Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Explicit expressions for the fourth-order susceptibility (⁴), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature T _c ⁽⁴⁾ = 2k _B ^–1 J/ln{1+2/[(z–1)^3/4–1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B is the Boltzmann constant. The temperatures at which ⁽⁴⁾ and the ordinary susceptibility (²) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.     

Decay of the Two-Point Function in One-Dimensional O(N) Spin Models with Long-Range Interactions

Using Griffiths and Lieb–Simon type inequalities, it is shown that the two-point function of ferromagnetic spin models with N components in one dimension decays like the interaction J(n)n ^– provided that 1N4 and T>T _c.     

The lattice covering time problem for sites visitedk-times

The problem of the covering time for sites visitedk-times is defined as the mean time taken by a random walk to visit each site of a lattice at leastk times. We performed the investigation using Monte Carlo simulations over one dimensional lattices, ofN sites, with periodic boundary conditions. Two different regions are investigated:Nk1 andkN1. In the former region, we obtain a behaviour of the typet _k/t ₁=a –Bk ^–0.35+A(k)N ^–0.75, (a <1.6). In the latter region we obtain two possible behaviours:t _k k ^0.95 andt _k k(lnk)^–0.5. Two formulas which have a very close behaviour.     

The effect of a constant number of particles on the pair correlation function and the density profile in a one-dimensional lattice gas

A one-dimensional lattice gas (Ising model) of lengthL and with nearest-neighbor couplingJ is considered in a canonical ensemble with fixed number of particlesN=L/2. Exact expressions and asymptotic forms for largeL are derived for the density-density correlation function, using periodic boundary conditions, and for the density (magnetization) profile, using antisymmetric boundary conditions. The density-density correlation function,g, assumes for temperaturesT> T, withT = 2J(_BlnL)^–1 and forL large, the formg(x) =g ^gc(x) +BL ^–1 +a(x)L ^–1 +O(L^–2) wherex is a distance between considered lattice sites,B is known from earlier work of Lebowitz and Percus,(1b) anda(x) decays exponentially forx . For TT, the correlation function and the density profile behave differently, the latter exhibiting a step in the middle of the interface.     

On the connection between the one-dimensionalS=1/2 Heisenberg chain and Haldane-Shastry model

Extra integrals of motion and the Lax representation are found for interacting spin systems with the HamiltonianH = (J/2) _{j, k=1,j k} ^N P(j – k) _{j k} , where one of the periods of the WeierstrassP function is equal toN. The Heisenberg and Haldane-Shastry chains appear as limiting cases of these systems at some values of the second period. The simplest eigenvectors and eigenvalues ofH corresponding to the scattering of two spin waves are presented explicitly for these finite-dimensional systems and for their infinite-dimensional version.     

Special energies and special frequencies

Special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators. Our investigation includes both this model and the random one-dimensional Schrödinger operator describing an alloy or its discrete analogue. Using the phase method we exactly determine a bilateral Lific asymptotic of the integrated density of statesk(E) at special energiesE _s , which is not only of the classical type exp(–c/|E–E _s |^1/2) but also exp(–c/|E–E _s |) is a typical behaviour. In addition, other asymptotics occur, e.g. |E–E _s | ^c , which show thatk(E) need not bek .     

EPR and Thermoluminescence in ZnO Single Crystals with Anionic Vacancies

The radiation defects created in hydrothermal ZnO–Li single crystals by irradiating them with electrons, protons, and highenergy ions have been investigated. The anionic vacancies (Fcenters) in ZnO are established to be the centers of radiationless recombination of the charge carriers with a photoionization energy of 2.3 eV (a signal of the photoEPR with the gfactor for the F⁺center: g = 1.9948 and g = 1.9963). The anionic vacancies in the form of the F and F⁺ states are a good reference of the electron and hole processes. The [F_Li]^×centers that correspond to the oxygen vacancies localized near the point defects Li_Zn are detected. In the temperature range 530–660 K, ZnO crystals display thermally stimulated processes such as the healing of anionic vacancies (530–630 K) and the disappearance of the [F_Li]^×centers (610–660 K).     

Parametric relaxation of spectra

We study the change of an quasienergy spectrum upon variation of the weight of a perturbation in the Floquet operatorF=F ₀e^–iV . Employing ideas from level dynamics and random matrix-theory we show that the distribution of nearest-neighbor spacings can display effectively irreversible behavior. Small deviations from equilibrium relax in a certain collision time which scales with the numberN of levels as _collN ^–3/2.     

On the indistinguishability of classical particles

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2)³ are reduced to points in the phase space, the partition functionTr{exp(–) for N identical bosons (fermions) approaches (2)^–3N(N!) ... d³r₁ d³p₁ ... d³r_N d³p_N exp(–H). The two factors, (2)^–3N and (N!)^–1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles.     

On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results

We consider the model of a 2D surface above a fixed wall and attracted toward it by means of a positive magnetic fieldh in the solid-on-solid (SOS) approximation when the inverse temperature is very large and the external fieldh is exponentially small in . We improve considerably previous results by Dinaburg and Mazel on the competition between the external field and the entropic repulsion with the wall, leading, in this case, to the phenomenon of layering phase transitions. In particular, we show, using the Pirogov-Sinai scheme as given by Zahradník, that there exists a unique critical valueh _k ^* () in the interval (1/4e ^–4k , 4e ^–4k ) such that, for allh(h _k+1 ^* ,h _k ^* ) and large enough, there exists a unique infinite-volume Gibbs state. The typical configurations are small perturbation of the ground state represented by a surface at heightk+1 above the wall. Moreover, for the same choice of the thermodynamic parameters, the influence of the boundary conditions of the Gibbs measure in a finite cube decays exponentially fast with the distance from the boundary. Whenh=h _k ^* () we prove instead the convergence of the cluster expansion for bothk andk+1 boundary conditions. This fact signals the presence of a phase transition. In the second paper of this series we will consider a Glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external fieldh. Using the results proven in this paper, we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free (absent).     

First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models

We investigate the large N limit of spectral measures of matrices which relate to the Gibbs measures of a number of statistical mechanical systems on random graphs. These include the Ising and Potts models on random graphs. For most of these models, we prove that the spectral measures converge almost surely and describe their limit via solutions to an Euler equation for isentropic flow with negative pressure p()=–3^–123.     

Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ _x,y const/¦x–y¦². For translation-invariant systems with well-defined lim _x x ² J _x =J ⁺ (possibly 0 or ) we establish: (1) There is no long-range order at inverse temperatures withJ ⁺1. (2) IfJ ⁺>q, then by sufficiently increasingJ ₁ the spontaneous magnetizationM is made positive. (3) In models with 0<J ⁺< the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM( _c )1/( _c J ⁺)^1/2. (4) For Ising (q=2) models withJ ⁺<, it is noted that the correlation function decays as xy()c()/|x–y|² whenever< _c . Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.     

Phase diagrams and magnetic properties of diluted Ising and Heisenberg magnets with competing interactions

Ising and Heisenberg magnets with nearest-neighbor ferromagnetic exchangeJ ₁ and next-nearest antiferromagnetic exchangeJ ₂ and randomly distributed frozen-in nonmagnetic impurities of arbitrary concentration 1–x are studied by several methods: systematic series expansions inx, 1–x and inverse temperature (1/T) as well as Monte Carlo simulation. Depending onRJ ₂/J ₁,T andx the model is in paramagnetic, ferromagnetic, antiferromagnetic or spin glass phases. The microscopic magnetic structures of all these phases are investigated and found to be more complicated than usually (e.g., the ferromagnetic state contains spins and clusters either aligned antiparallel or not aligned at all, when frustration effects make bonds ineffective). We suggest that the concentrationx _c of magnetic ions below which no (anti-)ferromagnetic long range order occurs depends onR continuously, andx _c 1 at the multicritical point (R _m ,T=0) where the order changes from ferromagnetic to antiferromagnetic. Our results for phase diagram, susceptibility etc. are compared to recent data on the Eu _x Sr_1–x S system and very good agreement is found.     

Spin glasses with long-range interaction at high temperatures

We study the Ising andN-vector spin glasses with exchange couplings J=(J _ij ;i, jZ ^d ), which are independent random variables with EJ_ij=0 andEJ ⁿ _ij ⁿ n!¦i–j¦ ^–nd , forn, some finite constant >0, and >1/2. For sufficiently small, we show that forE-a.a.J there is a weakly unique, extremal, infinite-volume Gibbs measure _J for which the expectation of a single (component of) spin vanishes and which has the cluster property inL ₂(E) with the same decay as interaction. This work is based on results and methods of Fröhlich and Zegarlinski.     

Correlation inequalities and the Kosterlitz-Thouless transition for anisotropic rotators

I give a proof of the Kosterlitz-Thouless transition for sufficiently anisotropic (J _zJ _x ^–1 =J _zJ _y ^–1 <2q ^–(JKT)^–J) two-dimensionalN-component rotators (N 3). The method is based on Wells' inequality and is related to mean field Gaussian inequalities.     

Complex temperature plane zeros in the mean-field approximation

We derive asymptotic expressions for the complex temperature plane zeros of the infinite-range Ising model in the scaling regime. The results also apply to high-dimensional, short-range Ising systems. For thenth zero in a system ofN spins, the leading asymptotic result ist _n (n/N)^1/2(–1 ±i).     

Cosets and ferromagnetic correlation inequalities

Consideration of subgroups of the group of all subsets ofN={1, 2, ...,n} with symmetric difference as the operation leads to new inequalities on correlations for a generalized Ising ferromagnet. An upper bound for the rate of change of ^R with respect toJ _S in terms of correlations and a new, brief proof for the monotonicity of ^R as a function ofJ _S are given.Supported by NSF GP 7946.     

Effect of disorder on relaxation in the one-dimensional Glauber model

We study the long-time relaxation of magnetization in a disordered linear chain of Ising spins from an initially aligned state. The coupling constants are ferromagnetic and nearest-neighbor only, taking valuesJ ₀ andJ ₁ with probabilitiesp and 1–p, respectively. The time evolution of the system is governed by the Glauber master equation. It is shown that for large timest, the magnetizationM(t) varies as [exp(–₀ t](t), where ₀ is a function of the stronger bond strengthJ ₀ only, and (t) decreases slower than an exponential. For very long times, we find that ln (t) varies as –t ^1/3. For low enough temperatures, there is an intermediate time regime when ln (t) varies as –t ^1/2. The results can be extended to more general probability distributions of ferromagnetic coupling constants, assuming thatM(t) can only increase if any bond in the chain is strengthened. If the coupling constants have a continuous distribution in which the probability density varies as a power law near some maximum valueJ ₀, we find that ln (t) varies as –t ^1/3(lnt)^2/3 for large times.     

Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a (k)a(l)= _k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA _n (q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA _n (q) as a product of matrices of the form(1–q ^jT)^±1 with 1jn andT a permutation matrix. In particular,A _n (q) is singular if and only ifq ^M=1 for some integerM of the formk ²–k, 2kn.     

Magnetic and electric hyperfine interactions in the system EuSm

Magnetic and electric hyperfine interactions in the system EuSm were investigated with the isomeric 11/2^– state in¹⁴⁵Eu by applying the time-differential perturbed angular distribution (TDPAD) method. The temperature dependence of paramagnetism was studied between 90 K and 1000 K by measuring the magnetic hyperfine interaction frequency _L=g_NN¹ (T)B_ext. The paramagnetic correction factor strictly follows the Curie-Weiss relation =1+C/(T-), withC=–50(2) K and =–29(5) K. This is compatible with a hyperfine field ofB _int(0)=–25(1) T, a valence of two for Eu in Sm, and antiferromagnetic order at low temperatures.The temperature dependence of the electric quandrupole coupling constant v_Q, investigated between 100 K and 400 K, can be reproduced by a linear temperature variation v_Q(T)=v_Q(0) (1-AT), with v_Q(0)=16.2(4) MHz andA=7.2(8)·10^–4 K^–1.The paramagnetic relaxation time _rel of the nuclear alignment is proportional to the temperature of the sample, with _rel T^–1=3.7(2) ·10^–9s K^–1.This leads to the Korringa relation _J T=const=5.1(5)·10^–11s K for the relaxation time of the 4f electronic spinJ. Assuming that the relaxation ofJ is mainly caused by exchange interaction between conduction electrons and localized 4f electrons at the Eu site, an exchange integral of |J _eff|=0.10(2) eV can be deduced.