Z₂ Gauge Systems, Wilson Loops, and Free Correlators

The previous page ended with the geometric reason why the three-dimensional Ising model cannot be self-dual in the same way as the two-dimensional Ising model. In two dimensions, domain walls are closed curves, and closed curves can also be produced by the high-temperature expansion of an ordinary spin model. In three dimensions, domain walls are closed surfaces. To get closed surfaces from a high-temperature expansion, the variables must live on links and the elementary interaction must live on plaquettes. That is the first appearance in these notes of a lattice gauge theory.

This page develops that idea in its simplest form: a $\mathbb Z_2$ gauge system. Its local variables are signs on links, its gauge-invariant local field strength is a plaquette product, and its most important extended observable is a Wilson loop. The Wilson loop is the gauge-theory analog of the disorder line in the two-dimensional Ising model: it is not built from a local order parameter, but from a defect supported on a curve.

The last part of the page turns back to the two-dimensional critical theory and records the elementary free correlators that will be used repeatedly in the coming conformal-field-theory pages. This juxtaposition is deliberate. A field theory is not only a Lagrangian. It is also a list of observables and their singularities.

Required background. Lesson 10 supplies the three-dimensional Ising/$\mathbb Z_2$-gauge duality and the continuum Majorana limit used here.

Helpful background. Lesson 7 develops order–disorder defects, while Lesson 8 explains how their branch cuts produce Ising fermions.

From closed surfaces to gauge links

Z₂ links and free-field normalizations

For the $\mathbb Z_2$ gauge model, link variables are

$$M_{x,\mu}=\pm1, \qquad \mu=1,2,3,$$

on the oriented link from $x$ to $x+\hat\mu$. Since $M^{-1}=M$, orientation signs are invisible in the $\mathbb Z_2$ formulas. The plaquette variable in the $\mu\nu$ plane is

$$M_{x,\mu\nu} = M_{x,\mu}M_{x+\hat\mu,\nu}M_{x+\hat\nu,\mu}M_{x,\nu}.$$

The gauge partition function is

$$Z_g(K_g)=\sum_{\{M_\ell=\pm1\}} \exp\left(K_g\sum_p M_p\right).$$

The duality relation to the three-dimensional nearest-neighbor Ising model is written, up to nonsingular normalization factors and boundary-sector subtleties, as

$$e^{-2K_\sigma}=\tanh K_g.$$

For two-dimensional continuum correlators, use

$$z=x+iy, \qquad \bar z=x-iy, \qquad \partial={1\over2}(\partial_x-i\partial_y), \qquad \bar\partial={1\over2}(\partial_x+i\partial_y),$$

with

$$\bar\partial {1\over z}=\pi\delta^{(2)}(z).$$

Overall constants in CFT two-point functions are conventional. We choose the common normalization

$$\langle \psi(z)\psi(w)\rangle={1\over z-w}, \qquad \langle \phi(z,\bar z)\phi(w,\bar w)\rangle =-\log\!\big(\mu^2|z-w|^2\big).$$

Here $\mu$ is an arbitrary inverse-length scale. Changing $\mu$ only adds a constant to the massless-scalar Green function, so correlators of derivatives and neutral vertex operators are unaffected.

The $\mathbb Z_2$ gauge model is obtained by putting an Ising-valued variable on every link rather than every site. Its elementary gauge-invariant object is not a nearest-neighbor product $\sigma_x\sigma_{x+\hat\mu}$, but a plaquette product

$$M_p=\prod_{\ell\in\partial p}M_\ell.$$

On a cubic lattice this is the product of the four link variables around an elementary square. The action rewards plaquettes with $M_p=+1$:

$$S_g[M]=-K_g\sum_p M_p, \qquad Z_g(K_g)=\sum_{\{M_\ell\}}e^{K_g\sum_p M_p}.$$

The high-temperature expansion is almost identical in spirit to the Ising high-temperature expansion, but one dimension higher geometrically. For each plaquette,

$$e^{K_gM_p}=\cosh K_g\left(1+t_g M_p\right), \qquad t_g=\tanh K_g.$$

Expanding the product over plaquettes selects a set $S$ of plaquettes. The contribution of $S$ contains

$$\prod_{p\in S}M_p =\prod_{p\in S}\prod_{\ell\in\partial p}M_\ell.$$

When we sum over a link variable $M_\ell=\pm1$, the answer vanishes unless $M_\ell$ appears an even number of times. Therefore a nonzero contribution requires every link to be contained in an even number of selected plaquettes. This is precisely the condition that the selected plaquettes form a closed surface:

$$\partial S=0.$$

Thus

$$Z_g(K_g) \propto \sum_{S:\,\partial S=0} (\tanh K_g)^{A(S)},$$

where $A(S)$ is the number of plaquettes in the surface. This matches the low-temperature expansion of the three-dimensional Ising model,

$$Z_\sigma^{\mathrm{low}}(K_\sigma) \propto \sum_{S:\,\partial S=0}e^{-2K_\sigma A(S)},$$

provided

$$\boxed{e^{-2K_\sigma}=\tanh K_g.}$$

The slogan is useful, but the derivation is better: the duality works because both sides sum over the same closed surfaces. The Ising model produces those surfaces as domain walls; the gauge theory produces them as plaquette excitations.

A $\mathbb Z_2$ gauge field assigns a sign $M_{x,\mu}$ to each link. The plaquette product is invariant under the local change $M_{x,\mu}\mapsto\eta_xM_{x,\mu}\eta_{x+\hat\mu}$, because every vertex sign appears twice around the plaquette.

Gauge redundancy is not an ordinary symmetry

The local transformation

$$M_{x,\mu}\mapsto \eta_xM_{x,\mu}\eta_{x+\hat\mu}, \qquad \eta_x=\pm1,$$

is a gauge redundancy. The plaquette variable is invariant because each $\eta_x$ appearing at a corner of the plaquette appears twice:

$$M_{x,\mu\nu}\mapsto M_{x,\mu\nu}.$$

This is different from a global Ising symmetry $\sigma_x\mapsto-\sigma_x$. A global symmetry can be spontaneously broken and can have a local order parameter. A gauge redundancy is a many-to-one description of the same physical configuration. Gauge-dependent quantities, such as a single link expectation value $\langle M_{x,\mu}\rangle$, are not physical observables. In a gauge-invariant formulation they vanish unless a gauge is fixed, and even after gauge fixing their value is not a direct diagnostic of a physical phase.

The local gauge-invariant objects are products around closed boundaries. The smallest one is the plaquette $M_p$. Larger ones are Wilson loops.

The fact that the dual of the three-dimensional Ising model is a gauge theory is already conceptually important. It says that a system with a perfectly ordinary local order parameter can be equivalent, after duality, to a system whose natural probes are nonlocal loop observables. Duality trades one notion of locality for another.

U(1) links and the continuum curl

The $\mathbb Z_2$ gauge system is the discrete cousin of a compact $U(1)$ lattice gauge theory. Replace each sign by a phase,

$$U_{x,\mu}=e^{i\theta_{x,\mu}}, \qquad \theta_{x,\mu}\sim\theta_{x,\mu}+2\pi.$$

When there is a smooth continuum gauge potential, one writes approximately

$$U_{x,\mu}=\exp\left(i a A_\mu\left(x+{a\over2}\hat\mu\right)\right),$$

where $a$ is the lattice spacing. The oriented plaquette product is

$$U_{x,\mu\nu} =U_{x,\mu}U_{x+\hat\mu,\nu}U^{-1}_{x+\hat\nu,\mu}U^{-1}_{x,\nu}.$$

Taking logarithms and expanding at small $a$ gives

$$\log U_{x,\mu\nu} =i\left[\theta_{x,\mu}+\theta_{x+\hat\mu,\nu}-\theta_{x+\hat\nu,\mu}-\theta_{x,\nu}\right] =i a^2 F_{\mu\nu}(x)+O(a^3),$$

with

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

Thus the plaquette is a small Wilson loop measuring the flux through an elementary square:

$$U_{x,\mu\nu}=\exp\left(i a^2F_{\mu\nu}+O(a^3)\right).$$

A compact Wilson action has the schematic form

$$S_{U(1)}=-K\sum_p\cos\theta_p, \qquad U_p=e^{i\theta_p}.$$

For weak fields,

$$1-\cos\theta_p={\theta_p^2\over2}+O(\theta_p^4) ={a^4\over2}F_{\mu\nu}^2+O(a^5),$$

so the lattice action becomes the Maxwell action in the continuum limit after the usual normalization of $K$ with $a$.

More explicitly, in $d$ dimensions the nonconstant part of the action is

$$S_{U(1)}-S_0 = {K a^{4-d}\over4} \int d^d x\,F_{\mu\nu}F_{\mu\nu}+\cdots.$$

Thus the standard normalization $(4g^2)^{-1}\int F^2$ is obtained by holding $K a^{4-d}=g^{-2}$ fixed in the smooth-field continuum limit. Compact configurations with order-one plaquette angles are not described by this Taylor expansion.

For a compact $U(1)$ link variable, the oriented product around a plaquette is the exponential of the discrete curl. At long wavelength it becomes $e^{ia^2F_{\mu\nu}}$, and the Wilson plaquette action reduces to the Maxwell term.

The $\mathbb Z_2$ model keeps only the signs $\pm1$ rather than a continuous phase. But the structural lesson is the same: a gauge field is naturally integrated along links, and its curvature is naturally integrated over plaquettes.

Wilson loops

For a closed lattice contour $C$, define the $\mathbb Z_2$ Wilson loop

$$\boxed{ W(C)=\prod_{\ell\in C}M_\ell. }$$

It is gauge invariant because the gauge signs at every vertex on the closed contour cancel pairwise. If the path were open, the product would transform at its two endpoints and would not be a gauge-invariant observable by itself.

For a compact $U(1)$ theory the Wilson loop is the holonomy

$$W_q(C)=\exp\left(iq\sum_{\ell\in C}\theta_\ell\right) \longrightarrow \exp\left(iq\oint_C A_\mu dx^\mu\right),$$

where $q$ is the charge of the probe. The $\mathbb Z_2$ Wilson loop is the same idea with only two possible phases.

In a compact $U(1)$ theory normalized so that the fundamental electric charge is one, $q$ is an integer. This ensures that the loop is unchanged when a link angle is shifted by $2\pi$.

The strong-coupling expansion of $\langle W(C)\rangle$ is one of the cleanest calculations in lattice gauge theory. Insert $W(C)$ into the high-temperature expansion:

$$\langle W(C)\rangle ={1\over Z_g} \sum_{\{M_\ell\}}\left(\prod_{\ell\in C}M_\ell\right) \prod_p \cosh K_g\left(1+t_gM_p\right).$$

After expanding in plaquettes, a link $\ell$ on the contour $C$ appears once from $W(C)$ and then as many times as selected plaquettes touch it. The sum over $M_\ell$ is nonzero only if the total power is even. Therefore the selected plaquettes no longer form a closed surface. They form a surface whose boundary is $C$:

$$\partial S=C.$$

Hence

$$\langle W(C)\rangle = {\sum_{S:\,\partial S=C} t_g^{A(S)}\over \sum_{S:\,\partial S=0}t_g^{A(S)}}.$$

At strong coupling, $t_g=\tanh K_g\ll1$, the leading contribution is the minimal-area surface spanning $C$:

$$\boxed{ \langle W(C)\rangle\sim (\tanh K_g)^{A_{\min}(C)} =\exp[-\sigma A_{\min}(C)], \qquad \sigma=-\log\tanh K_g+\cdots. }$$

This is an area law. In gauge-theory language, an area law for large Wilson loops is the hallmark of a confining phase for external test charges. Later in the course, when compact gauge fields, monopoles, and confinement reappear, this simple $\mathbb Z_2$ calculation will be the toy model to keep in mind.

In the strong-coupling expansion of a $\mathbb Z_2$ gauge theory, a Wilson-loop insertion changes the plaquette constraint from $\partial S=0$ to $\partial S=C$. The leading contribution is the smallest spanning surface, giving an area law.

Wilson loops as dual disorder loops

Under the three-dimensional Ising/gauge duality, the Wilson loop has a spin-system interpretation as a disorder loop. Choose an arbitrary surface $\Sigma$ in the spin system whose boundary is $C$. Define a defect by flipping the sign of all Ising bonds crossing $\Sigma$:

$$K_\sigma\sigma_x\sigma_y \longmapsto - K_\sigma\sigma_x\sigma_y \qquad \text{for bonds crossing }\Sigma.$$

The expectation value of this defect is a ratio of partition functions. If we deform $\Sigma$ without changing its boundary, the change can be undone by flipping spins in the region swept out by the deformation. On a simply connected lattice, and away from other insertions, only the boundary $C=\partial\Sigma$ is therefore physical. Nontrivial topology can leave additional global sector data. This is exactly the higher-dimensional analog of the Kadanoff–Ceva disorder line in the two-dimensional Ising model: the line or surface used to define the defect is a convention, while its endpoint or boundary is an observable insertion.

In the dual Ising description, a Wilson loop is represented by flipping bonds crossing a surface $\Sigma$. Moving $\Sigma$ is a change of variables; the invariant information is the boundary curve $C=\partial\Sigma$.

This construction also explains why it is often misleading to ask for “the” local order parameter of a gauge theory. Some phases are best diagnosed by extended probes. In the $\mathbb Z_2$ gauge theory, the large-loop behavior of $\langle W(C)\rangle$ distinguishes the strong-coupling area-law phase from the weak-coupling perimeter-law phase. In the dual spin language, the same transition is the ordinary Ising transition.

Free chiral correlators

The notes now shift from lattice duality back to the continuum fixed point of the two-dimensional Ising model. The reason is that the next step in the course is conformal invariance. For that, we need the singular behavior of free correlators.

At criticality, the Ising fermion splits into two chiral Majorana fields. In Euclidean coordinates, write them as

$$\psi(z), \qquad \bar\psi(\bar z).$$

Their massless equations of motion are

$$\bar\partial\psi=0, \qquad \partial\bar\psi=0,$$

away from operator insertions. The Green function for $\bar\partial$ is $1/z$, in the distributional sense

$$\bar\partial_z {1\over z-w}=\pi\delta^{(2)}(z-w).$$

With the normalization chosen above,

$$\boxed{ \langle \psi(z)\psi(w)\rangle={1\over z-w}, \qquad \langle \bar\psi(\bar z)\bar\psi(\bar w)\rangle={1\over \bar z-\bar w}, \qquad \langle\psi(z)\bar\psi(\bar w)\rangle=0. }$$

These formulas are the CFT version of the statement that the critical Ising fermion is free. The pole $1/(z-w)$ says that $\psi$ has holomorphic scaling weight $h=1/2$, while $\bar\psi$ has antiholomorphic scaling weight $\bar h=1/2$.

There is a Lorentzian version of the same singularity. Let

$$x^+=t+x, \qquad x^-=t-x.$$

In one common analytic-continuation convention, the time-ordered chiral pole is the boundary value

$${1\over x^+ + i0\,\operatorname{sgn}x^-} =\operatorname{PV}{1\over x^+}-i\pi\operatorname{sgn}(x^-)\delta(x^+).$$

This compact formula records two facts at once: the field is singular on a light ray, and the sign of the infinitesimal imaginary part remembers the time ordering. The Euclidean correlator is obtained by analytic continuation.

The elementary two-dimensional Green function is the pole $1/(z-w)$, with $\bar\partial(1/(z-w))=\pi\delta^{(2)}(z-w)$. The critical Majorana two-point functions have simple chiral poles, while the Ising energy field $\epsilon\sim i\psi\bar\psi$ has a two-point function proportional to $|z-w|^{-2}$.

Free boson and energy correlators

The massless scalar field in two Euclidean dimensions is another basic conformal field. With a standard normalization,

$$\langle \phi(z,\bar z)\phi(w,\bar w)\rangle =-\log\!\big(\mu^2|z-w|^2\big).$$

The arbitrary scale $\mu$ reflects the scalar zero-mode ambiguity; changing it shifts the correlator by a constant. The logarithm is the Green function of the two-dimensional Laplacian:

$$\partial\bar\partial \log |z|^2 =\pi\delta^{(2)}(z).$$

Because the scalar itself has a logarithmic two-point function, it is better to regard its derivatives and vertex operators as the primary local observables. For example,

$$\langle \partial\phi(z)\partial\phi(w)\rangle =-{1\over (z-w)^2},$$

up to the sign fixed by the normalization of $\phi$.

For the Ising model, the energy-density operator is the fermion bilinear. With a conventional normalization,

$$\epsilon(z,\bar z)\sim i\psi(z)\bar\psi(\bar z),$$

where the factor of $i$ is convention dependent and may be absorbed into the normalization of the Euclidean fields.

Using Wick contraction,

$$\langle \epsilon(z,\bar z)\epsilon(w,\bar w)\rangle \propto \langle \psi(z)\psi(w)\rangle \langle \bar\psi(\bar z)\bar\psi(\bar w)\rangle,$$

so

$$\boxed{ \langle \epsilon(z,\bar z)\epsilon(w,\bar w)\rangle \propto {1\over |z-w|^2}. }$$

Thus the energy field has scaling dimension

$$\Delta_\epsilon=h_\epsilon+\bar h_\epsilon=1, \qquad h_\epsilon=\bar h_\epsilon={1\over2}.$$

This should be compared with the spin field $\sigma$, whose two-point function at the critical point behaves as

$$\langle \sigma(z,\bar z)\sigma(w,\bar w)\rangle \propto {1\over |z-w|^{1/4}}.$$

The spin field therefore has scaling dimension $\Delta_\sigma=1/8$. Unlike the energy operator, it is not a local polynomial in the free fermion. It is a twist field for the fermion. That fact is the continuum echo of the order–disorder branch cut discussed earlier.

Summary

The three-dimensional Ising model is dual to a $\mathbb Z_2$ gauge theory because both theories have expansions in terms of closed surfaces. In the spin model those surfaces are domain walls. In the gauge model they are selected plaquettes in the high-temperature expansion.

The gauge variables live on links, and the elementary gauge-invariant field strength is the plaquette product. The local transformation $M_{x,\mu}\mapsto\eta_xM_{x,\mu}\eta_{x+\hat\mu}$ is a redundancy, not an ordinary global symmetry. The natural extended observable is the Wilson loop $W(C)=\prod_{\ell\in C}M_\ell$.

In the strong-coupling expansion, inserting $W(C)$ forces selected plaquettes to form a surface with boundary $C$, giving an area law at leading order. In the dual Ising model this Wilson loop is a disorder loop, represented by flipping bonds across a surface whose boundary is $C$.

At the two-dimensional critical Ising point, the continuum fermions are free chiral Majorana fields with two-point functions $1/(z-w)$ and $1/(\bar z-\bar w)$. The energy field is the bilinear $\epsilon\sim i\psi\bar\psi$ and has correlator $|z-w|^{-2}$. The spin field has dimension $1/8$ and is a twist field rather than a local fermion bilinear.

Common pitfalls

Gauge redundancy is not symmetry breaking. A gauge transformation is not a physical operation relating different states. It is a redundancy in the variables used to describe one state. Gauge-invariant quantities are closed products such as plaquettes and Wilson loops.

Orientation matters beyond $\mathbb Z_2$. In the $\mathbb Z_2$ theory, link orientation looks irrelevant because $M^{-1}=M$. In $U(1)$ or non-Abelian gauge theory, orientation is essential: reversing a link takes the inverse group element.

The coupling map is not the whole finite-volume identity. The duality relation $e^{-2K_\sigma}=\tanh K_g$ matches singular physics and surface weights, but exact finite-volume partition functions also contain normalization factors and possible topological sectors.

A Wilson loop is an extended probe. It is not the same as a local order parameter. Its large-loop behavior, area law versus perimeter law, diagnoses the gauge phase.

The chiral pole is a distribution. The expression $1/(z-w)$ is not an ordinary function at $z=w$. The contact term in $\bar\partial(1/(z-w))$ is what makes it the inverse of the chiral kinetic operator.

Exercises

Exercise 1: Plaquette gauge invariance

Show explicitly that the $\mathbb Z_2$ plaquette product

$$M_{x,\mu\nu} = M_{x,\mu}M_{x+\hat\mu,\nu}M_{x+\hat\nu,\mu}M_{x,\nu}$$

is invariant under

$$M_{x,\mu}\mapsto \eta_xM_{x,\mu}\eta_{x+\hat\mu}, \qquad \eta_x=\pm1.$$

Solution

Each link in the plaquette transforms as

$$M_{x,\mu}\mapsto \eta_xM_{x,\mu}\eta_{x+\hat\mu},$$ $$M_{x+\hat\mu,\nu}\mapsto \eta_{x+\hat\mu}M_{x+\hat\mu,\nu}\eta_{x+\hat\mu+\hat\nu},$$ $$M_{x+\hat\nu,\mu}\mapsto \eta_{x+\hat\nu}M_{x+\hat\nu,\mu}\eta_{x+\hat\mu+\hat\nu},$$

and

$$M_{x,\nu}\mapsto \eta_xM_{x,\nu}\eta_{x+\hat\nu}.$$

Multiplying all four transformed links gives the original product times

$$\eta_x^2\eta_{x+\hat\mu}^2\eta_{x+\hat\nu}^2\eta_{x+\hat\mu+\hat\nu}^2.$$

Since each $\eta$ is $\pm1$, every square is one. Therefore

$$M_{x,\mu\nu}\mapsto M_{x,\mu\nu}.$$

Exercise 2: Closed surfaces from link sums

Derive the closed-surface constraint in the high-temperature expansion of the $\mathbb Z_2$ gauge partition function

$$Z_g(K_g)=\sum_{\{M_\ell=\pm1\}} \exp\left(K_g\sum_pM_p\right).$$

Solution

Use

$$e^{K_gM_p}=\cosh K_g(1+t_gM_p), \qquad t_g=\tanh K_g.$$

Then

$$Z_g(K_g)=(\cosh K_g)^{N_p} \sum_{\{M_\ell\}} \prod_p(1+t_gM_p).$$

Expanding the product over plaquettes chooses a subset $S$ of plaquettes:

$$\prod_p(1+t_gM_p) =\sum_S t_g^{A(S)}\prod_{p\in S}M_p.$$

Now

$$\prod_{p\in S}M_p =\prod_\ell M_\ell^{n_\ell(S)},$$

where $n_\ell(S)$ is the number of selected plaquettes containing the link $\ell$. The sum over $M_\ell=\pm1$ vanishes unless $n_\ell(S)$ is even. Therefore every link must be touched by an even number of selected plaquettes. This is the condition that $S$ has no boundary:

$$\partial S=0.$$

Thus, up to the overall factor from the link sums and $(\cosh K_g)^{N_p}$,

$$Z_g(K_g)\propto \sum_{S:\partial S=0}t_g^{A(S)}.$$

Exercise 3: Wilson-loop boundary condition

Repeat the previous exercise with a Wilson-loop insertion and show that selected plaquettes must obey $\partial S=C$.

Solution

The numerator of $\langle W(C)\rangle$ is

$$\sum_{\{M_\ell\}}\left(\prod_{\ell\in C}M_\ell\right) \prod_p\cosh K_g(1+t_gM_p).$$

After expanding in plaquettes, a selected surface $S$ contributes

$$t_g^{A(S)} \prod_\ell M_\ell^{n_\ell(S)+\chi_C(\ell)},$$

where $\chi_C(\ell)=1$ if $\ell$ lies on $C$ and $0$ otherwise. The sum over $M_\ell$ is nonzero only if

$$n_\ell(S)+\chi_C(\ell)$$

is even for every link. Hence $n_\ell(S)$ is odd on links of $C$ and even elsewhere. This is exactly the statement that the boundary of the selected plaquette surface is $C$:

$$\partial S=C.$$

The leading strong-coupling contribution is the smallest such surface, so

$$\langle W(C)\rangle\sim t_g^{A_{\min}(C)}.$$

Exercise 4: Continuum curvature from a plaquette

For compact $U(1)$ link variables

$$U_{x,\mu}=\exp\left(i a A_\mu\left(x+{a\over2}\hat\mu\right)\right),$$

show that the oriented plaquette product satisfies

$$U_{x,\mu\nu}=\exp\left(i a^2F_{\mu\nu}(x)+O(a^3)\right).$$

Solution

The plaquette product is

$$U_{x,\mu\nu} =U_{x,\mu}U_{x+\hat\mu,\nu}U^{-1}_{x+\hat\nu,\mu}U^{-1}_{x,\nu}.$$

Taking the logarithm gives

$$\log U_{x,\mu\nu} =i a\left[ A_\mu\left(x+{a\over2}\hat\mu\right) +A_\nu\left(x+a\hat\mu+{a\over2}\hat\nu\right) -A_\mu\left(x+a\hat\nu+{a\over2}\hat\mu\right) -A_\nu\left(x+{a\over2}\hat\nu\right) \right].$$

Group the two $A_\nu$ terms and the two $A_\mu$ terms. Taylor expansion at their respective link midpoints gives

$$A_\nu\left(x+a\hat\mu+{a\over2}\hat\nu\right) -A_\nu\left(x+{a\over2}\hat\nu\right) =a\partial_\mu A_\nu(x)+O(a^2),$$

and

$$A_\mu\left(x+a\hat\nu+{a\over2}\hat\mu\right) -A_\mu\left(x+{a\over2}\hat\mu\right) =a\partial_\nu A_\mu(x)+O(a^2).$$

Shifting the derivative evaluation point from a link midpoint to $x$ changes only the displayed $O(a^2)$ remainders.

Therefore

$$\log U_{x,\mu\nu} =i a^2(\partial_\mu A_\nu-\partial_\nu A_\mu)+O(a^3) =i a^2F_{\mu\nu}+O(a^3).$$

Exponentiating gives the desired result.

Exercise 5: Scaling dimension of the Ising energy field

Using

$$\langle\psi(z)\psi(w)\rangle={1\over z-w}, \qquad \langle\bar\psi(\bar z)\bar\psi(\bar w)\rangle={1\over \bar z-\bar w},$$

show that the Ising energy operator $\epsilon\sim i\psi\bar\psi$ has scaling dimension $\Delta_\epsilon=1$.

Solution

By Wick contraction, the sign from exchanging the two middle fermion fields cancels the factor $i^2=-1$, so up to the chosen normalization,

$$\langle \epsilon(z,\bar z)\epsilon(w,\bar w)\rangle \propto \langle\psi(z)\psi(w)\rangle \langle\bar\psi(\bar z)\bar\psi(\bar w)\rangle.$$

Substituting the chiral two-point functions gives

$$\langle \epsilon(z,\bar z)\epsilon(w,\bar w)\rangle \propto {1\over (z-w)(\bar z-\bar w)} ={1\over |z-w|^2}.$$

For a scalar primary of scaling dimension $\Delta$, the two-point function scales as

$$\langle O(z,\bar z)O(w,\bar w)\rangle\propto {1\over |z-w|^{2\Delta}}.$$

Comparing with $|z-w|^{-2}$ gives

$$2\Delta_\epsilon=2, \qquad \Delta_\epsilon=1.$$

Equivalently, $\psi$ has weights $(1/2,0)$ and $\bar\psi$ has weights $(0,1/2)$, so $\epsilon\sim i\psi\bar\psi$ has weights $(1/2,1/2)$ and total dimension $1$.

References

• L. P. Kadanoff and H. Ceva, Determination of an Operator Algebra for the Two-Dimensional Ising Model, Physical Review B 3, 3918–3939 (1971). The two-dimensional order–disorder construction that motivates the disorder-loop language.
• J. B. Kogut, An Introduction to Lattice Gauge Theory and Spin Systems, Reviews of Modern Physics 51, 659–713 (1979). A detailed review of high-temperature expansions, spin systems, gauge systems, and duality.
• F. J. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, Journal of Mathematical Physics 12, 2259–2272 (1971). The classic spin/gauge duality reference.
• K. G. Wilson, Confinement of Quarks, Physical Review D 10, 2445–2459 (1974). Introduces Wilson’s lattice gauge theory and the Wilson-loop confinement criterion.

Further reading

• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer (1997). See the chapters on free fields and the Ising model for the continuum correlators.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers (1987). Useful for the broader perspective on discrete gauge systems, disorder variables, Wilson loops, and the relation between statistical mechanics and field theory.