# Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model

## Abstract

We study the two-dimensional (2D) Hubbard model using exact diagonalization for spin-1/2 fermions on the triangular and honeycomb lattices decorated with a single hexagon per site. In certain parameter ranges, the Hubbard model maps to a quantum compass model on those lattices. On the triangular lattice, the compass model exhibits collinear stripe antiferromagnetism, implying d-density wave charge order in the original Hubbard model. On the honeycomb lattice, the compass model has a unique, quantum disordered ground state that transforms nontrivially under lattice reflection. The ground state of the Hubbard model on the decorated honeycomb lattice is thus a 2D fermionic symmetry-protected topological phase. This state—protected by time-reversal and reflection symmetries—cannot be connected adiabatically to a free-fermion topological phase.

- Authors:

- Argonne National Lab. (ANL), Argonne, IL (United States). Advanced Photon Source (APS); Univ. of Alabama, Tuscaloosa, AL (United States)
- Princeton Univ., NJ (United States)
- Univ. of Alberta, Edmonton, AB (Canada); Canadian Inst. for Advanced Research, Toronto, ON (Canada)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- Natural Sciences and Engineering Research Council of Canada (NSERC); University of Alberta; Canadian Institute for Advanced Research (CIFAR); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1339548

- Alternate Identifier(s):
- OSTI ID: 1306704

- Grant/Contract Number:
- AC02-06CH11357; FG02-05ER46201; AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Volume: 117; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Chen, Cheng-Chien, Muechler, Lukas, Car, Roberto, Neupert, Titus, and Maciejko, Joseph. Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model. United States: N. p., 2016.
Web. doi:10.1103/PhysRevLett.117.096405.
```

```
Chen, Cheng-Chien, Muechler, Lukas, Car, Roberto, Neupert, Titus, & Maciejko, Joseph. Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model. United States. doi:10.1103/PhysRevLett.117.096405.
```

```
Chen, Cheng-Chien, Muechler, Lukas, Car, Roberto, Neupert, Titus, and Maciejko, Joseph. Thu .
"Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model". United States. doi:10.1103/PhysRevLett.117.096405. https://www.osti.gov/servlets/purl/1339548.
```

```
@article{osti_1339548,
```

title = {Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model},

author = {Chen, Cheng-Chien and Muechler, Lukas and Car, Roberto and Neupert, Titus and Maciejko, Joseph},

abstractNote = {We study the two-dimensional (2D) Hubbard model using exact diagonalization for spin-1/2 fermions on the triangular and honeycomb lattices decorated with a single hexagon per site. In certain parameter ranges, the Hubbard model maps to a quantum compass model on those lattices. On the triangular lattice, the compass model exhibits collinear stripe antiferromagnetism, implying d-density wave charge order in the original Hubbard model. On the honeycomb lattice, the compass model has a unique, quantum disordered ground state that transforms nontrivially under lattice reflection. The ground state of the Hubbard model on the decorated honeycomb lattice is thus a 2D fermionic symmetry-protected topological phase. This state—protected by time-reversal and reflection symmetries—cannot be connected adiabatically to a free-fermion topological phase.},

doi = {10.1103/PhysRevLett.117.096405},

journal = {Physical Review Letters},

number = ,

volume = 117,

place = {United States},

year = {2016},

month = {8}

}

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Works referenced in this record:

##
Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals

journal, November 2015

- Watanabe, Haruki; Po, Hoi Chun; Vishwanath, Ashvin
- Proceedings of the National Academy of Sciences, Vol. 112, Issue 47

##
Orbital Order in Mott Insulators of Spinless $p$ -Band Fermions

journal, April 2008

- Zhao, Erhai; Liu, W. Vincent
- Physical Review Letters, Vol. 100, Issue 16

##
Orbital Ordering and Frustration of $p$ -Band Mott Insulators

journal, May 2008

- Wu, Congjun
- Physical Review Letters, Vol. 100, Issue 20

##
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems

journal, September 2005

- Hernandez, Vicente; Roman, Jose E.; Vidal, Vicente
- ACM Transactions on Mathematical Software, Vol. 31, Issue 3

##
Compass models: Theory and physical motivations

journal, January 2015

- Nussinov, Zohar; van den Brink, Jeroen
- Reviews of Modern Physics, Vol. 87, Issue 1

##
A Krylov--Schur Algorithm for Large Eigenproblems

journal, January 2002

- Stewart, G. W.
- SIAM Journal on Matrix Analysis and Applications, Vol. 23, Issue 3

##
Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

journal, October 2009

- Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 80, Issue 15

##
Orbital Ordering in Frustrated Jahn-Teller Systems with 90° Exchange

journal, November 2002

- Mostovoy, M. V.; Khomskii, D. I.
- Physical Review Letters, Vol. 89, Issue 22

##
Symmetry-Protected Topological Phases of Quantum Matter

journal, March 2015

- Senthil, T.
- Annual Review of Condensed Matter Physics, Vol. 6, Issue 1

##
Symmetry protected topological orders and the group cohomology of their symmetry group

journal, April 2013

- Chen, Xie; Gu, Zheng-Cheng; Liu, Zheng-Xin
- Physical Review B, Vol. 87, Issue 15

##
Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

journal, December 2011

- Chen, Xie; Liu, Zheng-Xin; Wen, Xiao-Gang
- Physical Review B, Vol. 84, Issue 23

##
Featureless and nonfractionalized Mott insulators on the honeycomb lattice at 1/2 site filling

journal, September 2013

- Kimchi, I.; Parameswaran, S. A.; Turner, A. M.
- Proceedings of the National Academy of Sciences, Vol. 110, Issue 41

##
Phenomenological theory of unconventional superconductivity

journal, April 1991

- Sigrist, Manfred; Ueda, Kazuo
- Reviews of Modern Physics, Vol. 63, Issue 2

##
Detecting Topological Order in a Ground State Wave Function

journal, March 2006

- Levin, Michael; Wen, Xiao-Gang
- Physical Review Letters, Vol. 96, Issue 11

##
Classification of Interacting Electronic Topological Insulators in Three Dimensions

journal, February 2014

- Wang, C.; Potter, A. C.; Senthil, T.
- Science, Vol. 343, Issue 6171

##
Topological Entanglement Entropy

journal, March 2006

- Kitaev, Alexei; Preskill, John
- Physical Review Letters, Vol. 96, Issue 11

##
The quantum spin Hall effect and topological insulators

journal, January 2010

- Qi, Xiao-Liang; Zhang, Shou-Cheng
- Physics Today, Vol. 63, Issue 1, p. 33-38

##
Doubly degenerate orbital system in honeycomb lattice: Implication of orbital state in layered iron oxide

journal, July 2008

- Nasu, J.; Nagano, A.; Naka, M.
- Physical Review B, Vol. 78, Issue 2

##
Integer Quantum Hall Effect for Bosons

journal, January 2013

- Senthil, T.; Levin, Michael
- Physical Review Letters, Vol. 110, Issue 4

##
Braiding statistics approach to symmetry-protected topological phases

journal, September 2012

- Levin, Michael; Gu, Zheng-Cheng
- Physical Review B, Vol. 86, Issue 11

##
Myriad phases of the checkerboard Hubbard model

journal, October 2007

- Yao, Hong; Tsai, Wei-Feng; Kivelson, Steven A.
- Physical Review B, Vol. 76, Issue 16

##
Microscopic model for the boson integer quantum Hall effect

journal, October 2013

- Regnault, N.; Senthil, T.
- Physical Review B, Vol. 88, Issue 16

##
Exact realization of integer and fractional quantum Hall phases in models in

journal, July 2013

- Geraedts, Scott D.; Motrunich, Olexei I.
- Annals of Physics, Vol. 334

##
Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $\sigma $ models and a special group supercohomology theory

journal, September 2014

- Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 90, Issue 11

##
Density-wave states of nonzero angular momentum

journal, August 2000

- Nayak, Chetan
- Physical Review B, Vol. 62, Issue 8

##
Topological crystalline Bose insulator in two dimensions via entanglement spectrum

journal, November 2015

- Ware, Brayden; Kimchi, Itamar; Parameswaran, S. A.
- Physical Review B, Vol. 92, Issue 19

##
Symmetry protection of topological phases in one-dimensional quantum spin systems

journal, February 2012

- Pollmann, Frank; Berg, Erez; Turner, Ari M.
- Physical Review B, Vol. 85, Issue 7

##
Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order

journal, December 2014

- Burnell, F. J.; Chen, Xie; Fidkowski, Lukasz
- Physical Review B, Vol. 90, Issue 24

##
Symmetry-Protected Topological Orders in Interacting Bosonic Systems

journal, December 2012

- Chen, X.; Gu, Z. -C.; Liu, Z. -X.
- Science, Vol. 338, Issue 6114