skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

This content will become publicly available on October 7, 2020

Title: From eigenstate to Hamiltonian: Prospects for ergodicity and localization

Abstract

Here, this paper addresses the so-called inverse problem which consists in searching for (possibly multiple) parent target Hamiltonian(s), given a single quantum state as input. Starting from ψ 0, an eigenstate of a given local Hamiltonian Η 0, we ask whether or not there exists another parent Hamiltonian Η P for ψ 0, with the same local form as H 0. Focusing on one-dimensional quantum disordered systems, we extend the recent results obtained for Bose-glass ground states [M. Dupont and N. Laflorencie, Phys. Rev. B 99, 020202(R) (2019)2469-995010.1103/PhysRevB.99.020202] to Anderson localization, and the many-body localization (MBL) physics occurring at high energy. We generically find that any localized eigenstate is a very good approximation for an eigenstate of a distinct parent Hamiltonian, with an energy variance σ$$_P^2$$(L)=⟨$$H^2_P$$⟩ ψ0-(Η P)$$_{ψ_0}^2$$ vanishing as a power law of system size L. This decay is microscopically related to a chain-breaking mechanism, also signaled by bottlenecks of vanishing entanglement entropy. A similar phenomenology is observed for both Anderson and MBL. In contrast, delocalized ergodic many-body eigenstates uniquely encode the Hamiltonian in the sense that σ$$_P^2$$(L) remains finite at the thermodynamic limit, i.e., L → +∞. Finally, as a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of σ$$_P^2$$(L).

Authors:
ORCiD logo [1]; ORCiD logo [2];  [2]
  1. Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
  2. Université de Toulouse, CNRS, UPS (France)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division; French National Research Agency (ANR); CALMIP; GENCI
OSTI Identifier:
1580943
Grant/Contract Number:  
AC02-05CH11231; ANR-16-CE30-0023-02; ANR-11-IDEX-0002-02; 2018-P0677; 2019-P0677; 2018-A0030500225
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 100; Journal Issue: 13; Journal ID: ISSN 2469-9950
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Dupont, Maxime, Macé, Nicolas, and Laflorencie, Nicolas. From eigenstate to Hamiltonian: Prospects for ergodicity and localization. United States: N. p., 2019. Web. doi:10.1103/physrevb.100.134201.
Dupont, Maxime, Macé, Nicolas, & Laflorencie, Nicolas. From eigenstate to Hamiltonian: Prospects for ergodicity and localization. United States. doi:10.1103/physrevb.100.134201.
Dupont, Maxime, Macé, Nicolas, and Laflorencie, Nicolas. Mon . "From eigenstate to Hamiltonian: Prospects for ergodicity and localization". United States. doi:10.1103/physrevb.100.134201.
@article{osti_1580943,
title = {From eigenstate to Hamiltonian: Prospects for ergodicity and localization},
author = {Dupont, Maxime and Macé, Nicolas and Laflorencie, Nicolas},
abstractNote = {Here, this paper addresses the so-called inverse problem which consists in searching for (possibly multiple) parent target Hamiltonian(s), given a single quantum state as input. Starting from ψ0, an eigenstate of a given local Hamiltonian Η0, we ask whether or not there exists another parent Hamiltonian ΗP for ψ0, with the same local form as H0. Focusing on one-dimensional quantum disordered systems, we extend the recent results obtained for Bose-glass ground states [M. Dupont and N. Laflorencie, Phys. Rev. B 99, 020202(R) (2019)2469-995010.1103/PhysRevB.99.020202] to Anderson localization, and the many-body localization (MBL) physics occurring at high energy. We generically find that any localized eigenstate is a very good approximation for an eigenstate of a distinct parent Hamiltonian, with an energy variance σ$_P^2$(L)=⟨$H^2_P$⟩ψ0-(ΗP)$_{ψ_0}^2$ vanishing as a power law of system size L. This decay is microscopically related to a chain-breaking mechanism, also signaled by bottlenecks of vanishing entanglement entropy. A similar phenomenology is observed for both Anderson and MBL. In contrast, delocalized ergodic many-body eigenstates uniquely encode the Hamiltonian in the sense that σ$_P^2$(L) remains finite at the thermodynamic limit, i.e., L → +∞. Finally, as a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of σ$_P^2$(L).},
doi = {10.1103/physrevb.100.134201},
journal = {Physical Review B},
number = 13,
volume = 100,
place = {United States},
year = {2019},
month = {10}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on October 7, 2020
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Local Conservation Laws and the Structure of the Many-Body Localized States
journal, September 2013


Many-body mobility edge due to symmetry-constrained dynamics and strong interactions
journal, August 2015


Anderson localization and interactions in one-dimensional metals
journal, January 1988


Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain
journal, July 2016


Does a Single Eigenstate Encode the Full Hamiltonian?
journal, April 2018


Anderson transitions
journal, October 2008


Glassy properties of the Bose-glass phase of a one-dimensional disordered Bose fluid
journal, September 2019


Boson localization and the superfluid-insulator transition
journal, July 1989

  • Fisher, Matthew P. A.; Weichman, Peter B.; Grinstein, G.
  • Physical Review B, Vol. 40, Issue 1
  • DOI: 10.1103/PhysRevB.40.546

Detecting a many-body mobility edge with quantum quenches
journal, January 2016


Many-body localization edge in the random-field Heisenberg chain
journal, February 2015


Strong disorder renormalization group primer and the superfluid–insulator transition
journal, October 2013


Extended slow dynamical regime close to the many-body localization transition
journal, February 2016


Impurity scattering in 1D ring of interacting electrons
journal, November 1994

  • Bouzerar, Georges; Poilblanc, Didier
  • Journal de Physique I, Vol. 4, Issue 11
  • DOI: 10.1051/jp1:1994215

Purification and Many-Body Localization in Cold Atomic Gases
journal, November 2014


Local integrals of motion in many-body localized systems: Local integrals of motion in many-body localized systems
journal, May 2017

  • Imbrie, John Z.; Ros, Valentina; Scardicchio, Antonello
  • Annalen der Physik, Vol. 529, Issue 7
  • DOI: 10.1002/andp.201600278

The theory of impurity conduction
journal, April 1961


Critical Properties of the Superfluid—Bose-Glass Transition in Two Dimensions
journal, April 2015


Many-body localization: An introduction and selected topics
journal, September 2018


Unbounded Growth of Entanglement in Models of Many-Body Localization
journal, July 2012


Spin transport of weakly disordered Heisenberg chain at infinite temperature
journal, June 2016


Many-Body Localization in a Disordered Quantum Ising Chain
journal, September 2014


Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions
journal, July 2018


Area laws in a many-body localized state and its implications for topological order
journal, September 2013


Many-body localization from the perspective of Integrals of Motion: Many-body localization from the perspective of Integrals of Motion
journal, March 2017

  • Rademaker, Louk; Ortuño, Miguel; Somoza, Andres M.
  • Annalen der Physik, Vol. 529, Issue 7
  • DOI: 10.1002/andp.201600322

Entanglement holographic mapping of many-body localized system by spectrum bifurcation renormalization group
journal, March 2016


Colloquium : Many-body localization, thermalization, and entanglement
journal, May 2019


Quantum Critical Scaling of Dirty Bosons in Two Dimensions
journal, June 2015


Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations
journal, May 1983


Many-body localization phase transition
journal, November 2010


Emergence of Quantum Chaos in Finite Interacting Fermi Systems
journal, September 1997


Obtaining Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach
journal, June 2016


Determining a local Hamiltonian from a single eigenstate
journal, July 2019


Many-body localization in the Heisenberg X X Z magnet in a random field
journal, February 2008


Transport in a one-channel Luttinger liquid
journal, February 1992


Anderson Localization versus Delocalization of Interacting Fermions in One Dimension
journal, January 1998


From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics
journal, May 2016


Many-body localization as a large family of localized ground states
journal, January 2019


Theory of Superconductivity
journal, December 1957


Shift-invert diagonalization of large many-body localizing spin chains
journal, January 2018


Spectral statistics across the many-body localization transition
journal, January 2016


Effects of quenched disorder on spin-1/2 quantum XXZ chains
journal, February 1992


Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems
journal, June 2013


Ergodicity breaking in a model showing many-body localization
journal, February 2013


Many-Body Localization and Thermalization in Quantum Statistical Mechanics
journal, March 2015


Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System
journal, July 2016


Many-Body Mobility Edge in a Mean-Field Quantum Spin Glass
journal, November 2014


Bimodal entanglement entropy distribution in the many-body localization transition
journal, November 2016


Weak- versus strong-disorder superfluid—Bose glass transition in one dimension
journal, November 2017


Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality
journal, March 2014


Quantum statistical mechanics in a closed system
journal, February 1991


Chaos and quantum thermalization
journal, August 1994


Phenomenology of fully many-body-localized systems
journal, November 2014


Many-body localization and delocalization in large quantum chains
journal, November 2018

  • Doggen, Elmer V. H.; Schindler, Frank; Tikhonov, Konstantin S.
  • Physical Review B, Vol. 98, Issue 17
  • DOI: 10.1103/PhysRevB.98.174202