topological phases in floquet systemscmt-roy.physics.ucla.edu/sites/default/files/floquet... ·...

Topological Phases in Floquet Systems

Rahul Roy

University of California, Los Angeles

June 2, 2016

arXiv:1602.08089, arXiv:1603.06944

Post-doc: Fenner Harper

Rahul Roy (UCLA) Topological Phases in Floquet SystemsJune 2, 2016[1mm] arXiv:1602.08089, arXiv:1603.06944 1

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Outline

1 Introduction

2 Free Fermionic Systems

3 Interacting Systems in Class D


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Topological Insulators and SPT Phases

Bulk-boundary correspondence...of what?Ground statesHamiltoniansUnitaries (for some small time evolution):

U(t) = exp (−iHt)

‘Quasienergies’ εi (t) correspond to eigenvalues e iεi (t) of U(t).

0-π

0

π

t

ϵ(t)


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Unitary Evolutions

What if the Hamiltonian is time-varying?

U(t) = T[

exp

(−i∫ t

0H(t ′)dt ′

)]Without loss of generality, we will assume the spectrum contains bulkgaps at ε = 0 and ε = π, when t = T .

-π 0 π

-π

0

π

k

ϵ(T)


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Floquet Hamiltonians

Question

Given a gapped bulk unitary, are there any robust topological edge statesfor the different AZ classes that lie outside of the ones that result from aconstant Htop?

Consider the Floquet Hamiltonian (with T = 1 and branch cut at π)

HF (k) = −i ln [U(k , 1)] .

Let U(t) have eigenvalues e iε(t) and result from evolution due to

H(t) = H0 + H ′(t),

with H0 topologically trivial.

If HF is topologically non-trivial, then we expect edge modes.


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Rudner System

Question

Given a gapped bulk unitary, are there any robust edge states that can’tbe accounted for by a non-trivial HF ?

Consider the five-part drive of Rudner et al. [PRX 3, 031005 (2013)]

1 A →2 A ↑3 A ←4 A ↓5 Apply on-site energy


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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.

Ubulk(1) = I BUT Uedge(1) 6= I


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Topological Invariants

C1

C2

-π 0 π

-π

0

π

k

ϵ(1)

The number of edge modes in a gap is:

Nedge = C+ − C− + NU ≡W [U]

The winding number is defined by

W [U] =1

8π2

∫dtdkxdkyTr

(U−1∂tU

[U−1∂kxU,U

−1∂kyU]),

where U is an associated unitary cycle defined on a 3-torus.


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Other Classes

Class D in d = 1:

Jiang et al. [PRL 106, 220402 (2011)]Thakurathi et al. [PRB 88, 155133(2013)]

Question

Can we extend these results to other symmetry classes and dimensions,analogous to Kitaev’s periodic table of topological insulators andsuperconductors [Kitaev 2009]?


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Outline

1 Introduction




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Classification of Unitaries

Theorem

If G is the static topological insulator classification, then the classificationof Floquet topological insulators for a unitary with gaps at 0 and π isG × 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 (forinstance in classifying class AIII in various dimensions).


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Symmetries for Unitary Evolution

For any AZ symmetry class S, we:

1 Require that HF have the defining symmetries of Hamiltonians in S.

2 Require the instantaneous Hamiltonian H(t) to belong to the AZsymmetry class S.

The 10 AZ classes are based on:

Particle-hole symmetry,

P = KP ⇒ PHP−1 = −H∗

Time-reversal symmetry,

Θ = Kθ ⇒ θHθ−1 = +H∗

Chiral symmetry,

C ⇒ CHC−1 = −H


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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 requirement1.

For TRS, a natural time-dependent symmetry definition is

θH(t∗ + t)θ−1 = H∗(t∗ − t),

leading to a Floquet Hamiltonian HF that satisfies

θHF θ−1 = H∗F .

For chiral symmetry, there is no natural time-dependent symmetrycondition. However, the choice

CH(t∗ + t)C−1 = −H(t∗ − t),

lends itself to a topological classification of unitaries.


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Composition of Unitaries

For U1,U2 ∈ AZ class S, we define U1 ∗ U2 as the sequential evolution ofeach unitary in turn:

If S has no TRS and is not chiral, then

H(t) =

{H1(k, 2t) 0 ≤ t ≤ 1/2

H2(k, 2t − 1) 1/2 ≤ t ≤ 1.

Otherwise, we define the composition through

H(t) =

H2(k, 2t) 0 ≤ t ≤ 1/4

H1(k, 2t − 1/2) 1/4 ≤ t ≤ 3/4

H2(k, 2t − 1) 3/4 ≤ t ≤ 1

,

which ensures that U1 ∗ U2 ∈ S.


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Equivalence Classes of Unitaries

Denote space of gapped unitaries within the AZ symmetry class S asUSgWe write U1 ≈ U2 if U1 is homotopic to U2 within USgWe define U1 ∼ U2 ⇐⇒ ∃ trivial evolutions U0

n1 and U0n2 , such that

U1 ⊕ U0n1 ≈ U2 ⊕ U0

n2 ,

Finally, for pairs (U1,U2) and (U3,U4), we write

(U1,U2) ∼ (U3,U4) ⇐⇒ U1 ⊕ U4 ∼ U2 ⊕ U3.


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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), whoseeigenvalues have magnitude strictly less than π.

Theorem

Every unitary U ∈ US0,π can be continuously deformed to a composition ofa unitary loop L and a constant Hamiltonian evolution C , which we writeas U ≈ L ∗ C . L and C are unique up to homotopy.

We can classify any gappedunitary by separately classifyingthe loop component and theconstant Hamiltonian evolutioncomponent. 0 1

2 1-π

0

π

t

ϵ(t)


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Classification of Constant H Unitaries

Label the set of constant Hamiltonian evolutions in symmetry class Sas USC .

Label the set of static gapped Hamiltonians in symmetry class S,whose eigenvalues E satisfy 0 < |E | < π, by HS .

The set of gapped constant evolutions in USC is clearly in one-to-onecorrespondence with the set of static Hamiltonians in HS .

⇒ Constant Hamiltonian evolutions are classified according to the statictopological insulator periodic table [Kitaev 2009]


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K Theory Classification

To classify the loop unitary components, we construct a group as follows:

Take pairs (U1,U2) and consider the operation ‘+’ defined through

(U1,U2) + (U3,U4) = (U1 ⊕ U3,U2 ⊕ U4),

where ⊕ is the direct sum

We map the problem onto the more standard problem of finding thegroup of equivalence classes of Hermitian maps HU on S1 × X with aset of symmetries S ′:

U(k, t) ↔ HU(k, t) =

(0 U(k, t)

U†(k, t) 0

)[S ′ is a set of symmetry operators derived from S; X is the Brillouinzone].

Using Morita equivalence, we reduce this to a standard K group.


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K Theory Classification

For each symmetry class, we find:

K 1(S1 × X , {0} × X ) = K 0(X ) Class AK 2(S1 × X , {0} × X ) = K 1(X ) Class AIII

KR0,7(S1 × X , {0} × X ) = KR0,0(X ) Class AIKR0,0(S1 × X , {0} × X ) = KR0,1(X ) Class BDIKR0,1(S1 × X , {0} × X ) = KR0,2(X ) Class DKR0,2(S1 × X , {0} × X ) = KR0,3(X ) Class DIIIKR0,3(S1 × X , {0} × X ) = KR0,4(X ) Class AIIKR0,4(S1 × X , {0} × X ) = KR0,5(X ) Class CIIKR0,6(S1 × X , {0} × X ) = KR0,7(X ) Class CIKR0,5(S1 × X , {0} × X ) = KR0,6(X ) Class C


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Complete Classification of Unitaries

Combining the loop and constant evolution components, two-gappedunitaries are classified according to the following table:

S d = 0 1 2 3 4 5 6 7

A Z× Z ∅ Z× Z ∅ Z× Z ∅ Z× Z ∅AIII ∅ Z× Z ∅ Z× Z ∅ Z× Z ∅ Z× ZAI Z× Z ∅ ∅ ∅ Z× Z ∅ Z2 × Z2 Z2 × Z2

BDI Z2 × Z2 Z× Z ∅ ∅ ∅ Z× Z ∅ Z2 × Z2

D Z2 × Z2 Z2 × Z2 Z× Z ∅ ∅ ∅ Z× Z ∅DIII ∅ Z2 × Z2 Z2 × Z2 Z× Z ∅ ∅ ∅ Z× ZAII Z× Z ∅ Z2 × Z2 Z2 × Z2 Z× Z ∅ ∅ ∅CII ∅ Z× Z ∅ Z2 × Z2 Z2 × Z2 Z× Z ∅ ∅C ∅ ∅ Z× Z ∅ Z2 × Z2 Z2 × Z2 Z× Z ∅CI ∅ ∅ ∅ Z× Z ∅ Z2 × Z2 Z2 × Z2 Z× Z


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Bulk-Edge Correspondence

At an interface between two systems, the bulk topology may bemanifested as edge modes.

U1 U2

0-π

0

π

r

ϵ

If nC is the constant evolution invariant and nL is the unitary loopinvariant, then the number of edge modes in each gap is:

nπ = nL

n0 = nC + nL


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Outline

1 Introduction




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Class D Hamiltonians

General Class D Hamiltonian

H =i

4

∑ij

γiAijγj ,

where γ†i = γi are Majorana fermions and Aij is an antisymmetricmatrix.

In d=1, the static classification is Z2 [Kitaev 2001].


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Class D Unitaries in d=1

For a time dependent Hamiltonian, let

O(t) = T exp

(∫ t

0Aij(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∈ Z2 × Z2 for n > 2.

This agrees with the classification obtained using K-theory


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Non-trivial Unitary Loop

Consider a one-dimensional fermionic chain of length K that has atwo-state Hilbert space at each site (with annihilation operators ajand bj):

γ1a γ2

a γ3a γ4

a γ5a γ6

a

...

γ2 K-1a γ2 K

a

γ1b γ2

b γ3b γ4

b γ5b γ6

b

...

γ2 K-1b γ2 K

b

We define two sets of Majorana operators through

γa2j−1 = aj + a†j , γa2j =aj − a†j

i

γb2j−1 = bj + b†j , γb2j =bj − b†j

i,

which satisfy γ† = γ.


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Let

H1a = −∑j

(a†j aj −

1

2

)= − i

2

∑j

(γa2j−1γ

a2j

)H2a =

1

2

∑j

(−a†j aj+1 − a†j+1aj + ajaj+1 + a†j+1a

†j

)=

i

2

∑j

(γa2jγ

a2j+1

),

with H1b,H2b defined identically in terms of the b operators and γb

Majoranas.

H1a and H2a respectively correspond to the trivial and nontrivialphases of the 1D class D superconductor.


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Evolve the system withH1 = H1a + 2H1b for 0 ≤ t ≤ π and H2 = H2a + 2H2b for π ≤ t ≤ 2π.

H2a and H2b are topologically non-trivial while H1a,H1b are trivial.

The evolution by H1a pushes the Majorana mode of subsystem a toquasienergy ε = π,

Evolving the closed system until t = 2π leads to a unitary that is theidentity (up to an overall phase factor).


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Unitaries for Open and Closed Systems

Terms i2γ

a2Kγ

a1 from H2a and i

2γb2Kγ

b1 from H2b are absent in the open

system Hamiltonians.

With

d =1

2(γa2K + iγa1) , d† =

1

2(γa2K − iγa1) ,

e =1

2

(γb2K + iγb1

), e† =

1

2

(γb2K − iγb1

),

the unitary of the open system may be written

Uop(2π) = e[π2 (γa2Kγa1+2γb2Kγ

b1)]Ucl(2π)

= e[− iπ2 (d†d−dd†)−iπ(e†e−ee†)]Ucl(2π).


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Effective Unitary for the Edge

Effective Unitary for the edge:

Ueff(2π) = e[− iπ2 (d†d−dd†)−iπ(e†e−ee†)],

Define effective parity operator P̂L at the left edge of the open chain

Then,

{P̂L,Ueff} = 0.


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Many body picture

Bulk topological invariant: U0(T )U†π(T ) = e iνπIcorrectly predicts π Majorana modeagrees with the non-interacting invariant

Edge picture:

two fold degeneracy in spectrum of open systemhas effective unitary of the form described above


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Phase Transitions

Suppose there is phase transition at t0.i.e., no edge π Majorana modes for t < t0 but Majorana modes existfor t > t0.

At t0, 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). Thetwo states in each pair have opposite parity.

e†d† |ψ⟩

|ψ⟩

e† |ψ⟩

d† |ψ⟩

-π

π

-π

2

π

2

0

t0

ϵ(t0)


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Bulk Boundary correspondence

Suppose at some point beyond t0, we reconnect the edges with anedge Hamiltonian

H ′ = hd

(dd† − d†d

)+ he

(ee† − e†e

),

and choose hd and he so that the unitary at the end of the evolutionis IIf we reconnect with a π flux, hd , he change sign, the unitary goes to−I.

e†d† |ψ⟩

|ψ⟩

e† |ψ⟩

d† |ψ⟩

-π

π

-π

2

π

2

0

t0 T

ϵ(t)(a)

e†d† |ψ⟩

|ψ⟩

e† |ψ⟩

d† |ψ⟩

-π

π

-π

2

π

2

0

t0 T

ϵ(t)(b)


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Conclusions

Topological Floquet systems may be classified using K theory.

A gapped unitary evolution can be uniquely decomposed (up tohomotopy) into a loop component and a constant evolution.

The periodic table for free fermionic unitary evolutions may beobtained from the static periodic table through G → G × G .

Unitary loops in one-dimensional interacting systems in class D maybe classified by considering the effective edge unitary.

A topological invariant is given by the phase change under a fluxinsertion.


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