Ising CFT, Majorana Fermions, and Tricritical Extensions

The previous page turned the minimal-model machinery into the finite operator algebra of the Ising fixed point. The result was beautifully small:

$$[\mathbf 1],\qquad [\sigma],\qquad [\varepsilon],$$

with

$$[\sigma]\times[\sigma]=[\mathbf 1]+[\varepsilon], \qquad [\sigma]\times[\varepsilon]=[\sigma], \qquad [\varepsilon]\times[\varepsilon]=[\mathbf 1].$$

This page opens the next layer. The Ising CFT is not only a minimal model; it is also a free Majorana fermion theory. That second description explains why the energy operator is a fermion bilinear, why the order and disorder fields create branch cuts, and why the Ising model is the first example in a ladder of richer two-dimensional critical points.

The main new endpoint is the tricritical Ising model. In Landau language, it appears when both the quadratic and quartic terms are tuned so that the stabilizing interaction is $\phi^6$. In conformal language, it is the next unitary minimal model,

$$\mathcal M(4,5), \qquad c={7\over10},$$

The same critical theory also has a fermionic, spin-structure-dependent $N=1$ extension whose holomorphic supercurrent has weight $3/2$. Its OPE with itself closes on the stress tensor; equivalently, one mode of the supercurrent squares to a translation. This is the first hint of two-dimensional supersymmetry in the course.

There is an important distinction here. In the diagonal bosonic minimal model, the $(1,4)$ family gives a scalar field $\varepsilon''$ with weights $(3/2,3/2)$. In the fermionic $N=1$ organization, the holomorphic Virasoro representation of weight $3/2$ is joined to the vacuum representation and supplies $G(z)$. The scalar $\varepsilon''$ and the chiral supercurrent $G$ are therefore related by the chiral-algebra extension, but they are not the same local field.

The Majorana realization of the Ising fixed point

Normalization of fermion OPEs

We use the standard chiral Majorana normalization

$$\psi(z)\psi(w)\sim {1\over z-w}, \qquad \bar\psi(\bar z)\bar\psi(\bar w)\sim {1\over \bar z-\bar w}.$$

Some statistical-mechanics and lattice conventions differ by signs or factors such as $1/2$. Rescaling $\psi$ changes these coefficients but not the conformal weights, fusion rules, branch cuts, or central charge.

The critical Ising model is the minimal model

$$\mathcal M(3,4), \qquad c={1\over2}.$$

Its bosonic local scalar primary fields are

$$\mathbf 1:(h,\bar h)=(0,0), \qquad \sigma:(h,\bar h)=\left({1\over16},{1\over16}\right), \qquad \varepsilon:(h,\bar h)=\left({1\over2},{1\over2}\right).$$

The fermionic realization introduces a real left-moving Majorana field $\psi(z)$ and a real right-moving Majorana field $\bar\psi(\bar z)$. At the critical point they obey

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

A convenient Euclidean action near the Ising point is

$$S={1\over2\pi}\int d^2z\, \left(\psi\bar\partial\psi+\bar\psi\partial\bar\psi+i m\,\bar\psi\psi\right).$$

At $m=0$ the two chiralities decouple. The thermal perturbation is the fermion bilinear

$$\varepsilon(z, \bar z)=i\bar\psi(\bar z)\psi(z),$$

up to a conventional real normalization. Since $\psi$ has weights $(1/2,0)$ and $\bar\psi$ has weights $(0,1/2)$, the energy operator has

$$(h_\varepsilon,\bar h_\varepsilon)=\left({1\over2},{1\over2}\right), \qquad \Delta_\varepsilon=1.$$

This makes the mass parameter $m$ relevant in two dimensions. Changing the sign of $m$ exchanges the ordered and disordered phases; this is the continuum version of Kramers–Wannier duality.

The Ising fixed point can be viewed through bosonic order fields, dual disorder fields, or a fermionic spin description. The chiral Majorana fields are order–disorder composites, while $\varepsilon=i\bar\psi\psi$ is local in all descriptions.

The chiral fermion is not one of the three scalar primaries of the bosonic diagonal Ising model. It is a chiral field with spin $1/2$:

$$s=h-\bar h={1\over2}.$$

It is perfectly local in the fermionic theory, but it has nontrivial monodromy with the spin field $\sigma$. This is why it is better to think of the Ising CFT as having several closely related presentations: the bosonic Ising model, the dual bosonic Ising model, and the fermionic theory with a choice of spin structure.

Fermion OPEs and the stress tensor

The free Majorana OPE is

$$\boxed{ \psi(z)\psi(w) ={1\over z-w}+\text{regular}. }$$

The holomorphic stress tensor is

$$\boxed{ T(z)=-{1\over2}:\psi\partial\psi:(z). }$$

With this convention, the stress tensor has the OPE

$$T(z)\psi(w) \sim {{1\over2}\psi(w)\over (z-w)^2} +{\partial\psi(w)\over z-w}.$$

Thus $\psi$ is a chiral primary of weight

$$h_\psi={1\over2}.$$

The same free-field calculation gives

$$T(z)T(w) \sim {c/2\over(z-w)^4} +{2T(w)\over(z-w)^2} +{\partial T(w)\over z-w}, \qquad c={1\over2}.$$

The first subleading local operator in the $\psi\psi$ OPE is the stress tensor:

$$\boxed{ \psi(z)\psi(w) ={1\over z-w}+2(z-w)T(w)+\cdots. }$$

The coefficient $2$ follows from expanding

$$\psi(z)=\psi(w)+(z-w)\partial\psi(w)+\cdots$$

and using

$$:\partial\psi\,\psi:=2T.$$

The short-distance product of two chiral Majorana fields contains the identity pole and, at the next nontrivial order, the stress tensor. This is the free-fermion origin of $c=1/2$.

In radial quantization the fermion has a mode expansion

$$\psi(z)=\sum_r \psi_r z^{-r-1/2}.$$

The allowed values of $r$ depend on the spin structure around the origin. In the Neveu–Schwarz sector,

$$r\in\mathbb Z+{1\over2},$$

while in the Ramond sector,

$$r\in\mathbb Z.$$

The OPE $\psi(z)\psi(w)\sim 1/(z-w)$ is equivalent to

$$\boxed{ \{\psi_r,\psi_s\}=\delta_{r+s,0}. }$$

The spin fields $\sigma$ and $\mu$ are precisely the operators that change the fermion boundary condition. In that sense, they are not optional decorations of the free-fermion theory; they are the twist fields that make the Ising theory complete.

Order and disorder fields as fermion twist fields

The order field $\sigma$ and the disorder field $\mu$ have the same conformal weights,

$$(h,\bar h)=\left({1\over16},{1\over16}\right),$$

but they are not simultaneously local bosonic fields. A fermion circling either of them changes sign. Locally, their OPEs with a chiral Majorana field have square-root singularities:

$$\psi(z)\sigma(0,0) \sim {\text{phase}\over z^{1/2}}\mu(0,0)+\cdots,$$

and

$$\psi(z)\mu(0,0) \sim {\text{phase}\over z^{1/2}}\sigma(0,0)+\cdots.$$

The phases depend on where the branch cut is placed. The exponent does not. It is fixed by conformal weights:

$$h_\mu-h_\psi-h_\sigma ={1\over16}-{1\over2}-{1\over16} =-{1\over2}.$$

This is the continuum form of the lattice statement that the fermion is an order–disorder composite. In a shorthand notation,

$$\psi\sim \sigma\mu,$$

but the symbol $\sim$ hides a branch cut. A more honest statement is that taking an order field around a disorder field produces a sign, and the endpoint of that sign defect carries fermionic statistics.

A Majorana fermion has square-root OPEs with the order and disorder fields. The branch cut is the local CFT version of the lattice disorder line.

This explains the three common Ising operator lists:

$$(\mathbf 1,\sigma,\varepsilon), \qquad (\mathbf 1,\mu,\varepsilon), \qquad (\mathbf 1,\sigma,\mu,\psi,\bar\psi,\varepsilon).$$

The first is the ordinary bosonic Ising presentation. The second is the dual presentation. The third is the enlarged order–disorder–fermion presentation. It is extremely useful for calculations and for understanding duality, but its fields have nontrivial mutual locality data.

Tricriticality from a φ⁶ Landau theory

The ordinary Ising critical point is reached by tuning one relevant even parameter when the magnetic field is set to zero. In Landau language, write

$$V(\phi)=m^2\phi^2+\lambda\phi^4+g\phi^6+\cdots, \qquad g>0.$$

If $\lambda>0$, the transition occurs by tuning $m^2$ through zero. The stabilizing interaction near the transition is then $\phi^4$, and the infrared fixed point in two dimensions is the ordinary Ising CFT.

For $\lambda<0$, the mean-field transition is first order. With the normalization written here, coexistence of the minimum at $\phi=0$ with a nonzero minimum follows from $V=0$ and $V'=0$. Writing $u=\phi^2$ gives

$$u=-{\lambda\over2g}, \qquad m^2={\lambda^2\over4g}.$$

This first-order line terminates where the continuous line $m^2=0$, $\lambda>0$ terminates.

The tricritical point occurs when the quartic term is also tuned to zero:

$$m^2=0, \qquad \lambda=0, \qquad g>0.$$

Thus the $\mathbb Z_2$-symmetric tricritical point has codimension two in the even coupling space. If the magnetic field is also allowed, there are additional odd relevant directions, but the simplest tricritical tuning is already visible in the even potential.

The corresponding continuum action is schematically

$$S\sim\int d^2x\, \left[(\partial\phi)^2+m^2\phi^2+\lambda\phi^4+g\phi^6\right].$$

The symbol $\phi$ here is a Landau–Ginzburg field, not a free scalar. At the tricritical fixed point it flows to the leading spin primary of the tricritical Ising CFT.

For $V=m^2\phi^2+\lambda\phi^4+g\phi^6$, the continuous line $m^2=0$, $\lambda>0$ meets the first-order line $m^2=\lambda^2/(4g)$, $\lambda<0$, at the tricritical point. Reaching that endpoint requires tuning both even couplings.

Mean-field theory already captures the need for two tunings, but it does not give the correct two-dimensional exponents. The exact infrared fixed point is instead the minimal model $\mathcal M(4,5)$.

The tricritical Ising minimal model

The bosonic tricritical Ising CFT is the next unitary Virasoro minimal model after Ising:

$$\boxed{ \mathcal M(4,5), \qquad c=1-{6\over4\cdot5}={7\over10}. }$$

Its Kac weights are

$$h_{r,s}^{(4,5)}={(5r-4s)^2-1\over80},$$

with

$$1\le r\le3, \qquad 1\le s\le4, \qquad (r,s)\sim(4-r,5-s).$$

There are therefore

$${1\over2}(4-1)(5-1)=6$$

primary families. A standard naming convention is

$$\begin{array}{c|c|c|c} \text{field} & (r,s)\ \text{representative} & h & \mathbb Z_2\ \text{parity} \\ \hline \mathbf 1 & (1,1) & 0 & + \\ \varepsilon & (1,2) & 1/10 & + \\ \varepsilon' & (1,3) & 3/5 & + \\ \varepsilon'' & (1,4) & 3/2 & + \\ \sigma & (2,2) & 3/80 & - \\ \sigma' & (2,1) & 7/16 & - \end{array}$$

For the diagonal scalar theory, the full scaling dimension is $\Delta=2h$. Hence

$$\Delta_\sigma={3\over40}, \qquad \Delta_{\sigma'}={7\over8}, \qquad \Delta_\varepsilon={1\over5}, \qquad \Delta_{\varepsilon'}={6\over5}, \qquad \Delta_{\varepsilon''}=3.$$

The four fields $\sigma$, $\sigma'$, $\varepsilon$, and $\varepsilon'$ are relevant in two dimensions. The two even relevant fields correspond, in the Landau picture, to the two even tunings $m^2$ and $\lambda$.

The table is a table of Virasoro representations. For the diagonal bosonic theory, the $h=3/2$ representation is paired with its anti-holomorphic partner to make $\varepsilon''$ with $(h,\bar h)=(3/2,3/2)$. It becomes part of the extended chiral vacuum sector only in the fermionic $N=1$ theory.

The bosonic tricritical Ising model $\mathcal M(4,5)$ has six Virasoro primary families after the reflection identification. Its $h=3/2$ representation supplies the supercurrent only after passing to the fermionic $N=1$ chiral extension.

The contrast with ordinary Ising is instructive. Ordinary Ising has one even relevant scalar, $\varepsilon$, and one odd relevant scalar, $\sigma$. Tricritical Ising has a second even relevant scalar $\varepsilon'$. This is the CFT meaning of tricriticality: there is one more relevant even direction to tune.

The supercurrent and the square root of translations

The tricritical fixed point has more structure than a generic Virasoro minimal model. When its spin sectors are retained, it is the first nontrivial unitary $N=1$ superconformal minimal model. The extended chiral algebra contains a fermionic holomorphic field $G(z)$ of weight

$$h_G={3\over2}.$$

We will call it the supercurrent. Its defining OPEs are

$$\boxed{ T(z)G(w) \sim {{3\over2}G(w)\over(z-w)^2} +{\partial G(w)\over z-w}, }$$

and

$$\boxed{ G(z)G(w) \sim {2c/3\over(z-w)^3} +{2T(w)\over z-w}. }$$

The first formula says that $G$ is a primary field of weight $3/2$. The second says that the product of two supercurrents closes on the identity family: the leading pole is a central term, and the next singular term is the stress tensor.

In modes,

$$T(z)=\sum_{n\in\mathbb Z}L_n z^{-n-2}, \qquad G(z)=\sum_r G_r z^{-r-3/2}.$$

The supercurrent OPE is equivalent to

$$\boxed{ \{G_r,G_s\} =2L_{r+s}+{c\over3}\left(r^2-{1\over4}\right)\delta_{r+s,0}. }$$

The values of $r$ are half-integers in the Neveu–Schwarz sector and integers in the Ramond sector. In the Neveu–Schwarz sector on the plane, the global holomorphic supersymmetry mode

$$Q=G_{-1/2}=\oint {dz\over2\pi i}\,G(z)$$

obeys

$$\boxed{ \{Q,Q\}=2L_{-1}. }$$

Since $L_{-1}$ generates translations on the plane,

$$[L_{-1},O(z)]=\partial O(z),$$

this equation says, in a literal algebraic sense, that the holomorphic supercharge is a square root of translation. The full two-dimensional theory also has the anti-holomorphic relation $\{\bar G_{-1/2},\bar G_{-1/2}\}=2\bar L_{-1}$.

In the fermionic $N=1$ extension, the supercurrent OPE packages the superconformal algebra. The mode $Q=G_{-1/2}$ satisfies $\{Q,Q\}=2L_{-1}$, so the holomorphic supersymmetry generator squares to a translation.

The ordinary Ising model also has a chiral fermion of weight $1/2$, but a weight-$1/2$ fermion is not a supercurrent. Supersymmetry requires a weight-$3/2$ current whose self-OPE produces the stress tensor. This is why the tricritical Ising model, not the ordinary Ising model, is the first unitary minimal model with $N=1$ superconformal symmetry.

How the two descriptions fit together

There are now three related but distinct layers:

1. The ordinary Ising CFT $\mathcal M(3,4)$ has $c=1/2$ and a free Majorana realization. Its chiral fermion has $h=1/2$, and the thermal operator is $\varepsilon=i\psi\bar\psi$.

2. The tricritical Ising CFT $\mathcal M(4,5)$ has $c=7/10$ and describes the infrared endpoint of a $\mathbb Z_2$-symmetric $\phi^6$ Landau theory after two even tunings.

3. Its fermionic spin-CFT extension has an $N=1$ chiral algebra generated by $T(z)$ and a supercurrent $G(z)$ of weight $3/2$. The diagonal bosonic theory and this fermionic extension have related Virasoro data but different locality and spin-structure bookkeeping.

The first layer explains fermionization and order–disorder variables. The second explains tricriticality and the appearance of extra relevant fields. The third explains why the next page naturally continues from conformal currents to worldlines, gauge fixing, and reparametrization: the idea that a local current can generate a square root of translations is already a geometric idea.

Summary

The Ising fixed point $\mathcal M(3,4)$ has central charge $c=1/2$ and admits a free Majorana representation. At criticality,

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

with

$$\psi(z)\psi(w)\sim{1\over z-w}, \qquad T=-{1\over2}:\psi\partial\psi:.$$

The energy operator is the Majorana mass operator,

$$\varepsilon=i\bar\psi\psi,$$

and the order and disorder fields are twist fields for the fermion.

The tricritical Ising point is reached in a $\mathbb Z_2$-symmetric Landau theory by tuning both $m^2$ and $\lambda$ in

$$S\sim\int d^2x\, \left[(\partial\phi)^2+m^2\phi^2+\lambda\phi^4+g\phi^6\right].$$

Its exact CFT is

$$\mathcal M(4,5), \qquad c={7\over10}.$$

The bosonic model has six Virasoro primaries. Its fermionic extension has an $N=1$ superconformal current $G$ of weight $3/2$ satisfying

$$G(z)G(w) \sim {2c/3\over(z-w)^3}+{2T(w)\over z-w}.$$

The corresponding mode algebra contains

$$\{G_{-1/2},G_{-1/2}\}=2L_{-1},$$

so supersymmetry appears as a square root of translation.

Common pitfalls

The Majorana fields $\psi$ and $\bar\psi$ are chiral fermions, not scalar bosonic primaries of the diagonal Ising model. They are local in the fermionic theory but have branch cuts with $\sigma$ and $\mu$.

The disorder field $\mu$ has the same scaling dimension as $\sigma$, but $\sigma$ and $\mu$ are not simply two independent local scalar fields in one ordinary bosonic theory. Their mutual locality data matter.

The field $\phi$ in the Landau $\phi^6$ description is not a free scalar field. At the tricritical fixed point it flows to a linear combination whose leading piece is the tricritical spin primary.

For minimal models, the Kac weights $h_{r,s}$ are chiral weights. The full scaling dimension of a diagonal scalar primary is $\Delta=2h$.

A chiral field of weight $1/2$ is a fermion, not a supercurrent. The $N=1$ supercurrent has weight $3/2$ and its self-OPE produces the stress tensor.

The scalar Virasoro primary $\varepsilon''$ of the diagonal bosonic model is not itself the holomorphic current $G(z)$. Both involve the $h=3/2$ Virasoro representation, but $G$ belongs to the extended chiral vacuum sector of the fermionic theory.

Exercises

Majorana transformation law

Using the Majorana stress tensor

$$T(z)=-{1\over2}:\psi\partial\psi:(z)$$

and the OPE $\psi(z)\psi(w)\sim1/(z-w)$, derive the leading singular terms in $T(z)\psi(w)$.

Solution

Use Wick contraction inside

$$T(z)\psi(w)=-{1\over2}:\psi(z)\partial\psi(z):\psi(w).$$

There are two possible contractions:

$$\psi(z)\psi(w)\sim {1\over z-w}, \qquad \partial\psi(z)\psi(w)\sim \partial_z {1\over z-w}=-{1\over (z-w)^2}.$$

Keeping the signs from moving fermions through each other gives

$$T(z)\psi(w) \sim {{1\over2}\psi(w)\over(z-w)^2} +{\partial\psi(w)\over z-w}.$$

Thus $\psi$ is a primary of holomorphic weight $h=1/2$.

Energy-operator two-point function

Show that the Ising energy operator $\varepsilon=i\bar\psi\psi$ has the two-point function of a scalar primary of dimension $\Delta=1$.

Solution

Using the chiral OPEs,

$$\psi(z)\psi(0)\sim{1\over z}, \qquad \bar\psi(\bar z)\bar\psi(0)\sim{1\over\bar z},$$

we find, up to the overall sign fixed by the choice of $i$ in $\varepsilon$,

$$\langle \varepsilon(z,\bar z)\varepsilon(0)\rangle \propto \langle \psi(z)\psi(0)\rangle \langle \bar\psi(\bar z)\bar\psi(0)\rangle ={1\over z\bar z}={1\over |z|^2}.$$

A scalar primary of full scaling dimension $\Delta$ has two-point function proportional to $|z|^{-2\Delta}$. Therefore $\Delta=1$, or equivalently $(h,\bar h)=(1/2,1/2)$.

The tricritical Kac table

Compute the six distinct chiral weights of the tricritical Ising model $\mathcal M(4,5)$ from

$$h_{r,s}={(5r-4s)^2-1\over80}.$$

Solution

The allowed labels are

$$1\le r\le3, \qquad 1\le s\le4,$$

with the identification $(r,s)\sim(4-r,5-s)$. Evaluate representative entries:

$$h_{1,1}=0,$$ $$h_{1,2}={(5-8)^2-1\over80}={8\over80}={1\over10},$$ $$h_{1,3}={(5-12)^2-1\over80}={48\over80}={3\over5},$$ $$h_{1,4}={(5-16)^2-1\over80}={120\over80}={3\over2},$$ $$h_{2,2}={(10-8)^2-1\over80}={3\over80},$$

and

$$h_{2,1}={(10-4)^2-1\over80}={35\over80}={7\over16}.$$

The remaining entries are paired with these by the reflection identification. Thus the six weights are

$$0, \quad {1\over10}, \quad {3\over5}, \quad {3\over2}, \quad {3\over80}, \quad {7\over16}.$$

Codimension of tricritical tuning

Explain why a $\mathbb Z_2$-symmetric $\phi^6$ Landau theory has a tricritical point of codimension two in the even coupling space.

Solution

Take

$$V(\phi)=m^2\phi^2+\lambda\phi^4+g\phi^6, \qquad g>0.$$

For $\lambda>0$, the ordinary continuous transition is reached by tuning only $m^2$ to zero. The quartic term stabilizes the potential near the origin.

A tricritical point occurs when the quartic term is also absent at the transition, so the leading stabilizing interaction is $g\phi^6$. This requires

$$m^2=0, \qquad \lambda=0.$$

Thus two independent even parameters must be tuned. The tricritical point is therefore codimension two in the even coupling space.

A supercharge squares to translation

Assume a holomorphic supercurrent $G(z)$ satisfies

$$G(z)G(w) \sim {2c/3\over(z-w)^3} +{2T(w)\over z-w}.$$

Use the standard mode algebra to show that $Q=G_{-1/2}$ squares to a translation.

Solution

The OPE is equivalent to

$$\{G_r,G_s\} =2L_{r+s}+{c\over3}\left(r^2-{1\over4}\right)\delta_{r+s,0}.$$

Set $r=s=-1/2$. Then $r+s=-1$, and the central term vanishes because

$$r^2-{1\over4}={1\over4}-{1\over4}=0.$$

Therefore

$$\{G_{-1/2},G_{-1/2}\}=2L_{-1}.$$

Since $L_{-1}$ acts on local fields as

$$[L_{-1},O(z)]=\partial O(z),$$

it is the holomorphic translation generator. Hence $Q=G_{-1/2}$ is a square root of translation.

Relevant tricritical primaries

Which scalar primaries of the diagonal tricritical Ising model are relevant in two dimensions?

Solution

A scalar perturbation is relevant in two dimensions when its full scaling dimension satisfies

$$\Delta<2.$$

For diagonal scalar primaries, $\Delta=2h$. The six tricritical Ising chiral weights are

$$0, \quad {1\over10}, \quad {3\over5}, \quad {3\over2}, \quad {3\over80}, \quad {7\over16}.$$

The nontrivial full dimensions are

$${1\over5}, \quad {6\over5}, \quad 3, \quad {3\over40}, \quad {7\over8}.$$

The relevant scalar primaries are those with full dimension below $2$:

$$\varepsilon, \qquad \varepsilon', \qquad \sigma, \qquad \sigma'.$$

The field $\varepsilon''$ has $\Delta=3$ and is irrelevant as a scalar perturbation.

References

• A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Physics B 241 (1984), 333–380.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer, 1997, Chapters 7–12.
• D. Friedan, Z. Qiu, and S. Shenker, “Conformal invariance, unitarity, and critical exponents in two dimensions,” Physical Review Letters 52 (1984), 1575–1578.
• P. Ginsparg, “Applied Conformal Field Theory,” in Fields, Strings and Critical Phenomena, Les Houches Session XLIX, Elsevier, 1989, 1–168.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, 1987.