Invariance of Topological Indices Under Hilbert Space Truncation

Huang, Zhoushen; Zhu, Wei; Arovas, Daniel P.; Zhu, Jian -Xin; Balatsky, Alexander V.

doi:10.1103/PhysRevLett.120.016403

Title: Invariance of Topological Indices Under Hilbert Space Truncation

Journal Article · Fri Jan 05 00:00:00 EST 2018 · Physical Review Letters

DOI:https://doi.org/10.1103/PhysRevLett.120.016403· OSTI ID:1418765

^[1];

^[1]; Arovas, Daniel P. ^[2];

^[1];

^[3]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of California San Diego, La Jolla, CA (United States)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States); NORDITA, Stockholm (Sweden)

Here, we show that the topological index of a wave function, computed in the space of twisted boundary phases, is preserved under Hilbert space truncation, provided the truncated state remains normalizable. If truncation affects the boundary condition of the resulting state, the invariant index may acquire a different physical interpretation. If the index is symmetry protected, the truncation should preserve the protecting symmetry. We discuss implications of this invariance using paradigmatic integer and fractional Chern insulators, Z₂ topological insulators, and spin-1 Affleck-Kennedy-Lieb-Tasaki and Heisenberg chains, as well as its relation with the notion of bulk entanglement. As a possible application, we propose a partial quantum tomography scheme from which the topological index of a generic multicomponent wave function can be extracted by measuring only a small subset of wave function components, equivalent to the measurement of a bulk entanglement topological index.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-06NA25396; BES E3B7

OSTI ID:: 1418765

Alternate ID(s):: OSTI ID: 1415875

Report Number(s):: LA-UR-17-24060; PRLTAO; TRN: US1801298

Journal Information:: Physical Review Letters, Vol. 120, Issue 1; ISSN 0031-9007

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 4 works

Citation information provided by
Web of Science

References (76)

Rigorous results on valence-bond ground states in antiferromagnets Affleck, Ian; Kennedy, Tom; Lieb, Elliott H. Physical Review Letters, Vol. 59, Issue 7 https://doi.org/10.1103/PhysRevLett.59.799	journal	August 1987
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
Wave functions for fractional Chern insulators McGreevy, John; Swingle, Brian; Tran, Ky-Anh Physical Review B, Vol. 85, Issue 12 https://doi.org/10.1103/PhysRevB.85.125105	journal	March 2012
Tomography of Band Insulators from Quench Dynamics Hauke, Philipp; Lewenstein, Maciej; Eckardt, André Physical Review Letters, Vol. 113, Issue 4 https://doi.org/10.1103/PhysRevLett.113.045303	journal	July 2014
Variational Monte Carlo study of a chiral spin liquid in the extended Heisenberg model on the kagome lattice Hu, Wen-Jun; Zhu, Wei; Zhang, Yi Physical Review B, Vol. 91, Issue 4 https://doi.org/10.1103/PhysRevB.91.041124	journal	January 2015
Equivalent expression of $Z_{2}$ topological invariant for band insulators using the non-Abelian Berry connection Yu, Rui; Qi, Xiao Liang; Bernevig, Andrei Physical Review B, Vol. 84, Issue 7 https://doi.org/10.1103/PhysRevB.84.075119	journal	August 2011
Many-body physics with ultracold gases Bloch, Immanuel; Dalibard, Jean; Zwerger, Wilhelm Reviews of Modern Physics, Vol. 80, Issue 3 https://doi.org/10.1103/RevModPhys.80.885	journal	July 2008
Entanglement spectra and entanglement thermodynamics of Hofstadter bilayers Schliemann, John New Journal of Physics, Vol. 15, Issue 5 https://doi.org/10.1088/1367-2630/15/5/053017	journal	May 2013
Quantum orders and symmetric spin liquids Wen, Xiao-Gang Physical Review B, Vol. 65, Issue 16 https://doi.org/10.1103/PhysRevB.65.165113	journal	April 2002
Ultracold Atoms Out of Equilibrium Langen, Tim; Geiger, Remi; Schmiedmayer, Jörg Annual Review of Condensed Matter Physics, Vol. 6, Issue 1 https://doi.org/10.1146/annurev-conmatphys-031214-014548	journal	March 2015
Many-body density matrices for free fermions Cheong, Siew-Ann; Henley, Christopher L. Physical Review B, Vol. 69, Issue 7 https://doi.org/10.1103/PhysRevB.69.075111	journal	February 2004
Establishing non-Abelian topological order in Gutzwiller-projected Chern insulators via entanglement entropy and modular $S$ -matrix Zhang, Yi; Vishwanath, Ashvin Physical Review B, Vol. 87, Issue 16 https://doi.org/10.1103/PhysRevB.87.161113	journal	April 2013
Experimental reconstruction of the Berry curvature in a Floquet Bloch band Flaschner, N.; Rem, B. S.; Tarnowski, M. Science, Vol. 352, Issue 6289 https://doi.org/10.1126/science.aad4568	journal	May 2016
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C. Science, Vol. 314, Issue 5806, p. 1757-1761 https://doi.org/10.1126/science.1133734	journal	December 2006
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 Chern Number for an Extensive Partition of a Topological Ground State Fukui, Takahiro; Hatsugai, Yasuhiro Journal of the Physical Society of Japan, Vol. 83, Issue 11 https://doi.org/10.7566/JPSJ.83.113705	journal	November 2014
Classification of topological quantum matter with symmetries Chiu, Ching-Kai; Teo, Jeffrey C. Y.; Schnyder, Andreas P. Reviews of Modern Physics, Vol. 88, Issue 3 https://doi.org/10.1103/RevModPhys.88.035005	journal	August 2016
Topological Insulators and Topological Superconductors Bernevig, B. Andrei Princeton University Press https://doi.org/10.1515/9781400846733	book	January 2013
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
Theory of polarization of crystalline solids King-Smith, R. D.; Vanderbilt, David Physical Review B, Vol. 47, Issue 3 https://doi.org/10.1103/PhysRevB.47.1651	journal	January 1993
Robustness of the spin-Chern number Prodan, Emil Physical Review B, Vol. 80, Issue 12 https://doi.org/10.1103/PhysRevB.80.125327	journal	September 2009
Nonlocal Order in Gapless Systems: Entanglement Spectrum in Spin Chains Thomale, Ronny; Arovas, D. P.; Bernevig, B. Andrei Physical Review Letters, Vol. 105, Issue 11 https://doi.org/10.1103/PhysRevLett.105.116805	journal	September 2010
Topological aspects of the quantum spin-Hall effect in graphene: $Z_{2}$ topological order and spin Chern number Fukui, Takahiro; Hatsugai, Yasuhiro Physical Review B, Vol. 75, Issue 12 https://doi.org/10.1103/PhysRevB.75.121403	journal	March 2007
Quantum simulations with ultracold quantum gases Bloch, Immanuel; Dalibard, Jean; Nascimbène, Sylvain Nature Physics, Vol. 8, Issue 4 https://doi.org/10.1038/nphys2259	journal	April 2012
Fractional Chern Insulators with Strong Interactions that Far Exceed Band Gaps Kourtis, Stefanos; Neupert, Titus; Chamon, Claudio Physical Review Letters, Vol. 112, Issue 12 https://doi.org/10.1103/PhysRevLett.112.126806	journal	March 2014
The Resonating Valence Bond State in La2CuO4 and Superconductivity Anderson, P. W. Science, Vol. 235, Issue 4793 https://doi.org/10.1126/science.235.4793.1196	journal	March 1987
Topological insulators and superconductors: tenfold way and dimensional hierarchy Ryu, Shinsei; Schnyder, Andreas P.; Furusaki, Akira New Journal of Physics, Vol. 12, Issue 6 https://doi.org/10.1088/1367-2630/12/6/065010	journal	June 2010
The density-matrix renormalization group in the age of matrix product states Schollwöck, Ulrich Annals of Physics, Vol. 326, Issue 1 https://doi.org/10.1016/j.aop.2010.09.012	journal	January 2011
Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State Haldane, F. D. M. Physical Review Letters, Vol. 50, Issue 15 https://doi.org/10.1103/PhysRevLett.50.1153	journal	April 1983
Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State Hsieh, Timothy H.; Fu, Liang Physical Review Letters, Vol. 113, Issue 10 https://doi.org/10.1103/PhysRevLett.113.106801	journal	September 2014
Time-Reversal-Symmetry-Broken Quantum Spin Hall Effect Yang, Yunyou; Xu, Zhong; Sheng, L. Physical Review Letters, Vol. 107, Issue 6 https://doi.org/10.1103/PhysRevLett.107.066602	journal	August 2011
Physics of projected wavefunctions Gros, Claudius Annals of Physics, Vol. 189, Issue 1 https://doi.org/10.1016/0003-4916(89)90077-8	journal	January 1989
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
Surface states and topological invariants in three-dimensional topological insulators: Application to ${Bi}_{1 - x} {Sb}_{x}$ Teo, Jeffrey C. Y.; Fu, Liang; Kane, C. L. Physical Review B, Vol. 78, Issue 4, Article No. 045426 https://doi.org/10.1103/PhysRevB.78.045426	journal	July 2008
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
Quantized Hall conductance as a topological invariant Niu, Qian; Thouless, D. J.; Wu, Yong-Shi Physical Review B, Vol. 31, Issue 6 https://doi.org/10.1103/PhysRevB.31.3372	journal	March 1985
Convergent Theory for Effective Interaction in Nuclei Suzuki, K.; Lee, S. Y. Progress of Theoretical Physics, Vol. 64, Issue 6 https://doi.org/10.1143/PTP.64.2091	journal	December 1980
Inversion-symmetric topological insulators Hughes, Taylor L.; Prodan, Emil; Bernevig, B. Andrei Physical Review B, Vol. 83, Issue 24 https://doi.org/10.1103/PhysRevB.83.245132	journal	June 2011
Symmetry-protected fractional Chern insulators and fractional topological insulators Lu, Yuan-Ming; Ran, Ying Physical Review B, Vol. 85, Issue 16 https://doi.org/10.1103/PhysRevB.85.165134	journal	April 2012
Topological classification of gapped spin chains: Quantized Berry phase as a local order parameter Hirano, T.; Katsura, H.; Hatsugai, Y. Physical Review B, Vol. 77, Issue 9 https://doi.org/10.1103/PhysRevB.77.094431	journal	March 2008
Berry’s phase for energy bands in solids Zak, J. Physical Review Letters, Vol. 62, Issue 23 https://doi.org/10.1103/PhysRevLett.62.2747	journal	June 1989
Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology Legner, Markus; Neupert, Titus Physical Review B, Vol. 88, Issue 11 https://doi.org/10.1103/PhysRevB.88.115114	journal	September 2013
Topological Insulator Materials Ando, Yoichi Journal of the Physical Society of Japan, Vol. 82, Issue 10, Article No. 102001 https://doi.org/10.7566/JPSJ.82.102001	journal	October 2013
Non-Abelian Quantum Hall Effect in Topological Flat Bands Wang, Yi-Fei; Yao, Hong; Gu, Zheng-Cheng Physical Review Letters, Vol. 108, Issue 12 https://doi.org/10.1103/PhysRevLett.108.126805	journal	March 2012
Refined characterization of lattice Chern insulators by bulk entanglement spectrum Chiou, Dah-Wei; Kao, Hsien-Chung; Lin, Feng-Li Physical Review B, Vol. 94, Issue 23 https://doi.org/10.1103/PhysRevB.94.235129	journal	December 2016
Preroughening Transitions in Surfaces Rommelse, Koos; den Nijs, Marcel Physical Review Letters, Vol. 59, Issue 22 https://doi.org/10.1103/PhysRevLett.59.2578	journal	November 1987
Dynamical Quantum Phase Transitions: Role of Topological Nodes in Wave Function Overlaps Huang, Zhoushen; Balatsky, Alexander V. Physical Review Letters, Vol. 117, Issue 8 https://doi.org/10.1103/PhysRevLett.117.086802	journal	August 2016
Localization, wave-function topology, and the integer quantized Hall effect Arovas, Daniel P.; Bhatt, R. N.; Haldane, F. D. M. Physical Review Letters, Vol. 60, Issue 7 https://doi.org/10.1103/PhysRevLett.60.619	journal	February 1988
Adiabatic continuity, wave-function overlap, and topological phase transitions Gu, Jiahua; Sun, Kai Physical Review B, Vol. 94, Issue 12 https://doi.org/10.1103/PhysRevB.94.125111	journal	September 2016
Hidden topological order in integer quantum spin chains Girvin, S. M.; Arovas, Daniel P. Physica Scripta, Vol. T27 https://doi.org/10.1088/0031-8949/1989/T27/027	journal	January 1989
Topological invariant and the quantization of the Hall conductance Kohmoto, Mahito Annals of Physics, Vol. 160, Issue 2 https://doi.org/10.1016/0003-4916(85)90148-4	journal	April 1985
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
Spin-Orbit-Free Topological Insulators without Time-Reversal Symmetry Alexandradinata, A.; Fang, Chen; Gilbert, Matthew J. Physical Review Letters, Vol. 113, Issue 11 https://doi.org/10.1103/PhysRevLett.113.116403	journal	September 2014
Entanglement Spectrum of a Disordered Topological Chern Insulator Prodan, Emil; Hughes, Taylor L.; Bernevig, B. Andrei Physical Review Letters, Vol. 105, Issue 11 https://doi.org/10.1103/PhysRevLett.105.115501	journal	September 2010
Entanglement spectrum and Wannier center flow of the Hofstadter problem Huang, Zhoushen; Arovas, Daniel P. Physical Review B, Vol. 86, Issue 24 https://doi.org/10.1103/PhysRevB.86.245109	journal	December 2012
Quantum Spin-Hall Effect and Topologically Invariant Chern Numbers Sheng, D. N.; Weng, Z. Y.; Sheng, L. Physical Review Letters, Vol. 97, Issue 3 https://doi.org/10.1103/PhysRevLett.97.036808	journal	July 2006
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields Hofstadter, Douglas R. Physical Review B, Vol. 14, Issue 6 https://doi.org/10.1103/PhysRevB.14.2239	journal	September 1976
Physics of projected wavefunctions Claudius, Gros, ETH Zurich https://doi.org/10.3929/ethz-a-000494086	text	January 1988
Hidden Topological Order in Integer Quantum Spin Chains Girvin, S. M.; Arovas, Daniel P. Proceedings of Nobel Symposium 73, Physics of Low-dimensional Systems https://doi.org/10.1142/9789814541152_0027	conference	August 2014
Many-Body Physics with Ultracold Gases Bloch, Immanuel; Dalibard, Jean; Zwerger, Wilhelm arXiv https://doi.org/10.48550/arxiv.0704.3011	text	January 2007
Topological Classification of Gapped Spin Chains :Quantized Berry Phase as a Local Order Parameter Hirano, Takaaki; Katsura, Hosho; Hatsugai, Yasuhiro arXiv https://doi.org/10.48550/arxiv.0710.4198	text	January 2007
Entanglement Spectrum of a Disordered Topological Chern Insulator Prodan, Emil; Hughes, Taylor L.; Bernevig, B. Andrei arXiv https://doi.org/10.48550/arxiv.1005.5148	text	January 2010
The density-matrix renormalization group in the age of matrix product states Schollwoeck, Ulrich arXiv https://doi.org/10.48550/arxiv.1008.3477	text	January 2010
Inversion Symmetric Topological Insulators Hughes, Taylor L.; Prodan, Emil; Bernevig, B. Andrei arXiv https://doi.org/10.48550/arxiv.1010.4508	text	January 2010
Entanglement spectrum and Wannier center flow of the Hofstadter problem Huang, Zhoushen; Arovas, Daniel P. arXiv https://doi.org/10.48550/arxiv.1201.0733	text	January 2012
Establishing non-Abelian topological order in Gutzwiller projected Chern insulators via Entanglement Entropy and Modular S-matrix Zhang, Yi; Vishwanath, Ashvin arXiv https://doi.org/10.48550/arxiv.1209.2424	text	January 2012
Entanglement Spectra and Entanglement Thermodynamics of Hofstadter Bilayers Schliemann, John arXiv https://doi.org/10.48550/arxiv.1302.5517	text	January 2013
Topological Insulator Materials Ando, Yoichi arXiv https://doi.org/10.48550/arxiv.1304.5693	text	January 2013
Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology Legner, Markus; Neupert, Titus arXiv https://doi.org/10.48550/arxiv.1306.4008	text	January 2013
Tomography of band insulators from quench dynamics Hauke, Philipp; Lewenstein, Maciej; Eckardt, André arXiv https://doi.org/10.48550/arxiv.1401.8240	text	January 2014
Ultracold atoms out of equilibrium Langen, Tim; Geiger, Remi; Schmiedmayer, Jörg arXiv https://doi.org/10.48550/arxiv.1408.6377	text	January 2014
Classification of topological quantum matter with symmetries Chiu, Ching-Kai; Teo, Jeffrey C. Y.; Schnyder, Andreas P. arXiv https://doi.org/10.48550/arxiv.1505.03535	text	January 2015
Adiabatic continuity, wavefunction overlap and topological phase transitions Gu, Jiahua; Sun, Kai arXiv https://doi.org/10.48550/arxiv.1605.07627	text	January 2016
Calculation of reduced density matrices from correlation functions Peschel, Ingo arXiv https://doi.org/10.48550/arxiv.cond-mat/0212631	text	January 2002
Quantum Spin Hall Effect and Topologically Invariant Chern Numbers Sheng, D. N.; Weng, Z. Y.; Sheng, L. arXiv https://doi.org/10.48550/arxiv.cond-mat/0603054	text	January 2006
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells Bernevig, B. Andrei; Hughes, Taylor L.; Zhang, Shou-Cheng arXiv https://doi.org/10.48550/arxiv.cond-mat/0611399	text	January 2006