Critical properties of the anisotropic Ising model with competing interactions

The critical properties of the anisotropic Ising model with competing interactions have been investigated by Monte Carlo methods. The region of localization of the Lifshitz point on the phase diagram has been computed. Relations of the finite-size scaling theory are used to calculate the critical exponents of the heat capacity, susceptibility, and magnetization at various values of the competing interaction parameter J ₁. A crossover to a critical behavior characteristic of a multicritical point with increasing parameter J ₁ is shown to be present in the system.     

Phase transition in the 3d random field Ising model

We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,d _l for the theory isd _l 2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.     

Properties of the skeleton of aggregates grown on a Cayley tree

We discuss and analyze a family of trees grown on a Cayley tree, that allows for a variable exponent in the expression for the mass as a function of chemical distance, M(l)l ^dl . For the suggested model, the corresponding exponent for the mass of the skeleton,d _l ^s , can be expressed in terms ofd _l asd _l ^s = 1,d _l d _l ^c = 2;d _l ^s = d _l –1,d ₁ d _l ^c = 2, which implies that the tree is finitely ramified ford _l 2 and infinitely ramified whend _l 2. Our results are derived using a recursion relation that takes advantage of the one-dimensional nature of the problem. We also present results for the diffusion exponents and probability of return to the origin of a random walk on these trees.     

Generalized classical theory of magnetism

We consider an Ising model with Kac potential ^dK(¦x¦) which may have arbitrary sign, and show, following Gates and Penrose, that the free energy in the classical limit0+ can be obtained from a variational principle. When the Fourier transform of the potential has its maximum atp=0 one recovers the usual mean-field theory of magnetism. When the maximum occurs forp ₀0, however, one obtains an oscillatory or helicoidal phase in which the magnetization near the critical point oscillates with period 2/¦p ₀¦. An example with a potential possessing parameter-dependent oscillations is shown to exhibit crossover phenomena and a multicritical Lifshitz point in the classical limit.     

On Criticality for Competing Influences of Boundary and External Field in the Ising Model

We continue a study of Schonmann (1994), Schonmann and Shlosman (1996), and Greenwood and Sun (1997) regarding the competing influences of boundary conditions and external field for the Ising model. We find a critical point B ₀ in the competing influences for low temperature in dimension d 2A7E; 2.     

Reentrant behaviour in spin glasses

We present the phase diagram of thed-dimensional random bond Ising model as a representative system for spin glasses. We consider nearest neighbour ferromagnetic couplingsJ with concentration 1-p and impurity couplingsaJ (|a|1) with concentrationp. It is shown that for antiferromagnetic couplings, –1<a<0, the system quite generally exhibits reentrant behaviour, i.e. two phase transitions at finite temperatures, in certain ranges of the concentrationp. It is further argued that this behaviour is a quite common feature for spin glass systems characterized by competing interactions.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Coulombic and non-Coulombic contributions to the criticality of ionic fluids. An experimental approach

The recent discovery of liquid-liquid phase separations in electrolyte solutions with critical points near room temperature enables the systematic study of the critical behavior of ionic fluids. Depending on the nature of the molecular interactions, either sharp mean-field or Ising behavior is obtained in the temperature range down tot=(T–T _c )/T _c =10^–4 or less. Mean-field-like criticality is obtained with systems which in the framework of a simple corresponding states model are fairly close to the critical point of the restricted primitive model (RPM) of equally-sized charged spheres in a dielectric continuum. In these cases the phase separation is driven by the Coulombic forces (so-calledCoulombic phase separations). This type of unmixing occurs for 11 electrolytes in solvents of low dielectric constant. Simple mechanisms for unmixing suggested in the literature are discussed in relation to the available data. Some evidence for departures from the simple RPM prediction is found. The presence of additional short-range interactions leads to sharp Ising behavior. Examples are solutions of tetraalkylammonium salts in water and other highly structured solvents, where phase separation results from the peculiar solvophobic nature of ions (solvophobic phase separations). Previous speculations that this type of unmixing shows the tendency toward closed loops are confirmed by the first direct observation of a lower consolute point in an aqueous solution of propyl-tributylammonium iodide. By light scattering studies and measurements of the coexistence curve near the upper and lower consolute points Ising criticality is confirmed. A new mechanism for phase separation is reported for the system ethylammonium nitrate+octanol, where ion pairs are stabilized by hydrogen bonding beyond what is expected from the RPM. This comparatively subtle additional interaction (so-calledstricky ions) already changes the behavior of otherwise RPM-like systems from mean-field to Ising criticality. The results are discussed with particular emphasis on their implications for possible scenarios for explaining a mean-field critical point or crossover from mean-field to Ising behavior beyond the accessible temperature range.     

Quantum rotors with regular frustration and the quantum Lifshitz point

We have discussed the zero-temperature quantum phase transition in n-component quantum rotor Hamiltonian in the presence of regular frustration in the interaction. The phase diagram consists of ferromagnetic, helical and quantum paramagnetic phase, where the ferro-para and the helical-para phase boundary meets at a multicritical point called a (d,m) quantum Lifshitz point where (d,m) indicates that the m of the d spatial dimensions incorporate frustration. We have studied the Hamiltonian in the vicinity of the quantum Lifshitz point in the spherical limit and also studied the renormalisation group flow behaviour using standard momentum space renormalisation technique (for finite n). In the spherical limit ()one finds that the helical phase does not exist in the presence of any nonvanishing quantum fluctuation for m =d though the quantum Lifshitz point exists for all d > 1+m/2, and the upper critical dimensionality is given by d _u = 3 +m/2. The scaling behaviour in the neighbourhood of a quantum Lifshitz point in d dimensions is consistent with the behaviour near the classical Lifshitz point in (d+z) dimensions. The dynamical exponent of the quantum Hamiltonian z is unity in the case of anisotropic Lifshitz point (d>m) whereas z=2 in the case of isotropic Lifshitz point (d=m). We have evaluated all the exponents using the renormalisation flow equations along-with the scaling relations near the quantum Lifshitz point. We have also obtained the exponents in the spherical limit (). It has also been shown that the exponents in the spherical model are all related to those of the corresponding Gaussian model by Fisher renormalisation. Received: 23 December 1997 / Received in final form: 6 January 1998 / Accepted: 7 January 1998     

Study of the three-dimensional Ising model on film geometry with the cluster Monte Carlo method

Topological properties of clusters are used to extract critical parameters. This method is tested for the bulk properties ofd=2 percolation and thed=2, 3 Ising model. For the latter we obtain an accurate value of the critical temperatureJ/k _B T _c=0.221617(18). In the case of thed=3 Ising model with film geometry the critical value of the surface coupling at the special transitions is determined as J_1c/J=1.5004(20) together with the critical exponents ₁ ^m =0.237(5) and=0.461(15).     

Aspects of magnetic disorder

A brief review is given of the effects that quenched, magnetic disorder have on the magnetic properties of systems with short-range interactions. Of primary interest are random exchange, random anisotropy and random fields. Recent theoretical and experimental studies have begun to illuminate the unusual critical behavior that is seen in randomly diluted antiferromagnets in the presence of a uniform field, which is the most direct manner by which the random field problem may be approached. Considerable uncertainty still exists as to what is the lower critical dimensionalityd ₁ and the effective dimensionality ¯d ford-dimension Ising systems in the presence of a random field. This whole area appears to be one in which further insight might be gained through the application of microscopic probes such as NMR, Mössbauer Effect andSR.     

Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor interactions J ₁, prolonged next-nearest-neighbor interactions J _p and one-level next-nearest-neighbor interactions J _o . Vannimenus proved that the phase diagram of Ising model with J _o =0 contains a modulated phase, as found for similar models on periodic lattices, but the multicritical Lifshitz point is at zero temperature. Later Mariz et al. generalized this result for Ising model with J _o ≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with J _o =0. We consider Potts model with J _o ≠0 and show that for some values of J _o the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJ _o is nonzero, the paramagnetic phase found at high temperatures for J _o =0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for J _o =0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J ₁,J _p and J _o . At finite temperatures several interesting features are exhibited for typical values of J _o /J ₁.     

Monte Carlo renormalization of hard sphere polymer chains in two to five dimensions

A renormalization group for polymer chains with hard-core interaction is considered, where a chain ofN ₀ links of lengthl ₀ and hard-core diameterh ₀ is mapped onto a chain ofN ₁=N ₀/s links of lengthl ₁ and hard-core diameterh ₁. The lengthl ₁ is defined in terms of suitable interior distances of the original chain, andh ₁ is found from the condition that the end-to-end distance is left invariant. This renormalization group procedure is carried through by various Monte-Carlo methods (simple sampling is found advantageous for short enough chains or high dimensionalities, while dynamic methods involving kinkjumps or reptation are used else). Particular attention is paid to investigate systematic errors of the method by checking the dependence of the results on bothN ₀ ands. It is found that for dimensionalitiesd=2, 3 only the nontrivial fixed-point is stable, where upon iteration the ratio _k =h _k /l _k tends to nonzero fixed-point value ^*, while ford=4,5 the method converges to the gaussian fixed point with ^*=0. Taking both statistical and systematic errors into account, we estimate the exponentv asv=0.74±0.01 (d=2) andv =0.59±0.01 (d=3). The results are consistent with the expected crossover exponents =1/2 (d=3) and =1 (d=2), respectively.     

Characterisations of phase transitions in Ising spin systems

We construct aC*-algebraic formalism designed to provide a framework for the characterisation of phase transitions in a class of Ising spin systems: this class is large enough to include the rectangular lattice models, of arbitrary finite dimensionality, with nearest neighbour interactions. Using an extension of Onsager's transfer matrix formalism, we express properties of a Gibbs state of a system in terms of a contractive linear transformation, ₀, of a certain Hilbert space, the properties of ₀ being governed by the temperature as well as the interactions in the system. We obtain conditions on ₀ under which the system exhibits a phase transition characterised by (A) a thermodynamical singularity, (B) a change in symmetry, associated with theG-ergodic decomposition of Gibbs states, (C) a divergence of a correlation length (appropriately defined) at the critical point, and (D) scaling laws in the critical region. Applying our formalism to the rectangular two-dimensional Ising model with nearest neighbour interactions, we show that its phase transition possesses the properties (B) and (C), as well as (A).     

On the interface stiffness of the random field ising model

Recent results of Grinstein, Ma, Villain and Binder on interface roughening incontinuum andlattice random field Ising models are related by introducing an effective interface stiffness function {ei247-1}. Ford3 dimensions the continuum theory is shown to be valid for non-zero random field strengthh for all temperatures and on a length scaleL>l _d (h,T) _d (h,T). Ford=2 and smallT a smeared spin-glass transition occurs at ₂(h,T)h. It is argued, that for 3<d<5 interface roughening occurs only forh larger than a critical field strengthh _R (T).     

Critical properties and finite-size effects of the five-dimensional Ising model

Monte Carlo calculations of the thermodynamic properties (energy, specific heat, magnetization suceptibility, renormalized coupling) of the nearest-neighbour Ising ferromagnet on a five-dimensional hypercubic lattice are presented and analyzed. Lattices of linear dimensionsL=3, 4, 5, 6, 7 with periodic boundary conditions are studied, and a finite size scaling analysis is performed, further confirming the recent suggestion thatL does not scale with the correlation length (the temperature variation of which near the critical temperatureT _c is |1-T/T _c |^–1/2), but rather with a thermodynamic lengthl (withl|1-T/T _c |^–2/d ,d=5 here). The susceptibility (extrapolated to the thermodynamic limit) agrees quantitatively with high temperature series extrapolations of Guttmann. The problem of fluctuation corrections to the leading (Landau-like) critical behaviour is briefly discussed, and evidence given for a specific-heat singularity of the form |1-T/T _c |^1/2, superimposed on its leading jump.Dedicated to Prof. Dr. H.E. Müser on the occasion of his 60th birthday     

Conformal invariance and surface defects in the two-dimensional Ising model. Exact results

The surface critical behavior of the two-dimensional Ising model with homogeneous perturbations in the surface interactions is studied on the one-dimensional quantum version. A transfer-matrix method leads to an eigenvalue equation for the excitation energies. The spectrum at the bulk critical point is obtained using anL ^–1 expansion, whereL is the length of the Ising chain. It exhibits the towerlike structure which is characteristic of conformal models in the case of irrelevant surface perturbations (h _s /J _s 0) as well as for the relevant perturbationh _s =0 for which the surface is ordered at the bulk critical point leading to an extraordinary surface transition. The exponents are deduced from the gap amplitudes and confirmed by exact finite-size scaling calculations. Both cases are finally related through a duality transformation.     

On the equivalence of different order parameters and coexistence of phases for Ising ferromagnet. II

We investigate Ising spin systems with general ferromagnetic, translation invariant interactions,H=–J _BB,J _B0. We show that the critical temperatureT _i for the order parameterp _i defined as the temperature below whichp _i>0, is independent of the way in which the symmetry breaking interactions approach zero from above. Furthermore, all the equivalent correlation functions have the same critical exponents asT T_i from below, e.g. for pair interactions all the odd correlations have the same critical index as the spontaneous magnetization. The number of fluid and crystalline phases (periodic equilibrium states) coexisting at a temperatureT at which the energy is continuous is shown to be related to the number of symmetries of the interactions. This generalizes previous results for Ising spins with even (and non-vanishing nearest-neighbour) ferromagnetic interactions. We discuss some applications of these results to the triangular lattice with three body interactions and to the Ashkin-Teller model. Our results give the answer to the question raised by R.J. Baxter et al. concerning the equality of some critical exponents.Supported by NSF Grant PHY 77-22302     

Relaxation times in a finite Ising system with random impurities

Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM ²(t). In addition, universal cumulant ratios are calculated to second order in ^1/4, where =4–d andd<4 denotes the spatial dimension.