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arXiv:1612.08876

Critical behavior of the Ashkin-Teller model with a line defect: a Montecarlo study

G. Duchowney, Carlos Naón, Anı́bal Iucci

Published Dec 28, 2016• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1612.08876

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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 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 ϵϵϵ on the two sides of the junction. By varying ϵϵϵ, 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 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 Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.
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{\mathit{T}}_{\mathit{c}}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 investigate the short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.