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Title: Gapless Symmetry-Protected Topological Order

Journal Article · · Physical Review. X
 [1];  [1];  [2]
  1. Univ. of California, Berkeley, CA (United States)
  2. Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of Massachusetts, Amherst, MA (United States)

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.

Research Organization:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Organization:
USDOE Office of Science (SC); USDOE
Grant/Contract Number:
AC02-05CH11231
OSTI ID:
1412946
Alternate ID(s):
OSTI ID: 1485068
Journal Information:
Physical Review. X, Vol. 7, Issue 4; ISSN 2160-3308
Publisher:
American Physical SocietyCopyright Statement
Country of Publication:
United States
Language:
English

References (161)

Topological States in a One-Dimensional Fermi Gas with Attractive Interaction journal March 2015
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
Is the Composite Fermion a Dirac Particle? journal September 2015
Field Theories of Condensed Matter Physics book January 2013
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
Entanglement in many-body systems journal May 2008
Experimental Discovery of Weyl Semimetal TaAs journal July 2015
Density matrix formulation for quantum renormalization groups journal November 1992
Random antiferromagnetic quantum spin chains journal August 1994
O(3) Nonlinear σ Model and the Topological Distinction between Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions journal August 1988
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
Area laws in a many-body localized state and its implications for topological order journal September 2013
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
Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases journal April 2013
Isotropic to anisotropic transition in a fractional quantum Hall state journal August 2010
Symmetry-protected topological phases from decorated domain walls journal March 2014
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
Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations journal December 2011
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
Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications to Z 2 spin liquids journal April 2016
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
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
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
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
Dynamical Quantum Phase Transitions in Random Spin Chains journal May 2014
Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States journal July 2008
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
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
Topological crystalline insulator states in Pb1−xSnxSe journal September 2012
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
  • Kitaev, Alexei; Lebedev, Vladimir; Feigel’man, Mikhail
  • ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings https://doi.org/10.1063/1.3149495
conference January 2009
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
Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory journal September 2014
Stacked topological insulator built from bismuth-based graphene sheet analogues journal March 2013
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
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
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
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
Non-Abelian anyons and topological quantum computation journal September 2008
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
Quantum Physics in One Dimension book December 2003
Antiferromagnetism. The Triangular Ising Net journal June 1973
String order in dipole-blockaded quantum liquids other January 2014
Non-Abelian Anyons and Topological Quantum Computation text January 2007
Classification of topological insulators and superconductors in three spatial dimensions text January 2008
Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order text January 2009
Shannon and entanglement entropies of one- and two-dimensional critical wave functions text January 2009
Symmetry protection of topological order in one-dimensional quantum spin systems text January 2009
An Isotropic to Anisotropic Transition in a Fractional Quantum Hall State text January 2010
Correlated Topological Insulators and the Fractional Magnetoelectric Effect text January 2010
The density-matrix renormalization group in the age of matrix product states text January 2010
A number conserving theory for topologically protected degeneracy in one-dimensional fermions text January 2011
Symmetry protected topological orders and the group cohomology of their symmetry group text January 2011
Dynamical stability of the quantum Lifshitz theory in 2+1 Dimensions text January 2012
Parafermionic edge zero modes in Z_n-invariant spin chains text January 2012
Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases text January 2012
Symmetry protected topological phases from decorated domain walls text January 2013
Area laws in a many-body localized state and its implications for topological order text January 2013
The Hilbert-glass transition: new universality of temperature-tuned many-body dynamical quantum criticality text January 2013
Localization and topology protected quantum coherence at the edge of 'hot' matter preprint January 2013
Wave Function and Strange Correlator of Short Range Entangled states text January 2013
Stability of zero modes in parafermion chains text January 2014
Hidden order and flux attachment in symmetry protected topological phases: a Laughlin-like approach text January 2014
Quantum criticality of hot random spin chains text January 2014
Is the Composite Fermion a Dirac Particle? text January 2015
Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs text January 2015
Experimental discovery of Weyl semimetal TaAs text January 2015
Quantum Phase Transitions Between a Class of Symmetry Protected Topological States text January 2015
Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories text January 2015
The phase transitions between $Z_n\times Z_n$ bosonic topological phases in 1+1 D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases text January 2017
Criticality in self-dual sine-Gordon models text January 2002
Topological Order and Conformal Quantum Critical Points text January 2003
Topological invariants of time-reversal-invariant band structures text January 2006
Topological Insulators in Three Dimensions text January 2006
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells text January 2006
Entanglement in Many-Body Systems text January 2007
Magnetization and dimerization profiles of open spin ladders journal May 2003

Cited By (6)

Emergent mode and bound states in single-component one-dimensional lattice fermionic systems journal November 2019
Characterization of Topological States via Dual Multipartite Entanglement journal June 2018
Topologically Protected Long Edge Coherence Times in Symmetry-Broken Phases journal June 2019
Mapping topological to conformal field theories through strange correlators text January 2018
Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions text January 2018
Emergent mode and bound states in single-component one-dimensional lattice fermionic systems text January 2018

Figures / Tables (8)


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