Fermionic symmetry protected topological phases and cobordisms

Kapustin, Anton; Thorngren, Ryan; Turzillo, Alex; Wang, Zitao

doi:10.1007/JHEP12(2015)052

Title: Fermionic symmetry protected topological phases and cobordisms

Abstract

It has been proposed recently that interacting Symmetry Protected Topological Phases can be classified using cobordism theory. We test this proposal in the case of Fermionic SPT phases with $ {\mathrm{\mathbb{Z}}}_2 $ symmetry, where $ {\mathrm{\mathbb{Z}}}_2 $ is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known Fermionic SPT phases in space dimension D ≤ 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases

Authors:

Kapustin, Anton ^[1]; Thorngren, Ryan ^[2]; Turzillo, Alex ^[1]; Wang, Zitao ^[1]

California Inst. of Technology (CalTech), Pasadena, CA (United States)
Univ. of California, Berkeley, CA (United States)

Publication Date:: 2015-12-10

Research Org.:: California Institute of Technology (CalTech), Pasadena, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP)

OSTI Identifier:: 1599360

Grant/Contract Number:: SC0011632; FG02-92ER40701

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 12; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Topological Field Theories; Effective field theories; Topological States of Matter

Citation Formats


                    Kapustin, Anton, Thorngren, Ryan, Turzillo, Alex, and Wang, Zitao. Fermionic symmetry protected topological phases and cobordisms.  United States: N. p., 2015. 
Web.  doi:10.1007/JHEP12(2015)052.

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                    Kapustin, Anton, Thorngren, Ryan, Turzillo, Alex, & Wang, Zitao. Fermionic symmetry protected topological phases and cobordisms.  United States.  https://doi.org/10.1007/JHEP12(2015)052

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                    Kapustin, Anton, Thorngren, Ryan, Turzillo, Alex, and Wang, Zitao. Thu .  
"Fermionic symmetry protected topological phases and cobordisms".  United States.  https://doi.org/10.1007/JHEP12(2015)052.  https://www.osti.gov/servlets/purl/1599360.

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@article{osti_1599360,

  title        = {Fermionic symmetry protected topological phases and cobordisms},

  author       = {Kapustin, Anton and Thorngren, Ryan and Turzillo, Alex and Wang, Zitao},

  abstractNote = {It has been proposed recently that interacting Symmetry Protected Topological Phases can be classified using cobordism theory. We test this proposal in the case of Fermionic SPT phases with \( {\mathrm{\mathbb{Z}}}_2 \) symmetry, where \( {\mathrm{\mathbb{Z}}}_2 \) is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known Fermionic SPT phases in space dimension D ≤ 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases},

  doi          = {10.1007/JHEP12(2015)052},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 12,

  volume       = 2015,

  place        = {United States},

  year         = {2015},

  month        = {12}

}

Copy to clipboard

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Title: Fermionic symmetry protected topological phases and cobordisms

Abstract

Citation Formats

Pin cobordism and related topics journal, December 1969

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

Exotic structures on 4-manifolds detected by spectral invariants journal, February 1988

Periodic table for topological insulators and superconductors conference, January 2009

Topological phases of fermions in one dimension journal, February 2011

Topological insulators and superconductors: tenfold way and dimensional hierarchy journal, June 2010

Dimers on Surface Graphs and Spin Structures. I journal, July 2007

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

Generalizations of the Kervaire Invariant journal, March 1972

The Structure of the Spin Cobordism Ring journal, September 1967

A calculation ofPin + bordism groups journal, December 1990

Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect journal, April 2000

Interacting fermionic topological insulators/superconductors in three dimensions journal, May 2014

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model journal, November 2013

Pin Cobordism and Related Topics. text, January 1969

Periodic table for topological insulators and superconductors text, January 2009

Symmetry protected topological orders and the group cohomology of their symmetry group text, January 2011

Symmetry protected topological phases from decorated domain walls text, January 2013

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model text, January 2013

Interacting fermionic topological insulators/superconductors in three dimensions text, January 2014

Dimers on surface graphs and spin structures. I text, January 2006

Minimalist approach to the classification of symmetry protected topological phases journal, October 2018

Fermionic projected entangled-pair states and topological phases journal, December 2017

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries journal, July 2018

Gauge interactions and topological phases of matter journal, September 2016

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

Dimensional Hierarchy of Fermionic Interacting Topological Phases journal, November 2016

Surface field theories of point group symmetry protected topological phases journal, February 2018

Coupled wire model of symmetric Majorana surfaces of topological superconductors journal, October 2016

Fermion decoration construction of symmetry-protected trivial order for fermion systems with any symmetry and in any dimension journal, December 2019

Symmetry-protected topological phase transition in one-dimensional Kondo lattice and its realization with ultracold atoms journal, October 2017

Ground-state degeneracy of non-Abelian topological phases from coupled wires journal, June 2019

Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories journal, February 2019

Classification and surface anomaly of glide symmetry protected topological phases in three dimensions journal, July 2017

A web of 2d dualities: ${\bf Z}_2$ gauge fields and Arf invariants journal, January 2019

Anomalous Symmetry Protected Topological States in Interacting Fermion Systems journal, November 2019

Choreographed entanglement dances: Topological states of quantum matter journal, February 2019

Translation constraints on quantum phases with twisted boundary conditions journal, November 2019

Partial time-reversal transformation and entanglement negativity in fermionic systems journal, April 2017

Lattice models that realize Z n -1 symmetry-protected topological states for even n journal, January 2020

Anomaly indicators for topological orders with U ( 1 ) and time-reversal symmetry journal, October 2019

Ground-state degeneracy of non-Abelian topological phases from coupled wires text, January 2019

Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions text, January 2015

Classification of topological quantum matter with symmetries text, January 2015

Computational Power of Symmetry-Protected Topological Phases text, January 2016

Classification and surface anomaly of glide symmetry protected topological phases in three dimensions text, January 2017

Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases text, January 2017

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions text, January 2017

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions text, January 2017

Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases text, January 2017

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries text, January 2017

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups text, January 2017

Why are fractional charges of orientifolds compatible with Dirac quantization? text, January 2018

Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions text, January 2018

Remarks on the Green-Schwarz terms of six-dimensional supergravity theories text, January 2018

Free and Interacting Short-Range Entangled Phases of Fermions: Beyond the Ten-Fold Way text, January 2018

Crystalline topological phases as defect networks text, January 2018

Choreographed entangle dances: topological states of quantum matter text, January 2019

Translation constraints on quantum phases with twisted boundary conditions text, January 2019

Pin cobordism and related topics
journal, December 1969

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

Exotic structures on 4-manifolds detected by spectral invariants
journal, February 1988

Periodic table for topological insulators and superconductors
conference, January 2009

Topological phases of fermions in one dimension
journal, February 2011

Topological insulators and superconductors: tenfold way and dimensional hierarchy
journal, June 2010

Dimers on Surface Graphs and Spin Structures. I
journal, July 2007

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

Generalizations of the Kervaire Invariant
journal, March 1972

The Structure of the Spin Cobordism Ring
journal, September 1967

A calculation ofPin + bordism groups
journal, December 1990

Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect
journal, April 2000

Interacting fermionic topological insulators/superconductors in three dimensions
journal, May 2014

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model
journal, November 2013

Pin Cobordism and Related Topics.
text, January 1969

Periodic table for topological insulators and superconductors
text, January 2009

Symmetry protected topological orders and the group cohomology of their symmetry group
text, January 2011

Symmetry protected topological phases from decorated domain walls
text, January 2013

Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model
text, January 2013

Interacting fermionic topological insulators/superconductors in three dimensions
text, January 2014

Dimers on surface graphs and spin structures. I
text, January 2006

Minimalist approach to the classification of symmetry protected topological phases
journal, October 2018

Fermionic projected entangled-pair states and topological phases
journal, December 2017

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
journal, July 2018

Gauge interactions and topological phases of matter
journal, September 2016

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

Dimensional Hierarchy of Fermionic Interacting Topological Phases
journal, November 2016

Surface field theories of point group symmetry protected topological phases
journal, February 2018

Coupled wire model of symmetric Majorana surfaces of topological superconductors
journal, October 2016

Fermion decoration construction of symmetry-protected trivial order for fermion systems with any symmetry and in any dimension
journal, December 2019

Symmetry-protected topological phase transition in one-dimensional Kondo lattice and its realization with ultracold atoms
journal, October 2017

Ground-state degeneracy of non-Abelian topological phases from coupled wires
journal, June 2019

Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories
journal, February 2019

Classification and surface anomaly of glide symmetry protected topological phases in three dimensions
journal, July 2017

A web of 2d dualities: ${\bf Z}_2$ gauge fields and Arf invariants
journal, January 2019

Anomalous Symmetry Protected Topological States in Interacting Fermion Systems
journal, November 2019

Choreographed entanglement dances: Topological states of quantum matter
journal, February 2019

Translation constraints on quantum phases with twisted boundary conditions
journal, November 2019

Partial time-reversal transformation and entanglement negativity in fermionic systems
journal, April 2017

Lattice models that realize $Z_{n}$ -1 symmetry-protected topological states for even $n$
journal, January 2020

Anomaly indicators for topological orders with $U (1)$ and time-reversal symmetry
journal, October 2019

Ground-state degeneracy of non-Abelian topological phases from coupled wires
text, January 2019

Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions
text, January 2015

Classification of topological quantum matter with symmetries
text, January 2015

Computational Power of Symmetry-Protected Topological Phases
text, January 2016

Classification and surface anomaly of glide symmetry protected topological phases in three dimensions
text, January 2017

Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases
text, January 2017

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
text, January 2017

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions
text, January 2017

Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
text, January 2017

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
text, January 2017

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
text, January 2017

Why are fractional charges of orientifolds compatible with Dirac quantization?
text, January 2018

Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
text, January 2018

Remarks on the Green-Schwarz terms of six-dimensional supergravity theories
text, January 2018

Free and Interacting Short-Range Entangled Phases of Fermions: Beyond the Ten-Fold Way
text, January 2018

Crystalline topological phases as defect networks
text, January 2018

Choreographed entangle dances: topological states of quantum matter
text, January 2019

Translation constraints on quantum phases with twisted boundary conditions
text, January 2019