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 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.
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 report an effective functional form for the spin-spin correlation function of the 2D Ising model as a function of temperature and field. Although the Ising model has been well studied, no analytical result for the spin-spin correlation function exists for arbitrary magnetic fields and temperatures. We show the validity of our form by comparison with simulations using the Wolff algorithm, and obtain remarkable precision by including analytic corrections to scaling. Given recent interest in comparing biomembrane heterogeneity to Ising criticality, our spin-spin correlation function may be used as a predictive quantitative measure for FRET or NMR membrane experiments.