vF(s) defines the Fermi velocity

and the orientation of the Fermi surface.

The CRUCIAL role of the Fermi surface Bloch wavefunctions (apparently ignored in 1957-62)

Hqpij (kF (s))uj(s) = 0

Note: s = (s1,s2) ∈Sα is a two-dimensional parameterization of the Fermi surface manifold Sα : (it may be broken up into disjoint submanifolds)

vF (s) = vF (s)nF (s)

vF (s) = !−1∇kHqp(kF (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) ≡ nF (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 = (nF (s) · ∂µkF (s) × ∂νkF (s)) dsµ ∧ dsν

limω,T→0

σab(ω, T ) → e2

!2Dabδ(ω) +

1(2π)2

e2

! εabcKc

Drude term(symmetric)

Anomalous Hall term(antisymmetric)

Dab =∑

s

!vF (s)naF (s)nb

F (s) +∑s,s′

f(s, s′)naF (s)nb

F (s′) single quasiparticle term Fermi-liquid interaction term

vanishes unless time-reversal symmetry is broken!

“Anomalous Hall vector”:

K =∑α

Kα(modulo G)Kα =

12π

∫Sα

d2F kF +14π

∑i

Gαi

∫∂Si

α

dA

∂S 2

∂S1

∂S1

∂S 2

∂S 3

Sintegral of Fermi vector

weighted by Berry curvature

Berry phase aroundFS intersection with

BZ boundary

Before proceeding, (in case time runs out!) shown here is the final result for the DC conductivity of a Fermi-liquid, expressed completely in terms of FLT quantities

Involves the new “quantum geometry”of the Fermi surface (will explain later).Is ambiguous up to a reciprocal vector,which is a non-FLT quantized Hall edge-state effect

Geometry of a manifold induced by a differentiable quantum state:( U(1) gauge field + metric)

•

Aµ = −i〈Ψ(g)|∂µΨ(g)〉

g ≡ g1, g2, . . .

|DµΨ(g)〉 = |∂µΨ(g)〉 − iAµ(g)|Ψ(g)〉

〈Ψ(g)|DµΨ(g)〉 = 0

|Ψ(g)〉 → eiχ(g)|Ψ(g)〉|DµΨ(g)〉 → eiχ(g)|DµΨ(g)〉

〈DµΨ(g)|DνΨ(g)〉 = Gµν(g) + iFµν(g)

“Berry connection”

Covariant derivative

Fµν(g) = ∂µAν(g) − ∂νAµ(g)

regular derivative on manifold

parameterization of manifold:

U(1) Gauge transformation

Aµ(g) → Aµ(g) + ∂µχ(g)

Gauge invariant antisymmetric “Berry curvature”

(by construction!) Gauge invariant symmetric “quantum metric”

|Ψ(g)〉 =∑

i

Ψi(g)|i〉

|∂µΨ(g)〉 ≡∑

i

∂

∂gµΨi(g)|i〉

(real)

g-independent orthonormal basis

Meaning of the “quantum metric”:

gauge-invariant “quantum distance”: d(ΨA,ΨB)2 = 1 − |〈ΨA|ΨB〉|2

• “quantum distance” is dimensionless, in range [0,1]

• distance between gauge-equivalent states = 0

• distance between orthogonal states = 1

• it is symmetric, satisfies triangle identity, etc.

(Bures 1978, Provost and Vallee 1980)

d(Ψ(g),Ψ(g + δg))2 = Gµν(g)δgµδgν

The “quantum metric” tensor on the manifold is real symmetric positive (not necessarily definite)

The “quantum metric” is distinct from the Berry connection and curvature; it is related to fluctuations and uncertainty-principle bounds.

For infinitesimal displacements on the manifold:

U(1) Berry “gauge field” on the manifold

•

eiφΓ = exp i

∮ΓAµ(g)dgµ

= exp i

∫ΣFµν(g)dgµ ∧ dgν

F

AΣΓ=∂Σ

Berry’s phase for a closed directed path on the manifold can be obtained from the integral of the Berry curvature over any oriented 2-manifold bounded by the path.

12π

∫M

Fµν(g)dgµ ∧ dgν = C(1)(M)

The integral of Berry curvature over a closed 2-submanifold M gives the integer “Chern number” topological invariant of M (“first Chern class”),

Berry 1984

F

M

Application to the parameterized Hermitian eigenproblem.

Eigenstate n is differentiable provided it is non-degenerate: the metric and Berry gauge field that it induces may be singular at degeneracy points.

H(g)|Ψn(g)〉 = En(g)|Ψn(g)〉 En(g) ≤ En+1(g)

embedding type Berry gauge group

degeneracy manifold

codimensionsymmetry type multi-

plicity

Unitary(complex)

U(1) 3generic case(no symmetry)

1

Orthogonal(real)

Z(2)=SU(2)/SO(3) 2

antiunitary symmetry: H(g)*=UH(g)U-1 Tr U ≠ 0.

1

Symplectic(real quaternion)

SU(2) 5antiunitary symmetry:

H(g)*=UH(g)U-1 Tr U = 0.2

• If g-independent antiunitary symmetry is present, there are two other variant Berry gauge group types (”quantum metric” structure is unchanged)

• “multiplicity” = generic eigenvalue multiplicity (doublet in symplectic case).

• “codimension of degeneracy manifold” = number of parameters to vary to find an accidental degeneracy.

• The 3-manifold associated with a non-degenerate band is the Brillouin zone, k modulo G.

• Can choose unσ(k,r) to be periodic in reciprocal space (a gauge choice).

Application to one-electron Bloch states

Electronic bands in an ideal 3D perfectly-periodic structure:

〈rσ|Ψn(k)〉 = eik·runσ(k, r)

T (R)|Ψn(k)〉 = eik·R|Ψn(k)〉eiG·R = 1

(R are real-space lattice translations, G are reciprocal lattice vectors)

Bloch wavefunctions:

|unσ(k + G, r)|2 = |unσ(k, r)|2unσ(k, r + R) = unσ(k, r)

Aan(k) = −i

(∑σ

∫UC d3r u∗

nσ(k, r)∇akunσ(k, r)∑

σ

∫UC d3r u∗

nσ(k, r)unσ(k, r)

)Berry connection

H|Ψn(k)〉 = εn(k)|Ψn(k)〉εn(k + G) = εn(k)

• 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: (Marder 1999, Sundaram and Niu 1999)

Fab(r) = εabcBc(r)

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

Karplus and Luttinger 1954

!dka

dt= eEa(r) + eFab

drb

dt

!dra

dt= ∇a

kεn(k) + !Fabn (k)

dkb

dt

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

Anomalous Hall effect in metals with broken time-reversal symmetry

• Integrate by parts to get Fermi surface (∇kn0(k)) expressions ?

• If time-reversal symmetry is unbroken: K = 0, since then:

The ideal DC conductivity of a “non-interacting” metal with lattice translational symmetry at T=0 is easily evaluated in terms of occupied Bloch state properties using the Kubo formula:

limω,T→0

σab(ω, T ) → e2

!2Dabδ(ω) +

1(2π)2

e2

! εabcKc

Drude term(symmetric)

Anomalous Hall term(antisymmetric)

Dab =1

(2π)3∑

n

∫BZ

d3k n0n(k)∇a

k∇bkεn(k) n0

n(k) = Θ(µF − εn(k))

Fabn (k) = ∇a

kAbn(k) −∇b

kAan(k)

Fabn (k) = Fba

n (−k)εn(k) = εn(−k)

εabcKc =12π

∑n

∫BZ

d3k n0n(k)Fab

n (k)

Karplus and Luttinger 1954

Is the Hall conductivity a Fermi surface property?

• The quantized Hall effect is an apparent counter-example: a clean QHE system has no bulk electronic states at the Fermi energy.

• But dissipationless Hall conduction (transverse to electric fields) occurs through its “chiral” edge states, which ARE at the Fermi energy.

• In the QHE, the bulk Hall conductivity is interpreted as describing a “diamagnetic current density response” to electric fields in the interior of the sample, which does not contribute to transport (which occurs exclusively via edge states).

• There is a viewpoint that the “Hall conductivity” itself is not physically meaningful, and only “Hall conductances” (relating the currents flowing in leads attached to a device to voltage drops across them) are physical.

• Resolution: These viewpoints are too pessimistic: any non-quantized part of the bulk Hall conductivity is a true Fermi-surface transport property (e.g., the anomalous Hall effect in ferromagnetic metals).

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 demonstated in an explicit “graphene” model shown at the right.).

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

second-neighbor hopping t2eiφ, alternating onsite potential M.)

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

• The 2D Hall integer QHE conductivity formula for Bloch electrons with a (bulk) gap at the Fermi energy is just a sum of Chern invariants of filled bands (TKNN formula):

Chern invariants and 2D Quantized Hall Effect

σab2D =

e2

hεabν ν =

∑n

′C(1)

n

C(1)n =

12π

∫BZ

d2kFn(k)

sum over filled bands

This is just the (first) Chern invariant: the 2D Brillouin zone is is a compact 2-manifold (a Torus), as the Berry curvature is a periodic function in the BZ.

3D zero-field Quantized Hall Effect

• Families of lattice planes in a 3D periodic structure are indexed by a primitive reciprocal lattice vector G0 . Each plane is a 2D periodic system that could exhibit a 2D QHE with integer “filling factor” ν. This

adds up to a 3D Hall conductivity with “Hall vector” K = νG0 = GH, a reciprocal vector (in general, non-primitive).

• Such a system will have a gap at the Fermi level, with a number of completely-filled Bloch state bands. The “Hall vector” in this case is a sum of topological invariants of the non-degenerate filled bands (or groups of bands linked by degeneracies).

GH =∑

n

′Gn.

εabcGnc =12π

∫BZ

d3kFabn (k) (band n “Chern vector”)

(sum over filled bands)

a 3x3 antisymmetric matrix can always be brought to “symplectic diagonal form”

0 F 0−F 0 00 0 0

Intrinsic geometry on the Fermi surface manifolds:

d2F = Fµν(s)dsµ ∧ dsν

(ds)2 = Gµν(s)dsµdsν

Fermi surface Berry connection 1-form, Berry-curvature 2-form and quantum distance:

Fermi surface displacement 1-form, geometric area 2-form and k-space distance:

(dkF )2 = (∂µkF (s) · ∂νkF (s)) dsµdsνd2SF = (nF (s) · ∂µkF (s) × ∂νkF (s)) dsµ ∧ dsν

• The geometry of the the Fermi surface in the Brillouin zone obviously defines a “2-form” surface area element and a metric distance element on the surface.

• The new idea here is that the quantum geometry induced by the “internal” structure of the Bloch state leads to a complementary physically important “2-form” (Berry curvature) and a second metric-type quantity (“quantum distance”).

dA = Aµ(s)dsµ Aµ(s) ≡ Aa(kF (s))∂µkFa

Gµν(s) ≡ Gab(kF (s))∂µkFa∂νkFb

Fµν(s) ≡ Fab(kF (s))∂µkFa∂νkFb

dk = ∂µkF dsµ

The Fermi surface formulas (free Bloch electrons)

• Here Sα labels the distinct Fermi surface manifolds. These are free

electron formulas: ( the Drude weight will get a Fermi liquid theory correction, but the non-quantized (anomalous) Hall vector will not).

Kα =12π

∫Sα

d2F kF +14π

∑i

Gαi

∫∂Si

α

dAK =

∑α

Kα(modulo G)

Dab =∑α

Dabα

Dabα =

∫Sα

d2SF

(2π)3!vF na

F nbF

Drude weight tensor:

“Anomalous Hall vector”:

This term is ambiguous up to a reciprocal vector G. It involves the Berry phase around a closed path ∂Si along a discontinuity of the Fermi vector (where it jumps back by Gi into an arbitrarily chosen “Brillouin zone” ). It also guarantees that the expression for Kα (modulo G) is left invariant by a change of choice of “Brillouin zone”

∂S 2

∂S1

∂S1

∂S 2

∂S 3

S

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+

εabc

2π

∫BZ

d3k

∫ ∞

−∞

dω

2πeiωηTr

(∂G

∂ω∇b

kG−1G∇ckG−1

)

Gij(k,ω) = −i

∫ ∞

−∞dt e−iωt〈0|Ttcki(t), c†kj(0)|0〉

exact (interacting) T=0 propagator

limω,T→0

σabH (ω, T ) =

e2

!εabc

(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+

εabc

2π

∫BZ

d3k

∫ ∞

−∞

dω

2πeiωηTr

((∇b

k∂

∂ω(lnG))(G∇c

kG−1))

Drude tensor for a Fermi liquid:

• The Fermi vector is not (electromagnetic) gauge-invariant: Bloch states allow an arbitrary gauge change that shifts all Bloch vectors by a constant k→ k+ K0, δA = -eK0/ħ . Only differences kF(s)-kF(s’) are physically

meaningful.

• Kohn’s formula for the Drude weight (the effect of “twisted boundary conditions” on the total ground state energy density) here becomes

free-electron term Fermi-liquid interaction term

In a rotational- and Galilean-invariant system, Dab = (ħ2n/m)δab and vF = (ħ/meff)kF , and this becomes Landau’s famous formula relating bare and effective masses.

Dab =∑

s

!vF (s)naF (s)nb

F (s) +∑s,s′

f(s, s′)naF (s)nb

F (s′)

Dab =∂2

∂K0a∂K0

b

h[δkF (s) = K0]∣∣∣∣K0=0

Other developments:

• separate dissipationless Hall currents (with their own adiabatic conservation laws) on each distinct manifold (generalizes separate conservation law of each chiral Fermi point in 1D to 3D) . A separate chemical potential can be established on each manifold.

• Fermi surface with non-zero Chern numbers are connected by “wormholes” (Dirac degeneracy points that connect bands; see also discussions of “Fermi points” by Volovik). Charge can be transferred through the “wormhole”, so such connected Fermi surfaces must have the same chemical potential.

• Streda formula:(charge density induced by magnetic field is also controlled by the Hall vector K ): Application to “spin Hall effect” suggests that there is no intrinsic spin Hall effect or (dissipationless current response in general) without broken time-reversal symmetry (cancellation)(?)

• Non-Abelian SU(2) Berry connection when time-reversal and inversion are unbroken:

• Cannot propagate a consistent spin quantization axis from one point on the Fermi surface to all other points (spin-orbit coupling prevents it)

• Non-Abelian curvature Fiμν(s) and connection Aiμ(s) (with 3 SO(3) components i = 1,2,3) modify the usual Pontryagin form for Berry curvature if the direction Ω(s) of the “spin” (Kramers degeneracy coherent state) is fixed at each point on the Fermi surface by a k-dependent perturbation that resolves the doublet degeneracy. (If there is no spin orbit-coupling, Fiμν(s) vanishes, and a gauge choice Aiμ(s)= 0 can be made):

•••• Role of “quantum metric” (spatial localization of wave packets on Fermi

surface) : (related to “most localized Wannier orbitals in insulators”) ?

Fµν = F iµνΩi + S0ε

ijkΩiDµΩjDνΩk

DµΩi = ∂µΩi −Aiµ, S0 = 1

2

F iµν = ∂µAi

ν − ∂νAiµ + εijkAj

µAkν