The Berry Phase and Luttinger s Anomalous Velocity in Metals

Transcription

1 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, (2004) (cond-mat/ ) Talk presented at the IRG1 workshop Strongly Correlated Electronic Materials, Princeton Center for Complex Materials, January 28-29, 2005 work supported by NSF MRSEC DMR haldane@princeton.edu F. D. M. Haldane 2005 v 0.01

2 Hall effect in isotropic (cubic) metals: ρ xy = R 0 B z Hall effect in ferromagnetic metals with B parallel to a magnetization in the z-direction, and isotropy in the x-y plane: ρ xy = ρ xy s + R 0 B z The anomalous extra term is constant when H z 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 nonuniformities in the magnetization?

3 he hole density n h (solid circles) in x determined from R 0 at 400 K (one hole rresponds to n h = cm 3 ). The C is shown as open circles. (B) Curves of vs. H at 5 K in 3 samples (x values indiion value M s = 3.52, 3.72, 3.95 (10 5 A/m) espectively. (C) The resistivity ρ vs. T in content x indicated (a, b indicate different me x). Values of n h in all samples fall in (for x = 1, n h = cm 3 ). example of a very large AHE FIG. 2: Curves of the observed Hall resistivity ρ xy = R 0 B + R s µ 0 M vs. H (at temperatures indicated) in CuCr 2 Se 4 x Br x with x = 0.25 (Panel A) and x = 1.0 (B). In (A), the anomalous Hall coefficient R s changes sign below 250 K, becomes negative, and saturates to a constant value below 50 K. However, in (B), R s is always positive and rises to large values at low T (note difference in scale). Dissipationless Anomalous Hall Current in the Ferromagnetic Spinel CuCr 2 Se 4 x Br x. Wei-Li Lee 1, Satoshi Watauchi 2, V. L. Miller 2, R. J. Cava 2,3, and N. P. Ong 1,3 1 Department of Physics, 2 Department of Chemistry, 3 Princeton Materials Institute, Princeton University, New Jersey 08544, U.S.A. T=5K

4 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!

5 The DC conductivity tensor can be divided into a symmetric Ohmic (dissipative) part and an antisymmetric non-dissipative Hall part: σ ab = σ ab Ohm + σ ab Hall 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) σ ab Hall = e2 1 V D nk F ab n (k)θ(ε F ε n (k)) This is given in terms of the total Berry curvature of occupied states with band index n and Bloch vector k.

6 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) σ xy = e2 σ ab = e2 1 2π ν 1 (2π) 2 ɛabc K c ν = an integer(2d) TKNN formula K = a reciprocal vector G (3D) In 3D G = νg 0, where G 0 indexes a family of lattice planes with a 2D QHE. 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!

7 2D zero-field Quantized Hall Effect FDMH, Phys. Rev. Lett. 61, 2015 (1988). 2D quantized Hall effect: σ xy = νe 2 /h. In the absence of interactions between the particles, ν must be an integer. There are no currentcarrying 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 t 1, complex secondneighbor hopping t 2 e iφ, alternating onsite potential M.)

8 Berry curvature is analog of a magnetic field in Bloch space. Vector potential = Berry connection, curl of Vector potential = Berry curvature. F ab n (k) = a ka b n(k) b ka a n(k) The Berry connection here derives from the k-dependence of the periodic part of the Bloch wavefunction A a n(k) = i UC e iφ(γ) = exp i dv u n(k, x) a ku n (k, x) The Berry phase factor is the analog of a Bohm-Aharonov factor for a closed path in k-space da n

9 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: (Sundaram and Niu 1999) dk a dt dra dt = ee a (r) + ef ab dr b dt = a kε n (k) + F ab n (k) dk b dt write magnetic flux density as an antisymmetric tensor F ab (r) = ɛ abc B c (r) Note the anomalous velocity term! (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. Covariant notation k a, ra is used here to emphasize the duality between k- space and r-space, and expose metric dependence or independence (a {x,y,z }).

10 Current flow as a Bloch wavepacket is accelerated k, t k+δk, t+δt regular flow 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. x x

11 2D case: Bohm-Aharonov in k-space σ xy = e2 1 (2π) 2 σ xy = e2 d 2 k k A(k)n(k) 1 (2π) 2 FS σ xy = e2 h dk F A(k) ( Φ Berry F 2π ) 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!

12 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!)

13 Application of formula to composite fermion Fermi surface at ν = 1/m a composite fermion is modeled as an electron laterally displaced from the center of the m-vortex that is bound to it. 2π l 2 1 m Berry phase for moving composite quasiparticle around Fermi-surface σ xy AHE = e2 h 1 m Flux enclosed by path of displaced electron around vortex: independent of Fermi surface shape!

14 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 K α (modulo G) Γ 1 S Γ 2 K α = 1 2π ( S α d 2 S F Fk F + integral of Fermi vector weighted by Berry curvature on FS r G i i=1 Γ i α da ) Berry phase around FS intersection with BZ boundary Γ 3 Γ 1 Γ 2 This is ambiguous up to a reciprocal vector, which is a non-flt quantized Hall edge-state contribution

15 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.

16 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

17 The new FLT ingredient: quasiparticle wavefunctions Quasiparticle states on the Fermi surface (only) have Bloch wavefunctions which are naturally defined by the singleparticle Green s function of the interacting system, without reference to any fictional non-interacting precursor. Ψ qp σ (r; s, Ω) = e ik F (s) r u σ (r; s, Ω) Here s is a 2-component parameterization of the Fermi surface, and the 3-component unit vector Ω is a coherentstate parameterization of 2-fold spin degeneracy if timereversal and inversion symmetry are unbroken. The periodic factor u σ (r;s,ω) is the source of the new FLT ingredients.

18 Local Bloch-state basis: R are lattice translations and G are the reciprocal lattice vectors: U(G) = exp i(ig r); 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 T (R) = exp(ip R/ ) T (R) k, x, σ = e ik R k, x, σ U(G) k, x, σ = e ig x k, x, σ S z k, x, σ = σ k, x, σ (k,x) creates an electron in this state. k + G, x + R, σ = k, x, σ

19 The single-particle Green s function: Ψ σ(k, x, t) = e iht Ψ σ(k, x)e iht, H = H µn G σσ (ω, k, x, x ) = G 1 lim Σ σσ (ω, k, x, T 0 x ) = Σ 0 σσ (k, x, x ) + Δ σσ' (E,k,x,x') is positive-definite Hermitian for E 0, vanishes when E = 0;! -1 σσ' (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 = E F u) lim T 0 σ UC dt 2πi e iωt T t {Ψ σ (k, x, 0), Ψ σ(k, x, t)} T σσ (ω, k, x, x ) = ω11 σσ (x, x ) Σ σσ (ω, k, x, x ) de σσ (E, k, x, x ) ω E(1 + iɛ + ) d 3 x G 1 σσ (0, k F (s), x, x )u σ (x, s, Ω) = 0 *In this talk, the issue of the Kohn-Luttinger BCS instability will be swept under the rug.

20 All properties of the FLT fixed point except the Landau function derive from the one-electron Green s function: k-space geometry: Fermi vector k F (s), and outward normal n F (s). energetics: Fermi speed v F (s), and Landau function f(s,ω,s',ω') = f 0 (s,s') + f ij (s,s')ω i Ω' j, intrinsic magnetic moment µ i a (s)ω i. NEW... Hilbert space geometry: Berry gauge potential A μ quantum distance d Q (s,s'). (s,ω) and Non fixed-point properties: quasiparticle residue Z(s) and asymptotic lowtemperature inelastic scattering path length l(s,t). v F (s)n F (s) = u(s, Ω) k G 1 (0, k F (s)) u(s, Ω) l(s, T ) = v F (s)/γ(πk B T, s) µ i a(s)ω i = u(s, Ω) µ ea u(s, Ω) Γ(E, s) = π 1 u(s, Ω) (E, k F (s)) u(s, Ω) Z(s) = u(s, Ω) ω G 1 (0, k F (s)) u(s, Ω)

21 The Landau ground state energy density functional d 2 S F = (ˆn F (s) µ k F (s) ν k F (s)) ds µ ds ν s α S α d 2 S F (2π) 3 simplified notation for Fermi surface area sum (S α means primitive region e.g., in BZ) The Landau functional gives the change in energy density if the Fermi surface is changed from k F to k F + δk F. Change in electron density δk F component normal to original Fermi surface δn[δk F ] = s δk F (s) δk F (s) ˆn F (s) δk F (s) Change in energy density δh[δk F ] = 1 2 v F (s) s ( δk F (s) ) s,s f(s, s )δk F (s)δk F (s ) Fermi speed Landau function coupling different Fermi surface points

22 Bures-Uhlmann Quantum distance between two (pure) states in Hilbert space. non-degenerate (orthogonal or unitary) case: d Q (Ψ A, Ψ B ) = (1 Ψ A Ψ B ) 1/2 doubly-degenerate (symplectic) case: d Q (Ψ A, Ψ B )) = ( 1 ( 2 d 2 Ω 4π Ψ A(Ω) Ψ B (Ω ) 2 ) 1/2 ) 1/2 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.

23 Berry connection (gauge fields on Fermi surface) without spin-splitting, there are three distinct Fermi surface gauge fields: i σ i σ UC UC d 3 x u σ(x, s, Ω) Ω i u σ(x, s, Ω) = SA i (s, Ω) d 3 x u σ(x, s, Ω) s µ u σ(x, s, Ω) = A 0 µ(s) + A i µ(s)ω i SA i (s, Ω) : standard Berry spin conection (S = 1 2 ) ( / Ω i )A j (s, Ω) = ɛ ijk Ω k A 0 µ(s), µ = 1, 2: a Z(2) Berry gauge field ( µ A 0 ν ν A 0 ν = 0.) A i µ(s): a SO(3) non-abelian Berry gauge field: Its non-abelian Berry curvature field is: F i µν(s) = µ A i ν ν A i µ + Sɛ ijk A j µa k ν

24 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 spinorbit 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)

25 An exact formula for the T=0 DC Hall conductivity: While the Kubo formula gives the conductivity tensor as a currentcurrent 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 QED 3 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. G ij (k, ω) = i K a = lim η 0 + ɛ abc 2π exact (interacting) T=0 propagator BZ dt e iωt 0 T t {c ki (t), c kj (0)} 0 lim ω,t 0 σab H (ω, T ) = e2 d 3 k ɛ abc (2π) 2 K c {c ki, c kj } = δ kk δ ij (PBC, discretized k) antisymmetric part of conductivity tensor ( ) dω G 2π eiωη Tr ω b kg 1 G c kg 1 agrees with Kubo for free electrons, but is quite generally EXACT at T=0 for interacting Bloch electrons with local current conservation (gauge invariance).

26 K a = lim η 0 + ɛ abc 2π BZ d 3 k ( ) dω 2π eiωη Tr ( b k ω (ln G))(G c kg 1 ) 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.

27 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 dk F a = k ab (s)dn b F n F is outward normal vector k ab (s) is the FS radius of curvature field: In a suitable coordinate basis: n a F nb F = diag(0,0,1). k ab (s) = diag(k 1 (s),k 2 (s),0) principal radii K(s) = (k 1 k 2 ) 1 (Gaussian curvature) σ ab Ohmic = e2 l(s)n a F (s)n b F (s) s σ ab H = e2 s l 2 (s)ɛ abc K(s)k cd (s) ebd *can be related to Ong s 2D formula!

28 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 spindegenerate Fermi surface? role of quantum distance? (approach weak localization by adding disorder to FLT, not interactions to disordered free electrons?)