Tricritical Ising Model Research Articles

Lattice approach to excited TBA boundary flows: tricritical Ising model

We show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields ξ and η which play the role of the perturbing boundary fields ϕ1,3 and ϕ1,2 inducing the renormalization group flow between boundary fixed points. The excitations are completely classified in terms of (m,n) systems and quantum numbers but the string content changes by certain mechanisms along the flow. For our prototypical example, we identify these mechanisms and the induced map between the relevant finitized Virasoro characters. We also solve the boundary TBA equations numerically to determine the flows for the leading excitations.

String theory on G2 manifolds based on Gepner construction

We study the type II string theories compactified on manifolds of G 2 holonomy of the type (Calabi–Yau 3 -fold×S 1)/ Z 2 where CY 3 sectors realized by the Gepner models. We construct modular invariant partition functions for G 2 manifold for arbitrary Gepner models of the Calabi–Yau sector. We note that the conformal blocks contain the tricritical Ising model and find extra massless states in the twisted sectors of the theory when all the levels k i of minimal models in Gepner constructions are even.

Ising tricriticality and the dilute A3model

Some universal amplitude ratios appropriate to the 2,1 perturbation of the c = 7/10 minimal field theory, the subleading magnetic perturbation of the tricritical Ising model, are explicitly demonstrated in the dilute A3 model, in regime 1.

CFT description of string theory compactified on non-compact manifolds with G2 holonomy

We construct modular invariant partition functions for strings propagating on non-compact manifolds of G_2 holonomy. Our amplitudes involve a pair of N=1 minimal models M_m, M_{m+2} (m=3,4,...) and are identified as describing strings on manifolds of G_2 holonomy associated with A_{m-2} type singularity. It turns out that due to theta function identities our amplitudes may be cast into a form which contain tricritical Ising model for any m. This is in accord with the results of Shatashvili and Vafa. We also construct a candidate partition function for string compactified on a non-compact Spin(7) manifold.

Consistent superconformal boundary states

We propose a supersymmetric generalization of Cardy's equation for consistent N = 1 superconformal boundary states. We solve this equation for the superconformal minimal models (p/p + 2) with p odd, and thereby provide a classification of the possible superconformal boundary conditions. In addition to the Neveu-Schwarz (NS) and Ramond boundary states, there are states. The NS and boundary states are related by a Z2 `spin-reversal' transformation. We treat the tricritical Ising model as an example, and in an appendix we discuss the (non-superconformal) case of the Ising model.

Excited TBA equations I: Massive tricritical Ising model

We consider the massive tricritical Ising model M(4,5) perturbed by the thermal operator phi_{1,3} in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels (r,s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massive thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A_4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime III. The complete classification of excitations, in terms of (m,n) systems, is precisely the same as at the conformal tricritical point. Our methods also apply on a torus but we first consider (r,s) boundaries on the cylinder because the classification of states is simply related to fermionic representations of single Virasoro characters chi_{r,s}(q). We study the TBA equations analytically and numerically to determine the conformal UV and free particle IR spectra and the connecting massive flows. The TBA equations in Regime IV and massless RG flows are studied in Part II.

Random defect lines in conformal minimal models

We analyze the effect of adding quenched disorder along a defect line in the 2D conformal minimal models using replicas. The disorder is realized by a random applied magnetic field in the Ising model, by fluctuations in the ferromagnetic bond coupling in the Tricritical Ising model and Tricritical Three-state Potts model (the $\phi_{12}$ operator), etc.. We find that for the Ising model, the defect renormalizes to two decoupled half-planes without disorder, but that for all other models, the defect renormalizes to a disorder-dominated fixed point. Its critical properties are studied with an expansion in $\eps \propto 1/m$ for the mth Virasoro minimal model. The decay exponents $X_N=\frac{N}{2}(1-\frac{9(3N-4)}{4(m+1)^2}+ \mathcal{O}(\frac{3}{m+1})^3)$ of the Nth moment of the two-point function of $\phi_{12}$ along the defect are obtained to 2-loop order, exhibiting multifractal behavior.This leads to a typical decay exponent $X_{\rm typ}={1/2} (1+\frac{9}{(m+1)^2}+\mathcal{O}(\frac{3}{m+1})^3)$. One-point functions are seen to have a non-self-averaging amplitude. The boundary entropy is larger than that of the pure system by order 1/m^3. As a byproduct of our calculations, we also obtain to 2-loop order the exponent $\tilde{X}_N=N(1-\frac{2}{9\pi^2}(3N-4)(q-2)^2+\mathcal{O}(q-2)^3)$ of the Nth moment of the energy operator in the q-state Potts model with bulk bond disorder.

Universal amplitude ratios of the renormalization group: two-dimensional tricritical Ising model.

The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.

Splitting of Heterogeneous Boundaries in a System of the Tricritical Ising Model Coupled to 2-Dim Gravity

We study disk amplitudes whose boundaries have heterogeneous matter states in a system of $(4,5)$ conformal matter coupled to 2-dim gravity. They are analysed by using the 3-matrix chain model in the large $N$ limit. Each of the boundaries is composed of two or three parts with distinct matter states. From the obtained amplitudes, it turns out that each heterogeneous boundary loop splits into several loops and we can observe properties in the splitting phenomena that are common to each of them. We also discuss the relation to boundary operators.

Universal ratios in the 2D tricritical ising model

We consider the universality class of the two-dimensional tricritical Ising model. The scaling form of the free energy leads to the definition of universal ratios of critical amplitudes which may have experimental relevance. We compute these universal ratios by a combined use of results coming from perturbed conformal field theory, integrable quantum field theory, and numerical methods.

The wrong kind of gravity

The KPZ formula [V.G. Knizhnik, A.M. Polyakov, and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819] shows that coupling central charge c≤1 spin models to 2D quantum gravity dresses the conformal weights to get new critical exponents, where the relation between the original and dressed weights depends only on c. At the discrete level the coupling to 2D gravity is effected by putting the spin models on annealed ensembles of Φ 3 planar random graphs or their dual triangulations, where the connectivity fluctuates on the same time-scale as the spins. Since the sole determining factor in the dressing is the central charge, one could contemplate putting a spin model on a quenched ensemble of 2D gravity graphs with the “wrong” c value. We might then expect to see the critical exponents appropriate to the c value used in generating the graphs. In such cases the KPZ formula could be interpreted as giving a continuous line of critical exponents which depend on this central charge. We note that rational exponents other than the KPZ values can be generated using this procedure for the Ising, tricritical Ising and 3-state Potts models.

Tricritical Ising model with a boundary

We study the integrable and supersymmetric massive Φ(1,3) deformation of the tricritical Ising model in the presence of a boundary. We use constraints from supersymmetry in order to compute the exact boundary S-matrices, which turn out to depend explicitly on the topological charge of the supersymmetry algebra. We also solve the general boundary Yang-Baxter equation and show that in appropriate limits the general reflection matrices go over the supersymmetry preserving solutions. Finally, we briefly discuss the possible connection between our reflection matrices and boundary perturbations within the framework of perturbed boundary conformal field theory.

TRICRITICAL ISING MODEL NEAR CRITICALITY

The most relevant thermal perturbation of the continuous d=2 minimal conformal theory with c=7/10 (Tricritical Ising Model) is treated here. This model describes the scaling region of the ϕ6 universality class near the tricritical point. The problematic IR divergences of the naive perturbative expansion around conformal theories are dealt within the OPE approach developed at all orders by the authors. The main result is a description of the short distance behavior of correlators that is compared with existing long distance expansion (form factors approach) related to the integrability of the model.

Fusion rules in N=1 superconformal minimal models

The generalization to N=1 superconformal minimal models of the relation between the modular transformation matrix and the fusion rules in rational conformal field theories, the Verlinde theorem, is shown to provide complete information about the fusion rules, including their fermionic parity. The results for the superconformal Tricritical Ising and Ashkin-Teller models agree with the known rational conformal formulation. The Coulomb gas description of correlation functions in the Ramond sector of N=1 minimal models is also discussed and a previous formulation is completed.

Interaction of boundaries with heterogeneous matter states in matrix models

We study disk amplitudes whose boundary conditions on matter configurations are not restricted to homogeneous ones. They are examined in the two-matrix model as well as in the three-matrix model for the case of the tricritical Ising model. Comparing these amplitudes, we demonstrate relations between degrees of freedom of matter states in the two models. We also show that they have a simple geometrical interpretation in terms of interactions of the boundaries. It plays an important role that two parts of a boundary with different matter states stick to each other. We also find two closed sets of Schwinger-Dyson equations which determine disk amplitudes in the three-matrix model.

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

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 tricritical 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 tricritical 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 tricritical 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.

Vacuum expectation values from a variational approach

In this letter we propose to use an extension of the variational approach known as Truncated Conformal Space to compute numerically the Vacuum Expectation Values of the operators of a conformal field theory perturbed by a relevant operator. As an example we estimate the VEV's of all (UV regular) primary operators of the Ising model and of some of the Tricritical Ising Model when perturbed by any choice of relevant primary operators. We compare our results with some other independent predictions.

The Relation between Mixed Boundary States in Two- and Three-Matrix Models

We discuss the relation among some disk amplitudes with non-trivial boundary conditions in two-dimensional quantum gravity. They are obtained by the two-matrix model as well as the three-matirx model for the case of the tricritical Ising model. We examine them for simple spin configurations, and find that a finite number of insertions of the different spin states cannot be observed in the continumm limit. We also find a closed set of eight Schwinger-Dyson equations which determines the disk amplitudes in the three-matrix model.

FORM FACTORS AND CORRELATION FUNCTIONS OF THE STRESS-ENERGY TENSOR IN MASSIVE DEFORMATION OF THE MINIMAL MODELS (En)1⊗ (En)1/(En)2

The magnetic deformation of the Ising model, and the thermal deformations of both the tricritical Ising model and the tricritical Potts model, are governed by an algebraic structure based on the Dynkin diagram associated with the exceptional algebras En (respectively for n = 8, 7, 6). We make use of these underlying structures as well as the discrete symmetries of the models to compute the matrix elements of the stress-energy tensor and its two-point correlation function by means of the spectral representation method.

BOUNDARY S MATRIX FOR THE TRICRITICAL ISING MODEL

The tricritical Ising model perturbed by the subleading energy operator [Formula: see text] was known to be an integrable scattering theory of massive kinks,14 and in fact it preserves supersymmetry. We consider here the model defined on the half-plane with a boundary and compute the associated factorizable boundary S matrix. The conformal boundary conditions of this model are identified and the corresponding S matrices are found. We also show how some of these S matrices can be perturbed and generate “flows” between different boundary conditions.