Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point

We consider the Farey fraction spin chain in an external field h. Using ideas from dynamical systems and functional analysis, we show that the free energy f in the vicinity of the second-order phase transition is given, exactly, by $$f\sim\frac{t}{\log t}-\frac{1}{2}\frac{h^2}{t}\quad\mbox{for }h^2\ll t\ll1.$$ Here $t=\lambda_{G}\log(2)(1-\frac{\beta}{\beta_{c}})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, λ _G . It follows that λ _G determines the amplitude of both the specific heat and susceptibility singularities. To our knowledge, there is only one other microscopically defined interacting model for which the free energy near a phase transition is known as a function of two variables. Our results confirm what was found previously with a cluster approximation, and show that a clustering mechanism is in fact responsible for the transition. However, the results disagree in part with a renormalisation group treatment.     

Phase Transition in Ferromagnetic Ising Models with Non-uniform External Magnetic Fields

In this article we study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field. We show that the Ising model on the hypercubic lattice with a summable magnetic field has a first-order phase transition and, for any positive (resp. negative) and bounded magnetic field, the model does not present the phase transition phenomenon whenever lim?inf?h _i >0, where \(\mathbf{h}=(h_{i})_{i\in \mathbb{Z}^{d}}\) is the external magnetic field.     

Recognition ability of the fully connected Hopfield neural network under a persistent stimulus field

We investigate the pattern recognition ability of the fully connected Hopfield model of a neural network under the influence of a persistent stimulus field. The model considers a biased training with a stronger contribution to the synaptic connections coming from a particular stimulated pattern. Within a mean-field approach, we computed the recognition order parameter and the full phase diagram as a function of the stimulus field strength h, the network charge α and a thermal-like noise T. The stimulus field improves the network capacity in recognizing the stimulated pattern while weakening the first-order character of the transition to the non-recognition phase. We further present simulation results for the zero temperature case. A finite-size scaling analysis provides estimates of the transition point which are very close to the mean-field prediction.     

Kinetics of a mixed spin-1/2 and spin-3/2 Ising ferrimagnetic model

We present a study, within a mean-field approach, of the kinetics of a mixed ferrimagnetic model on a square lattice in which two interpenetrating square sublattices have spins that can take two values, , alternated with spins that can take the four values, . We use the Glauber-type stochastic dynamics to describe the time evolution of the system with a crystal-field interaction in the presence of a time-dependent oscillating external magnetic field. The nature (continuous and discontinuous) of transition is characterized by studying the thermal behaviors of average order parameters in a period. The dynamic phase transition points are obtained and the phase diagrams are presented in the reduced magnetic field amplitude (h) and reduced temperature (T) plane, and in the reduced temperature and interaction parameter planes, namely in the (h, T) and (d, T) planes, d is the reduced crystal-field interaction. The phase diagrams always exhibit a tricritical point in (h, T) plane, but do not exhibit in the (d, T) plane for low values of h. The dynamic multicritical point or dynamic critical end point exist in the (d, T) plane for low values of h. Moreover, phase diagrams contain paramagnetic (p), ferromagnetic (f), ferrimagnetic (i) phases, two coexistence or mixed phase regions, (f+p) and (i+p), that strongly depend on interaction parameters.     

Spin-glass and spin-crystal phase transition of a regular Ising model on the Cayley tree

A regular Ising model with nearest-neighbour interaction of +J and -J on a Cayley tree is reported to show a phase transition from the paramagnetic phase to the spin-glass and spin-crystal phase in a uniform external field below a critical value h_c.     

Transition order and dynamics of a model with competing exchange and dipolar interactions

We performed Monte Carlo simulation of phase transitions from isotropic stripe phase with short-range order to long-range stripe phase in a model with competing ferromagnetic exchange and antiferromagnetic dipolar interactions on triangular lattice. We calculated phase diagram for different values of exchange and dipolar interaction constants ratio, η. We also determined the order of the transitions to stripe phases AFh of different stripe widths h: first-order phase transition was found to transitions into AF1 and AF2 phases, while transitions to AF3 and AF4 phases were of the second order. In the phase diagram the tricritical point was determined at the AF2 and AF3 phase boundary. We observed the peak of nematic phase at the transition region to the AF1 phase, but found it metastable at low values of η. We have also found that in AF1 phase spin relaxation corresponds to the Ising model dynamics. In phases AF3 and AF4 the dynamics slows down, and stripe domain growth with time is proportional to logt.     

Dynamic magnetic hysteresis behavior and dynamic phase transition in the spin-1 Blume-Capel model

The nature (time variation) of response magnetization m(wt) of the spin-1 Blume-Capel model in the presence of a periodically varying external magnetic field h(wt) is studied by employing the effective-field theory (EFT) with correlations as well as the Glauber-type stochastic dynamics. We determine the time variations of m(wt) and h(wt) for various temperatures, and investigate the dynamic magnetic hysteresis behavior. We also investigate the temperature dependence of the dynamic magnetization, hysteresis loop area and correlation near the transition point in order to characterize the nature (first- or second-order) of the dynamic transitions as well as obtain the dynamic phase transition temperatures. The hysteresis loops are obtained for different reduced temperatures and we find that the areas of the loops are decreasing with the increasing of the reduced temperatures. We also present the dynamic phase diagrams and compare the results of the EFT with the results of the dynamic mean-field approximation. The phase diagrams exhibit many dynamic critical points, such as tricritical (•), zero-temperature critical (Z), triple (TP) and multicritical (A) points. According to values of Hamiltonian parameters, besides the paramagnetic (P), ferromagnetic (F) fundamental phases, one coexistence or mixed phase region, (F+P) and the reentrant behavior exist in the system. The results are in good agreement with some experimental and theoretical results.     

The effective-field theory studies of critical phenomena in a mixed spin-1 and spin-2 Ising model on honeycomb and square lattices

An effective-field theory with correlations has been used to study critical behaviors of a mixed spin-1 and spin-2 Ising system on a honeycomb and square lattices in the absence and presence of a longitudinal magnetic field. The ground-state phase diagram of the model is obtained in the longitudinal magnetic field (h) and a single-ion potential or crystal-field interaction (Δ) plane. The thermal behavior of the sublattice magnetizations of the system are investigated to characterize the nature of (continuous and discontinuous) of the phase transitions and obtain the phase transition temperature. The phase diagrams are presented in the (Δ/|J|, k_BT/|J|) plane. The susceptibility, internal energy and specific heat of the system are numerically examined and some interesting phenomena in these quantities are found due to the absence and presence of the applied longitudinal magnetic field. Moreover, the system undergoes second- and first-order phase transition; hence, the system gives a tricritical point. The system also exhibits reentrant behavior.     

Dynamic phase transitions and dynamic phase diagrams in the kinetic spin-5/2 Blume-Capel model in an oscillating external magnetic field: Effective-field theory and the Glauber-type stochastic dynamics approach

Using an effective field theory with correlations, we study a kinetic spin-5/2 Blume-Capel model with bilinear exchange interaction and single-ion crystal field on a square lattice. The effective-field dynamic equation is derived by employing the Glauber transition rates. First, the phases in the kinetic system are obtained by solving this dynamic equation. Then, the thermal behavior of the dynamic magnetization, the hysteresis loop area and correlation are investigated in order to characterize the nature of the dynamic transitions and to obtain dynamic phase transition temperatures. Finally, we present the phase diagrams in two planes, namely (T/zJ, h₀/zJ) and (T/zJ, D/zJ), where T absolute temperature, h₀, the amplitude of the oscillating field, D, crystal field interaction or single-ion anisotropy constant and z denotes the nearest-neighbor sites of the central site. The phase diagrams exhibit four fundamental phases and ten mixed phases which are composed of binary, ternary and tetrad combination of fundamental phases, depending on the crystal field interaction parameter. Moreover, the phase diagrams contain a dynamic tricritical point (T), a double critical end point (B), a multicritical point (A) and zero-temperature critical point (Z).     

Ground-state phase diagrams of Ising metamagnet in a mixed longitudinal and transverse magnetic field

The ground-state magnetic properties of a two-sublattice Ising metamagnet in a mixed longitudinal and transverse magnetic field are studied within the effective-field theory. A parameter j₂=J₂/J₁ is introduced, which reflects the strength ratio of spin coupling between adjacent planes and in each plane. In addition to the second-order transition lines, the first-order transition lines are also presented, since the ground-state energy can be calculated numerically. The ground-state phase diagrams in h_x–h_z are presented. The results show that when j₂<0 the phase transition of the system is always first-order for h_x<2.751, and when −1000?j₂<0 it is always second-order for h_x>4.36. For the given h_x (0<h_x<14.71), the longitudinal critical magnetic field increases as j₂ decreases. The reentrant phenomenon occurs in the range of j₂<−11.89, h_x>14.71. There is no fourth-order critical point in the phase diagrams given by using EFT as found by using mean field theory (MFT).     

Anomalous behavior of the surface tension at the interface of the metallic and insulating phases in the vicinity of the metal-insulator phase transition in a magnetic field

Mean-field equations describing the metal-insulator (MI) transition are formulated. They involve two coupled order parameters characterizing this transition: (i) a scalar order parameter describing the density change accompanying the transition from the insulating state to the metallic one and (ii) an order parameter (a two-component vector) describing the electron density in the metallic or semimetallic phase affected by the applied magnetic field. Two components of this vector correspond to different possible spin states of electrons in the applied magnetic field. The transition in the density of metallic and insulating phases being a first order phase transition is treated in terms of the Cahn-Hilliard-type gradient expansion. The transition in the electron density is a second order phase described by the Ginzburg-Landau-type functional. The coupling of these two parameters is described by the term linearly dependent on the electron density n in the metal with the proportionality factor being a function of the density of the metallic phase. The derived equations are solved in the case of the MI interface in the presence of both parallel and perpendicular uniform magnetic fields. The calculated surface tension Σ_mi between the metallic and insulating phases has a singular behavior. In the limit of zero electron density n ? 0, Σ_mi ～ n ^3/2. Near the MI transition point T _c(h) in the applied magnetic field, Σ_mi ～ [T - T _c(h)]^3/2. The singular behavior of the surface tension at the MI interface results in the clearly pronounced hysteresis accompanying the transition from the insulating to metallic state and vice versa.     

Quantum phase transition in lattice model of unconventional superconductors

In this paper we shall introduce a lattice model of unconventional superconductors (SC) like d-wave SC in order to study quantum phase transition at vanishing temperature (T). Finite-T counterpart of the present model was proposed previously with which SC phase transition at finite T was investigated. The present model is a noncompact U(1) lattice-gauge-Higgs model in which the Higgs boson, the Cooper-pair field, is put on lattice links in order to describe d-wave SC. We first derive the model from a microscopic Hamiltonian in the path-integral formalism and then study its phase structure by means of the Monte Carlo simulations. We calculate the specific heat, monopole densities and the magnetic penetration depth (the gauge-boson mass). We verified that the model exhibits a second-order phase transition from normal to SC phases. Behavior of the magnetic penetration depth is compared with that obtained in the previous analytical calculation using XY model in four dimensions. Besides the normal to SC phase transition, we also found that another second-order phase transition takes place within the SC phase in the present model. We discuss physical meaning of that phase transition.     

Spin-peierls systems in magnetic field: Phase transition from dimerized to soliton lattice state

The phase diagram for the spin-Peierls system in high magnetic field is studied. The line of transition from dimerized (D) to soliton lattice (SL) phase is found, and it is shown that the SL phase is stable in fields h;h_c. Magnetization vs h is calculated as well as the sound velocity u_s of the domain walls oscillations. The possible experimental verification of the proposed theory is discussed also.     

Phase front motion in uniaxial ferromagnet

It is well known that in the uniaxial ferromagnet in the presence of an external magnetic field perpendicular to the easy axis (h_x) a continuous phase transition occurs for a critical value of this field. There are metastable and stable states if one includes a small field parallel to the easy axis (h_z). The motion of the relaxation front of the metastable state is investigated. It is found that an “interphase wall of Neel-type” exists, its velocity is proportional to h_z and increases when the critical point is approached.     

The quantum compass chain in a transverse magnetic field

We study the magnetic behaviors of a spin-1/2 quantum compass chain (QCC) in a transverse magnetic field, by means of the analytical spinless fermion approach and numerical Lanczos method. In the absence of the magnetic field, the phase diagram is divided into four gapped regions. To determine what happens by applying a transverse magnetic field, using the spinless fermion approach, critical fields are obtained as a function of exchanges. Our analytical results show, the field-induced effects depend on in which one of the four regions the system is. In two regions of the phase diagram, the Ising-type phase transition happens in a finite field. In another region, we have identified two quantum phase transitions (QPT)s in the ground state magnetic phase diagram. These quantum phase transitions belong to the universality class of the commensurate-incommensurate phase transition. We also present a detailed numerical analysis of the low energy spectrum and the ground state magnetic phase diagram. In particular, we show that the intermediate state (h _c1 < h < h _c2) is gapful, describing the spin-flop phase.     

Magnetic properties of Ising metamagnet in both external longitudinal and transverse fields

The phase diagrams of a two-sublattice Ising metamagnet at finite temperature in a mixed longitudinal field and a transverse magnetic field are investigated by the use of an effective-field theory (EFT) with correlations. In addition to the second-order transition lines, the first-order transition lines are also presented in the phase diagrams, since the Gibbs free energy can be calculated numerically. The results show that there is no fourth-order critical line in the phase diagrams given by using EFT as found by using mean-field theory (MFT). The tricritical lines and their projection in the t−h_x plane obtained by using EFT are also quite different from those by using MFT. Only one type of phase diagram is obtained by using EFT while three kinds of phase diagrams are obtained by using MFT, which indicates that only the first kind of phase diagrams obtained by using MFT is reliable. Furthermore, it is shown that the region of first-order transitions increases as the transverse magnetic field h_x decreases.     

The sphericalp-spin interaction spin-glass model

The relaxational dynamics for local spin autocorrelations of the sphericalp-spin interaction spin-glass model is studied in the mean field limit. In the high temperature and high external field regime, the dynamics is ergodic and similar to the behaviour in known liquid-glass transition models. In the static limit, we recover the replica symmetric solution for the long time correlation. This phase becomes unstable on a critical line in the (T, h) plane, where critical slowing down is observed with a cross-over to power law decay of the correlation function ∝t ^?ν, with an exponent ν varying along the critical line. For low temperatures and low fields, ergodicity in phase space is broken. For small fields the transition is discontinuous, and approaching this transition from above, two long time scales are seen to emerge. This dynamical transition lies at a somewhat higher temperature than the one obtained within replica theory. For larger fields the transition becomes continuous at some tricritical point. The low temperature phase with broken ergodicity is studied within a modified equilibrium theory and alternatively for adiabatic cooling across the transition line. This latter scheme yields rather detailed insight into the formation and structure of the ergodic components.     

Effect of an electric field on ferroelastic phase transitions

The effect of an external electric field on the proper ferroelastic phase transitions described in a two-dimensional representation of the D _4h class is analyzed theoretically. The electric field induces a Lifshitz invariant. It is shown that this invariant does not lead to the formation of an incommensurate phase. The phase diagram for commensurate transitions occurring under the action of an electric field changes significantly. The second-order phase transition from the initial phase is split by an electric field of specific orientation into a sequence of two second-order phase transitions with close temperatures.     

A real space description of magnetic field induced melting in the charge ordered manganites: I. The clean limit

We study the melting of charge order in the half doped manganites using a model thatincorporates double exchange, antiferromagnetic superexchange, and Jahn-Teller couplingbetween electrons and phonons. We primarily use a real space Monte Carlo technique tostudy the phase diagram in terms of applied field (h) and temperature(T),exploring the melting of charge order with increasing h and its recovery ondecreasing h.We observe hysteresis in this response, and discover that the “field melted” highconductance state can be spatially inhomogeneous even without extrinsic disorder. Thehysteretic response plays out in the background of field driven equilibrium phaseseparation. Our results, exploring h, T, and the electronic parameter space, are backedup by analysis of simpler limiting cases and a Landau framework for the field response.This paper focuses on our results in the “clean” systems, a companion paper studies theeffect of cation disorder on the melting phenomena.     

Dynamic phase transitions in the kinetic spin-1 Blume-Capel model on the Bethe lattice

The stationary states of the kinetic spin-1 Blume-Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field (h) and crystal field (D) interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures (T) for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the (kT/J,h/J) and (kT/J,D/J) planes for the coordination number q=4. In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed.