Anomalous Hall Effect, Dissipationless Currents, and Berry Phases: a New Topological Ingredient in the Fermi-Liquid Theory of

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 [email protected] © F. D. M. Haldane 2004 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 (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? Is Fermi-liquid theory a valid description of a T=0 phase of matter?

• Q: Does Fermi liquid theory describe a valid ground state of interacting ? 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. Traditional elements of Fermi-liquid theory, as applied to metals: • 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

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

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- propagator (Green’s function) creates an electron in a state with Bloch vector k and “internal” state i ck† i (i ∈spin + position in the unit cell).

∞ iωt G (k, ω; T ) = i dt e− T c (t), c† (0) ij − " t{ ki kj }#T Propagator !−∞

1 (1) Inverse propagator Gij− (k, ω; T ) = (ω + µ(T ))δij Hij (k) Σij(k, ω; T ) − − Positive Hermitian ∆ij(k, E; T ) = ∆ji(k, E; T )∗ 0 self-energy matrix ≥ spectral function 1 ∞ ∆ij(k, E; T ) Σij(k, ω; T ) = lim dE Self-energy matrix η 0+ π ω E(1 + iη) → !−∞ − “imaginary part of self energy vanishes at the at T=0” Fundamental Fermi-liquid property: lim ∆ij(k, E; T ) = 0 E,T 0 (consequence of Pauli principle) → qp qp 1 ij (k) = ( ji (k))∗ = lim Gij− (k, ω; T ) H H − ω,T 0 Defines a Hermitian “single quasiparticle Hamiltonian”→ The CRUCIAL role of the Fermi surface Bloch wavefunctions (apparently ignored in 1957-62)

qp ij (kF (s))uj(s) = 0 H 1 qp vF (s) = !− k (kF (s)) ∇ H • 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 defines the Fermi velocity • vF(s) and the orientation of the Fermi surface. vF (s) = vF (s)nˆ F (s)

1 2 Note: s = (s ,s ) ∈Sα is a two-dimensional parameterization of the Fermi surface manifold Sα : (it may be broken up into disjoint submanifolds) The Landau ground state energy density functional 2 µ ν d SF = (nˆ F (s) ∂µkF (s) ∂ν kF (s)) ds ds 2 · × ∧ d SF simplified notation for Fermi surface area ≡ (2π)3 sum (S means primitive region e.g., in BZ) s α Sα α ! ! " • The Landau functional gives the change in energy density if the Fermi surface is changed from kF to kF + δkF .

Change in electron density δkF component normal to original Fermi surface

δn[δkF ] = δkF! (s) δkF! (s) nˆ F (s) δkF (s) s ≡ · ! Change in energy density 2 1 1 ! ! ! δh[δkF ] = 2 !vF (s) δkF (s) + 2 f(s, s")δkF (s)δkF (s") s ! " # !s,s! Fermi speed Landau function coupling different Fermi surface points 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

2 2 ab e ab 1 e abc lim σ (ω, T ) D δ(ω) + % Kc ω,T 0 → 2 (2π)2 vanishes unless → ! ! time-reversal symmetry is Drude term Anomalous Hall term broken! (symmetric) (antisymmetric)

ab a b a b D = !vF (s)nˆF (s)nˆF (s) + f(s, s!)nˆF (s)nˆF (s!) s ! single quasiparticle term !s,s! Fermi-liquid interaction term “Anomalous Hall vector”: 1 2 1 Kα = d kF + Gαi d K = K (modulo G) 2π F 4π i A α !Sα i !∂Sα α " ! ∂S1 integral of Fermi vector Berry phase around S ∂S 2 weighted by Berry FS intersection with curvature BZ boundary

2 Involves the new “quantum geometry” ∂S of the Fermi surface (will explain later). Is ambiguous up to a reciprocal vector,which 3 ∂S ∂S1 is a non-FLT quantized Hall edge-state effect •

Geometry of a manifold induced by a differentiable quantum state: ( U(1) gauge field + metric) g-independent orthonormal basis parameterization of manifold: Ψ(g) = Ψi(g) i | ! | ! g g1, g2, . . . (real) i ! ∂ ≡ { } ∂ Ψ(g) Ψ (g) i | µ ! ≡ ∂gµ i | ! regular derivative on manifold i ! µ = i Ψ(g) ∂µΨ(g) “Berry connection” A − " | # DµΨ(g) = ∂µΨ(g) i µ(g) Ψ(g) Covariant derivative | ! | ! − A | !

Ψ(g) eiχ(g) Ψ(g) | ! → | ! D Ψ(g) eiχ(g) D Ψ(g) U(1) Gauge transformation | µ ! → | µ ! (g) (g) + ∂ χ(g) Aµ → Aµ µ

µν (g) = ∂µ ν (g) ∂ν µ(g) Gauge invariant antisymmetric “Berry curvature” F A − A Ψ(g) D Ψ(g) = 0 DµΨ(g) Dν Ψ(g) = µν (g) + i µν (g) ! | µ " ! | " G F (by construction!) Gauge invariant symmetric “quantum metric” Meaning of the “quantum metric”:

2 2 gauge-invariant “quantum distance”: d(ΨA, ΨB) = 1 ΨA ΨB − |" | #| (Bures 1978, Provost and Vallee 1980)

• “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.

For infinitesimal displacements on the manifold: d(Ψ(g), Ψ(g + δg))2 = (g)δgµδgν Gµν 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. U(1) Berry “gauge field” on the manifold F

iφΓ µ e = exp i µ(g)dg Γ A Γ=∂Σ ! Σ A = exp i (g)dgµ dgν Fµν ∧ !Σ Berry’s phase for a closed directed path on the manifold can be obtained from the integral of the Berry curvature Berry 1984 over any oriented 2-manifold bounded by the path. F 1 (g)dgµ dgν = C(1)(M) 2π Fµν ∧ !M The integral of Berry curvature over a closed 2-submanifold M gives the integer M “Chern number” topological invariant of M (“first Chern class”), • Application to the parameterized Hermitian eigenproblem.

H(g) Ψ (g) = E (g) Ψ (g) En(g) En+1(g) | n ! n | n ! ≤ Eigenstate n is differentiable provided it is non-degenerate: the metric and Berry gauge field that it induces may be singular at degeneracy points.

Berry gauge degeneracy multi- embedding type manifold symmetry type group codimension plicity generic case Unitary U(1) 3 1 (complex) (no symmetry) Orthogonal Z(2)= antiunitary symmetry: 2 - 1 (real) SU(2)/SO(3) H(g)*=UH(g)U 1 Tr U ≠ 0. Symplectic antiunitary symmetry: SU(2) 5 - 2 (real quaternion) H(g)*=UH(g)U 1 Tr U = 0. • 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. Application to one-electron Bloch states

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

ik R H Ψn(k) = εn(k) Ψn(k) T (R) Ψ (k) = e · Ψ (k) | ! | ! | n ! | n ! iG R εn(k + G) = εn(k) e · = 1 (R are real-space lattice translations, G are reciprocal lattice vectors) Bloch wavefunctions: ik r unσ(k, r + R) = unσ(k, r) rσ Ψn(k) = e · unσ(k, r) u (k + G, r) 2 = u (k, r) 2 ! | " | nσ | | nσ | • The 3-manifold associated with a non-degenerate band is the Brillouin zone, k modulo G.

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

3 a d r u∗ (k, r) unσ(k, r) a (k) = i σ UC nσ ∇k Berry connection An − d3r u (k, r)u (k, r) !" σ# UC n∗σ nσ $ " # Semiclassical dynamics of Bloch electrons 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) b write magnetic flux density dka dr as an antisymmetric tensor ! = eEa(r) + eFab c dt dt Fab(r) = !abcB (r) a dr a ab dkb Note the “anomalous velocity” term! ! = εn(k) + ! (k) dt ∇k Fn dt (in addition to the group velocity) Karplus and Luttinger 1954

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

a • Covariant notation ka, r is used here to emphasize the duality between k- space and r-space, and expose metric dependence or independence (a ∈{x,y,z }). Current flow as a Bloch wavepacket is accelerated k, t x regular flow k+δk, t+δt x “anomalous” flow • 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 Hall effect in metals with broken time-reversal symmetry

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

2 2 ab e ab 1 e abc lim σ (ω, T ) D δ(ω) + % Kc ω,T 0 → 2 (2π)2 → ! !

Drude term Anomalous Hall term Karplus and Luttinger (symmetric) (antisymmetric) 1954 1 Dab = d3k n0 (k) a b ε (k) 0 (2π)3 n ∇k∇k n nn(k) = Θ(µF εn(k)) n BZ − ! " abc 1 3 0 ab ab a b b a ! Kc = d k n (k) (k) n (k) = k n(k) k n(k) 2π n Fn F ∇ A − ∇ A n BZ ! " ∇ n0(k)) • Integrate by parts to get Fermi surface ( k expressions ? • If time-reversal symmetry is unbroken: K = 0, since then: ab ba εn(k) = εn( k) (k) = ( k) − Fn Fn − 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 FDMH, Phys. Rev. Lett. 61, 2015 (1988).

• 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 iφ second-neighbor hopping t2e , alternating onsite potential M.) Chern invariants and 2D Quantized Hall Effect

• 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):

e2 ! (1) ab ab ν = Cn sum over filled bands σ2D = " ν h n !

This is just the (first) 1 C(1) = d2k (k) Chern invariant: the 2D n 2π Fn Brillouin zone is is a !BZ 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 0 adds up to a 3D Hall conductivity with “Hall vector” K = νG = 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 = Gn. (sum over filled bands) n ! 1 !abcG = d3k ab(k) (band n “Chern vector”) nc 2π Fn !BZ a 3x3 antisymmetric matrix can always be brought 0 0 F0 0 to “symplectic diagonal form”  −0F 0 0    Intrinsic geometry on the Fermi surface manifolds: Fermi surface displacement 1-form, geometric area 2-form and k-space distance: µ dk = ∂µkF ds 2 µ ν d SF = (nˆ F (s) ∂µkF (s) ∂ν kF (s)) ds ds (dk )2 = (∂ k (s·) ∂ k (s×)) dsµdsν ∧ F µ F · ν F Fermi surface Berry connection 1-form, Berry-curvature 2-form and quantum distance: µ a d = (s)ds µ(s) (kF (s))∂µkF a A Aµ A ≡ A 2 µ ν ab d = (s)ds ds µν (s) (kF (s))∂µkF a∂ν kF b F Fµν ∧ F ≡ F (ds)2 = (s)dsµdsν (s) ab(k (s))∂ k ∂ k Gµν Gµν ≡ G F µ F a ν F b • 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”). 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).

Drude weight tensor: 2 ab d SF a b ab ab Dα = !vF nˆF nˆF D = Dα 3 Sα (2π) α ! !

“Anomalous Hall vector”: 1 2 1 Kα = d kF + Gαi d K = K (modulo G) 2π S F 4π ∂Si A α ! α i ! α α " !

∂S1 This term is ambiguous up to a reciprocal S ∂S 2 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 ∂S 2 into an arbitrarily chosen “Brillouin zone” ). It also guarantees that the expression for Kα (modulo G) is left invariant by a change 3 ∂S ∂S1 of choice of “Brillouin zone” 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.

∞ iωt Gij(k, ω) = i dt e− 0 Tt cki(t), c† (0) 0 c , c† = δ δ − " | { kj }| # { ki kj} kk! ij !−∞ exact (interacting) T=0 propagator (PBC, discretized k) 2 abc ab e # antisymmetric part lim σH (ω, T ) = Kc ω,T 0 (2π)2 of conductivity tensor → !

!abc 3 ∞ dω iωη ∂G b 1 c 1 Ka = lim d k e Tr G− G G− + k k η 0 2π BZ 2π ∂ω ∇ ∇ → ! !−∞ " # agrees with Kubo for free electrons, but is quite generally EXACT at T=0 for interacting Bloch electrons with local current conservation (gauge invariance). !abc 3 ∞ dω iωη b ∂ c 1 Ka = lim d k e Tr ( (ln G))(G G− ) + k k η 0 2π BZ 2π ∇ ∂ω ∇ → ! !−∞ " #

• 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”. 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 0 0 k→ k+ K , δA = -eK /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 ∂2 Dab = h[δk (s) = K0] ∂K0∂K0 F a b !K0=0 ! ! ! ab a b a b D = !vF (s)nˆF (s)nˆF (s) + f(s, s!)nˆF (s)nˆF (s!)

s s,s! fr!ee-electron term Fermi-liquid! interaction term

ab 2 ab eff In a rotational- and Galilean-invariant system, D = (ħ n/m)δ and vF = (ħ/m )kF , and this becomes Landau’s famous formula relating bare and effective masses. 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)

i • Non-Abelian curvature F μν(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 i is no spin orbit-coupling, F μν(s) vanishes, and a gauge choice Aiμ(s)= 0 can be made): = i Ωi + S !ijkΩiD ΩjD Ωk • Fµν Fµν 0 µ ν D Ωi = ∂ Ωi i , S = 1 • µ µ − Aµ 0 2 i = ∂ i ∂ i + !ijk j k • Fµν µAν − ν Aµ AµAν • Role of “quantum metric” (spatial localization of wave packets on Fermi surface) : (related to “most localized Wannier orbitals in insulators”) ?