Fermionic tensor networks for higher-order topological insulators from charge pumping
Abstract
We apply the charge-pumping argument to fermionic tensor network representations of $$\textit{d}$$-dimensional topological insulators (TIs) to obtain tensor network states (TNSs) for ($$\textit{d}$$ + 1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair state (PEPS) for a Chern insulator starting from a matrix product state (MPS) in $$\textit{d}$$ = 1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in $$\textit{d}$$ = 2 for the obstructed atomic insulator (OAI) of the quadrupole model. The ($$\textit{d}$$ + 1)-dimensional TNSs obtained in this way have a constant bond dimension inherited from the $$\textit{d}$$-dimensional TNSs in all but one spatial direction, making them candidates for numerical applications. From the $$\textit{d}$$-dimensional models, we identify gapped next-nearest-neighbor Hamiltonians interpolating between the trivial and OAI phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an MPS and a PEPS with a constant bond dimension equal to 2, respectively.
- Publication Date:
- Research Org.:
- Princeton Univ., NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC); European Research Council (ERC); European Union’s Horizon 2020; German Research Foundation (DFG); National Science Foundation (NSF); US Department of the Navy, Office of Naval Research (ONR); Packard Foundation; Schmidt Fund for Innovative Research; US Army Research Office (ARO); Agence Nationale de la Recherché (ANR)
- OSTI Identifier:
- 1605502
- Alternate Identifier(s):
- OSTI ID: 1802869
- Grant/Contract Number:
- SC0016239; 636201; EXC-2111–390814868; ANR-17-CE30-0013-01; ANR-16- CE30-0025; DMR 1643312; 404513; N00014-14-1-0330; W911NF-12-1-0461
- Resource Type:
- Published Article
- Journal Name:
- Physical Review B
- Additional Journal Information:
- Journal Name: Physical Review B Journal Volume: 101 Journal Issue: 11; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Materials Science; Physics
Citation Formats
Hackenbroich, Anna, Bernevig, B. Andrei, Schuch, Norbert, and Regnault, Nicolas. Fermionic tensor networks for higher-order topological insulators from charge pumping. United States: N. p., 2020.
Web. doi:10.1103/PhysRevB.101.115134.
Hackenbroich, Anna, Bernevig, B. Andrei, Schuch, Norbert, & Regnault, Nicolas. Fermionic tensor networks for higher-order topological insulators from charge pumping. United States. https://doi.org/10.1103/PhysRevB.101.115134
Hackenbroich, Anna, Bernevig, B. Andrei, Schuch, Norbert, and Regnault, Nicolas. Thu .
"Fermionic tensor networks for higher-order topological insulators from charge pumping". United States. https://doi.org/10.1103/PhysRevB.101.115134.
@article{osti_1605502,
title = {Fermionic tensor networks for higher-order topological insulators from charge pumping},
author = {Hackenbroich, Anna and Bernevig, B. Andrei and Schuch, Norbert and Regnault, Nicolas},
abstractNote = {We apply the charge-pumping argument to fermionic tensor network representations of $\textit{d}$-dimensional topological insulators (TIs) to obtain tensor network states (TNSs) for ($\textit{d}$ + 1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair state (PEPS) for a Chern insulator starting from a matrix product state (MPS) in $\textit{d}$ = 1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in $\textit{d}$ = 2 for the obstructed atomic insulator (OAI) of the quadrupole model. The ($\textit{d}$ + 1)-dimensional TNSs obtained in this way have a constant bond dimension inherited from the $\textit{d}$-dimensional TNSs in all but one spatial direction, making them candidates for numerical applications. From the $\textit{d}$-dimensional models, we identify gapped next-nearest-neighbor Hamiltonians interpolating between the trivial and OAI phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an MPS and a PEPS with a constant bond dimension equal to 2, respectively.},
doi = {10.1103/PhysRevB.101.115134},
journal = {Physical Review B},
number = 11,
volume = 101,
place = {United States},
year = {2020},
month = {3}
}
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Works referenced in this record:
Observation of a phononic quadrupole topological insulator
journal, January 2018
- Serra-Garcia, Marc; Peri, Valerio; Süsstrunk, Roman
- Nature, Vol. 555, Issue 7696
Soliton excitations in polyacetylene
journal, August 1980
- Su, W. P.; Schrieffer, J. R.; Heeger, A. J.
- Physical Review B, Vol. 22, Issue 4
Entanglement in a second-order topological insulator on a square lattice
journal, December 2018
- Wang, Qiang; Wang, Da; Wang, Qiang-Hua
- EPL (Europhysics Letters), Vol. 124, Issue 5
Projected Entangled-Pair States Can Describe Chiral Topological States
journal, December 2013
- Wahl, T. B.; Tu, H. -H.; Schuch, N.
- Physical Review Letters, Vol. 111, Issue 23
Reduced density matrices and entanglement entropy in free lattice models
journal, December 2009
- Peschel, Ingo; Eisler, Viktor
- Journal of Physics A: Mathematical and Theoretical, Vol. 42, Issue 50
Renormalization and tensor product states in spin chains and lattices
journal, December 2009
- Cirac, J. Ignacio; Verstraete, Frank
- Journal of Physics A: Mathematical and Theoretical, Vol. 42, Issue 50
Fermionic projected entangled-pair states and topological phases
journal, December 2017
- Bultinck, Nick; Williamson, Dominic J.; Haegeman, Jutho
- Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 2
Higher-order topological insulators
journal, June 2018
- Schindler, Frank; Cook, Ashley M.; Vergniory, Maia G.
- Science Advances, Vol. 4, Issue 6
String Order and Symmetries in Quantum Spin Lattices
journal, April 2008
- Pérez-García, D.; Wolf, M. M.; Sanz, M.
- Physical Review Letters, Vol. 100, Issue 16
Compression of correlation matrices and an efficient method for forming matrix product states of fermionic Gaussian states
journal, August 2015
- Fishman, Matthew T.; White, Steven R.
- Physical Review B, Vol. 92, Issue 7
Establishing non-Abelian topological order in Gutzwiller-projected Chern insulators via entanglement entropy and modular -matrix
journal, April 2013
- Zhang, Yi; Vishwanath, Ashvin
- Physical Review B, Vol. 87, Issue 16
Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
journal, July 2008
- Li, Hui; Haldane, F. D. M.
- Physical Review Letters, Vol. 101, Issue 1
Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States
journal, June 2006
- Verstraete, F.; Wolf, M. M.; Perez-Garcia, D.
- Physical Review Letters, Vol. 96, Issue 22
Corner states of light in photonic waveguides
journal, September 2019
- El Hassan, Ashraf; Kunst, Flore K.; Moritz, Alexander
- Nature Photonics, Vol. 13, Issue 10
Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators
journal, March 2020
- Takahashi, Ryo; Tanaka, Yutaro; Murakami, Shuichi
- Physical Review Research, Vol. 2, Issue 1
Observation of quadrupole transitions and edge mode topology in an LC circuit network
journal, January 2019
- Serra-Garcia, Marc; Süsstrunk, Roman; Huber, Sebastian D.
- Physical Review B, Vol. 99, Issue 2
Fermionic projected entangled pair states
journal, May 2010
- Kraus, Christina V.; Schuch, Norbert; Verstraete, Frank
- Physical Review A, Vol. 81, Issue 5
Distilling Gaussian States with Gaussian Operations is Impossible
journal, September 2002
- Eisert, J.; Scheel, S.; Plenio, M. B.
- Physical Review Letters, Vol. 89, Issue 13
PEPS as ground states: Degeneracy and topology
journal, October 2010
- Schuch, Norbert; Cirac, Ignacio; Pérez-García, David
- Annals of Physics, Vol. 325, Issue 10
Gaussian Transformations and Distillation of Entangled Gaussian States
journal, September 2002
- Fiurášek, Jaromír
- Physical Review Letters, Vol. 89, Issue 13
Calculation of reduced density matrices from correlation functions
journal, March 2003
- Peschel, Ingo
- Journal of Physics A: Mathematical and General, Vol. 36, Issue 14
Tensor-product representations for string-net condensed states
journal, February 2009
- Gu, Zheng-Cheng; Levin, Michael; Swingle, Brian
- Physical Review B, Vol. 79, Issue 8
Observation of higher-order topological acoustic states protected by generalized chiral symmetry
journal, December 2018
- Ni, Xiang; Weiner, Matthew; Alù, Andrea
- Nature Materials, Vol. 18, Issue 2
Acoustic higher-order topological insulator on a kagome lattice
journal, December 2018
- Xue, Haoran; Yang, Yahui; Gao, Fei
- Nature Materials, Vol. 18, Issue 2
Quantization of particle transport
journal, May 1983
- Thouless, D. J.
- Physical Review B, Vol. 27, Issue 10
Finitely correlated states on quantum spin chains
journal, March 1992
- Fannes, M.; Nachtergaele, B.; Werner, R. F.
- Communications in Mathematical Physics, Vol. 144, Issue 3
Topological entanglement entropy of spin liquids and lattice Laughlin states
journal, August 2011
- Zhang, Yi; Grover, Tarun; Vishwanath, Ashvin
- Physical Review B, Vol. 84, Issue 7
Strong and fragile topological Dirac semimetals with higher-order Fermi arcs
journal, January 2020
- Wieder, Benjamin J.; Wang, Zhijun; Cano, Jennifer
- Nature Communications, Vol. 11, Issue 1
Photonic quadrupole topological phases
journal, June 2019
- Mittal, Sunil; Orre, Venkata Vikram; Zhu, Guanyu
- Nature Photonics, Vol. 13, Issue 10
Anyons in an exactly solved model and beyond
journal, January 2006
- Kitaev, Alexei
- Annals of Physics, Vol. 321, Issue 1
Chiral Projected Entangled-Pair State with Topological Order
journal, March 2015
- Yang, Shuo; Wahl, Thorsten B.; Tu, Hong-Hao
- Physical Review Letters, Vol. 114, Issue 10
Area Laws in Quantum Systems: Mutual Information and Correlations
journal, February 2008
- Wolf, Michael M.; Verstraete, Frank; Hastings, Matthew B.
- Physical Review Letters, Vol. 100, Issue 7
Matrix product state algorithms for Gaussian fermionic states
journal, December 2019
- Schuch, Norbert; Bauer, Bela
- Physical Review B, Vol. 100, Issue 24
Classifying quantum phases using matrix product states and projected entangled pair states
journal, October 2011
- Schuch, Norbert; Pérez-García, David; Cirac, Ignacio
- Physical Review B, Vol. 84, Issue 16
Symmetries and boundary theories for chiral projected entangled pair states
journal, September 2014
- Wahl, Thorsten B.; Haßler, Stefan T.; Tu, Hong-Hao
- Physical Review B, Vol. 90, Issue 11
Entanglement spectrum and boundary theories with projected entangled-pair states
journal, June 2011
- Cirac, J. Ignacio; Poilblanc, Didier; Schuch, Norbert
- Physical Review B, Vol. 83, Issue 24
Solving gapped Hamiltonians locally
journal, February 2006
- Hastings, M. B.
- Physical Review B, Vol. 73, Issue 8
Colloquium: Topological insulators
journal, November 2010
- Hasan, M. Z.; Kane, C. L.
- Reviews of Modern Physics, Vol. 82, Issue 4, p. 3045-3067
Higher-order topology in bismuth
journal, July 2018
- Schindler, Frank; Wang, Zhijun; Vergniory, Maia G.
- Nature Physics, Vol. 14, Issue 9
Characterization of Gaussian operations and distillation of Gaussian states
journal, September 2002
- Giedke, Géza; Ignacio Cirac, J.
- Physical Review A, Vol. 66, Issue 3
Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators
journal, December 2017
- Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L.
- Physical Review B, Vol. 96, Issue 24
Gapped topological kink states and topological corner states in honeycomb lattice
journal, January 2020
- Yang, Yuting; Jia, Ziyuan; Wu, Yijia
- Science Bulletin
Topological quantum chemistry
journal, July 2017
- Bradlyn, Barry; Elcoro, L.; Cano, Jennifer
- Nature, Vol. 547, Issue 7663
Reflection-Symmetric Second-Order Topological Insulators and Superconductors
journal, December 2017
- Langbehn, Josias; Peng, Yang; Trifunovic, Luka
- Physical Review Letters, Vol. 119, Issue 24
String-net condensation: A physical mechanism for topological phases
journal, January 2005
- Levin, Michael A.; Wen, Xiao-Gang
- Physical Review B, Vol. 71, Issue 4
Topolectrical-circuit realization of topological corner modes
journal, September 2018
- Imhof, Stefan; Berger, Christian; Bayer, Florian
- Nature Physics, Vol. 14, Issue 9
Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"
journal, October 1988
- Haldane, F. D. M.
- Physical Review Letters, Vol. 61, Issue 18
Fault-tolerant quantum computation by anyons
journal, January 2003
- Kitaev, A. Yu.
- Annals of Physics, Vol. 303, Issue 1
Fermionic matrix product states and one-dimensional topological phases
journal, February 2017
- Bultinck, Nick; Williamson, Dominic J.; Haegeman, Jutho
- Physical Review B, Vol. 95, Issue 7
Modewise entanglement of Gaussian states
journal, May 2003
- Botero, Alonso; Reznik, Benni
- Physical Review A, Vol. 67, Issue 5
Trace index and spectral flow in the entanglement spectrum of topological insulators
journal, November 2011
- Alexandradinata, A.; Hughes, Taylor L.; Bernevig, B. Andrei
- Physical Review B, Vol. 84, Issue 19
Quantized electric multipole insulators
journal, July 2017
- Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L.
- Science, Vol. 357, Issue 6346
Visualization of Higher-Order Topological Insulating Phases in Two-Dimensional Dielectric Photonic Crystals
journal, June 2019
- Xie, Bi-Ye; Su, Guang-Xu; Wang, Hong-Fei
- Physical Review Letters, Vol. 122, Issue 23
Entanglement Spectrum of Topological Insulators and Superconductors
journal, April 2010
- Fidkowski, Lukasz
- Physical Review Letters, Vol. 104, Issue 13
Compactly supported Wannier functions and algebraic -theory
journal, March 2017
- Read, N.
- Physical Review B, Vol. 95, Issue 11
Exact matrix product states for quantum Hall wave functions
journal, December 2012
- Zaletel, Michael P.; Mong, Roger S. K.
- Physical Review B, Vol. 86, Issue 24
Fractional quantum Hall physics in topological flat bands
journal, November 2013
- Parameswaran, Siddharth A.; Roy, Rahul; Sondhi, Shivaji L.
- Comptes Rendus Physique, Vol. 14, Issue 9-10
Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension
journal, November 2015
- Dubail, J.; Read, N.
- Physical Review B, Vol. 92, Issue 20
A quantized microwave quadrupole insulator with topologically protected corner states
journal, March 2018
- Peterson, Christopher W.; Benalcazar, Wladimir A.; Hughes, Taylor L.
- Nature, Vol. 555, Issue 7696
Fermionic projected entangled pair states and local gauge theories
journal, December 2015
- Zohar, Erez; Burrello, Michele; Wahl, Thorsten B.
- Annals of Physics, Vol. 363