# Competing orders, non-linear sigma models, and topological terms in quantum magnets

https://arxiv.org/abs/cond-mat/0510459

# Anomalies of discrete symmetries in various dimensions and group cohomology

## Good Review Article on Anderson Localization

Perturbation theory approaches to Anderson and Many-Body
Localization: some lecture notes (https://arxiv.org/pdf/1710.01234.pdf) , and the reference therein.

Mathematica formulation of Anderson Localization: A short introduction to Anderson localizationhttps://faculty.math.illinois.edu/~dirk/preprints/localization3.pdf.

## GAP calculation

gap> G :=SpaceGroupBBNWZ(“Fd-3m”);
SpaceGroupOnRightBBNWZ( 3, 7, 5, 2, 4 )
gap> GroupCohomology(G,2);
[ 2, 2 ]
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,3);
[ 2, 2, 2 ]
gap> GroupCohomology(G,3,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,4);
[ 2, 2, 2, 2, 2, 12 ]

gap> GroupCohomology(G,4,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

gap> GroupCohomology(G,5);
[ 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,5,2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

gap> G :=SpaceGroupBBNWZ(“Fd-3c”);
SpaceGroupOnRightBBNWZ( 3, 7, 5, 2, 3 )
gap> GroupCohomology(G,2);
[ 2, 2 ]
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2 ]
gap> GroupCohomology(G,3);
[ 2, 2 ]
gap> GroupCohomology(G,3,2);
[ 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,4);
[ 2, 2, 12 ]
gap> GroupCohomology(G,4,2);
[ 2, 2, 2, 2 ]

gap> G := SpaceGroupBBNWZ(“R-3m”);
SpaceGroupOnRightBBNWZ( 3, 5, 5, 1, 1 )
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,2);
[ 2, 2, 2 ]
gap> G := SpaceGroupBBNWZ(“P4132”);
SpaceGroupOnRightBBNWZ( 3, 7, 3, 1, 2 )
gap> GroupCohomology(G,2,2);
[ 2 ]
gap> GroupCohomology(G,2);
[ 2 ]
gap> G := SpaceGroupBBNWZ(“P4332”);
SpaceGroupOnRightBBNWZ( 3, 7, 3, 1, 2 )
gap> GroupCohomology(G,2,2);
[ 2 ]
gap> GroupCohomology(G,2);
[ 2 ]
gap> G := SpaceGroupBBNWZ(“Fm-3m”);
SpaceGroupOnRightBBNWZ( 3, 7, 5, 2, 1 )
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2, 2, 2 ]
gap> GroupCohomology(G,2);
[ 2, 2 ]

gap> G := SpaceGroupBBNWZ(“F-43m”);
SpaceGroupOnRightBBNWZ( 3, 7, 4, 2, 1 )
gap> GroupCohomology(G,2,2);
[ 2, 2, 2, 2 ]
gap> GroupCohomology(G,2);
[ 2 ]

## Good book on field theory

It is Fields and Rings by Kaplansky!

it seems to be a small pamphlet but is concisely and very well composed. Only about 70 pages on Field Theory but almost all major results are there. What’s more, the exercises there are so good that I hav seen many people recommending them, including our Prof. Jacob who puts quite a few in his problem sets. The book is also what Prof. Jacob covered in his first eight week lectures.

I think the Ring Theory part, which is unusually put after Field Theory in the book, is also excellent lecture note. Anyway,  Kaplansky is a big guy in Algebra!

There may be other good books on Field Theory, such as Dummit’s and Rotman’s.

## Talk by Dominic

https://arxiv.org/abs/1612.00846

Topological cystalline insulator (Liang Fu, 2011)

Interacting SPTs/SETs

Internal symmetry (charge conservation, Z^T_2, etc.)

Bosonic SPTs: Group cohomology classification $H^{n+1}(G,U(1))$ [Chen et. al. 2011]

Bosonic SETs $G$-crossed braided tensor categories [Barkeshli, et.al,2014]

0) How do you gauge internal symmetry?

1. How do you”gauge”spatial symmetry”?
2. Classification of phases ($U(1)$) of phases)

If we have a U(1) symmetry $\vec{A}$,

$S_{SPT} \rightarrow S_{SPT}[\vec{A}]\xrightarrow{integrate out SPT} \frac{n}{2\pi} \int \varepsilon_{\mu\nu\lambda} A_\mu \partial_\nu A_\lambda$

Hall conductance, integer

$H^3(U(1),U(1))=\mathbb{Z}$

$S_{SPT}\xrightarrow{gauge}S_{SPT}[A]\xrightarrow{integrate out SPT}$Topological term, gauge field

[Dijgraaf Witten,1991] $H^{d+1}(G,U(1))$

What is a gauge field: it is a connection on a principal $G$-bundle. A physical picture of this: branch cuts, across which a particle state $|\psi\rangle$ becomes $g|\psi\rangle$, where $g$ is element of the group. (In case of $\mathbb{R}^n$, the group is GL). Branch cuts always have codim=1.

Examples (pass)

“Topological crystalline liquid”: topological part of it (TQFT) does not care about lattice. (liquid in the sense that it doesn’t care about the lattice)

E.g. $G=1$, $X=\mathbb{R}^2$, $M = \mathbb{R}^2$, $f(z) = z^2$,

Gauge field: $M\rightarrow BG$, Classifying space: $BG=EG/G$

“Crystalline gauge field” $M\rightarrow X//G=(X\times EG)/G$

$H^{d+1}(BG,U(1))=H^{d+1}(G,U(1))$, $H^{d+1}(X//G,U(1))$: equivariant cohomology (crystalline SPTs)