Squaring the fermion: The threefold way and the fate of zero modes
Abstract
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 boson systems, which serves as a playground to reveal the role of topology in bosonic phases and their localized midgap boundary modes. Finally, we determine the symmetry classes inherited from the fermionic tenfold-way classification, unveiling an elegant threefold-way topological classification of noninteracting bosons. We illustrate our main findings in one- and two-dimensional bosonic lattice and field-theory models.
- Authors:
-
[1];
[2];
[3];
[4];
[2];
[1]
- Indiana Univ., Bloomington, IN (United States)
- Dartmouth College, Hanover, NH (United States)
- Univ. of Calgary, AB (Canada)
- Dartmouth College, Hanover, NH (United States), SUNY Polytechnic Inst., Utica, NY (United States)
- Publication Date:
- Research Org.:
- Indiana Univ., Bloomington, IN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)
- OSTI Identifier:
- 1786998
- Grant/Contract Number:
- SC0020343; PHY-1620541; OIA-1921199
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review. B
- Additional Journal Information:
- Journal Volume: 102; Journal Issue: 12; 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
Citation Formats
Xu, Qiao-Ru, Flynn, Vincent P., Alase, Abhijeet, Cobanera, Emilio, Viola, Lorenza, and Ortiz, Gerardo. Squaring the fermion: The threefold way and the fate of zero modes. United States: N. p., 2020.
Web. doi:10.1103/physrevb.102.125127.
Xu, Qiao-Ru, Flynn, Vincent P., Alase, Abhijeet, Cobanera, Emilio, Viola, Lorenza, & Ortiz, Gerardo. Squaring the fermion: The threefold way and the fate of zero modes. United States. https://doi.org/10.1103/physrevb.102.125127
Xu, Qiao-Ru, Flynn, Vincent P., Alase, Abhijeet, Cobanera, Emilio, Viola, Lorenza, and Ortiz, Gerardo. Wed .
"Squaring the fermion: The threefold way and the fate of zero modes". United States. https://doi.org/10.1103/physrevb.102.125127. https://www.osti.gov/servlets/purl/1786998.
@article{osti_1786998,
title = {Squaring the fermion: The threefold way and the fate of zero modes},
author = {Xu, Qiao-Ru and Flynn, Vincent P. and Alase, Abhijeet and Cobanera, Emilio and Viola, Lorenza and Ortiz, Gerardo},
abstractNote = {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 boson systems, which serves as a playground to reveal the role of topology in bosonic phases and their localized midgap boundary modes. Finally, we determine the symmetry classes inherited from the fermionic tenfold-way classification, unveiling an elegant threefold-way topological classification of noninteracting bosons. We illustrate our main findings in one- and two-dimensional bosonic lattice and field-theory models.},
doi = {10.1103/physrevb.102.125127},
journal = {Physical Review. B},
number = 12,
volume = 102,
place = {United States},
year = {Wed Sep 16 00:00:00 EDT 2020},
month = {Wed Sep 16 00:00:00 EDT 2020}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Integrable model of a -wave bosonic superfluid
journal, November 2019
- Lerma-Hernández, Sergio; Dukelsky, Jorge; Ortiz, Gerardo
- Physical Review Research, Vol. 1, Issue 3
What is a particle-conserving Topological Superfluid? The fate of Majorana modes beyond mean-field theory
journal, September 2016
- Ortiz, Gerardo; Cobanera, Emilio
- Annals of Physics, Vol. 372
Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle
journal, February 1978
- Youla, D.; Kazanjian, N.
- IEEE Transactions on Circuits and Systems, Vol. 25, Issue 2
Fractional total-charge Eigenvalues for a fermion in a finite one-dimensional box
journal, August 1984
- Roy, S. M.; Singh, Virendra
- Physics Letters B, Vol. 143, Issue 1-3
Generalization of Bloch's theorem for arbitrary boundary conditions: Theory
journal, November 2017
- Alase, Abhijeet; Cobanera, Emilio; Ortiz, Gerardo
- Physical Review B, Vol. 96, Issue 19
Majorana flat bands in -wave gapless topological superconductors
journal, April 2014
- Deng, Shusa; Ortiz, Gerardo; Poudel, Amrit
- Physical Review B, Vol. 89, Issue 14
Symmetry and Topology in Non-Hermitian Physics
journal, October 2019
- Kawabata, Kohei; Shiozaki, Ken; Ueda, Masahito
- Physical Review X, Vol. 9, Issue 4
Diagonalization of the quadratic boson Hamiltonian with zero modes
journal, January 1986
- Colpa, J. H. P.
- Physica A: Statistical Mechanics and its Applications, Vol. 134, Issue 2
Solitons in Polyacetylene
journal, June 1979
- Su, W. P.; Schrieffer, J. R.; Heeger, A. J.
- Physical Review Letters, Vol. 42, Issue 25
Many-Body Characterization of Particle-Conserving Topological Superfluids
journal, December 2014
- Ortiz, Gerardo; Dukelsky, Jorge; Cobanera, Emilio
- Physical Review Letters, Vol. 113, Issue 26
Solitons with fermion number ½
journal, June 1976
- Jackiw, R.; Rebbi, C.
- Physical Review D, Vol. 13, Issue 12
Diagonalization of the quadratic boson Hamiltonian with zero modes
journal, January 1986
- Colpa, J. H. P.
- Physica A: Statistical Mechanics and its Applications, Vol. 134, Issue 2
The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics
journal, November 1962
- Dyson, Freeman J.
- Journal of Mathematical Physics, Vol. 3, Issue 6
Bosonic quadratic Hamiltonians
journal, December 2017
- Dereziński, Jan
- Journal of Mathematical Physics, Vol. 58, Issue 12
Random Matrix Theory and Chiral Symmetry in QCD
journal, December 2000
- Verbaarschot, J. J. M.; Wettig, T.
- Annual Review of Nuclear and Particle Science, Vol. 50, Issue 1
Search for Majorana Fermions in Superconductors
journal, April 2013
- Beenakker, C. W. J.
- Annual Review of Condensed Matter Physics, Vol. 4, Issue 1
New directions in the pursuit of Majorana fermions in solid state systems
journal, June 2012
- Alicea, Jason
- Reports on Progress in Physics, Vol. 75, Issue 7
Quantal Phase Factors Accompanying Adiabatic Changes
journal, March 1984
- Berry, M. V.
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 392, Issue 1802
Dirac Hamiltonians for bosonic spectra
journal, July 2020
- Kumar, P. Sathish; Herbut, Igor F.; Ganesh, R.
- Physical Review Research, Vol. 2, Issue 3
Topological symmetry classes for non-Hermitian models and connections to the bosonic Bogoliubov–de Gennes equation
journal, September 2018
- Lieu, Simon
- Physical Review B, Vol. 98, Issue 11
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
Diagonalization of the quadratic boson hamiltonian
journal, September 1978
- Colpa, J. H. P.
- Physica A: Statistical Mechanics and its Applications, Vol. 93, Issue 3-4
Unpaired Majorana fermions in quantum wires
journal, October 2001
- Kitaev, A. Yu
- Physics-Uspekhi, Vol. 44, Issue 10S
Multiband -wave topological superconductors: Role of dimensionality and magnetic field response
journal, May 2013
- Deng, Shusa; Ortiz, Gerardo; Viola, Lorenza
- Physical Review B, Vol. 87, Issue 20
Topological chiral magnonic edge mode in a magnonic crystal
journal, May 2013
- Shindou, Ryuichi; Matsumoto, Ryo; Murakami, Shuichi
- Physical Review B, Vol. 87, Issue 17
Single Band Motion of Conduction Electrons in a Uniform Magnetic Field
journal, October 1955
- Harper, P. G.
- Proceedings of the Physical Society. Section A, Vol. 68, Issue 10
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
Bound States in Quantum Field Theory
journal, October 1951
- Gell-Mann, Murray; Low, Francis
- Physical Review, Vol. 84, Issue 2
Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect
journal, April 2000
- Read, N.; Green, Dmitry
- Physical Review B, Vol. 61, Issue 15
Simple scheme for surface-band calculations. I
journal, May 1981
- Lee, D. H.; Joannopoulos, J. D.
- Physical Review B, Vol. 23, Issue 10
Periodic table for topological bands with non-Hermitian symmetries
journal, June 2019
- Zhou, Hengyun; Lee, Jong Yeon
- Physical Review B, Vol. 99, Issue 23
Topological edge states for disordered bosonic systems
journal, March 2018
- Peano, Vittorio; Schulz-Baldes, Hermann
- Journal of Mathematical Physics, Vol. 59, Issue 3
Exact solution of corner-modified banded block-Toeplitz eigensystems
journal, April 2017
- Cobanera, Emilio; Alase, Abhijeet; Ortiz, Gerardo
- Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 19
Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
journal, August 2020
- Flynn, Vincent P.; Cobanera, Emilio; Viola, Lorenza
- New Journal of Physics, Vol. 22, Issue 8
Signature and Spectral Flow of J-Unitary $${\mathbb{S}^1}$$ S 1 -Fredholm Operators
journal, September 2013
- Schulz-Baldes, Hermann
- Integral Equations and Operator Theory, Vol. 78, Issue 3
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields
journal, September 1976
- Hofstadter, Douglas R.
- Physical Review B, Vol. 14, Issue 6
Majorana Modes in Time-Reversal Invariant -Wave Topological Superconductors
journal, January 2012
- Deng, Shusa; Viola, Lorenza; Ortiz, Gerardo
- Physical Review Letters, Vol. 108, Issue 3
Equivalence of topological insulators and superconductors
journal, October 2015
- Cobanera, Emilio; Ortiz, Gerardo
- Physical Review B, Vol. 92, Issue 15
Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions
journal, August 2016
- Alase, Abhijeet; Cobanera, Emilio; Ortiz, Gerardo
- Physical Review Letters, Vol. 117, Issue 7
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
Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures
journal, January 1997
- Altland, Alexander; Zirnbauer, Martin R.
- Physical Review B, Vol. 55, Issue 2
Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
journal, December 2018
- Cobanera, Emilio; Alase, Abhijeet; Ortiz, Gerardo
- Physical Review B, Vol. 98, Issue 24
Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions
text, January 2016
- Alase, Abhijeet; Cobanera, Emilio; Ortiz, Gerardo
- arXiv
What is a particle-conserving Topological Superfluid?
text, January 2016
- Ortiz, Gerardo; Cobanera, Emilio
- arXiv
Topological edge states for disordered bosonic systems
text, January 2017
- Peano, Vittorio; Schulz-Baldes, Hermann
- arXiv
Random Matrix Theory and Chiral Symmetry in QCD
text, January 2000
- Verbaarschot, J. J. M.; Wettig, T.
- arXiv
Works referencing / citing this record:
Spin-Space Groups and Magnon Band Topology
text, January 2021
- Corticelli, A.; Moessner, R.; McClarty, P. A.
- arXiv