arXiv subjects

Bertrand Berche

Publications and source records attributed to Bertrand Berche.

At least 109 records · Page 6Linked to original sources

arXiv2004

Bond dilution in the 3D Ising model: a Monte Carlo study

We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent $α$. According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, between the pure model limit and the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted $q=4$ Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is governed by the same universality class as the site-diluted model.

cond-mat.dis-nn↗

arXiv2004

Numerical investigation of logarithmic corrections in two-dimensional spin models

The analysis of correlation function data obtained by Monte Carlo simulations of the two-dimensional 4-state Potts model, XY model, and self-dual disordered Ising model at criticality are presented. We study the logarithmic corrections to the algebraic decay exhibited in these models. A conformal mapping is used to relate the finite-geometry information to that of the infinite plane. Extraction of the leading singularity is altered by the expected logarithmic corrections, and we show numerically that both leading and correction terms are mutually consistent.

cond-mat.stat-mech↗

arXiv2004

Topological phase transitions in liquid crystals -- Transiciones de fase topológicas en cistales l{'ı}quidos

By means of Monte Carlo simulations, using finite size scaling and conformal transformations, evidences of a topological phase transition in liquid crystals in two dimensions are reported. A paramagnetic phase appears at high temperature whereas a quasi-long-range-order (QLRO) phase is founded at low temperatures. It is determined that the correlation function exponent, at very low temperatures, is linear with the temperature, a typical behavior of spin waves. This contradicts predictions that for systems with non-abelian group of global symmetry the spin waves are not relevant. Aditionally, the implications that would have the presence of impurities in this type of transitions is discussed. ----- Mediante simulaciones de Monte Carlo, utilizando escalamiento de tamaño finito y transformaciones conforme, se reportan evidencias de una transición de fase topológica en cristales l{\'ı}quidos en dos dimensiones. A altas temperaturas se presenta una fase paramagnética mientras que a bajas temperaturas se encuentra un fase de cuasi-largo-orden (QLRO). Se encuentra que a muy bajas temperaturas el exponente de la función de correlación de la fase QLRO es lineal con la temperatura teniéndose de esta manera un comportamiento t{\'ı}pico de ondas de esp{\'ı}n. Esto último contradice predicciones de que para sistemas con grupo de simetr{\'ı}a global no abeliano las ondas de esp{\'\}n no son relevantes. Adicionalmente se discute que implicaciones tendr{\'ı}a la presencia de impurezas en este tipo de transiciones.

cond-mat.other↗

arXiv2002

Bulk and surface properties in the critical phase of the two-dimensional XY model

Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent $η_σ(T)$ of the order parameter correlation function and its surface analogue $η_\|(T)$ as a function of the temperature in the critical (low-temperature) phase of the model.

cond-mat.stat-mech↗

arXiv2002

Surface properties at the Kosterlitz-Thouless transition

Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with free and mixed fixed-free boundary conditions. Using a Schwarz-Christoffel conformal mapping, we deduce the exponent eta of the order parameter correlation function and its surface equivalent eta_parallel at the Kosterlitz-Thouless transition temperature. The well known value eta(T_{KT}) = 1/4 is easily recovered even with systems of relatively small sizes, since the shape effects are encoded in the conformal mapping. The exponent associated to the surface correlations is similarly obtained eta_1(T_{KT}) ~= 0.54.

cond-mat.dis-nn↗

arXiv2002

Phase transitions in two-dimensional random Potts models

The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.

cond-mat.dis-nn↗

arXiv2001

Susceptibility amplitude ratio in the two-dimensional three-state Potts model

We analyze Monte Carlo simulation and series-expansion data for the susceptibility of the three-state Potts model in the critical region. The amplitudes of the susceptibility on the high- and the low-temperature sides of the critical point as extracted from the Monte Carlo data are in good agreement with those obtained from the series expansions and their (universal) ratio compares quite well with a recent quantum field theory prediction by Delfino and Cardy.

cond-mat.stat-mech↗

arXiv2001

Quenched bond dilution in two-dimensional Potts models

We report a numerical study of the bond-diluted 2-dimensional Potts model using transfer matrix calculations. For different numbers of states per spin, we show that the critical exponents at the random fixed point are the same as in self-dual random-bond cases. In addition, we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin-spin correlation function in the cylinder geometry. We show that the behaviour is fully compatible with the one observed in the random bond case, confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder: dilution, self-dual binary random-bond, self-dual continuous random bond.

cond-mat.dis-nn↗

arXiv2001

Evidence for softening of first-order transition in 3D by quenched disorder

We study by extensive Monte Carlo simulations the effect of random bond dilution on the phase transition of the three-dimensional 4-state Potts model which is known to exhibit a strong first-order transition in the pure case. The phase diagram in the dilution-temperature plane is determined from the peaks of the susceptibility for sufficiently large system sizes. In the strongly disordered regime, numerical evidence for softening to a second-order transition induced by randomness is given. Here a large-scale finite-size scaling analysis, made difficult due to strong crossover effects presumably caused by the percolation fixed point, is performed.

cond-mat.stat-mech↗

arXiv2000

Universality and multifractal behaviour of spin-spin correlation functions in disordered Potts models

We report a transfer matrix study of the random bond $q-$state Potts model in the vicinity of the Ising model $q=2$. We draw attention to a precise determination of magnetic scaling dimensions in order to compare with perturbative results. Universality is checked by the computation of the spin-spin correlation function decay exponent obtained with different types of probability distributions of the coupling strengths. Our numerical data, compared to perturbative results for the second moment of the correlation function, obtained with both replica symmetry and replica symmetry breaking schemes, are conclusively in favour of the replica symmetric calculations. The multifractal behaviour of higher moments as well as that of typical correlation functions are also investigated and a comparison is made with the perturbative expansions. Finally, the shape of the correlation function probability distribution is analyzed.

cond-mat.stat-mech↗

arXiv1999

Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries

We present a numerical study of 2D random-bond Potts ferromagnets. The model is studied both below and above the critical value $Q_c=4$ which discriminates between second and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques which were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes which is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.

cond-mat.stat-mech↗

arXiv1999

Critical Behavior of the Random Potts Chain

We study the critical behavior of the random q-state Potts quantum chain by density matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains ($L \leq 16$) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar (Phys. Rev. Lett.{\bf 76}, 3001 (1996)). We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.

cond-mat.stat-mech↗

arXiv1999

Boundary critical behaviour of two-dimensional random Potts models

We consider random q-state Potts models for $3\le q \le 8$ on the square lattice where the ferromagnetic couplings take two values $J_1>J_2$ with equal probabilities. For any q the model exhibits a continuous phase transition both in the bulk and at the boundary. Using Monte Carlo techniques the surface and the bulk magnetizations are studied close to the critical temperature and the critical exponents $β_1$ and $β$ are determined. In the strip-like geometry the critical magnetization profile is investigated with free-fixed spin boundary condition and the characteristic scaling dimension, $β_1/ν$, is calculated from conformal field theory. The critical exponents and scaling dimensions are found monotonously increasing with q. Anomalous dimensions of the relevant scaling fields are estimated and the multifractal behaviour at criticality is also analyzed.

cond-mat.stat-mech↗

arXiv1998

Tests of Conformal Invariance in Randomness Induced Second-Order Phase Transitions

The conformal covariance of correlation functions is checked in the second-order transition induced by random bonds in the two-dimensional 8-state Potts model. The decay of correlations is obtained {\it via} transfer matrix calculations in a cylinder geometry, and large-scale Monte Carlo simulations give access to the correlations and the profiles inside a square with free or fixed boundary conditions, respectively. In both geometries, conformal transformations constrain the form of the spatial dependence, leading to accurate determinations of the order parameter scaling index in good agreement with previous independent determinations obtained through standard techniques. The energy density exponent is also computed.

cond-mat.stat-mech↗

arXiv1998

Second-Order Phase Transition Induced by Deterministic Fluctuations in Aperiodic Eight-State Potts Models

We investigate the influence of aperiodic modulations of the exchange interactions between nearest-neighbour rows on the phase transition of the two-dimensional eight-state Potts model. The systems are studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm for different aperiodic sequences. The transition point is located through duality relations, and the critical behaviour is investigated using FSS techniques at criticality. While the pure system exhibits a first-order transition, we show that the deterministic fluctuations resulting from the aperiodic coupling distribution are liable to modify drastically the physical properties in the neighbourhood of the transition point. For strong enough fluctuations of the sequence under consideration, a second-order phase transition is induced. The exponents $β/ν$, $γ/ν$ and $(1-α)/ν$ are obtained at the new fixed point and crossover effects are discussed. Surface properties are also studied.

cond-mat.stat-mech↗

arXiv1998

The McCoy-Wu Model in the Mean-field Approximation

We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory. In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents $β\approx 3.6$ and $β_1 \approx 4.1$ in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent $α\approx -3.1$. The samples reduced critical temperature $t_c=T_c^{av}-T_c$ has a power law distribution $P(t_c) \sim t_c^ω$ and we show that the difference between the values of the critical exponents in the pure and in the random system is just $ω\approx 3.1$. Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.

cond-mat.stat-mech↗

arXiv1998

Finite-Size Scaling Study of the Surface and Bulk Critical Behavior in the Random-Bond 8-state Potts Model

The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We find that a belonging of the model to the 2D Ising model universality class can be conclusively ruled out, and the dimensions of the relevant bulk and surface scaling fields are found to take the values $y_h=1.849$, $y_t=0.977$, $y_{h_s}=0.54$, to be compared to their Ising values: 15/8, 1, and 1/2.

cond-mat.stat-mech↗

arXiv1997

Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model

We investigate the critical behavior of the two-dimensional 8-state Potts model with an aperiodic distribution of the exchange interactions between nearest-neighbor rows. The model is studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm. The transition point is located through duality relations, and the critical behavior is investigated using FSS techniques at criticality. For strong enough fluctuations of the aperiodic sequence under consideration, a second order phase transition is found. The exponents $β/ν$ and $γ/ν$ are obtained at the new fixed point.

cond-mat.stat-mech↗