Symmetry and Topology inNon-Hermitian Physics
Kohei Kawabata(The University of Tokyo)
ID rId2

ID rId2

Pseudo-Hermitian Hamiltonians in Quantum Physics
December 9th, 2021

Collaborators

Nobuyuki Okuma(Kyoto)

Ken Shiozaki(Kyoto)

Shinsei Ryu(Princeton)

Masatoshi Sato(Kyoto)

PHHQP

1/35

Pseudo-Hermitian Hamiltonians in Quantum Physics
PT symmetry and pseudo-Hermicity are important forreal spectra of non-Hermitian Hamiltonians.
Other symmetry in non-Hermitian physics?
(charge conjugation, chiral, …)☆ We develop general symmetry classification in non-Hermitian physics.
(internal)[Hermitian][non-Hermitian]

10-fold

38-fold

Altland & Zirnbauer,
PRB 55, 1142 (1997)

KK, Shiozaki, Ueda & Sato,
PRX 9, 041015 (2019)

Topological phases of matter

2/35☆ Topological phases:
Phases of matter characterized by topology of the wavefunctions
・The first example: integer quantum Hall effect (1980s)
TKNN, PRL 49, 405 (1982)
Phases are characterized by the top. inv. (Chern #)Robust gapless edge states appear (bulk-edge correspondence)

Topological phases are everywhere, even in band theory!
・Z2 topological insulator (TI)
Quantum spin Hall effect (helical edge states)

Kane & Mele, PRL 95, 146802 (2005)

・Topological superconductor (TSC)
Majorana edge states→ topological quantum computation?

Kitaev, Phys.-Usp. 44, 131 (2001)

Periodic table for TIs and TSCs

3/35

General and comprehensive theoretical framework of TIs and TSCs:
Periodic table based on spatial dimension and symmetry

Quantum Hall insulator

Kitaev/Majorana chain
Quantum spin Hall insulator

Schnyder, Ryu, Furusaki & Ludwig, PRB 78, 195125 (2008)
Kitaev, AIP Conf. Proc. 1134, 22 (2009)Non-Hermitian physics

4/35

Despite the enormous success, the existing framework oftopological phases is confined to Hermitian systems at equilibrium.
Richer properties appear in non-Hermitian systems!
☆ Non-Hermiticity arises from dissipation, i.e., exchanges of energy orparticles with an environment.
・Photonic lattices with gain/loss
Unidirectional light transport

Regensburger et al.,Nature 488, 167 (2012)

・Finite-lifetime quasiparticles
Bulk Fermi arc due to nonHermitian self-energy

Kozii & Fu, arXiv:
1708.05841

Non-Hermitian topological systems 5/35
・Topological laser

New laser with high efficiency due to the interplay of
non-Hermiticity and topology.

Bandres et al., Science 359, eaar4005 (2018)

Zhao et al., Science 365, 1163 (2019)

・Non-Hermitian skin effectAnomalous localization of an extensive number of eigenstates due to non-Hermiticity.

Xiao et al., Nat. Phys.
16, 761 (2020)

Helbig et al., Nat. Phys.
16, 747 (2020)

Weidemann et al.,Science 368, 311 (2020)

Non-Hermitian topological systems 5/35
・Topological laser

New laser with high efficiency due to the interplay ofnon-Hermiticity and topology.

・While conventional physics focuses on Hermitian systems,nnon-Hermiticity is natural and everywhere.
・Unconventional phenomena occur as a result of theminterplay
betweennon-Hermiticityandtopology.
Zhao etal., Science 365, 1163 (2019)
Bandres et al., Science359, eaar4005(2018)
・Non-Hermitian skin effect

topology
non-Hermiticity
Anomalous
localization of an extensive number of eigenstates
due to non-Hermiticity.
interplay

Xiao et al., Nat. Phys.
16, 761 (2020)

Helbig et al., Nat. Phys.
16, 747 (2020)

Weidemann et al.,Science 368, 311 (2020)

☆ Non-Hermiticity gives rise to unique topological phasesthat have no Hermitian analogs.
Characterization? Bulk-boundary correspondence? Physical phenomena?

・Classification
Two types of complex-energy gaps (point and line gaps)

・Skin effect: new type of bulk-boundary correspondence
Intrinsic non-Hermitian topology leads to the skin effectSymmetry-protected skin effect・Topological field theory
Universal description of non-Hermitian topological phenomena
The skin effect is a signature of a non-Hermitian anomaly

Outline
1. Introduction
2. Symmetry in non-Hermitian physics
3. Topology in non-Hermitian physics
4. Non-Hermitian skin effects
5. Topological field theory

Symmetry inNon-Hermitian Physics
ID rId2

Kawabata, Higashikawa, Gong,
Ashida & Ueda, Nat. Commun. 10, 297 (2019)
ID rId2
Kawabata, Shiozaki, Ueda & Sato, PRX 9, 041015 (2019)10-fold symmetry class

7/35☆ 3-fold symmetry class by Wigner & Dyson

time reversal±1anti-unitary(with complex conjugation)☆ 10-fold symmetry class by Altland & Zirnbauer

Wigner (1959)
Dyson, J. Math. Phys.
3, 1199 (1962)

Altland & Zirnbauer,
PRB 55, 1142 (1997)

particle holeanti-unitarychiral(sublattice)unitary・Universality
Random matrix theory, topological classification, ……

Hermitian

Non-Hermitian?

Hermitian

Non-Hermitianunification

ramification

Symmetry ramification (1)9/35☆ Symmetry ramifies (bifurcates) in non-Hermitian systems.
KK, Shiozaki, Ueda & Sato,
PRX 9, 041015 (2019).

Hermitian TRS

Non-Hermitian TRS
time reversal

reciprocity

Two types of symmetry appear due to the distinction of complex
conjugation and transpose operation in non-Hermitian systems!

Symmetry ramification (2)10/35☆ Symmetry ramifies (bifurcates) in non-Hermitian systems.
KK, Shiozaki, Ueda & Sato,
PRX 9, 041015 (2019).

Hermitian CS=SLS

Non-Hermitian CS/SLS
sublatticechiralTwo types of symmetry appear due to the distinction of complex
conjugation and transpose operation in non-Hermitian systems!

Hermitian

Non-Hermitian

unification

ramification

Symmetry unification (1)11/35☆ Antiunitary symmetries distinct in Hermitian systems
are unified in non-Hermitian systems.
KK et al., Nat. Commun.

10, 297 (2019)

Two antiunitary symmetries are distinct for HermitianIf we allow non-Hermitian, they are equivalent!

one-to-one mapping(wavefunctions are invariant)Symmetry unification (2)12/35☆ Antiunitary symmetries distinct in Hermitian systems
are unified in non-Hermitian systems.
KK et al., Nat. Commun.

10, 297 (2019)Symmetry unification (2)12/35☆ Antiunitary symmetries distinct in Hermitian systems
are unified in non-Hermitian systems.
KK et al., Nat. Commun.

10, 297 (2019)

38-fold symmetry class

13/35

・Hermitian case: 10 classes (AZ symmetry class)time reversal, particle hole, and chiral (=sublattice)・Non-Hermitian case: 38 classes
10-fold non-Hermitian AZ symmetry class

KK, Shiozaki, Ueda & Sato,
PRX 9, 041015 (2019).

10-fold non-Hermitian AZ symmetry class(Hermitian conjugate of the AZ class)22-fold non-Hermitian AZ symmetry class with sublattice symmetry(NOT equivalent to chiral symmetry)

10 + 10 + 22 – 4 = 38 symmetry classes
unification

38-fold symmetry class

13/35

・Hermitiancase: 10 classes(AZleadsymmetryclass)physics:
New symmetry
classes
to newtime reversal, particle hole, and chiral (=sublattice)・Non-Hermitian random matrix theory

Hamazaki, KK, Kura & Ueda, PRRKK,2,Shiozaki,023286Ueda(2020)& Sato,・Non-Hermitian case: 38 classes

PRX 9, 041015 (2019).

・Non-HermitianAnderson
10-fold
non-Hermitian AZ symmetry
classlocalization
KK & Ryu, PRL 126, 166801 (2021)
・Non-Hermitian topological phases

10-fold non-Hermitian AZ symmetry class(gapped) KK, Shiozaki, Ueda & Sato, PRX 9, 041015 (2019)(gapless) KK, Bessho & Sato, PRL 123, 066405 (2019)(Hermitian conjugate of the AZ class)・Non-Hermitianskin effects
22-fold
non-Hermitian AZ symmetry
class with sublattice symmetry
chiral symmetry)
Okuma, KK, Shiozaki &(NOTSato,equivalent
PRL 124, to086801(2020)
10 + 10 + 22 – 4 = 38 symmetry classes
unification

Topology in
Non-Hermitian Physics
ID rId2

ID rId2
Gong, Ashida, Kawabata, Takasan,Higashikawa & Ueda, PRX 8, 031079 (2019)

Kawabata, Shiozaki, Ueda & Sato, PRX 9, 041015 (2019)
Kawabata, Bessho & Sato, PRL 123, 066405 (2019)

Complex-energy gaps (1)14/35

An “energy gap” is needed to define a topological phase.
However, a non-Hermitian extension of an “energy gap”
is nontrivial since the spectrum is complex.
Energy gap in Hermitian systems:

Hermitian・Energy regions where states are
forbidden to be present.
・They should be point-like (0D)since the real spectrum is 1D.

Since the complex spectrum is 2D (real and imaginary),such vacant regions can be either 0D or 1D!

Complex-energy gaps (2)15/35
KK, Shiozaki, Ueda & Sato,
PRX 9, 041015 (2019)

Point gap (0D)
Two types ofcomplex-energy gaps!

Hermitian (0D)

Gong, Ashida, KK et al.,
PRX 8, 031079 (2018)

Line gap (1D)Esaki, Sato, Hasebe & Kohmoto,
PRB 84, 205128 (2011)
Shen, Zhen & Fu, PRL 120, 146402 (2018)
KK et al., Nat. Commun. 10, 297 (2019)

NOTE: The definition that should be adopted depends on the individual
physical situations that we are interested in.

Point gap: unitary flattening

16/35

Unitary flattening for a point gap:
A non-Hermitian Hamiltonian with a point gap can be continuously
deformed into a unitary matrix while having symmetry and the point gap.
non-Hermitian

unitary

HermitianClassification is well established!

unitary matrix

Line gap: Hermitian flattening

17/35

Hermitian flattening for a line gap:
A non-Hermitian Hamiltonian with a line gap can be continuously
deformed into a Hermitian matrix while having symmetry and the line gap.
non-Hermitian

HermitianClassification is well established!

Hermitian matrix

KK, Shiozaki, Ueda & Sato, PRX 9, 041015 (2019)

Classification of
non-Hermitian topological phases

38-fold symmetry class, 2 types of complex gaps

Intrinsic non-Hermitian topology

19/35Line-gap topology: stability of Hermitian topology against non-Hermiticity
Point-gap topology: intrinsic non-Hermitian topology
・Hatano-Nelson model

Hatano & Nelson, PRL 77, 570 (1996)

asymmetric hopping

Winding of complex energy!

Im E

1.0
0.5
0.0-0.5(ill-defined in Hermitian systems)-1.0-3-2-10Re E

123Gong, Ashida, KK et al., PRX 8, 031079 (2018)☆ Physical consequences of intrinsic non-Hermitian topology?
bulk-boundary correspondence, topological field theory, ……

Topological Origin ofNon-Hermitian Skin Effects
ID rId2

rId2
Okuma, Kawabata,IDShiozaki& Sato, PRL 124, 086801 (2020)

Kawabata, Okuma & Sato, PRB 101, 195147 (2020)
Kawabata, Sato & Shiozaki, PRB 102, 205118 (2020)

Bulk-boundary correspondence (BBC)20/35

Emergence of boundary states due to nontrivial bulk topology.
Non-Hermiticity alters the nature of BBC!
e.g. Non-Hermitian (non-reciprocal) SSH model

asymmetric hopping
Sublattice symmetry:winding number (integer, Z)☆ Zero modes appear, but are localized only at one edge.
Lee, PRL 116, 133903 (2016); Yao & Wang, PRL 121, 086803 (2018)

Non-Hermitian skin effect

21/35
2

Non-Hermitian skin effect
All the eigenstates are localized at the edges!
The bulk Hamiltonian strongly depends on
the boundary conditions!

1OBC (arcs)

0-1

PBC (loops)

-2
-2

-1

0

1

2

Modified bulk-boundary correspondence
Zero modes are predicted byrather than

Yao & Wang, PRL 121, 086803 (2018); Kunst et al., PRL 121, 026808 (2018)

BBC in non-Hermitian systems

22/35

The modified BBC is developed for line-gap topology.
Non-Hermitian Hamiltonian with a line gap
Hermitian Hamiltonian
The modified BBC210line gap(→ Hermitian)-1-2-2-1012☆ BBC for point-gap topology?
Non-Hermitian Hamiltonian with a point gap
Unitary Hamiltonian
Not necessarily deformed into a Hermitian one
Intrinsic to non-Hermitian Hamiltonians
NO modified BBCpoint gap(→ unitary)

Hatano-Nelson model

23/35

・Hatano-Nelson model
asymmetric hopping
Topological invariant (winding #)Hatano & Nelson, PRL 77, 570 (1996)

・Under PBC,
・Under OBC, however,the skin effect occurs
no point gap is openGong, Ashida, KK et al.,
PRX 8, 031079 (2018)

Skin effect as point-gap topology
Is it possible to haveNo!

24/35

with point-gap topology?
is always topologically trivial for a point gap.
(can be nontrivial for a line gap)

Then, what does point-gap topology imply?
Point-gap topology leads to the non-Hermitian skin effect!
(intrinsic NH topology)(intrinsic NH phenomenon): skin effect: no skin effectOkuma, KK, Shiozaki & Sato, PRL 124, 086801 (2020)
cf. K. Zhang, Z. Yang & C. Fang, PRL 125, 126402 (2020).
Same conclusion, but different approach(BBC for intrinsic non-Hermitian topology)

Skin effect

Point-gap topology

Classification?

Classification

Symmetry-protectedskin effects?

Symmetry-protected
topological phases

Z2 skin effect protected by TRS (1)26/35

・Z2 topological phase in Hermitian systems+
Haldane modelZ invariant (Chern #)chiral edge states=
time-reversed partner

Kane-Mele modelZ2 invariant with TRS
helical edge states

・Z2 skin effect in non-Hermitian systems=+
localized at left (+W)
localized at right (-W)
Hatano-Nelson modelZ invariant (point gap)

time-reversed partner(reciprocal)

localized at both!

Reciprocal skin effect with
the Z2 invariant protectedby time-reversal symmetry

Z2 skin effect protected by TRS (2)(symmetry-preserving perturbation)

Z2 topological, Kramers degeneracy
skin effect!

reciprocal skin effect!
(Kramers pair)

27/35

Topological Field Theory ofNon-Hermitian Systems
ID rId2

ID rId2
Kawabata, Shiozaki& Ryu, PRL 126, 216405 (2021)

Topological field theory

28/35☆ Topological phenomena are universally described by field theory.
・(2+1)-D Chern-Simons theory: quantum Hall effect

Oh, Science 340, 153 (2013)is gauge dependent in the presence of a boundary (anomaly)
Compensated by an anomaly at the boundary, i.e., chiral edge states.
(field-theoretic understanding of bulk-boundary correspondence)・Axion electrodynamics for (3+1)-D topological insulators

Qi, Hughes & Zhang, PRB 78, 195424 (2008)

Derivation of topological field theory
The topological action

29/35

is derived from a microscopic Hamiltonian(gauge potential)

Let us integrate out the matter degrees of freedom (i.e., )effective action:
The topological invariant is given by the Green’s function:e.g. Chern insulator(2+1)-D Chern-Simons theory☆ An energy gap ensures the well-defined topological action(matter)

Breakdown of the spacetime formulation
Can we have a topological action from a non-Hermitian Hamiltonian?
(e.g., Hatano-Nelson model)
No, we do not have a well-defined action!
Point gap: open (gapped)
Line gap: closed (gapless)

1.0
0.5
0.0
-0.5

Divergence of the action-1.0-3-2-10123(The system looks like a metal rather than an insulator)

The above formulation assumes the Gibbs state for equilibrium,and focuses on the ground state.
Intrinsic non-Hermitian topology: out of equilibrium
The Gibbs state and the ground state are no longer relevant……

Space formulation

31/35“Time” should play a special role out of equilibrium
Let us Fourier transform
Spatial field theory

KK, Shiozaki & Ryu, PRL 126, 216305 (2021)

generating function of the Green’s function
captures all physical information!
☆ We have a well-defined action for intrinsic non-Hermitian topology!
(Green’s function)[(d-1)+1]-D Hermitian systems(d+0)-D Non-Hermitian systems

One dimension
For32/35

in 1D (such as the Hatano-Nelson model),(1+0)-D Chern-Simons theory!
cf. (0+1)-D CS theory for 0D Hermitian systems
・Current:
Nonreciprocal transport due to non-Hermiticity!
Application: directional amplification (laser)

Wanjura et al., Nat. Commun. 11, 3149 (2020)

Skin effect as an anomaly

33/35

1D non-Hermitian systems:
Gauge transformation:
The action transforms as
NOT gauge invariant! (anomaly)
To retain gauge invariance, additional system is needed at the boundary:(phase of the wavefunction)

is gauge invariant
describes a pair of charges at the boundary: skin modes☆ The non-Hermitian skin effect is a signature of an anomaly!

Three dimensions: chiral magnetic effect
Generally, d-dimensional non-Hermitian systems are described bythe (d+0)-dimensional Chern-Simons theory (d: odd)
In three dimensions, we have the (3+0)-D Chern-Simons theory:cf. (2+1)-D CS theory for 2D Hermitian systems (QHE)

・Current:
electric fieldchiral magnetic effect!
non-Hermiticity

Fukushima et al., PRD78, 074033 (2008)

・Application: directional laser controlled by a magnetic field
cf. lattice realization: Bessho & Sato, PRL 127, 196404 (2021)

Chiral magnetic skin effect35/35

・(2+1)-D Chern-Simons theory (quantum Hall effect)
Boundary exhibits a chiral anomaly:(axial current)・(3+0)-D Chern-Simons theory for 3D non-Hermitian systems
Boundary also exhibits a chiral anomaly (t → y, E → B):
The number of the skin modes is given by the number of the fluxes

skin modeszyxSummary・38-fold internal symmetry in non-Hermitian physics.
・Non-Hermiticity gives rise to unique topological phases without
Hermitian analogs.
・Intrinsic non-Hermitian topology leads to the skin effects.
・Spatial field theory for intrinsic non-Hermitian topology; the skineffect is a signature of a non-Hermitian anomaly.