We investigate the scaling properties of the spin interfaces in the Ashkin–Teller model. These interfaces are a very simple instance of lattice curves coexisting with a fluctuating degree of freedom, which renders the analytical determination of their exponents very difficult. One of our main findings is the construction of boundary conditions which ensure that the interface still satisfies the Markov property in this case. Then, using a novel technique based on the transfer matrix, we compute numerically the left-passage probability, and our results confirm that the spin interface is described by a Schramm–Loewner evolution (SLE) in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of the Ashkin–Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.
We investigate the scaling properties of the spin interfaces in the Ashkin–Teller model. These interfaces are a very simple instance of lattice curves coexisting with a fluctuating degree of freedom, which renders the analytical determination of their exponents very difficult. One of our main findings is the construction of boundary conditions which ensure that the interface still satisfies the Markov property in this case. Then, using a novel technique based on the transfer matrix, we compute numerically the left-passage probability, and our results confirm that the spin interface is described by a Schramm–Loewner evolution (SLE) in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of the Ashkin–Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.
We present an extensive study of interfaces defined in the Z_4 spin lattice representation of the Ashkin-Teller (AT) model. In particular, we numerically compute the fractal dimensions of boundary and bulk interfaces at the Fateev-Zamolodchikov point. This point is a special point on the self-dual critical line of the AT model and it is described in the continuum limit by the Z_4 parafermionic theory. Extending on previous analytical and numerical studies [10,12], we point out the existence of three different values of fractal dimensions which characterize different kind of interfaces. We argue that this result may be related to the classification of primary operators of the parafermionic algebra. The scenario emerging from the studies presented here is expected to unveil general aspects of geometrical objects of critical AT model, and thus of c=1 critical theories in general.
We consider two critical semi-infinite subsystems with different critical exponents and couple them through their surfaces. The critical behavior at the interface, influenced by the critical fluctuations of the two subsystems, can be quite rich. In order to examine the various possibilities, we study a system composed of two coupled Ashkin-Teller models with different four-spin couplings
ϵ on the two sides of the junction. By varying
ϵ, some bulk and surface critical exponents of the two subsystems are continuously modified, which in turn changes the interface critical behavior. In particular, we study the marginal situation, for which magnetic critical exponents at the interface vary continuously with the strength of the interaction parameter. The behavior expected from scaling arguments is checked by density matrix renormalization group calculations.
We report on the numerical measures on different spin interfaces and Fortuin-Kasteleyn (FK) cluster boundaries in the Askhin-Teller (AT) model. For a general point on the AT critical line, we find that the fractal dimension of a generic spin cluster interface can take one of four different possible values. In particular we found spin interfaces whose fractal dimension is d(f)=3/2 all along the critical line. Furthermore, the fractal dimension of the boundaries of FK clusters was found to satisfy all along the AT critical line a duality relation with the fractal dimension of their outer boundaries. This result provides clear numerical evidence that such duality, which is well known in the case of the O(n) model, exists in an extended conformal field theory.
We study magnetic critical behavior in the Ashkin-Teller model with an asymmetric defect line. This system is represented by two Ising lattices of spins
σ and
τ interacting through a four-spin coupling
ε. In addition, the couplings between
σ-spins are modified along a particular line, whereas couplings between
τ-spins are kept unaltered. This problem has been previously considered by means of analytical field-theoretical methods and by numerical techniques, with contradictory results. For
ε>0 field-theoretical calculations give a magnetic critical exponent corresponding to
σ-spins which depends on the defect strength only (it is independent of
ε), while
τ-spins magnetization decay with the universal Ising value
1/8. On the contrary, numerical computations based on density matrix renormalization (DMRG) give, for
ε>0 similar scaling behaviors for
σ and
τ spins, which depend on both
ε and defect intensity. In this paper we revisit the problem by performing a direct Montecarlo simulation. Our results are in well agreement with DMRG computations. We also discuss some possible sources for the disagreement between numerical and analytical results.
It is widely believed that the critical properties of several planar lattice models, like the Eight Vertex or the Ashkin-Teller models, are well described by an effective Quantum Field Theory obtained as formal scaling limit. On the basis of this assumption several extended scaling relations among their indices were conjectured. We prove the validity of some of them, among which the ones by Kadanoff, [13], and by Luther and Peschel, [16]. 1 Introduction and main results Integrable models in statistical mechanics, like the Ising or the Eight vertex (8V) models in two dimensions, provide conceptual laboratories for the understanding of phase transitions. Integrability is however a rather delicate property requiring very special features, and it is usually lost in more realistic models.