Surface magnetic order in the Ising model: A Cayley tree approximation

N. R. da Silva, J.M. Silva

Luigi Accardi, Farrukh Maksutovich Mukhamedov, Mansoor Khajibaevich Saburov

In the present paper wes tudy forward Quantum Markov Chains (QMC)associated with Ising model on Cayley tree of order k. Using the tree structure of graphs, we give a construction of Quantum Markov Chains on a Cayley tree. By means of such construction we prove the existance of a phase transition for the Ising model on a Cayley tree of order k in QMC scheme. By the phase transition we mean the existance of two not quasi equivelant QMC for the given family of interaction operators.

Cécile Monthus, Thomas Garel

The 'cavity-mean-field' approximation developed for the random transverse field Ising model on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) has been found to reproduce the known exact result for the surface magnetization in d = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the present paper, we propose to extend these ideas in finite dimensions d > 1 via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations corresponds to the transfer matrix for a directed polymer in a random medium of transverse dimension D = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011 arXiv: 1110.3145 ). We present numerical results of the non-linear transfer approach in dimensions d = 2 and 3. In both cases, we find that the critical point is governed by infinite disorder scaling. In particular, exactly at criticality, the one-point surface magnetization scales as , where ω c ( d ) coincides with the droplet exponent ω DP ( D = d − 1) of the corresponding directed polymer model, with ω c ( d = 2) = 1/3 and . The distribution P ( v ) of the positive random variable v of order O(1) presents a power-law singularity near the origin with a ( d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-law decay with x s = ω c (1 + a ) independently of the order k .

Anthony John Guttmann, Glenn M. Torrie, Stuart G. Whittington

Ala Ouguergouz

Farrukh Maksutovich Mukhamedov, Abdessatar Barhoumi, Abdessatar Souissi +1 more

In this paper, we consider the Ising-XY model with competing interactions on the Cayley tree of order two. This model can be seen as a non-commutative (i.e. J - XY -interactions on next-neighbor vertices) perturbation of the classical Ising model on the Cayley tree. For the considered model we establish the existence of three translation-invariant quantum Markov chains. We notice that if the XY -interactions vanish, i.e. J = 0, then one gets the Ising model. If the classical Ising model vanishes in the considered model, then we obtain XY -model for which it turns out there exists only one translation invariant QMC.