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arXiv:cond-mat/9705038 · Physical Review Letters

Critical Behavior of Random-Bond Potts Models

John Cardy, Jesper Lykke Jacobsen

Published Nov 24, 1997• 169 citations
arXivHTMLPDF

Abstract

The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the $q$-state Potts model. For large $q$ a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for $d>2$ such systems exhibit a tricritical point with an exponent $\ensuremath{\nu}$ related to those of the random-field model by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\nu}}_{\mathrm{RF}}/(2\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{RF}}\ensuremath{-}{\ensuremath{\beta}}_{\mathrm{RF}})$. In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio $\ensuremath{\beta}/\ensuremath{\nu}$ which varies continuously with $q$, and a weakly varying exponent $\ensuremath{\nu}\ensuremath{\approx}1$. We find strong evidence for the multiscaling of the correlation functions.

DOI
10.1103/physrevlett.79.4063

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•Jan 1, 2015•0 citations
The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the qqq-state Potts model. For large qqq a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for d>2d>2d>2 such systems exhibit a tricritical point with an exponent \ensuremathν\ensuremath{\nu}\ensuremathν related to those of the random-field model by \ensuremathν=\ensuremathνRF/(2\ensuremath−\ensuremathαRF\ensuremath−\ensuremathβRF)\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\nu}}_{\mathrm{RF}}/(2\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{RF}}\ensuremath{-}{\ensuremath{\beta}}_{\mathrm{RF}})\ensuremathν=\ensuremathνRF​/(2\ensuremath−\ensuremathαRF​\ensuremath−\ensuremathβRF​). In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio \ensuremathβ/\ensuremathν\ensuremath{\beta}/\ensuremath{\nu}\ensuremathβ/\ensuremathν which varies continuously with qqq, and a weakly varying exponent \ensuremathν\ensuremath≈1\ensuremath{\nu}\ensuremath{\approx}1\ensuremathν\ensuremath≈1. We find strong evidence for the multiscaling of the correlation functions.
PART I: DISORDER The effect of quenched impurities on systems which undergo first-order phase transitions is studied within the framework of the q-state Potts model. For large q a mapping to the random field Ising model is introduced which provides a simple physical explanation of the absence of any latent heat in two dimensions, and suggests that in higher dimensions such systems should exhibit a tricritical point with a correlation length exponent related to the exponents of the random field model by \nu = \nu_RF/(2-\alpha_RF-\beta_RF). A phase diagram unifying pure, percolative and non-trivial random behaviour is proposed. In two dimensions we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a magnetic exponent \beta/\nu which varies continuously with q, and a weakly varying correlation length exponent \nu ~ 1. For q>4 the first-order transitions of the pure model are softened due to the impurities, and the resulting universality class is different from that of the pure Ising model. We find strong evidence for multiscaling of the correlation functions, as expected for such random systems. PART II: FRUSTRATION Exact results for conformational statistics of compact polymers are derived from the two-flavour fully packed loop model on the square lattice. This loop model exhibits a two-dimensional manifold of critical fixed points each one characterised by an infinite set of geometrical scaling dimensions. We calculate these dimensions exactly by mapping the loop model to an interface model whose scaling limit is described by a Liouville field theory. The formulae for the central charge and the first few scaling dimensions are compared to numerical transfer matrix results and excellent agreement is found. Compact polymers are identified with a particular point in the phase diagram of the loop model, and the non-mean field value of the conformational exponent \gamma = 117/112 is calculated for the first time. We also present the very precise numerical estimate \kappa = 1.472801(10) for the connective constant. Interacting compact polymers are described by a line of fixed points along which \gamma and \kappa vary continuously. Our identification of compact polymers with a critical model has bearings on the cooperativity of protein folding thermodynamics. Namely, in the large chain limit, the absence of an energy gap separating the denatured states from the native (compact) ones implies that homopolymer collapse is a one-state process. We comment on the prospects of modeling specific sequence (amino acid) information through the imposition of quenched disorder.
We study the effect of varying strength $\ensuremath{\delta}$ of bond randomness on the phase transition of the three-dimensional Potts model for large qqq. The cooperative behavior of the system is determined by large correlated domains in which the spins point in the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder $\ensuremath{\delta}>{\ensuremath{\delta}}_{t}$ this percolating cluster coexists with a percolating cluster of noncorrelated spins. Such a coexistence is only possible in more than two dimensions. We argue and check numerically that ${\ensuremath{\delta}}_{t}$ is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.10(2)$ and ${\ensuremath{\nu}}_{t}=0.67(4)$. We claim these exponents are qqq independent for sufficiently large qqq. In the second-order transition regime the critical exponents ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.60(2)$ and ${\ensuremath{\nu}}_{t}=0.73(1)$ are independent of the strength of disorder.
We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the q-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as q-2 grows from small to large values. This theory exhibits a q-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.
Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.
Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents beta\\betabeta and nu\\nunu vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to 128times128128 \\times 128128times128 spins, each of which is represented by three colors taking values pm1\\pm 1pm1, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents, nu\\nunu and beta\\betabeta, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound 2/Dlenu2/D \\le \\nu2/Dlenu, where DDD is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in (1+1)(1+1)(1+1) dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.