We consider the Ashkin–Teller model on the square lattice, which is represented by two Ising models (σ and τ) having a four-spin coupling of strength, , between them. We introduce an asymmetric defect line in the system along which the couplings in the σ Ising model are modified. In the Hamiltonian version of the model we study the scaling behavior of the critical magnetization at the defect, both for σ and for τ spins, by means of density matrix renormalization. For > 0 we observe identical scaling for σ and τ spins, whereas for < 0 one model becomes locally ordered and the other locally disordered. This is different from the critical behavior of the uncoupled model ( = 0) and is in contradiction with the results of recent field-theoretical calculations.
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.
We consider the critical spin-spin correlation function of the Ashkin-Teller and Baxter models. By using path-integral techniques in the continuum description of these models in terms of fermion fields, we show that the correlation decays with distance with the same critical exponent as the Ising model. The procedure is straightforwardly extended to take into account the presence of a line defect. Thus we find that in these altered models the critical index of the magnetic correlation on the defect coincides with the one of the defective two-dimensional Ising or Bariev model.
The critical behavior of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness \ensuremath{\varphi}=(\ensuremath{\alpha}/\ensuremath{\nu}) varies continuously and is positive so that the randomness is relevant, and different critical behavior is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four-state Potts models. The critical behavior of the disordered model is compared with the critical behavior of an anisotropic Ashkin-Teller model, which is used as a reference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat C is changed dramatically. Our results favor a logarithmic type divergence at
Tc, C\ensuremath{\sim}lnL for the random-bond Ashkin-Teller and four-state Potts models and C\ensuremath{\sim}ln lnL for the random-bond Ising model.
We consider the critical spin–spin correlation function of the 2D Ising model with a line defect whose strength is an arbitrary function of the position. By using path integral techniques in the continuum description of this model in terms of fermion fields, we obtain an analytical expression for the correlator as a functional of the position-dependent coupling. Thus, our result provides one of the few analytical examples that allows one to illustrate the transit of a magnetic system from scaling to non-scaling behavior in a critical regime. We also show that the non-scaling behavior obtained for the spin correlator along a nonuniformly altered line of an Ising model remains unchanged in the Ashkin–Teller model.
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 consider the critical spin–spin correlation function of the 2D Ising model with a line defect whose strength is an arbitrary function of the position. By using path integral techniques in the continuum description of this model in terms of fermion fields, we obtain an analytical expression for the correlator as a functional of the position-dependent coupling. Thus, our result provides one of the few analytical examples that allows one to illustrate the transit of a magnetic system from scaling to non-scaling behavior in a critical regime. We also show that the non-scaling behavior obtained for the spin correlator along a nonuniformly altered line of an Ising model remains unchanged in the Ashkin–Teller model.