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Physical Review Letters

Effects of Lattice Compressibility on Critical Behavior

George Allen Baker Jr., J W Essam

Published Mar 2, 1970• 137 citations
Publisher

Abstract

We present a model of a spin system on a compressible lattice whose solution we are able to express in terms of that for the Ising model on a rigid lattice. We find that the critical indices are renormalized when observed both at constant pressure and at constant volume. In particular the infinite Ising-model specific-heat peak is rounded into dynamic instability.

DOI
10.1103/physrevlett.24.447

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We present a model of a spin system on a compressible lattice whose solution we are able to express in terms of that for the Ising model on a rigid lattice. We find that the critical indices are renormalized when observed both at constant pressure and at constant volume. In particular the infinite Ising-model specific-heat peak is rounded into dynamic instability.
The critical behavior of a compressible Ising system of the Baker-Essam type is examined for positive pressures. In addition to satisfying the assumptions of Fisher's theory of renormalization of critical exponents, the model, allows one to calculate the temperature region around Tc{T}_{c}Tc​ where the renormalized exponents should show up. The region is $\ensuremath{\Delta}{T}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\sim}{(P{J}_{1}^{2})}^{\frac{1}{\ensuremath{\alpha}}}$ for small pressures, where J1{J}_{1}J1​ is the spin-lattice coupling and $\ensuremath{\alpha}$ is the specific-heat exponent of the corresponding rigid Ising system.
Within the droplet model for the Ising spin system it is shown that the magnetic specific heat Cp{C}_{p}Cp​ and the susceptibility ${\ensuremath{\chi}}_{p}$ for constant vanishing pressure and magnetic field remain finite in a compressible lattice as the critical point Tc{T}_{c}Tc​ is approached from below. This regularization of Cp{C}_{p}Cp​ and ${\ensuremath{\chi}}_{p}$ is caused by the long-range indirect spin-spin interaction, which is a consequence of the dependence of the exchange integral on the lattice spacing.
Extensive Monte Carlo simulations in the semi-grand-canonical ensemble are used to study the critical behavior of a three-dimensional compressible Ising model with antiferromagnetic interactions under constant volume conditions. Elastic forces between spins are introduced by the Stillinger-Weber potential and energy parameters are chosen in such a way that antiparallel spin ordering is favored, analogous to the antiferromagnetic coupling in the rigid Ising Hamiltonian. All the quantities analyzed strongly indicate that the system remains in the universality class of the standard (rigid) three-dimensional Ising model, in contrast with theoretical predictions.
Assuming the existence of certain limits we give an exact solution of a two-dimensional elastic Ising model with quadratic coupling. The main results are these: (I) If the lattice is slightly stretched, thermodynamic instability occurs for temperatures in a neighborhood of the critical point. (II) Outside this neighborhood, ${\ensuremath{\kappa}}_{T}$, $\ensuremath{\alpha}$, and Cp{C}_{p}Cp​ diverge. (III) The nearest-neighbor spin-spin correlation function is nonzero for any pressure when TTT becomes infinite. (IV) Antiferromagnetic as well as ferromagnetic behavior can occur for weak coupling at any pressure. (V) For strong coupling, at 0 pressure the model can have a Curie and N\'eel point or no critical point.
The Landau-Ginsburg free-energy functional (effective Hamiltonian) is derived for compressible Ising systems where the Ising spins are coupled with quantum phonon fields and a transverse field is present. In the zero transverse field case, the phonon dynamical modes also disappear from the effective Hamiltonian, which then compares with those studied by Sak and De Moura et al. in the non-quantum compressible Ising systems. In the presence of dynamics in the spin system (compressible transverse Ising system), the phonon modes get effectively coupled in the Hamiltonian and while at non-zero temperatures this does not affect the critical behaviour (which remains the same as in the non-quantum case), the presence of these dynamical modes effectively increases the lattice dimensionality by unity at zero-temperature transitions (T c =0) where one observes the critical behaviour corresponding to a d+1 dimensional non-quantum compressible Ising model. The critical behaviour of a cooperative Jahn-Teller phase transition when T c to 0 is also discussed.