Entanglement Phase Transition
Induced by theNon-Hermitian Skin Effect
ID rId2

Kohei Kawabata (Princeton University)
Collaborators: Tokiro Numasawa (UTokyo)
Collaborators: Shinsei Ryu (Princeton)
ID rId2

arXiv:2206.05384

Non-Hermi6an physics

1/13

Conven&onally, physicists focused on Hermi&an systems at equilibrium.
Richer proper8es appear in non-Hermi8an systems!
☆ Non-Hermiticity arises from dissipation, i.e., exchanges of energy orparticles with an environment.
El-Ganainy et al., Nat. Phys. 14, 11 (2018)
・Photonic lattices with gain/loss
Unidirec(onal light transport

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

・Finite-lifetime quasiparticles
Bulk Fermi arc due to nonHermi(an self-energy

Kozii & Fu, arXiv:
1708.05841

Non-Hermitian skin effect

2/13☆ Non-Hermitian skin effect Lee, PRL 116, 133903 (2016); Yao & Wang, PRL 121, 086803 (2018)
Localiza(on of an extensive number of eigenmodes due to non-Hermi(an topology

・Photonic la?ce

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

・Mechanical metamaterials

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

Entanglement dynamics3/13☆ Entanglement gives important informa8on on the quantum dynamics.

closed quantum systems

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

Steady-state entanglement is propor1onal to thevolume of the subsystem (volume law)
(Related to thermalization)

open quantum systems

Chan et al., PRB 99, 224307 (2019);
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.

Mo1va1on and Results
The nature of entanglement in open quantum systems
has remained largely elusive.
We show that the non-Hermi8an skin eﬀect plays an importantrole in the entanglement dynamics of open quantum systems.
(1) The skin effect prohibits the entanglement propagation andthermalization, leading to the area law for the steady state.
(2) The skin effect even triggers a new type of entanglement
phase transition characterized by a new nonunitary CFT.

Skin dynamics
closed quantum systems
diﬀusion

5/13
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 thermaliza8on)!
area law
Kawabata, Numasawa & Ryu, arXiv:2206.05384

Hatano-Nelson model

6/13

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

7/13

Hermi7an case(a)

Non-Hermitian case1.0(b)
timetime

0.8

0.6

0.4

0.20sitesite

Clear signature of the skin eﬀect!

Plots of the evolu&ons of the local par&cle numbers

Entanglement suppression
The skin eﬀect greatly suppresses the entanglement growth!

volume law

area law

8/13

Symplec1c Hatano-Nelson model

9/13

In the Hatano-Nelson model, inﬁnitesimal non-Hermi<city leads to the skin eﬀect.
Can we have a continuous phase transition due to the skin effect?
Symplectic (helical) generalization of the Hatano-Nelson model:

SOCNHCompe;;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 law
Oγ/JReciprocal skin eﬀect(a) γ = 0.0(b) γ = 0.4

10/13(c) γ = 0.5(d) γ = 0.81.0

↑

time

0.8

0.6

0.4

0.2

0

1.0

↓

time

0.8

0.6

0.4

0.20

site

sitesitesite

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

Entanglement phase transition
The skin eﬀect induces an entanglement phase transi8on!

volume law

area law

11/13

Nonequilibrium quantum cri1cality

12/13

The steady-state entanglement entropy at the critical point

The entanglement entropy is well ﬁXed with the CFT descrip8onCalabrese & Cardy, J. Stat. Phys. P06002 (2004)

However, the central charge is parameter dependent
nonunitary (irra8onal) CFTCritical skin effect

13/13

What is the origin of the nonequilibrium quantum cri&cality?
Scale invariance of the skin modes!
☆ Around the critical point, the localization length diverges.

☆ At the cri8cal point, the skin modes decay with the power law.

The power-law decay arises from an excep8onal point!
Single-particle Schrödinger equation

Jordan matrix (nondiagonalizable)☆ New type of cri8cal phenomena unique to open quantum systems.

SummaryarXiv:2206.05384

・The skin effect prohibits the entanglement growth and thermalization,
leading to the area law of the entanglement entropy.
・The skin effect triggers a new type of entanglement phase transition
that is characterized by an anomalous nonunitary CFT.

cacrit iΔ/Jl

volume law

area lawOγ/J