Invariance of Topological Indices Under Hilbert Space Truncation
Abstract
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, Z _{2} topological insulators, and spin1 AffleckKennedyLiebTasaki 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.
 Authors:

 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)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22); USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1418765
 Alternate Identifier(s):
 OSTI ID: 1415875
 Report Number(s):
 LAUR1724060
Journal ID: ISSN 00319007; PRLTAO; TRN: US1801298
 Grant/Contract Number:
 AC5206NA25396; BES E3B7
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review Letters
 Additional Journal Information:
 Journal Volume: 120; Journal Issue: 1; Journal ID: ISSN 00319007
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Material Science
Citation Formats
Huang, Zhoushen, Zhu, Wei, Arovas, Daniel P., Zhu, Jian Xin, and Balatsky, Alexander V. Invariance of Topological Indices Under Hilbert Space Truncation. United States: N. p., 2018.
Web. doi:10.1103/PhysRevLett.120.016403.
Huang, Zhoushen, Zhu, Wei, Arovas, Daniel P., Zhu, Jian Xin, & Balatsky, Alexander V. Invariance of Topological Indices Under Hilbert Space Truncation. United States. doi:10.1103/PhysRevLett.120.016403.
Huang, Zhoushen, Zhu, Wei, Arovas, Daniel P., Zhu, Jian Xin, and Balatsky, Alexander V. Fri .
"Invariance of Topological Indices Under Hilbert Space Truncation". United States. doi:10.1103/PhysRevLett.120.016403. https://www.osti.gov/servlets/purl/1418765.
@article{osti_1418765,
title = {Invariance of Topological Indices Under Hilbert Space Truncation},
author = {Huang, Zhoushen and Zhu, Wei and Arovas, Daniel P. and Zhu, Jian Xin and Balatsky, Alexander V.},
abstractNote = {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 spin1 AffleckKennedyLiebTasaki 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.},
doi = {10.1103/PhysRevLett.120.016403},
journal = {Physical Review Letters},
number = 1,
volume = 120,
place = {United States},
year = {2018},
month = {1}
}
Figures / Tables:
Works referenced in this record:
Rigorous results on valencebond ground states in antiferromagnets
journal, August 1987
 Affleck, Ian; Kennedy, Tom; Lieb, Elliott H.
 Physical Review Letters, Vol. 59, Issue 7
Topological insulators and superconductors
journal, October 2011
 Qi, XiaoLiang; Zhang, ShouCheng
 Reviews of Modern Physics, Vol. 83, Issue 4
Wave functions for fractional Chern insulators
journal, March 2012
 McGreevy, John; Swingle, Brian; Tran, KyAnh
 Physical Review B, Vol. 85, Issue 12
Tomography of Band Insulators from Quench Dynamics
journal, July 2014
 Hauke, Philipp; Lewenstein, Maciej; Eckardt, André
 Physical Review Letters, Vol. 113, Issue 4
Variational Monte Carlo study of a chiral spin liquid in the extended Heisenberg model on the kagome lattice
journal, January 2015
 Hu, WenJun; Zhu, Wei; Zhang, Yi
 Physical Review B, Vol. 91, Issue 4
Equivalent expression of ${\mathbb{Z}}_{2}$ topological invariant for band insulators using the nonAbelian Berry connection
journal, August 2011
 Yu, Rui; Qi, Xiao Liang; Bernevig, Andrei
 Physical Review B, Vol. 84, Issue 7
Manybody physics with ultracold gases
journal, July 2008
 Bloch, Immanuel; Dalibard, Jean; Zwerger, Wilhelm
 Reviews of Modern Physics, Vol. 80, Issue 3
Entanglement spectra and entanglement thermodynamics of Hofstadter bilayers
journal, May 2013
 Schliemann, John
 New Journal of Physics, Vol. 15, Issue 5
Quantum orders and symmetric spin liquids
journal, April 2002
 Wen, XiaoGang
 Physical Review B, Vol. 65, Issue 16
Ultracold Atoms Out of Equilibrium
journal, March 2015
 Langen, Tim; Geiger, Remi; Schmiedmayer, Jörg
 Annual Review of Condensed Matter Physics, Vol. 6, Issue 1
Manybody density matrices for free fermions
journal, February 2004
 Cheong, SiewAnn; Henley, Christopher L.
 Physical Review B, Vol. 69, Issue 7
Establishing nonAbelian topological order in Gutzwillerprojected Chern insulators via entanglement entropy and modular $\mathcal{S}$ matrix
journal, April 2013
 Zhang, Yi; Vishwanath, Ashvin
 Physical Review B, Vol. 87, Issue 16
Experimental reconstruction of the Berry curvature in a Floquet Bloch band
journal, May 2016
 Flaschner, N.; Rem, B. S.; Tarnowski, M.
 Science, Vol. 352, Issue 6289
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
journal, December 2006
 Bernevig, B. A.; Hughes, T. L.; Zhang, S.C.
 Science, Vol. 314, Issue 5806, p. 17571761
Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in NonAbelian Fractional Quantum Hall Effect States
journal, July 2008
 Li, Hui; Haldane, F. D. M.
 Physical Review Letters, Vol. 101, Issue 1
Entanglement Chern Number for an Extensive Partition of a Topological Ground State
journal, November 2014
 Fukui, Takahiro; Hatsugai, Yasuhiro
 Journal of the Physical Society of Japan, Vol. 83, Issue 11
Classification of topological quantum matter with symmetries
journal, August 2016
 Chiu, ChingKai; Teo, Jeffrey C. Y.; Schnyder, Andreas P.
 Reviews of Modern Physics, Vol. 88, Issue 3
Calculation of reduced density matrices from correlation functions
journal, March 2003
 Peschel, Ingo
 Journal of Physics A: Mathematical and General, Vol. 36, Issue 14
Theory of polarization of crystalline solids
journal, January 1993
 KingSmith, R. D.; Vanderbilt, David
 Physical Review B, Vol. 47, Issue 3
Robustness of the spinChern number
journal, September 2009
 Prodan, Emil
 Physical Review B, Vol. 80, Issue 12
Nonlocal Order in Gapless Systems: Entanglement Spectrum in Spin Chains
journal, September 2010
 Thomale, Ronny; Arovas, D. P.; Bernevig, B. Andrei
 Physical Review Letters, Vol. 105, Issue 11
Topological aspects of the quantum spinHall effect in graphene: ${\mathrm{Z}}_{2}$ topological order and spin Chern number
journal, March 2007
 Fukui, Takahiro; Hatsugai, Yasuhiro
 Physical Review B, Vol. 75, Issue 12
Quantum simulations with ultracold quantum gases
journal, April 2012
 Bloch, Immanuel; Dalibard, Jean; Nascimbène, Sylvain
 Nature Physics, Vol. 8, Issue 4
Fractional Chern Insulators with Strong Interactions that Far Exceed Band Gaps
journal, March 2014
 Kourtis, Stefanos; Neupert, Titus; Chamon, Claudio
 Physical Review Letters, Vol. 112, Issue 12
The Resonating Valence Bond State in La2CuO4 and Superconductivity
journal, March 1987
 Anderson, P. W.
 Science, Vol. 235, Issue 4793
Topological insulators and superconductors: tenfold way and dimensional hierarchy
journal, June 2010
 Ryu, Shinsei; Schnyder, Andreas P.; Furusaki, Akira
 New Journal of Physics, Vol. 12, Issue 6
The densitymatrix renormalization group in the age of matrix product states
journal, January 2011
 Schollwöck, Ulrich
 Annals of Physics, Vol. 326, Issue 1
Nonlinear Field Theory of LargeSpin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the OneDimensional EasyAxis Néel State
journal, April 1983
 Haldane, F. D. M.
 Physical Review Letters, Vol. 50, Issue 15
Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State
journal, September 2014
 Hsieh, Timothy H.; Fu, Liang
 Physical Review Letters, Vol. 113, Issue 10
TimeReversalSymmetryBroken Quantum Spin Hall Effect
journal, August 2011
 Yang, Yunyou; Xu, Zhong; Sheng, L.
 Physical Review Letters, Vol. 107, Issue 6
Physics of projected wavefunctions
journal, January 1989
 Gros, Claudius
 Annals of Physics, Vol. 189, Issue 1
Quantized Hall Conductance in a TwoDimensional Periodic Potential
journal, August 1982
 Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.
 Physical Review Letters, Vol. 49, Issue 6
Surface states and topological invariants in threedimensional topological insulators: Application to ${\text{Bi}}_{1x}{\text{Sb}}_{x}$
journal, July 2008
 Teo, Jeffrey C. Y.; Fu, Liang; Kane, C. L.
 Physical Review B, Vol. 78, Issue 4, Article No. 045426
Colloquium: Topological insulators
journal, November 2010
 Hasan, M. Z.; Kane, C. L.
 Reviews of Modern Physics, Vol. 82, Issue 4, p. 30453067
Quantized Hall conductance as a topological invariant
journal, March 1985
 Niu, Qian; Thouless, D. J.; Wu, YongShi
 Physical Review B, Vol. 31, Issue 6
Convergent Theory for Effective Interaction in Nuclei
journal, December 1980
 Suzuki, K.; Lee, S. Y.
 Progress of Theoretical Physics, Vol. 64, Issue 6
Inversionsymmetric topological insulators
journal, June 2011
 Hughes, Taylor L.; Prodan, Emil; Bernevig, B. Andrei
 Physical Review B, Vol. 83, Issue 24
Symmetryprotected fractional Chern insulators and fractional topological insulators
journal, April 2012
 Lu, YuanMing; Ran, Ying
 Physical Review B, Vol. 85, Issue 16
Topological classification of gapped spin chains: Quantized Berry phase as a local order parameter
journal, March 2008
 Hirano, T.; Katsura, H.; Hatsugai, Y.
 Physical Review B, Vol. 77, Issue 9
Berry’s phase for energy bands in solids
journal, June 1989
 Zak, J.
 Physical Review Letters, Vol. 62, Issue 23
Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology
journal, September 2013
 Legner, Markus; Neupert, Titus
 Physical Review B, Vol. 88, Issue 11
Topological Insulator Materials
journal, October 2013
 Ando, Yoichi
 Journal of the Physical Society of Japan, Vol. 82, Issue 10, Article No. 102001
NonAbelian Quantum Hall Effect in Topological Flat Bands
journal, March 2012
 Wang, YiFei; Yao, Hong; Gu, ZhengCheng
 Physical Review Letters, Vol. 108, Issue 12
Refined characterization of lattice Chern insulators by bulk entanglement spectrum
journal, December 2016
 Chiou, DahWei; Kao, HsienChung; Lin, FengLi
 Physical Review B, Vol. 94, Issue 23
Preroughening Transitions in Surfaces
journal, November 1987
 Rommelse, Koos; den Nijs, Marcel
 Physical Review Letters, Vol. 59, Issue 22
Dynamical Quantum Phase Transitions: Role of Topological Nodes in Wave Function Overlaps
journal, August 2016
 Huang, Zhoushen; Balatsky, Alexander V.
 Physical Review Letters, Vol. 117, Issue 8
Localization, wavefunction topology, and the integer quantized Hall effect
journal, February 1988
 Arovas, Daniel P.; Bhatt, R. N.; Haldane, F. D. M.
 Physical Review Letters, Vol. 60, Issue 7
Adiabatic continuity, wavefunction overlap, and topological phase transitions
journal, September 2016
 Gu, Jiahua; Sun, Kai
 Physical Review B, Vol. 94, Issue 12
Hidden topological order in integer quantum spin chains
journal, January 1989
 Girvin, S. M.; Arovas, Daniel P.
 Physica Scripta, Vol. T27
Topological invariant and the quantization of the Hall conductance
journal, April 1985
 Kohmoto, Mahito
 Annals of Physics, Vol. 160, Issue 2
Entanglement spectrum of a topological phase in one dimension
journal, February 2010
 Pollmann, Frank; Turner, Ari M.; Berg, Erez
 Physical Review B, Vol. 81, Issue 6
SpinOrbitFree Topological Insulators without TimeReversal Symmetry
journal, September 2014
 Alexandradinata, A.; Fang, Chen; Gilbert, Matthew J.
 Physical Review Letters, Vol. 113, Issue 11
Entanglement Spectrum of a Disordered Topological Chern Insulator
journal, September 2010
 Prodan, Emil; Hughes, Taylor L.; Bernevig, B. Andrei
 Physical Review Letters, Vol. 105, Issue 11
Entanglement spectrum and Wannier center flow of the Hofstadter problem
journal, December 2012
 Huang, Zhoushen; Arovas, Daniel P.
 Physical Review B, Vol. 86, Issue 24
Quantum SpinHall Effect and Topologically Invariant Chern Numbers
journal, July 2006
 Sheng, D. N.; Weng, Z. Y.; Sheng, L.
 Physical Review Letters, Vol. 97, Issue 3
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields
journal, September 1976
 Hofstadter, Douglas R.
 Physical Review B, Vol. 14, Issue 6
Figures / Tables found in this record: