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Title: 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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1103/PhysRevX.7.041048

Citation Metrics:
Cited by: 51 works
Citation information provided by
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Figures / Tables:

FIG. 1 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 » to fully packed loop configurations (defined below), closing the gap and giving a c = 1 edge mode. (d) Doing both yields a gapless SPT with c = 1 + 1 = 2 edge modes.« less

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Works referenced in this record:

Topological States in a One-Dimensional Fermi Gas with Attractive Interaction
journal, March 2015


Is the Composite Fermion a Dirac Particle?
journal, September 2015


Antiferromagnetism. The Triangular Ising Net
journal, July 1950


Many-body systems with non-abelian statistics
journal, May 1992


Quantum orders and symmetric spin liquids
journal, April 2002


Number conserving theory for topologically protected degeneracy in one-dimensional fermions
journal, October 2011


Classification of symmetry enriched topological phases with exactly solvable models
journal, April 2013


Conformal invariance and universality in finite-size scaling
journal, May 1984


Bipartite Rokhsar–Kivelson points and Cantor deconfinement
journal, June 2004


Quantum criticality and deconfinement in phase transitions between valence bond solids
journal, June 2004


Correlated topological insulators and the fractional magnetoelectric effect
journal, May 2011


Symmetry protected topological orders and the group cohomology of their symmetry group
journal, April 2013


Topological phases with generalized global symmetries
journal, April 2016


Experimental observation of Weyl points
journal, July 2015


Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states
journal, December 2012


Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases
journal, September 2016


Bulk-edge correspondence in entanglement spectra
journal, November 2011


Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with Z 2 symmetry
journal, May 2014


Density matrix formulation for quantum renormalization groups
journal, November 1992


Random antiferromagnetic quantum spin chains
journal, August 1994


Complete classification of one-dimensional gapped quantum phases in interacting spin systems
journal, December 2011


Time-Reversal Symmetric U ( 1 ) Quantum Spin Liquids
journal, March 2016


Topological phases of fermions in one dimension
journal, February 2011


Local Magnetization in Critical Ising Model with Boundary Magnetic Field
journal, August 1994


Topological phases of one-dimensional fermions: An entanglement point of view
journal, February 2011


Nonabelions in the fractional quantum hall effect
journal, August 1991


Boundary conditions, fusion rules and the Verlinde formula
journal, October 1989


Topological paramagnetism in frustrated spin-1 Mott insulators
journal, May 2015


Colloquium : Area laws for the entanglement entropy
journal, February 2010


Majorana zero modes in one-dimensional quantum wires without long-ranged superconducting order
journal, November 2011


Isotropic to anisotropic transition in a fractional quantum Hall state
journal, August 2010


Symmetry-protected topological phases from decorated domain walls
journal, March 2014

  • Chen, Xie; Lu, Yuan-Ming; Vishwanath, Ashvin
  • Nature Communications, Vol. 5, Issue 1
  • DOI: 10.1038/ncomms4507

Dynamical stability of the quantum Lifshitz theory in 2+1 dimensions
journal, February 2013


Criticality in self-dual sine-Gordon models
journal, September 2002


Quantum Spin Hall Effect in Graphene
journal, November 2005


Topology and criticality in the resonating Affleck-Kennedy-Lieb-Tasaki loop spin liquid states
journal, May 2014


Phase diagrams of lattice gauge theories with Higgs fields
journal, June 1979


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
  • DOI: 10.1088/1751-8121/aa7782

Construction of a paired wave function for spinless electrons at filling fraction ν = 2 5
journal, February 2007


Topological Insulators in Three Dimensions
journal, March 2007


Superconductivity and the Quantum Hard-Core Dimer Gas
journal, November 1988


Random transverse field Ising spin chains
journal, July 1992


Experimental evidence for the Haldane gap in a spin-1 nearly isotropic, antiferromagnetic chain
journal, January 1986


Symmetry Protection of Critical Phases and a Global Anomaly in 1 + 1 Dimensions
journal, January 2017


Stability of zero modes in parafermion chains
journal, October 2014


Topological invariants of time-reversal-invariant band structures
journal, March 2007


Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality
journal, March 2014


Gapless symmetry-protected topological phase of fermions in one dimension
journal, June 2015


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
  • DOI: 10.1103/PhysRevB.80.184421

Classification and description of bosonic symmetry protected topological phases with semiclassical nonlinear sigma models
journal, April 2015


Symmetry protection of topological phases in one-dimensional quantum spin systems
journal, February 2012


Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates
journal, May 2011


Fully packed loop model on the honeycomb lattice
journal, February 1994


Gapless topological superconductors: Model Hamiltonian and realization
journal, July 2015


Parafermionic edge zero modes in Z n -invariant spin chains
journal, November 2012


Localization-protected quantum order
journal, July 2013


Quantum phase transitions between a class of symmetry protected topological states
journal, July 2015


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
  • DOI: 10.1088/0305-4470/29/16/001

Entanglement entropy and conformal field theory
journal, December 2009


Edge magnetic field in the xxz spin- chain
journal, March 1998


String order in dipole-blockaded quantum liquids
journal, September 2014


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
  • DOI: 10.1038/nphys1606

Quasi-topological phases of matter and topological protection
journal, May 2013


Quantum spin liquids: a review
journal, November 2016


Quantum Spin Hall Insulator State in HgTe Quantum Wells
journal, November 2007


Quantum Criticality of Hot Random Spin Chains
journal, May 2015


Protected boundary states in gapless topological phases
journal, June 2013


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
  • DOI: 10.1126/science.1133734

Dynamical Quantum Phase Transitions in Random Spin Chains
journal, May 2014


Hidden order and flux attachment in symmetry-protected topological phases: A Laughlin-like approach
journal, May 2015


Entanglement Entropy of 2D Conformal Quantum Critical Points: Hearing the Shape of a Quantum Drum
journal, August 2006


SU ( 2 ) 1 chiral edge modes of a critical spin liquid
journal, May 2016


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
  • DOI: 10.1038/ncomms8341

Detecting Topological Order in a Ground State Wave Function
journal, March 2006


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
  • DOI: 10.1038/nature06843

Topological crystalline insulator states in Pb1−xSnxSe
journal, September 2012

  • Dziawa, P.; Kowalski, B. J.; Dybko, K.
  • Nature Materials, Vol. 11, Issue 12
  • DOI: 10.1038/nmat3449

Classification of Interacting Electronic Topological Insulators in Three Dimensions
journal, February 2014


Topological Entanglement Entropy
journal, March 2006


Classification and analysis of two-dimensional Abelian fractional topological insulators
journal, September 2012


Theory of the half-filled Landau level
journal, March 1993


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
  • DOI: 10.1063/1.3149495

Coexisting Edge States and Gapless Bulk in Topological States of Matter
journal, March 2015


Topological phases and the quantum spin Hall effect in three dimensions
journal, May 2009


Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach
journal, September 2012


Emergent particle-hole symmetry in the half-filled Landau level
journal, August 2016


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
  • DOI: 10.1103/PhysRevB.92.165125

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
  • DOI: 10.1038/nmat3570

Symmetry-Protected Topological Orders in Interacting Bosonic Systems
journal, December 2012


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
  • DOI: 10.1007/BF02097239

The density-matrix renormalization group in the age of matrix product states
journal, January 2011


Vertex Operators in the Fractional Quantum hall Effect
journal, February 1991


Fractional Topological Insulators in Three Dimensions
journal, December 2010


Pyrochlore photons: The U ( 1 ) spin liquid in a S = 1 2 three-dimensional frustrated magnet
journal, February 2004


Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order
journal, October 2009


Anomalies and entanglement renormalization
journal, September 2017


Topological order and conformal quantum critical points
journal, April 2004


Wave Function and Strange Correlator of Short-Range Entangled States
journal, June 2014


Conformal invariance and surface critical behavior
journal, November 1984


Colloquium: Topological insulators
journal, November 2010


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
  • DOI: 10.1088/0305-4470/29/20/007

Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain
journal, June 2016


Discovery of a Weyl fermion semimetal and topological Fermi arcs
journal, July 2015


Classification of topological insulators and superconductors in three spatial dimensions
journal, November 2008


Stripe melting and a transition between weak and strong symmetry protected topological phases
journal, May 2016


Symmetric Z 2 spin liquids and their neighboring phases on triangular lattice
journal, April 2016


Braiding statistics approach to symmetry-protected topological phases
journal, September 2012


Wave functions of symmetry-protected topological phases from conformal field theories
journal, March 2016


Statistics of Two-Dimensional Lattices with Four Components
journal, September 1943


Effects of interactions on the topological classification of free fermion systems
journal, April 2010


Gapless fermions and quantum order
journal, December 2002


Ising anyons in frustration-free Majorana-dimer models
journal, September 2016


Works referencing / citing this record:

Emergent mode and bound states in single-component one-dimensional lattice fermionic systems
journal, November 2019


Full commuting projector Hamiltonians of interacting symmetry-protected topological phases of fermions
journal, October 2018


Characterization of Topological States via Dual Multipartite Entanglement
journal, June 2018


Topologically Protected Long Edge Coherence Times in Symmetry-Broken Phases
journal, June 2019


Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.