Relativistic Fields, Spinors, and the Ising Continuum Limit

The previous page produced a surprise: a two-dimensional Ising model written in terms of ordinary commuting spins contains a natural fermionic field. The field is not inserted by hand. It is the point-split product of an order operator and a disorder operator, and its fermionic sign comes from the branch cut between them.

This page translates that observation into the language of continuum field theory. We first recall how fields are classified by their transformation laws under rotations: scalars, vectors, tensors, and spinors. Then we return to the Ising corner field and show how its four lattice components become the two real components of a continuum Majorana fermion. The same continuum limit identifies the Ising energy operator with the fermion mass operator. As a quick payoff, the famous logarithmic singularity of the two-dimensional Ising specific heat follows from one line of field theory.

Required background. This lesson builds directly on order, disorder, and the Ising fermion. The earlier discussions of critical propagators and the upper critical dimension and two-dimensional disorder lines are useful context, but the rotation-representation review below is self-contained.

Fields as rotation representations

Euclidean spinor basis and Ising mass sign

Most of this page is Euclidean. Spatial indices $i,j$ are summed with $\delta_{ij}$, and the two-dimensional coordinates are

$$z=x^1+i x^2, \qquad \partial={1\over2}(\partial_1-i\partial_2), \qquad \bar\partial={1\over2}(\partial_1+i\partial_2).$$

For the two-dimensional Dirac equation we use Hermitian Euclidean gamma matrices

$$\gamma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad \gamma_2=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}, \qquad \{\gamma_i,\gamma_j\}=2\delta_{ij}.$$

Overall signs of the Euclidean mass term are convention-dependent. The invariant statement is that the Ising mass is proportional to the deviation from criticality and changes sign under Kramers–Wannier duality.

A continuum field is not defined only by its equation of motion. It is also defined by how it transforms under spacetime symmetries. In Euclidean field theory, the local rotation group is $\mathrm{SO}(d)$, or more precisely its double cover $\operatorname{Spin}(d)$ when spinors are present. A field with components $\Phi_A(x)$ transforms as

$$\Phi'_A(x')=D(R)_{AB}\Phi_B(x), \qquad x'=Rx,$$

where $D(R)$ is a finite-dimensional representation carried by the field indices. A scalar has $D(R)=1$. A vector has $D(R)=R$. A spinor has a matrix $S(R)$ satisfying

$$S(R)^{-1}\gamma_i S(R)=R_{ij}\gamma_j.$$

This is the first difference between writing down a field and writing down a collection of components. A two-component object is not automatically a spinor; it is a spinor only if rotations act by the spin representation.

Scalars, vectors, and spinors differ by the representation attached to the field index. A spinor is a representation of the double cover of the rotation group; a $2\pi$ rotation acts as $-1$.

A free scalar field is the simplest example. In Euclidean signature one may write

$$S_0[\phi] ={1\over2}\int d^d x\, \left[(\partial_i\phi)^2+m^2\phi^2\right],$$

so the classical equation is

$$(-\partial^2+m^2)\phi=0.$$

After analytic continuation to Lorentzian signature this becomes the Klein–Gordon equation. The field carries no internal rotation index; all angular momentum of a scalar excitation is orbital.

A vector field carries one rotation index:

$$A_i'(x')=R_{ij}A_j(x).$$

A tensor field carries several such indices. For example,

$$B_{ij}'(x')=R_{ik}R_{j\ell}B_{k\ell}(x).$$

These transformation laws are local. They do not yet say which field equations are correct or how many physical polarizations propagate. Those are dynamical questions.

Vector fields and transversality

A massive vector field can be described by the Euclidean Proca action

$$S[A]=\int d^d x\, \left[ {1\over4}F_{ij}F_{ij}+{m^2\over2}A_iA_i \right], \qquad F_{ij}=\partial_iA_j-\partial_jA_i.$$

Varying $A_i$ gives

$$-\partial_jF_{ji}+m^2A_i=0.$$

Expanding $F_{ji}$,

$$\partial_jF_{ji} =\partial^2A_i-\partial_i\partial_jA_j,$$

so the equation is

$$(-\partial^2+m^2)A_i+\partial_i\partial_jA_j=0.$$

Taking a divergence gives

$$m^2\partial_iA_i=0.$$

Therefore, for $m\ne0$,

$$\boxed{\partial_iA_i=0.}$$

The vector equation plus this constraint is equivalent to a Klein–Gordon equation for the transverse components:

$$(-\partial^2+m^2)A_i=0, \qquad \partial_iA_i=0.$$

In momentum space, the decomposition is especially transparent:

$$A_i(k)=A_i^{\perp}(k)+A_i^{\parallel}(k), \qquad k_iA_i^{\perp}(k)=0,$$

with transverse projector

$$P_{ij}^{\perp}(k)=\delta_{ij}-{k_i k_j\over k^2}.$$

A vector field decomposes into components transverse and longitudinal to the momentum. The massive Proca equation imposes $\partial_iA_i=0$ on shell, leaving the spin-one polarizations.

This is a useful warning for later gauge theory. A vector field has a vector index, but not every component of that index represents an independent physical polarization. Gauge redundancy, constraints, and equations of motion all matter. In the Ising problem the analogous lesson is that the four corner labels of the lattice fermion are not four independent particles. They are a convenient lattice package whose long-distance content is a smaller spinor field.

Spinors and half-angle rotations

Spinors are not ordinary tensors. In two Euclidean dimensions, rotations are labeled by an angle $\theta$. A vector rotates with phases $e^{\pm i\theta}$ in the complex basis, while a spinor rotates with phases $e^{\pm i\theta/2}$. Thus a two-component spinor may be written schematically as

$$\Psi=\begin{pmatrix}u\\v\end{pmatrix}, \qquad S_\theta(u)=e^{i\theta/2}u, \qquad S_\theta(v)=e^{-i\theta/2}v.$$

The important point is

$$S_{2\pi}(u)=-u, \qquad S_{2\pi}(v)=-v.$$

A spinor changes sign under a full $2\pi$ rotation. Only after a $4\pi$ rotation does it return to itself.

A two-dimensional spinor carries half-angle phases. This is the continuum version of the sign found by taking the Ising order–disorder composite once around its branch point.

The Euclidean two-dimensional Dirac equation is a first-order equation,

$$(\gamma_i\partial_i-m)\Psi=0.$$

With the gamma matrices stated above and $\Psi=(u,v)^T$, this is equivalent to

$$\boxed{ (\partial_1+i\partial_2)u=m v, \qquad (\partial_1-i\partial_2)v=m u. }$$

At $m=0$ these equations split into holomorphic and antiholomorphic pieces:

$$\bar\partial u=0, \qquad \partial v=0.$$

Thus one component depends only on $z$, and the other depends only on $\bar z$. This is the continuum seed of the Ising conformal field theory: the critical fermion separates into left-moving and right-moving Majorana components.

There is no contradiction between the first-order Dirac equation and the second-order Klein–Gordon equation. Squaring the operator gives

$$(\gamma_i\partial_i+m)(\gamma_j\partial_j-m)\Psi =(\partial^2-m^2)\Psi,$$

again up to Euclidean sign conventions. Each spinor component obeys a second-order massive wave equation, but the spinor is constrained by the stronger first-order equation.

The Ising corner field as a spinor

Recall the point-split order–disorder composite from the previous page. Put a spin at an original-lattice site $x$ and a disorder endpoint at a neighboring dual site $x+e_a$, where the four half-lattice vectors have angles

$$\theta_a={\pi\over4}+{\pi\over2}(a-1), \qquad a=1,2,3,4.$$

The four corner fields are

$$\chi_a(x)=\sigma_x\mu_{x+e_a}.$$

Because moving the disorder endpoint once around the spin crosses the branch cut once, the corner field is antiperiodic:

$$\chi_{a+4}(x)=-\chi_a(x).$$

This antiperiodicity is exactly the lattice form of spinorial behavior. A scalar function of a continuous direction would have integer angular harmonics, whereas an antiperiodic function has half-integer harmonics. Here, however, the direction is sampled at only four corners. The corner-label space is therefore four-dimensional, with four independent Fourier characters. A convenient set of representatives is

$$s\in\left\{+{1\over2},-{1\over2},+{3\over2},-{3\over2}\right\} \qquad (s\ \mathrm{mod}\ 4).$$

A convenient schematic expansion is

$$\chi_a(x) = e^{i\theta_a/2}u(x) + e^{-i\theta_a/2}v(x) + e^{3i\theta_a/2}u_{3/2}(x) + e^{-3i\theta_a/2}v_{3/2}(x).$$

The phases in this formula depend on the convention for drawing the disorder cut and numbering the corners. The content does not: the critical long-distance projection retains the spin $+1/2$ and $-1/2$ combinations as the two Majorana components. The two remaining lattice combinations, represented by $s=\pm3/2$, are subleading in that projection. They are not the beginning of extra independent corner harmonics: higher-spin continuum contributions arise through lattice-spacing-suppressed derivatives and descendants.

The four Ising corner fields are diagonalized into four antiperiodic corner characters. The critical spin $\pm1/2$ pair becomes the two components of the Majorana fermion, obeying a massive Dirac equation away from criticality.

The local lattice equation derived from moving a disorder endpoint across a bond now becomes a continuum equation. The local move contains the factor

$$e^{-2K\sigma_i\sigma_j} =\cosh 2K-(\sinh 2K)\sigma_i\sigma_j.$$

When inserted into a corner correlator, the first term keeps the spin insertion where it was, while the second term transfers the spin across the crossed bond. The result is a linear relation among nearby corner fields. In the continuum limit one expands

$$\chi_a(x+\delta)=\chi_a(x)+\delta_i\partial_i\chi_a(x)+O(a^2).$$

At the self-dual coupling $K_c$, the zeroth-order part of the relation cancels. This cancellation is the lattice reason the fermion is massless at criticality. Away from $K_c$, the uncancelled zeroth-order term becomes the mass:

$$m\propto K-K_c.$$

After projecting onto the spin $\pm1/2$ harmonics, the continuum equations are

$$(\partial_1+i\partial_2)u=m v, \qquad (\partial_1-i\partial_2)v=m u.$$

This is the two-dimensional Majorana equation. The word “Majorana” means that the fermion is real: the Ising model supplies one real fermionic degree of freedom rather than a complex Dirac fermion with an independent conserved $U(1)$ charge.

Mass, duality, and the two phases

The Kramers–Wannier relation is

$$\sinh 2K\,\sinh 2K^*=1.$$

At the self-dual point,

$$K^*=K=K_c, \qquad \sinh 2K_c=1.$$

Expanding around criticality gives

$$K^*-K_c=-(K-K_c)+O((K-K_c)^2).$$

Therefore the continuum mass changes sign under duality:

$$m(K^*)=-m(K)+O(m^2).$$

This sign is not a harmless convention once the phases and boundary conditions are fixed. The two signs of the Majorana mass correspond to the two sides of the Ising transition. On one side the order variable has long-range order; on the dual side the disorder variable does. The mass gap is controlled by $|m|$, so the correlation length behaves as

$$\xi\sim {1\over |m|}\sim {1\over |K-K_c|}.$$

Thus the two-dimensional Ising correlation-length exponent is

$$\nu=1.$$

This agrees with the exact solution and is much sharper than mean-field theory. The deeper reason is that the critical Ising model is not an interacting scalar theory at long distances; it is a free massless Majorana fermion plus nontrivial spin and disorder fields.

Energy operator and the specific heat

The lattice energy is the operator conjugate to the coupling. For the nearest-neighbor square-lattice model,

$$Z(K)=\sum_{\{\sigma\}}\exp\!\left(K\sum_{\langle ij\rangle}\sigma_i\sigma_j\right),$$

define the bond energy

$$\epsilon_{ij}=\sigma_i\sigma_j.$$

The dimensionless free energy $F=-\log Z$ satisfies

$$-{\partial F\over\partial K} =\left\langle\sum_{\langle ij\rangle}\epsilon_{ij}\right\rangle,$$

and

$$-{\partial^2 F\over\partial K^2} =\sum_{\langle ij\rangle,\langle k\ell\rangle} \left\langle \epsilon_{ij}\epsilon_{k\ell}\right\rangle_c.$$

Up to conventional powers of the temperature, this is the specific heat. Translational invariance turns it into an integral of the connected energy–energy correlator:

$$C\propto \sum_x \langle\epsilon(0)\epsilon(x)\rangle_c.$$

The continuum identification is

$$\boxed{\epsilon(x)\sim i\bar\psi(x)\psi(x).}$$

For a real Euclidean Majorana field this notation is shorthand for the rotationally invariant fermion bilinear; in components it is proportional to $u v$ after a conventional choice of phases. The factor of $i$ is also convention-dependent. The scaling dimension, however, is not convention-dependent.

Equivalently, the energy operator is the operator multiplying the fermion mass in the action. Since the Majorana field has scaling dimension $1/2$, the bilinear has scaling dimension

$$\Delta_\epsilon=1.$$

At criticality,

$$\langle\epsilon(x)\epsilon(0)\rangle_c\sim {1\over |x|^{2\Delta_\epsilon}} ={1\over |x|^2}.$$

Therefore the specific heat diverges logarithmically:

$$C\sim \int_a^\xi 2\pi r\,dr\,{1\over r^2} \sim \log {\xi\over a} \sim \log {1\over |K-K_c|}.$$

The same result appears in momentum space as the one-loop fermion bubble

$$C\sim \int_{|m|}^{1/a}{d^2k\over k^2} \sim \log {1\over |m|a}.$$

The Ising energy density is the continuum fermion mass operator $i\bar\psi\psi$. Its two-point function has scaling $1/|x|^2$, so the integrated energy–energy correlator gives the logarithmic specific-heat singularity.

This is a clean example of the power of the continuum limit. Once the correct scaling field is known, a thermodynamic singularity follows from dimensional analysis. The exact lattice solution fixes the nonuniversal amplitude and the regular terms, but the logarithm is already visible from the Majorana field theory.

Summary

A continuum field is classified not only by its equation of motion but also by its rotation representation. Scalars transform trivially, vectors transform by the ordinary rotation matrix, and spinors transform by the double cover. The essential spinor fact is $S(2\pi)=-1$.

The Ising order–disorder composite has precisely this sign. Its four corner components are antiperiodic under a full rotation of the disorder endpoint around the order insertion. Their four independent corner characters may be represented by spins $\pm1/2$ and $\pm3/2$ modulo $4$. In the scaling limit the leading $\pm1/2$ combinations form a real two-component fermion. The local lattice transfer relation becomes the massive two-dimensional Dirac equation

$$(\partial_1+i\partial_2)u=m v, \qquad (\partial_1-i\partial_2)v=m u, \qquad m\propto K-K_c.$$

At $K=K_c$, the mass vanishes and the two components become holomorphic and antiholomorphic Majorana fields. The energy operator is the fermion mass bilinear $i\bar\psi\psi$, whose scaling dimension is $1$. Hence the specific heat is logarithmically singular:

$$C\sim \log {1\over |K-K_c|}.$$

Common pitfalls

The four Ising corner fields are not four independent continuum fermions, nor do four samples support an unlimited independent harmonic tower. They form a four-dimensional lattice-label space; the long-distance spin $\pm1/2$ projection gives one Majorana fermion, while derivative descendants account for higher continuum-spin corrections.

The statement $\chi_{a+4}=-\chi_a$ is not ordinary anticommutation of microscopic spins. It is monodromy of mixed order–disorder correlation functions. The original Ising variables are still commuting classical spins.

The sign of the continuum mass depends on phase conventions, but its oddness under Kramers–Wannier duality is physical. The two signs encode the ordered and disordered phases.

The Feynman-bubble picture of the specific heat should be understood as a continuum scaling argument. The lattice cutoff $a$ and the normalization of $\epsilon$ determine nonuniversal constants, while the logarithmic divergence is universal.

Exercises

Exercise 1: Proca transversality

Derive the transversality condition for a massive vector field from the Proca equation

$$-\partial_jF_{ji}+m^2A_i=0, \qquad F_{ij}=\partial_iA_j-\partial_jA_i.$$

Then show that the transverse momentum-space projector is

$$P_{ij}^{\perp}(k)=\delta_{ij}-{k_i k_j\over k^2}.$$

Solution

Taking a divergence of the Proca equation gives

$$-\partial_i\partial_jF_{ji}+m^2\partial_iA_i=0.$$

The first term vanishes because $\partial_i\partial_j$ is symmetric in $i,j$, while $F_{ji}$ is antisymmetric:

$$\partial_i\partial_jF_{ji}=0.$$

For $m\ne0$, this implies

$$\partial_iA_i=0.$$

In momentum space, the longitudinal part of a vector is proportional to $k_i$. The projector onto this direction is

$$P_{ij}^{\parallel}={k_i k_j\over k^2}.$$

Therefore the projector onto the orthogonal subspace is

$$P_{ij}^{\perp}=\delta_{ij}-P_{ij}^{\parallel} =\delta_{ij}-{k_i k_j\over k^2}.$$

It obeys

$$k_iP_{ij}^{\perp}=k_j-{k^2k_j\over k^2}=0,$$

so $P^{\perp}A$ is transverse.

Exercise 2: Half-angle rotation and corner characters

Let a two-dimensional spinor transform under rotations by

$$S_\theta(u)=e^{i\theta/2}u, \qquad S_\theta(v)=e^{-i\theta/2}v.$$

Show that a $2\pi$ rotation gives a minus sign. Then explain why a continuously defined antiperiodic angular field has half-integer harmonics, and why its values at four corner directions contain only four independent characters, which can be labeled by $s=\pm1/2,\pm3/2$ modulo $4$.

Solution

For $\theta=2\pi$,

$$e^{i\theta/2}=e^{i\pi}=-1, \qquad e^{-i\theta/2}=e^{-i\pi}=-1.$$

Thus

$$S_{2\pi}(u)=-u, \qquad S_{2\pi}(v)=-v.$$

Now suppose a function of the corner angle has an angular harmonic $e^{is\theta}$. Under $\theta\mapsto\theta+2\pi$ it transforms as

$$e^{is(\theta+2\pi)}=e^{2\pi i s}e^{is\theta}.$$

Antiperiodicity requires

$$e^{2\pi i s}=-1.$$

Therefore

$$s\in {1\over2}+\mathbb Z.$$

Thus a continuously defined antiperiodic angular field has half-integer harmonics. On the four corner angles $\theta_a=\theta_0+\pi a/2$, however,

$$e^{i(s+4)\theta_a}=e^{4i\theta_0}e^{is\theta_a},$$

so $s$ and $s+4$ give the same corner character up to an $a$-independent phase. There are only four independent characters. One may choose the representatives

$$s=+{1\over2},\ -{1\over2},\ +{3\over2},\ -{3\over2} \qquad (\mathrm{mod}\ 4).$$

Exercise 3: Euclidean Dirac components

Using

$$\gamma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad \gamma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \Psi=\begin{pmatrix}u\\v\end{pmatrix},$$

show that

$$(\gamma_i\partial_i-m)\Psi=0$$

is equivalent to

$$(\partial_1+i\partial_2)u=m v, \qquad (\partial_1-i\partial_2)v=m u.$$

What happens at $m=0$?

Solution

Compute

$$\gamma_1\partial_1\Psi =\begin{pmatrix}\partial_1 v\\ \partial_1 u\end{pmatrix}, \qquad \gamma_2\partial_2\Psi =\begin{pmatrix}-i\partial_2 v\\ i\partial_2 u\end{pmatrix}.$$

Therefore

$$\gamma_i\partial_i\Psi =\begin{pmatrix} (\partial_1-i\partial_2)v\\ (\partial_1+i\partial_2)u \end{pmatrix}.$$

The equation $\gamma_i\partial_i\Psi=m\Psi$ gives

$$(\partial_1-i\partial_2)v=m u, \qquad (\partial_1+i\partial_2)u=m v.$$

These are the desired equations, just written in the opposite order.

At $m=0$ they become

$$(\partial_1+i\partial_2)u=0, \qquad (\partial_1-i\partial_2)v=0.$$

Since

$$\bar\partial={1\over2}(\partial_1+i\partial_2), \qquad \partial={1\over2}(\partial_1-i\partial_2),$$

we get

$$\bar\partial u=0, \qquad \partial v=0.$$

Thus $u$ is holomorphic and $v$ is antiholomorphic.

Exercise 4: The duality-odd Ising mass

Use the Kramers–Wannier relation

$$\sinh 2K\,\sinh 2K^*=1$$

to show that near the self-dual point,

$$K^*-K_c=-(K-K_c)+O((K-K_c)^2).$$

Why does this imply that the Ising fermion mass is duality-odd?

Solution

Set

$$K=K_c+\delta K, \qquad K^*=K_c+\delta K^*.$$

At criticality,

$$\sinh 2K_c=1.$$

Expanding to first order,

$$\sinh 2K =1+2\cosh 2K_c\,\delta K+O(\delta K^2),$$

and similarly

$$\sinh 2K^* =1+2\cosh 2K_c\,\delta K^*+O((\delta K^*)^2).$$

Multiplying and using $\sinh 2K\sinh 2K^*=1$ gives

$$1+2\cosh 2K_c(\delta K+\delta K^*)+O(\delta K^2)=1.$$

Therefore

$$\delta K^*=-\delta K+O(\delta K^2),$$

or

$$K^*-K_c=-(K-K_c)+O((K-K_c)^2).$$

The continuum mass is proportional to the relevant perturbation away from criticality:

$$m\propto K-K_c.$$

Since duality reverses $K-K_c$, it reverses $m$. Thus the mass term is duality-odd.

Exercise 5: The logarithmic specific heat

Assume the energy operator at the critical Ising point has scaling dimension $\Delta_\epsilon=1$. Show that the integrated energy–energy correlator diverges logarithmically. Then express the answer in terms of the correlation length $\xi$ and the mass $m$.

Solution

At criticality, an operator of scaling dimension $\Delta_\epsilon$ has two-point function

$$\langle\epsilon(x)\epsilon(0)\rangle_c\sim {1\over |x|^{2\Delta_\epsilon}}.$$

For $\Delta_\epsilon=1$,

$$\langle\epsilon(x)\epsilon(0)\rangle_c\sim {1\over |x|^2}.$$

The specific heat is proportional to the integral of this correlator. With a UV cutoff $a$ and an IR cutoff $\xi$,

$$C\sim \int_a^\xi 2\pi r\,dr\,{1\over r^2} =2\pi\int_a^\xi {dr\over r} =2\pi\log {\xi\over a},$$

up to the normalization of $\epsilon$.

The mass gap cuts off correlations at

$$\xi\sim {1\over |m|}.$$

Since $m\propto K-K_c$,

$$C\sim \log {1\over |m|a} \sim \log {1\over |K-K_c|}$$

plus nonuniversal regular terms.

References and further reading

• 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 order–disorder construction and operator algebra.
• B. Kaufman, Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis, Physical Review 76, 1232–1243 (1949). The classic spinor solution of the two-dimensional Ising model.
• T. D. Schultz, D. C. Mattis, and E. H. Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions, Reviews of Modern Physics 36, 856–871 (1964). A standard fermionic formulation.
• B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model. Detailed exact results for the lattice model.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, especially the Ising-model chapters. A clear account of the fields $\sigma$, $\mu$, $\psi$, $\bar\psi$, and $\epsilon$.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena. A field-theoretic treatment of critical phenomena, scaling fields, and the Ising universality class.
• A. M. Polyakov, Gauge Fields and Strings. See the discussions of statistical mechanics, disorder variables, and the two-dimensional Ising Dirac equation.