Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Delocalization transitions of low-dimensional manifolds

Stefan Grotehans, Reinhard Lipowsky

Published Jun 1, 1992• 9 citations
Publisher

Abstract

The behavior of low-dimensional manifolds such as interfaces, membranes, or polymers in an external potential is studied in d=${\mathit{d}}_{\mathrm{\ensuremath{\parallel}}}$+${\mathit{d}}_{\mathrm{\ensuremath{\perp}}}$ dimensions. If the potential has several minima, the manifolds can undergo nontrivial transitions from a localized to a delocalized state at a characteristic temperature T=${\mathit{T}}_{\mathrm{*}}$. For interfaces in d=1+1, several universality classes must be distinguished which can be determined exactly by transfer-matrix methods. The same classification also applies to (self-intersecting) polymers in d=1+${\mathit{d}}_{\mathrm{\ensuremath{\perp}}}$. Within a functional renormalization-group approach, the critical behavior is governed by a line of fixed points in close analogy with unbinding transitions. However, in contrast to unbinding, this fixed point line now contains the trivial fixed point at which one can determine the spectrum of eigenperturbations exactly. The number of relevant eigenperturbations increases monotonically with the decay exponent \ensuremath{\tau}=${\mathit{d}}_{\mathrm{\ensuremath{\parallel}}}$/\ensuremath{\zeta} where the roughness exponent \ensuremath{\zeta} characterizes the delocalized manifold.

DOI
10.1103/physreva.45.8644

Related papers

Quantum wetting transitions in two dimensions: An alternative path to non-universal interfacial singularities

Paweł Jakubczyk, M. Napiórkowski, Federico Benitez

We consider two-dimensional (d = 2) systems with short-ranged microscopic interactions, where interface unbinding (wetting) transitions occur in the limit of vanishing temperature T .For T = 0 the transition is characterized by non-universal critical properties analogous to those established for thermal wetting transitions in d = 3, albeit with a redefined capillary parameter ω.Within a functional renormalization-group treatment of an effective interfacial model, we compute the finite-temperature phase diagram, exhibiting a line of interface unbinding transitions, terminating at T = 0 with an interfacial quantum-critical point.We identify distinct scaling regimes, reflecting the interplay between quantum and thermal interfacial fluctuations.A crossover line marking the onset of the quantum-critical regime is described by the d = 3 interfacial correlationlength exponent ν || .This potentially opens another way to investigate the non-universal character of ν || .On the other hand, the emergent interfacial quantum-critical regime shows no signatures of non-universality.

arXiv:1501.05664•Europhysics Letters (EPL)•Apr 1, 2015•2 citations

Connecting polymers to the quantum Hall plateau transition

Joel E. Moore

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.

arXiv:cond-mat/0104033•Physical review. B, Condensed matter•Dec 13, 2001•3 citations

Scaling regimes and functional renormalization for wetting transitions

Reinhard Lipowsky, Michael Ellis Fisher

Interface unbinding transitions, such as those arising in wetting phenomena, are studied in d dimensions with general interactions. Three scaling regimes must be distinguished: a mean-field (MF), a weak-fluctuation (WFL), and a strong-fluctuation (SFL) regime. A simple picture clarifies the origin and nature of the different regimes and correctly describes the MF and WFL critical behavior. In the SFL regime, however, this picture fails, as do more elaborate perturbative methods. To overcome this an approximate functional renormalization group is introduced: it acts as a nonlinear map in the space of interaction potentials, V(l), for two interfaces at a separation l. The formulation is exact to first order in V and embodies the correct scaling behavior at a continuous unbinding transition. In the SFL regime, it reveals two nontrivial fixed point potentials, ${V}_{0}^{\mathrm{*}}$(l) and ${V}_{c}^{\mathrm{*}}$(l), which describe, respectively, the completely delocalized phase and the critical manifold for the unbinding transition. On approaching the upper boundary dimension, ${d}_{u}$=3, these fixed points do not coalesce with the standard Gaussian fixed point but, rather, mutually annihilate leaving a line of novel ``drifting'' fixed points. For d<3, sufficiently long-ranged perturbations cause crossover to the WFL and MF regimes. Thus the functional renormalization-group approach yields a unified description of all scaling regimes.

Physical review. B, Condensed matter•Aug 1, 1987•208 citations

Delocalization transition in an exactly solvable many-body system in Two Dimensions

R. Ganesh, Saugata Ghosh

We study delocalization transition in a many body system in two dimension. We identify the presence of a complex vector potential that gives rise to this transition.

arXiv:cond-mat/0603169•Mar 7, 2006•0 citations

Localization-delocalization transitions in a two-dimensional quantum percolation model: von Neumann entropy studies

Longyan Gong, Peiqing Tong

In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold ${p}_{q}$. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of ${p}_{q}$ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold ${p}_{q}<1$ in the two-dimensional quantum percolation problem.

arXiv:0911.5531•Physical Review B•Nov 20, 2009•19 citations

Frustration and disorder in discrete lattice models

Jesper Lykke Jacobsen

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.

•Sep 11, 1998•0 citations
The behavior of low-dimensional manifolds such as interfaces, membranes, or polymers in an external potential is studied in d=d\ensuremath∥{\mathit{d}}_{\mathrm{\ensuremath{\parallel}}}d\ensuremath∥​+d\ensuremath⊥{\mathit{d}}_{\mathrm{\ensuremath{\perp}}}d\ensuremath⊥​ dimensions. If the potential has several minima, the manifolds can undergo nontrivial transitions from a localized to a delocalized state at a characteristic temperature T=T∗{\mathit{T}}_{\mathrm{*}}T∗​. For interfaces in d=1+1, several universality classes must be distinguished which can be determined exactly by transfer-matrix methods. The same classification also applies to (self-intersecting) polymers in d=1+d\ensuremath⊥{\mathit{d}}_{\mathrm{\ensuremath{\perp}}}d\ensuremath⊥​. Within a functional renormalization-group approach, the critical behavior is governed by a line of fixed points in close analogy with unbinding transitions. However, in contrast to unbinding, this fixed point line now contains the trivial fixed point at which one can determine the spectrum of eigenperturbations exactly. The number of relevant eigenperturbations increases monotonically with the decay exponent \ensuremath{\tau}=d\ensuremath∥{\mathit{d}}_{\mathrm{\ensuremath{\parallel}}}d\ensuremath∥​/\ensuremath{\zeta} where the roughness exponent \ensuremath{\zeta} characterizes the delocalized manifold.
We consider two-dimensional (d = 2) systems with short-ranged microscopic interactions, where interface unbinding (wetting) transitions occur in the limit of vanishing temperature T .For T = 0 the transition is characterized by non-universal critical properties analogous to those established for thermal wetting transitions in d = 3, albeit with a redefined capillary parameter ω.Within a functional renormalization-group treatment of an effective interfacial model, we compute the finite-temperature phase diagram, exhibiting a line of interface unbinding transitions, terminating at T = 0 with an interfacial quantum-critical point.We identify distinct scaling regimes, reflecting the interplay between quantum and thermal interfacial fluctuations.A crossover line marking the onset of the quantum-critical regime is described by the d = 3 interfacial correlationlength exponent ν || .This potentially opens another way to investigate the non-universal character of ν || .On the other hand, the emergent interfacial quantum-critical regime shows no signatures of non-universality.
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.
Interface unbinding transitions, such as those arising in wetting phenomena, are studied in d dimensions with general interactions. Three scaling regimes must be distinguished: a mean-field (MF), a weak-fluctuation (WFL), and a strong-fluctuation (SFL) regime. A simple picture clarifies the origin and nature of the different regimes and correctly describes the MF and WFL critical behavior. In the SFL regime, however, this picture fails, as do more elaborate perturbative methods. To overcome this an approximate functional renormalization group is introduced: it acts as a nonlinear map in the space of interaction potentials, V(l), for two interfaces at a separation l. The formulation is exact to first order in V and embodies the correct scaling behavior at a continuous unbinding transition. In the SFL regime, it reveals two nontrivial fixed point potentials, V0∗{V}_{0}^{\mathrm{*}}V0∗​(l) and Vc∗{V}_{c}^{\mathrm{*}}Vc∗​(l), which describe, respectively, the completely delocalized phase and the critical manifold for the unbinding transition. On approaching the upper boundary dimension, du{d}_{u}du​=3, these fixed points do not coalesce with the standard Gaussian fixed point but, rather, mutually annihilate leaving a line of novel ``drifting'' fixed points. For d<3, sufficiently long-ranged perturbations cause crossover to the WFL and MF regimes. Thus the functional renormalization-group approach yields a unified description of all scaling regimes.
In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold pq{p}_{q}pq​. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of pq{p}_{q}pq​ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold pq<1{p}_{q}<1pq​<1 in the two-dimensional quantum percolation problem.
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.