Ising CFT, Majorana Fermions, and Tricritical Extensions

The Ising operator algebra has three chiral primary families,

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

with

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

The Ising fixed point has a free Majorana realization. It explains the energy operator as a fermion bilinear and the order and disorder fields as twists that create branch cuts. The local fermion equations and OPEs are simple; recovering the bosonic Ising theory also requires the appropriate global combination of fermion sectors.

The main new endpoint is the tricritical Ising model. A $\phi^6$ Landau potential illustrates its two even tunings at zero magnetic field. The mean-field picture organizes these tunings; its exponents are replaced in two dimensions by those of 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.

Helpful background. Free fermion correlators supply the normalized Majorana fields, scaling dimensions fix the thermal coupling, and minimal models and fusion supply the Kac labels and chiral fusion rules used below.

The Majorana realization of the Ising fixed point

Normalization of fermion OPEs

We retain the unit-pole Majorana normalization of the earlier free-correlator calculation:

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

The two components anticommute with one another. Their relative phase and the energy sign are fixed by the continuum dictionary below, so they cannot be changed independently of the signed thermal coupling.

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 the two Euclidean chiral components of a Majorana system, $\psi(z)$ and $\bar\psi(\bar z)$. They are independent Grassmann fields in the Euclidean functional integral, not pointwise complex conjugates; the Majorana reality condition refers to the Lorentzian continuation and is encoded Euclideanly by an antisymmetric kinetic operator and a Pfaffian. At the critical point they obey

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

In the independent fields $u,v$ of the Ising continuum limit, the normalized components are

$$\psi=\sqrt{2\pi}\,u, \qquad \bar\psi=i\sqrt{2\pi}\,v.$$

The inherited Euclidean action near the Ising point is therefore

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

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

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

with unit two-point normalization, as checked in the exercises. The factor $1/(2\pi)$ in the action gives the chiral kernel $\bar\partial/\pi$ when the Grassmann quadratic form is written with its factor $1/2$; its inverse is $1/z$. This component-action and OPE normalization is developed in Di Francesco, Mathieu and Sénéchal 1997, §5.3.2, pp. 129–132.

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. Comparing with the earlier thermal deformation gives

$$S=S_*+\tau\int d^2z\,\varepsilon, \qquad m=2\pi\tau.$$

In this inherited convention, positive $m$ and positive $\tau$ are on the ordered side. Changing their sign exchanges ordered and disordered phases under Kramers–Wannier duality. Reversing the Grassmann order without a minus sign would reverse this thermal calibration. The signed Ising OPE uses the same energy field and $C_{\sigma\sigma\varepsilon}=-1/2$; a simultaneous rephasing of energy and thermal coupling describes the same physics.

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

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

For example, writing $s=z-w$, the single contractions give $:\psi(z)\partial\psi(z):\,\psi(w)\sim-\partial\psi(z)/s-\psi(z)/s^2$. Multiplication by $-1/2$ and expansion about $w$ produce both displayed poles, including the full coefficient of $\partial\psi$.

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 double contractions contribute $1/[4(z-w)^4]$, so the central coefficient is $c/2=1/4$. These free-field Ward OPEs are worked out in Di Francesco, Mathieu and Sénéchal 1997, §5.3.2, pp. 131–132, Eqs. 5.96–5.100.

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 stress tensor here is a descendant in the identity family, rather than an additional primary family in the fusion product.

In radial quantization the fermion has a mode expansion

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

The mode indices label fermionic sectors. In the Neveu–Schwarz (NS) sector,

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

while in the Ramond sector,

$$r\in\mathbb Z.$$

Periodicity also depends on the conformal frame. On a cylinder of circumference $L$ with coordinate $w\sim w+iL$, the standard-frame fermion is antiperiodic in NS and periodic in R. The map to the plane gives

$$z=e^{2\pi w/L}, \qquad \psi_{\rm cyl}(w)=\left(\frac{2\pi z}{L}\right)^{1/2}\psi(z).$$

The square-root frame factor changes sign around the cylinder. Consequently the plane component in the expansion above is single-valued in NS and changes sign around the origin in R. The local $1/(z-w)$ OPE pole holds in either sector; a global Ramond-state two-point function has additional dependence on the insertions that define that state. See Di Francesco, Mathieu and Sénéchal 1997, §§6.4.1–6.4.2, pp. 168–170, Eqs. 6.96–6.109.

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 $\mu$ change the local fermion monodromy and connect the corresponding sectors. Global equivalence to bosonic Ising additionally requires the appropriate spin-structure combination and fermion-parity projection; adjoining twists to a fixed-spin-structure theory is not by itself that construction. The Ising torus sector combination illustrates this extra information in Di Francesco, Mathieu and Sénéchal 1997, §12.4.1, pp. 454–455, Eqs. 12.72–12.74 and continuation.

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:

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

and

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

The phases of $a,b$ depend on the branch-cut and field-phase choices. The exponent does not: it is fixed by conformal weights,

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

The square-root OPEs are stated up to multiplicative factors in Di Francesco, Mathieu and Sénéchal 1997, §12.2.2, p. 445, Eq. 12.26. Their magnitudes can be fixed in the unitary fermionic presentation with orthonormal Ramond twist ground states. The Clifford relation gives $\psi_0^2=1/2$, and the leading OPE terms give $\psi_0|\sigma\rangle=a|\mu\rangle$, $\psi_0|\mu\rangle=b|\sigma\rangle$. With radial adjoint $\psi_0^\dagger=\psi_0$, this implies

$$ab=\frac12, \qquad b=a^*, \qquad |a|=|b|=\frac1{\sqrt2}.$$

The zero-mode relation is described in Di Francesco, Mathieu and Sénéchal 1997, §6.4.1, p. 169. This argument fixes the amplitude, while leaving the phase convention explicit.

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 this means a point-split product projected onto the chiral fermion channel, with its separation-dependent power and branch choice included. It is not an ordinary coincident product of two mutually local bosonic fields. The branch cut records the lattice disorder line, whose endpoint binds to an order insertion to give the fermionic observable.

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 nonidentity relevant even parameter when the magnetic field is set to zero. To organize tricritical tuning, consider the mean-field potential

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

In this section, $m^2$ names a signed Landau coefficient and can be negative; it is not the square of the Majorana mass used earlier. The statements about transition curves below refer to the displayed polynomial truncated at $\phi^6$, with zero magnetic field.

If $\lambda>0$, the mean-field continuous transition occurs by tuning this coefficient through zero. The quartic term stabilizes the potential near the transition. With fluctuations included, the corresponding two-dimensional critical behavior is ordinary Ising; the bare location of its critical surface need not remain $m^2=0$.

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

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

At this value the potential factorizes as

$$V(\phi)=g\phi^2\left(\phi^2+\frac{\lambda}{2g}\right)^2,$$

so the origin and the two nonzero minima are degenerate global minima. The mean-field first-order line terminates where the continuous line $m^2=0$, $\lambda>0$ terminates. This is the same coexistence calculation as in fixed points and tricriticality, with the potential coefficients translated to the normalization above.

The mean-field 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 its leading scaling component is the spin primary of the tricritical Ising CFT, with a nonuniversal normalization.

Mean-field theory captures the need for two tunings, but it does not give the correct two-dimensional exponents. At the renormalized tricritical point the two even relevant scaling fields are combinations of bare couplings, and the exact critical theory is $\mathcal M(4,5)$. The Landau–Ginzburg interpretation and its limitations are developed in Di Francesco, Mathieu and Sénéchal 1997, §7.4.7, pp. 231–235.

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

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

chiral irreducible primary families. Their diagonal bulk pairing gives the six scalar families below, with the conventional field names and spin-reversal parity. The weights and fusion data are given in Di Francesco, Mathieu and Sénéchal 1997, §7.4.3, pp. 222–224, Tables 7.2–7.3.

Field	Kac representative	Chiral weight $h$	Spin-reversal parity
$\mathbf 1$	$(1,1)$	$0$	Even
$\varepsilon$	$(1,2)$	$1/10$	Even
$\varepsilon'$	$(1,3)$	$3/5$	Even
$\varepsilon''$	$(1,4)$	$3/2$	Even
$\sigma$	$(2,2)$	$3/80$	Odd
$\sigma'$	$(2,1)$	$7/16$	Odd

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 nonidentity fields $\sigma$, $\sigma'$, $\varepsilon$, and $\varepsilon'$ are relevant in two dimensions. The two even relevant fields correspond to the two renormalized even tunings illustrated by $m^2$ and $\lambda$ in the Landau picture. The identity only shifts the vacuum energy.

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 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 theory admits a fermionic chiral extension joining its vacuum and weight-$3/2$ Virasoro representations, together with compatible bulk spin sectors. This extension is the first nontrivial unitary $N=1$ superconformal minimal model; it is additional structure beyond the diagonal bosonic theory. The superconformal series and its tricritical realization are described in Di Francesco, Mathieu and Sénéchal 1997, §7.4.3, pp. 223–225, Eqs. 7.87–7.88. The extended chiral algebra contains a fermionic holomorphic field $G(z)$ of weight

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

We call it the supercurrent. The $(1,4)$ representation self-fuses into the identity family. With $G$ normalized so that its self-OPE has coefficient $2$ multiplying $T$, 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

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

To see the central coefficient, choose a branch regular near $w\ne0$. The inner counterclockwise residue of $z^{r+1/2}(2c/3)/(z-w)^3$ gives $(c/3)(r^2-1/4)w^{r-3/2}$. The outer mode integral against $w^{s+1/2}$ selects $r+s=0$. The simple $2T$ pole similarly gives $2L_{r+s}$. The graded sum of the two radial orderings produces the anticommutator.

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 $Q^2=L_{-1}$: the holomorphic supercharge squares to a plane translation. Its action on a field uses the graded commutator, with an anticommutator when the field is fermionic. The full two-dimensional fermionic extension also has the anti-holomorphic NS relation $\{\bar G_{-1/2},\bar G_{-1/2}\}=2\bar L_{-1}$.

The ordinary Ising model also has a chiral fermion of weight $1/2$, but that field is not a supercurrent. The $N=1$ algebra requires a weight-$3/2$ current with the stated self-OPE. The tricritical fermionic extension realizes the first nontrivial member of the unitary $N=1$ series; its preceding endpoint has $c=0$ and is trivial.

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,

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

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 diagonal bosonic model has six scalar Virasoro primary families. 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 NS 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

Using the unit-normalized plane vacuum correlators, show that the Ising energy operator $\varepsilon=i\psi\bar\psi$ has the unit two-point function of a scalar primary of dimension $\Delta=1$.

Solution

Using the chiral OPEs,

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

the factor $i^2=-1$ multiplies the four-fermion correlator. Moving $\bar\psi(\bar z)$ past $\psi(0)$ supplies a second minus sign; mixed-chirality contractions vanish. Thus

$$\langle \varepsilon(z,\bar z)\varepsilon(0)\rangle =-\langle\psi(z)\bar\psi(\bar z)\psi(0)\bar\psi(0)\rangle$$

and hence

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

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

In the truncated mean-field potential at zero magnetic field, explain why a $\mathbb Z_2$-symmetric $\phi^6$ Landau theory has a tricritical point of codimension two in the even coupling space. Here $m^2$ is the signed quadratic Landau coefficient.

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

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

In the Neveu–Schwarz sector, use the standard mode algebra to show that $Q=G_{-1/2}$ squares to a holomorphic plane 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 nonidentity scalar primaries of the diagonal tricritical Ising model are relevant in two dimensions? The identity perturbation only shifts the vacuum energy.

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

• Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal Field Theory, Springer, 1997. DOI.