Yadin Y. Goldschmidt

Finite size scaling effects in dynamics

We calculate the linear relaxation time for a finite size system with a cubic geometry and “model A” dynamics both above T c in a 4 − e expansion and below T c in a 2+e expansion, and express the results in a scaling form. The universal scaling functions are obtained to one-loop order. We use the method of the effective hamiltonian for the homogeneous modes. The quantum mechanical hamiltonian is supersymmetric. The large n limit (infinitely many spin components) is also considered both above and below T c .

Large q expansions for q-state gauge-matter Potts models in lagrangian form

Abstract We consider the lagrangian form of a q -state generalization of Ising gauge theories with matter fields in d = 3 and 4 dimensions. The theory is exactly soluble in the limit q → ∞ and corrections are easily calculable in power series in 1 q 1 d . Extrapolating the series for the free energies and latent heats by the method of Pade approximants, we have constructed the phase diagrams for all values of q . Our results agree well with known results for pure spin systems and, for the case q = 2, with Ising Monte Carlo data.

Dynamical relaxation in finite size systems: Non-linear and linear decay of the magnetization

We calculate analytically the non-linear time-dependent equation of state for the magnetization in a finite size system. The calculation is done to one-loop order in the q ≠ 0 modes. The equation is obtained in an expansion in powers of e 1/2 where e = 4− d up to o(e). The stochastic equation of state is solved numerically at the critical point and the results are compared with a recent Monte Carlo simulation. The solution displays the full crossover from the bulk power law decay to the finite size exponential relaxation. We also present more explicitly the results obtained for the linear relaxation time in an earlier paper, in a form which enables a direct comparison with simulations. To simplify the calculation we exploit the supersymmetry of the corresponding quantum mechanical hamiltonian.

Nonequilibrium critical behavior in unidirectionally coupled stochastic processes.

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.

Magnets with random uniaxial anisotropy: Thermodynamic properties in the large-N limit

Abstract We show that the random-axis model lends itself to a systematic large- N calculation. The model shows different behavior below and above four dimensions. The equation of state is derived and discussed in terms of “Arrott” plots. Higher-order terms in the disorder, when summed, have a crucial effect on the susceptibility which is found to be finite below four dimensions (and above four dimensions for strong disorder). A spin-glass to paramagnetic phase transition is characterized by the vanishing of the Edwards-Anderson order parameter, which differs from zero in the spin-glass phase. A cusp in the specific-heat and susceptibility is seen across the transition. The cross-over exponent and other exponents of interest are calculated. Above four dimensions a third phase appears for weak disorder and low-temperature ferromagnetic in nature. The transverse and longitudinal susceptibilities are discussed. Whereas the ferromagnetic transition is characterized by mean-field exponents, the ferromagnetic to spin-glass exponents are equal to their counterparts in the non-random system in d − 2 dimensions. This is shown to originate from an effective random field proportional to the EA order parameter. The flow equations in the large- N limit are also discussed.

Phase transitions of the flux-line lattice in high-temperature superconductors with weak columnar and point disorder

We study the effects of weak point and columnar disorder on the vortex-lattice phase transitions in high-temperature superconductors. The combined effect of thermal fluctuations and of quenched disorder is investigated using a simplified cage model. For point disorder we use the mapping to a directed polymer in a disordered medium in 2+1 dimensions. For columnar disorder the problem is mapped into a quantum particle in a harmonic and a random potential. We use the variational approximation to show that point and columnar disorder have opposite effect on the position of the melting line as is observed experimentally. For point disorder, replica symmetry breaking plays a role at the transition into a vortex glass at low temperatures. {copyright} {ital 1997} {ital The American Physical Society}

Monte Carlo studies of the ising spin-glass in a transverse field

The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P(q) is defined and calculated for the present quantum case. The P(q) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.

Application of statistical mechanics to combinatorial optimization problems: The chromatic number problem andq-partitioning of a graph

Methods of statistical mechanics are applied to two important NP-complete combinatorial optimization problems. The first is the chromatic number problem, which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color. The second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections. Both models are mapped into a frustrated Potts model, which is related to theq- state Potts spin glass. For the first problem, we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation. The quenched model is also discussed. For the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq=3 and 4 Potts spin glass using Parisis replica symmetry breaking. We also perform some numerical simulations to test the theoretical result and obtain very good agreement.

The XY model with random p-fold anisotropy: Dynamics and statics near two dimensions

Abstract The phase structure and renormalization group behavior of the XY model with p -fold random anisotropy is studied via time-dependent Langevin formulation. The statics and dynamics of the model are derived in two dimensions and extended to 2 + ϵ dimensions without the use of replicas. The connection with previous replica treatments is discussed. In two dimensions and above we show that a would-be spin glass phase is destroyed at large distances by the topological defects viz. vortices. The mechanism is: random interactions generate random Dzyaloshinskii-Moriya bond interactions which unbind the vortices and this in turn renders the glass phase paramagmetic at large scales. In two dimensions and p 2 > 8 there is a re-entrant transition from an intermediate XY phase into the glass phase. In 2 + ϵ dimensions and p 2 > 8 there is a transition governed by the pure XY fixed point from a paramagnetic phase into a low-temperature glass phase, paramagnetic at large length scales.

A Kosterlitz-Thouless phase transition associated with the supersymmetric sine-Gordon theory

A renormalization group analysis of the supersymmetric sine-Gordon theory near β2 = 4π yields a flow diagram characteristic of a Kosterlitz-Thouless transition, which is expected to manifest itself in certain correlation functions of the theory. The partition function of the theory is equal to 1 to any order in perturbation, in agreement with Zuminos theorem. Thus without explicitly breaking supersymmetry the theory is not equivalent to a classical gas of particles as is the case for the ordinary sine-Gordon.