Quantum Geometry in Conducting Systems

Quantum geometry in Conducting systems

F. Duncan M. Haldane Princeton University.

• Formal discussion of quantum distance and Berry curvature • Application to the Anomalous Hall effect in 3D Ferromagnets, and 2D composite Fermions. • Fermi surface geometry. geometry of a manifold of quantum states

Ψ(g) = Ψ (g) i | ! i | ! i ! continuous family of orthonormal ﬁxed basis states parameterized (g-independent) by d real parameters

gµ, µ 1, 2, . . . d { ∈ } • Berry gauge ambiguity: nothing physical changes if we make a g-dependent gauge change Ψ(g) eiχ(g) Ψ(g) | ! → | ! Examples of manifolds of states: • bands of Bloch states (manifold = Brillouin zone) • Fermi liquid quasiparticles (manifold = Fermi surface in 2D or 3D) • eigenstates of a family of non-degenerate Hamiltonians (manifold = parameter space of Hamiltonians) • Coherent states (spin coherent states, Landau level “guiding center” coherent states, etc.) Hilbert-space distance

d (1, 2)2 = 1 Ψ(1) Ψ(2) p, p 1 p − |" | #| ≥ d(12) = d(21) 0 Berry gauge invariant (absolute value) ≥ • d(12) = 0 iff 1 = 2 obeys the triangle inequality, positivity, etc. d(12) + d(23) d(13) • ≥ fundamental property • dimensionless: dp(1,2) = 1 for orthogonal states of a distance • case p=1 is Bures-Uhlmann distance, p=2 is Hilbert-Schmidt distance, p=∞ is classical (“trivial distance” dij = 1 for i≠j) • Eigenstates of Hamiltonian are stationary (only phase changes with time): with this definition of distance, the speed of motion in Hilbert space of a non-eigenstate is 1/2 (p/2) 2 2 2 vp = (∆H)rms (∆H) H H ! energy≡ variance" # − " # • a generic quantum state on a manifold induces both a Riemannian metric (through its distance property) and a U(1) gauge field (the “Berry connection”)(“unitary” case)

(Hermitian) (symmetric) (antisymmetric) DµΨ(g) Dν Ψ(g) = µν (g) + i µν (g) ! | " RiemannG FBerry covariant metric curvature derivative • If the states are eigenstates of a family of time- reversal-invariant Hamiltonians, the Berry gauge ﬁeld is Z(2) (“orthogonal” case, without spin- orbit coupling, or SU(2) (“symplectic” case):

Ψ(g) eiχ(g) Ψ(g) Transform the same way | ! → | ! with a gauge change D Ψ(g) eiχ(g) D Ψ(g) | µ ! → | µ ! Ψ(g) D Ψ(g) = 0 gauge-invariant relation ! | µ " (parallel transport) • after a gauge transformation: + ∂ χ(g) not gauge invariant Aµ → Aµ µ • The metric and Berry curvature are gauge-invariant = Re.( ∂ Ψ ∂ Ψ ) Gµν ! µ | ν " − AµAν = ∂ ∂ Fµν µAν − ν Aµ • We can also make a GR-like covariant tensor formulation by using the metric to obtain the Christoffel connection: • D ∂ Γσ µVν ≡ µVν − µν Vσ = D D unchanged Fµν µAν − ν Aµ (antisymmetric) Unitary case: Berry phase and Chern invariant:

• For a closed path Γ on the manifold: iφ(Γ) µ geometrical U(1) e = exp i µdg Berry phase factor Γ A • For a closed 2-surface! Σ on the manifold: topological 1 µ ν dg dg µν = C1(Σ) (1st) Chern class 2π ∧ F integer invariant !Σ (These are the analogs of the Bohm-Aharonov phase and the Dirac monopole quantization) Orthogonal case: • Vanishing Berry curvature = 0 Fµν • Berry phase factor: topological Z(2) η(Γ) = exp i dgµ = 1 Berry phase factor Aµ ± !Γ Symplectic case: As above, plus Chern invariant • topological 1 dgµ dgν dgσ dgτ ρ = C ( ) (2nd) Chern class 24π2 ∧ ∧ ∧ µνστ 2 M4 !M4 integer invariant ρ a a + a a + a a µνστ ≡ Fµν Fστ Fµτ Fνσ FµσFτν integral over (O(3) vector dot product) closed 4-d surface 2D manifolds. • There is only one independent 2-form (volume element) of an (orientable) 2-d manifold; all others can be related to it. Various choices are possible. (g) = (g)(det (g))1/2! antisymmetric Fµν F G µν symbol • since µ ν i µ ν is positive Hermitian, is a dimensionlessG ± F pseudoscalar with the F bounds 2 0 ( (g)) 1 (unitary case) ≤ F ≤ 0 a(g) a(g) 1 (symplectic case) ≤ F F ≤ In general λν ν κλ = δ µν µκ νλ GµλG µ G ≥ G F F • If the Berry curvature diverges at some point, so does the Riemann metric. • These divergences are associated with “Dirac points” where bands touch linearly - like “wormholes” in GR though which quantized magnetic ﬂux passes.

distance 1 between “close” points History:

• The Riemann metric structure was explored (in the mathematical context of coherent states) by Provost and Vallee (1980) who noted in passing that there was also an antisymmetric part that might also be worth studying .....(now known as Berry curvature (!)) Not much tangible physics has emerged from the metric, until recently what appears to be developing into a fundamental characterization of localization lengths in band insulators has been formulated (Marzani and Vanderbilt, (1997) Resta and Sorella (1999), Souza ) • The Berry curvature, following from the Berry phase (1984), the TKNN (1984) treatment of the QHE, followed by Simon’s (1984) mathematical explanation continues to play a major role in modern physics. Application to Bloch states ik (R+ri) ψR,i(k) = e · ui(k) • amplitude for a particle to be on i’th site at position R + ri in unit cell R. Note that the Bloch factor depends on the assumed changing the set ri changes the relative location of site i in the unit cell. quantum metric • Manifold is the Brillouin zone k mod G. Berry connection is a a (k) = i u∗(k) u (k) A − i ∇k i “mean position of pari ticle relative to unit • ! cell” ra = i a a(k) − ∇k − A [ra, rb] = i ab(k) non-commuting F coordinates! • What physical properties are influenced by the “quantum geometry” of the Bloch states at the Fermi level? • NOT static ground state properties; only properties that involve acceleration of particles at the Fermi surface by applied uniform electric fields, chemical potential and thermal gradients, etc. • unaccelerated particles are not influenced by these effects, but they can have a profound effect on transport. Hall effect in ferromagnetic metals: isotropic (cubic) case Hall effect in ferromagnetic metals with B parallel to a magnetization in the z-direction, and isotropy in the x-y plane:

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

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h e t t e t e B u o n m b ( H p ) e t s ρ r r e f a s o m y n s m w h t i f e f t i o t e n r , p f f e i c 2 w p e H n m h c a 0 t y , o o m n a g i C r e a i u 5 y c t t e r d p a a a 3 c - b e l s a f − K p n r s . r ( o e n c a m 1 o h - n l e v u 3 C K 4 e t l a e i m d n o ) m j h u 5 r n w e P s S t s f t h t t c s a n n r o t i e r e L a y o ( A n n r h u a t n n 2 a w p e a r x s u − ) e u 4 f d o b s r c d l a c t l o p a e m r H l a h r t e . e i 9 ( a v d ρ l r h n ( f ) . q 0 h i r r t t d o i v l m h 2 u p o e 3 [ e d g t p i p m e J n i C r n e i a l m ‘ s s J a s i l f e − a n r A n ( o n o 1 h a l i e 3 S n i K o e e d e t s i l u c c W m m e r c n e o e ) n d e n i . c e c a e s g n 2 a n e o h H g u m o e 4 n m Y c d n c n o s 0 s c l o 5 t L s ) H g t n 9 v s ( L n ) s i 0 t u 4 e r t m T e e r u p 3 w [ e o e a r H u ff r i c t ‘ a s u i e n n c u c e f e c , t S y n A o t a a d s ] t s h 7 l c a r r s . e ) t / n d e i . i b p s . l e h m h f . o s f r n e s 0 u a 5 e f ) s A , e t r r s r 4 v e a T m a r r s o r e a e m l ff g o e a g o E r . c , i s e e E l s y m 3 A e t r s t ] f n m u e t e a o i a ‘ . e s ∼ o r : r e i f . g e C e e c t m A a e 8 o v t v l b a s o n w 4 o e s r t r g b d n c n a . o i a e p s 3 1 s ) e i n a o t s e ∼ o a e e t C s ) t ) t e r o t ( n o w 4 e s b d e t b T e , H u ) e i h n o t e m e l n n h e n , n 0 r r l s e o n T / C s n J T n n i H e e l d r e o i 9 p r t ( h 0 r l e n i T / , t m i r r n s r a t d e a r t n , t s o o o s c e n a t y i d d . m s b o o l t t t y s n s b o d l s r o r a t m t H s i s ) 2 2 e e e h I I S l , s i , i t u - 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T n a l v u n e fi : t H n n n b n E e i c k e l e r a 5 d a r d s . s d t l t i m e - t m n e i s a f e 5 L v n L i 9 R s = d n s o t d n d n d r a e r d l e d e e e n r n r b y ) t s t ffi s o t t - - - - n ------s , f i o b M : t fi 0 n , o t fl l d = . i e i e a s . o . c a o v n d ( o o s h a K i v b u ( y e , = s ff y l s b a l × ( i a r e n a n g s n n m n H 2 a e o r c i y u w ) M P o y e C t m B e n l a n p u n ) v r e n D b y l c E t s e d ρ e i c W a 0 i A R x y t t n o a , l n g t t x t e e U e a e g v r g c u o s e l 7 v i i r 1 e a t t o . l R ρ a m b T ρ r fi w i ( ( m ρ ( w h r l b c ( i τ J w c a h L A a a t 2 k h a r r l t o h u ) f e e c n n m a i t e s n . h n , d x " e p a d n l n 7 a r 1 e e i 1 1 t h r l u s o d e i 6 s o n n x " x " x " 0 y . d m c a a e s f H v m e = e y a e e s a t 0 r h i a o h n c e , 2 H a d f n o o e e b u a , o c w l e t e i 7 n a 8 1 r s , t y y y e r i t T I I r g e v t µ d d n w u y r e d r b a t o H y r e i s v 2 f r d d s l e s e e e e t a n d t . T e e d n n s d ( b t a i h - e r c h , ) m n / d n n i h E 2 i s t l 2 e h i 5 i w u a d e t c d e t l l e i m e - n t d s 0 t n r ∼ 1 f 3 e t v s s i e h 9 R n e i l b d i 0 s F s o p c R s y m l 2 a e s e n r , e ρ o i d 0 S e e e r 0 e n e t n r b s n e ffi h a e c l s M d y T s t A 9 n 0 ρ n a t t f c e o m b e m g l t o c b i B e i t s t a n e M p 0 r fl e d t 2 t i d = d c f s i e s 4 n e e o s t l r t e t e , n , s . o R c e e s c a . p ) v 0 n a ) n n x t i ρ s s o o o n t h s c e i i 4 s u ) r u o e ( , C e i o e y p o ff o e r c y . 2 l r t n s a r × ( i o b ) e c . a e n r r r x n m ) r T α t i t o p α a o n a o t u ) s y t ( . m p n o a t a d m e r r e d a c ( l s a d o l y o u a b o t P o K s A a s c p s t d c e i u r n l e 5 e ( r a m . b p 3 t e , m c r o o u l n , n ( t t x p r s H s g n a m e o r e b h n h l l d o J n m B o t h a r C = a h e e t t e s e d i l i h d t c s N ( e p s , o W − h a p t a i H c ) s o w s R e x v p i p n d k n o N n l r n f e d t u t 7 . o t e a r e r g t r b e r y o A e a o o r g o e e e ( t r c o h e n µ t p u t a s u e e s r r r f a c i i 1 e m h o α a l a t L r e n a f g t o s a ) E c ) c i n d 3 e a h y n w a e o u 1 e e d i i F m r a t a C a a e t h - C i i o r f p e i r c t f n m d , i a . s 0 e m z s n e , ρ t a 0 d n a n n C n ) f n w o h s d g e t i i l h e . r n v fi t c n c s i h i s o a s t 0 r H t e - o d l p s r e o s s h er v i t e o e r = i n a a e i x " r l n o i a e i l n n g s r i d w c e o d e A e c g o m i i x g t n h a l o v d i l e i a c t e d 2 e c e e a n s d z l f i t g S w e n n v e t y s 4 ( x e o i b t µ i α h t n o s a e d e n J o s a d e ( r e u g o g e 2 ) S o h f o u r o e i a l e u e r e d n s n s t 9 s a e t e u o ( ρ m c d i p h t e m n i d . n i e i m i . r h g m . d d n t t l ρ s o . n a n r h t p n e B w n c a = l n o τ a h t b v s n r c n d h ) s s 2 s n d 0 = t e l d e . r , s r h a p i h e e h e , s 0 s 4 c t i R e s f h o i 2 a n e a c f , . e s d t l e h a s o t e i 0 u H d a o s t i e s o e 0 s e s n t 2 − e m . x 3 r T s t t d e T e i J e A e s i e c y lar i p e e h i I s e d t i m l U e h f m n t o H fi l s o ρ h n u , u s e r i n a t n ) h M p i i v t i r , 2 a t o t t n s t i i p f s x 1 e a I d o . 4 e n i A r d w m z n t t y A m g t H o c a h s e i , a r h . r a i m t r a x o t a n p i c s s e . a n n 8 f t h o n t B p e t i n e u e a c r u T G o i g o o a e C e . y y m y e n m o y c n i h r r d a 2 h a y t p m i l e o t n e l h n l ) e c h H a e n 0 e n a t e m ) t T 0 f a r c ) l u n o l n p B 5 a d , u c ρ t n s r o t h t d e l t n o = w r s a p n a e o r o i t t d a i a c e f s e d n l t fi a n d d i p p s e 8 b x h = s o c a t v A ( g n m r o d a e t e E u a r 5 e d r e e n h t s v . s p m , l c i y H d o r h o u A n u o r ( ( s i 5 t e e a e , 1 v a m l e h u g g o h r r n o n o S e s e r - a h m t t r o a t e t e o h 2 a r s e l r H 2 m , b t t e s b h e i B p l 4 e s ( r e a e s r 3 b r e c − h a r m a o e t R c a r a n ) H c i s t u n N t r ( i m h e d o i u v r n e a n p n 1 c n e a r m p d t t e n 4 n s h a u o t . o t i n e y , s m o u A t o o E l e e a 1 s e e d ( t H X n c o µ l d i a t e n l o a r v s a r t , ) d i e a l m o e 1 d y ) 0 l h ( d R e , e . i u e e o n t s m a e s e r p u a e n t e h e a n ) i y l f a 3 c l l a s s i n r p d e a s o . . e u 1 r F d i B m F 4 r i r e 2 d a a s e e r . t c M ) o e o H n r r c s n d , s R o f t n u o t a n P i , a U , e ge d b u - 0 e a s t r t 0 , ( s 7 e a n a i λ e o i , p f o n 0 s s e t = t l I l u r o l ffi 1 T k s s , v fi t c w c b e i n s e t g t 1 n r e a t s H a 1 n i . l p s r c e l a I i e n i l r 0 ) p g i s . o + n a r E n m a c n k i h i o B e s i h n ) s v i m t i A w n n l i n b n S e t e o l l 5 . l o e α 0 e d e . t u i i x g t r o t o t s i l n t a a a i y s l e t e e l O t m r - s h o z d i t s t d g 0 . r n t e c e t h s 4 s a . n i ) λ e e u y i h i t ) t o n d r t e n H g , e o s n ρ o t ( × h i e u H h A ) 2 u b T h ( r t a e l c e h u t p t t n e a a . e h C , d n s t t n i , g i p e = d n s - C r o k e t e u i o o l u t i o l i x " i e h u a p n A d c . i a y e a g s e u a e . e , 5 E . s d c i c ( r y ρ . n v . e a n e A a n h o i s z n c = t e a f g n τ p y r r v s b a l S e r s e t d s AHE r c p d d e o e u l n n - a i t B e s B , s r r i × d n h 0 r l I l e a N u M a e 1 n v i w s 2 v r m t a t , d d n d fi t t t a ) i i a e s e l b l i c e i ρ n c d r x , d l n t o i H d m H , C e n e s e v y e 0 n h t e n t 2 e n . r t r , e e d t i T , e h e d o l 3 y c t o u e c L C s t i v d e ) d l A p I r n e t i S x e m h c r o H fi l i i ( r s s . s n h b r a g s K e m t g e g ‡ o h t o a c u i a t p a t d ρ o t i r o t s 5 . ρ d r t t w 1 t n a r , . e e E e 2 t A v d g e i y h n t e o , i u g m R e a H G h e n h s f h o a y r e a i r a h s r o r s y i a u 2 h o i c e s s r n b a 0 t 8 f r 0 x " h o e r h e , t W t n S r e a h J e s n v T ( g o v , a a t e . m e t s n e b i i i w e a l 2 n l s d i J E S a i e c e a y . e h = d i α y i ) n i e σ e F t t a l n s t y m e s s v n e p o i n H k i 0 a t r i e s n b e S a e d ) t g n l y i 5 o b c ρ t r K w A f c y e e e H t d t l t s t e l n i c w r e g e H t t e g " ( e d r g h i t r fi a a e t t i c N H u g t n i l b l h i i d i w i e a e b h e x t l i o i e v l r e n y d e n 4 a e w i h v m g a T s s r n h y R y i i g n a e n m y , f l l e t . c e v n r d e h r m p n o s m n r n o ( a e . w r − e a t u i v c u ( 4 i i a f l e s o n t e e s e c e S a t h . s i a t t ( t m e o a s t e d o r l d H z H l l s − s ) 0 g b e r l e o λ n u | o t B a A p g u a a n w a r = n y l p e x 1 ( u a m u c i e t a , c a H c m i n a t ρ r m h n , O e o a t v r e e B t t t x ` r v . i a e m t l a c m i h 1 e 2 E s p l B a h s r p s o o s s o o fi t m e d d t o i l a s e e e m i H a n l t a r i i x e y e B t l o r r l m r a e ) t a e m i ) d t h r e 1 a A l p h f e r d w R e l i h t s n u M e n s n f n a e s p v t d a u r r o q . s o l h e e e t a g o y + e s l f e a i l A ) e . s s e c p d e s . e o s r d h e y a e i a 2 A t e o d e a x ( r s w e 4 p e f t c s n s ) t t e o s d r - - - - - t - h s o | , . . . h v o t a o u r d u r T p 1 t c a e t t t a t o , u - x i e 1 l i t i e p h u 0 H K e t e t e e i i v I l i ffi 1 m ( ( e i 2 s , t e r e w 6 l d i i n e g l n / o s . t n i T h e e c c l r i l n 2 1 a s a n c o p v e d e o l s t . r m n ) E h i ∗ s f B L i ) d d h d τ v n y k y w o g 0 e b n e e e l s ) s ) s t t . r r - - - - - e , , . , s a t t m s i t 0 . r c e i e n o h 2 ( r l h t t a C g i p d i l i p A y e a e e 5 c i ( a n A o z c e r l r e s u s B d n 0 r l e a w t a t n t a ) e ρ n l i H C e r r t i , o y s t

v d ) d x

c i s

a g K m t

ρ t i

. r t n

i y

i u

R e

a e

a s

i 2 h

s 0

x "

, S

s y v

i s

b a e i b

K w

t s e ( g

i a t i i l i e a b l i e r n e n 4 i g T n y R y g n l e . e v s n e − c u i n S h . a m a H l s ) 0 g o | o u w e x ( c i a a ρ n O a t B m t m i 1 E s B o s o e d d s e m l i x y t m A e w t M n n d a r q l e e e a y + i ) d h y a 2 x e s s t t ------| , . . . • 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.) various explanations of the anomalous Hall effect • lntrinsic dissipationless antisymmetric part of the conductivity tensor of the ideal periodic material (Karplus- Luttinger term) • Magnetic “skew” and “sidejump” scattering from impurities (or inhomogeneous textures of the ferromagetic order parameter), so amplitudes for spin-orbit scattering to “left” and “right” (determined relative to vF×S) are inequivalent (violate so-called “detailed balance” )

In different regimes of temperature and purity, either of these mechanisms may dominate. In many systems, the controversial Karplus-Luttinger mechanism dominates. Physical origin of Berry curvature in Ferromagnetic bands • In a naive Stoner-type theory (neglecting spin-orbit coupling) of ferromagnetic metals, the bands are “exchange-split” into bands of “majority” and “minority” spin carriers. • In this picture, the majority and minority spin Fermi surfaces are independent, and can intersect:

↑and ↓Fermi even though weak, SOC surfaces intersect dominates near “avoided intersections” of the Fermi surface, where it causes i s rapid variation of i s quasiparticle spin with kF without spin-orbit coupling

t -

Berry curvature due to spin rotations:

J : (

• g-dependent spin direction: Ω ˆ (g) S (g) = Ωˆ (g) ∂ Ωˆ (g) ∂ Ωˆ (g) Fµν 4π · µ × ν • The Berry phase accumulated as a spin-S rotates is S times the solid angle enclosed by the path of its direction Ω on the unit sphere. • (Here “g” is position on the Fermi surface, S= ½) 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) write magnetic ﬂux density as an antisymmetric tensor

Karplus and Luttinger 1954 Note the “anomalous velocity” term! (in addition to the group velocity) • 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 }). • A useful way to write the semiclassical dynamics: c !abcB (r)

b b (e/!)Fab(r) δa d r aV (r) ! a −ab = ∇a δ (k) dt kb εn(k) ! b F " ! " ! ∇k "

b [ka, kb] [ka, r ] aH(r, k) i a a b ∇a − [r , kb] [r , r ] H(r, k) ! " ! ∇k " commutators of variables (symplectic form, Poisson brackets) H(r, k) = εn(k) + V (r) determinant (Jacobian) of the symplectic form : c ab eB (r) modifies phase space det . . . = 1 + !abc (k) | | F ! volume integral ! " (will use later) 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. The DC conductivity tensor can be divided into a symmetric Ohmic (dissipative) part and an antisymmetric non-dissipative Hall part:

ab ab ab σ = σOhm + σHall

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) 2 ab e 1 ab σHall = n (k)Θ(εF εn(k)) ! VD F − !nk 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)

e2 1 TKNN formula σxy = ν ν = an integer(2D) ! 2π 2 ab e 1 abc σ = # Kc K = a reciprocal vector G (3D) ! (2π)2

In 3D G = νG0, where G0 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! 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    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! • 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!) 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 demonstrated in an explicit “graphene” model shown at the right.). • Electronic states are “simple” Bloch states! (real first-neighbor hopping t1, complex second- iφ neighbor hopping t2e , alternating onsite potential M.) 2D “graphene” bandstructure

( ,xIA ( ,xIA Xt -$ 'KXt -$-A A B 'K tr iv& E -A I iv& Breaking either tr inversion (I) or B A E I time-reversal (T) two distinct “Dirac points” symmetry opens a (at corners of hexagonal “mass gap” at Brillouin zone) Dirac points.) A B E E E k-space

Dirac Break onlyT: mA = mB points EF same sign Berry curvature EF near A and B points

Break only I: mA = -mB ( ( densityIA of states ( opposite sign Berry curvature ,x IA massive case massive case (massless),xIAXt,xXt-$(bulk insulator) (bulk metal) near A and B points 'K 'KXt-A-$-A-$ 'Ktr iv& -A E tr iv&I tr iv&E I ( E I ,xIA Xt 'K -A-$ tr iv& E I • Intrinsic (Karplus Luttinger) Hall conductivity interpolates between quantized Hall conductance from edge states

quantized (0) non-quantized (AHE) gap quantized (1)

non-quantized (AHE)

A r quantizedJ (0) Graphene model with second neighbor hopping is very useful!

• Quantum Hall effect with simple Bloch states • Used for anomalous Hall effect studies (Nagaosa), add disorder etc. • used for testing/developing fundamental band- stucture formulas for orbital magnetization (Vanderbilt) • Quantum Spin Hall effect (Kane and Mele) • Analog system for photonic edge states (Haldane and Raghu) • graphene edge states (zigzag edge) edge a!) 4\4\ {-f /7ffi a.AV 19" )z / \

broken T

k E I I I ! I I I broken I

gapless case I

I I 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”:

Γ1 integral of Fermi vector Berry phase around 2 S Γ weighted by Berry FS intersection with curvature on FS BZ boundary

Γ2 This is ambiguous up to a reciprocal vector, which is a non-FLT quantized Hall edge-state contribution 3 Γ Γ1 3

0.80 400,000 K-points 0.78 )

1,000,000 y R 0.76 800 2,000,000 ( y EF g

T = 0 K r 0.74 e

T = 300 K n E

) 0.72 1 -

m 0.70 3 c

# (

700 4000 y x ) 0.80 s " t i

400,000 K-points n 0.78 ) u y c 1,000,000 i 2000 R m 0.76 (

800 2,000,000 o t y E

a F g

T = 0 K (

) 0.74 e k n 600 T = 300 K ( z 0 E # ) 0.72 - 1 -

m 0.000 0.002 0.004 0.006 0.008 0.010 0.70 c N $ P N # $ H P N $ H ( ! (Ry) (0,0,0) (1,0,0) 1/2(1,1,1) 1/2(1,1,0) (0,0,0) (0,0,1) 1/2(1,0,1) (0,0,0) 1/2(1,1,1)1/2(1,0,1) 700 4000 y x ) s " t i n u c FIG. 2: Anomalous Hall effect vs. δ with different numbers FIGi .2030:0 Band structure near Fermi surface (upper figure) and m

z o of k points in full Brillouin zone. Here δ is introduced by Berryt curvature Ω (k) (lower ﬁgure) along symmetry lines. 2 a (

adding δ to the denominator in Eq.(4). The solid lines were ) k

600 (

z 0 obtained by an adaptive mesh reﬁnement method. # H- (001) $(101) 0.000 Firs0t.00P2 rincip0.0l0e4s Calc0.u00l6ation 0o.0f08Anom0.a01l0ous Hall Conductivity in Ferromagnetic bcc Fe

$ H P N $ H N $ 5 P N distribution) and a slightly smaller value of σxy = 734 5 10 ! (Ry) (0,0,0) (1,0,0) 1/2(1,1,1) 1/2(1,1,0) (0,0,0) (0,0,1) 1/2(1,0,1) (0,0,0) 1/2(1,1,1)1/2(1,0,1) (Ωcm) 1 at room temperaYtuugruei (Y3a0o01,2K,3),.LO. uKrlerinesmualnt1i,sAin. H. MacDonald1, Jairo Sinova4,1, − 4 T. Jungwirth5,1, Di1ng-sheng Wang3, Enge Wang2,3, Qian Niu1 4 10 fair agreement with the val1ue 1032 (Ωcm)− extracted Department of Physics, University of Texas, Austin, Texas 78712 3 from Dheer’s da2Itnateornnatiiroonnal wChenistekrerfosr[Q21u]anattumroSotmructuerme,pCehiner- se Academy of Sciences, Beijing 100080, China 3 10 FIG. 2: Anomalous Hall effect v3s. δ with different numbers FIG. 3: Band structure near Fermi surface (upper figure) and ature. Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China 2 z 2 10 of k points in full Brillouin4Dzeopnaret.menHt eorfePδhysiiscs,inTterxoads uAc&eMd UbnyiversityBe, CrryollegceurvStaaturetion, TXΩ77(8k4)3-(4l2o4w2 earndfigure) along symmetry lines. 2 The slow convergen4ce is caused by the appearance of adding δ to the denominator inInsEtiqtu.t(e4o)f. PThhyseics oAliSdCRli,nCeuskwrovearrenická 10, 162 53 Praha 6, Czech Republic 1 1 large contributions to Ωz of opposite sign which occur 10

(Dated: June 22, 2006) obtained by an adaptive mesh refinement method. in very small regions of k-space. Spin-orbit effects are 0 H(001) $(101)0 small except whenWtehpeeyrfmormixastfiarsttesprtinhcaiptlews ocuallcdulaottihonerowf itshee anomalous Hall effect in ferromagnetic bcc Fe. cond-mat/0307337Our theory identifies an in trinsic contribution to the anomalous Hall conductivity and relates it to 1 -1 -10 be degenerate orthneae k-rslypacdege Beernerry pahte,ase aondf ocevcupeniedthen,Bloch tshotatsees. The theory is able to account for both dc and Phys. Rev. Lett. 92, 037204 (2004) 2 distributmioixne)d asntadteas wsliimlglahgctnolenytotr-soimbptuiactaellleHnraelalvrcaloylnudceuactnoivcfietiσleinsxgwy itc=honot7ra3idb4jus-table parameters. -2-10 5 105 1 z (Ωcm)− at room temperature (300 K). Our result is in 3 tions to Ω . Only when the Fermi surface lies in a spin- -10 4 PACS numbers: 75.47.-m, 71.15.-m1, 72.15.Eb, 78.20.Ls -3 4 10 fair agreoermbitenintdwucietdh gtahpeisvtahleuree a10la3r2ge(cΩocnmtri)b−utioenx.trTahcitsecdan from Dhbeeerseen’s daintaFigon. 3irwohern wehtheiskBerersr[y21cur] avtatrurooemalotngemlinesper-in 3 103 In many ferromagnets the Hall resistivity, ρH , exhibits tested, and that the intrinsic contribution initially pro- k-space is compared with energy bands near EF and in ature. (000) 2 an anomalous contribution proportional to the magne- posed b$y Karplus and Luttinger has been discountHed.(100) 2 10 The sFiglow. ct4iozwantvherioenregoeitfntcisheecoimsmpaactearuiraedsl,ediwinbthayddttheihtioenianttpoersectionptehaeraorndcineoafroytfhe Several years ago, the scattering free contribution of Fermi surface with the central (010) plane in the Brillouin 1 1 large contribcuonttiroinbustiotno pΩrozpoortfioonaplptostihteeaspipglined wmhagicnheticocfieclud,r Karplus and Luttinger was rederived in a semiclassical 10

zone.ρH = R0B + Rs4πM [1–3]. The anomalous Hall effect framework of wavepacket motion in Bloch bands by tak- in very small regions of k-space. Spin-orbit effects are FIG. 4: (010) plane Fermi-surface (solid lines) and Berry cu0r- In (oArHdEer) htoas fpulratyhederanunimdpeorsrtaanntdrotleheinrtohleeinovfesstpiginat-ioornbit ing into accounzt Berry phase effects [8, 9]. According to 0 vature Ω (k) (color map). Ωz is in atomic units. small exceptawndhechnartahcteeyrizmatioxn sotfaitteinsertahnat telwectoruonldfeorrtohmeargwneistse this work, t−he AHC in the scatterin−g free limit is a sum of 1 coupling in the AHE, we artificially varied the speed of -1 -10 be degenerlighatte,btehcoaerurseneaebyRrsclyhisandegugsuinaegllnerythateate,lsepasinta-ndoonrebeviotrdceneoruthen,poflinmgagsnttihorteundsegeth Berry curvatures (see eqs.(2) and (6) below) of the occu- 2 Thisla r2calculationger than the o rsampleddinary con ALLstant Rstates0. Alt hbelooughwth etheeffe ct pied Bloch states [10]. Recently, Jungwirth et al. [11, 12] -2-10 mixed sξtatecs−w.ilAl scsohnotwrinbiunteFing.ea5r,lσyxycaisnlineacelinrgincoξnftorribsmu-all ∝Fzermihas bleeevnelr e(unnecessarcognized for myor ewtork!)han a cbutentu rshoy [2ws] it is still appwherlied tehis& pisictaurpeoosfittihvee AinHfiEnittoes(iImII,aMl.n)IVn ftehreroumpapg-er panel in tions to cΩoup. liOngn,lybuwthneont tfohre lFaregremciosuuprlifnagc.e Flioers iirnona, snpoinnl-in- 3 somewhat poorly understood, a circumstance reflected netiFcigse.6m,icwoendsuhcotworsreasnudltfsoufonrd tvheeryimgaogodinaargyreepmaernttof ωσ a-3s-10 eahoritiwes abveoidedcome sFiermignific asurfacent for ξ intersections/ξ > 1/2, wh ich means xy orbit inducedbygtahpe ciosnttrhoevreersiaal laanrdgesomcoentimestr0ibcoutnfusingion. Tliterhisatcurane witha feuxnpecrtiimonenot.f f(rIeIqI,uMenn)cVy ftehrarotmaargenientsaagrreeeumnuesnutalw, ith earlier thpratotovidehnethsep theisnu-bo jrdominantebcti.t iPnrteevriaocu contributionstsiothneoinretiircoanl wcoar nktonho atthesbfae iKLlterde atoted however, because they are strongly disordered and have be seen in Fig. 3 where the Berry curvature along lines in calculations[23]. Experimental results [24] are in excel- pertuerxbpalatinvetlhye. magnitude of the observed effect even in well extremely strong spin-orbit interactions. In this Letter, k-space isformcomula.pared with energy bands near E and in lent agreement below 1.7 eV but become smaller at higher So unfadrerswtoeodhmavaeterdiailsscluikseseFde [4o]n. ly the dc-AF HE. It is we rep$o(r0t0o0n) an evaluation of th e intrinsic AHC in Ha (100) Fig. 4 where it is compared with the intersection of the clasesnicetrrgainesi.tiIonn tmheetallofwererrompaangneelto, fbtchceFefi.gOuruer,ctahlceur-eal part of straightKfoarpwluasrdantdoLeuxttienngdero[5u]rpcioanleceurleadtitohne tthoeotrheeticaacl iHn-all latiothneisHbaslledcoondsupicnt-idveintysi,tyobfutnacintieodnaflrtohmeortyhaenidmtahgeinary part Fermi sucrafsaecebveywstuigtsahintigtohntehofecteKhnisutbreffaoelcf(to0,rb1my0u)slhapowl[a2i2ng]eahiponwptrshopaeincB-ho:rbililtocuouin- LAPbWy amKethraomd.erTs-hKe rcolonsieg argerleaetmioennt, bisetwsheoenwntheaosrya function zone. pling in Bloch bands can give rise to an anomalous Hall anFdIGexp. er4:imen(01t0)thaptlawnee findFermleai-dssurufsatcoe c(osnocludelid linthaest) and Berry cur- conductivity (AHC)e2 in dfer3kromagnetic crystals. Their con- of frequency. The dc limit result, σ(ω = 0)xy = 750.8 In order to further understand the role of spin-orbit the AHC in 1traznsition metal ferromagnets is intrinsic in σ(ω)xy = (fn,k fn!,k) (7) va(Ωtucrme) Ω, i(ske)ssen(coltoiarllymaidenp). ticaΩlz tios itnhaattoombtaicinedunitsfr.om clusion was questio¯h ned(2bπy)3Smit [6], who argued that Rs − coupling in the AHE, we aVGrtificially varied−the speed of origin, except− possibly at low-temper−ature in highly con- must vanish in a!periodic lantt=icne!. Smit proposed an al- Eq.(6). Despite the small discrepancy with theory in the "" ductive samples. light, therebytercnhataivnegminecghatImnhisemψ,ssnpkkienwv-oxscraψbttnietrkicnog,ψuinplwkinhvigcyhssψtprninek-norgbtiht dc limit, the experimental point [21] seems to agree 2 ! ! We begin our discussion by briefly reviewing the semi- coupling causes spi$n pola|riz|ed el2ec%t$rons to| be|2scatt%e,red • ξ c− . As shown in ×Fig. 5(,ωσnxy isωnlinea) (rωi+n iξ&)for small classicarathletrrawnspelol rwt theoith rtyh.eBoyviencludingrall trenthed oBfertrhyephafresequency de- ∝ preferentially to one side! by impurities. The skew scat- where & is a positive infinitesimal. In the upper panel in coupling, but not for large coup−ling. −For iron, nonlin- correction to the group velocity [8, 9], one can derive the tering mechanism predicts an anomalous Hall resistivity Fig.6, we show results for the imaginary part of ωσ as earities become significant for ξ/ξ > 1/2, which means following equations of motion: xy linearly proportional to the 0longitudinal resistivity; this a function of frequency that are in agreement with earlier is in accord with experiment in some cases, but an ap- that the spin-orbit interaction in iron cannot be treated 1 ∂εn(k) · calculatioxns[2· c =3]. Experimenk Ωnt(akl),results [(214)] are in excel- perturbativeplyr.oximately quadratic proportionality is more common. ¯h ∂k − × Later, Berger [7] proposed yet another mechanism, the lent agreem. ent below 1.7 eV but become smaller at higher So far we have discussed only the dc-AHE. It is ¯h k = eE e x·c B, side jump, in which the trajectories of scattered elec- energies. In the− low−er pa×nel of the figure, the real part of straightforwatrrodnstsohieftxttoenonde soidueractaimlcpuulraittyiosnitestobetchaueseaocf sHpianl-l wherthee ΩHnalisl tchoenBdeurrcyticvuirtvya,tuorebtoaf itnheedBlofrcohmstattehedei-maginary part case by usingortbithecoKuplingubo. foThermsuidela j[ump22] ampecpharonismach:does predict fined by a quadratic dependence of the AHC on the longitudi- by a Kramers-Kronig relation, is shown as a function nal resistivity. However, because of inevitable difficulties e2 d3k of frequΩenn(kc)y.= TImhe dkcunlkimit rkeusnuk lt, , σ(ω (=2) 0)xy = 750.8 in modeling impurity scattering in real materials, it has 1 − #∇ |×| ∇ % σ(ω)xy = (fn,k fn!,k) (7) (Ωcm)− , is essentially identical to that obtained from not be¯hen possib(l2eπto)3compare quant−itatively with experi- with u being the periodic part of the Bloch wave in the VG n=n nk ment. It !appears to us "t"hat! the AHE has generally been ntEh qb.a(n6d).. WDeeswpililtbeetihnteersemsteadllinditshcerceapseanofcyB w=it0h, theory in the regardedImas aψn enxktrivnxsicψeffneckt duψe snolkelyvtyo imψpnukrity scat- fodr cwhliicmh iεtn,(kt)hies juesxtptehreimbaenndteanlerpgyo.inTthe di[s2tr1ib] us-eems to agree $ | | ! % $ ! | | %, • tering×, even th(ωough thiωs n)o2tion h(aωs +nevie&r)b2een critically tiornatfuhnecrtiowneslaltiwsfiietshthtehBeolotzvmearnanlleqturaetniodn woifthththee frequency de- n! − n − • The new Fermi-surface Berry curvature formula suggests a different - intrinsic - way to think about Fermi surface geometry in Fermi liquid theory..... Fermi surface of a noble metal (silver):

(b)

Abstract view of the same surface (and orbits) as a compact manifold (a) of quasiparticle states (with genus g = 4, G conventional view as a surface in “open-orbit dimension” d = 3). the Brillouin zone, periodically repeated in k-space Dimension of Bravais lattice of reciprocal lattice vectors G corresponding to k-space De Haas-Van Alpen effect allows extremal cross-sections to be experimentally displacements associated with periodic open determined orbits on the manifold. Ingredients of Fermi-liquid theory on a Fermi-surface manifold k-space geometry kinematic parameters k-space metric inelastic Fermi vector mean free path direction of renormalization Fermi velocity factor Hilbert-space geometry Hilbert-space metric Berry gauge ﬁelds: Z(2) + SO(3) { U(1) Fermi surface NEW spin degeneracy quasiparticle energy parameters quasiparticle coordinate: manifold coordinate (d=2) Fermi speed

quasiparticle Landau functions magnetic moment spin coherent-state coupling pairs of quasiparticle states direction Quasiparticles “live” only on the Fermi surface.

• This leads to a 5-dimensional symplectic (phase space) structure: • 3 real space + 2 k-space • 2 pairs + 1 “chiral” unpaired real space direction at each point on the Fermi- surface manifold • the unpaired direction is the local Fermi velocity direction. Physical signiﬁcance of “Hilbert space geometry”

Hilbert-space metric if both spatial inversion and time-reversal symmetry Berry gauge ﬁelds: are present* Z(2) + SO(3) { U(1) otherwise* * assumes spin-orbit coupling

• The Hilbert-space metric and the Berry gauge fields modify the ballistic behavior of quasiparticles which are accelerated by quasi- uniform electromagnetic fields, chemical potential and thermal gradients, strain fields, etc. • Hilbert space geometric effects are completely omitted in a single-band approximation that also neglects spin-orbit coupling (like a one-band Hubbard model). “intrinsic” picture of Fermi surface as an abstract 2-manifold

• “homology group” has a basis of G (=genus) pairs of non- trivial paths (that don’t cut the manifold in half). Only members of the same pair intersect. Open orbits (along which kF increases by a reciprocal lattice vector G ) are non-trivial. two different ways to view a genus-2 Fermi surface of “open-orbit dimension” dG=2

!! !! !!

!" !" !" #" #! !! #" !! #! !"

!! !" !" Standard “schema” (homology group) for Fermi surface manifolds: k-space images dG=0 dG=1 dG=2 dG=3 G2 G2 G1 G3 G1 G1

a 2-manifold of vF G genus g has g conjugate d =2, pairs of elementary G1 chiral non-trivial closed G2 G1 paths (homology G2 group generators) G1 normal case: the first dG pairs of generators G2 couple a closed-orbit “zone chiral (quasi-1d) case: boundary” with an “open orbit” the first pair of generators are both “open orbits” AND dG =0,1,2, or 3 is the dimension of the Bravais lattice of “zone boundaries.” reciprocal vectors generated by “open orbits”. (If chiral quasi-1D Fermi surfaces are present, dG =0,1,or 2) • 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 ): Streda formula • the Hall conductance (linear response of transverse electric current density to electric field) also describes linear response of electron density n to magnetic flux density

∂n e K = ∂B h 2π !µ,T =0 • Xiao, Shi and Niu! (2005) note that ! Fermi ! occupation factor eBa n = d3k 1 + ! bc n(k) abc F ! " ! # • Fermi surface sheets with non-zero chern number (total Fermi surface Chern number must vanish, but individual pieces can have non-zero Chern number)

t-5 q I

B'fer-: ;%\-lz#

(= -l Ch+.n N.^,".b4,.,. C=t\ Wo-M, tr'o',[ *- I • As in one dimension, each distinct piece of the Fermi surface has its own “adiabatic conservation law” in the low-T limit, in the absence of large-momentum-transfer scattering processes. • Pieces of FS with non-zero Chern number only have such a conservation law as a group with zero total Chern number: charge can be “pumped” between them through the “wormhole” that connects them! Application of formula to a composite fermion is modeled composite fermion Fermi as an electron laterally displaced from the center of the m-vortex surface at ν = 1/m that is bound to it.

2π 1 "2 × m Berry phase for moving Flux enclosed by path composite quasiparticle of displaced electron around Fermi-surface around vortex: e2 1 independent of σxy = AHE h m Fermi surface shape! • 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. 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

exact (interacting) T=0 propagator (PBC, discretized k) 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. • The fundamental Luttinger (1961) theorem relating the non-quantized part of the electron density to the Fermi surface volume now has a “partner”. (in fact, its derivative w.r.t. B)