Topological Phases in Floquet Systems

Transcription

1 Rahul Roy University of California, Los Angeles June 2, 2016 arxiv: , arxiv: Post-doc: Fenner Harper

2 Outline 1 Introduction 2 Free Fermionic Systems 3 Interacting Systems in Class D

3 Topological Insulators and SPT Phases Bulk-boundary correspondence...of what? Ground states Hamiltonians Unitaries (for some small time evolution): U(t) = exp ( iht) Quasienergies ɛ i (t) correspond to eigenvalues e iɛ i (t) of U(t). π ϵ(t) 0 -π 0 t

4 Unitary Evolutions What if the Hamiltonian is time-varying? [ ( t )] U(t) = T exp i H(t )dt 0 Without loss of generality, we will assume the spectrum contains bulk gaps at ɛ = 0 and ɛ = π, when t = T. π ϵ(t) 0 -π -π 0 π k

5 Floquet Hamiltonians Question Given a gapped bulk unitary, are there any robust topological edge states for the different AZ classes that lie outside of the ones that result from a constant H top? Consider the Floquet Hamiltonian (with T = 1 and branch cut at π) H F (k) = i ln [U(k, 1)]. Let U(t) have eigenvalues e iɛ(t) and result from evolution due to H(t) = H 0 + H (t), with H 0 topologically trivial. If H F is topologically non-trivial, then we expect edge modes.

6 Rudner System Question Given a gapped bulk unitary, are there any robust edge states that can t be accounted for by a non-trivial H F? Consider the five-part drive of Rudner et al. [PRX 3, (2013)] 1 A 2 A 3 A 4 A 5 Apply on-site energy

7 Rudner System After one complete cycle: Bulk particles return to their original position. Particles on B-sites on top edge move two units to the left. Particles on A-sites on bottom edge move two units to the right. U bulk (1) = I BUT U edge (1) I

8 Topological Invariants π ϵ(1) 0 C 2 The number of edge modes in a gap is: C 1 N edge = C + C + N U W [U] -π -π 0 π k The winding number is defined by W [U] = 1 8π 2 dtdk x dk y Tr ( U 1 t U [ U 1 kx U, U 1 ky U ]), where U is an associated unitary cycle defined on a 3-torus.

9 Other Classes Class D in d = 1: Jiang et al. [PRL 106, (2011)] Thakurathi et al. [PRB 88, (2013)] Question Can we extend these results to other symmetry classes and dimensions, analogous to Kitaev s periodic table of topological insulators and superconductors [Kitaev 2009]?

10 Outline 1 Introduction 2 Free Fermionic Systems 3 Interacting Systems in Class D

11 Classification of Unitaries Theorem If G is the static topological insulator classification, then the classification of Floquet topological insulators for a unitary with gaps at 0 and π is G G. Some elements of the periodic table had been filled earlier, notably: Class D in d = 1. Class AIII in d = 2, 3. Class A in d = 2. Most elements were missing, however, and there were some errors (for instance in classifying class AIII in various dimensions).

12 Symmetries for Unitary Evolution For any AZ symmetry class S, we: 1 Require that H F have the defining symmetries of Hamiltonians in S. 2 Require the instantaneous Hamiltonian H(t) to belong to the AZ symmetry class S. The 10 AZ classes are based on: Particle-hole symmetry, Time-reversal symmetry, Chiral symmetry, P = KP PHP 1 = H Θ = Kθ θhθ 1 = +H C CHC 1 = H

13 Symmetries for Unitary Evolution For PHS, we can satisfy both symmetry requirements 1 and 2. For TRS and Chiral symmetry, requirements 1 and 2 are incompatible: a generic H(t) satisfying requirement 2 does not satisfy requirement 1. For TRS, a natural time-dependent symmetry definition is θh(t + t)θ 1 = H (t t), leading to a Floquet Hamiltonian H F that satisfies θh F θ 1 = H F. For chiral symmetry, there is no natural time-dependent symmetry condition. However, the choice CH(t + t)c 1 = H(t t), lends itself to a topological classification of unitaries.

14 Composition of Unitaries For U 1, U 2 AZ class S, we define U 1 U 2 as the sequential evolution of each unitary in turn: If S has no TRS and is not chiral, then { H 1 (k, 2t) 0 t 1/2 H(t) = H 2 (k, 2t 1) 1/2 t 1. Otherwise, we define the composition through H 2 (k, 2t) 0 t 1/4 H(t) = H 1 (k, 2t 1/2) 1/4 t 3/4 H 2 (k, 2t 1) 3/4 t 1, which ensures that U 1 U 2 S.

15 Equivalence Classes of Unitaries Denote space of gapped unitaries within the AZ symmetry class S as U S g We write U 1 U 2 if U 1 is homotopic to U 2 within U S g We define U 1 U 2 trivial evolutions U 0 n 1 and U 0 n 2, such that U 1 U 0 n 1 U 2 U 0 n 2, Finally, for pairs (U 1, U 2 ) and (U 3, U 4 ), we write (U 1, U 2 ) (U 3, U 4 ) U 1 U 4 U 2 U 3.

16 Effective Decomposition of Unitaries Defn: Unitary loops : U(k, 0) = U(k, 1) = I. Defn: Const H Evolution: U(k, t) = e ih(k)t for some H(k), whose eigenvalues have magnitude strictly less than π. Theorem Every unitary U U0,π S can be continuously deformed to a composition of a unitary loop L and a constant Hamiltonian evolution C, which we write as U L C. L and C are unique up to homotopy. We can classify any gapped unitary by separately classifying 0 the loop component and the constant Hamiltonian evolution -π component ϵ(t) π t

17 Classification of Constant H Unitaries Label the set of constant Hamiltonian evolutions in symmetry class S as U S C. Label the set of static gapped Hamiltonians in symmetry class S, whose eigenvalues E satisfy 0 < E < π, by H S. The set of gapped constant evolutions in UC S is clearly in one-to-one correspondence with the set of static Hamiltonians in H S. Constant Hamiltonian evolutions are classified according to the static topological insulator periodic table [Kitaev 2009]

18 K Theory Classification To classify the loop unitary components, we construct a group as follows: Take pairs (U 1, U 2 ) and consider the operation + defined through (U 1, U 2 ) + (U 3, U 4 ) = (U 1 U 3, U 2 U 4 ), where is the direct sum We map the problem onto the more standard problem of finding the group of equivalence classes of Hermitian maps H U on S 1 X with a set of symmetries S : ( U(k, t) H U (k, t) = 0 U(k, t) U (k, t) 0 [S is a set of symmetry operators derived from S; X is the Brillouin zone]. Using Morita equivalence, we reduce this to a standard K group. )

19 K Theory Classification For each symmetry class, we find: K 1 (S 1 X, {0} X ) = K 0 (X ) Class A K 2 (S 1 X, {0} X ) = K 1 (X ) Class AIII KR 0,7 (S 1 X, {0} X ) = KR 0,0 (X ) Class AI KR 0,0 (S 1 X, {0} X ) = KR 0,1 (X ) Class BDI KR 0,1 (S 1 X, {0} X ) = KR 0,2 (X ) Class D KR 0,2 (S 1 X, {0} X ) = KR 0,3 (X ) Class DIII KR 0,3 (S 1 X, {0} X ) = KR 0,4 (X ) Class AII KR 0,4 (S 1 X, {0} X ) = KR 0,5 (X ) Class CII KR 0,6 (S 1 X, {0} X ) = KR 0,7 (X ) Class CI KR 0,5 (S 1 X, {0} X ) = KR 0,6 (X ) Class C

20 Complete Classification of Unitaries Combining the loop and constant evolution components, two-gapped unitaries are classified according to the following table: S d = A Z Z Z Z Z Z Z Z AIII Z Z Z Z Z Z Z Z AI Z Z Z Z Z 2 Z 2 Z 2 Z 2 BDI Z 2 Z 2 Z Z Z Z Z 2 Z 2 D Z 2 Z 2 Z 2 Z 2 Z Z Z Z DIII Z 2 Z 2 Z 2 Z 2 Z Z Z Z AII Z Z Z 2 Z 2 Z 2 Z 2 Z Z CII Z Z Z 2 Z 2 Z 2 Z 2 Z Z C Z Z Z 2 Z 2 Z 2 Z 2 Z Z CI Z Z Z 2 Z 2 Z 2 Z 2 Z Z

21 Bulk-Edge Correspondence At an interface between two systems, the bulk topology may be manifested as edge modes. π ϵ 0 U 1 U 2 -π 0 r If n C is the constant evolution invariant and n L is the unitary loop invariant, then the number of edge modes in each gap is: n π = n L n 0 = n C + n L

22 Outline 1 Introduction 2 Free Fermionic Systems 3 Interacting Systems in Class D

23 Class D Hamiltonians General Class D Hamiltonian H = i γ i A ij γ j, 4 where γ i = γ i are Majorana fermions and A ij is an antisymmetric matrix. In d=1, the static classification is Z 2 [Kitaev 2001]. ij

24 Class D Unitaries in d=1 For a time dependent Hamiltonian, let ( t ) O(t) = T exp A ij (t )dt. O(t) belongs to the special orthogonal group, SO(2n). H(0) and H(π) have the full symmetry of a 0d Class D Hamiltonian. = 1d Floquet cycles in Class D can be characterized by an invariant Z 2 Z 2 for n > 2. This agrees with the classification obtained using K-theory 0

25 Non-trivial Unitary Loop Consider a one-dimensional fermionic chain of length K that has a two-state Hilbert space at each site (with annihilation operators a j and b j ): γ 1 a γ 2 a γ 3 a γ 4 a γ 5 a γ 6 a a a γ 2 K-1 γ 2 K γ 1 b γ 2 b γ 3 b γ 4 b γ 5 b γ 6 b b b γ 2 K-1 γ 2 K We define two sets of Majorana operators through γ a 2j 1 = a j + a j, γa 2j = a j a j i which satisfy γ = γ. γ b 2j 1 = b j + b j, γb 2j = b j b j i,

26 Non-trivial Unitary Loop Let H 1a = ( a j a j 1 ) = i ( γ a 2 2 2j 1 γ2j) a j j H 2a = 1 ( ) a j 2 a j+1 a j+1 a j + a j a j+1 + a j+1 a j = i 2 j j ( γ a 2j γ2j+1 a ), with H 1b, H 2b defined identically in terms of the b operators and γ b Majoranas. H 1a and H 2a respectively correspond to the trivial and nontrivial phases of the 1D class D superconductor.

27 Non-trivial Unitary Loop Evolve the system with H 1 = H 1a + 2H 1b for 0 t π and H 2 = H 2a + 2H 2b for π t 2π. H 2a and H 2b are topologically non-trivial while H 1a, H 1b are trivial. The evolution by H 1a pushes the Majorana mode of subsystem a to quasienergy ɛ = π, Evolving the closed system until t = 2π leads to a unitary that is the identity (up to an overall phase factor).

28 Unitaries for Open and Closed Systems Terms i 2 γa 2K γa 1 from H 2a and i 2 γb 2K γb 1 from H 2b are absent in the open system Hamiltonians. With d = 1 2 (γa 2K + iγa 1), d = 1 2 (γa 2K iγa 1), ( ( γ2k b + iγb 1 γ2k b iγb 1 e = 1 2 ), e = 1 2 the unitary of the open system may be written U op (2π) = e [ π 2 (γ a 2K γa 1 +2γb 2K γb 1)] Ucl (2π) ), = e [ iπ 2 (d d dd ) iπ(e e ee )] Ucl (2π).

29 Effective Unitary for the Edge Effective Unitary for the edge: U eff (2π) = e [ iπ 2 (d d dd ) iπ(e e ee )], Define effective parity operator ˆP L at the left edge of the open chain Then, { ˆP L, U eff } = 0.

30 Many body picture Bulk topological invariant: U 0 (T )U π(t ) = e iνπ I correctly predicts π Majorana mode agrees with the non-interacting invariant Edge picture: two fold degeneracy in spectrum of open system has effective unitary of the form described above

31 Phase Transitions Suppose there is phase transition at t 0. i.e., no edge π Majorana modes for t < t 0 but Majorana modes exist for t > t 0. At t 0, each bulk eigenstate ψ becomes part of a multiplet { ψ, d ψ, e ψ, d e ψ } after the transition. These form two pairs separated by quasienergy π (modulo 2π). The two states in each pair have opposite parity. ϵ(t 0) π π π 2 -π e d ψ d ψ e ψ ψ t 0

32 Bulk Boundary correspondence Suppose at some point beyond t 0, we reconnect the edges with an edge Hamiltonian ) ( ) H = h d (dd d d + h e ee e e, and choose h d and h e so that the unitary at the end of the evolution is I If we reconnect with a π flux, h d, h e change sign, the unitary goes to I. ϵ(t) π (a) ϵ(t) π (b) π 2 e d ψ d ψ π 2 e d ψ d ψ π 2 e ψ ψ - π 2 e ψ ψ -π t 0 T -π t 0 T

33 Conclusions Topological Floquet systems may be classified using K theory. A gapped unitary evolution can be uniquely decomposed (up to homotopy) into a loop component and a constant evolution. The periodic table for free fermionic unitary evolutions may be obtained from the static periodic table through G G G. Unitary loops in one-dimensional interacting systems in class D may be classified by considering the effective edge unitary. A topological invariant is given by the phase change under a flux insertion.