Lattice Dirac Equations and Euclidean Spinors

The previous page identified the continuum limit of the two-dimensional Ising model with a Majorana fermion. This is not just a poetic statement that “fermions appear.” The order–disorder composite satisfies local lattice difference equations, and the long-wavelength limit of those equations is a Dirac equation. To see why the continuum spinor transformation law is already encoded on the lattice, it helps to separate three ideas: finite-difference equations, light-cone spinor kinematics, and Euclidean half-angle phases.

This page develops those three ideas carefully. We also use the end of the page to look ahead: Kramers–Wannier duality is special in two dimensions because both the high-temperature and low-temperature expansions are organized by closed one-dimensional objects. In three dimensions the low-temperature objects are surfaces, so the dual theory is not another ordinary Ising spin model. It is a $\mathbb Z_2$ gauge system, which is the subject of the next page.

Required background. The order–disorder construction and its continuum Majorana limit are developed in lessons 8 and 9. This lesson supplies the finite-difference and light-cone calculations that connect the lattice relation to the continuum Dirac equation.

From disorder endpoints to corner spinors

Lattice spacing and light-cone momenta

The first lattice example is Euclidean and one-dimensional. We set the lattice spacing to $a=1$ until the continuum limit is taken, and define

$$\Delta_{\mathrm{lat}}\phi_x=\phi_{x+1}-2\phi_x+\phi_{x-1}.$$

Thus the positive lattice operator is $-\Delta_{\mathrm{lat}}+M^2$. With nonzero lattice spacing, the physical continuum mass is $m=M/a$.

For two-dimensional Lorentzian spinors we use light-cone momenta

$$p_+=p_0+p_1, \qquad p_-=p_0-p_1,$$

so a boost of rapidity $\eta$ acts as $p_+\mapsto e^{\eta}p_+$ and $p_-\mapsto e^{-\eta}p_-$. After Wick rotation, we use Euclidean complex momenta schematically denoted by the same symbols,

$$p_+=p_1+i p_2, \qquad p_-=p_1-i p_2.$$

The notation is economical, but the meaning changes: Lorentzian boosts become Euclidean rotations.

The disorder operator is most cleanly defined by changing the signs of a set of bonds. If $P_{x^*}$ denotes a defect line ending at the dual-lattice point $x^*$, then a normalized partition function with such defect insertions defines the corresponding disorder correlation function:

$${Z(P_{x_1^*},\ldots,P_{x_n^*})\over Z} = \langle \mu(x_1^*)\cdots \mu(x_n^*)\rangle.$$

The path of a defect line is not itself physical; only its endpoint is physical, up to the sign picked up when an order field crosses the line. A local fermionic object is obtained by putting an order field and a disorder endpoint next to each other. If $e_a$ labels one of the four half-lattice directions from an original-lattice site to a neighboring dual-lattice site, define the corner field

$$\psi_a(x)=\sigma(x)\mu(x+e_a).$$

Moving the disorder endpoint once around the spin crosses the branch cut once. Therefore the corner label is antiperiodic:

$$\boxed{\psi_{a+4}(x)=-\psi_a(x).}$$

This is the lattice ancestor of a spinor sign. A continuously defined periodic angular function has integer harmonics, while an antiperiodic one has half-integer harmonics. The corner field samples only four directions, so it has exactly four independent Fourier characters. If the four corners are labeled by angles $\theta_a=\pi a/2$, one may write

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

Equivalently, in a slightly different convention for numbering the corners, the phases look like $e^{\pm i\pi a/4}$ and $e^{\pm 3i\pi a/4}$. The convention changes the names of the components, not the content. These are the four classes $s=\pm1/2,\pm3/2$ modulo $4$. The critical scaling projection retains the $s=\pm1/2$ combinations as the continuum Majorana components; the other two lattice combinations are subleading. Further continuum-spin contributions come from lattice-spacing-suppressed derivatives and descendants, not from additional independent corner-label harmonics.

A corner field is a point-split order–disorder composite. Taking the disorder endpoint once around the order insertion gives $\psi_{a+4}=-\psi_a$. Its four antiperiodic corner characters are represented by $s=\pm1/2,\pm3/2$ modulo $4$; the critical pair gives the continuum components $u$ and $v$.

The local Ising identities from the previous page give linear relations among nearby corner fields. The continuum Dirac equation arises by expanding those relations in powers of the lattice spacing and projecting onto the $s=\pm1/2$ modes. Before doing that for spinors, let us warm up with the scalar case.

Lattice Green functions as continuum equations

A finite-difference equation is a lattice version of a differential equation only when the relevant fields vary slowly from site to site. The simplest example is

$$(2+M^2)\phi_x-\phi_{x+1}-\phi_{x-1}=j_x.$$

Equivalently,

$$(-\Delta_{\mathrm{lat}}+M^2)\phi_x=j_x.$$

Fourier-transform with

$$\phi_x=\int_{-\pi}^{\pi}{dp\over 2\pi}\,e^{ipx}\phi(p), \qquad j_x=\int_{-\pi}^{\pi}{dp\over 2\pi}\,e^{ipx}j(p).$$

Since $\phi_{x+1}$ contributes $e^{ip}\phi(p)$ and $\phi_{x-1}$ contributes $e^{-ip}\phi(p)$, the equation becomes

$$\left[M^2+2(1-\cos p)\right]\phi(p)=j(p).$$

For small momentum,

$$2(1-\cos p)=p^2+O(p^4),$$

so the Green function is

$$G(p)={1\over M^2+2(1-\cos p)} \simeq {1\over p^2+M^2}.$$

Restoring the lattice spacing $a$, write $x=na$ and $p=a k$. If $M=ma$, then

$${1\over a^2}\left(2\phi_n-\phi_{n+1}-\phi_{n-1}\right)+m^2\phi_n \longrightarrow (-\partial_x^2+m^2)\phi(x).$$

Thus the continuum equation is

$$\boxed{(-\partial_x^2+m^2)\phi(x)=j(x).}$$

The finite-difference operator has Fourier symbol $M^2+2(1-\cos p)$. At small momentum it becomes $M^2+p^2$, giving the continuum Green function of $-\partial_x^2+m^2$.

The assumptions are not decorative. We need

$$|\phi_{x+1}-\phi_x|\ll |\phi_x|, \qquad |j_{x+1}-j_x|\ll |j_x|, \qquad M^2\ll1.$$

The first condition says the field is smooth on the lattice scale. The second says the source does not inject lattice-scale momentum. The third says the correlation length is much larger than the lattice spacing:

$$\xi\sim {1\over M}\gg1.$$

This is the elementary model for what happens in the Ising fermion problem. A local lattice relation becomes a differential equation only near criticality, where the correlation length is large. Far from criticality the exact lattice equation is still true, but it should not be replaced by a continuum Dirac equation.

The Dirac equation in light-cone variables

The two-dimensional Dirac equation is a first-order square root of the massive scalar dispersion relation. In Lorentzian signature the mass shell is

$$p_0^2-p_1^2=m^2.$$

Using light-cone momenta,

$$p_+=p_0+p_1, \qquad p_-=p_0-p_1,$$

this becomes

$$p_+p_-=m^2.$$

In a chiral basis, the Dirac equation for a two-component spinor $(u_+,u_-)$ may be written as

$$\boxed{ p_+u_-=m u_+, \qquad p_-u_+=m u_-. }$$

Eliminating either component gives the scalar mass shell. For example, applying $p_+$ to the second equation and using the first gives

$$p_+p_-u_+=m^2u_+.$$

Thus each component satisfies the second-order equation, but the spinor also satisfies a stronger first-order relation between its components.

A useful way to remember the transformation law is to solve the first-order equations by square roots. Up to an overall normalization,

$$u_+\sim \sqrt{p_+}, \qquad u_-\sim \sqrt{p_-}.$$

Under a boost of rapidity $\eta$,

$$p_+\mapsto e^{\eta}p_+, \qquad p_-\mapsto e^{-\eta}p_-.$$

Therefore

$$\boxed{ u_+\mapsto e^{\eta/2}u_+, \qquad u_-\mapsto e^{-\eta/2}u_-. }$$

This is the Lorentzian spinor representation in its most economical form. Vectors scale by $e^{\pm\eta}$ in light-cone components; spinors scale by the square roots $e^{\pm\eta/2}$.

In two Lorentzian dimensions the massive shell is $p_+p_-=m^2$. A boost rescales $p_+$ and $p_-$ oppositely, while the spinor components transform by the square-root factors $e^{\pm\eta/2}$. After Wick rotation, the same square-root logic becomes the Euclidean half-angle phase.

This explains why a first-order equation naturally produces spinors. The field components are not just two unrelated functions; they transform as square roots of light-cone momentum components.

Wick rotation and Euclidean spinors

The Ising scaling limit is usually discussed in Euclidean signature. A Wick rotation replaces the Lorentzian energy by an imaginary Euclidean momentum. Schematically,

$$p_0\mapsto i p_2.$$

The light-cone components become complex conjugate Euclidean momenta. With the convention

$$p_+=p_1+i p_2, \qquad p_-=p_1-i p_2,$$

write

$$p_+=|p|e^{i\theta}, \qquad p_-=|p|e^{-i\theta}.$$

A Euclidean rotation by angle $\alpha$ shifts $\theta\mapsto\theta+\alpha$. Therefore

$$p_+\mapsto e^{i\alpha}p_+, \qquad p_-\mapsto e^{-i\alpha}p_-.$$

The spinor components transform by square roots:

$$\boxed{ u_+\mapsto e^{i\alpha/2}u_+, \qquad u_-\mapsto e^{-i\alpha/2}u_-. }$$

A $2\pi$ rotation gives

$$u_+\mapsto -u_+, \qquad u_-\mapsto -u_-.$$

That is the continuum version of the corner-field identity $\psi_{a+4}=-\psi_a$.

There is also a useful reality distinction. In Lorentzian signature one can choose a real representation of the $1+1$-dimensional Clifford algebra and impose a Majorana condition on the field. In Euclidean signature, the rotation weights $e^{\pm i\alpha/2}$ are complex phases, and the local chiral fields are usually treated as independent Grassmann variables:

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

Reflection positivity relates the two after a reflection, but one should not impose a pointwise condition such as $\bar\psi=\psi^*$ inside holomorphic Euclidean calculations. In the Euclidean path integral, the Majorana nature is encoded by a real antisymmetric fermion operator and a Pfaffian rather than by ordinary complex conjugation of the two chiral fields.

For the Ising model, the mass term is the relevant perturbation away from the critical point. Near criticality,

$$m(K)\propto K-K_c, \qquad \xi^{-1}=|m(K)|.$$

Depending on conventions, one may call the physical mass gap $|m|$ and reserve the sign of $m$ for distinguishing the ordered and disordered phases. Kramers–Wannier duality reverses the sign of $K-K_c$ to leading order, and hence reverses the sign of $m$.

Why three dimensions lead to gauge variables

In two dimensions, Kramers–Wannier duality has a striking geometric simplicity. The low-temperature expansion is a sum over closed domain-wall loops on the dual lattice, while the high-temperature expansion is a sum over closed even subgraphs on the original lattice. Both are one-dimensional closed objects. This is why an ordinary spin system can be dual to another ordinary spin system.

In three dimensions the first statement changes. Domain walls in an ordered Ising configuration are codimension-one objects, hence two-dimensional closed surfaces. If $S$ is a closed domain-wall surface and $A(S)$ is its area in plaquettes, the low-temperature expansion has the schematic form

$$Z_{\mathrm{Ising}}^{\mathrm{low}}(K) \propto \sum_{S\,\mathrm{closed\ surfaces}} e^{-2K A(S)}.$$

The high-temperature expansion of the ordinary spin model is still built from products of bonds. Expanding

$$e^{K\sigma_x\sigma_y} =\cosh K\left(1+\tanh K\,\sigma_x\sigma_y\right)$$

and summing over spins forces an even number of occupied bonds at each site. Thus

$$Z_{\mathrm{Ising}}^{\mathrm{high}}(K) \propto \sum_{P\,\mathrm{even\ subgraphs}} (\tanh K)^{L(P)}.$$

In three dimensions, then, the two expansions involve different kinds of objects:

$$\begin{aligned} \text{low-temperature Ising}&:\ \text{closed surfaces},\\ \text{high-temperature Ising}&:\ \text{closed even-subgraph loop networks}. \end{aligned}$$

So the dual of the three-dimensional Ising model cannot be another ordinary nearest-neighbor Ising spin model.

In three dimensions the low-temperature Ising expansion is a sum over closed surfaces, while the high-temperature spin expansion is a sum over closed even-subgraph loop networks. Closed surfaces reappear as the high-temperature expansion of a $\mathbb Z_2$ gauge theory, with the dual relation $e^{-2K}=\tanh K_g$.

To get a high-temperature expansion made of closed surfaces, put $\mathbb Z_2$ variables on links rather than sites. Let

$$U_\ell=\pm1$$

be a $\mathbb Z_2$ gauge field on each link, and define the plaquette product

$$U_p=\prod_{\ell\in\partial p} U_\ell.$$

The gauge action is

$$Z_{\mathrm{gauge}}(K_g)= \sum_{\{U_\ell=\pm1\}} \exp\left(K_g\sum_p U_p\right).$$

Its high-temperature expansion is

$$\prod_p\left(1+\tanh K_g\,U_p\right).$$

When one sums over the link variables, a nonzero contribution requires each link to be contained in an even number of selected plaquettes. That condition means the selected plaquettes form closed surfaces. Therefore

$$Z_{\mathrm{gauge}}^{\mathrm{high}}(K_g) \propto \sum_{S\,\mathrm{closed\ surfaces}} (\tanh K_g)^{A(S)}.$$

This matches the low-temperature expansion of the three-dimensional Ising model if

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

This is the higher-dimensional lesson: duality preserves the fluctuating geometric objects, but the variables needed to represent those objects may change. In two dimensions, closed curves can be represented by either spin domain walls or high-temperature spin graphs. In three dimensions, closed surfaces are naturally represented by plaquette excitations of a gauge theory. The next page develops this $\mathbb Z_2$ gauge system and introduces Wilson loops.

Summary

The Ising order–disorder composite is a lattice spinor because its corner label is antiperiodic under a full turn. Its four independent corner characters can be represented by $s=\pm1/2,\pm3/2$ modulo $4$; the leading $s=\pm1/2$ combinations become the two components of the continuum Majorana field.

A continuum equation is obtained from a lattice equation only in the long-wavelength regime. For a scalar lattice Green function, the symbol $M^2+2(1-\cos p)$ becomes $M^2+p^2$ when $p\ll1$. The analogous Ising corner-field difference equations become the Dirac equation near criticality, where $\xi\gg a$.

The two-dimensional Dirac equation is the square root of the mass shell. In light-cone variables,

$$p_+u_-=m u_+, \qquad p_-u_+=m u_-.$$

Lorentz boosts scale $p_\pm$ by $e^{\pm\eta}$ and spinors by $e^{\pm\eta/2}$. After Wick rotation, Euclidean rotations scale $p_\pm$ by $e^{\pm i\theta}$ and spinors by $e^{\pm i\theta/2}$, giving the $2\pi$ minus sign.

Finally, Kramers–Wannier duality becomes structurally different in three dimensions. The low-temperature Ising expansion contains closed surfaces, not closed loops. Those surfaces are naturally produced by the high-temperature expansion of a $\mathbb Z_2$ gauge theory, with dual relation $e^{-2K}=\tanh K_g$.

Common pitfalls

The antiperiodicity $\psi_{a+4}=-\psi_a$ is not an extra assumption imposed on the Ising model. It is the branch-cut sign of the mixed order–disorder operator.

A finite-difference equation is not automatically a differential equation. The continuum approximation requires small lattice momentum, slowly varying sources, and a correlation length much larger than the lattice spacing.

The Euclidean spinor phases $e^{\pm i\theta/2}$ are not the same as Lorentzian boost factors $e^{\pm\eta/2}$, but they are related by Wick rotation. Confusing the two is a quick way to lose factors of $i$.

The sign of the Ising Majorana mass is convention-dependent unless the duality convention is fixed. The mass gap is $|m|$; the sign distinguishes the two phases.

Four corner values do not define an infinite collection of independent half-integer modes. They define four character classes modulo $4$; higher-spin continuum corrections enter through derivatives and other descendants.

In three dimensions, the dual of the Ising model is not another ordinary nearest-neighbor spin model. The mismatch is geometric: surfaces are dual to plaquette excitations of a gauge theory, not to site-spin high-temperature loop networks.

Exercises

Exercise 1: The lattice Green function

Fourier-transform the lattice equation

$$(2+M^2)\phi_x-\phi_{x+1}-\phi_{x-1}=j_x$$

and show that the long-wavelength Green function is $G(p)\simeq 1/(p^2+M^2)$.

Solution

Use

$$\phi_x=\int_{-\pi}^{\pi}{dp\over2\pi}\,e^{ipx}\phi(p), \qquad j_x=\int_{-\pi}^{\pi}{dp\over2\pi}\,e^{ipx}j(p).$$

Then

$$\phi_{x+1}=\int_{-\pi}^{\pi}{dp\over2\pi}\,e^{ipx}e^{ip}\phi(p), \qquad \phi_{x-1}=\int_{-\pi}^{\pi}{dp\over2\pi}\,e^{ipx}e^{-ip}\phi(p).$$

Substituting gives

$$\left(2+M^2-e^{ip}-e^{-ip}\right)\phi(p)=j(p).$$

Since $e^{ip}+e^{-ip}=2\cos p$,

$$\left(M^2+2(1-\cos p)\right)\phi(p)=j(p).$$

Thus

$$G(p)={\phi(p)\over j(p)}={1\over M^2+2(1-\cos p)}.$$

For $p\ll1$,

$$2(1-\cos p)=p^2+O(p^4),$$

so

$$G(p)\simeq {1\over p^2+M^2}.$$

Exercise 2: Light-cone Dirac kinematics

Starting from

$$p_+u_-=m u_+, \qquad p_-u_+=m u_-,$$

Show that nonzero solutions require $p_+p_-=m^2$. Then show that under a boost of rapidity $\eta$, the spinor components transform as $u_+\mapsto e^{\eta/2}u_+$ and $u_-\mapsto e^{-\eta/2}u_-$.

Solution

Apply $p_-$ to the first equation:

$$p_-p_+u_-=m p_-u_+.$$

Using the second equation, $p_-u_+=m u_-$, this becomes

$$p_+p_-u_-=m^2u_-.$$

For $u_-\ne0$,

$$p_+p_-=m^2.$$

The same conclusion follows by eliminating $u_-$ instead.

A boost acts as

$$p_+\mapsto e^\eta p_+, \qquad p_-\mapsto e^{-\eta}p_-.$$

Since the spinor components can be chosen as square roots,

$$u_+\sim \sqrt{p_+}, \qquad u_-\sim \sqrt{p_-},$$

they transform as

$$u_+\mapsto e^{\eta/2}u_+, \qquad u_-\mapsto e^{-\eta/2}u_-.$$

Exercise 3: The Euclidean spinor sign

In Euclidean two-dimensional momentum space, set

$$p_+=p_1+i p_2, \qquad p_-=p_1-i p_2.$$

If $p_+=|p|e^{i\theta}$ and $p_-=|p|e^{-i\theta}$, show that a rotation by $2\pi$ changes the sign of the spinor components $u_+\sim\sqrt{p_+}$ and $u_-\sim\sqrt{p_-}$.

Solution

A rotation by angle $\alpha$ shifts the polar angle:

$$\theta\mapsto\theta+\alpha.$$

Therefore

$$p_+\mapsto e^{i\alpha}p_+, \qquad p_-\mapsto e^{-i\alpha}p_-.$$

Taking square roots gives

$$u_+\mapsto e^{i\alpha/2}u_+, \qquad u_-\mapsto e^{-i\alpha/2}u_-.$$

For $\alpha=2\pi$,

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

Hence

$$u_+\mapsto -u_+, \qquad u_-\mapsto -u_-.$$

This is the Euclidean spinor sign under a full rotation.

Exercise 4: Why 3D spin duality changes form

Explain why the high-temperature expansion of the ordinary Ising model is a sum over closed even subgraphs in any dimension. Then explain why, in three dimensions, this cannot be directly dual to the low-temperature Ising expansion.

Solution

For each nearest-neighbor bond,

$$e^{K\sigma_x\sigma_y}=\cosh K\left(1+\tanh K\,\sigma_x\sigma_y\right).$$

Expanding the product over bonds selects a subset of occupied bonds. At a given site $x$, the spin $\sigma_x$ appears once for each occupied bond incident on $x$. The sum over $\sigma_x=\pm1$ vanishes unless the power of $\sigma_x$ is even. Therefore every vertex must have even degree. A set of bonds with even degree at every vertex is a closed even subgraph. It can contain several loops meeting at even-valence vertices, so it need not be a disjoint union of simple polygons.

This statement is independent of dimension. In three dimensions the high-temperature spin expansion is still made of one-dimensional even-subgraph loop networks.

The low-temperature expansion is different. Domain walls separate regions of opposite spin. In $d$ dimensions domain walls are codimension-one objects. In three dimensions they are two-dimensional closed surfaces. Thus the high-temperature ordinary spin expansion contains one-dimensional loop networks, while the low-temperature expansion contains closed surfaces. Since the fluctuating objects have different dimension, the dual theory cannot be another ordinary nearest-neighbor spin model.

Exercise 5: Gauge-theory surfaces

For the $\mathbb Z_2$ gauge partition function

$$Z_{\mathrm{gauge}}(K_g)= \sum_{\{U_\ell=\pm1\}} \exp\left(K_g\sum_p U_p\right), \qquad U_p=\prod_{\ell\in\partial p}U_\ell,$$

show that the high-temperature expansion is a sum over closed surfaces with weight $(\tanh K_g)^{A(S)}$.

Solution

For each plaquette,

$$e^{K_g U_p}=\cosh K_g\left(1+\tanh K_g\,U_p\right).$$

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

$$\prod_{p\in S}U_p = \prod_{p\in S}\prod_{\ell\in\partial p}U_\ell.$$

When we sum over a link variable $U_\ell=\pm1$, the result vanishes unless $U_\ell$ appears an even number of times. Thus every link must belong to an even number of selected plaquettes. This is precisely the condition that the selected plaquettes form a closed surface, possibly with several connected components.

If $A(S)$ is the number of selected plaquettes, the weight is

$$(\tanh K_g)^{A(S)}.$$

Thus

$$Z_{\mathrm{gauge}}^{\mathrm{high}}(K_g) \propto \sum_{S\,\mathrm{closed\ surfaces}}(\tanh K_g)^{A(S)}.$$

Matching this with the low-temperature Ising surface expansion gives the dual relation

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

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 classic order–disorder operator construction.
• B. Kaufman, Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis, Physical Review 76, 1232–1243 (1949). The spinor solution of the square-lattice 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 derivation of the two-dimensional Ising solution.
• F. J. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, Journal of Mathematical Physics 12, 2259–2272 (1971). Duality between spin and gauge systems.
• J. B. Kogut, An Introduction to Lattice Gauge Theory and Spin Systems, Reviews of Modern Physics 51, 659–713 (1979). A broad review of lattice spin/gauge duality and strong-coupling expansions.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory. Useful for the continuum Majorana theory and Ising conformal fields.
• A. M. Polyakov, Gauge Fields and Strings. See the discussions of statistical mechanics, disorder variables, gauge systems, and the Ising Dirac equation.