Rigorous Analysis of Discontinuous Phase Transitions via Mean-Field Bounds

 We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice ℤ ^d . Our essential assumption is that these models satisfy the condition of reflection positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean- field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on ℤ ^d undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with q≥3, the r-component cubic models with r≥4 and the O(N)-nematic liquid-crystal models with N≥3. Received: 22 July 2002 / Accepted: 12 January 2003 Published online: 5 May 2003 RID="⋆" ID="⋆" ? Copyright rests with the authors. Reproduction of the entire article for non-commercial purposes is permitted without charge. Communicated by J. Z.Imbrie     

Phase transitions and critical phenomena in a three-dimensional site-diluted Potts model

The phase transitions and critical phenomena in the three-dimensional (3D) site-diluted q-state Potts models on a simple cubic lattice are explored. We systematically study the phase transitions of the models for q=3 and q=4 on the basis of Wolff high-effective algorithm by the Monte–Carlo (MC) method. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.00–0.65. It is shown that introducing of weak disorder (p∼0.95) into the system is sufficient to change the first order phase transition into a second order one for the 3D 3-state Potts model, while for the 3D 4-state Potts model, such a phase transformation occurs when introducing strong disorder (p∼0.65). Results for 3D pure 3-state and 4-state Potts models (p=1.00) agree with conclusions of mean field theory. The static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the exponent of the correlation radius ν are calculated for the samples on the basis of finite-size scaling theory.     

Potts model on complex networks

We consider the general p-state Potts model on random networks with a given degree distribution (random Bethe lattices). We find the effect of the suppression of a first order phase transition in this model when the degree distribution of the network is fat-tailed, that is, in more precise terms, when the second moment of the distribution diverges. In this situation the transition is continuous and of infinite order, and size effect is anomalously strong. In particular, in the case of p = 1, we arrive at the exact solution, which coincides with the known solution of the percolation problem on these networks.Received: 3 December 2003, Published online: 17 February 2004PACS: 05.10.-a Computational methods in statistical physics and nonlinear dynamics - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.50. + q Lattice theory and statistics (Ising, Potts, etc.) - 87.18.Sn Neural networks     

Boundary critical behaviour of two-dimensional random Potts models

Using extensive Monte Carlo simulations, transfer matrix techniques and conformal invariance, ferromagnetic random q-state Potts models for are studied in the vicinity of the critical temperature. In particular the surface and bulk magnetization exponents and are found monotonically increasing with q. At the critical temperature, different moments (n) of the magnetization profiles are calculated which are all found to accurately follow predictions of conformal invariance. The critical correlation functions show multifractal behaviour, the decay exponents of the different moments both in the volume and at the surface, are n-dependent. Received 4 June 1999     

Complete Analyticity of the 2D Potts Model above the Critical Temperature

We investigate the complete analyticity (CA) of the two-dimensional q-state Potts model for large values of q. We are able to prove it for every temperature , provided we restrict ourselves to nice subsets, their niceness depending on the temperature T. Contrary to this restricted complete analyticity (RCA), the full CA is known to fail for some values of the temperature above . Our proof is based on Pirogov-Sinai theory and cluster expansions for the Fortuin-Kasteleyn representation, which are available for the Potts model at all temperatures, provided q is large enough. Received: 16 July 1996 / Accepted: 8 January 1997     

A DMRG study of the -symmetric Heisenberg chain

The spin one-half Heisenberg chain with U q [ SU (2)] symmetry is studied via density-matrix renormalization. Ground-state energy and q-symmetric correlation functions are calculated for the non-Hermitian case with integer r. This gives bulk and surface exponents for (para)fermionic correlations in the related Ising and Potts models. The case of real q corresponding to a diffusion problem is treated analytically. Received: 18 February 1998 / Accepted: 17 March 1998     

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs. It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type. For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1. We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. Received: 27 March 2000 / Accepted: 7 December 2000     

Magnetization and configurational anisotropy in magnetic clusters: Monte Carlo simulation

Based on the Monte Carlo simulation, the magnetic properties of the clusters, e.g. magnetization, Curie temperature, hysteresis, coercivity, natural angle and energy distribution etc., have been calculated. It has been found that, for the pure ferromagnetic cluster, the T^3/2 Bloch law is well satisfied at low temperature (T < 0.5 T_C) and B_sur is equal to 3 B_bulk. Meanwhile, there are clear indications that B increases drastically with the reducing atomic number Nwhich is consistent with the experimental facts. The results have been evalucted using the Bloch exponent law in the approximate crystalline approximation. It has also been demonstrated that the size dependence of the Curie temperature can be described by finite-size scaling theory. The investigation of the hysteresis and the spin configurations in different magnetization processes reveals the existence of an easy magnetization direction and anisotropy. The thermal coercivity for the clusters with zero and finite uniaxial anisotropy matches the experimental results well. The simulated results for the natural angle and energy distribution in the clusters prove further the existence of the configurational anisotropy in the clusters. It has been discussed that the natural angle and energy distribution influence the hysteresis of a cluster.Received: 10 September 2003, Published online: 15 March 2004PACS: 75.75. + a Magnetic properties of nanostructures - 75.40.Mg Numerical simulation studies - 75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects - 75.40.Cx Static properties (order parameter, static susceptibility, heat capacities, critical exponents, etc.)     

Quark mass dependence of the nuclear forces

We investigate the behavior of the nuclear force as a function of the light-quark masses m _q in the framework of chiral effective field theory at next-to-leading order. The unknown m _q -dependent short-range contribution is estimated by means of dimensional analysis. We calculate various observables for different values of m _q . We found no new bound states and a larger deuteron binding energy, MeV, in the chiral limit.Received: 30 September 2002, Published online: 22 October 2003PACS: 11.30.Rd Chiral symmetries - 13.75.Cs Nucleon-nucleon interactions (including antinucleons, deuterons, etc.) - 21.30.Cb Nuclear forces in vacuum - 21.30.Fe Forces in hadronic systems and effective interactions     

Effects of the screening breakdown in the diffusion-limited aggregation model

Several models based on the diffusion-limited aggregation (DLA) model were proposed and their scaling properties explored by computational and theoretical approaches. In this paper, we consider a new extension of the on-lattice DLA model in which the unitary random steps are replaced by random flights of fixed length. This procedure reduces the screening for particle penetration present in the original DLA model and, consequently, generates new pattern classes. The patterns have DLA-like scaling properties at small length of the random flights. However, as the flight size increases, the patterns are initially round and compact but become fractal for sufficiently large clusters. Their radius of gyration and number of particles at the cluster surface scale asymptotically as in the original DLA model. The transition between compact and fractal patterns is characterized by wavelength selection, and 1/k noise was observed far from the transition.Received: 2 March 2004, Published online: 14 December 2004PACS: 05.40.Fb Random walks and Levy flights - 05.50. + q Lattice theory and statistics (Ising, Potts, etc.) - 05.10.Ln Monte Carlo methods     

Charge density plateaux and insulating phases in the $\mathsf{t-J}$ model with ladder geometry

We discuss the occurrence and the stability of charge density plateaux in ladder-like t-J systems (at zero magnetization M = 0) for the cases of 2- and 3-leg ladders. Starting from isolated rungs at zero leg coupling, we study the behaviour of plateaux-related phase transitions by means of first order perturbation theory and compare our results with Lanczos diagonalizations for t-J ladders (N = 2 × 8) with increasing leg couplings. Furthermore we discuss the regimes of rung and leg couplings that should be favoured for the appearance of the charge density plateaux.Received: 28 July 2003, Published online: 8 December 2003PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.) - 71.27. + a Strongly correlated electron systems; heavy fermions - 75.10.-b General theory and models of magnetic ordering - 75.10.Jm Quantized spin models     

Critical exponents of Manhattan Hamiltonian walks in two dimensions,from Potts andO(n) models

We consider a set of Hamiltonian circuits filling a Manhattan lattice, i.e., a square lattice with alternating traffic regulation. We show that the generating function (with fugacityz) of this set is identical to the critical partition function of aq-state Potts model on an unoriented square lattice withq ^1/2 =z. The set of critical exponents governing correlations of Hamiltonian circuits is derived using a Coulomb gas technique. These exponents are also found to be those of an O(n) vector model in the low-temperature phase withn =q ^1/2 =z. The critical exponents in the limitz = 0 are then those of spanning trees (q= 0) and of dense polymers (n=0,T < T_c), corresponding to a conformal theory with central chargeC = –2. This shows that the Manhattan orientation and the Hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of Hamiltonian walks.     

Coupled Potts models: Self-duality and fixed point structure

We consider q-state Potts models coupled by their energy operators. Restricting our study to self-dual couplings, numerical simulations demonstrate the existence of non-trivial fixed points for 2 ⩽ q ⩽ 4. These fixed points were first predicted by perturbative renormalisation group calculations. Accurate values for the central charge and the multiscaling exponents of the spin and energy operators are calculated using a series of novel transfer matrix algorithms employing clusters and loops. These results compare well with those of the perturbative expansion, in the range of parameter values where the latter is valid. The criticality of the fixed-point models is independently verified by examining higher eigenvalues in the even sector, and by demonstrating the existence of scaling laws from Monte Carlo simulations. This might be a first step towards the identification of the conformal field theories describing the critical behaviour of this class of models.     

Phase transitions in a cluster molecular field approximation

Cluster molecular field approximations represent a substantial progress over the simple Weiss theory where only one spin is considered in the molecular field resulting from all the other spins. In this work we discuss a systematic way of improving the molecular field approximation by inserting spin clusters of variable sizes into a homogeneously magnetised background. The density of states of these spin clusters is then computed exactly. We show that the true non-classical critical exponents can be extracted from spin clusters treated in such a manner. For this purpose a molecular field finite size scaling theory is discussed and effective critical exponents are analysed. Reliable values of critical quantities of various Ising and Potts models are extracted from very small system sizes. Received 30 September 2002 / Received in final form 25 November 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: pleim@theorie1.physik.uni-erlangen.de     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation

We develop a new analysis of the order-disorder transition in ferromagnetic Potts models for large numberq of spin states. We use the Pirogov-Sinaï theory which we adapt to the Fortuin-Kasteleyn representation of the models. This theory applies in a rather direct way in our approach and leads to a system of non-interacting contours with small activities. As a consequence, simpler and more natural techniques are found, allowing us to recover previous results on the bulk properties of the model (which then extend to non-integer values ofq) and to deal with non-translation invariant boundary conditions. This will be applied in a second part of this work to study the behaviour of the interfaces at the transition point.Laboratoire Propre du CNRS: LP 7061     

Investigation of the critical behavior of the three-dimensional weakly diluted Potts model

The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q ⁻⁴⁷ (resp. q ⁻⁴⁶). Finally, we compute chromatic roots for strips of widths 9≤m≤12 with free boundary conditions and locate roughly the limiting curves.     

Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.