Symmetry classification oftypical quantum entanglement
ID rId2

Kohei Kawabata(Princeton → UTokyo ISSP)
ID rId2

Phys. Rev. B 108, 085109 (2023)

Collaborators

Yuhan Liu(UChicago)

Jonah Kudler-Flam(IAS, Princeton)

Outline
1. Introduction
2. Symmetry classification of typicalquantum entanglement
3. Weingarten calculus
4. Connection with mesoscopic
condensed-matter physics

Quantum entanglement

1/24

Quantum entanglement lies at the heart of quantum physics.
Entanglement characterizes quantum phases of matter:・Quantum critical phenomena
・Topological phases
Entanglement also provides a foundation of thermalization or lack thereof.

Kaufman et al., Science 353, 794 (2016)

Lukin et al., Science 364, 256 (2019)

Typical entanglement: Page curve

2/24☆ Entanglement entropy of typical states in chaotic (nonintegrable) systems:

maximal and proportional to the volume ofthe subsystem (volume law)thermalization☆ Relevant to black-hole physics (“Page curve”)
Page, PRL 71, 1291 (1993)

Bianchi et al., PRX
Quantum 3, 030201 (2022)☆ Entanglement of typical Gaussian states in free fermions“free-fermion (Gaussian) Page curve”

single-particle quantum chaos and thermalization in free fermions
Liu et al., PRB 97, 245126 (2018); Bianchi et al., PRB 103, L241118 (2021)

Quantum chaos

3/24☆ Quantum chaos manifests itself in spectral statistics.

・Integrable quantum systems: Poisson statistics

Berry & Tabor, Proc. R. Soc. A
356, 375 (1977)

・Nonintegrable quantum systems: random-matrix statistics
quantum chaos & thermalization

Bohigas et al., PRL 52, 1 (1984)

Altland-Zirnbauer symmetry

4/24☆ Quantum chaos (random matrices) is classified by the tenfold AZ symmetry.

time reversal
particle holechiralAltland & Zirnbauer, PRB 55, 1142 (1997)

e.g., Tenfold symmetry classification of the SYK model
symmetry-enriched behavior of quantum chaos
You et al., PRB 95, 115150 (2017)
Cotler et al., JHEP 2017, 118☆ AZ symmetry is also relevant to the physics of free fermions.

・Anderson localization and transition
・Topological insulators and superconductors

Motivation
Typical entanglement characterizes quantum chaos.
Quantum chaos (spectral statistics) is classified according tothe tenfold Altland-Zirnbauer symmetry.
How does symmetry affect typical entanglement?
In general, little has been understood about the role ofsymmetry in entanglement theory.

Results☆ We develop the tenfold classification of typical entanglement(free-fermion Page curves) based on AZ symmetry!

Relevant to symmetry-enriched quantum chaos and thermalizationin free fermions.

Liu, Kudler-Flam & Kawabata, PRB 108, 085109 (2023)

Outline
2. Symmetry classification of typicalquantum entanglement

Altland-Zirnbauer (AZ) symmetry

7/24

・Random free fermion (two-body complex Sachdev-Ye-Kitaev model)

Altland-Zirnbauer (AZ) symmetry classification

Altland & Zirnbauer, PRB 55, 1142 (1997)

time reversal:(antiunitary)

particle hole:(antiunitary)chiral:(unitary)
unitary matrices

Tenfold internal symmetry classes for Hermitian matrices
Relevant to band insulators and superconductors

Random matrix theory

8/24☆ AZ symmetry classifies the universality classes of Hermitian random matrices.
・Time-reversal symmetry changes the bulk spectral correlations.

Threefold spectral correlations(Wigner & Dyson)

Atas et al., PRL 110, 084101 (2013)

・Particle-hole or chiral symmetry changes the spectral correlations
around zero eigenvalue.

cf. chiral symmetry breaking in QCD

Verbaarschot, PRL 72, 2531 (1994)

cf. stability of zero modes in superconductors

Beenakker, RMP 87, 1037 (2015)

Random matrix theory

8/24

β: bulk (time-reversal symmetry)

α: zero (chiral or particle-hole symmetry)

Topological insulators

9/24☆ Periodic table of topological insulators and superconductors

Schnyder, Ryu, Furusaki & Ludwig, PRB 78, 195125 (2008); NJP 12, 065010 (2010)
Kitaev, AIP Conf. Proc. 1134, 22 (2009)

Free-fermion Page curve (1)10/24

We study typical entanglement entropy of random free fermionsSingle-particle eigenstates are randomly chosen by the Haar measure
・Free-fermion (Gaussian) Page curve
Liu, Chen & Balents, PRB 97, 245126 (2018)
Bianchi, Hackl & Kieburg, PRB 103, L241118 (2021)
Bianchi et al., PRX Quantum 3, 030201 (2022)

Free-fermion Page curve was studied without symmetry (i.e., class A or D)
Effect of symmetry?
We develop the tenfold classification of free-fermion
Page curve based on AZ symmetry!
Liu, Kudler-Flam & Kawabata, PRB 108, 085109 (2023)

Free-fermion Page curve (2)11/24☆ We numerically calculate the average and variance of entanglement entropy.
(we focus on the half filling and half bipartition)(a) 0.2

A(b) 0.25

0.1

AI

0.20

AII

-0.0

AIII
BDI-0.1

CII
D

-0.2

DIII
-0.3
-0.4
0.00

CCI
0.05

0.10

A

BDI, CIAII
AIII

0.15

AI, AIII, D, CBDI
CII

0.10

A, DIII, CII
0.05
0.000AI

DDIII
C50100AIICI200250150

Tenfold different universal behavior!
☆ We derive the tenfold Page curves analytically, by the Weingarten calculus
for different symmetry classes.

Standard (Wigner-Dyson) class (1)12/24☆ Time-reversal symmetry

Class A: no symmetry → unitary U(N)Class AI: TRS with +1 → orthogonal O(N)Class AII: TRS with -1 → symplectic Sp(N)(classifying spaces of single-particle eigenstates)
We take eigenstates Haar-randomly from U(N), O(N), and Sp(N)The leading term ∝ N does not change even in the presence of TRS.
The constant term does depend on TRS.

Standard (Wigner-Dyson) class (2)13/24

・Constant term of the average entanglement entropy

TRS with +1 decreases average entanglement
TRS with -1 increases average entanglement
・Variance of entanglement entropy

TRS with +1 (-1) increases (decreases) the variance of entanglement entropy
Consistent with the level repulsion in random matrices

Chiral class

14/24☆ Chiral symmetry(depending on time-reversal symmetry)
・Average entanglement entropy
The volume-law term does not change.
The constant term does depend on chiral symmetry:

・Variance of entanglement entropy(twice larger than the standard class)Bogoliubov-de Gennes (BdG) class

15/24☆ Particle-hole symmetry

Classifying spaces:

・Classes D & C(random-matrix index for zero eigenvalue)・Classes CI & DIII

Liu, Kudler-Flam & Kawabata, PRB 108, 085109 (2023)☆ We develop the tenfold classification of free-fermion Page curves
based on AZ symmetry!

Relevant to symmetry-enriched quantum chaos and thermalizationin free fermions.

Outline

3. Weingarten calculus

Weingarten calculus (1)17/24

The analytical approaches in the literature are not straightforward togeneralize in the presence of symmetry.
We use the symmetry-enriched versions of Weingarten calculus.
Collins, Int. Math. Res. Not. 2003, 953 (2003)
Collins & Śniady, Commun. Math. Phys. 264, 773 (2006)

As a special feature of free fermions, the entanglement entropy is
obtained from the single-particle correlation matrix: Peschel, J. Phys. A 36, L205 (2003)(many-body eigenstate)eigenspectrum of(i.e., single-particle entanglement spectrum)
average entanglement
density of the ES

Weingarten calculus (2)18/24

The average density is obtained from the resolvent:resolventThus, the calculations of average entanglement reduce tosystematically calculable by the Weingarten calculus☆ Depending on different classifying spaces in different symmetry classes,different types of the Weingarten functions are relevant.

Tenfold different values of typical entanglement entropy

Entanglement spectra
Standard class (A, AI & AII):
Chiral class (AIII, BDI & CII):
BdG class (D, DIII, C & CI):

Time-reversal symmetry:global scaling of the entanglement spectrum
Chiral and particle-hole symmetries:
singular peaks at the center of the entanglement spectrum(entanglement zero modes λ = 1/2)

19/24

Sachdev-Ye-Kitaev model

20/24

Numerical results of the density of the single-particle entanglement spectrum
in the quadratic SYK model (N=100)(a)1.0(b)

0.8

1.0
0.8

0.6

0.6

0.4

0.4
D

0.2
0.0
0.0

C
0.2

0.4

0.6

0.8

1.0

DIII

0.2
0.0
0.0

CI
0.2

0.4

0.6

0.8

1.0

The singular peaks and dips are controlled by particle-hole symmetry

Outline

4. Connection with mesoscopic
condensed-matter physics

Anderson localization

21/24☆ Typical entanglement has a similarity to mesoscopic transport phenomena!
・Anderson localization and transition

disorder induces localization of coherent waves

Anderson, PR 109, 1492 (1958)

Anderson localization also leads to a continuous phase transition[scaling theory]: conductance: length scale
Abrahams et al., PRL 42, 673 (1979)

Weak localization (quantum correction) 22/24
Perturbative expansion of the beta function around
Gor’kov et al., JETP Lett. 30, 228 (1979)
Altshuler et al., PRB 22, 5142 (1980)
Hikami et al., PTP 63, 707 (1980)

Drude term(semiclassical)

Quantum correction(relevant to Anderson transition)

Average entanglement entropy (our present result):

Same dependence! Originating from the same theoretical mechanism?

Universal conductance fluctuations

23/24

・Universal conductance fluctuations

Conductance fluctuations in
the diffusive regime:
Lee & Stone, PRL 55, 1622 (1985)
Altshuler, JETP Lett. 41, 648 (1985)
Washburn & Webb, Adv.
Phys. 35, 375 (1986)

Variance in entanglement entropy:

Same 1/β dependence!

We find that typical quantum entanglement has a similarity tomesoscopic transport phenomena in old condensed matter physics.
There, the transmission probability seems to hold a parallel role tothe single-particle entanglement spectrum.
Mesoscopic transport phenomena: field theory of nonlinear sigma modeltarget space: classified by AZ symmetry
Efetov, Supersymmetry in Disorder and Chaos (1996)

We may have a nonlinear sigma model description of typical entanglement.

SummaryPhys. Rev. B 108, 085109 (2023)

・We develop the tenfold classification of typical quantum entanglement(free-fermion Page curve) based on Altland-Zirnbauer symmetry.
・It is relevant to characterization of symmetry-enriched quantum chaos.
・Typical entanglement exhibits a similarity to mesoscopic transport.