Duality and form factors in the thermally deformed two-dimensional tricritical Ising model

Abstract

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional E_7E7 Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation operators, \sigma(x)σ(x) and \sigma'(x)σ′(x), in either phase are accompanied by associated disorder operators, \mu(x)μ(x) and \mu'(x)μ′(x). Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For \sigma(x)σ(x) and \sigma'(x)σ′(x), the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest \mathbb{Z}_2ℤ2 odd particles. Similarly, the equations for \mu(x)μ(x) and \mu'(x)μ′(x) have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions are eventually not independent; instead, the parameters of the solutions for \sigma(x)/\sigma'(x)σ(x)/σ′(x) are fixed in terms of those for \mu(x)/\mu'(x)μ(x)/μ′(x). We use the truncated conformal space approach to confirm numerically the derived expressions of the matrix elements as well as the validity of the \DeltaΔ-sum rule as applied to the off-critical correlators. We employ the derived form factors of the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.

Form factors of the tricritical three-state Potts model in its scaling limit

“…While the second solution is associated with the disorder operator µ, the first one is acceptable: its contribution to the ∆-theorem is given in table 12 and it is compatible with the conformal weight of the operator. Notice that convergence is slower than in the leading case, as it generally happens when one considers less relevant operators-see also [22]. Solving the system of equations, together with equation (63), yields the results of table 11.…”

“…Similarly, one defines the components of the subleading order operator Z to be Z and Z. As it happens in the case of the Ising model [21], the tricritical Ising model [22] and the three-state Potts model [23,24], in view of the self-duality of the lattice model, we can introduce a leading disorder operator (µ, μ) associated with the leading-order operator (σ, σ). In complete analogy to its leading counterpart, (ζ, ζ) is the subleading disorder operator associated with the subleading order operator (Z, Z).…”

“…the only contribution that needs to be evaluated numerically in equation (22) is that of F µ l l (θ). Indeed, because of C-invariance, the FFs of the operator μ are read from that of µ:…”

Variations on vacuum decay: the scaling Ising and tricritical Ising field theories

“…In the paramagnetic phase there is a single vacuum and all 7 excitations are topologically trivial. We recently examined this model in detail [15], and the interested reader is referred to this work for more details and further references.…”

“…These include vacuum degeneracy unrelated to any spontaneous symmetry breaking, and also phases with three degenerate vacua. The scaling tricritical Ising field theory has recently been revisited by the authors of the present paper in relation, in particular, of two topics: the study of the Kramers-Vannier duality using the form factor bootstrap and the integrability of the model [15], and also the confinement phenomenon of the kink excitations into mesons [16]. We note that kink confinement is another facet of lifting vacuum degeneracy, and so the study in this paper can also be viewed as complementary and a natural extension of [16].…”