Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Hsieh, Timothy H.; Fu, Liang

doi:10.1103/physrevlett.113.106801

Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Journal Article · Thu Sep 04 00:00:00 EDT 2014 · Physical Review Letters

DOI:https://doi.org/10.1103/physrevlett.113.106801· OSTI ID:1505721

Hsieh, Timothy H. ^[1]; Fu, Liang ^[2]

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

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.

View Accepted Manuscript (DOE)

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

Sponsoring Organization:: USDOE; USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

Grant/Contract Number:: SC0010526

OSTI ID:: 1505721

Alternate ID(s):: OSTI ID: 1180117

Journal Information:: Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 10 Vol. 113; ISSN 0031-9007

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (35)

Ground state entanglement and geometric entropy in the Kitaev model Hamma, Alioscia; Ionicioiu, Radu; Zanardi, Paolo Physics Letters A, Vol. 337, Issue 1-2 https://doi.org/10.1016/j.physleta.2005.01.060	journal	March 2005
The birth of topological insulators Moore, Joel E. Nature, Vol. 464, Issue 7286 https://doi.org/10.1038/nature08916	journal	March 2010
Percolation, quantum tunnelling and the integer Hall effect Chalker, J. T.; Coddington, P. D. Journal of Physics C: Solid State Physics, Vol. 21, Issue 14 https://doi.org/10.1088/0022-3719/21/14/008	journal	May 1988
Calculation of reduced density matrices from correlation functions Peschel, Ingo Journal of Physics A: Mathematical and General, Vol. 36, Issue 14 https://doi.org/10.1088/0305-4470/36/14/101	journal	March 2003
Bipartite entanglement and entropic boundary law in lattice spin systems Hamma, Alioscia; Ionicioiu, Radu; Zanardi, Paolo Physical Review A, Vol. 71, Issue 2 https://doi.org/10.1103/PhysRevA.71.022315	journal	February 2005
Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces Wen, X. G.; Niu, Q. Physical Review B, Vol. 41, Issue 13 https://doi.org/10.1103/PhysRevB.41.9377	journal	May 1990
Classification of topological insulators and superconductors in three spatial dimensions Schnyder, Andreas P.; Ryu, Shinsei; Furusaki, Akira Physical Review B, Vol. 78, Issue 19 https://doi.org/10.1103/PhysRevB.78.195125	journal	November 2008
Entanglement spectrum of a topological phase in one dimension Pollmann, Frank; Turner, Ari M.; Berg, Erez Physical Review B, Vol. 81, Issue 6 https://doi.org/10.1103/PhysRevB.81.064439	journal	February 2010
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang Physical Review B, Vol. 82, Issue 15 https://doi.org/10.1103/PhysRevB.82.155138	journal	October 2010
Bulk-edge correspondence in entanglement spectra Chandran, Anushya; Hermanns, M.; Regnault, N. Physical Review B, Vol. 84, Issue 20 https://doi.org/10.1103/PhysRevB.84.205136	journal	November 2011
Quasiparticle statistics and braiding from ground-state entanglement Zhang, Yi; Grover, Tarun; Turner, Ari Physical Review B, Vol. 85, Issue 23 https://doi.org/10.1103/PhysRevB.85.235151	journal	June 2012
Geometric proof of the equality between entanglement and edge spectra Swingle, Brian; Senthil, T. Physical Review B, Vol. 86, Issue 4 https://doi.org/10.1103/PhysRevB.86.045117	journal	July 2012
Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states Dubail, J.; Read, N.; Rezayi, E. H. Physical Review B, Vol. 86, Issue 24 https://doi.org/10.1103/PhysRevB.86.245310	journal	December 2012
Momentum polarization: An entanglement measure of topological spin and chiral central charge Tu, Hong-Hao; Zhang, Yi; Qi, Xiao-Liang Physical Review B, Vol. 88, Issue 19 https://doi.org/10.1103/PhysRevB.88.195412	journal	November 2013
Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States Li, Hui; Haldane, F. D. M. Physical Review Letters, Vol. 101, Issue 1 https://doi.org/10.1103/PhysRevLett.101.010504	journal	July 2008
Entanglement Spectrum of Topological Insulators and Superconductors Fidkowski, Lukasz Physical Review Letters, Vol. 104, Issue 13 https://doi.org/10.1103/PhysRevLett.104.130502	journal	April 2010
Entanglement Gap and a New Principle of Adiabatic Continuity Thomale, R.; Sterdyniak, A.; Regnault, N. Physical Review Letters, Vol. 104, Issue 18 https://doi.org/10.1103/PhysRevLett.104.180502	journal	May 2010
Entanglement Entropy and Entanglement Spectrum of the Kitaev Model Yao, Hong; Qi, Xiao-Liang Physical Review Letters, Vol. 105, Issue 8 https://doi.org/10.1103/PhysRevLett.105.080501	journal	August 2010
Entanglement Spectra of Complex Paired Superfluids Dubail, J.; Read, N. Physical Review Letters, Vol. 107, Issue 15 https://doi.org/10.1103/PhysRevLett.107.157001	journal	October 2011
General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States Qi, Xiao-Liang; Katsura, Hosho; Ludwig, Andreas W. W. Physical Review Letters, Vol. 108, Issue 19 https://doi.org/10.1103/PhysRevLett.108.196402	journal	May 2012
Boundary-Locality and Perturbative Structure of Entanglement Spectra in Gapped Systems Alba, Vincenzo; Haque, Masudul; Läuchli, Andreas M. Physical Review Letters, Vol. 108, Issue 22 https://doi.org/10.1103/PhysRevLett.108.227201	journal	May 2012
Quantized Hall Conductance in a Two-Dimensional Periodic Potential Thouless, D. J.; Kohmoto, M.; Nightingale, M. P. Physical Review Letters, Vol. 49, Issue 6 https://doi.org/10.1103/PhysRevLett.49.405	journal	August 1982
Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly" Haldane, F. D. M. Physical Review Letters, Vol. 61, Issue 18 https://doi.org/10.1103/PhysRevLett.61.2015	journal	October 1988
Z2 Topological Order and the Quantum Spin Hall Effect Kane, C. L.; Mele, E. J. Physical Review Letters, Vol. 95, Issue 14, Article No. 146802 https://doi.org/10.1103/PhysRevLett.95.146802	journal	September 2005
Topological Entanglement Entropy Kitaev, Alexei; Preskill, John Physical Review Letters, Vol. 96, Issue 11 https://doi.org/10.1103/PhysRevLett.96.110404	journal	March 2006
Detecting Topological Order in a Ground State Wave Function Levin, Michael; Wen, Xiao-Gang Physical Review Letters, Vol. 96, Issue 11 https://doi.org/10.1103/PhysRevLett.96.110405	journal	March 2006
Colloquium: Topological insulators Hasan, M. Z.; Kane, C. L. Reviews of Modern Physics, Vol. 82, Issue 4, p. 3045-3067 https://doi.org/10.1103/RevModPhys.82.3045	journal	November 2010
Topological insulators and superconductors Qi, Xiao-Liang; Zhang, Shou-Cheng Reviews of Modern Physics, Vol. 83, Issue 4 https://doi.org/10.1103/RevModPhys.83.1057	journal	October 2011
Topological Orders in Rigid States Wen, X. G. International Journal of Modern Physics B, Vol. 04, Issue 02 https://doi.org/10.1142/S0217979290000139	journal	February 1990
Classification of topological insulators and superconductors in three spatial dimensions Schnyder, Andreas P.; Ryu, Shinsei; Furusaki, Akira arXiv https://doi.org/10.48550/arxiv.0803.2786	text	January 2008
Boundary-locality and perturbative structure of entanglement spectra in gapped systems Alba, Vincenzo; Haque, Masudul; Laeuchli, Andreas M. arXiv https://doi.org/10.48550/arxiv.1107.1726	text	January 2011
Quasi-particle Statistics and Braiding from Ground State Entanglement Zhang, Yi; Grover, Tarun; Turner, Ari arXiv https://doi.org/10.48550/arxiv.1111.2342	text	January 2011
Edge state inner products and real-space entanglement spectrum of trial quantum Hall states Dubail, J.; Read, N.; Rezayi, E. H. arXiv https://doi.org/10.48550/arxiv.1207.7119	text	January 2012
Momentum polarization: an entanglement measure of topological spin and chiral central charge Tu, Hong-Hao; Zhang, Yi; Qi, Xiao-Liang arXiv https://doi.org/10.48550/arxiv.1212.6951	text	January 2012
Calculation of reduced density matrices from correlation functions Peschel, Ingo arXiv https://doi.org/10.48550/arxiv.cond-mat/0212631	text	January 2002

Cited By (17)

Learning phase transitions by confusion van Nieuwenburg, Evert P. L.; Liu, Ye-Hua; Huber, Sebastian D. Nature Physics, Vol. 13, Issue 5 https://doi.org/10.1038/nphys4037	journal	February 2017
Momentum-space entanglement after a quench in one-dimensional disordered fermionic systems Lundgren, Rex; Liu, Fangli; Laurell, Pontus Physical Review B, Vol. 100, Issue 24 https://doi.org/10.1103/physrevb.100.241108	journal	December 2019
Entanglement spectrum of a random partition: Connection with the localization transition Vijay, Sagar; Fu, Liang Physical Review B, Vol. 91, Issue 22 https://doi.org/10.1103/physrevb.91.220101	journal	June 2015
Analytic expression for the entanglement entropy of a two-dimensional topological superconductor Borchmann, Jan; Pereg-Barnea, T. Physical Review B, Vol. 95, Issue 7 https://doi.org/10.1103/physrevb.95.075152	journal	February 2017
Topological invariant for two-dimensional open systems Zheng, Jun-Hui; Hofstetter, Walter Physical Review B, Vol. 97, Issue 19 https://doi.org/10.1103/physrevb.97.195434	journal	May 2018
Entanglement polarization for the topological quadrupole phase Fukui, Takahiro; Hatsugai, Yasuhiro Physical Review B, Vol. 98, Issue 3 https://doi.org/10.1103/physrevb.98.035147	journal	July 2018
Decoding quantum criticality from fermionic/parafermionic topological states Wang, Zi-Qi; Zhu, Guo-Yi; Zhang, Guang-Ming Physical Review B, Vol. 98, Issue 15 https://doi.org/10.1103/physrevb.98.155139	journal	October 2018
Reconstructing entanglement Hamiltonian via entanglement eigenstates Zhu, W.; Huang, Zhoushen; He, Yin-Chen Physical Review B, Vol. 99, Issue 23 https://doi.org/10.1103/physrevb.99.235109	journal	June 2019
Invariance of Topological Indices Under Hilbert Space Truncation Huang, Zhoushen; Zhu, W.; Arovas, Daniel P. Physical Review Letters, Vol. 120, Issue 1 https://doi.org/10.1103/physrevlett.120.016403	journal	January 2018
Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines Cheng, Song; Chen, Jing; Wang, Lei Entropy, Vol. 20, Issue 8 https://doi.org/10.3390/e20080583	journal	August 2018
Entanglement Spectrum of a Random Partition: Connection with the Localization Transition Vijay, Sagar; Fu, Liang arXiv https://doi.org/10.48550/arxiv.1412.4733	text	January 2014
Learning phase transitions by confusion van Nieuwenburg, Evert P. L.; Liu, Ye-Hua; Huber, Sebastian D. arXiv https://doi.org/10.48550/arxiv.1610.02048	text	January 2016
Invariance of topological indices under Hilbert space truncation Huang, Zhoushen; Zhu, W.; Arovas, Daniel P. arXiv https://doi.org/10.48550/arxiv.1705.05949	text	January 2017
Topological invariant for two-dimensional open systems Zheng, Jun-Hui; Hofstetter, Walter arXiv https://doi.org/10.48550/arxiv.1710.03119	text	January 2017
Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines Cheng, Song; Chen, Jing; Wang, Lei arXiv https://doi.org/10.48550/arxiv.1712.04144	text	January 2017
Decoding quantum criticalities from fermionic/parafermionic topological states Wang, Zi-Qi; Zhu, Guo-Yi; Zhang, Guang-Ming arXiv https://doi.org/10.48550/arxiv.1802.04542	text	January 2018
Reconstructing Entanglement Hamiltonian via Entanglement Eigenstates Zhu, W.; Huang, Zhoushen; He, Yin-chen arXiv https://doi.org/10.48550/arxiv.1806.08060	text	January 2018

Figures / Tables (2)

FIG. 1(p. 3)

FIG. 2(p. 4)

Similar Records

Tensor network implementation of bulk entanglement spectrum

Journal Article · Sun Aug 24 20:00:00 EDT 2014 · Physical Review. B, Condensed Matter and Materials Physics · OSTI ID:1505786

Entanglement spectrum of a random partition: Connection with the localization transition

Journal Article · Sun Jun 14 20:00:00 EDT 2015 · Physical Review. B, Condensed Matter and Materials Physics · OSTI ID:1505793

Related Subjects

75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY

Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Citation Formats

References (35)

Cited By (17)

Figures / Tables (2)

Similar Records

Related Subjects