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

@article{Putrov2016BraidingSA, title={Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions}, author={Pavel Putrov and Juven C. Wang and Shing-Tung Yau}, journal={arXiv: Strongly Correlated Electrons}, year={2016} }

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories 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 spin TQFTs are either higher-form gauge…

## Figures and Tables from this paper

## 79 Citations

Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms

- Physics, Mathematics
- 2018

We formulate a family of spin Topological Quantum Field Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf–Witten TQFTs. They are obtained by gauging G -equivariant invertible…

Topological quantum field theory for Abelian topological phases and loop braiding statistics in
(3+1)
-dimensions

- Physics, MathematicsPhysical Review B
- 2019

Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1d, it is well known that the Chern-Simons theory captures all the…

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

- Physics, Mathematics
- 2018

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…

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

- Physics
- 2017

We study Abelian braiding statistics of loop excitations in three-dimensional (3D) gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic…

Non-Abelian three-loop braiding statistics for 3D fermionic topological phases

- MedicineNature communications
- 2021

This work systematically study the so-called three-loop braiding statistics for 3D interacting fermion systems and discoveries are made that can only be realized in fermionic systems, giving rise to an alternative way to classify FSPT phases.

Quantum statistics and spacetime topology: Quantum surgery formulas

- Physics, MathematicsAnnals of Physics
- 2019

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where…

Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

- PhysicsPhysical Review B
- 2018

We analyze $2+1d$ and $3+1d$ Bosonic Symmetry Protected Topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk we study an…

Classification of
3+1D
Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions

- PhysicsPhysical Review X
- 2019

We call a topological order of 3+1-dimensional bosonic systems an all-boson (AB) topological order if all emergent point-like excitations are bosons. It was shown that AB topological orders,…

Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions

- Physics, Mathematics
- 2021

Twisted BF Theories in Five Dimensions Zhi-Feng Zhang and Peng Ye∗ School of Physics, Sun Yat-sen University, Guangzhou, 510275, China (Dated: Friday 10th December, 2021) Abstract Topological orders…

Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

- Physics, MathematicsPhysical Review D
- 2019

We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short-/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal…

## References

SHOWING 1-10 OF 107 REFERENCES

Topological invariants for gauge theories and symmetry-protected topological phases

- Physics
- 2015

We study the braiding statistics of particlelike and looplike excitations in two- (2D) and three-dimensional (3D) gauge theories with finite, Abelian gauge group. The gauge theories that we consider…

Wilson operator algebras and ground states of coupled BF theories

- Physics
- 2017

The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between…

Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

- Physics, Mathematics
- 2016

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered…

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

- Physics, Mathematics
- 2015

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group G and a 4-cocycle twist ω 4 of G 's cohomology group H 4…

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

- Physics, MedicinePhysical review letters
- 2015

Gauge fields are used to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology and show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravityactions describe the beyond-group-cohomology SPTs.

Quantum field theory of non-abelian strings and vortices

- Physics
- 1991

We develop an operator formalism for investigating the properties of non-abelian cosmic strings (and vortices) in quantum field theory. Operators are constructed that introduce classical string…

Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory

- Physics, Mathematics
- 2017

Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been…

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

- Physics
- 2014

In topologically ordered quantum states of matter in 2+1D (space-time dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property which is…

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory

- Physics, Mathematics
- 2014

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the…

Topological Gauge Theories of Antisymmetric Tensor Fields

- Physics
- 1991

Abstract We introduce a new class of topological gauge field theories in any dimension, based on anti-symmetric tensor fields, and discuss the BRST-quantization of these reducible systems as well as…