Confinement in the tricritical Ising model (original) (raw)

Thermodynamic Bethe ansatz for the subleading magnetic perturbation of the tricritical Ising model

Nuclear Physics B, 1998

We give further support to Smirnov's conjecture on the exact kink S-matrix for the massive Quantum Field Theory describing the integrable perturbation of the c = 0.7 minimal Conformal Field theory (known to describe the tri-critical Ising model) by the operator φ 2,1. This operator has conformal dimensions (7 16 , 7 16) and is identified with the subleading magnetic operator of the tri-critical Ising model. In this paper we apply the Thermodynamic Bethe Ansatz (TBA) approach to the kink scattering theory by explicitly utilising its relationship with the solvable lattice hard hexagon model. Analytically examining the ultraviolet scaling limit we recover the expected central charge c = 0.7 of the tri-critical Ising model. We also compare numerical values for the ground state energy of the finite size system obtained from the TBA equations with the results obtained by the Truncated Conformal Space Approach and Conformal Perturbation Theory.

The particle spectrum of the tricritical Ising model with spin reversal symmetric perturbations

Journal of Statistical Mechanics-theory and Experiment, 2008

We analyze the evolution of the particle spectrum of the tricritical Ising model by varying the couplings of the energy and vacancy density fields. The particle content changes from the spectrum of a supersymmetric theory (either of an exact or a spontaneously broken supersymmetric theory) to the spectrum of seven particles related to the underlying E7 structure. In the low

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

SciPost physics, 2022

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 7 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, σ(x) and σ (x), in either phase are accompanied by associated disorder operators, µ(x) and µ (x). Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For σ(x) and σ (x), the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest 2 odd particles. Similarly, the equations for µ(x) and 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 σ(x)/σ (x) are fixed in terms of those for 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 ∆-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.

Analytic properties of the free energy: the tricritical Ising model

2007

We investigate the tricritical Ising model in complex magnetic field in order to characterize the analytic structure of its free energy. By supplementing analytic methods with the truncation of conformal space technique we obtain nonperturbative data even if the field theories we consider are not integrable. The existence of edge singularities analogous to the Lee-Yang points in the Ising field theory is confirmed. A surprising result, due to the conformal dimensions of the operators involved, is the appearance of two branching points which seems appealing to identify with a pair of complex conjugate spinodal singularities.

Low temperature expansion for the Ising model

Physical Review Letters, 1992

We compute the weak coupling expansion for the energy of the three dimensional Ising model through 48 excited bonds. We also compute the magnetization through 40 excited bonds. This was achieved via a recursive enumeration of states of fixed energy on a set of finite lattices. We use a linear combination of lattices with a generalization of helical boundary conditions to eliminate finite volume effects.

Quench dynamics of the Ising field theory in a magnetic field

SciPost physics, 2018

We numerically simulate the time evolution of the Ising field theory after quenches starting from the E 8 integrable model using the Truncated Conformal Space Approach. The results are compared with two different analytic predictions based on form factor expansions in the pre-quench and post-quench basis, respectively. Our results clarify the domain of validity of these expansions and suggest directions for further improvement. We show for quenches in the E 8 model that the initial state is not of the integrable pair state form. We also construct quench overlap functions and show that their high-energy asymptotics are markedly different from those constructed before in the sinh/sine-Gordon theory, and argue that this is related to properties of the ultraviolet fixed point.

Exact (d) map (+)&(-) boundary flow in the tricritical Ising model

J Stat Mech Theory Exp, 2004

The integrable perturbation of the degenerate boundary condition (d) by the ϕ 1,3 boundary field generates a renormalization group flow down to the superposition of Cardy boundary states (+)&(−). Exact Thermodynamic Bethe Ansatz (TBA) equations for all the excited states are derived here extending the results of [1] to this case. As an intermediate step, the non-Cardy boundary conformal sector (+)&(−) is also described as the scaling limit of an A 4 lattice model with appropriate integrable boundary conditions and produces the first example of superposition of finitized Virasoro characters.

Exact ϕ1,3 boundary flows in the tricritical Ising model

Nuclear Physics B, 2003

We consider the tricritical Ising model on a strip or cylinder under the integrable perturbation by the thermal ϕ 1,3 boundary field. This perturbation induces five distinct renormalization group (RG) flows between Cardy type boundary conditions labelled by the Kac labels (r, s). We study these boundary RG flows in detail for all excitations. Exact Thermodynamic Bethe Ansatz (TBA) equations are derived using the lattice approach by considering the continuum scaling limit of the A 4 lattice model with integrable boundary conditions. Fixing the bulk weights to their critical values, the integrable boundary weights admit a thermodynamic boundary field ξ which induces the flow and, in the continuum scaling limit, plays the role of the perturbing boundary field ϕ 1,3. The excitations are completely classified, in terms of string content, by (m, n) systems and quantum numbers but the string content changes by either two or three well-defined mechanisms along the flow. We identify these mechanisms and obtain the induced maps between the relevant finitized Virasoro characters. We also solve the TBA equations numerically to determine the boundary flows for the leading excitations.