22.1. Tricritical Ising model with even perturbations Consider the TIM deformed by its even fields (x) (energy density field) and t(x)

(vacancy density field) with conformal weights given by 2 = 1/10 and 4 = 3/5 respectively. The action of the perturbed model can be formally written as A = A0 +g2  ε(x)d2x+g4  t(x)d2x,

where A0 is the action of the TIM at its conformal point, whereas g2 and g4 are the two coupling constants of our theory. The theory depends upon the dimensionless combinations η± = g4/|g2| 49 , where η+ if g2 > 0 and η− if g2 < 0.
The model is integrable both when g2 = 0 (this is the supersymmetry line of the model) and g4 = 0 (this is the integrable line associated to the E7 structure). In more details: the g4 > 0 half line consists of a massless RG flow to the Ising critical point in which the supersymmetry is spontaneously broken, while the g4 < 0 half line consists of first-order phase transition points with unbroken supersymmetry. Along the g4 = 0 line the particle spectrum of the model and its exact S-matrix are related to the E7 Lie algebra, where the high- and low-temperature phases are related by duality.

a. Use the identification ε ∼: φ2 :, t ∼: φ4 : to express the perturbed action (22.6.1)
in terms of a Landau–Ginzburg potential and study the evolution of this potential by moving the couplings. Show that the different shapes of the potential match with those shown in Figure 22.6.

b. Use the self-duality of the model to argue that the operator t(x) is local with respect to the low-temperature kinks.

c. Using the FFPT nearby the low-temperature axis to show that that the kink excitations do not get confined once the E7 integrable theory is perturbed by the operator t(x).

d. From the shape of the potential nearby the vertical negative axis of Figure 22.6, argue that the operator (x) is non-local with respect to the kinks relative to the first-order phase line of the model.