Berry-phase Approach to Electric Polarization and Charge Fractionalization. Dennis P. Clougherty Department of Physics University of Vermont

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1 Berry-phase Approach to Electric Polarization and Charge Fractionalization Dennis P. Clougherty Department of Physics University of Vermont

2 Outline Quick Review Berry phase in quantum systems adiabatic theorem, spin-1/2 in B, analogies Semiclassical dynamics Berry connection, curvature, LDOS 1D dimer model: fractionally charged kinks Rice-Mele, Su-Schrieffer-Heeger Fractionally charged vortices in 2D dimer model

3 Adiabatic transport λ 2 H = H(λ 1, λ 2 ) H i = H f ψ n (T )=ψ n (0)e i R T 0 dt n(t) e iγ n γ n = C A d λ C λ 1 γ n = i T 0 Berry phase ψ n dψ n dt dt A = iψ n λ ψ n Berry connection

4 Spin-1/2 in B H = µ B(t) B(t) =B ˆn(t) µ S C B z θ B H ˆn(t); ± = ± ˆn(t); ± γ = i C d B ˆn B ˆn B y B x

5 Spin-1/2 in B d B = B sin θ dφ ˆφ A φ = i(cos θ 2,e iφ sin θ 2 ) 1 B sin θ γ = 2π 0 φ A φ B sin θdφ = π(1 cos θ) γ = Ω C 2 θ, φ ˆn = geometry of parameter space cos θ 2 e iφ sin θ 2 cos θ 2 e iφ sin θ 2 C

6 Berry curvature γ n = γ n = C S A d λ Ω dλ 1 dλ 2 Ω = λ 2 λ 1 A 2 λ 2 A 1 S C λ 1

7 Analogies Berry connection A j = iψ ψ λ j Magnetic potential A Berry curvature Ω ij = A j λ i λ j A i Magnetic field B = A γ = Geometric phase S A = S Ω Φ = Magnetic flux S A ds = S B d S Chern number S Ω =2π (integer) Monopole charge S B d S = hc e (integer)

8 Applications Berry phase Interference Energy spectrum Polarization in crystals Berry curvature spin dynamics electron dynamics in crystals fermion fractionalization Chern numbers quantum Hall quantum charge pumps topological insulators

9 Ashcroft+Mermin INCOMPLETE--missing terms due to Berry curvature

10 Semiclassical dynamics r-space k-space a x wave packet made of Bloch states Ψ = e ik r u n ( k) parameter space: k,r Ω kk αβ = A kβ H is k dependent A kα k α k β Ω kx αβ = A xβ Sundaram+Niu, PRB 1999 A kα k α x β Non-zero for broken inversion or time-reversal symmetries π a k π a

11 Modified equations of motion ẋ α = kα Ω kx αβẋ β Ω kk αβ k β Ω kt α k α = xα + Ω xx αβẋ β + Ω xk k αβ β + Ω xt α Example: anomalous velocity k = ee Sundaram+Niu, PRB 1999 r = k k Ω( k) transverse current in FM metals Ω α = αβγ Ω kk βγ

12 Modified DOS k V 1 V V t = r r + k k r Liouville s theorem breaks down D(r, k) = 1 (1 + Ωkr (2π) d αα) Xiao et al. 2006

13 Peierls Theorem H e = t 0 /2 j (c j c j+1 + H.c.) H e ph = α j (u j+1 u j )(c j c j+1 + H.c.) H ph = K/2 j (u j+1 u j ) 2 αu j+1 u j = t 1 2 ( 1)j u j u j+1 Bersuker and Polinger: Band JT effect

14 Kink soliton in SSH t 1 B x A domain wall between phases + topological solitons

15 Rice-Mele model H = j [ t 0 2 (c j c j+1 + H.c) H = k + t 1 2 ( 1)j (c j c j+1 + H.c)+ ( 1) j c j c j] h(k, t 1, ) σ maps into spin-1/2 problem h(k, t 1, ) =(t 0 cos k 2, t 1 sin k 2, )

16 LDOS as Form Factor t 0 sin 2 k 2 Ω kx = xt 1 4( 2 + t 2 0 cos2 k 2 + t2 1 sin2 k 2 )3/2 δn(x) = π π dk 2π Ω kx Density Theory Numerical Lattice

17 2 Kinks on a ring t 1 +

18 Q = dx δn(x) Soliton Charge dx dk = Ω kx 2π = 1 πt 1 t 0 π tan t π2 t 2 0 /4 = 1 t1 irrational fraction π tan 1, t 0 t 1, 1 2, 0 the SSH result

19 2D Dimer Model H e = ij (t ij e iθ ij c j c i + H.c.)+ i i c i c i Δ -Δ t(1+m y ) t(1-m y ) t(1+m x ) t(1-m x ) Hou, Chamon & Mudry, PRL (2007), Jackiw & Pi, PRL (2007) Chamon et al, arxiv: , Seradjeh, Weeks & Franz, PRB 77, (2008)

20 Non-Abelian Berry Curvature H = cos k x γ 1 + cos k y γ 2 + γ 3 + m x sin k x γ 4 + m y sin k y γ 5 H 2 I E(k) =± cos 2 k x + cos 2 k y + m 2 x sin 2 k x + m 2 y sin 2 k y + 2 H 4 4 matrix. Thus, double degeneracy Need to generalize to non-abelian case

21 Polarization Turn on vortex slowly: λ =0 m vanishes λ =1 m fully turned on Polarization change P x (1) = e [dk] dλ(ω ky yω kx λ Ω kx yω ky λ + Ω kx k y Ω yλ ) Q = dr P V

22 Fractional vortex charge Q !!!!!!!!! 0.1 Q = n e 2 ( m 2 )!/m m(r)e inθ = m x + im y

23 Conclusions New way to calculate inhomogeneous polarization with Berry curvature Fermion fractionalization can be understood as a Berry curvature effect

24 Acknowledgments Di Xiao, ORNL Junren Shi, ICQS Beijing Qian Niu, UT Austin

25 Outline Quick Review Berry phase in quantum systems adiabatic theorem, spin-1/2 in B, analogies Semiclassical dynamics Berry connection, curvature, LDOS 1D dimer model: fractionally charged kinks Rice-Mele, Su-Schrieffer-Heeger Fractionally charged vortices in 2D dimer model