Gapless Symmetry-Protected Topological Order
Abstract
We introduce exactly solvable gapless quantum systems in d dimensions that support symmetry-protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls “decorated” with dimension ( d - 1 ) SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, these edge modes are robust against arbitrary symmetry-preserving local perturbations near the edges. In two dimensions, we construct wave functions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.
- Authors:
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1412946
- Alternate Identifier(s):
- OSTI ID: 1485068
- Grant/Contract Number:
- AC02-05CH11231
- Resource Type:
- Published Article
- Journal Name:
- Physical Review. X
- Additional Journal Information:
- Journal Name: Physical Review. X Journal Volume: 7 Journal Issue: 4; Journal ID: ISSN 2160-3308
- Publisher:
- American Physical Society
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Scaffidi, Thomas, Parker, Daniel E., and Vasseur, Romain. Gapless Symmetry-Protected Topological Order. United States: N. p., 2017.
Web. doi:10.1103/PhysRevX.7.041048.
Scaffidi, Thomas, Parker, Daniel E., & Vasseur, Romain. Gapless Symmetry-Protected Topological Order. United States. https://doi.org/10.1103/PhysRevX.7.041048
Scaffidi, Thomas, Parker, Daniel E., and Vasseur, Romain. Wed .
"Gapless Symmetry-Protected Topological Order". United States. https://doi.org/10.1103/PhysRevX.7.041048.
@article{osti_1412946,
title = {Gapless Symmetry-Protected Topological Order},
author = {Scaffidi, Thomas and Parker, Daniel E. and Vasseur, Romain},
abstractNote = {We introduce exactly solvable gapless quantum systems in d dimensions that support symmetry-protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls “decorated” with dimension ( d - 1 ) SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, these edge modes are robust against arbitrary symmetry-preserving local perturbations near the edges. In two dimensions, we construct wave functions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.},
doi = {10.1103/PhysRevX.7.041048},
journal = {Physical Review. X},
number = 4,
volume = 7,
place = {United States},
year = {2017},
month = {11}
}
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Figures / Tables:
FIG. 1: Representative states of each order in the 2d example. (a) Trivial: Paramagnetic spins on a triangular lattice with fluctuating domain walls. (b) SPT: Decorating the domain walls, giving a SPTwith a c = 1 edge mode. (c) Gapless trivial: Tuning the domain walls to criticality by restricting themmore »
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Works referenced in this record:
Topological States in a One-Dimensional Fermi Gas with Attractive Interaction
journal, March 2015
- Ruhman, Jonathan; Berg, Erez; Altman, Ehud
- Physical Review Letters, Vol. 114, Issue 10
The phase transitions between Z n × Z n bosonic topological phases in 1 + 1D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases
journal, June 2017
- Tsui, Lokman; Huang, Yen-Ta; Jiang, Hong-Chen
- Nuclear Physics B, Vol. 919
Is the Composite Fermion a Dirac Particle?
journal, September 2015
- Son, Dam Thanh
- Physical Review X, Vol. 5, Issue 3
Antiferromagnetism. The Triangular Ising Net
journal, July 1950
- Wannier, G. H.
- Physical Review, Vol. 79, Issue 2
Many-body systems with non-abelian statistics
journal, May 1992
- Blok, B.; Wen, X. G.
- Nuclear Physics B, Vol. 374, Issue 3
Quantum orders and symmetric spin liquids
journal, April 2002
- Wen, Xiao-Gang
- Physical Review B, Vol. 65, Issue 16
Number conserving theory for topologically protected degeneracy in one-dimensional fermions
journal, October 2011
- Sau, Jay D.; Halperin, B. I.; Flensberg, K.
- Physical Review B, Vol. 84, Issue 14
Classification of symmetry enriched topological phases with exactly solvable models
journal, April 2013
- Mesaros, Andrej; Ran, Ying
- Physical Review B, Vol. 87, Issue 15
Conformal invariance and universality in finite-size scaling
journal, May 1984
- Cardy, J. L.
- Journal of Physics A: Mathematical and General, Vol. 17, Issue 7
Bipartite Rokhsar–Kivelson points and Cantor deconfinement
journal, June 2004
- Fradkin, Eduardo; Huse, David A.; Moessner, R.
- Physical Review B, Vol. 69, Issue 22
Quantum criticality and deconfinement in phase transitions between valence bond solids
journal, June 2004
- Vishwanath, Ashvin; Balents, L.; Senthil, T.
- Physical Review B, Vol. 69, Issue 22
Correlated topological insulators and the fractional magnetoelectric effect
journal, May 2011
- Swingle, B.; Barkeshli, M.; McGreevy, J.
- Physical Review B, Vol. 83, Issue 19
Symmetry protected topological orders and the group cohomology of their symmetry group
journal, April 2013
- Chen, Xie; Gu, Zheng-Cheng; Liu, Zheng-Xin
- Physical Review B, Vol. 87, Issue 15
Topological phases with generalized global symmetries
journal, April 2016
- Yoshida, Beni
- Physical Review B, Vol. 93, Issue 15
Experimental observation of Weyl points
journal, July 2015
- Lu, L.; Wang, Z.; Ye, D.
- Science, Vol. 349, Issue 6248
Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states
journal, December 2012
- Dubail, J.; Read, N.; Rezayi, E. H.
- Physical Review B, Vol. 86, Issue 24
Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases
journal, September 2016
- Tarantino, Nicolas; Fidkowski, Lukasz
- Physical Review B, Vol. 94, Issue 11
Bulk-edge correspondence in entanglement spectra
journal, November 2011
- Chandran, Anushya; Hermanns, M.; Regnault, N.
- Physical Review B, Vol. 84, Issue 20
Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with symmetry
journal, May 2014
- Gu, Zheng-Cheng; Levin, Michael
- Physical Review B, Vol. 89, Issue 20
Density matrix formulation for quantum renormalization groups
journal, November 1992
- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19
Random antiferromagnetic quantum spin chains
journal, August 1994
- Fisher, Daniel S.
- Physical Review B, Vol. 50, Issue 6
O(3) Nonlinear Model and the Topological Distinction between Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions
journal, August 1988
- Haldane, F. D. M.
- Physical Review Letters, Vol. 61, Issue 8
Complete classification of one-dimensional gapped quantum phases in interacting spin systems
journal, December 2011
- Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 84, Issue 23
Time-Reversal Symmetric Quantum Spin Liquids
journal, March 2016
- Wang, Chong; Senthil, T.
- Physical Review X, Vol. 6, Issue 1
Topological phases of fermions in one dimension
journal, February 2011
- Fidkowski, Lukasz; Kitaev, Alexei
- Physical Review B, Vol. 83, Issue 7
Local Magnetization in Critical Ising Model with Boundary Magnetic Field
journal, August 1994
- Chatterjee, R.; Zamolodchikov, A.
- Modern Physics Letters A, Vol. 09, Issue 24
Topological phases of one-dimensional fermions: An entanglement point of view
journal, February 2011
- Turner, Ari M.; Pollmann, Frank; Berg, Erez
- Physical Review B, Vol. 83, Issue 7
Nonabelions in the fractional quantum hall effect
journal, August 1991
- Moore, Gregory; Read, Nicholas
- Nuclear Physics B, Vol. 360, Issue 2-3
Boundary conditions, fusion rules and the Verlinde formula
journal, October 1989
- Cardy, John L.
- Nuclear Physics B, Vol. 324, Issue 3
Topological paramagnetism in frustrated spin-1 Mott insulators
journal, May 2015
- Wang, Chong; Nahum, Adam; Senthil, T.
- Physical Review B, Vol. 91, Issue 19
Colloquium : Area laws for the entanglement entropy
journal, February 2010
- Eisert, J.; Cramer, M.; Plenio, M. B.
- Reviews of Modern Physics, Vol. 82, Issue 1
Majorana zero modes in one-dimensional quantum wires without long-ranged superconducting order
journal, November 2011
- Fidkowski, Lukasz; Lutchyn, Roman M.; Nayak, Chetan
- Physical Review B, Vol. 84, Issue 19
Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases
journal, April 2013
- Hung, Ling-Yan; Wen, Xiao-Gang
- Physical Review B, Vol. 87, Issue 16
Isotropic to anisotropic transition in a fractional quantum Hall state
journal, August 2010
- Mulligan, Michael; Nayak, Chetan; Kachru, Shamit
- Physical Review B, Vol. 82, Issue 8
Symmetry-protected topological phases from decorated domain walls
journal, March 2014
- Chen, Xie; Lu, Yuan-Ming; Vishwanath, Ashvin
- Nature Communications, Vol. 5, Issue 1
Dynamical stability of the quantum Lifshitz theory in 2+1 dimensions
journal, February 2013
- Hsu, Benjamin; Fradkin, Eduardo
- Physical Review B, Vol. 87, Issue 8
Criticality in self-dual sine-Gordon models
journal, September 2002
- Lecheminant, P.; Gogolin, Alexander O.; Nersesyan, Alexander A.
- Nuclear Physics B, Vol. 639, Issue 3
Quantum Spin Hall Effect in Graphene
journal, November 2005
- Kane, C. L.; Mele, E. J.
- Physical Review Letters, Vol. 95, Issue 22
Topology and criticality in the resonating Affleck-Kennedy-Lieb-Tasaki loop spin liquid states
journal, May 2014
- Li, Wei; Yang, Shuo; Cheng, Meng
- Physical Review B, Vol. 89, Issue 17
Phase diagrams of lattice gauge theories with Higgs fields
journal, June 1979
- Fradkin, Eduardo; Shenker, Stephen H.
- Physical Review D, Vol. 19, Issue 12
Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum
journal, July 2017
- Cho, Gil Young; Shiozaki, Ken; Ryu, Shinsei
- Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 30
Construction of a paired wave function for spinless electrons at filling fraction
journal, February 2007
- Simon, Steven H.; Rezayi, E. H.; Cooper, N. R.
- Physical Review B, Vol. 75, Issue 7
Topological Insulators in Three Dimensions
journal, March 2007
- Fu, Liang; Kane, C. L.; Mele, E. J.
- Physical Review Letters, Vol. 98, Issue 10, Article No. 106803
Superconductivity and the Quantum Hard-Core Dimer Gas
journal, November 1988
- Rokhsar, Daniel S.; Kivelson, Steven A.
- Physical Review Letters, Vol. 61, Issue 20
Random transverse field Ising spin chains
journal, July 1992
- Fisher, Daniel S.
- Physical Review Letters, Vol. 69, Issue 3
Experimental evidence for the Haldane gap in a spin-1 nearly isotropic, antiferromagnetic chain
journal, January 1986
- Buyers, W. J. L.; Morra, R. M.; Armstrong, R. L.
- Physical Review Letters, Vol. 56, Issue 4
Symmetry Protection of Critical Phases and a Global Anomaly in Dimensions
journal, January 2017
- Furuya, Shunsuke C.; Oshikawa, Masaki
- Physical Review Letters, Vol. 118, Issue 2
Stability of zero modes in parafermion chains
journal, October 2014
- Jermyn, Adam S.; Mong, Roger S. K.; Alicea, Jason
- Physical Review B, Vol. 90, Issue 16
Topological invariants of time-reversal-invariant band structures
journal, March 2007
- Moore, J. E.; Balents, L.
- Physical Review B, Vol. 75, Issue 12, Article No. 121306(R)
Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications to spin liquids
journal, April 2016
- Lu, Yuan-Ming; Vishwanath, Ashvin
- Physical Review B, Vol. 93, Issue 15
Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality
journal, March 2014
- Pekker, David; Refael, Gil; Altman, Ehud
- Physical Review X, Vol. 4, Issue 1
Gapless symmetry-protected topological phase of fermions in one dimension
journal, June 2015
- Keselman, Anna; Berg, Erez
- Physical Review B, Vol. 91, Issue 23
Shannon and entanglement entropies of one- and two-dimensional critical wave functions
journal, November 2009
- Stéphan, Jean-Marie; Furukawa, Shunsuke; Misguich, Grégoire
- Physical Review B, Vol. 80, Issue 18
Classification and description of bosonic symmetry protected topological phases with semiclassical nonlinear sigma models
journal, April 2015
- Bi, Zhen; Rasmussen, Alex; Slagle, Kevin
- Physical Review B, Vol. 91, Issue 13
Symmetry protection of topological phases in one-dimensional quantum spin systems
journal, February 2012
- Pollmann, Frank; Berg, Erez; Turner, Ari M.
- Physical Review B, Vol. 85, Issue 7
Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates
journal, May 2011
- Wan, Xiangang; Turner, Ari M.; Vishwanath, Ashvin
- Physical Review B, Vol. 83, Issue 20
Fully packed loop model on the honeycomb lattice
journal, February 1994
- Blöte, H. W. J.; Nienhuis, B.
- Physical Review Letters, Vol. 72, Issue 9
Gapless topological superconductors: Model Hamiltonian and realization
journal, July 2015
- Baum, Yuval; Posske, Thore; Fulga, Ion Cosma
- Physical Review B, Vol. 92, Issue 4
Parafermionic edge zero modes in Z n -invariant spin chains
journal, November 2012
- Fendley, Paul
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2012, Issue 11
Localization-protected quantum order
journal, July 2013
- Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim
- Physical Review B, Vol. 88, Issue 1
Quantum phase transitions between a class of symmetry protected topological states
journal, July 2015
- Tsui, Lokman; Jiang, Hong-Chen; Lu, Yuan-Ming
- Nuclear Physics B, Vol. 896
Critical behaviour of the fully packed loop model on the square lattice
journal, August 1996
- Batchelor, M. T.; Blöte, H. W. J.; Nienhuis, B.
- Journal of Physics A: Mathematical and General, Vol. 29, Issue 16
Entanglement entropy and conformal field theory
journal, December 2009
- Calabrese, Pasquale; Cardy, John
- Journal of Physics A: Mathematical and Theoretical, Vol. 42, Issue 50
Edge magnetic field in the xxz spin- chain
journal, March 1998
- Affleck, Ian
- Journal of Physics A: Mathematical and General, Vol. 31, Issue 12
String order in dipole-blockaded quantum liquids
journal, September 2014
- Weimer, Hendrik
- New Journal of Physics, Vol. 16, Issue 9
Mott physics and band topology in materials with strong spin–orbit interaction
journal, March 2010
- Pesin, Dmytro; Balents, Leon
- Nature Physics, Vol. 6, Issue 5
Quasi-topological phases of matter and topological protection
journal, May 2013
- Bonderson, Parsa; Nayak, Chetan
- Physical Review B, Vol. 87, Issue 19
Quantum spin liquids: a review
journal, November 2016
- Savary, Lucile; Balents, Leon
- Reports on Progress in Physics, Vol. 80, Issue 1
Quantum Spin Hall Insulator State in HgTe Quantum Wells
journal, November 2007
- Konig, M.; Wiedmann, S.; Brune, C.
- Science, Vol. 318, Issue 5851, p. 766-770
Quantum Criticality of Hot Random Spin Chains
journal, May 2015
- Vasseur, R.; Potter, A. C.; Parameswaran, S. A.
- Physical Review Letters, Vol. 114, Issue 21
Protected boundary states in gapless topological phases
journal, June 2013
- Matsuura, Shunji; Chang, Po-Yao; Schnyder, Andreas P.
- New Journal of Physics, Vol. 15, Issue 6
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. 1757-1761
Dynamical Quantum Phase Transitions in Random Spin Chains
journal, May 2014
- Vosk, Ronen; Altman, Ehud
- Physical Review Letters, Vol. 112, Issue 21
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
Hidden order and flux attachment in symmetry-protected topological phases: A Laughlin-like approach
journal, May 2015
- Ringel, Zohar; Simon, Steven H.
- Physical Review B, Vol. 91, Issue 19
Entanglement Entropy of 2D Conformal Quantum Critical Points: Hearing the Shape of a Quantum Drum
journal, August 2006
- Fradkin, Eduardo; Moore, Joel E.
- Physical Review Letters, Vol. 97, Issue 5
chiral edge modes of a critical spin liquid
journal, May 2016
- Poilblanc, Didier; Schuch, Norbert; Affleck, Ian
- Physical Review B, Vol. 93, Issue 17
Localization and topology protected quantum coherence at the edge of hot matter
journal, July 2015
- Bahri, Yasaman; Vosk, Ronen; Altman, Ehud
- Nature Communications, Vol. 6, Issue 1
Detecting Topological Order in a Ground State Wave Function
journal, March 2006
- Levin, Michael; Wen, Xiao-Gang
- Physical Review Letters, Vol. 96, Issue 11
A topological Dirac insulator in a quantum spin Hall phase
journal, April 2008
- Hsieh, D.; Qian, D.; Wray, L.
- Nature, Vol. 452, Issue 7190, p. 970-974
Topological crystalline insulator states in Pb1−xSnxSe
journal, September 2012
- Dziawa, P.; Kowalski, B. J.; Dybko, K.
- Nature Materials, Vol. 11, Issue 12
Classification of Interacting Electronic Topological Insulators in Three Dimensions
journal, February 2014
- Wang, C.; Potter, A. C.; Senthil, T.
- Science, Vol. 343, Issue 6171
Topological Entanglement Entropy
journal, March 2006
- Kitaev, Alexei; Preskill, John
- Physical Review Letters, Vol. 96, Issue 11
Classification and analysis of two-dimensional Abelian fractional topological insulators
journal, September 2012
- Levin, Michael; Stern, Ady
- Physical Review B, Vol. 86, Issue 11
Theory of the half-filled Landau level
journal, March 1993
- Halperin, B. I.; Lee, Patrick A.; Read, Nicholas
- Physical Review B, Vol. 47, Issue 12
Periodic table for topological insulators and superconductors
conference, January 2009
- Kitaev, Alexei; Lebedev, Vladimir; Feigel’man, Mikhail
- ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings
Coexisting Edge States and Gapless Bulk in Topological States of Matter
journal, March 2015
- Baum, Yuval; Posske, Thore; Fulga, Ion Cosma
- Physical Review Letters, Vol. 114, Issue 13
Topological phases and the quantum spin Hall effect in three dimensions
journal, May 2009
- Roy, Rahul
- Physical Review B, Vol. 79, Issue 19
Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach
journal, September 2012
- Lu, Yuan-Ming; Vishwanath, Ashvin
- Physical Review B, Vol. 86, Issue 12
Emergent particle-hole symmetry in the half-filled Landau level
journal, August 2016
- Mulligan, Michael; Raghu, S.; Fisher, Matthew P. A.
- Physical Review B, Vol. 94, Issue 7
Particle-hole symmetry and the composite Fermi liquid
journal, October 2015
- Barkeshli, Maissam; Mulligan, Michael; Fisher, Matthew P. A.
- Physical Review B, Vol. 92, Issue 16
Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear models and a special group supercohomology theory
journal, September 2014
- Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 90, Issue 11
Stacked topological insulator built from bismuth-based graphene sheet analogues
journal, March 2013
- Rasche, Bertold; Isaeva, Anna; Ruck, Michael
- Nature Materials, Vol. 12, Issue 5
Symmetry-Protected Topological Orders in Interacting Bosonic Systems
journal, December 2012
- Chen, X.; Gu, Z. -C.; Liu, Z. -X.
- Science, Vol. 338, Issue 6114
Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains
journal, July 1992
- Kennedy, Tom; Tasaki, Hal
- Communications in Mathematical Physics, Vol. 147, Issue 3
The density-matrix renormalization group in the age of matrix product states
journal, January 2011
- Schollwöck, Ulrich
- Annals of Physics, Vol. 326, Issue 1
Vertex Operators in the Fractional Quantum hall Effect
journal, February 1991
- Fubini, S.; LÜTken, C. A.
- Modern Physics Letters A, Vol. 06, Issue 06
Fractional Topological Insulators in Three Dimensions
journal, December 2010
- Maciejko, Joseph; Qi, Xiao-Liang; Karch, Andreas
- Physical Review Letters, Vol. 105, Issue 24
Pyrochlore photons: The spin liquid in a three-dimensional frustrated magnet
journal, February 2004
- Hermele, Michael; Fisher, Matthew P. A.; Balents, Leon
- Physical Review B, Vol. 69, Issue 6
Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order
journal, October 2009
- Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 80, Issue 15
Anomalies and entanglement renormalization
journal, September 2017
- Bridgeman, Jacob C.; Williamson, Dominic J.
- Physical Review B, Vol. 96, Issue 12
Topological order and conformal quantum critical points
journal, April 2004
- Ardonne, Eddy; Fendley, Paul; Fradkin, Eduardo
- Annals of Physics, Vol. 310, Issue 2
Wave Function and Strange Correlator of Short-Range Entangled States
journal, June 2014
- You, Yi-Zhuang; Bi, Zhen; Rasmussen, Alex
- Physical Review Letters, Vol. 112, Issue 24
Conformal invariance and surface critical behavior
journal, November 1984
- Cardy, John L.
- Nuclear Physics B, Vol. 240, Issue 4
Colloquium: Topological insulators
journal, November 2010
- Hasan, M. Z.; Kane, C. L.
- Reviews of Modern Physics, Vol. 82, Issue 4, p. 3045-3067
Operator spectrum and exact exponents of the fully packed loop model
journal, October 1996
- Kondev, Jané; Gier, Jan de; Nienhuis, Bernard
- Journal of Physics A: Mathematical and General, Vol. 29, Issue 20
Unconventional Surface Critical Behavior Induced by a Quantum Phase Transition from the Two-Dimensional Affleck-Kennedy-Lieb-Tasaki Phase to a Néel-Ordered Phase
journal, February 2017
- Zhang, Long; Wang, Fa
- Physical Review Letters, Vol. 118, Issue 8
Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain
journal, June 2016
- Fendley, Paul
- Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 30
Discovery of a Weyl fermion semimetal and topological Fermi arcs
journal, July 2015
- Xu, S. -Y.; Belopolski, I.; Alidoust, N.
- Science, Vol. 349, Issue 6248
Classification of topological insulators and superconductors in three spatial dimensions
journal, November 2008
- Schnyder, Andreas P.; Ryu, Shinsei; Furusaki, Akira
- Physical Review B, Vol. 78, Issue 19
Stripe melting and a transition between weak and strong symmetry protected topological phases
journal, May 2016
- You, Yizhi; You, Yi-Zhuang
- Physical Review B, Vol. 93, Issue 19
Symmetric spin liquids and their neighboring phases on triangular lattice
journal, April 2016
- Lu, Yuan-Ming
- Physical Review B, Vol. 93, Issue 16
Braiding statistics approach to symmetry-protected topological phases
journal, September 2012
- Levin, Michael; Gu, Zheng-Cheng
- Physical Review B, Vol. 86, Issue 11
Wave functions of symmetry-protected topological phases from conformal field theories
journal, March 2016
- Scaffidi, Thomas; Ringel, Zohar
- Physical Review B, Vol. 93, Issue 11
Statistics of Two-Dimensional Lattices with Four Components
journal, September 1943
- Ashkin, J.; Teller, E.
- Physical Review, Vol. 64, Issue 5-6
Effects of interactions on the topological classification of free fermion systems
journal, April 2010
- Fidkowski, Lukasz; Kitaev, Alexei
- Physical Review B, Vol. 81, Issue 13
Gapless fermions and quantum order
journal, December 2002
- Wen, X. G.; Zee, A.
- Physical Review B, Vol. 66, Issue 23
Ising anyons in frustration-free Majorana-dimer models
journal, September 2016
- Ware, Brayden; Son, Jun Ho; Cheng, Meng
- Physical Review B, Vol. 94, Issue 11
Works referencing / citing this record:
Emergent mode and bound states in single-component one-dimensional lattice fermionic systems
journal, November 2019
- He, Yuchi; Tian, Binbin; Pekker, David
- Physical Review B, Vol. 100, Issue 20
Full commuting projector Hamiltonians of interacting symmetry-protected topological phases of fermions
journal, October 2018
- Tantivasadakarn, Nathanan; Vishwanath, Ashvin
- Physical Review B, Vol. 98, Issue 16
Characterization of Topological States via Dual Multipartite Entanglement
journal, June 2018
- Zhang, Yu-Ran; Zeng, Yu; Fan, Heng
- Physical Review Letters, Vol. 120, Issue 25
Topologically Protected Long Edge Coherence Times in Symmetry-Broken Phases
journal, June 2019
- Parker, Daniel E.; Vasseur, Romain; Scaffidi, Thomas
- Physical Review Letters, Vol. 122, Issue 24
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