Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Hsieh, Timothy H.; Fu, Liang

doi:10.1103/physrevlett.113.106801

Title: Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Abstract

A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. We show that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce an extensive partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wave function by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between the topological phase transition and the entanglement criticality is rigorously established for integer quantum Hall states.

Authors:

Hsieh, Timothy H. ^[1]; Fu, Liang ^[1]

Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Dept. of Physics

Publication Date:: Thu Sep 04 00:00:00 EDT 2014

Research Org.:: Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

OSTI Identifier:: 1505721

Alternate Identifier(s):: OSTI ID: 1180117

Grant/Contract Number:: SC0010526

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review Letters

Additional Journal Information:: Journal Volume: 113; Journal Issue: 10; Journal ID: ISSN 0031-9007

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats


                    Hsieh, Timothy H., and Fu, Liang. Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State.  United States: N. p., 2014. 
Web.  doi:10.1103/physrevlett.113.106801.

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                    Hsieh, Timothy H., & Fu, Liang. Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State.  United States.  https://doi.org/10.1103/physrevlett.113.106801

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                    Hsieh, Timothy H., and Fu, Liang. Thu .  
"Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State".  United States.  https://doi.org/10.1103/physrevlett.113.106801.  https://www.osti.gov/servlets/purl/1505721.

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@article{osti_1505721,

  title        = {Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State},

  author       = {Hsieh, Timothy H. and Fu, Liang},

  abstractNote = {A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. We show that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce an extensive partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wave function by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between the topological phase transition and the entanglement criticality is rigorously established for integer quantum Hall states.},

  doi          = {10.1103/physrevlett.113.106801},

  journal      = {Physical Review Letters},

  number       = 10,

  volume       = 113,

  place        = {United States},

  year         = {Thu Sep 04 00:00:00 EDT 2014},

  month        = {Thu Sep 04 00:00:00 EDT 2014}

}

Copy to clipboard

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https://doi.org/10.1103/physrevlett.113.106801

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Cited by: 42 works

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Figures / Tables:

FIG. 1: (color online). Extensive partition of a topological state yields a topological phase transition in the bulk entanglement Hamiltonian H _A of a subsystem. The horizontal and vertical sequences represent two ways of realizing the transition: the horizontal sequence denotes geometrically tuning the partition towards quantum criticality at themore »

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Title: Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Abstract

Citation Formats

Figures / Tables:

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order journal, October 2010

Topological insulators and superconductors journal, October 2011

The birth of topological insulators journal, March 2010

Momentum polarization: An entanglement measure of topological spin and chiral central charge journal, November 2013

General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States journal, May 2012

Quantized Hall Conductance in a Two-Dimensional Periodic Potential journal, August 1982

Bipartite entanglement and entropic boundary law in lattice spin systems journal, February 2005

Z2 Topological Order and the Quantum Spin Hall Effect journal, September 2005

Colloquium: Topological insulators journal, November 2010

Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States journal, July 2008

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states journal, December 2012

Topological Orders in Rigid States journal, February 1990

Bulk-edge correspondence in entanglement spectra journal, November 2011

Classification of topological insulators and superconductors in three spatial dimensions journal, November 2008

Detecting Topological Order in a Ground State Wave Function journal, March 2006

Ground state entanglement and geometric entropy in the Kitaev model journal, March 2005

Boundary-Locality and Perturbative Structure of Entanglement Spectra in Gapped Systems journal, May 2012

Topological Entanglement Entropy journal, March 2006

Entanglement Gap and a New Principle of Adiabatic Continuity journal, May 2010

Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly" journal, October 1988

Geometric proof of the equality between entanglement and edge spectra journal, July 2012

Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces journal, May 1990

Calculation of reduced density matrices from correlation functions journal, March 2003

Entanglement Spectrum of Topological Insulators and Superconductors journal, April 2010

Percolation, quantum tunnelling and the integer Hall effect journal, May 1988

Entanglement spectrum of a topological phase in one dimension journal, February 2010

Entanglement Entropy and Entanglement Spectrum of the Kitaev Model journal, August 2010

Entanglement Spectra of Complex Paired Superfluids journal, October 2011

Quasiparticle statistics and braiding from ground-state entanglement journal, June 2012

Classification of topological insulators and superconductors in three spatial dimensions text, January 2008

Boundary-locality and perturbative structure of entanglement spectra in gapped systems text, January 2011

Quasi-particle Statistics and Braiding from Ground State Entanglement text, January 2011

Edge state inner products and real-space entanglement spectrum of trial quantum Hall states text, January 2012

Momentum polarization: an entanglement measure of topological spin and chiral central charge text, January 2012

Calculation of reduced density matrices from correlation functions text, January 2002

Learning phase transitions by confusion journal, February 2017

Momentum-space entanglement after a quench in one-dimensional disordered fermionic systems journal, December 2019

Entanglement spectrum of a random partition: Connection with the localization transition journal, June 2015

Analytic expression for the entanglement entropy of a two-dimensional topological superconductor journal, February 2017

Topological invariant for two-dimensional open systems journal, May 2018

Entanglement polarization for the topological quadrupole phase journal, July 2018

Decoding quantum criticality from fermionic/parafermionic topological states journal, October 2018

Invariance of Topological Indices Under Hilbert Space Truncation journal, January 2018

Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines journal, August 2018

Entanglement Spectrum of a Random Partition: Connection with the Localization Transition text, January 2014

Learning phase transitions by confusion text, January 2016

Invariance of topological indices under Hilbert space truncation text, January 2017

Topological invariant for two-dimensional open systems text, January 2017

Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines text, January 2017

Decoding quantum criticalities from fermionic/parafermionic topological states text, January 2018

Reconstructing Entanglement Hamiltonian via Entanglement Eigenstates text, January 2018

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
journal, October 2010

Topological insulators and superconductors
journal, October 2011

The birth of topological insulators
journal, March 2010

Momentum polarization: An entanglement measure of topological spin and chiral central charge
journal, November 2013

General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
journal, May 2012

Quantized Hall Conductance in a Two-Dimensional Periodic Potential
journal, August 1982

Bipartite entanglement and entropic boundary law in lattice spin systems
journal, February 2005

$Z_{2}$ Topological Order and the Quantum Spin Hall Effect
journal, September 2005

Colloquium: Topological insulators
journal, November 2010

Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
journal, July 2008

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states
journal, December 2012

Topological Orders in Rigid States
journal, February 1990

Bulk-edge correspondence in entanglement spectra
journal, November 2011

Classification of topological insulators and superconductors in three spatial dimensions
journal, November 2008

Detecting Topological Order in a Ground State Wave Function
journal, March 2006

Ground state entanglement and geometric entropy in the Kitaev model
journal, March 2005

Boundary-Locality and Perturbative Structure of Entanglement Spectra in Gapped Systems
journal, May 2012

Topological Entanglement Entropy
journal, March 2006

Entanglement Gap and a New Principle of Adiabatic Continuity
journal, May 2010

Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"
journal, October 1988

Geometric proof of the equality between entanglement and edge spectra
journal, July 2012

Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces
journal, May 1990

Calculation of reduced density matrices from correlation functions
journal, March 2003

Entanglement Spectrum of Topological Insulators and Superconductors
journal, April 2010

Percolation, quantum tunnelling and the integer Hall effect
journal, May 1988

Entanglement spectrum of a topological phase in one dimension
journal, February 2010

Entanglement Entropy and Entanglement Spectrum of the Kitaev Model
journal, August 2010

Entanglement Spectra of Complex Paired Superfluids
journal, October 2011

Quasiparticle statistics and braiding from ground-state entanglement
journal, June 2012

Classification of topological insulators and superconductors in three spatial dimensions
text, January 2008

Boundary-locality and perturbative structure of entanglement spectra in gapped systems
text, January 2011

Quasi-particle Statistics and Braiding from Ground State Entanglement
text, January 2011

Edge state inner products and real-space entanglement spectrum of trial quantum Hall states
text, January 2012

Momentum polarization: an entanglement measure of topological spin and chiral central charge
text, January 2012

Calculation of reduced density matrices from correlation functions
text, January 2002

Learning phase transitions by confusion
journal, February 2017

Momentum-space entanglement after a quench in one-dimensional disordered fermionic systems
journal, December 2019

Entanglement spectrum of a random partition: Connection with the localization transition
journal, June 2015

Analytic expression for the entanglement entropy of a two-dimensional topological superconductor
journal, February 2017

Topological invariant for two-dimensional open systems
journal, May 2018

Entanglement polarization for the topological quadrupole phase
journal, July 2018

Decoding quantum criticality from fermionic/parafermionic topological states
journal, October 2018

Invariance of Topological Indices Under Hilbert Space Truncation
journal, January 2018

Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
journal, August 2018

Entanglement Spectrum of a Random Partition: Connection with the Localization Transition
text, January 2014

Learning phase transitions by confusion
text, January 2016

Invariance of topological indices under Hilbert space truncation
text, January 2017

Topological invariant for two-dimensional open systems
text, January 2017

Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
text, January 2017

Decoding quantum criticalities from fermionic/parafermionic topological states
text, January 2018

Reconstructing Entanglement Hamiltonian via Entanglement Eigenstates
text, January 2018