====== Ising model ======

{{tag>3d}}

The Ising model is an interacting CFT in three dimensions. For related models in other dimensions, see [[#other dimensions]]. The Ising model is the $N=1$ case of the [[on_model]].

The Ising CFT also goes by the name "Wilson-Fisher fixed point" (in any $d < 4$). Another terminological difficulty: "Ising model" can mean either the lattice Ising model (discussed below) or the CFT itself. Even when the lattice model is tuned to criticality, these are different objects. To disambiguate, one can refer to the "Ising CFT" or the "critical Ising model".

===== Operator content =====

The Ising CFT has two relevant operators, conventionally denoted $\sigma$ and $\epsilon$. Their scaling dimensions are given by
$$
\Delta_\sigma = 0.5181489(10)\text{ and }
\Delta_\epsilon = 1.412625(10)\text.
$$
The nonvanishing [[operator product expansion|OPE]] coefficients are
$$
\lambda_{\sigma\sigma\epsilon} = 1.0518537(41)\text{ and }
\lambda_{\epsilon\epsilon\epsilon} = 1.532435(19)
$$

All above data are from [[https://arxiv.org/pdf/1603.04436.pdf]].

Both relevant operators have spin $l=0$. The stress tensor of course has $l=2$ and $\Delta = 3$. The lowest-lying irrelevant operator is $\epsilon'$ (also $l=0$), with $\Delta_{\epsilon'} \approx 3.8$.
===== Lattice model =====

The lattice Ising model is usually defined on a three-dimensional rectangular lattice. A configuration labels site $i$ by a spin $s_i$. The action is given by
\[
S = - J \sum_{\langle i j \rangle} s_i s_j
\]
and the partition function is of course the sum over all possible spin configurations $Z = \sum_s e^{-S(s)}$.

On this lattice, there is a critical J ($J_c \approx 2.7$) at which a second-order phase transition appears. This lattice model is not, of course, itself a CFT. Rather, once the coupling has been tuned to $J_c$, it flows to the Ising CFT in the IR. It is not unique in this respect; see for example [[#scalar field theory]].

==== Other lattice geometries ====

An Ising model can of course be defined on any graph.  There are many lattices with 3d translational symmetries (although discrete) but different local structure. Typically, an Ising model on these lattices will again have a critical point that flows to the Ising CFT in the IR.

==== Hamiltonian system ====

One option is to keep the same graph, but change the geometry. This yields the anisotropic Ising model.
\[
S = \sum_r \left[ J_x s_r s_{r+\hat x} + J_y s_r s_{r+\hat y} + J_z s_r s_{r+\hat z} \right]
\]
The case where $J_y = J_z \gg J_x$ is termed the Hamiltonian limit, as it is connected by the Suzuki-Trotter expansion (and the [[transfer matrix|transfer matrix]]) to a quantum mechanical system, usually termed the "transverse-field Ising model".
\[
H = -\mu \sum_i \sigma_x(i) -J \sum_{\langle i j \rangle} \sigma_z(i) \sigma_z(j)
\]
Note that the sum now goes over all sites on a two-dimensional "spatial" lattice. This model has a critical point at $\mu / J \sim 3$.

===== Scalar field theory =====

Scalar field theory, in the continuum, exhibits a second-order transition in the $\phi^4$ coupling. At this transition, the theory flows to the Ising CFT in the IR. This is true in both two and three dimensions. In 2d, see for instance [[https://arxiv.org/abs/0902.0045|0902.0045]] for a lattice measurement of the critical coupling.

On the lattice, there's a second (essentially unrelated) way to extract the Ising model from scalar field theory. Taking $m^2\rightarrow -\infty$ and $\lambda\rightarrow\infty$ simultaneously, two deep and narrow potential wells are created, and the fields are fixed to sit in one of these wells. In the limit, this recovers the lattice Ising model, with a coupling set by $\lambda/m^2$. Tuning that lattice Ising coupling yields the Ising CFT.


===== Liquid-gas transtiion =====

The liquid-gas transition in 3d is generally believed to belong to the Ising universality class.  See for instance [[https://arxiv.org/pdf/1202.5369.pdf|this]] determination of the critical exponents via molecular dynamics simulation.

===== Computational methods =====

==== Epsilon expansion ====

[[https://arxiv.org/pdf/cond-mat/9803240.pdf]]
==== Lattice simulations ====

The standard [[https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm|Metropolis-Hastings algorithm]] works for the Ising model. HMC is not available due to the discrete degrees of freedom of the standard lattice model.

Near the quantum phase transition one encounters "critical slowing down". This is not exponentially bad, but still inconvenient, so it's better to use the [[https://doi.org/10.1103/PhysRevLett.58.86|Swendsen-Wang algorithm]] ([[https://en.wikipedia.org/wiki/Swendsen%E2%80%93Wang_algorithm|Wikipedia]]) for serious work. Note that that algorithm can be generalized to speed the mixing of scalar field theory as well.
==== Renormalization group ====

[[https://arxiv.org/abs/cond-mat/9803240]]
==== Conformal bootstrap ====

See [[https://arxiv.org/pdf/2007.14315.pdf|this review]] by Slava Rychkov.

See also [[https://arxiv.org/pdf/1203.6064.pdf]] and [[https://arxiv.org/pdf/1403.4545.pdf]] and [[https://arxiv.org/pdf/1603.04436.pdf]].
===== Physical realizations =====

As mentioned [[#liquid-gas transition|above]], some properties of the liquid-gas transition are believed to be described by the Ising model.

[[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.35.4823]]

It is sometimes conjectured that the thermodynamic behavior of the critical point of QCD is described by the Ising model.
===== Other dimensions =====
Lattice Ising models can be defined in any number of dimensions. The critical point of such a theory is referred to as an Ising CFT in that number of dimensions. Only in three dimensions does an interacting theory result. The other possibilities are described below.

==== One dimension ====

As usual, in one dimension the lattice Ising model is exactly solvable. This is most easily done through the [[transfer matrix|transfer matrix]]. The corresponding quantum Hamiltonian is a two-level system.

==== Two dimensions ====

The two-dimensional lattice Ising model was solved exactly by Onsager.

The Hamiltonian limit of this model is a spin chain (referred to as the "transverse Ising model"):
\[
H = -\mu \sum_r \sigma_x(r) - J \sum_r \sigma_z(r) \sigma_z(r+1)\text.
\]
This model can be transformed to a quadratic theory of fermion fields, and thence solved, via a Jordan-Wigner transformation.
==== Higher dimensions ====

Critical points of lattice $\phi^4$ and Ising models are believed to be trivial in $d \ge 4$ dimensions.

In $d > 4$ dimensions this was [[https://projecteuclid.org/journalArticle/Download?urlid=cmp%2F1103921614|proven]] by Michael Aizenman.

==== Fractional dimensions ====

[[https://arxiv.org/abs/cond-mat/9802018]]

[[https://hal.archives-ouvertes.fr/jpa-00210418/document]]
===== External links =====
  * [[https://arxiv.org/abs/0902.0045|An improved lattice measurement of the critical coupling in $\phi^4_2$ theory]] (Schaich)
  * [[https://projecteuclid.org/journalArticle/Download?urlid=cmp%2F1103921614|Geometric analysis of $\phi^4$ fields and Ising models]] (Aizenman)