Surface tension and phase transition for lattice systems (original) (raw)
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Surface tension and phase coexistence for general lattice systems
Journal of Statistical Physics, 1983
We investigate the surface tension between coexisting phases of general discrete lattice systems. In particular the different phases need not be connected by any symmetry. We prove the positivity of the surface tension in the low-temperature regime where the Pirogov-Sinai theory of first-order phase transitions is valid: finite-range Hamiltonian having a finite number of periodic ground states. We give a brief description (with some extensions) of this theory.
ON THE SURFACE TENSION OF LATTICE SYSTEMS
Annals of The New York Academy of Sciences, 1980
There are various microscopic definitions of the surface tension /I-'? in the literature, but it is far from obvious (or known), in general, that they are all A proof5 that for the two-dimensional king model (on a square lattice with nearest-neighbor interactions), many different definitions give the same answer as that obtained explicitly by Onsager is therefore encouraging. Here, we use a "grand definition of surface tension that seems natural to us. It is particularly simple when the two pure phases are related to each other by a symmetry of the Hamiltonian, as is the case for the Ising models we shall consider. To make things easy, we deal first with the simplest cases and leave all generalizations to the end.
Phase transitions in anisotropic lattice spin systems
Communications in Mathematical Physics, 1978
A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity is not proved, but for which the rest of the proof is valid.
Phase transitions in spin systems with frustration
Theoretical and Mathematical Physics, 1980
A study is made of ferromagnetic spin models frustrated by random ant[ferromagnetic couplings. It is shown that a random system without frustration is equivalent for any number of dimensions to a regular system without ant[ferromagnetic couplings, in the case of two and three dimensions it is shown that the completely frustrated models on square and cubic lattices are identical to the corresponding periodic, completely frustrated models. The explicit form of the local transformation of each of the considered models into the equivalent model is found. It is shown that for the completely frustrated twodimensional Ising model on a square lattice there is no phase transition. Consideration is also given to the general case of the partly frustrated two-dimensional Ising model on a square lattice~ and lower bounds are obtained for the ground-state energy.
Quantum Phase Transitions in Spin Systems
150 Years Of Quantum Many-Body Theory - A Festschrift in Honour of the 65th Birthdays of John W Clark, Alpo J Kallio, Manfred L Ristig and Sergio Rosati, 2001
We discuss the influence of strong quantum fluctuations on zero-temperature phase transitions in a two-dimensional spin-half Heisenberg system. Using a high-order coupled cluster treatment, we study competition of magnetic bonds with and without frustration. We find that the coupled cluster treatment is able to describe the zero-temperature transitions in a qualitatively correct way, even if frustration is present and other methods such as quantum Monte Carlo fail.
Spin systems on hierarchical lattices. Introduction and thermodynamic limit
Physical Review B, 1982
A number of exactly soluble models in statistical mechanics can be produced with the use of spins interacting with nearest neighbors on a hierarchical lattice. A general definition and several examples of such lattices are given, and the topological properties of one of these, the "diamond" lattice, are discussed in detail. It is shown that the free energy has a well-defined thermodynamic limit for a large class of discrete spin models on hierarchical lattices.
First-order transition in the hexagonal-close-packed lattice with vector spins
First-order transition in the hexagonal-close-packed lattice with vector spins, 1992
By extensive Monte Carlo simulations, we show that the magnetic phase transition is of first order in the hexagonal-close-packed lattice with vector (XY and Heisenberg) spins interacting via isotropic antiferromagnetic nearest-neighbor bonds. Despite the infinite ground-state degeneracy, the system is shown to retain only the collinear spin configurations at finite temperature, in agreement with a theoretical conjecture.
Finite-temperature properties of frustrated classical spins coupled to the lattice
2005
We present extensive Monte Carlo simulations for a classical antiferromagnetic Heisenberg model with both nearest (J1) and next-nearest (J2) exchange couplings on the square lattice coupled to the lattice degrees of freedom. The Ising-like phase transition, that appears for J2/J1 > 1/2 in the pure spin model, is strengthened by the spin-lattice coupling, and is accompanied by a lattice deformation from a tetragonal symmetry to an orthorhombic one. Evidences that the universality class of the transition does not change with the inclusion of the spin-lattice coupling are reported. Implications for Li2VOSiO4, the prototype for a layered J1−J2 model in the collinear regime, are also discussed.
Spin systems on hierarchical lattices. II. Some examples of soluble models
Physical Review B Condensed Matter, 1984
Several examples are given of soluble models of phase-transition phenomena utilizing classical discrete spin systems with nearest-neighbor interaction on hierarchical lattices. These include critical exponents which depend continuously on a parameter, the Potts model on a lattice with two different coupling constants, surface tension, and excess free energy of a line of defects. In each case we point out similarities and differences with a corresponding Bravais-lattice model.
A quantum phase transition may occur in a system at zero temperature when a controlling parameter is tuned towards a critical point. An important question is whether such a critical point exists in a particular system and how stable it is. Here, we identify the critical point of a quantum phase transition as a singular point in the affine algebraic variety of the characteristic equation for the Hamiltonian describing the system, with an unstable critical point being associated with an isolated singular point which has a finite Tjurina number. The theory is illustrated by studying a model system of zero-dimensional (finite) Heisenberg spin chain with an impurity, which exhibits a nontrivial first-order quantum phase transition. Both analytical and numerical calculations show that the quantum phase transition always exists when the impurity has a Z_2Z_2Z2 symmetry but only remains in systems with an even number of spin sites when the Z2Z_2Z_2 symmetry is broken.