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arXiv:cond-mat/0104033 · Physical review. B, Condensed matter

Connecting polymers to the quantum Hall plateau transition

Joel E. Moore

Published Dec 13, 2001• 3 citations
arXivHTMLPDF

Abstract

A mapping is developed between the quantum Hall plateau transition and two-dimensional self-interacting lattice polymers. This mapping is exact in the classical percolation limit of the plateau transition, and diffusive behavior at the critical energy is shown to be related to the critical exponents of a class of chiral polymers at the \ensuremath{\theta} point. The exact critical exponents of the chiral polymer model on the honeycomb lattice are found, verifying that this model is in the same universality class as a previously solved model of polymers on the Manhattan lattice. The mapping is obtained by averaging analytically over the local random potentials in a previously studied lattice model for the classical plateau transition. This average generates a weight on chiral polymers associated with the classical localization length exponent $\ensuremath{\nu}=\frac{4}{3}.$ We discuss the differences between the classical and quantum transitions in the context of polymer models and use numerical results on higher-moment scaling laws at the quantum transition to constrain possible polymer descriptions. Some properties of the polymer models are verified by transfer matrix and Monte Carlo studies.

DOI
10.1103/physrevb.65.035307

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A mapping is developed between the quantum Hall plateau transition and two-dimensional self-interacting lattice polymers. This mapping is exact in the classical percolation limit of the plateau transition, and diffusive behavior at the critical energy is shown to be related to the critical exponents of a class of chiral polymers at the \ensuremath{\theta} point. The exact critical exponents of the chiral polymer model on the honeycomb lattice are found, verifying that this model is in the same universality class as a previously solved model of polymers on the Manhattan lattice. The mapping is obtained by averaging analytically over the local random potentials in a previously studied lattice model for the classical plateau transition. This average generates a weight on chiral polymers associated with the classical localization length exponent \ensuremathν=43.\ensuremath{\nu}=\frac{4}{3}.\ensuremathν=34​. We discuss the differences between the classical and quantum transitions in the context of polymer models and use numerical results on higher-moment scaling laws at the quantum transition to constrain possible polymer descriptions. Some properties of the polymer models are verified by transfer matrix and Monte Carlo studies.
We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We solve the problem of the spin quantum Hall transition on random networks using a mapping to classical percolation that focuses on the boundary of percolating clusters. Using tools of two-dimensional quantum gravity, we compute critical exponents that characterize this transition and confirm that these are related to the exponents for the regular (square) network through the KPZ relation. Our results demonstrate the relevance of the geometric randomness of the networks and support conclusions of numerical simulations of random networks for the integer quantum Hall transition.
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.
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 derive the exact critical couplings (x∗,ya∗)({x}^{*}{,y}_{a}^{*})(x∗,ya∗​), where ya∗/x∗=1+2=1.533…{y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}ya∗​/x∗=1+2​​=1.533… , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the O(n)O\left(n\right)O(n) loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.