Entanglement Phase Transi0on
Induced by theNon-Hermi0an Skin Eﬀect
ID rId2

Kohei Kawabata (Princeton → UTokyo)
ID rId2

Phys. Rex. X 13, 021007 (2023)

Collaborators

Shinsei Ryu(Princeton)

Tokiro Numasawa(UTokyo, ISSP)

Outline
1. Introduc,on
2. Entanglement suppression induced by thenon-Hermi,an skin eﬀect
3. Entanglement phase transi,on induced by
the non-Hermi,an skin eﬀect
4. Puriﬁca,on induced by the non-Hermi,anskin eﬀect in LindbladiansNon-Hermitian physics

1/30

Despite the enormous success, the exis1ng framework of
topological phases is conﬁned to Hermi<an systems at equilibrium.
Richer properties appear in non-Hermitian systems!
☆ Non-Hermi1city arises from dissipa<on, i.e., exchanges of energy orpar1cles with an environment.
El-Ganainy et al., Nat. Phys. 14, 11 (2018)
・Photonic la*ces with gain/loss
Unidirectional light transport

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

・Finite-lifetime quasiparticles
Bulk Fermi arc due to nonHermi8an self-energy

Kozii & Fu, arXiv:
1708.05841

Non-Hermitian topological systems
・Topological laserNew laser with high eﬃciency due to the interplay ofnon-Hermi0city and topology.

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

・Exceptional point

2/30

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

Nondiagonalizable gapless point (Jordan matrix)

Zhen et al., Nature 525, 354 (2015)

Zhou et al., Science 359, 1009 (2018)

Non-HermiAan skin eﬀect☆ Non-Hermi<an skin eﬀect

3/30

Lee, PRL 116, 133903 (2016); Yao & Wang, PRL 121,086803 (2018); Kunst et al., PRL 121, 026808 (2018)

Localization of an extensive number of eigenmodes due to non-Hermitian topology

・Mechanical metamaterials

Brandenbourger et al., Nat. Commun. 10, 4608 (2019)・Photonic lattice

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

・Electrical circuits

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

・Active matter

Palacios et al., Nat. Commun. 12, 4691 (2021)

Skin effect in quantum physics

4/30

Skin eﬀect has been observed also in recent quantum experiments.
・Quantum walk (single photons)

・Ultracold atoms

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

Liang et al., PRL 129, 070401 (2022)

Skin effect in quantum physics

4/30

Non-Hermitian topology is relevant even to more generic open quantum
systems that are not characterized by Hamiltonians.
Song, Yao & Wang, PRL 123, 170401 (2019)

・Master equation
Non-Hermi*ansuperoperator(Liouvillian)
Chiral damping due to the skin eﬀect
Slowdown of relaxation due to the skin effectHaga et al., PRL 127, 070402 (2021); Mori et al., PRL 125, 230604 (2020)

Motivation and Results
Despite the significance of the skin effect for non-Hermitian topology,
its impact on genuine quantum nature has remained unclear.
We show that the non-Hermi<an skin eﬀect plays an importantrole in the entanglement dynamics of open quantum systems.
(1) The skin eﬀect prohibits the entanglement propaga<on andthermaliza<on, leading to the area law for the steady state.
(2) The skin eﬀect even triggers a new type of entanglement
phase transi<on characterized by a new nonunitary CFT.

Outline
2. Entanglement suppression induced by thenon-Hermi,an skin eﬀectComplex-energy gaps (1)6/30

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 Hermi1an systems:

Hermitian・Energy regions where states areforbidden 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)7/30
KK, Shiozaki, Ueda & Sato,
PRX 9, 041015 (2019)

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

Hermi<an (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 ﬂaNening

8/30

Unitary ﬂa[ening for a point gap:
A non-Hermi1an Hamiltonian with a point gap can be con1nuously
deformed into a unitary matrix while having symmetry and the point gap.
non-Hermitian

unitaryHermitianClassification is well established!

unitary matrix

Line gap: Hermitian flattening

9/30

Hermi<an ﬂa[ening for a line gap:
A non-Hermi1an Hamiltonian with a line gap can be con1nuously
deformed into a Hermi<an matrix while having symmetry and the line gap.
non-Hermi1anHermitianClassification is well established!

Hermitian matrix

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

Classification ofnon-Hermitian topological phases

38-fold symmetry class, 2 types of complex gaps

Intrinsic non-Hermitian topology

11/30

Line-gap topology: stability of Hermitian topology against non-Hermiticity
Point-gap topology: intrinsic non-Hermi<an topology
・Hatano-Nelson model

Hatano & Nelson, PRL 77, 570 (1996)

asymmetric hopping

Winding of complex energy!
(ill-defined in Hermitian systems)
Gong, Ashida, KK et al., PRX 8, 031079 (2018)☆ Skin eﬀect: bulk-boundary correspondence for point-gap topology
Emergence of an extensive number of boundary modes!
Okuma, KK, Shiozaki & Sato, PRL 124, 086801 (2020); Zhang, Yang & Fang, PRL 125, 126402 (2020)

Entanglement dynamics

12/30☆ Entanglement gives important information on the quantum dynamics.

closed quantum systems

Calabrese & Cardy, J. Stat. Phys. P04010 (2005)

Steady-state entanglement is proportional to thevolume of the subsystem (volume law)
(Related to thermaliza:on)

open quantum systems

Skinner et al., PRX 9, 031009 (2019);
Li et al., PRB 98, 205136 (2018)

Entanglement phase transition as a competition between
unitary dynamics and quantum measurements
volume law ↔ area law☆ In 1+1 D, conformal field theory describes the entanglement dynamics.

Skin dynamics
closed quantum systems
diﬀusion

13/30
Calabrese & Cardy, J. Stat. Phys. P04010 (2005)

quantum diffusion (thermalization)→ entanglement propagation
volume law

quasiparticles

open quantum systems with the skin eﬀectskin eﬀectskin eﬀect→ no diﬀusion (no thermaliza<on)!
area law
Kawabata, Numasawa & Ryu, PRX 13, 021007 (2023)

Hatano-Nelson model

14/30

We confirm the entanglement suppression for the Hatano-Nelson model
Hatano & Nelson, PRL 77, 570 (1996)

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

Nonunitary quantum dynamics:

・Quantum walk (single photons)

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

・Ultracold atoms

Liang et al., PRL 129, 070401 (2022)

Skin eﬀect

15/30

Hermi,an case(a)

Non-Hermitian case1.0(b)
timetime

0.8

0.6

0.4

0.20

site

site

Clear signature of the skin eﬀect!

Plots of the evolu1ons of the local par1cle numbers

Skin current

16/30

charge current:

Nonequilibrium steady state with a nonzero current
Thermal equilibrium state:cf. Bloch theorem

Watanabe, J. Stat. Phys. 177, 717 (2019)

Entanglement suppression
The skin effect greatly suppresses the entanglement growth!

volume law

area law

17/30

Outline

3. Entanglement phase transition induced by
the non-Hermitian skin effectSymplecAc Hatano-Nelson model

18/30

In the Hatano-Nelson model, infinitesimal non-Hermiticity leads to the skin effect.
Can we have a phase transition due to the skin effect?
Symplec@c (helical) generaliza@on of the Hatano-Nelson model:

SOC
NH

Compe@@on between non-Hermi@city andspin-orbit coupling!
volume law

Okuma, Kawabata, Shiozaki & Sato, PRL 124, 086801 (2020)
Kawabata, Okuma & Sato, PRB 101, 195147 (2020)
Kawabata & Ryu, PRL 126, 166801 (2021)

cacr
it i

Characterized by a Z2 topological invariant

Δ/Jlno skinZ2 skinarea lawOγ/J

Z2 skin eﬀect protected by TRS† (1)19/30

・Z2 topological phase in Hermi<an systems+Haldane modelZ invariant (Chern #)chiral edge states=:me-reversed partner

Kane-Mele modelZ2 invariant with TRS
helical edge states

・Z2 skin eﬀect in non-Hermi<an systems=+localized at leD (+W)
localized at right (-W)
Hatano-Nelson modelZ invariant (point gap)

:me-reversed partner(reciprocal)

localized at both!

Reciprocal skin eﬀect with
the Z2 invariant protectedby Lme-reversal symmetry†Z2 skin effect protected by TRS† (2)(symmetry-preserving perturba*on)

Okuma, KK, Shiozaki & Sato, PRL 124, 086801 (2020)

Z2 topological, Kramers degeneracy
skin eﬀect!

20/30

reciprocal skin effect!
(Kramers pair)Reciprocal skin effect(a) γ = 0.0(b) γ = 0.421/30(c) γ = 0.5(d) γ = 0.8
1.0↑

time0.8

0.6

0.40.201.0↓

time

0.8

0.6

0.4

0.2

0

site

sitesitesite

Reciprocal skin effect!
Up spins are localized toward the right edge,
and down spins are localized toward the lec edge.
Okuma, Kawabata, Shiozaki & Sato, PRL 124, 086801 (2020); Kawabata, Okuma & Sato, PRB 101, 195147 (2020)

Spin current

The nonequilibrium steady state is characterized bya nonzero spin current!

22/30

Entanglement phase transiAon
The skin effect induces an entanglement phase transition!

volume law

area law23/30

Nonequilibrium quantum criticality

24/30

The steady-state entanglement entropy at the cri<cal point

The entanglement entropy is well ﬁ[ed with the CFT descrip<onCalabrese & Cardy, J. Stat. Phys. P06002 (2004)

However, the central charge is parameter dependent
nonunitary (irra<onal) CFT

CriAcal skin eﬀect

25/30

What is the origin of the nonequilibrium quantum criticality?
Scale invariance of the skin modes!
☆ Around the cri<cal point, the localiza<on length diverges.

☆ At the cri<cal point, the skin modes decay with the power law.

The power-law decay arises from an excep<onal point!
Single-par=cle Schrödinger equa=on

Jordan matrix (nondiagonalizable)☆ New type of critical phenomena unique to open quantum systems.

Periodic boundary conditions

26/30

The skin eﬀect is crucial for the universality class.
Entanglement dynamics under the PBC:

The CFT scaling is well, but the central charge is a constant!
Mechanisms of the nonequilibrium quantum cri<cality:
OBC: cri<cal skin eﬀect (scale-invariant skin modes due to an EP)
PBC: no skin effect occurs; real-complex spectral transition

The scale invariance at the critical point looks similar to thecritical phenomena at thermal equilibrium.

However, our phase transi<ons provide a new nonequilibrium
quantum cri<cality unique to open quantum systems.
(mechanism: an excep1onal point and concomitantscale-invariant skin modes)
New universality class of nonequilibrium quantum phase transi1onin open quantum systems.

Outline

4. Puriﬁca,on induced by the non-Hermi,anskin eﬀect in LindbladiansLiouvillian skin eﬀect

28/30☆ The skin effect occurs also in the quantum master equation.
Song, Yao & Wang, PRL 123, 170401 (2019)
Znidaric, PRE 92, 042143 (2015);
Haga et al., PRL 127, 070402 (2021)

dissipa@ve hopping

Ini1al state: completely mixed state(single-parAcle subspace)Open quantum dynamics subject to the skin effect?

Purification

29/30

The skin eﬀect induces puriﬁca<on!

dynamics

steady stateSuppression of von Neumann entropy 30/30
The skin eﬀect also suppresses von Neumann entropy!

dynamics

steady state

Puriﬁca1on can arise also in the condi1onal dynamics under
Gullans & Huse, PRX 10, 041020 (2020)
quantum measurements.
The skin eﬀect induces puriﬁca<on even in the master equa<on!
(averaged over mul1ple quantum trajectories)

SummaryPhys. Rev. X 13, 021007 (2023)

・The skin eﬀect prohibits the entanglement growth and thermaliza1on,leading to the area law of the entanglement entropy.
・The skin eﬀect triggers a new type of entanglement phase transi1onthat is characterized by an anomalous nonunitary CFT.
・The skin eﬀect leads to the puriﬁca1on also in the master equa1on.

cacrit iΔ/Jl

volume lawarea lawOγ/J