Non-Abelian Berry phase and topological spin-currents

Transcription

1 Non-Abelian Berry phase and topological spin-currents Clara Mühlherr University of Constance January 0, 017

2 Reminder Non-degenerate levels Schrödinger equation Berry connection: ^H() j n ()i = E n j n ()i = ( 1 ; : : : ; r ): set of parameters non-degenerate state j n ()i for any Berry phase along C: n (C) gauge invariant, but A n = i h n ()j@ ()i n (C) = C A n () d j n ()i! e i() j n ()i : A n ()! A n () ()

3 Reminder Non-degenerate levels Berry curvature: A n () d = B n ()d 1 d F B n () gauge invariant explicit form: B (n) A(n) () holds if [ A (n) ] (); A (n) () = 0! for Abelian gauge elds, i.e. non-degenerate levels

4 Occurrence of degeneracies In physics we can not avoid degeneracies, for example: Kramer's theorem We have to deal with degenerate states ) non-abelian gauge theory

5 Non-Abelian Gauge Theory Gauge elds in presence of degeneracies time-dependent Schrödinger equation H((t)) j n (t)i j n(t)i m n degenerate sublevels in subspace H n (t) set of parameters, varied in time (0) = i adiabatic theorem: By varying (t), subspaces do not cross each other j n;m (0)i H n are mapped to j n;m 0(T )i H n! (T ) = f

6 Non-Abelian Gauge Theory Gauge elds in presence of degeneracies j a (t)i set of basis functions of subspace H n choose locally H((t)) j a (t)i = 0 unitary U(t) maps those solutions to functions j a(t)i H n j a(t)i = U ab (t) j b (t)i ; j a(0)i! = j a (0)i j a(t)i remain normalized ) h bj@ t ai = 0, (U t U) ba = h b j@ t a i A ab A(t): gauge potential, depends on geometry of subspace n

7 Non-Abelian Gauge Theory Gauge elds in presence of degeneracies Gauge potential A ab = (U t U) ba ) U(t) = Pexp A( )d arbitrary transformation R(t) SO(m n ): j a (t)i! R(t) j a (t)i t R(t)R 1 (t) + R(t)A(t)R 1 (t) U(t)! R(t)U(t)R 1 (t) ) eigenvalues of mapping U : H n! H n between degenerate sublevels are gauge invariant, i.e. potentially observable m n = 1: U(t) = e i adds a simple phase factor

8 Non-Abelian Gauge Theory Gauge elds in presence of degeneracies Generalize gauge potential A(t) to parameter space M, where dim(m) = r: = ( 1 ; : : : ; r ) ) A T = i = h j@ j i Loop in parameter space ( i = f ) ) mapping between degenerate sates: Wilson loop U = Pexp A n d non-degenerate levels (m n = 1): U! Berry phase

9 Non-Abelian Gauge Theory Gauge elds in presence of degeneracies What is the curvature of A in case of existing degeneracies? non-degenerate levels: B (n) () = more A(n) () B = [D ; D ] D A : covariant derivative (D! R()D ) ) B = [D ; D ] = [@ A A ] A [A ; A ]

10 Dissipationless Quantum Spin Current at Room Temperature Degeneracies in real insulators Consider Si, Ge, GaAs, InSb Top of VB: P = levels diamond structure, inversion symmetry, rotational symmetry ) terms in Hamiltonian: k and (k S) ) eective Luttinger Hamiltonian for holes ) H 0 = (( ~ m )k (k S) = H 0 (k) 1= : Luttinger parameters de.wikipedia.org/wiki/galliumarsenid

11 Dissipationless Quantum Spin Current at Room Temperature Dispersion relation ) H 0 (k) = (( ~ m )k (k S) diagonal in helicity basis ( = ^k S) eigenenergies ( ( E (k) = ~ k m ) ) ~ k m Science 05, (00) P =! S = 1 ;! = 1 ; ) light and heavy holes: LH and HH band 4-fold degeneracy lifted for k 6= 0

12 Dissipationless Quantum Spin Current at Room Temperature Eect of electric eld Introduce potential V (x) = ee x ) H = H 0 + V (x) Diagonalization of H 0 (k), k = k(sin cos ; sin sin ; cos ): rotate by R(k) = exp(is y ) exp(is x ) ) R(k)(k S)R y (k) = ks z ( ) ) H ~ = ~ k m S z + R(k)V (x)r y (k) x = i@ k! transformation gives V ( ~ D) ~D = i@ k + ir(k)@ k R y (k) = i@ k ~A ~A: pure gauge potential (F ij = i[d i ; D j ] = 0)

13 Dissipationless Quantum Spin Current at Room Temperature Pure gauge potential ~A Spin =-matrices Sx = p p 1 1 p p ; Sy = i p ip i p i i p ; Sz = 1 1 ~A dk = p cos d' (sin d' + i d) p (sin d' i d) 1 cos d' sin d' + i d sin d' i d p 1 cos d' (sin d' + i d) p (sin d' i d) cos d'

14 Dissipationless Quantum Spin Current at Room Temperature Approximations Adiabatic Approximation: Neglect interband transitions (o-block-diagonal matrix elements of A) ~ lll ) A 0 dk = llll existing degeneracies! A 0 non-abelian Abelian approximation: neglect o-diagonal elements ) A 0 A = S z cos d ^= Dirac monopole at k = 0, strength eg given by S z return to helicity basis ) A A = R y (k)a 0 A R(k) + kr y (k) R(k) lll

15 Dissipationless Quantum Spin Current at Room Temperature Topological invariants Non-trivial gauge connection A A in helicity basis ) non-vanishing curvature ^= eld strength (monopole eg = ) F ij i[d i ; D j ] = " ijk k k k reminder: H e = ~ k + V (x) m and x i = D i = i@=@k i A i (k) non-trivial commutation relations ) equations of motion [k i ; k j ] = 0; [x i ; k j ] = i ij ; [x i ; x j ] = i F ij ~ _ k i = ee i and _x i = ~k i m + F ij _k j

16 Dissipationless Quantum Spin Current at Room Temperature Topological term in equation of motion _x i = ~k i m + e ~ F ij E j topological term F ij = i [ D i ; D j ] = ijk k k k ) noncollinearity of velocity and momentum ^= Lorentz force in momentum space real-space trajectory: shift perpendicular to S ) spin current perpendicular to S and E Science 05, (00)

17 Dissipationless Quantum Spin Current at Room Temperature Spin current in AA Dene: electric eld in z-direction, spin parallel to x-axis ) spin current in y-direction j x y for heavy holes: j xh y = ~ = =;k _y S x n (k) = ~ n (k) : lling of holes in band = =;k _y k x k n (k) use _y = F y j _k j = k x ee z k ~ and S x = k kx analogous for light holes ( = =! = 1=) ) j x y = j xh y + j xl y = ee z 6 (9k H F + k L F ) (T = 0)

18 Dissipationless Quantum Spin Current at Room Temperature Non-Abelian corrections Like in AA: heavy holes: j xh y = ~ = =;k _y k x k n (k) _y = F y j _k j Eect of non-abelian [ gauge connection: ] F ij = i D i ; D j i j )F ij = ijk [ j i + i A i ; A j ( 7 ) kk k correction factor of only in the LH band ) spin current j x y = ee z 1 (k H F k L F )

19 Dissipationless Quantum Spin Current at Room Temperature Spin conductivity Quanten Hall Eect: Generalization to 4 dimensions: electric eld E induces SU() spin current j i = i E (; = 1; ; ; 4; i = 1; ; ) i : t0 Hooft tensor : dissipationless transport coecient restriction of E and j i ) dissipationless response to -dim. subspace: j i j = S ijk E k S : spin conductivity

20 Dissipationless Quantum Spin Current at Room Temperature Spin conductivity: p-gaas j i j = S ijk E k spin current including non-abelian corrections j x y = ee z 1 (k H F k L F ) = ~ e SE z ) expression for spin conductivity determined purely by gauge curvature in momentum space independent of mean free path or relaxation rates nite temperature: modication only by Fermi distribution function n (k) E HH LH 0:1 ev 0:05 ev ) eect of same order at room temperature for n = cm : S O, for n = cm : S > O

21 Experimental Setups Detection of spin current E; J z k ^z ) j x y Science 05, (00) A) Electric transport ferromagnet attached to +y magnetization M along x lead between +y and y measure I(+x)=I( ) B) Polarization of light quantum well structure sandwiches by p-and n-gaas recombination with e emission of + polarized light

22 Experimental Setups Literature S. Murakami, N. Nagosa, and S.C. Zhang, Science 05, (00) F. Wilczek, and A. Zee, Am. Phys. Soc. 5 4 (1984) E. Laenen, NIKHEF, nikhef.nl/45/ftip/