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Physical review. B, Solid state

Asymmetrical Ising model

Huey Wen Huang

Published Jul 1, 1975• 10 citations
Publisher

Abstract

A model is introduced, which shows a discontinuous or first-order phase transition in temperature coordinate. As temperature increases, its long-range order undergoes a discontinuity at the condensation point, but contrary to the conventional assumption it does not vanish after the phase change. The long-range order remains nonzero at all finite temperatures and vanishes only at infinite temperature.

DOI
10.1103/physrevb.12.216

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Quantum Phase Transitions

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Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.

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Comment on “Effect of long-range interactions on the critical behavior of the continuous Ising model”

Silvano Romano

Available mathematical results entail the absence of a proper ordering transition taking place at finite temperature in the case $d=1,$ $\ensuremath{\sigma}=1.1,$ where the authors of the article commented on find evidence of such a transition. Moreover, in contrast to the findings of the authors of the article, the case $d=2,$ $\ensuremath{\sigma}=0$ entails a ground-state energy per particle diverging logarithmically with increasing sample size, hence an infinite transition temperature.

Physical review. B, Condensed matter•Jul 1, 2000•1 citation

First-Order Phase Transition in a Rigorously Solvable Magnetic System

Martin Blume, Richard E. Watson

We have considered the paramagnetic—ferromagnetic transition for the Ising system with ``infinite range'' interactions (i.e., each spin interacts equally with all other spins) and in the presence of a single-ion anisotropy term: H=−2jΣ lim i>i SizSjz+ΔΣ lim i Siz2−gμHΣ lim iSiz. Kittel and Shore1 have shown that this system (in the absence of the term in Δ) is exactly soluble, and leads rigourously, in the limit of an infinite number of particles, to the equations of the molecular-field approximation, and to a second-order phase transition. The single-ion term can cause the transition to become first order2,3 in that there is a discontinuity in the magnetization as a function of temperature. We have solved for the relevant thermodynamic parameters as a function of temperature for the case of ions with spin one. The development of the first-order transition as a function of the number of particles will be discussed. This work will be submitted for publication in the Physical Review.

Journal of Applied Physics•Mar 1, 1967•14 citations
A model is introduced, which shows a discontinuous or first-order phase transition in temperature coordinate. As temperature increases, its long-range order undergoes a discontinuity at the condensation point, but contrary to the conventional assumption it does not vanish after the phase change. The long-range order remains nonzero at all finite temperatures and vanishes only at infinite temperature.
We consider an Ising model where longitudinal components of every pair of spins have antiferromagnetic interaction of the same magnitude. When subjected to a transverse magnetic field at zero temperature, the system undergoes a phase transition of second order to an ordered phase and if the temperature is now increased, there is another phase transition to disordered phase. We provide derivation of these features by perturbative treatment up to the second order and argue that the results are nontrivial and not derivable from the known results about related models.
Long-range order in the spin van der Waals model is considered when the number of spins is finite and also when infinite. We show explicitly that a finite system cannot support long-range order. An infinite system at high temperatures is found to be dominated by the entropy of degenerate states of the system and, as a result, the system behaves essentially like an ideal system. In an infinite system at low temperatures, long-range order exists, fully reflecting the spin symmetry of the Hamiltonian. For the XY-like regime (J≳Jz ), mx is finite but mz vanishes, where mx and mz are reduced order parameters for the transverse and longitudinal directions, respectively. For the Ising-like regime (J<Jz), mx vanishes but mz is finite. The isotropic interaction (Jz = J) behaves as a singularity and it must be considered separately. A physical interpretation of the behavior of long-range order is offered using the geometry of spin space.
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
Available mathematical results entail the absence of a proper ordering transition taking place at finite temperature in the case d=1,d=1,d=1, $\ensuremath{\sigma}=1.1,$ where the authors of the article commented on find evidence of such a transition. Moreover, in contrast to the findings of the authors of the article, the case d=2,d=2,d=2, $\ensuremath{\sigma}=0$ entails a ground-state energy per particle diverging logarithmically with increasing sample size, hence an infinite transition temperature.
We have considered the paramagnetic—ferromagnetic transition for the Ising system with ``infinite range'' interactions (i.e., each spin interacts equally with all other spins) and in the presence of a single-ion anisotropy term: H=−2jΣ lim i>i SizSjz+ΔΣ lim i Siz2−gμHΣ lim iSiz. Kittel and Shore1 have shown that this system (in the absence of the term in Δ) is exactly soluble, and leads rigourously, in the limit of an infinite number of particles, to the equations of the molecular-field approximation, and to a second-order phase transition. The single-ion term can cause the transition to become first order2,3 in that there is a discontinuity in the magnetization as a function of temperature. We have solved for the relevant thermodynamic parameters as a function of temperature for the case of ions with spin one. The development of the first-order transition as a function of the number of particles will be discussed. This work will be submitted for publication in the Physical Review.