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Register a new userSymmetry-respecting real-space renormalization for the quantum Ashkin-Teller model
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Added by
Stephen D. Bartlett
Publication date
2015
fields
Physics
and research's language is
English
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We use a simple real-space renormalization group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the Ising model. Our investigation explores this approach in more generality, in a model where the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically-intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.
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The two-dimensional ferromagnetic anisotropic Ashkin-Teller model is investigated through a real-space renormalization-group approach. The critical frontier, separating five distinct phases, recover all the known exacts results for the square lattice. The correlation length $( u_T)$ and crossover $(phi)$ critical exponents are also calculated. With the only exception of the four-state Potts critical point, the entire phase diagram belongs to the Ising universality class.
We have investigated the three-color Ashkin-Teller model (3AT), on the Wheatstone bridge hierarchical lattice, by means of a Migdal-Kadanoff renormalization group approach. We have obtained the exact recursion relations for the renormalized couplings, which have been used to investigate the phase diagram and to study the corresponding critical points. The phase diagram, represented in terms of the dual transmissivity vector, presents four magnetic phases and nine critical points. We have also numerically calculated the correlation length ($ u_T$) and crossover ($phi$) critical exponents, which show that seven of the critical points are in the Potts model universality class ($q=2$, 4 e 8). The remaining critical points are in a universality class which may belong to the Baxters line. Our results are exact on the hierarchical lattice used in the present work and the phase diagram can be considered as an approximation to more realistic Bravais lattices.
The universal critical point ratio $Q$ is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller (AT) model on the square lattice. A leading-order expansion of the ratio $Q$ in the presence of a non-vanishing thermal field is found from finite-size scaling and the corresponding expression is fitted to the accurate perturbative transfer-matrix data calculations for the $Ltimes L$ square clusters with $Lleq 9$.
The thermodynamic and retrieval properties of the Ashkin-Teller neural network model storing an infinite number of patterns are examined in the replica-symmetric mean-field approximation. In particular, for linked patterns temperature-capacity phase diagrams are derived for different values of the two-neuron and four-neuron coupling strengths. This model can be considered as a particular non-trivial generalisation of the Hopfield model and exhibits a number of interesting new features. Some aspects of replica-symmetry breaking are discussed.
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 epsilon, on the two sides of the junction. By varying epsilon, 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 DMRG calculations.
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