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3D Unoriented loop models and the $\mathrm{RP}^{n-1}$ sigma models

Pablo Serna

Published Jul 28, 2021• 0 citations
OpenAlex

Abstract

We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by $\mathrm{RP}^{n-1}$ sigma field theories. We use Monte Carlo simulations, with systems sizes up to $1800\times1800\times1800$, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit $n\rightarrow1$ of $\mathrm{RP}^{n-1}$ sigma field theory. Classic critical exponents include $\nu=0.918(5)$ and $\eta=-0.091(9)$. We also provide estimates of the scaling dimension of the 4-legs watermelon correlators, particularly for $n=1$ we obtain the value $x_4=1.292(8)$.

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3D Unoriented loop models and the $\mathrm{RP}^{n-1}$ sigma model

Pablo Serna

We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by $\mathrm{RP}^{n-1}$ sigma field theories. We use Monte Carlo simulations, with systems sizes up to $1800\times1800\times1800$, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit $n\rightarrow1$ of $\mathrm{RP}^{n-1}$ sigma field theory. Estimates of critical exponents include $ν=0.918(5)$ and $η=-0.091(9)$. We also study the scaling dimension of the 4-leg watermelon correlators, particularly for $n=1$ we obtain the value $x_4=1.292(8)$.

arXiv:2107.13366•Jul 28, 2021•0 citations

Loop models with crossings

Adam Nahum, Pablo Serna, A. M. Somoza +1 more

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.

arXiv:1303.2342•Physical Review B•May 28, 2013•54 citations

High-precision Monte Carlo study of the 3DXY-universality class

M Hasenbusch, T Török

We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.

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Finite size scaling study of lattice models in the three-dimensional Ising universality class

Martin Hasenbusch B

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter $D$, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.

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Monte Carlo study of an improved clock model in three dimensions

Martin Hasenbusch B

We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with ${Z}_{8}$ symmetry. At the transition, with increasing length scale, $O(2)$ symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to $L=512$. The field variable requires less memory and the updates are faster than for a model with $O(2)$ symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional $XY$-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.

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Universality class of the two-dimensional polymer collapse transition

Adam Nahum

The nature of the θ point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur for an exactly solvable model. We use a representation of the problem via the CP^{N-1}σ model in the limit N→1 to determine the stability of this critical point. First we prove that the Duplantier-Saleur (DS) critical exponents are robust, so long as the polymer does not cross itself: They can arise in a generic lattice model and do not require fine-tuning. This resolves a longstanding theoretical question. We also address an apparent paradox: Two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned (and not strictly in the DS universality class). Next we allow the polymer to cross itself, as appropriate, e.g., to the quasi-two-dimensional case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply. The exponents in the case with crossings will be those of the generic tricritical O(n) model at n=0 and different from the case without crossings. We also discuss interesting features of the operator content of the CP^{N-1} model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of N (or, equivalently, any value of the loop fugacity). Also we argue that for any value of N the CP^{N-1} model has a marginal odd-parity operator that is related to the winding angle.

arXiv:1510.09223•Physical review. E•May 6, 2016•12 citations
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).
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).
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 $\ensuremath{\sigma}$ model on real projective space (${\mathbb{RP}}^{n\ensuremath{-}1}$). 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 $\ensuremath{\Theta}$ 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.
We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.
We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.
We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with Z8{Z}_{8}Z8​ symmetry. At the transition, with increasing length scale, O(2)O(2)O(2) symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to L=512L=512L=512. The field variable requires less memory and the updates are faster than for a model with O(2)O(2)O(2) symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional XYXYXY-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.
The nature of the θ point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur for an exactly solvable model. We use a representation of the problem via the CP^{N-1}σ model in the limit N→1 to determine the stability of this critical point. First we prove that the Duplantier-Saleur (DS) critical exponents are robust, so long as the polymer does not cross itself: They can arise in a generic lattice model and do not require fine-tuning. This resolves a longstanding theoretical question. We also address an apparent paradox: Two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned (and not strictly in the DS universality class). Next we allow the polymer to cross itself, as appropriate, e.g., to the quasi-two-dimensional case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply. The exponents in the case with crossings will be those of the generic tricritical O(n) model at n=0 and different from the case without crossings. We also discuss interesting features of the operator content of the CP^{N-1} model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of N (or, equivalently, any value of the loop fugacity). Also we argue that for any value of N the CP^{N-1} model has a marginal odd-parity operator that is related to the winding angle.