Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

We study the quantized topological terms in a weak-coupling gauge theory with gauge group Gg and a global symmetry Gs in d space-time dimensions. We show that the quantized topological terms are classified by a pair (G,νd), where G is an extension of Gs by Gg and νd an element in group cohomology Hd(G,R/Z). When d=3 and/or when Gg is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by Hd(G,R/Z), where G/Gg=Gs. We also apply our theory to a simple case Gs=Gg=Z2, which leads to 12 different SET phases in 2+1 dimensions [(2+1)D], where quasiparticles have different patterns of fractional Gs=Z2 quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry Gs, which may lead to different fractionalizations of Gs quantum numbers and different fractional statistics [if in (2+1)D].

Note:

13 pages, 2 figures, PRB accepted version with added clarification on obtaining G_s charge for a given PSG with non-trivial topological terms. arXiv admin note: text overlap with arXiv:1301.7675

71.27.+a
02.40.Re
quantization: topological
gauge field theory: weak coupling
statistics: fractional
phase: topological
symmetry: global
quantum number
cohomology
any-dimensional

References(71)

Figures(2)

[1]

On the theory of phase transitions

L.D. Landau

- Zh.Eksp.Teor.Fiz. 7 (1937) 19-32,
- Phys.Z.Sowjetunion 11 (1937) 26,
- Ukr.J.Phys. 53 (2008) 25
•
DOI:
- 10.1016/B978-0-08-010586-4.50034-1

[2]

Zh. Ekaper. Teoret

V.L. Ginzburg
,
L.D. Landau

- Fizika 20 (1950) 1064

[3]

Statistical Physics Course of Theoretical Physics Vol 5 London, 1958)

L.D. Landau
,
E.M. Lifschitz

[4]

Vacuum Degeneracy of Chiral Spin States in Compactified Space

X.G. Wen
(
- Santa Barbara, KITP
)

- Phys.Rev.B 40 (1989) 7387-7390
•
DOI:
- 10.1103/PhysRevB.40.7387

[5]

Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces

X.G. Wen
(
- UC, Santa Barbara
)
,
Q. Niu
(
- UC, Santa Barbara
)

- Phys.Rev.B 41 (1990) 9377-9396
•
DOI:
- 10.1103/PhysRevB.41.9377

[6]

Topological Order in Rigid States

X.G. Wen
(
- Santa Barbara, KITP
)

- Int.J.Mod.Phys.B 4 (1990) 239
•
DOI:
- 10.1142/S0217979290000139

[7]

Detecting Topological Order in a Ground State Wave Function

Michael Levin
(
- MIT
)
,
Xiao-Gang Wen
(
- MIT
)

- Phys.Rev.Lett. 96 (2006) 110405
•
e-Print:
- cond-mat/0510613
•
DOI:
- 10.1103/PhysRevLett.96.110405

[8]

Topological entanglement entropy

Alexei Kitaev
(
- Caltech, IQI and
- Microsoft, Redmond
)
,
John Preskill
(
- Caltech, IQI
)

- Phys.Rev.Lett. 96 (2006) 110404
•
e-Print:
- hep-th/0510092
•
DOI:
- 10.1103/PhysRevLett.96.110404

[9]

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Xie Chen
(
- MIT
)
,
Zheng Cheng Gu
(
- Santa Barbara, KITP
)
,
Xiao Gang Wen
(
- MIT
)

- Phys.Rev.B 82 (2010) 155138
•
e-Print:
- 1004.3835
•
DOI:
- 10.1103/PhysRevB.82.155138

[10]

String net condensation: A Physical mechanism for topological phases

Michael A. Levin
(
- MIT, LNS
)
,
Xiao-Gang Wen
(
- MIT, LNS
)

- Phys.Rev.B 71 (2005) 045110
•
e-Print:
- cond-mat/0404617
•
DOI:
- 10.1103/PhysRevB.71.045110

[11]

Renormalization group transformations on quantum states

- Phys.Rev.Lett. 94 (2005) 140601
•
e-Print:
- quant-ph/0410227
•
DOI:
- 10.1103/PhysRevLett.94.140601

[12]

Entanglement Renormalization

G. Vidal
(
- Queensland U. and
- Caltech, IQI
)

- Phys.Rev.Lett. 99 (2007) 22, 220405
•
e-Print:
- cond-mat/0512165
•
DOI:
- 10.1103/PhysRevLett.99.220405

[13]

Equivalence of the resonating valence bond and fractional quantum Hall states

V. Kalmeyer
(
- Stanford U., Phys. Dept. and
- LLNL, Livermore
)
,
R.B. Laughlin
(
- Stanford U., Phys. Dept. and
- LLNL, Livermore
)

- Phys.Rev.Lett. 59 (1987) 2095-2098,
- Reprinted in *Wilczek, F. (ed.): Fractional statistics and anyon superconductivity* 308-311
•
DOI:
- 10.1103/PhysRevLett.59.2095

[14]

Chiral Spin States and Superconductivity

X.G. Wen
(
- Santa Barbara, KITP
)
,
Frank Wilczek
(
- Santa Barbara, KITP
)
,
A. Zee
(
- Santa Barbara, KITP
)

- Phys.Rev.B 39 (1989) 11413-11423,
- Also in *Wilczek, F. (ed.): Fractional statistics and anyon superconductivity* 312-322
•
DOI:
- 10.1103/PhysRevB.39.11413