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Physical review. B, Condensed matter

First- and second-order phase transitions in a system with multispin interactions

Amos Maritan, Attilio L. Stella, Carlo Vanderzande

Published Jan 1, 1984• 20 citations
Publisher

Abstract

A quantum model with $n$-spin interactions is studied in the ground state. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ the model is solved exactly by mean-field theory and undergoes a first-order transition. The transition is expected to become second order for $n<{n}_{c}$. With exploitation of self-duality and suitably improved mean-field approximations, a recently proposed criterion leads to the conjecture ${n}_{c}=3$ in one dimension. For the two-dimensional case the transition appears to be definitely of first order for $n=3 (2<{n}_{c}<3)$.

DOI
10.1103/physrevb.29.519

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Cambridge University Press eBooks•Jun 22, 1995•40 citations
A quantum model with nnn-spin interactions is studied in the ground state. When n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ the model is solved exactly by mean-field theory and undergoes a first-order transition. The transition is expected to become second order for n<ncn<{n}_{c}n<nc​. With exploitation of self-duality and suitably improved mean-field approximations, a recently proposed criterion leads to the conjecture nc=3{n}_{c}=3nc​=3 in one dimension. For the two-dimensional case the transition appears to be definitely of first order for n=3(2<nc<3)n=3 (2<{n}_{c}<3)n=3(2<nc​<3).
An Ising-type model including nnn-spin nearest-neighbor interactions is introduced. The ground-state properties of its Hamiltonian version are studied in one and two dimensions. The model is self-dual in one dimension. The mean-field theory indicates that for n>ncn>{n}_{c}n>nc​ the transition may become first order. Finite-size scaling suggests that nc=4{n}_{c}=4nc​=4 in one dimension. The critical exponents as functions of nnn are estimated.
The two-dimensional Ising model in the rectangular lattice is generalised to include m-spin interactions in one direction and two-spin interactions in the other. This model is self-dual and the critical line is the same as in the conventional Ising model with m=2. The m=3 model is solved in a generalised mean-field approximation where the transition is first order. Using a phenomenological renormalisation group approach, approximate values of the thermal and magnetic exponents y t and y h are obtained for m=3 and m=4. The m=3 results together with symmetry arguments suggest that this model belongs to the same class of universality as the Baxter-Wu and four-state ferromagnetic Potts models. When m=4 the transition is probably first order.
Introduces a family of D-dimensional q-state quantum Potts models with multi-spin interactions, which are self-dual for any D. These models are introduced using their (d=D+1)-dimensional classical formulations and the transfer matrix technique. The self-duality is proved on the quantum Hamiltonian. In the limit q to infinity the ground state energy is obtained exactly and the phase transition is first-order. A 1/q expansion allows one to get an approximate expression for the line in the (q,D)-plane separating regions where the transition changes from first- to second-order.
A conjecture, based on a comparison of different mean-field-like approximations, is proposed for determining the first- or second-order character of phase transitions. The qqq-state Potts model and several gauge theories [Z2{Z}_{2}Z2​, U(N)\mathrm{U}(N)U(N), SU(N)\mathrm{SU}(N)SU(N), SO(3)] on a lattice are considered. For Z2{Z}_{2}Z2​ gauge theory, in particular, a critical dimensionality dc=3.66{d}_{c}=3.66dc​=3.66 is predicted, whereas for the Potts model qc=2.75{q}_{c}=2.75qc​=2.75 in d=3d=3d=3, qc=2.4{q}_{c}=2.4qc​=2.4 in d=4d=4d=4, and ${q}_{c}\ensuremath{\propto}\mathrm{exp}[\frac{1}{(d\ensuremath{-}1)}]$ for $d\ensuremath{\rightarrow}1$ are obtained.
First- and second-order phase transitions in the isotropic Heisenberg model with multispin exchange were investigated with the help of the quantum Monte Carlo method. The calculations were performed for a system with spin S = 1/2 and N = L3 spins (L = 8, 12, and 16) on simple cubic lattices with periodic boundary conditions. A relation between the pair and multispin exchange constants for which the second- order phase transitions are replaced by first-order phase transitions is established. First-order phase transitions have not been studied previously by the quantum Monte Carlo method. Phase diagrams were constructed for the model under study. The computational results are compared with the method of Green’s functions and the molecular-field theory.
In the preceding two chapters, we have discussed various types of order that can occur in nature and how the ordering process can be quantified by the introduction of order parameters. We also developed a formalism for dealing with the thermodynamics of ordered states. In this chapter, we will use mean-field theory to study phase transitions and the properties of various ordered phases. Mean-field theory is an approximation for the thermodynamic properties of a system based on treating the order parameter as spatially constant. It is a useful description if spatial fluctuations are not important. It becomes an exact theory only when the range of interactions becomes infinite. It, nevertheless, makes quantitatively correct predictions about some aspects of phase transitions (e.g. critical exponents) in high spatial dimensions where each particle or spin has many nearest neighbors, and it makes qualitatively correct predictions in physical dimensions. Mean-field theory has the enormous advantage of being mathematically simple, and it is almost invariably the first approach taken to predict phase diagrams and properties of new experimental systems. Before proceeding, let us review some simple facts about phase transitions. At high temperatures, there is no order, and the order parameter 〈φ〉 is zero. At a critical temperature, T c , order sets in so that, for temperatures below T c , 〈φ〉 is nonzero. If 〈φ〉 rises continuously from zero, as shown in Fig. 4.0.1a, the transition is second order .