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arXiv:1303.2342 · Physical Review B

Loop models with crossings

Adam Nahum, Pablo Serna, A. M. Somoza, M. Ortuño

Published May 28, 2013• 54 citations
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Abstract

The universal behavior of two-dimensional loop models can change dramatically when loops are allowed to cross. We study models with crossings both analytically and with extensive Monte Carlo simulations. Our main focus (the ``completely packed loop model with crossings'') is a simple generalization of well-known models that shows an interesting phase diagram with continuous phase transitions of a new kind. These separate the unusual ``Goldstone'' phase observed previously from phases with short loops. Using mappings to ${\mathbb{Z}}_{2}$ lattice gauge theory, we show that the continuum description of the model is a replica limit of the $\ensuremath{\sigma}$ model on real projective space (${\mathbb{RP}}^{n\ensuremath{-}1}$). This field theory sustains ${\mathbb{Z}}_{2}$ point defects, which proliferate at the transition. In addition to studying the new critical points, we characterize the universal properties of the Goldstone phase in detail, comparing renormalization group (RG) calculations with numerical data on systems of linear size up to $L={10}^{6}$ at loop fugacity $n=1$. (Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their $\ensuremath{\Theta}$ point; we use the RG treatment of a perturbed $\ensuremath{\sigma}$ model to resolve some perplexing features in the previous literature on trails. Finally, one of the phase transitions considered here is a close analog of those in disordered electronic systems---specifically, Anderson metal-insulator transitions---and provides a simpler context in which to study the properties of these poorly understood (central-charge-zero) critical points.

DOI
10.1103/physrevb.87.184204

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Frustration and disorder in discrete lattice models

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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.

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The universal behavior of two-dimensional loop models can change dramatically when loops are allowed to cross. We study models with crossings both analytically and with extensive Monte Carlo simulations. Our main focus (the completely packed loop model with crossings'') is a simple generalization of well-known models that shows an interesting phase diagram with continuous phase transitions of a new kind. These separate the unusual Goldstone'' phase observed previously from phases with short loops. Using mappings to Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory, we show that the continuum description of the model is a replica limit of the \ensuremathσ\ensuremath{\sigma}\ensuremathσ model on real projective space (RPn\ensuremath−1{\mathbb{RP}}^{n\ensuremath{-}1}RPn\ensuremath−1). This field theory sustains Z2{\mathbb{Z}}_{2}Z2​ point defects, which proliferate at the transition. In addition to studying the new critical points, we characterize the universal properties of the Goldstone phase in detail, comparing renormalization group (RG) calculations with numerical data on systems of linear size up to L=106L={10}^{6}L=106 at loop fugacity n=1n=1n=1. (Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ point; we use the RG treatment of a perturbed \ensuremathσ\ensuremath{\sigma}\ensuremathσ model to resolve some perplexing features in the previous literature on trails. Finally, one of the phase transitions considered here is a close analog of those in disordered electronic systems---specifically, Anderson metal-insulator transitions---and provides a simpler context in which to study the properties of these poorly understood (central-charge-zero) critical points.
We consider the statistical mechanics of a class of models involving close-packed loops with fugacity nnn on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretizations of $C{P}^{n\ensuremath{-}1}$ $\ensuremath{\sigma}$ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the $\ensuremath{\sigma}$ model, and we discuss the relationship between loop properties and $\ensuremath{\sigma}$ model correlators. On large scales, loops are Brownian in an ordered phase and have a nontrivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n=1,2,3n=1,2,3n=1,2,3 and first order transitions for $n\ensuremath{\ge}4$. We also give a renormalization-group treatment of the $C{P}^{n\ensuremath{-}1}$ model that shows how a continuous transition can survive for values of nnn larger than (but close to) 2, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU(n)\mathrm{SU}(n)SU(n) quantum magnets in (2+1) dimensions, Anderson localization in symmetry class CCC, and the statistics of random curves in three dimensions.
3D Unoriented loop models and the RPn−1\mathrm{RP}^{n-1}RPn−1 sigma model
We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theories. We use Monte Carlo simulations, with systems sizes up to 1800×1800×18001800\times1800\times18001800×1800×1800, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit n→1n\rightarrow1n→1 of RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theory. Estimates of critical exponents include ν=0.918(5)ν=0.918(5)ν=0.918(5) and η=−0.091(9)η=-0.091(9)η=−0.091(9). We also study the scaling dimension of the 4-leg watermelon correlators, particularly for n=1n=1n=1 we obtain the value x4=1.292(8)x_4=1.292(8)x4​=1.292(8).
3D Unoriented loop models and the RPn−1\mathrm{RP}^{n-1}RPn−1 sigma models
We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theories. We use Monte Carlo simulations, with systems sizes up to 1800×1800×18001800\times1800\times18001800×1800×1800, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit n→1n\rightarrow1n→1 of RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theory. Classic critical exponents include ν=0.918(5)\nu=0.918(5)ν=0.918(5) and η=−0.091(9)\eta=-0.091(9)η=−0.091(9). We also provide estimates of the scaling dimension of the 4-legs watermelon correlators, particularly for n=1n=1n=1 we obtain the value x4=1.292(8)x_4=1.292(8)x4​=1.292(8).
We study the phase diagram and critical behaviors of three-dimensional lattice Z2{\mathbb{Z}}_{2}Z2​-gauge NNN-vector models, in which an NNN-component real field is minimally coupled with Z2{\mathbb{Z}}_{2}Z2​-gauge link variables. These models are invariant under global O(N\mathrm{O}(NO(N) and local Z2{\mathbb{Z}}_{2}Z2​ transformations. They present three phases characterized by the spontaneous breaking of the global O(N\mathrm{O}(NO(N) symmetry and by the different topological properties of the Z2{\mathbb{Z}}_{2}Z2​-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the N=2N=2N=2 model. In this case, continuous transitions can be observed along both transition lines where the NNN-component spins order, in the regimes of small and large inverse gauge coupling KKK. Even though these continuous transitions belong to the same XYXYXY universality class, their critical modes turn out to be different. When the gauge variables are disordered (small KKK), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large KKK), the order-parameter field is the gauge-dependent NNN-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.
We use Majorana operators to study entanglement dynamics under random free fermion unitary evolution and projective measurements in one dimension. For certain choices of unitary evolution, namely, those which swap neighboring Majorana operators, and measurements of neighboring Majorana bilinears, one can map the evolution to the statistical model of completely packed loops with crossings (CPLC) and study the corresponding phase diagram. We generalize this model using the language of fermionic Gaussian states to a general free fermion unitary evolution acting on neighboring Majorana operators and numerically compute its phase diagram. We find that both the Goldstone and area-law phases persist in this new phase diagram, but with a shifted phase boundary. One important qualitative aspect of the new phase boundary is that even for the case of commuting measurements, the Goldstone phase persists up to a finite nonzero measurement rate. This is in contrast with the CPLC, in which noncommuting measurements are necessary for realizing the Goldstone phase. We also numerically compute the correlation length critical exponent at the transition, which we find to be near to that of the CPLC, and give a tentative symmetry-based explanation for some differences in the phase transition line between the CPLC and generalized models.
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.