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Title: Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions

Abstract

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1 dimensions are explored. Many of our TQFTs are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf–Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf–Witten description and all fermionic TQFTs (namely the spin TQFTs) are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs. The acquired link invariants include not only the familiar Aharonov–Bohm linking number, but also Milnor triple linking number in 3 dimensions, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 4 dimensions. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(–Brown–Kervaire), Sato–Levine and others. We propose amore » new relation between the fermionic SPT partition function and the Rokhlin invariant. As an example, we can use these invariants and other physical observables, including ground state degeneracy, reduced modular $$S^{xy}$$ and $$T^{xy}$$ matrices, and the partition function on $$RP^3$$ manifold, to identify all $$\nu \in Z_8$$ classes of 2+1 dimensional gauged $$Z_2$$-Ising-symmetric $$Z^f_2$$-fermionic Topological Superconductors (realized by stacking v layers of a pair of chiral and anti-chiral -wave superconductors [ p + ip and p - ip], where boundary supports non-chiral Majorana–Weyl modes) with continuum spin-TQFTs.« less

Authors:
 [1];  [2];  [3]
  1. Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
  2. Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences; Harvard Univ., Cambridge, MA (United States)
  3. Harvard Univ., Cambridge, MA (United States)
Publication Date:
Research Org.:
Institute for Advanced Study, Princeton, NJ (United States)
Sponsoring Org.:
USDOE Office of Science (SC); National Science Foundation (NSF)
OSTI Identifier:
1533513
Alternate Identifier(s):
OSTI ID: 1576033
Grant/Contract Number:  
SC0009988; PHY-1306313; PHY-0937443; DMS-1308244; DMS-0804454; DMS-1159412
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Annals of Physics
Additional Journal Information:
Journal Volume: 384; Journal Issue: C; Journal ID: ISSN 0003-4916
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Putrov, Pavel, Wang, Juven, and Yau, Shing-Tung. Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions. United States: N. p., 2017. Web. doi:10.1016/j.aop.2017.06.019.
Putrov, Pavel, Wang, Juven, & Yau, Shing-Tung. Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions. United States. https://doi.org/10.1016/j.aop.2017.06.019
Putrov, Pavel, Wang, Juven, and Yau, Shing-Tung. Tue . "Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions". United States. https://doi.org/10.1016/j.aop.2017.06.019. https://www.osti.gov/servlets/purl/1533513.
@article{osti_1533513,
title = {Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions},
author = {Putrov, Pavel and Wang, Juven and Yau, Shing-Tung},
abstractNote = {Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1 dimensions are explored. Many of our TQFTs are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf–Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf–Witten description and all fermionic TQFTs (namely the spin TQFTs) are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs. The acquired link invariants include not only the familiar Aharonov–Bohm linking number, but also Milnor triple linking number in 3 dimensions, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 4 dimensions. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(–Brown–Kervaire), Sato–Levine and others. We propose a new relation between the fermionic SPT partition function and the Rokhlin invariant. As an example, we can use these invariants and other physical observables, including ground state degeneracy, reduced modular $S^{xy}$ and $T^{xy}$ matrices, and the partition function on $RP^3$ manifold, to identify all $\nu \in Z_8$ classes of 2+1 dimensional gauged $Z_2$-Ising-symmetric $Z^f_2$-fermionic Topological Superconductors (realized by stacking v layers of a pair of chiral and anti-chiral -wave superconductors [ p + ip and p - ip], where boundary supports non-chiral Majorana–Weyl modes) with continuum spin-TQFTs.},
doi = {10.1016/j.aop.2017.06.019},
url = {https://www.osti.gov/biblio/1533513}, journal = {Annals of Physics},
issn = {0003-4916},
number = C,
volume = 384,
place = {United States},
year = {2017},
month = {7}
}

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