Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona

Transcription

1 Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013

2 Continuous symmetries and conservation laws Noether s theorem If a system has a continuous symmetry, then there are corresponding quantities whose values are conserved in time. To every differentiable symmetry generated by local actions corresponds a conserved current. Ex.: spatial translational symmetry -> momentum conservation rotational symmetry -> angular momentum conservation time translational symmetry -> energy conservation...

3 Continuous symmetries and conservation laws Noether s theorem If a system has a continuous symmetry, then there are corresponding quantities whose values are conserved in time. To every differentiable symmetry generated by local actions corresponds a conserved current. Ex.: spatial translational symmetry -> momentum conservation rotational symmetry -> angular momentum conservation time translational symmetry -> energy conservation... In quantum mechanics : symmetries as unitary operators -> U + HU=H Ex.: spatial translational symmetry -> U = exp[-i P x] rotational symmetry -> U = exp[-i J φ]...

4 Discrete symmetries and degeneracies C : charge conjugation, q -> -q P : parity, (x,y,z) -> -(x,y,z) T : time reversal, t -> -t PH : Particle <-> Hole symmetry (with SC) SLS : Chiral / sublattice symmetry (Dirac, graphene...) Ex.: time-reversal symmetry -> twofold Kramers degeneracy (i) no spin : invert momentum -> complex conjugation, T = -i K, T 2 =1 (ii) spin 1/2 : invert momentum and spin -> T = -i σ y K, T 2 =-1!! T and PH are antiunitary operators : T[a φ 1 > + b φ 2 >]= a* Tφ 1 > + b* Tφ 2 > <Tφ 2 Tφ 1 > = <φ 2 φ 1 >

5 Symmetries and classification of states of matter Phase transition / spontaneous symmetry breaking Phase with broken symmetry : order parameter Classification with symmetry of order parameter Bulk properties! Table from K Binder, Rep Prog Phys 87

6 Outline Random matrix theory : can we guess something about apparently intractable problems only from basic symmetry considerations? General classification of quantum-mechanical systems (Hamiltonians or S-matrix) based on discrete symmetries : I. T-reversal II. particle-hole III. chiral Classification extended to topological phases of matter bulk topology generates boundary states

7 Random Matrix Theory: (i) Wigner Problem : excitation spectrum of heavy nuclei many-body problem; do not know Hamiltonian Solution : write Hamiltonian as random matrix Example : <H ij >=0, P(H) = exp{-β Tr[H 2 ]} ~ Gaussian ensembles ask that the ensemble is stationary under unitary transformation H -> H = U + H U

8 Random Matrix Theory: (ii) Dyson s 3-fold way The irreducible representations of a group by unitary matrices fall into three classes (...) real, complex and pseudoreal (quaternionic). -> three ensembles that are stationary under H -> H = U + H U P(H) = exp{-β Tr[H 2 ]} β=1: orthogonal ensemble of real symmetric H β=2: unitary ensemble of complex hermitian H β=4: symplectic ensemble of real quaternionic H

9 Random Matrix Theory: (ii) Dyson 3-fold way Extend the theory to unitary matrices S Three ensembles defined by invariance under β=1: Circular orthogonal ensemble β=2: Circular unitary ensemble β=4: Circular symplectic ensemble U, V are arbitrary unitary matrices, W is a real quaternionic unitary matrix (symplectic), W R = σ y W T σ y is the dual of W

10 Random Matrix Theory of Quantum Transport I in Conductance as transmission I out (Landauer, Büttiker, Fisher-Lee, Imry...) Chaotic cavity -> S as a unitary random matrix β=1: TRS and SRS β=2: no TRS β=4: TRS, no SRS (Jalabert-Pichard-Beenakker, Baranger-Mello...)

11 Random Matrix Theory of Quantum Transport I in Conductance as transmission I out (Landauer, Buttiker, Fisher-Lee, Imry...) Chaotic cavity -> S as a unitary random matrix β=1: T 2 =1 β=2: T=0 β=4: T 2 =-1 (Jalabert-Pichard-Beenakker, Baranger-Mello...)

12 Random Matrix Theory of Quantum Transport *S-matrix is unitary *If unitary is the only constraint - broken TRS ->all elements have the same distribution -> conductance Conductance quantum: Baranger and Mello, arxiv:cond-mat/

13 Random Matrix Theory of Quantum Transport *S-matrix is unitary *Systems with TRS and SRS -> S is symmetric, use representation blue pairing for i=j red pairing for all -> <U U U U> ~ <U U> <U U> (sum over all poss. pairing) ->average over randomly distributed variables Baranger and Mello, arxiv:cond-mat/

14 Random Matrix Theory of Quantum Transport ->Enhanced reflection probability ->Reduced conductance a.k.a. weak localization Baranger and Mello, arxiv:cond-mat/

15 Random Matrix Theory of Quantum Transport *S-matrix is unitary *If SRS is broken but TRS is preserved ->S-matrix is antisymmetric ->diagonal elements are zero ->reduced reflection ->enhanced conductance, a.k.a. weak antilocalization β=1: T 2 =+1 β=2: T 2 =0 β=4: T 2 =-1 Note: sign of correction to conductan related to T 2 =+1 (β=1), -1 (β=4)

16 RMT of Quantum Transport: microscopic justifications Efetov 82, 83 : -> Supermatrix sigma-model diffusive systems have the same correlation functions as RMT Altshuler-Shklovskii 86 : -> impurity Green s functions equivalence with RMT works up to the Thouless energy E ~h/t e (inverse ergodic time) Richter-Sieber; Haake et al.; PJ and Whitney; Brouwer-Rahav : -> trajectory-based semiclassics chaotic ballistic systems have RMT properties

17 Random Matrix Theory: (iii) Chiral symmetry Chiral symmetric Hamiltonian operator J Verbaarschot -> three new ensemble of RMT (also vs. TRS/SRS or T2=-1,0,1) Examples : QCD Dirac operator Lattice models with sublattice symmetry, e.g. graphene

18 Random Matrix Theory: (iv) Particle-hole symmetry Quantum coherent metal (N) in contact with a superconductor (S) Andreev reflection of electron into hole Fermi energy of S sets E=0 -> breaking of energy translational symmetry -> Four new ensembles of Bogoliubov-de Gennes H M Zirnbauer defined by presence/absence of TRS and/or SRS Particle (E>0) - hole (E<0) symmetry spin space Nambu space A Altland

19 Symmetry classes - the 10-fold way Dyson s 3-fold way + particle-hole symmetry Time-reversal symmetry Particle-hole symmetry Antiunitary symmetries P 2,T 2 = -1,0,1 3x3=9 and two possibilities for P=T=0 -> 10-fold way

20 Topological quantum numbers Quantum mechanical state = complex wavefunction Move the particle around closed contour -> single-valuedness requires?: Can we contract the curve down to a single point? What happens inside the loop? Existence of topological QM numbers depends on the answer to such questions.

21 Topological quantum numbers : flux quantization with SC Landau theory of superconductivity - macroscopic wavefunction Gauge-invariant current Take toroidal SC pierced by B-field Contour well inside SC : Meissner effect -> C C C -> -> flux quantization (regardless of exact shape, impurities aso.)

22 Topological phase of matter : (i) the integer quantum hall effect 2D electrons in strong magnetic field H=(p-eA)2/2m ~1D harmonic oscillator with ~continuous spectrum -> quantized Landau levels... E von Klitzing B=0 B=0

23 Topological phase of matter : (i) the integer quantum hall effect Not sensitive to impurities, disorder etc. -> topological protection

24 Topological phase of matter : (i) the integer quantum hall effect Chiral edge states and the IQHE (Halperin 82, Büttiker 88) ~2D electrons in magnetic field + confining potential Landau levels move up at edge Number of edge states fixed by chemical potential Not sensitive to perturbation Velocity of edge states v y ~d x V -> states are chiral (broken TRS) y x

25 Topological phase of matter : (i) the integer quantum hall effect Chiral edge states and the IQHE (Halperin 82, Büttiker 88) Edge states do not backscatter! (where can they go?) -> perfect transmission -> Hall conductance quantized as G=n 2e 2 /h -> n=0,1,2,3,...: Z-topological insulator topological # = # of edge states = # of occupied LL in bulk Topological protection from gap between LL s

26 Topological phase of matter : (ii) quantum spin Hall insulator Main points : I. insulating bulk, conducting edges II. helical edge states - chirality depends on spin

27 Topological insulators : bulk insulators with surface/edge states Topological insulator : definition Material that is insulating in the bulk but carries metallic (i.e. extended) states at its boundary Band theory predicts a gap at the Fermi energy -> bulk band insulator Yet, the bulk carries a topological quantum number n Connecting the bulk to the outside (with n=0) generates gapless edge states F Bloch

28 Topological phase of matter : (ii) quantum spin Hall insulator Graphene : honeycomb lattice ~2 sublattices A and B ~2 atoms per unit cell ~2 bands E=pv F S k y k x

29 Topological phase of matter : (ii) quantum spin Hall insulator Low-energy graphene Hamiltonian σ x,y act in sublattice space τ z act in valley space 2-fold spin degeneracy... K K

30 Topological phase of matter : (ii) quantum spin Hall insulator Low-energy graphene Hamiltonian σ x,y act in sublattice space τ z act in valley space 2-fold spin degeneracy Add spin-orbit interaction K K -> generate gap -> gap has opposite sign in different valleys S z acts in spin space (Kane and Mele 05)

31 Topological phase of matter : (ii) quantum spin Hall insulator K K Create two Kramers pairs of helical edge states No breaking of TRS Topological number n=0,1 Z 2 -topological insulator (Kane and Mele 05)

32 Topological phase of matter : (ii) quantum spin Hall insulator K K Create two Kramers pairs of helical edge states No breaking of TRS Topological number n=0,1 Z 2 -topological insulator Edge states still robust against impurities Backscattering forbidden by TRS Expl. #1 (technical) Presence of TRS -> S-matrix is self-dual (Kane and Mele 05) -> no off-diagonal elements for 2x2 matrix

33 Topological phase of matter : (ii) quantum spin Hall insulator K K Create two Kramers pairs of helical edge states No breaking of TRS Topological number n=0,1 Z 2 -topological insulator Edge states still robust against impurities Backscattering forbidden by TRS (Qi and Zhang, 10) Expl. #2 (handwaving) phase of π between clockwise and counterclockwise processes -> destructive interference

34 Topological insulators : experimental realizations Integer quantum hall state Z-topological insulator d=2, broken TRS -> symmetry class A Quantum spin Hall state HgTe/CdTe quantum wells ~semiconductors with large SOI and opposite gaps (...) (König et al. 07) Z 2 topological insulator d=2, T 2 =-1 (TRS but no SRS) -> AIII

35 Topological insulators with TRS : key ingredient K K

36 Classification of nontrivial topological states vs. symmetries Answer given by homotopy group of band structure as map from the Brillouin Zone to Hamiltonian space

37 Random matrix theory and topological insulators Manuscripts with RMT in title published cited

38 Random matrix theory and topological insulators Manuscripts with RMT in title topological insulator in title published cited