Mini-symposium on non-Hermitian quantum mechanics
2022/07/04 @ U Tokyo (Hongo campus)

Correlation effects on
Z classification of non-Hermitian topology

by Tsuneya Yoshida (U. Tsukuba)with Yasuhiro Hatsugai (U. Tsukuba)

・TY-Hatsugai, PRB 104, 075106 (2021)
c
・TY-Hatsugai, arXiv:2205.09333

Introduction
Novel topological phenomena in
non-Hermitian systems
Exceptional point
non-Hermitian topology:
band touching for both Re(E) and Im(E)
Im(E)

Re(E)
ky

Non-Hermitian skin effects

H. Shen, B. Zhen, and L. Fu PRL (2018)

Amplitudekx

Non-Hermitian topology
Most of eigenstates are localized at the boundary

S. Yao and Z. Wang PRL (2018)

Sites

Unique phenomena in non-Hermitian systems

Non-interacting case(band theory)

Dissipation in cold atoms
One-particle loss

Y. Takasu et al., PTEP(2020)

Two-particle loss

T. Tomita et al, PRA(2019)

Interaction can be tuned in experiments

Cold atoms:
platforms of strong interactions vs. non-Hermitian topology

〜nH topology vs. strong correlations〜
At the non-interacting level, classification of nH topology is done
Z. Gong, et al., PRX (2018)
K. Kawabata et al., PRX (2019)
H. Zhou and J.Y Lee PRB (2019)

Fate of point-gap topology under correlations?
Reduction of Z classification:(1D, with chiral symm.)
Free fermions

Correlations

Fidkowski-Kitaev
PRB(2010)

Contents〜Fate of point-gap topology under strong correlations〜

Part1: 0D with chiral symmetry: Z→Z2
TY-Hatsugai, PRB 104, 075106 (2021)

Part2: 1D with spin-parity symmetry: ZxZ→Z(synthetic/spatial dim.)

TY-Hatsugai, arXiv:2205.09333

cf.) line-gap topo.: Xi et al, Sci. Bull. (2021)

Warm up (Hermitian):
Reduction of Z classification:(1D, with chiral symm.)
Free fermionsFidkowski-Kitaev
PRB(2010)

Correlations

Correlation

one-bodyparameter

Non-interacting
W=4 (topological)~SSH chain×4

W=0 (trivial)
gap-closingPart 1~0-dimensional system with chiral symmetry~

TY-Hatsugai, PRB 104, 075106 (2021)

Overview
0D, chiral symmetryNon-interacting case

24nH point-gap topology
is characterized by a Z-invariant

02

interaction

Correlated case

gap-closing

Topological states with

N0Ch=0 and N0Ch=2
are connected without gap closing

Free fermions

Correlations

N0Ch=0N0Ch=2one-bodyparam.

TY-Hatsugai, PRB 104, 075106 (2021)

Gong et al., PRX (2018)

Definition of point-gap<gap: open>

Kawabata et al., PRX (2019)

ImE<gap: closed>

ReE
Reference energy
Eref=0: energy eigenvalues

ImE

ReE
Reference energy
Eref=0Hamiltonian (one-body)

AAc

Re[ε]

spinless fermionBBc<topological invariant>Im[ε]Gap-closing(point-gap)

Many-body Hamiltonian
fermion with spin 1/2
r=1/2

Acc

UBc

Chiral symmetry (many-body)

If U=0,cc
Hatsugai, JPSJ (2006)
Gurarie, PRB (2011)

2

Non-Hermitian case:4
Correlation

02???
one-body
parameter
Trivial(non-interacting)

Topological(non-interacting)

TY-Hatsugai, PRB 104, 075106 (2021)

Spectral flow β:0→1@U=0

U
N0Ch=0
0

β=0→1
β:0→1
U=0

gap-closing

0U=0

1

β
one-body
param.

1β
U=0

N0Ch=2ImE
ReE

β=0× β=0.5▲ β=1

gap-closing@β=0.5

Spectral flow β:0→1@U=1

α =0.5U01
gap-closing

β:0→10U=1β

one-bodyparam.

1βImE

ReE

β=0× β=0.533▲ β=1

No gap-closing@U=1

Ugap-closing(ii)cf.) gap-closing:

(iii)

(i)

10

N0Ch=0βN0Ch=2

one-bodyparam.

In the presence of interactions,

N0Ch=0 and N0Ch=2
can be smoothly connectedwithout gap-closing !

Z→Z2

Perspective fromc topo. invariants

correlated

freeNon-interacting
・H is quadratic Hamiltonian

Correlated
Many-body chiral symm.

ccchiral symm. of
one-body Hami.
cZ2-invariant
cZ-invariant# of negative eigenvalues ofc

β:0→1
U=1

Summary of part 1

Point-gap topology for 0D AIII
Z→Z2

β:0→1
β=0→1
U=0

U=0

TY-Hatsugai, PRB 104, 075106 (2021)

Part 2~1-dimensional system with spin-parity symmetry~

TY-Hatsugai, arXiv:2205.09333

Overview~ 1D topology with spin-parity and charge U(1) symmetry ~Synthetic dimension
freecorrelated

ZxZZ

freecorrelated

freecorrelated

constraint on spin-flipping terms(spin-parity)
one-body: forbidden
two-body: allowed
Spatial dimension
c ZxZ→ZReduction:

Destruction ofca skin effect by correlations

sitesite

TY-Hatsugai, arXiv:2205.09333

Thank you!