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Monte-Carlo simulation study of problems of quantum field theory and critical phenomena.

Jae-Kwon. Kim

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

In chapter one, we explain briefly the continuum limit, scaling, and high temperature expansion of critical phenomena, Monte Carlo algorithms and fitting. In chapter two, different continuum limits of the Ising model in dimensions (D) 2, 3 and 4 are investigated numerically. The data indicate that triviality occurs for D = 4 and fails for D < 4 in each limit. In chapter three, a relation between the critical exponents of the leading and confluent scaling terms is derived using the finite size scaling argument. We also determine the new scaling variable of the 4D Ising model based on a new Monte Carlo simulation data. In chapter four, a Monte Carlo study of two dimensional diluted Ising systems is reported. It is shown that regular dilution does not affect critical exponents, while a random one does, with critical exponents varying continuously with impurity concentration. The importance of fluctuations in producing such effects is emphasized. In chapter five, a different point of view regarding the critical exponent of the specific heat of the 3D Ising model is presented. Based on the analysis of high temperature expansion, finite size scaling and Monte Carlo data in the symmetric phase of the 3D Ising model, it is shown that logarithmic scaling behavior of specific heat is more consistent than power scaling behavior.

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H.G. Ballesteros, L. A. Fernández, Víctor Martín-Mayor +4 more

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Critical phenomena in the Random Ising Model.

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The modern techniques of field theory applied to critical phenomena, are briefly discussed, with particular emphasis on the use of the e-expansion to extract the asymptotic behaviour in the critical region. The dimensional regularisation and minimal subtraction method of renormalisation, Introduced by 't Hooft and Veltman, is outlined. These methods, along with a Monte Carlo simulation on a highly parallel computer, are utilised in extracting information pertaining to the behaviour near criticality of the random, site-diluted Ising model. The universal ratio of the susceptibility amplitude above and below the critical temperature is determined to one higher order than previously in an ɛ-expansion, within the framework of the replica formalism. The calculation is performed using a Taylor expansion, in the number of replicas, of the transverse propagator appearing In the Hamiltonian, about the longitudinal. This higher order correction goes in the right direction for agreement with experiment. An extension of the formalism to embrace the presence of two sets of replica spins is used to derive the o(ɛ¹/²) term in an expansion of the additional static - correlation function, C⁽ˢ⁾(q), arising from the non-interchangeability of the thermal and configurational averages In the random model. The appearance of C(sl(q) in the structure factor Implies that results of neutron scattering experiments to measure the susceptibility amplitude ratio must be re-Interpreted. Finally, analysis of the properties of block spin variables In a Monte Carlo simulation Is used to determine the phase diagram and critical exponents In two-dimensions. Possible Interpretations of the observed evolution of the exponent ratio, ß/υ, under Increasing dilution are discussed.

•Jan 1, 1986•0 citations
In chapter one, we explain briefly the continuum limit, scaling, and high temperature expansion of critical phenomena, Monte Carlo algorithms and fitting. In chapter two, different continuum limits of the Ising model in dimensions (D) 2, 3 and 4 are investigated numerically. The data indicate that triviality occurs for D = 4 and fails for D < 4 in each limit. In chapter three, a relation between the critical exponents of the leading and confluent scaling terms is derived using the finite size scaling argument. We also determine the new scaling variable of the 4D Ising model based on a new Monte Carlo simulation data. In chapter four, a Monte Carlo study of two dimensional diluted Ising systems is reported. It is shown that regular dilution does not affect critical exponents, while a random one does, with critical exponents varying continuously with impurity concentration. The importance of fluctuations in producing such effects is emphasized. In chapter five, a different point of view regarding the critical exponent of the specific heat of the 3D Ising model is presented. Based on the analysis of high temperature expansion, finite size scaling and Monte Carlo data in the symmetric phase of the 3D Ising model, it is shown that logarithmic scaling behavior of specific heat is more consistent than power scaling behavior.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
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\alphaα . 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, starting from the pure model limit down to the neighbourhood of 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 to check the stability of the disorder fixed point. 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 indeed governed by the same universality class as the site-diluted model.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
The modern techniques of field theory applied to critical phenomena, are briefly discussed, with particular emphasis on the use of the e-expansion to extract the asymptotic behaviour in the critical region. The dimensional regularisation and minimal subtraction method of renormalisation, Introduced by 't Hooft and Veltman, is outlined. These methods, along with a Monte Carlo simulation on a highly parallel computer, are utilised in extracting information pertaining to the behaviour near criticality of the random, site-diluted Ising model. The universal ratio of the susceptibility amplitude above and below the critical temperature is determined to one higher order than previously in an ɛ-expansion, within the framework of the replica formalism. The calculation is performed using a Taylor expansion, in the number of replicas, of the transverse propagator appearing In the Hamiltonian, about the longitudinal. This higher order correction goes in the right direction for agreement with experiment. An extension of the formalism to embrace the presence of two sets of replica spins is used to derive the o(ɛ¹/²) term in an expansion of the additional static - correlation function, C⁽ˢ⁾(q), arising from the non-interchangeability of the thermal and configurational averages In the random model. The appearance of C(sl(q) in the structure factor Implies that results of neutron scattering experiments to measure the susceptibility amplitude ratio must be re-Interpreted. Finally, analysis of the properties of block spin variables In a Monte Carlo simulation Is used to determine the phase diagram and critical exponents In two-dimensions. Possible Interpretations of the observed evolution of the exponent ratio, ß/υ, under Increasing dilution are discussed.