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Reservoir-induced topological

• rder & quantized transport in
• pen systems

Michael Fleischhauer & Dominik Linzner

• Dept. of Physics & research center OPTIMAS

Technische Universität Kaiserslautern Bad Honnef 09.05.2016

topological states

exo$c quantum states

Abelian & non-Abelian anyons

but: in general no protection against losses

protected edge states & edge transport

topological protec$on

topological order in the steady state

• f an open system ??

• pen dynamics drives the system to a steady state

steady state: attractor

• pen-system dynamics

Re[λ] d dtρ = L ρ

gapped open systems

Lρλ = −λ ρλ

parameter space

λ = 0

damping gap robustness of the steady state

• utline
• topological invariants & open systems
• Su-Schrieffer-Heeger model & Thouless pump
• quantized topological transport in open spin chain

with interactions

• detection of topological invariant

• utline
• topological invariants & open systems
• Su-Schrieffer-Heeger model & Thouless pump
• quantized topological transport in open spin chain

with interactions

• detection of topological invariant

• utline
• topological invariants & open systems
• Su-Schrieffer-Heeger model & Thouless pump
• quantized topological transport in open spin chain

with interactions

• detection of topological invariant

• utline
• topological invariants & open systems
• Su-Schrieffer-Heeger model & Thouless pump
• quantized topological transport in open spin chain

with interactions

• detection of topological invariant

topological invariants & open systems

topology

Möbius strip:

locally indistinguishable

topology

Möbius strip:

differ by global properties !

topological invariants: geometric phases Zak (Berry) phase

|u0

ki = eikx0|uki

φ0

Zak = φZak + 2π

a x0

choice of origin matters

Zak PRL (1989)

φZak = Z π/a

−π/a

dk huk|i∂k|uki C 6= 0

Chern number

C = i 2π Z Z

BZ

d2k n h∂kyuk|∂kxuki h∂kxuk|∂kyuki

• ∈ Z

no global gauge

a x0

geometric phases for density matrices Uhlmann connection

ρ = w w†

w → w U

gauge degree of freedom: U(N)

• O. Viyuela, et al. Phys. Rev. Lett. (2014)
• Z. Huang, D. P. Arovas, Phys. Rev. Lett. (2014)

w† → U † w†

U(1) Uhlmann phase

eiφ = I dλ Tr ⇥ w∂λw†⇤

Berry phases for density matrices Furthermore without constraints: trivial global gauge

w = √ρ

finite-T state of a Chern insulator

• J. C. Budich, S. Diehl 1501.04135:

H(k) = X

j

dj(k) ˆ σj

d1 = sin(kx) d2 = 3 sin(ky) d3 = 1 − cos(kx) − cos(ky)

C = 1 2π Z dky ✓∂φ(ky) ∂ky ◆ 6= C0 = 1 2π Z dkx ✓∂φ(kx) ∂kx ◆

φ(kx)

φ(ky)

• 3
• 2
• 1

1 2 3 kx

• 3
• 2
• 1

1 2 3

fU

kx

• 3
• 2
• 1

1 2 3 ky

• 3
• 2
• 1

1 2 3

fU

ky

non-interacting fermions Gaussian systems

C.E. Bardyn, et al. New J. Phys (2013)

H = X

ij

hij ˆ c(†)

i ˆ

cj

Lj ∼ α ˆ c†

j + β ˆ

cj

covariance matrix

ρ ∼ exp ⇢ − i 2wj Γjkwk

• Γjk ∼ Im Tr
• ρwiwj}

density matrix

wi ∼ ˆ ci ± ˆ c†

i

non-interacting fermions

ρ ∼ exp ⇢ − i 2wj Γjkwk

• topological classification in terms of Γij

γ(k) = ✓ γ11 γ12 γ21 γ22 ◆

∼ (1 + ~ n(k) · ~ )

(I) closing of the damping gap (criticality) (II) closing of the purity gap = gap of effective Hamiltonian

Heff = i X

jk

Γjkwjwk

topological phase transition beyond Gaussian systems ??

polarization

Thouless, Kohmoto, Nightingale, den Nijs (TKNN) PRL (1982)

topology quantized bulk transport Zak phase & Polarization

King-Smith, Vanderbilt PRB (1983)

P = Z dx w∗(x) x w(x)

∆P = a 2π ∆φZak

w(x)

quantization of Hall conductance

x

y

C = 1 2π Z 2π/a dky ∂kyφZak(ky)

~ E

dky = Eydt

σxy = jx Ey = dP dt 1 Ey = dP dky = C

Su-Schrieffer-Heeger model & Thouless pump

Su, Schrieffer, Heeger, PRL (1979) D.J. Thouless, PRB (1983) Attala et al. (I. Bloch), Nature Physics (2013)

SSH Model: free fermions on a superlattice with inversion symmetry

t1

t2

H = −t1 X

• dd

ˆ ciˆ c†

i+1 − t2

X

even

ˆ ciˆ c†

i+1 + h.a.

à half filling = band insulator of lower sub-band Eg ∼ t1 − t2

SSH Rice Mele Hamiltonian

locally indistinguishable topological phase transition

φZak = π

φZak = 0

H = −t1 X

• dd

ˆ c†

i ˆ

ci+1 − t2 X

even

ˆ c†

i ˆ

ci+1 + ∆ X

• dd

ˆ c†

i ˆ

ci − X

even

ˆ c†

i ˆ

ci ! h.a.

Inversion symmeric SSH symmetry breaking term

t1 = t2

M.J. Rice & E.J. Mele, PRL(1982)

Rice-Mele Hamiltonian breaking inversion symmetry & Thouless pump

H = −t1 X

• dd

ˆ c†

i ˆ

ci+1 − t2 X

even

ˆ c†

i ˆ

ci+1 + ∆ X

• dd

ˆ c†

i ˆ

ci − X

even

ˆ c†

i ˆ

ci ! h.a.

H = −t1 X

• dd

ˆ c†

i ˆ

ci+1 − t2 X

even

ˆ c†

i ˆ

ci+1 + ∆ X

• dd

ˆ c†

i ˆ

ci − X

even

ˆ c†

i ˆ

ci ! h.a.

breaking inversion symmetry & Thouless pump Rice-Mele Hamiltonian

H = −t1 X

• dd

ˆ c†

i ˆ

ci+1 − t2 X

even

ˆ c†

i ˆ

ci+1 + ∆ X

• dd

ˆ c†

i ˆ

ci − X

even

ˆ c†

i ˆ

ci ! h.a.

breaking inversion symmetry & Thouless pump Rice-Mele Hamiltonian

H = −t1 X

• dd

ˆ c†

i ˆ

ci+1 − t2 X

even

ˆ c†

i ˆ

ci+1 + ∆ X

• dd

ˆ c†

i ˆ

ci − X

even

ˆ c†

i ˆ

ci ! h.a.

breaking inversion symmetry & Thouless pump

∆P = a 2π ∆φZak

Rice-Mele Hamiltonian

quantized topological transport in an open spin chain

• D. Linzner, F. Grusdt, M. Fleischhauer, arxiv:1605.00756

model

˙ ρ = Lρ = X

j,µ

⇣ 2Lµ

j ρLµ† j − Lµ† j Lµ j ρ − ρLµ† j Lµ j

⌘

j j + 1

j − 1

LA

j

LB

j

Lindblad generators

LB

j =

√ 1 − ε h (1 − λ) ⇣ ˆ σL,j+1 + ˆ σ+

R,j

⌘ + (1 + λ) ⇣ ˆ σ+

L,j+1 + ˆ

σR,j ⌘i LA

j =

√ 1 + ε h (1 − λ) ⇣ ˆ σL,j + ˆ σ+

R,j

⌘ + (1 + λ) ⇣ ˆ σ+

L,j + ˆ

σR,j ⌘i

LB

j =

√ 1 − ε h (1 − λ) ⇣ ˆ σL,j+1 + ˆ σ+

R,j

⌘ + (1 + λ) ⇣ ˆ σ+

L,j+1 + ˆ

σR,j ⌘i LA

j =

√ 1 + ε h (1 − λ) ⇣ ˆ σL,j + ˆ σ+

R,j

⌘ + (1 + λ) ⇣ ˆ σ+

L,j + ˆ

σR,j ⌘i

model

action of Lindblad generators

λ = +1

j j + 1

λ = −1

j + 1 j

symmetries

j j + 1 j − 1

LA

j

LB

j

LB

j =

√ 1 − ε h (1 − λ) ⇣ ˆ σL,j+1 + ˆ σ+

R,j

⌘ + (1 + λ) ⇣ ˆ σ+

L,j+1 + ˆ

σR,j ⌘i

LA

j =

√ 1 + ε h (1 − λ) ⇣ ˆ σL,j + ˆ σ+

R,j

⌘ + (1 + λ) ⇣ ˆ σ+

L,j + ˆ

σR,j ⌘i

• r

λ = 0

ε = 0

inversion symmetry

σz

j → −σz j

hσz

j i = 0

λ = 0

particle-hole symmetry

σz

R → −σz L

hσz

Ri + hσz Li = 0

steady-state Thouless pump

polarization in finite system with PBC

• R. Resta PRL 80, 1800 (1998)

P = 1 2π Im ln ⌧ exp n i2π L X

j

jˆ nj

• j

j + 1

steady-state Thouless pump

periodic cycle in parameter space steady state is a pure state (dark state)

ε

λ 1 −1

(i)

(ii) (iii)

(iv)

1

−1

P = ⌥1 2 ✓1 2 + λ 1 + λ2 ◆

steady-state Thouless pump

winding !!

periodic cycle in parameter space

1/2

−1/2

parent Hamiltonian

steady-state Thouless pump

H = X

µ

L†

µLµ

= Rice-Mele Hamiltonian: winding à quantized bulk transport

H = − X

j

⇣ t1 ˆ σ−

L,jˆ

σ+

R,j + t2 ˆ

σ−

L,j+1ˆ

σ+

R,j + h.a.

⌘ +∆ X

j

⇣ ˆ σ+

L,jˆ

σ−

L,j − ˆ

σ+

R,jˆ

σ−

R,j

⌘

t1 = 2Γ(1 + ε)(1 − λ2) t2 = 2Γ(1 − ε)(1 − λ2) ∆ = 8Γλ

∆P = a 2π ∆φZak

topological invariant = Zak phase / Chern number

steady-state Thouless pump

inner part of parameter space

Lµ = L†

µ

λ = 0

LA

j → LA j +

p Γ(1 + ε) ⇣ ˆ σ+

L,jˆ

σ+

R,j − ˆ

σ−

L,jˆ

σ−

R,j

⌘ LB

j

→ LB

j +

p Γ(1 − ε) ⇣ ˆ σ+

L,j+1ˆ

σ+

R,j − ˆ

σ−

L,j+1ˆ

σ−

R,j

⌘

˙ ρ = Lρ = X

j,µ

⇣ 2Lµ

j ρLµ† j − Lµ† j Lµ j ρ − ρLµ† j Lµ j

⌘

totally mixed state is also steady state ! à lift degeneracy by (generic) nonlinear term

ε

λ 1 −1 1 −1

steady-state Thouless pump

TEBD simulations in inner part of parameter space Zak phase undefined

winding defines topological invariant

robustness

robustness

Hamiltonian disorder homogeneous local losses

ε

λ

robust to disorder and losses

symmetry protected topological order

symmetry-protected topology

inversion symmetric axes polarization constant & jumps at sigularity

λ = 0 ε = 0

symmetry-protected topology

Inversion symmetry

"

• 1

1

P

• 1/4

1/4

Pλ=0(ε) = Pλ=0(−ε) + 1 2

λ = 0 hσz

j i = 0

"

• 1
• 0.5

0.5 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

h (<L,j

x +<R,j x )2i

h (<L,j

x +<R,j-1 x

)2i

topological singularity

topological singularity

damping spectrum (4 sites) no closing of damping gap

Lρν = −νρν

ρν(t) = ρν(0) e−νt

Re[ν]

eigenvalues of the density matrix (4 sites)

à no degeneracies in the density matrix !

no closing of generalized “purity” gap

detection of topological invariant

detection of top. invariants in interacting systems

topological polarons

• F. Grusdt, N. Yao, D. Abanin, M.F., E. Demler, arxiv:1512.03407

auxiliary system

coupling of spin chain to closed fermion system

H = −t X

j

ˆ c†

j

 1 − 1 2✏ ⇣ ˆ x

Lj + ˆ

x

Rj

⌘2 ˆ cj+1 + h.a. − ⌘ X

j

ˆ c†

jˆ

cj ˆ z

j

j j + 1

j − 1

effective Hamiltonian of auxiliary system

tunneling rates staggered potential

H = −t X

j

ˆ c†

j

 1 − 1 2✏ ⇣ ˆ x

Lj + ˆ

x

Rj

⌘2 ˆ cj+1 + h.a. − ⌘ X

j

ˆ c†

jˆ

cj ˆ z

j

perturbative limit à no effect on open spin system mean-field dynamics in auxiliary system

∆j = ηhˆ σz

j i

H = −t1 X

• dd

ˆ c†

i ˆ

ci+1 − t2 X

even

ˆ c†

i ˆ

ci+1 − X

j

∆jˆ c†

jˆ

cj + h.a.

induced Thouless pump (Rice-Mele)

∆ t1 − t2 ε

ε

λ

summary

notion of topological order in open systems beyond Gaussian systems open problem quantized polarization winding to classify topology also in non-Gaussian systems reservoir induced topological polarization winding in open interacting spin chain

• robust to Hamiltonian perturbation, dephasing & losses
• symmetry protected topological order
• nature of topological singularity unlcear

(no damping gap nor purity gap closing)

• detection scheme

SFB TR 49

thanks to

2010

Dominik Linzner Contributors:

Fabian Grusdt Lukas Wawer