In fall 2023, Harvard’s Thursday Seminar was on topological quantum field theories and physics.

Summary

In the seminar, we have been discussing finite gauge theory as a fully-extended TQFT. Today, we will apply our understanding of gauge theory to a famous lattice model in statistical mechanics: the Ising model. We follow Freed and Teleman:

Topological dualities in the Ising model

See also the talks (https://youtu.be/pfILFQDQw5s?feature=shared, https://youtu.be/I4KgqaPyTsw?feature=shared, https://youtu.be/OVabkt7Zfjk?feature=shared) given by the authors.

Let $G$ be a finite group. We have discussed finite $G$-gauge theory, its two extreme boundary conditions, its operators, and its duality with Turaev-Viro theory for the category $\text{Rep}(G)$. In this talk, we will discuss the analogous phases, operators, and duality for the Ising model.

• $\mathscr{F}_G$ finite $G$ gauge theory in 3d

• boundary theories $\mathscr{B}_H$

  • $e\subset G$ Dirichlet
  • $G\subset G$ Neumann

• line operators

  • Wilson lines $\chi\colon G\to U(1)$
  • ‘t Hooft lines $\gamma\subset G$

• electromagnetic duality

  • $\mathscr{F}A \leftrightarrow \mathscr{F}{A^\vee}$ for $A$ abelian
  • $\mathscr{F}_G\leftrightarrow$ Turaev-Viro for $\text{Rep}(G)$ in general

• $G$-Ising model in 2d

• lattice model at inverse temperature $\beta$

  • low temperature (ordered)
  • high temperature (disordered)

• operators

  • order operators
  • disorder operators

• Kramers-Wannier duality

  • $\beta\leftrightarrow\beta^\vee$ temperature exchange
  • new dual for nonabelian $G$ defined by LHS

The conjectured relationship between the two sides above is that the Ising lattice model in a certain phase has a low energy field theory given by a boundary theory $\mathscr{B}_H$ to finite gauge theory. Taking the endpoints of the line operators in gauge theory reproduces the point operators in the Ising model, and electromagnetic duality in 3d reproduces Kramers-Wannier duality on the 2d boundary.

The Ising Model in 2d Stat Mech, $G=\Z/2$

The data of the Ising model includes

• $(Y,\Gamma)$ a lattice 2-manifold; i.e. a pair of a compact 2-manifold $Y$ and a smoothly embedded finite graph $\Gamma\hookrightarrow Y$
• $G$ a group of spins, for now $G=\Z/2\ni ~\uparrow,\downarrow$
• $\beta\sim1/T$ inverse temperature
• $\sigma\colon \text{Vert}(\Gamma)\to G$ a spin configuration, so $\sigma\in\text{Maps}(\text{Vert}(\Gamma),G) = \mathcal{S}_{(Y,\Gamma)}$
• energy function (i.e. scalar-valued Hamiltonian) $E(\sigma)=-J\sum_{\langle i j\rangle} \sigma_i\sigma_j$
• partition function: $I_{(Y,\Gamma)}=\sum_{\sigma\in\mathcal{S}_{(Y,\Gamma)}} e^{-\beta E(\sigma)}$

In the energy function, $J$ is some positive coupling constant, which we will set to 1 for the rest of the talk. The notation means to sum over nearest-neighbor vertices (i.e. those connected by an edge) and $\sigma_i$ means the value of the spin configuration at site $i$.

The Ising model in 1d was a problem given to Ising by his advisor Lenz. They were interested in finding out whether local interactions could lead to large spin correlations, like those found in ferromagnets. Note that the energy function above wants neighboring spins to align—configurations with fewer spin misalignments are energetically favorable.

Ising solved this model and found that in 1d there was no phase transition. However, Peierls showed that there is a phase transition in 2d, and this persists in higher dimensions. The Ising model is one of the simplest models to feature a phase transition.