Exponents and logarithms of the Potts model through a 1D quantum hamiltonian

Exact solution of the one-dimensional quantumq-state Potts model for finite chains and subsequent finite-size analysis yields values for the thermal critical exponenty=1/v forq=3 andq=4. By taking into account the shift of the critical point due to the finite size we obtainy=1.48±0.01 forq=4 in good agreement with they=3/2 conjectured by den Nijs. Using the conjecture of den Nijs we find evidence for a logarithmic factor forq=4 but not forq=3.     

The two-dimensional ANNNI model in the CVM approximation

The phase diagram of the two-dimensional ANNNI model is investigated by the cluster variation method (CVM). We confirm the stability of the disordered phase down toT=0 and the absence of a Lifshitz point at finite temperature forK<1/2, whereK the ratio of the second to the first neighbor pair interactions. Two different modulation regimes for the correlation functions of the disordered phase are separated by a disorder line along which theq vector of the susceptibility maximum undergoes a lock-in transition. The study in reciprocal space of the stability of the disordered phase allows us to define a critical line in the phase diagram along which theq vector characterizing the instability is incommensurate. Finally, we show the existence of another Lifshitz point forK tending to infinity.     

Absence of the partially ordered phase in theq-state models

We investigate theq-state models called (N ,N ) model using an infinitesimal Migdal-Kadanoff renormalization-group method. We distinguish two cases namely the isotropic model and the anisotropic model. The first one presents a critical value ofq,q _c such that forq _c we obtain an Ashkin-Teller phase diagrams while forq>q _c the partially ordered phase disappears then the model exhibits only phase transition between ferromagnetic phase and disordered one. The phase diagrams in the second case are qualitatively similar to one obtained forZ(6) model for all values ofq.     

The potts model on bethe lattices

We analyze the properties of theq-state ferromagnetic Potts model for realq. The nature of the phase transition at the critical point is first-order forq2, and second-order forq=2. The random-bond percolation limitq1, and its second-order-like transition, are not related to the previous behaviour since they arise from non-stable phases of the system. It is suggested that this property characterizes the model on high-dimensional lattices, too.Supported by MPI and CNR     

Excitations in Heisenberg spin glasses

Within the RPA approach forT=0, the excitations of the Heisenberg spin glass system Eu _x Sr_1–x S are studied by numerical methods, using a continued fraction algorithm. Both the density of statesg(E) and also the spectral functionS(q,E) are calculated for systems with (16)³ sites, withx=0.4, 0.5, and 0.6 (spin glass phase), and also forx0.7 (ferromagnetic phase). Forq-vectors within the (1,1,1) plane,S(q,E) shows magnon peaks even in the spin glass phase, over the whole range ofq. However, these peaks are quite broad, and there is considerable intensity at small energies even for largeq, leading to a finite intercept ofg(E) forE0. Over a large temperature range, the specific heat is approximately linear inT forx0.7.     

Critical phenomena with convergent series expansions in a finite volume

Linked cluster expansions are generalized from an infinite to a finite volume. They are performed to 20th order in the expansion parameter to approach the critical region from the symmetric phase. A new criterion is proposed to distinguish first- from second-order transitions within a finite-size scaling analysis. The criterion applies also to other methods for investigating the phase structure, such as Monte Carlo simulations. Our computational tools are illustrated with the example of scalar (O(N) models with four- and six-point couplings forN=1 andN=4 in three dimensions. It is shown how to localize the tricritical line in these models. We indicate some further applications of our methods to the electroweak transition as well as to models for superconductivity.     

Exact results for the Potts model in two dimensions

By considering the zeros of the partition function, we establish the following results for the Potts model on the square, triangular, and honeycomb lattices: (i) We show that there exists only one phase transition; (ii) we give an exact determination of the critical point; (iii) we prove the exponential decay of the correlation functions, in one direction at least, for all temperatures above the critical point. The results are established forq 4, whereq is the number of components.Work supported by the Fond. National Suisse de la Recherche Scientifique.Work supported in part by NSF Grant No. DMR 76-20643.     

Dynamics of the infinite-ranged Potts model

We formulate a theory of single-spin-flip dynamics for the infinite-rangeq-state Potts model. We derive a Fokker-Planck equation, without diffusive term, from a phenomenological master equation. It describes the approach to equilibrium of the time-dependent probability density and thus generalizes Griffiths' (1966) result for the Ising model. We particularly compare the dynamic evolutions ofq=2 andq=3 systems when sinusoidal external fields are applied. In the caseq=2 we find evidence of a nonequilibrium phase transition and forq=3 period doubling bifurcations are observed, yielding a good estimate of Feigenbaum's universal exponent.     

Plasmons and their damping in a doped semiconductor superlattice

The complex zeroes of dielectric response function of a doped GaAs superlattice are computed to study the frequencies and damping rates of oscillations in coupled electron-hole plasma. The real part of a complex zero describes the plasma frequency, whereas imaginary part of it yields the damping rate. Strong scattering of charge carriers from random impurity potentials in a doped GaAs superlattice gives rise to a large value of damping rate which causes over-damping of plasma oscillations of coupled electron-hole gas below q_c, a critical value of wave vector component (q) along the plane of a layer of electrons (holes). The plasma oscillations which correspond to electrons gas enter into over-damped regime for the case of weak coupling between layers. Whereas, plasma oscillations which belong to hole gas go to over-damped regime of oscillations for both strong as well as weak coupling between layers. The damping rate shows strongq-dependence forq < q_c, whereas it weakly depends onq forq ≥q _c . The damping rate exhibits a sudden change atq =q _c , indicating a transition from non-diffusive regime (where collective excitation can be excited) to diffusive regime (over-damped oscillations).     

λφ 4 q-Renormalization program

A regularization scheme for quantum field theories given in aq-mutator algebra for the internal momentum space in a loop integration is constructed. We show Feynman integrals that are finite forq 1but diverse asq 1. Using this regularization scheme, we propose a renormalization program in q-mutator space (q-renormalization program) for thef ⁴ theory as an example, up to some one-loop diagrams. This work paves the way to obtaining physically measurable quantities from quantum field theories over spaces that neither commute nor anticommute.     

Random Skew Plane Partitions and the Pearcey Process

We study random skew 3D partitions weighted by q ^vol and, specifically, the q → 1 asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.     

Critical surface wave velocity near phase transitions

The temperature dependent Rayleigh wave velocity is discussed for crystals with a surface layer of depth equal to the correlation length, ξ, with special reference to a critical region near phase transitions. For finite qξ the Rayleigh wave velocity reflects the specific critical properties of the surface layer. Under such conditions the temperature dependence of Rayleigh waves cannot be predicted on the temperature dependence of constants. A phenomenological analysis is made for qξ ≦? 1. Experimental results on SrTiO₃ show bulk-dominated critical Rayleigh wave velocities due to the short correlation length in this material.     

Correlation and anticorrelation of fractal aggregates. A model for «spinodal-type» aggregation

Summary We present a simple phenomenological model, based on mass conservation arguments, in order to describe aggregation in dense solutions. It has been previously shown that fordiffusion-limited processes scattered intensity distributions exhibit a peak atq≠0 which grows in time and moves to smaller and smallerq vectors. According to the model, each aggregate is surrounded by a depletion region whose size depends on the aggregation kinetics. The form factor for such a cluster should satisfy local mass conservation and hence exhibit a pronounced depression atq=0. the initial stages of aggregation, when the clusters are far apart, are accounted on the basis of a form factor alone. The model shows good agreement with the data and also explains the experimental evidence that the peak does not appear inreaction-limited conditions. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

The thermodynamics and the phase diagram of random field Ising model (RFIM) on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field (B). A Gaussian distribution of random field (hi) with zero mean and variance hi2 = HR2F is considered. The free-energy (F ), the magnetization (M) and the order parameter (q) are investigated for several values of coordination number (z). The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature TC = J/k and when HRF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (HRF) and finite z. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field HRF increases. The ferromagnetic (FM)-paramagnetic (PM) phase boundary is clearly observed only when z →∞. While FM-PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (χ) shows a sharp cusp at TC in a small random field for finite z and rounded different peaks on increasing HRF.     

New critical dimensions forq-deformed bosonic string

Applying the mass shell condition forq-deformed open and closed strings, new critical space-time dimensions are derived. Supported by the Algerian Ministry of High Education and Research under contract N^oD2501/17/01/93. Permanent address.     

Monte Carlo calculations on the gauge Potts model

Theq-state gauge Potts modelP _q ind-dimensions has been studied using Monte Carlo techniques. Ford=2 no phase transitions were detected. TheP ₂ model ford=3 shows a second order phase transition. On the other hand, all thed=3 (q2) andd=4 cases studied show first order phase transitions. In these cases, it was possible to estimate transition coupling parameters as well as latent heat. For selected cases, a study of the behavior of the Wilson loop factor was done.Partially supported by CONICET, CIC Pcia. de Buenos Aires and SUBCYT, Argentina     

Order parameters forn-vector spin glasses ind dimensions

We report Monte Carlo simulations of the time-dependent behavior of Edwards-Anderson spin glasses with Gaussian nearest-neighbor exchange, for both spin dimensionalityn and space dimensionalityd from 2 up to 6. A (nearly) logarithmic decay of the Edwards-Anderson order parameter with time is observed for alln and alld, similar to earlier studies forn=1. But the Monte Carlo data forn>1 suggest stronger than those forn=1 that all order parameters considered vanish in thermal equilibrium for nonzero temperature, because the decay forn>1 is faster at the temperatures of interest. For Heisenberg spins (n=3) no significant dependence of the Edwards-Anderson order parameterq on the size of the lattice was observed ford=2,3 and 4, whereas ford=5 and 6,q was smaller for smaller systems (in contrast to thed=5 Ising case). These results are the first Monte Carlo indication of a change in the bulk behavior of Heisenberg spin glasses at dimensionalityd=4. Quenching the system to zero temperature and then applying a field we find that the order parameter , measuring the alignment with respect to the state at zero field, is destroyed by a sufficiently strong magnetic field, for all observedn andd.Sonderforschungsbereich 125 Aachen-Jülich-Köln, FRG     

Exact results on the random Potts model

We present a number of exact results on the random-bond,q-state Potts model. The quenched model on any finite planar graph or lattice is shown to obey a duality relation for general type of bond-randomness. In the annealed case, the solution of the model reduces to that of the regular (nonrandom) Potts model on the corresponding lattice. Explicit knowledge of the critical parameters of theq-state Potts model in two dimensions allows us to evaluate exactly the phase diagram of the annealed model on the square, triangular and honeycomb lattices. We discuss the behavior near the (random) critical point and comment on the relationship between the quenched and annealed systems. The exact phase diagram of the annealed system is obtained for the bond-diluted model and the spin-glass model with and without dilutions.Work supported in part by NSF grant No. DMR-78-18808     

Composition of kinetic momenta: The(sl(2)) casecase

The tensor products of (restricted and unrestricted) finite dimensional irreducible representations of(sl(2)) are considered forq a root of unity. They are decomposed into direct sums of irreducible and/or indecomposable representations.     

Scaling for the discrete Mathieu equation

The scaling of the total width of the band for the discrete Mathieu equation is studied in the critical region near the transition between localized and extended states, for the special case in which there is one period of the modulation forp lattice spacings. A general expression for the bandwidthW in the critical region is found. At the critical point an analytic expression forpW is found which agrees to one part in 10⁸ with the result deduced from numerical work.