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arXiv:0709.1322 · Physical Review B

Scaling properties at the interface between different critical subsystems: The Ashkin-Teller model

Péter Lajkó, Loïc Turban, Ferenc Iglói

Published Dec 20, 2007• 3 citations
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Abstract

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.

DOI
10.1103/physrevb.76.224423

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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 study magnetic critical behavior in the Ashkin-Teller model with an asymmetric defect line. This system is represented by two Ising lattices of spins σσσ and τττ interacting through a four-spin coupling εεε. In addition, the couplings between σσσ-spins are modified along a particular line, whereas couplings between τττ-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>0ε> 0ε>0 field-theoretical calculations give a magnetic critical exponent corresponding to σσσ-spins which depends on the defect strength only (it is independent of εεε), while τττ-spins magnetization decay with the universal Ising value 1/81/81/8. On the contrary, numerical computations based on density matrix renormalization (DMRG) give, for ε>0ε> 0ε>0 similar scaling behaviors for σσσ and τττ spins, which depend on both εεε 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 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.
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.
A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The d=2d=2d=2 square lattice and the d=3d=3d=3 bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for d=3d=3d=3 are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.
Abstract Two replicas of a 2D Ising model are coupled by frustrated spin-spin interactions. It is known that this inter-layer coupling is marginal and that the bulk critical behavior belongs to the Ashkin–Teller (AT) universality class, as the J 1 − J 2 Ising model. In this work, the surface critical behavior is studied numerically by tensor renormalization-group calculations. The bond-weight tensor renormalization group algorithm is extended to tackle systems with boundaries. It is observed that the two-fold degeneracy of the surface magnetic scaling dimension of the AT model is lifted in the frustrated two-layer Ising model (F2LIM). The splitting is explained by the breaking of the Z 2 -symmetry under spin reversal of a single Ising replica in the F2LIM. The two distinct surface magnetic scaling dimensions x 1 s and x 2 s of the F2LIM satisfies a simple duality relation x 1 s = 1 / 4 x 2 s .
Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.