Gauging is a ubiquitous tool in many-body physics. It allows one to construct highly entangled topological phases of matter from relatively simple phases and to relate certain characteristics of the two. Here we develop a gauging procedure for general submanifold symmetries of Pauli Hamiltonians, including symmetries of fractal type. We show a relation between the pre- and postgauging models and use this to construct short-range entangled phases with fractal-like symmetries, one of which is mapped to the cubic code by the gauging.Gauging is a ubiquitous tool in many-body physics. It allows one to construct highly entangled topological phases of matter from relatively simple phases and to relate certain characteristics of the two. Here we develop a gauging procedure for general submanifold symmetries of Pauli Hamiltonians, including symmetries of fractal type. We show a relation between the pre- and postgauging models and use this to construct short-range entangled phases with fractal-like symmetries, one of which is mapped to the cubic code by the gauging.

arXiv:1603.05182Physical review. B./Physical review. BOct 17, 2016168 citations

Gauge Fields and Geometric Phases

David S. Simon

Undergraduate texts in physicsJan 1, 20250 citations

Quantized topological terms in weakly coupled gauge theories and their connection to symmetry protected topological phases

Ling-Yan Hung, Xiao-Gang Wen

We consider a weakly coupled gauge theory where charged particles all have large gaps (ie no Higgs condensation to break the gauge "symmetry") and the field strength fluctuates only weakly. We ask what kind of topological terms can be added to the Lagrangian of such a weakly coupled gauge theory. In this paper, we systematically construct quantized topological terms which are generalization of the Chern-Simons terms and $F\wedge F$ terms, in space-time dimensions $d$ and for any gauge groups (continuous or discrete), using each element of the topological cohomology classes $H^{d+1}(BG,\Z)$ on the classifying space $BG$ of the gauge group $G$ . In 3 $d$ or for finite gauge groups above 3 $d$ , the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories. Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group $G$ are also classified by the same $H^{d+1}(BG,\Z)$ . (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group $G$ and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.We consider a weakly coupled gauge theory where charged particles all have large gaps (ie no Higgs condensation to break the gauge "symmetry") and the field strength fluctuates only weakly. We ask what kind of topological terms can be added to the Lagrangian of such a weakly coupled gauge theory. In this paper, we systematically construct quantized topological terms which are generalization of the Chern-Simons terms and $F\wedge F$ terms, in space-time dimensions $d$ and for any gauge groups (continuous or discrete), using each element of the topological cohomology classes $H^{d+1}(BG,\Z)$ on the classifying space $BG$ of the gauge group $G$. In 3$d$ or for finite gauge groups above 3$d$, the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories. Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group $G$ are also classified by the same $H^{d+1}(BG,\Z)$. (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group $G$ and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.

arXiv:1211.2767Nov 12, 20125 citations

Gauge Theory and Geometric Phase in Quantum Systems

WORLD SCIENTIFIC eBooksAug 1, 20090 citations

Time-dependent gauge transformations and Berry's phase

Theoretical Physics

Annals of PhysicsSep 1, 19920 citations

Anomaly cascade in ( 2+1 )-dimensional fermionic topological phases

Daniel Bulmash, Maissam Barkeshli

We develop a theory of anomalies of fermionic topological phases of matter in ( $2+1$ )D with a general fermionic symmetry group ${G}_{f}$ . In general, ${G}_{f}$ can be a nontrivial central extension of the bosonic symmetry group ${G}_{b}$ by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the ${G}_{f}$ symmetry, which we dub the anomaly cascade: (i) An ${\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An ${\mathcal{H}}^{2}({G}_{b},kerr)$ obstruction to extending the ${G}_{b}$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An ${\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known ${\mathcal{H}}^{4}({G}_{b},U(1[))]$ obstruction to developing a consistent theory of ${G}_{b}$ symmetry defects for the fermion parity gauged theory. We describe how the ${\mathcal{H}}^{2}$ obstruction can be canceled by anomaly inflow from a bulk ( $3+1$ )D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.We develop a theory of anomalies of fermionic topological phases of matter in ($2+1$)D with a general fermionic symmetry group ${G}_{f}$. In general, ${G}_{f}$ can be a nontrivial central extension of the bosonic symmetry group ${G}_{b}$ by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the ${G}_{f}$ symmetry, which we dub the anomaly cascade: (i) An ${\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An ${\mathcal{H}}^{2}({G}_{b},kerr)$ obstruction to extending the ${G}_{b}$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An ${\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known ${\mathcal{H}}^{4}({G}_{b},U(1[))]$ obstruction to developing a consistent theory of ${G}_{b}$ symmetry defects for the fermion parity gauged theory. We describe how the ${\mathcal{H}}^{2}$ obstruction can be canceled by anomaly inflow from a bulk ($3+1$)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.

arXiv:2109.10922Physical review. B./Physical review. BApr 15, 202219 citations

F\wedge F

terms, in space-time dimensions

d

and for any gauge groups (continuous or discrete), using each element of the topological cohomology classes

H^{d+1}(BG,\Z)

on the classifying space

BG

of the gauge group

G

. In 3

d

or for finite gauge groups above 3

d

, the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories. Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group

G

are also classified by the same

H^{d+1}(BG,\Z)

. (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group

G

and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.

We develop a theory of anomalies of fermionic topological phases of matter in (

2+1

)D with a general fermionic symmetry group

{G}_{f}

. In general,

{G}_{f}

can be a nontrivial central extension of the bosonic symmetry group

{G}_{b}

by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the

{G}_{f}

symmetry, which we dub the anomaly cascade: (i) An

{\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})

obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An

{\mathcal{H}}^{2}({G}_{b},kerr)

obstruction to extending the

{G}_{b}

group structure of the symmetry permutations to the fermion parity gauged theory, where

r

is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An

{\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})

obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known

{\mathcal{H}}^{4}({G}_{b},U(1[))]

obstruction to developing a consistent theory of

{G}_{b}

symmetry defects for the fermion parity gauged theory. We describe how the

{\mathcal{H}}^{2}

obstruction can be canceled by anomaly inflow from a bulk (

3+1

)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.