Localization and universality in non-Hermitian many-body systems - - PowerPoint PPT Presentation

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Oct 21, 2023 163 likes •467 views

Localization and universality in non-Hermitian many-body systems Ryusuke Hamazaki R.H., K. Kawabata and M. Ueda. Phys. Rev. Lett. 123, 090603 (2019) R.H., K. Kawabata, N. Kura and M. Ueda. Phys. Rev. Research, to appear

SLIDE 1

Localization and universality in   non-Hermitian many-body systems 

Ryusuke Hamazaki

R.H., K. Kawabata and M. Ueda. 

Phys. Rev. Lett. 123, 090603 (2019)

  R.H., K. Kawabata, N. Kura and M. Ueda. 

Phys. Rev. Research, to appear

[arXiv:1904.13082] iTHEMS QFT Seminar

Kohei Naoto

Prof. Ueda

Sorry, this talk is mainly based on numerical results not about quantum field theory

SLIDE 2

Introduction Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians ・Many-body localization in Hermitian systems

Universality in non-Hermitian random matrices

R.H., K. Kawabata and M. Ueda. 

Phys. Rev. Lett. 123, 090603 (2019)

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

Outline

SLIDE 3

Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians

Universality in non-Hermitian random matrices

R.H., K. Kawabata and M. Ueda. 

Phys. Rev. Lett. 123, 090603 (2019)

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

Outline

Introduction

・Many-body localization in Hermitian systems

SLIDE 4

Thermalization of isolated quantum systems

unitary time evolution

?

Dating back to J. von Neumann in 1929

microcanonical ensemble Observation of thermalization dynamics  with ultra cold atomic gases  (almost isolated, highly tunable systems)

cf)

S. Trotzky et al., Nature Physics (2012)

Rapid development in these two decades

Deriving stat. mech. from quant. mech.

SLIDE 5

Localization transition in quantum many-body systems

Generic translation invariant systems thermalize

unitary time evolution Delocalized (ergodic) phase:  thermalization

Possibility of two phases in disordered systems

disorder  strength

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e.g.)

MBL phase:  no thermalization

transition many conserved  quantities (many-body localized)

SLIDE 6

Theory of many-body localization:   level-spacing statistics

Delocalized phase:  thermalization

disorder  strength

MBL phase:  no thermalization

transition

Level spacing distributions

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many-body eigenvalues

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Universality of random matrices:  complexity of eigenstates

RMT

uncorrelated eigenstates:  many symmetry sectors

Poisson

SLIDE 7

Non-Hermitian many-body localization Threefold way in non-Hermitian random matrices

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

Outline

Introduction

・Many-body localization in Hermitian systems ・Non-Hermitian Hamiltonians

R.H., K. Kawabata and M. Ueda. 

Phys. Rev. Lett. 123, 090603 (2019)

SLIDE 8

Dynamics of open quantum systems

Real systems are attached to external environment How does it affect stat. mech. of many-body dynamics? State-of-the-art experiments in e.g., cold atoms Controllable dissipation, real-time measurement

Increase of particle lifetime with measurement (quantum Zeno effect) Measurement strength Lifetime

cf) YS. Patil et al., PRL (2015)

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A class of open systems is described by non-Hermitian Hamiltonian

SLIDE 9

Time-reversal symmetry and real-complex transition

If , eigenvalues of are either real or  form a complex-conjugate pair

Time-reversal operator with

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e.g.)

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complex  conjugate real

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Two real eigenvalues collide and   become imaginary (real-complex transition)

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: complex in general

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: right/left eigenstate

SLIDE 10

Introduction Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians ・Many-body localization in Hermitian systems

Universality in non-Hermitian random matrices

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

Outline

R.H., K. Kawabata and M. Ueda. 

Phys. Rev. Lett. 123, 090603 (2019)

SLIDE 11

Review: Hatano-Nelson model 

N. Hatano and D. R. Nelson, Phys. Rev. Lett., 77, 570 (1996)

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Localization & time-reversal symmetry

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Single-particle problem

・One-particle spectra ・No interaction effect

a particle with asymmetric hopping on a disordered potential non-Hermiticity localization

Localization & real-complex transitions occur  for the single-particle level

weak disorder: delocalized ↔ complex eigenvalues strong disorder: localized ↔ real eigenvalues

SLIDE 12

Aim of this study

What physics emerges if all of non-Hermiticity, disorder and interactions are present? Our finding Novel real-complex transition appears,   which we argue is triggered by   non-Hermitian extension of the MBL

R.H., K. Kawabata and M. Ueda. Phys. Rev. Lett. 123, 090603 (2019)

SLIDE 13

Interacting hardcore bosons   with disorder and asymmetric  hopping terms (1D, PBC)

・For , disordered XXZ model (MBL when ) ・For , Hatano-Nelson model

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← Time-reversal symmetry ˆ H = ˆ H∗

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Interacting model with asymmetric hopping and disorder 

[Phys. Rev. Lett. 123, 090603 (2019)]

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SLIDE 14

Spectra of the model

[Phys. Rev. Lett. 123, 090603 (2019)]

Spectra

Complex many-body eigenvalues are suppressed   as we increase disorder strength! 

cf) Eigenvalues are either real or form complex-conjugate pairs  due to time-reversal symmetry

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SLIDE 15

5 10 15 10−6 10−5 10−4 10−3 10−2 10−1 100

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Real-complex transition of the many-body eigenvalues 

[Phys. Rev. Lett. 123, 090603 (2019)]

Ratio of eigenvalues with nonzero imaginary parts

real- dominant

complex- dominant

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Real-complex transitions!

average 

disorder

SLIDE 16

Complex-eigenvalue phase

Consequence: dynamical stability transition 

[Phys. Rev. Lett. 123, 090603 (2019)]

Real part of energy

Real-eigenvalue phase Small disorder Large disorder

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Dynamically unstable Energy changes dramatically Dynamically stable Energy oscillates a little

SLIDE 17

Non-Hermitian MBL: level-spacing statistics

[Phys. Rev. Lett. 123, 090603 (2019)]

Level-spacing statistics:

Natural extension of   the Hermitian case!

Ginibre 2D Poisson

(delocalized phase)

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(in complex plane) random matrix ensemble   for non-Hermitian matrices

Ginibre → Poisson

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(in real axis)

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large

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(localized phase)

1D Poisson 1D RMT = Wigner-Dyson

Coincidence of real-complex and MBL transitions?

SLIDE 18

101

6 × 100

10−1

Non-Hermitian MBL: entanglement entropy

[Phys. Rev. Lett. 123, 090603 (2019)]

Entanglement entropy of eigenstates 

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reduced density matrix for

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average over eigenstates

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・Consider right eigenstates Volume to area law Non-Hermitian MBL  transition at

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volume law area law

Coincidence of real-complex and MBL transitions?

from top to bottom

SLIDE 19

Coincidence of two transition points

[Phys. Rev. Lett. 123, 090603 (2019)]

Adjacent eigenstates do not mix by  non-Hermitian perturbations only in MBL phase

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(delocalized) (MBL)

cf) M. Serbyn et al., PRX (2015)

→ Coalescence process is suppressed in the MBL phase

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delocalized

Non-Hermitian

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Hermitian

× ×

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MBL cf) time-reversal symmetry

SLIDE 20

Introduction Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians ・Many-body localization in Hermitian systems

Universality in non-Hermitian random matrices

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

Outline

R.H., K. Kawabata and M. Ueda. 

Phys. Rev. Lett. 123, 090603 (2019)

SLIDE 21

Universality of Hermitian random matrices

Universal statistics of   random matrices

random variables taken from (real Gaussian) (real Bernoulli)

…

e.g.) level-spacing distributions

eigenvalue

1 2 3 0.6 1.2

Non-random but   complicated systems may 

bey the same universality

Nuclei Quantum chaos Mesoscopic  systems

SLIDE 22

Random matrices and symmetry

Dyson’s threefold way: time-reversal symmetry

Different level-spacing distributions (e.g., peak structure)

1 2 3 0.6 1.2

Also appear in e.g. chaotic systems   with the same symmetry

: eigenvalue : Kramers pair of eigenvalues

SLIDE 23

Symmetry classification and universality  in non-Hermitian random matrices

Ginibre’s threefold way: time-reversal symmetry

level-spacing distributions in the complex plane ← Three classes lead to the same universality  unlike the Hermitian threefold way  (symmetry only creates nonlocal correlations)

SLIDE 24

Motivation of our work

Does the threefold different universality appear  even for non-Hermitian random matrices? Our finding Yes, consider transposition symmetry  instead of time-reversal symmetry!

R.H., K. Kawabata, N. Kura and M. Ueda. arXiv:1904.13082

SLIDE 25

Role of transposition symmetry

Transposition differs from complex conjugation!

[D. Bernard and A. LeClair (2002); U. Magnea (2008); Kawabata et al. (2018)]

: eigenvalue : Kramers pair of eigenvalues

SLIDE 26

Threefold way in non-Hermitian random matrices

Transposition alters interactions between eigenvalues, just as  the Hermitian case 

Universal level-spacing distributions

(main) Gaussian  (inset) Bernoulli  cf) peak structure

R.H. et al., arXiv:1904.13082

SLIDE 27

Other non-Hermitian symmetries

Only transposition symmetry alters   the level repulsions,   leading to distinct universality of p(s)

R.H. et al., arXiv:1904.13082

TRS: time-reversal symmetry, PHS: particle-hole symmetry  CS: chiral symmetry, SLS: sub lattice symmetry nonlocal correlations

SLIDE 28

Other non-Hermitian symmetries

R.H. et al., arXiv:1904.13082

Only transposition symmetry alters   the level repulsions,   leading to distinct universality of p(s)

TRS: time-reversal symmetry, PHS: particle-hole symmetry  CS: chiral symmetry, SLS: sub lattice symmetry nonlocal correlations

Only three universal level-spacing statistics exist among all non-Hermitian symmetry classes

(38 types)

SLIDE 29

Spectra of the Lindblad many-body operator

Liouvillian spectra {λα}

Only (i) keeps the transposition symmetry

R.H. et al., arXiv:1904.13082

class AI† class A

The universality appears!

SLIDE 30

Conclusion

Real-complex transition can occur due to  non-Hermitian extension of the MBL transition,   characterized by the level-spacing statistics Threefold universality classes for non-Hermitian random matrices are found by considering transposition symmetry,  not complex-conjugation as Ginibre’s classes

Localization and universality in non-Hermitian many-body systems

Ryusuke Hamazaki

R.H., K. Kawabata and M. Ueda.

R.H., K. Kawabata, N. Kura and M. Ueda.

[arXiv:1904.13082] iTHEMS QFT Seminar

Sorry, this talk is *mainly based on numerical results *not about quantum field theory

Introduction Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians ・Many-body localization in Hermitian systems

Universality in non-Hermitian random matrices

R.H., K. Kawabata and M. Ueda.

R.H., K. Kawabata, N. Kura and M. Ueda. arXiv:1904.13082

Outline

Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians

Universality in non-Hermitian random matrices

R.H., K. Kawabata and M. Ueda.

R.H., K. Kawabata, N. Kura and M. Ueda. arXiv:1904.13082

Outline

Introduction

・Many-body localization in Hermitian systems

Thermalization of isolated quantum systems

unitary time evolution

?

Dating back to J. von Neumann in 1929

microcanonical ensemble Observation of thermalization dynamics with ultra cold atomic gases (almost isolated, highly tunable systems)

Rapid development in these two decades

Deriving stat. mech. from quant. mech.

Localization transition in quantum many-body systems

Generic translation invariant systems thermalize

unitary time evolution Delocalized (ergodic) phase: thermalization

Possibility of two phases in disordered systems

disorder strength

e.g.)

MBL phase: no thermalization

transition many conserved quantities (many-body localized)

Theory of many-body localization: level-spacing statistics

Delocalized phase: thermalization

disorder strength

MBL phase: no thermalization

transition

Level spacing distributions

many-body eigenvalues

Universality of random matrices: complexity of eigenstates

RMT

uncorrelated eigenstates: many symmetry sectors

Poisson

Non-Hermitian many-body localization Threefold way in non-Hermitian random matrices

R.H., K. Kawabata, N. Kura and M. Ueda. arXiv:1904.13082

Outline

Introduction

・Many-body localization in Hermitian systems ・Non-Hermitian Hamiltonians

R.H., K. Kawabata and M. Ueda.

Dynamics of open quantum systems

Real systems are attached to external environment How does it affect stat. mech. of many-body dynamics? State-of-the-art experiments in e.g., cold atoms Controllable dissipation, real-time measurement

Increase of particle lifetime with measurement (quantum Zeno effect) Measurement strength Lifetime

A class of open systems is described by non-Hermitian Hamiltonian

Time-reversal symmetry and real-complex transition

If , eigenvalues of are either real or form a complex-conjugate pair

Time-reversal operator with

e.g.)

complex conjugate real

Two real eigenvalues collide and become imaginary (real-complex transition)

: complex in general

: right/left eigenstate

Introduction Non-Hermitian many-body localization

・Non-Hermitian Hamiltonians ・Many-body localization in Hermitian systems

Universality in non-Hermitian random matrices

R.H., K. Kawabata, N. Kura and M. Ueda. arXiv:1904.13082

Outline

R.H., K. Kawabata and M. Ueda.

Review: Hatano-Nelson model

Localization & time-reversal symmetry

Single-particle problem

・One-particle spectra ・No interaction effect

a particle with asymmetric hopping on a disordered potential non-Hermiticity localization

Localization & real-complex transitions occur for the single-particle level

weak disorder: delocalized ↔ complex eigenvalues strong disorder: localized ↔ real eigenvalues

Aim of this study

What physics emerges if all of non-Hermiticity, disorder and interactions are present? Our finding Novel real-complex transition appears, which we argue is triggered by non-Hermitian extension of the MBL

R.H., K. Kawabata and M. Ueda. Phys. Rev. Lett. 123, 090603 (2019)

Localization and universality in   non-Hermitian many-body systems 

R.H., K. Kawabata and M. Ueda. 

  R.H., K. Kawabata, N. Kura and M. Ueda. 

Sorry, this talk is mainly based on numerical results not about quantum field theory

R.H., K. Kawabata and M. Ueda. 

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

R.H., K. Kawabata and M. Ueda. 

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

microcanonical ensemble Observation of thermalization dynamics  with ultra cold atomic gases  (almost isolated, highly tunable systems)

unitary time evolution Delocalized (ergodic) phase:  thermalization

disorder  strength

MBL phase:  no thermalization

transition many conserved  quantities (many-body localized)

Theory of many-body localization:   level-spacing statistics

Delocalized phase:  thermalization

disorder  strength

MBL phase:  no thermalization

Universality of random matrices:  complexity of eigenstates

uncorrelated eigenstates:  many symmetry sectors

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

R.H., K. Kawabata and M. Ueda. 

If , eigenvalues of are either real or  form a complex-conjugate pair

complex  conjugate real

Two real eigenvalues collide and   become imaginary (real-complex transition)

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

R.H., K. Kawabata and M. Ueda. 

Review: Hatano-Nelson model 

Localization & real-complex transitions occur  for the single-particle level

What physics emerges if all of non-Hermiticity, disorder and interactions are present? Our finding Novel real-complex transition appears,   which we argue is triggered by   non-Hermitian extension of the MBL

Interacting hardcore bosons   with disorder and asymmetric  hopping terms (1D, PBC)

Interacting model with asymmetric hopping and disorder 

Complex many-body eigenvalues are suppressed   as we increase disorder strength! 

cf) Eigenvalues are either real or form complex-conjugate pairs  due to time-reversal symmetry

Real-complex transition of the many-body eigenvalues 

average 

Consequence: dynamical stability transition 

Natural extension of   the Hermitian case!

(in complex plane) random matrix ensemble   for non-Hermitian matrices

Entanglement entropy of eigenstates 

・Consider right eigenstates Volume to area law Non-Hermitian MBL  transition at

Adjacent eigenstates do not mix by  non-Hermitian perturbations only in MBL phase

R.H., K. Kawabata, N. Kura and M. Ueda.  arXiv:1904.13082

R.H., K. Kawabata and M. Ueda. 