Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions

Wang, Juven; Ohmori, Kantaro; Putrov, Pavel; Zheng, Yunqin; Wan, Zheyan; Guo, Meng; Lin, Hai; Gao, Peng; Yau, Shing-Tung

doi:10.1093/ptep/pty051

Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions

Juven Wang
(
- Princeton, Inst. Advanced Study and
- Harvard U., CMSA
)
,
Kantaro Ohmori
(
- Princeton, Inst. Advanced Study
)
,
Pavel Putrov
(
- Princeton, Inst. Advanced Study
)
,
Yunqin Zheng
(
- Princeton U.
)
,
Zheyan Wan
(
- Hefei, CUST
)

Jan 16, 2018

52 pages

Published in:

PTEP 2018 (2018) 5, 053A01

Published: May 1, 2018

e-Print:

1801.05416 [cond-mat.str-el]

DOI:

10.1093/ptep/pty051

View in:

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Abstract: (Oxford University Press)

Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFTs. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological states (with fermion parity

\mathbb{Z}_2^f

) of symmetry groups

\mathbb{Z}_4\times \mathbb{Z}_2

and

(\mathbb{Z}_4)^2

in 3+1D, also

\mathbb{Z}_2

and

(\mathbb{Z}_2)^2

in 2+1D. Gauging the last three cases begets non-Abelian spin TQFTs (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with a boundary, such that the bulk and boundary are fully gapped and short- or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condense, but only fuzzy-composite objects of extended operators can end (e.g., a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g., 0-form/higher-form/“composite” breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some such LRE systems

Note:

59 pages, 8 figures, 7 tables. v3: Elaborate spin TQFTs and fermionic topological orders (3+1D/2+1D) derived from Cobordism classifications of fermionic SPTs. More clarifications/Refs added. PTEP version

A13 Other topics in mathematical physics
A63 Quantum many-body systems
B00 Lattice gauge field theories
B37 Various models of field theory
I46 Strong correlations
field theory: topological
vacuum state: topological
operator: surface
boson: operator
string: excited state

References(138)

Figures(9)

[1]

Can quantum mechanical description of physical reality be considered complete?

- Phys.Rev. 47 (1935) 777-780
•
DOI:
- 10.1103/PhysRev.47.777

[2]

Zoo of quantum-topological phases of matter

Xiao-Gang Wen

- Rev.Mod.Phys. 89 (2017) 4, 041004
•
e-Print:
- 1610.03911
•
DOI:
- 10.1103/RevModPhys.89.041004

[3]

Fractional Statistics and Anyon Superconductivity

Wilczek F.

•
DOI:
- 10.1142/SDCMP

[4]

Freedman M. H.
,
Kitaev A.
,
Larsen M. J.
,
Wang Z.

- Bull.Am.Math.Soc. 40 (2003) 31
•
DOI:
- 10.1090/S0273-0979-02-00964-3)

[4]

DOI:
- 10.1090/S0273-0979-02-00964-3

[5]

Lecture Notes for Physics 219: Quantum Computation

Preskill J.

[6]

Non-Abelian anyons and topological quantum computation

Chetan Nayak
(
- UC, Santa Barbara
)
,
Steven H. Simon
(
- Bell Labs
)
,
Ady Stern
(
- Weizmann Inst.
)
,
Michael Freedman
(
- UC, Santa Barbara
)
,
Sankar Das Sarma
(
- Maryland U.
)

- Rev.Mod.Phys. 80 (2008) 1083-1159
•
e-Print:
- 0707.1889
•
DOI:
- 10.1103/RevModPhys.80.1083

[7]

Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions

Pavel Putrov
(
- Princeton, Inst. Advanced Study
)
,
Juven Wang
(
)
,
Shing-Tung Yau
(
- Harvard U., Dept. Math. and
- Harvard U., Phys. Dept.
)

- Annals Phys. 384 (2017) 254-287
•
e-Print:
- 1612.09298
•
DOI:
- 10.1016/j.aop.2017.06.019

[8]

Braiding statistics of loop excitations in three dimensions

Chenjie Wang
(
- Chicago U.
)
,
Michael Levin
(
- Chicago U.
)

- Phys.Rev.Lett. 113 (2014) 8, 080403
•
e-Print:
- 1403.7437
•
DOI:
- 10.1103/PhysRevLett.113.080403

[9]

Generalized Modular Transformations in (3+1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding

- Phys.Rev.X 4 (2014) 3, 031048
•
e-Print:
- 1404.1062
•
DOI:
- 10.1103/PhysRevX.4.031048

[10]

Non-Abelian string and particle braiding in topological order: Modular SL(3, $\mathbb{Z}$ ) representation and (3+1) -dimensional twisted gauge theory

- Phys.Rev.B 91 (2015) 3, 035134
•
e-Print:
- 1404.7854
•
DOI:
- 10.1103/PhysRevB.91.035134

[11]

Quantum Field Theory and the Jones Polynomial

Edward Witten
(
- Princeton, Inst. Advanced Study
)

- Commun.Math.Phys. 121 (1989) 351-399,
•
DOI:
- 10.1007/BF01217730

[12]

Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond

- Phys.Rev.Lett. 114 (2015) 3, 031601
•
e-Print:
- 1405.7689
•
DOI:
- 10.1103/PhysRevLett.114.031601

[13]

Appearance of Gauge Structure in Simple Dynamical Systems

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

- Phys.Rev.Lett. 52 (1984) 2111-2114,
- IN *SHAPERE, A. (ED.), WILCZEK, F. (ED.): GEOMETRIC PHASES IN PHYSICS* 141-144. (PHYS. REV. LETT. 52 (1984) 2111-2114) AND CALIF. UNIV. SANTA BARBARA - NSF-ITP-84-051 (84,REC.MAY) 13 P. (SEE BOOK INDEX)
•
DOI:
- 10.1103/PhysRevLett.52.2111