Bulk-edge correspondence in nonHermitian systems & physics sensitive
vs. insensitive to the boundarycondition

Ken ImuraIIS, UTokyo

Takahiro Orito, Yositake Takane
Hiroshima UniversityThe outline
1) Bulk-edge correspondence (BEC) in Hermitian
topological systems: BEC-1
sensitive2) BEC in non-Hermitian topological systems: BEC1’
3) A new type of BEC in non-Hermitian systems
exhibiting skin effect: BEC-2
4) Non-Hermitian wave-packet dynamics

insensitivePhysics sensitive vs. insensitive to theboundary condition
in non-Hermitian systems

1) Bulk-edge correspondence in Hermitian topologicalsystems: BEC-1
A prototype: the SSH modelor the Su-Schrieffer-Heeger modelobc, in real space(Bulk) energy bands:

pbc

Bloch Hamiltonian in k-space, pbc;
energy spectrum/dispersion

The Bloch Hamiltonian:

Asboth, et al., Lecture Notes in Physics, 919 (2016)

The bulk-boundary correspondence
Edge states, (topological) phasetransition
no edge stateobc(a pair of) states

The winding number: ~ Berry/geometric phase
1 for
0 forpbc

2) BBC in non-Hermitian topological systems: BBC-1’- Non-Hermitian system (in particular, the HatanoNelson type model) is sensitive to boundary
conditions; so is a topological insulator…
A prototype: Hatano-Nelson model
anisotropic/non-reciprocal hopping:

The spectrum:

pbc: complex
obc: realpbc
obc

The wave function:
pbc: plane-wave likeobc: skin effectNon-Hermitian topological insulator
A paradigmatic example:
SSH×Hatano-Nelson model- anisotropic/non-reciprocal
hopping (Hatano-Nelson)- alternating hopping (SSH)(Apparent) breakdown of the bulk-edge correspondence:
OBCPBCgap closing under obc

gap closing under obc

The underlying reason: non-hermitian skin effectYao & Wang, PRL 2018

To recover the bulk-edge correspondence in non-Hermitian
topological systems

two recipes are known:
Imura & Takane, PRB 2019; PTEP 2020
Recipe 1 (our recipe):
Problem: the bulk (at least, under the standard pbc) was“wrong”boundary/edge

pbc- choose to modify the bulk
- modify/generalize the periodic boundary condition
generalized pbc (gpbc)
Recipe 2: an alternative scenario- the so-called “non-Bloch” approach
Yao & Wang, PRL 2018
- abandons pbc- completes a BEC within the obc Yokomizo & Murakami, PRL 2019

~ the underlying idea ~The generalized PBC:

e.g., case of Hatano-Nelson model

The boundary Hamiltonian:Correspondingly, generalized Bloch Hamiltonian, generalized Blochstates, and the generalized Brillouin zone

Remarks:- Eigen wave functions under obc arecomposed of solutions withobc- The (naive) periodic boundary
condition: b=1 automatically selects those solutions with b=1
inadequate for describing topological phase transitions under obc

Application to our paradigmatic example:

the SSH×Hatano-Nelson model
The generalized PBC:

b: free parameter

The boundary Hamiltonian:Generalized Bloch Hamiltonian, generalized Bloch states:

generalized Brillouin zone

To identify different topological phasesa pair of chiral winding numbers
where- Mapping:Yao & Wang, PRL 2018

- The trajectory ofMeasure of how many times
winding around the origin- Hermitian limit:

originx

TI: w=1

A pair of E=0edge states

OI: w=0

No edge state

Bulk-edge correspondencePhase diagram
b: free parameter

OI: w=0

TI: w=1: using the free parameter b chosen on this path, the BEC
is recovered !!

Remarks:

meaning of the phase boundaries

Phase boundaries areat the zeros:cf.

Are they all physically meaningful?
Do they all correspond to the physical phase boundary
under obc?
obc: open boundary condition
Bulk gap closing under obc
At the gap closing, bulk states appear at E=0
The bulk condition:satisfied at E=0

whereYao & Wang, PRL 2018
Yokomizo & Murakami, PRL 2019

obc

Spectrum:obc vs. pbc
Phase boundaries inthe generalizedparameter space:- The gap closingunder obc are atpbc
log bspecifying
gpbc
pbc
b=1

- A spinout: gap closing
under pbc

An alternative scenario: the so-called non-Bloch approach
Generalized Bloch
Hamiltonian:- secular equation:quartic equation for- fundamental solutions:wherebulk solutions evanescent modes
The eigenstates under obc are superpositions of thefundamental solutions:

The “bulk condition”:
at m=2

Yao & Wang,
PRL 2018
Yokomizo & Murakami,
PRL 2019

The generalized PBC~ the recipe ~Imura & Takane, PRB
2019; PTEP 2020

To recover the BEC in the paradigmatic non-Hermitian model,
specifying the parameter:
must be chosen so as to satisfy

Physics sensitive vs. insensitive to the boundary condition:4) Non-Hermitian wave-packet dynamics
Let us consider again the HatanoNelson model:- asymmetric/non-reciprocal hopping:- Hatano-Nelson×Aubry-Andre model:

Aubry-Andre model: quasi-periodic disorder

Aubry & Andre, AIPS ’80

an irrational constante.g., (chosen typically to be)disorder configuration- Wave-packetdynamics:

sample average- initial wave packet:
localized at asingle siteSimulation of the wave-packet dynamics:(a) Hermitian case- clean limit: W=0- weakly disordered case: W=1.0

pbc: periodic
boundary condition- initial wave packet:[type of disorder: quasi-periodic
potential disorder (AA model)]- Case of stronger disorder: W=3.0(localized phase)

Simulation of the wave-packet
dynamics: (a) Hermitian case(continued)[type of disorder: quasiperiodic potential disorder(AA model)]Notes on Localization-delocalization transition in the AA model- in 1D such a transition usually does not occur; cf. Anderson ’58- Aubry-Andre model is an exceptional case:
Localization length:atcf. Longhi, PRB ’21

Simulation of the wave-packet dynamics:(b) non-Hermitian case- clean limit: W=0- (weakly) disordered case: W=1.0

model: Hatano-Nelson ×
Aubry-Andre model;[type of disorder: quasi-periodic potentialdisorder (AA model)]pbc
periodic boundary condition

- Case of stronger disorder: W=3.0(still in the extended phase)- Case of even stronger disorder(localized phase): W=4.0

Simulation of the wave-packetdynamics: (b) non-Hermitian case(continued)[type of disorder: quasiperiodic potential disorder(AA model)]- phase diagram: delocalizationlocalization transition
delocalized

localized

pbc
periodic boundary condition

Localization-delocalization transitionin the Hatano-Nelson×Aubry-Andre model
wave function:

spectrum (eigenvalues):
complexreal

delocalized
localized

Hatano & Nelson,
PRL ’96- Localization length in the Aubry-Andre model:*in the Hermitian case:- in the Non-Hermitian case:at- phase diagram
delocalized

delocalization point:

localized

IPR: inverse participation ratio

Comparison of periodic vs. open
boundary conditions- clean limit: W=0

case of pbc:
the periodic boundary conditions

Simulation of the wavepacket dynamics:(b) non-Hermitian casegaussian- case of weak disorder: W=1.0

gaussianComparison of periodic vs. open
boundary conditions- clean limit: W=0

case of obc:
the open boundary conditions- case of weak disorder: W=1.0

Simulation of the wavepacket dynamics:(b) non-Hermitian case

pbc vs. obc(continued)

Simulation of the wave-packet dynamics:(b) non-Hermitian case

case of pbc:
the periodic boundary conditions- Case of stronger disorder: W=3.0 (still in the extended phase)- Case of even stronger disorder: W=4.0 (localized phase)pbc vs. obc(continued)

Simulation of the wave-packet dynamics:(b) non-Hermitian case

case of obc:
the open boundary conditions- Case of stronger disorder: W=3.0 (still in the extended phase)- Case of even stronger disorder: W=4.0 (localized phase)Summary1) Bulk-edge correspondence (BEC) in Hermitian
topological systems: BEC-1sensitive2) BEC in non-Hermitian topological systems: BEC1’
3) A new type of BEC in non-Hermitian systems
exhibiting skin effect: BEC-2
4) Non-Hermitian wave-packet dynamics

insensitive

Physics sensitive vs. insensitive to theboundary condition
in non-Hermitian systems

Summary/conclusions:Physics sensitive vs. insensitive to the boundary condition
1) The topological insulator:- the underlying mechanism: bulk-boundary correspondence (BBC)- BBC superficially breaks down in non-Hermitian systems;
two possible way out: one modifies, the other abandons pbc2) The Hatano-Nelson model:- 1D tight-binding model with non-reciprocal (asymmetric) hopping- Localization/delocalization transition in 1D- Non-Hermitian skin effect; a topological interpretation
Take-home message:
Nature of the localization/delocalization transition in Hatano-Nelson model
multifaceted/upgradedi) complex vs. real spectra;ii) delocalization-localization transition;iii) skin effect vs. no skin effect (obc)iv) topological transition;

A quadruple phase transition!

From the viewpoint of bulk-boundary correspondence:- Bulk-boundary correspondence (BBC) multi-faceted:

obc vs. pbc
skin effectthe original BBC doublet

winding number

The Hatano-Nelson quadruplet

IPR: inverse
participation ratio

ratio of complex
eigenvalues

The triplet of bulk phase transitions

3) Wave-packet dynamics- Hermitian case: quantum interference- Non-Hermitian casei) Clean limit: pseudo-classical diffusive dynamicsii) Disorder enhances quantum interference(Hatano-Nelson bulk triplet)4) On-going studies on :i) the effects of interaction; many-body caseii) entanglement/entropy dynamics
Orito & Imura, Phys. Rev. B 105, 024303 (2022)
Considering also…
+ wave-packet & entanglement/entropy dynamicsi) wave-packet dynamics
COMvelocity

wave packetspreadingii) entanglement/entropy dynamics
Maximal entanglement entropy

V=0

The Hatano-Nelson “sextuplet”

V=2

The modified periodic boundary condition
specified by a real positive parameter: b

to be added toIn a transformed basis:- A similarity transformation:- Under the transformed basis:standard pbc

The meaning of the modified periodic boundary condition(continued)- back in the original basis:

The coefficientswhereare determined by the eigenvalue equation:

Consistency of the two approches:
1) our modified periodic boundary condition

vs.

with 2) the so-called generalized Brillouin zone approach
Yokomizo & Murakami, arXiv:1902…

- So far, b has been a free parameter- We can choose our b to be the one corresponding to
the generalized Brillouin zone at the bottom of the band

ourw=1 region
gap closingBulk-edge correspondence visualized
under the modified periodic boundary condition- The modified periodic
boundary condition (mpbc):- continuous change of theboundary condition:
mpbc: purely bulk spectrum

obc: bulk+edge spectrum

smooth evolution = BEC

Proof: smoothly connected
non-Hermitian regime

Hermitian limit- phase boundaries in the non-hermitian regime:

Recall:

BEC firmly established
Ryu-Hatsugai line
- The phase boundaries in thehermitian limit:

hermitianline: b=1