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Journal of Physics A Mathematical and General

Spin - spin correlation function for the free-fermion model: crossover from two-dimensional Ising to Pokrovsky - Talapov critical behaviour

S. B. Rutkevich

Published Jun 7, 1997• 3 citations
Publisher

Abstract

We calculate the spin - spin correlation function for a (1 + 1)-dimensional free-fermion model displaying crossover from the two-dimensional Ising to Pokrovsky - Talapov critical behaviour. In the Ising limit our results reduce to the well known exact representations reported by Wu et al . Correspondence between the obtained correlation function and an exactly solvable classical Hamiltonian system is established.

DOI
10.1088/0305-4470/30/11/017

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We calculate the spin - spin correlation function for a (1 + 1)-dimensional free-fermion model displaying crossover from the two-dimensional Ising to Pokrovsky - Talapov critical behaviour. In the Ising limit our results reduce to the well known exact representations reported by Wu et al . Correspondence between the obtained correlation function and an exactly solvable classical Hamiltonian system is established.
The two{dimensional (2D) Ising model is reviewed as a theory of free fermions on a lattice.The discussion includes the fermionization procedure based on the mirror{ordered factorization of the density matrix, Gaussian fermionic integral representation for partition function, the momentum{ space analysis and Onsager's result, the eective continuum{limit eld theories and the critical{ point singularities.The emergence of long{range fermionic correlations in a nonzero magnetic eld and the behaviour of the speci c heat along the critical isotherm are commented.Attention is given to the choice of rational computational devices.
We study the triangular-lattice Ising model with dipolar interactions, inspired by its realisation in artificial arrays of nanomagnets. We show that a classical spin-liquid forms at intermediate temperatures, and that its behaviour can be tuned by temperature and/or a small lattice distortion between a string Luttinger liquid and a domain-wall-network state. At low temperature there is a transition into a magnetically ordered phase, which can be first-order or continous with a crossover in the critical behaviour between Pokrovsky-Talapov and 2D-Ising universality. When the Pokrovsky-Talapov criticality dominates, the transition is essentially of the Kasteleyn type.
The spin-spin correlation functions of the random one-dimensional Ising model are calculated exactly and found to have a probability distribution. It is then possible to define two characteristic lengths, one to describe the behaviour of the mean and the other of the standard deviation of the probability distribution. The existence of these two characteristic lengths then suggests that the thermodynamic behaviour of the system is considerably more complex than for the corresponding nonrandom system.
The spin-spin correlation function in the ground state of some two-dimensional fully frustrated spin models are investigated. The correlations are long ranged and decay as $\frac{1}{{r}^{\ensuremath{\eta}}}$, where rrr is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the FFF model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the FFF model above the critical temperature and the Gaussian model above the multicritical point is discussed.