Free and interacting short-range entangled phases of fermions: Beyond the tenfold way

Chen, Yu-An; Kapustin, Anton; Turzillo, Alex; You, Minyoung

doi:10.1103/PhysRevB.100.195128

Title: Free and interacting short-range entangled phases of fermions: Beyond the tenfold way

Abstract

Here, we extend the periodic table of phases of free fermions in the tenfold way symmetry classes to a classification of free fermionic phases protected by an arbitrary on-site unitary symmetry $$\hat{G}$$ in an arbitrary dimension. The classification is described as a function of the real representation theory of $$\hat{G}$$ and the data of the original periodic table. We also systematically study in low dimensions the relationship between the free invariants and the invariants of short-range entangled interacting phases of fermions. Namely we determine whether a given symmetry protected phase of free fermions is destabilized by sufficiently strong interactions or it remains stable even in the presence of interactions. Finally, we also determine which interacting fermionic phases cannot be realized by free fermions. Examples of both destabilized free phases and intrinsically interacting phases are common in all dimensions.

Authors:

^[1]; Kapustin, Anton ^[1];

^[1]; You, Minyoung ^[1]

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

Publication Date:: Mon Nov 18 00:00:00 EST 2019

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

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

OSTI Identifier:: 1597474

Grant/Contract Number:: SC0011632; PHY-1607611

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review B

Additional Journal Information:: Journal Volume: 100; Journal Issue: 19; Journal ID: ISSN 2469-9950

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Symmetry protected topological states; Band structure methods; Symmetries in condensed matter

Citation Formats


                    Chen, Yu-An, Kapustin, Anton, Turzillo, Alex, and You, Minyoung. Free and interacting short-range entangled phases of fermions: Beyond the tenfold way.  United States: N. p., 2019. 
Web.  doi:10.1103/PhysRevB.100.195128.

Copy to clipboard


                    Chen, Yu-An, Kapustin, Anton, Turzillo, Alex, & You, Minyoung. Free and interacting short-range entangled phases of fermions: Beyond the tenfold way.  United States.  https://doi.org/10.1103/PhysRevB.100.195128

Copy to clipboard


                    Chen, Yu-An, Kapustin, Anton, Turzillo, Alex, and You, Minyoung. Mon .  
"Free and interacting short-range entangled phases of fermions: Beyond the tenfold way".  United States.  https://doi.org/10.1103/PhysRevB.100.195128.  https://www.osti.gov/servlets/purl/1597474.

Copy to clipboard


                    
@article{osti_1597474,

  title        = {Free and interacting short-range entangled phases of fermions: Beyond the tenfold way},

  author       = {Chen, Yu-An and Kapustin, Anton and Turzillo, Alex and You, Minyoung},

  abstractNote = {Here, we extend the periodic table of phases of free fermions in the tenfold way symmetry classes to a classification of free fermionic phases protected by an arbitrary on-site unitary symmetry $\hat{G}$ in an arbitrary dimension. The classification is described as a function of the real representation theory of $\hat{G}$ and the data of the original periodic table. We also systematically study in low dimensions the relationship between the free invariants and the invariants of short-range entangled interacting phases of fermions. Namely we determine whether a given symmetry protected phase of free fermions is destabilized by sufficiently strong interactions or it remains stable even in the presence of interactions. Finally, we also determine which interacting fermionic phases cannot be realized by free fermions. Examples of both destabilized free phases and intrinsically interacting phases are common in all dimensions.},

  doi          = {10.1103/PhysRevB.100.195128},

  journal      = {Physical Review B},

  number       = 19,

  volume       = 100,

  place        = {United States},

  year         = {Mon Nov 18 00:00:00 EST 2019},

  month        = {Mon Nov 18 00:00:00 EST 2019}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1103/PhysRevB.100.195128

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 9 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Twisted Equivariant Matter
journal, March 2013

Freed, Daniel S.; Moore, Gregory W.
Annales Henri Poincaré, Vol. 14, Issue 8
DOI: 10.1007/s00023-013-0236-x

Fermionic SPT phases in higher dimensions and bosonization
journal, October 2017

Kapustin, Anton; Thorngren, Ryan
Journal of High Energy Physics, Vol. 2017, Issue 10
DOI: 10.1007/JHEP10(2017)080

Surface states and topological invariants in three-dimensional topological insulators: Application to ${Bi}_{1 - x} {Sb}_{x}$
journal, July 2008

Teo, Jeffrey C. Y.; Fu, Liang; Kane, C. L.
Physical Review B, Vol. 78, Issue 4, Article No. 045426
DOI: 10.1103/PhysRevB.78.045426

Spin topological field theory and fermionic matrix product states
journal, September 2018

Kapustin, Anton; Turzillo, Alex; You, Minyoung
Physical Review B, Vol. 98, Issue 12
DOI: 10.1103/PhysRevB.98.125101

Classification of Topological Insulators and Superconductors
conference, January 2009

Schnyder, Andreas P.; Ryu, Shinsei; Furusaki, Akira
ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings
DOI: 10.1063/1.3149481

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
journal, March 2015

Ando, Yoichi; Fu, Liang
Annual Review of Condensed Matter Physics, Vol. 6, Issue 1
DOI: 10.1146/annurev-conmatphys-031214-014501

Equivalent topological invariants for one-dimensional Majorana wires in symmetry class $D$
journal, August 2013

Budich, Jan Carl; Ardonne, Eddy
Physical Review B, Vol. 88, Issue 7
DOI: 10.1103/PhysRevB.88.075419

Characteristic Classes. (AM-76)
book, December 1974

Milnor, John; Stasheff, James D.
Princeton University Press
DOI: 10.1515/9781400881826

Spin TQFTs and fermionic phases of matter
journal, October 2016

Gaiotto, Davide; Kapustin, Anton
International Journal of Modern Physics A, Vol. 31, Issue 28n29
DOI: 10.1142/S0217751X16450445

Topological Crystalline Insulators
journal, March 2011

Fu, Liang
Physical Review Letters, Vol. 106, Issue 10
DOI: 10.1103/PhysRevLett.106.106802

Interacting fermionic symmetry-protected topological phases in two dimensions
journal, May 2017

Wang, Chenjie; Lin, Chien-Hung; Gu, Zheng-Cheng
Physical Review B, Vol. 95, Issue 19
DOI: 10.1103/PhysRevB.95.195147

Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with $Z_{2}$ symmetry
journal, May 2014

Gu, Zheng-Cheng; Levin, Michael
Physical Review B, Vol. 89, Issue 20
DOI: 10.1103/PhysRevB.89.201113

Clifford modules
journal, July 1964

Atiyah, M. F.; Bott, R.; Shapiro, A.
Topology, Vol. 3
DOI: 10.1016/0040-9383(64)90003-5

Antiferromagnetic topological insulators
journal, June 2010

Mong, Roger S. K.; Essin, Andrew M.; Moore, Joel E.
Physical Review B, Vol. 81, Issue 24
DOI: 10.1103/PhysRevB.81.245209

Characters and cohomology of finite groups
journal, December 1961

Atiyah, M. F.
Publications mathématiques de l'IHÉS, Vol. 9, Issue 1
DOI: 10.1007/BF02698718

Classification of topological quantum matter with symmetries
journal, August 2016

Chiu, Ching-Kai; Teo, Jeffrey C. Y.; Schnyder, Andreas P.
Reviews of Modern Physics, Vol. 88, Issue 3
DOI: 10.1103/RevModPhys.88.035005

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
journal, April 2017

Bhardwaj, Lakshya; Gaiotto, Davide; Kapustin, Anton
Journal of High Energy Physics, Vol. 2017, Issue 4
DOI: 10.1007/JHEP04(2017)096

Topological Classification of Crystalline Insulators through Band Structure Combinatorics
journal, December 2017

Kruthoff, Jorrit; de Boer, Jan; van Wezel, Jasper
Physical Review X, Vol. 7, Issue 4
DOI: 10.1103/PhysRevX.7.041069

Topological Insulators and Topological Superconductors
book, January 2013

Bernevig, B. Andrei
Princeton University Press
DOI: 10.1515/9781400846733

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
DOI: 10.1103/PhysRevB.84.235128

Symmetry Classes of Disordered Fermions
journal, May 2005

Heinzner, P.; Huckleberry, A.; Zirnbauer, M. R.
Communications in Mathematical Physics, Vol. 257, Issue 3
DOI: 10.1007/s00220-005-1330-9

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)
DOI: 10.1103/PhysRevB.75.121306

Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory
journal, March 2018

Wang, Qing-Rui; Gu, Zheng-Cheng
Physical Review X, Vol. 8, Issue 1
DOI: 10.1103/PhysRevX.8.011055

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $σ$ models and a special group supercohomology theory
journal, September 2014

Gu, Zheng-Cheng; Wen, Xiao-Gang
Physical Review B, Vol. 90, Issue 11
DOI: 10.1103/PhysRevB.90.115141

Topological defects and gapless modes in insulators and superconductors
journal, September 2010

Teo, Jeffrey C. Y.; Kane, C. L.
Physical Review B, Vol. 82, Issue 11
DOI: 10.1103/PhysRevB.82.115120

Fermionic symmetry protected topological phases and cobordisms
journal, December 2015

Kapustin, Anton; Thorngren, Ryan; Turzillo, Alex
Journal of High Energy Physics, Vol. 2015, Issue 12
DOI: 10.1007/JHEP12(2015)052

Topological phases of fermions in one dimension
journal, February 2011

Fidkowski, Lukasz; Kitaev, Alexei
Physical Review B, Vol. 83, Issue 7
DOI: 10.1103/PhysRevB.83.075103

Spin Tqfts and Fermionic Phases of Matter
conference, November 2016

Gaiotto, Davide; Kapustin, Anton
The Memorial Workshop Devoted to the 85th Birthday of V N Gribov, Gribov-85 Memorial Volume: Exploring Quantum Field Theory
DOI: 10.1142/9789813141704_0044

Stiefel-Whitney classes of real representations of finite groups
journal, November 1989

Gunarwardena, J.; Kahn, B.; Thomas, C.
Journal of Algebra, Vol. 126, Issue 2
DOI: 10.1016/0021-8693(89)90309-8

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

Ryu, Shinsei; Schnyder, Andreas P.; Furusaki, Akira
New Journal of Physics, Vol. 12, Issue 6
DOI: 10.1088/1367-2630/12/6/065010

Classification of Topological Insulators and Superconductors
text, January 2009

Schnyder, Andreas P.; Ryu, Shinsei; Furusaki, Akira
arXiv
DOI: 10.48550/arxiv.0905.2029

Topological Defects and Gapless Modes in Insulators and Superconductors
text, January 2010

Teo, Jeffrey C. Y.; Kane, C. L.
arXiv
DOI: 10.48550/arxiv.1006.0690

Topological Crystalline Insulators
text, January 2010

Fu, Liang
arXiv
DOI: 10.48550/arxiv.1010.1802

Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D
text, January 2013

Budich, Jan Carl; Ardonne, Eddy
arXiv
DOI: 10.48550/arxiv.1306.4459

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
text, January 2015

Ando, Yoichi; Fu, Liang
arXiv
DOI: 10.48550/arxiv.1501.00531

Classification of topological quantum matter with symmetries
text, January 2015

Chiu, Ching-Kai; Teo, Jeffrey C. Y.; Schnyder, Andreas P.
arXiv
DOI: 10.48550/arxiv.1505.03535

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
text, January 2016

Bhardwaj, Lakshya; Gaiotto, Davide; Kapustin, Anton
arXiv
DOI: 10.48550/arxiv.1605.01640

Topological invariants of time-reversal-invariant band structures
text, January 2006

Moore, J. E.; Balents, L.
arXiv
DOI: 10.48550/arxiv.cond-mat/0607314

Works referencing / citing this record:

Generic “unnecessary” quantum critical points with minimal degrees of freedom
journal, January 2020

Jian, Chao-Ming; Xu, Cenke
Physical Review B, Vol. 101, Issue 3
DOI: 10.1103/physrevb.101.035118

Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom
text, January 2019

Jian, Chao-Ming; Xu, Cenke
arXiv
DOI: 10.48550/arxiv.1907.08613

Similar Records in DOE PAGES and OSTI.GOV collections:

Classification of (2 + 1)D invertible fermionic topological phases with symmetry

Journal Article Barkeshli, Maissam ; Chen, Yu-An ; Hsin, Po-Shen ; ... - Physical Review. B

Here we provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups G_f and general values of the chiral central charge c_-. Here G_f is a central extension of a bosonic symmetry group G_b by fermion parity, (-1)^F, specified by a second cohomology class [ω₂]∈more »« less
https://doi.org/10.1103/physrevb.105.235143

Full Text Available
Squaring the fermion: The threefold way and the fate of zero modes

Journal Article Xu, Qiao-Ru ; Flynn, Vincent P. ; Alase, Abhijeet ; ... - Physical Review. B

In this work, we investigate topological properties and classification of mean-field theories of stable bosonic systems. Of the three standard classifying symmetries, only time reversal represents a real symmetry of the many-boson system, while the other two, particle-hole and chiral, are simply constraints that manifest as symmetries of the effective single-particle problem. For gapped systems in arbitrary space dimension, we establish three fundamental no-go theorems that prove the absence of parity switches, symmetry-protected-topological quantum phases, and localized bosonic zero modes under open boundary conditions. We then introduce a squaring, kernel-preserving map connecting noninteracting Hermitian theories of fermions and stable bosonmore »« less
https://doi.org/10.1103/physrevb.102.125127

Full Text Available
Fermionic matrix product states and one-dimensional short-range entangled phases with antiunitary symmetries

Journal Article Turzillo, Alex ; You, Minyoung - Physical Review. B

We extend the formalism of matrix product states (MPS) to describe one-dimensional gapped systems of fermions with both unitary and antiunitary symmetries. Additionally, systems with orientation-reversing spatial symmetries are considered. The short-ranged entangled phases of such systems are classified by three invariants, which characterize the projective action of the symmetry on edge states. We give interpretations of these invariants as properties of states on the closed chain. The relationship between fermionic MPS systems at a renormalization group fixed point and equivariant algebras is exploited to derive a group law for the stacking of fermionic phases. Furthermore, the result generalizes knownmore »« less
Cited by 13
https://doi.org/10.1103/physrevb.99.035103

Full Text Available
Nonlocal Finite-Depth Circuits for Constructing Symmetry-Protected Topological States and Quantum Cellular Automata

Journal Article Stephen, David T. ; Dua, Arpit ; Lavasani, Ali ; ... - PRX Quantum

Whether a given target state can be prepared by starting with a simple product state and acting with a finite-depth quantum circuit is a key question in condensed matter physics and quantum information science. It underpins classifications of topological phases, as well as the understanding of topological quantum codes, and has obvious relevance for device implementations. Traditionally, this question assumes that the quantum circuit is made up of unitary gates that are geometrically local. Inspired by the advent of noisy intermediate-scale quantum devices, we reconsider this question with k-local gates, i.e. gates that act on no more than k degreesmore »« less
https://doi.org/10.1103/PRXQuantum.5.010304
Fermionic symmetry protected topological phases and cobordisms

Journal Article Kapustin, Anton ; Thorngren, Ryan ; Turzillo, Alex ; ... - Journal of High Energy Physics (Online)

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
Cited by 187
https://doi.org/10.1007/JHEP12(2015)052

Full Text Available

Similar Records

Title: Free and interacting short-range entangled phases of fermions: Beyond the tenfold way

Abstract

Citation Formats

Twisted Equivariant Matter journal, March 2013

Fermionic SPT phases in higher dimensions and bosonization journal, October 2017

Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1−xSbx journal, July 2008

Spin topological field theory and fermionic matrix product states journal, September 2018

Classification of Topological Insulators and Superconductors conference, January 2009

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials journal, March 2015

Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D journal, August 2013

Characteristic Classes. (AM-76) book, December 1974

Spin TQFTs and fermionic phases of matter journal, October 2016

Topological Crystalline Insulators journal, March 2011

Interacting fermionic symmetry-protected topological phases in two dimensions journal, May 2017

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

Clifford modules journal, July 1964

Antiferromagnetic topological insulators journal, June 2010

Characters and cohomology of finite groups journal, December 1961

Classification of topological quantum matter with symmetries journal, August 2016

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter journal, April 2017

Topological Classification of Crystalline Insulators through Band Structure Combinatorics journal, December 2017

Topological Insulators and Topological Superconductors book, January 2013

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

Symmetry Classes of Disordered Fermions journal, May 2005

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

Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory journal, March 2018

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory journal, September 2014

Topological defects and gapless modes in insulators and superconductors journal, September 2010

Fermionic symmetry protected topological phases and cobordisms journal, December 2015

Topological phases of fermions in one dimension journal, February 2011

Spin Tqfts and Fermionic Phases of Matter conference, November 2016

Stiefel-Whitney classes of real representations of finite groups journal, November 1989

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

Classification of Topological Insulators and Superconductors text, January 2009

Topological Defects and Gapless Modes in Insulators and Superconductors text, January 2010

Topological Crystalline Insulators text, January 2010

Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D text, January 2013

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials text, January 2015

Classification of topological quantum matter with symmetries text, January 2015

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter text, January 2016

Topological invariants of time-reversal-invariant band structures text, January 2006

Generic “unnecessary” quantum critical points with minimal degrees of freedom journal, January 2020

Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom text, January 2019

Twisted Equivariant Matter
journal, March 2013

Fermionic SPT phases in higher dimensions and bosonization
journal, October 2017

Surface states and topological invariants in three-dimensional topological insulators: Application to ${Bi}_{1 - x} {Sb}_{x}$
journal, July 2008

Spin topological field theory and fermionic matrix product states
journal, September 2018

Classification of Topological Insulators and Superconductors
conference, January 2009

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
journal, March 2015

Equivalent topological invariants for one-dimensional Majorana wires in symmetry class $D$
journal, August 2013

Characteristic Classes. (AM-76)
book, December 1974

Spin TQFTs and fermionic phases of matter
journal, October 2016

Topological Crystalline Insulators
journal, March 2011

Interacting fermionic symmetry-protected topological phases in two dimensions
journal, May 2017

Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with $Z_{2}$ symmetry
journal, May 2014

Clifford modules
journal, July 1964

Antiferromagnetic topological insulators
journal, June 2010

Characters and cohomology of finite groups
journal, December 1961

Classification of topological quantum matter with symmetries
journal, August 2016

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
journal, April 2017

Topological Classification of Crystalline Insulators through Band Structure Combinatorics
journal, December 2017

Topological Insulators and Topological Superconductors
book, January 2013

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

Symmetry Classes of Disordered Fermions
journal, May 2005

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

Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory
journal, March 2018

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $σ$ models and a special group supercohomology theory
journal, September 2014

Topological defects and gapless modes in insulators and superconductors
journal, September 2010

Fermionic symmetry protected topological phases and cobordisms
journal, December 2015

Topological phases of fermions in one dimension
journal, February 2011

Spin Tqfts and Fermionic Phases of Matter
conference, November 2016

Stiefel-Whitney classes of real representations of finite groups
journal, November 1989

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

Classification of Topological Insulators and Superconductors
text, January 2009

Topological Defects and Gapless Modes in Insulators and Superconductors
text, January 2010

Topological Crystalline Insulators
text, January 2010

Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D
text, January 2013

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
text, January 2015

Classification of topological quantum matter with symmetries
text, January 2015

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
text, January 2016

Topological invariants of time-reversal-invariant band structures
text, January 2006

Generic “unnecessary” quantum critical points with minimal degrees of freedom
journal, January 2020

Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom
text, January 2019