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

arXiv:cond-mat/0405515

Quantum phase transitions of fractons (a fractal scaling theory for the FQHE)

Wellington da Cruz

Published May 21, 2004• 2 citations
arXivHTMLPDF

Abstract

We consider the quantum phase transitions of fractons in correspondence with the quantum phase transitions of the fractional quantum Hall effect-FQHE. We have that the Hall states can be modelled by fractons, known as charge-flux systems which satisfy a fractal distribution function associated with a fractal von Neumann entropy. In our formulation, the universality classes of the fractional quantum Hall transitions, are considered as fractal sets of dual topological quantum numbers filling factors labelled by the Hausdorff dimension $h$ ($1 < h < 2$) of the quantum paths of fractons. In this way we have defined, associated to these universality classes, a scaling exponent as $κ=1/h$, such that when $h$ runs into its interval of definition, we obtain $ 1 \gtrsim κ\gtrsim 0.5 $. The behavior of this scaling exponent, topological in character, is in agreement with some experimental values claimed in the literature. Thus, according to our approach we have a fractal scaling theory for the FQHE which distinguishes diverse universality classes for the fractional quantum Hall transitions.

DOI
10.48550/arxiv.cond-mat/0405515

Related papers

Entanglement measure for the universal classes of fractons

Wellington da Cruz, S. Shelly Sharma

We introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between ocuppation-numbers of fractons in the lowest Landau levels and the rest of the many-body system of particles. This definition came as an entropy of the probability distribution {\it à la} Shannon. Fractons are charge-flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or Hausdorff dimension defined within the interval $1 < h < 2$ and associated with the fractal quantum curves of such objects. They carry rational or irrational values of spin and the spin-statistics connection takes place in this fractal approach to the fractional spin particles. We take into account the fractal von Neumann entropy associated with the fractal distribution function which each universal class of fractons satisfies. We consider the fractional quantum Hall effect-FQHE given that fractons can model Hall states. According to our formulation entanglement between occupaton-numbers in this context increases with the universality classes of the quantum Hall transitions considered as fractal sets of dual topological quantum numbers filling factors. We verify that the Hall states have stronger entanglement between ocuppation-numbers and so we can consider this resource for fracton quantum computing.

arXiv:quant-ph/0310064•Oct 10, 2003•0 citations

Casimir amplitudes in topological quantum phase transitions

M. A. R. Griffith, M. A. Contínentino

Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.

Physical review. E•Jan 9, 2018•19 citations

Fractons and Luttinger liquids

Wellington da Cruz

We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter = ν + 1 which is identified with the controlling parameter e -2φ of the Luttinger model. In this way, a bosonic system characterized by a fractal index i f [ h ] = i f [2] = 1 is considered in a conformal field theory with central charge c [ν = 0] = 1 = i f [2] with a compactified radius R = 1/ 1/2 = 1. Thus, we have a mapping of a bosonic theory to a fermionic one and vice versa, i.e. the duality symmetry = 3- h of the universal class h of fractons defined in the interval 1< h <2 is satisfied.

arXiv:cond-mat/0006503•Journal of Physics Condensed Matter•Oct 18, 2000•3 citations

Classical and Quantum Phase Transitions in Multiscale Media: Universality and Critical Exponents in the Fractional Ising Model

Lewis, Joshua M., Lincoln D. Carr

Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with $q$-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising model for $0 < q \leq 2$ ($q = 2$ is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as $ν$, $δ$, and $η$, but also define the Hausdorff dimension $H_D$ of the system tied geometrically to the anomalous dimension $η$. We discover that for classical systems, $H_D$ is precisely equal to the fractional order $q$. In contrast, for quantum systems, $H_D$ deviates from this direct equivalence, scaling more gradually, driven by additional degrees of freedom introduced by quantum fluctuations. These results reveal how fractional derivatives fundamentally modify the fractal geometry of many-body interactions, directly influencing the universal symmetries of the system and overcoming traditional dimensional restrictions on phase transitions. Specifically, we find that for $q < 1$ in the classical regime and $q < 2$ in the quantum regime, fractional interactions allow phase transitions in one dimension. This work establishes fractional derivatives as a powerful tool for engineering critical behavior, offering new insights into the geometry of multiscale systems and opening avenues for exploring tunable quantum materials on NISQ devices.

arXiv:2501.14134•Jan 23, 2025•0 citations

Quantum criticality from Ising model on fractal lattices

Beni Yoshida, Aleksander Kubica

We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is $\log 3/ \log 2 \approx 1.585$, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum $3$-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from $ε$-expansion that the transition becomes first-order for $D > 1.3$. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for $D\in[1,3]$. We find that only the correlation length exponent $ν$ interpolates the values of integer-dimensional models. This implies that, contrary to a generally held belief, the universality class of quantum phase transition may not be uniquely determined by symmetry and spatial dimension of the system. This work initiates studies on quantum critical phenomena on graphs and networks which may be of significant importance in the context of quantum networks and communication.

arXiv:1404.6311•Apr 25, 2014•2 citations

Fractonic Fractional Quantum Hall Effect

Justin Schirmann, Peru d’Ornellas, Stahl, Charles +1 more

In non-interacting systems, disorder can drive a trivial phase into a topological one. However little is known how to construct a fractional quantum Hall ground-state, a paradigmatic topologically ordered state, that exists both in crystalline and disordered lattices and is qualitatively different to known topological phases. Here, we propose a general method for building such a phase. This is done by coupling quantum wires placed aperiodically in real-space, where the spatial positioning allows us to tune the inter-wire couplings. We call the emergent phase the Fractonic Fractional Quantum Hall Effect as it displays a rich interplay of fractional quantum Hall physics with fractonic constraints, formed by coupling differently-fractionalised wires into a globally gapped phase. The ground state has an exponential degeneracy in system size, a signature of the emergence of fractons. It displays a rich phenomenology of excitations, which can either behave like anyons confined to move in one dimension (lineons), multiples of which can then hop between two wires (s-lineons) or be free to travel across the system (C-anyons), depending on the multiplicity. Both the ground state degeneracy and mutual statistics are directly determined by the real-space positions of the wires, which can be disordered. Our method provides an analytically solvable pathway to non-crystalline fractional quantum Hall effects and fractonic theories in two-dimensions, examples of which were lacking.

arXiv:2504.18337•Apr 25, 2025•0 citations
We consider the quantum phase transitions of fractons in correspondence with the quantum phase transitions of the fractional quantum Hall effect-FQHE. We have that the Hall states can be modelled by fractons, known as charge-flux systems which satisfy a fractal distribution function associated with a fractal von Neumann entropy. In our formulation, the universality classes of the fractional quantum Hall transitions, are considered as fractal sets of dual topological quantum numbers filling factors labelled by the Hausdorff dimension hhh (1<h<21 < h < 21<h<2) of the quantum paths of fractons. In this way we have defined, associated to these universality classes, a scaling exponent as κ=1/hκ=1/hκ=1/h, such that when hhh runs into its interval of definition, we obtain 1≳κ≳0.51 \gtrsim κ\gtrsim 0.51≳κ≳0.5. The behavior of this scaling exponent, topological in character, is in agreement with some experimental values claimed in the literature. Thus, according to our approach we have a fractal scaling theory for the FQHE which distinguishes diverse universality classes for the fractional quantum Hall transitions.
We introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between ocuppation-numbers of fractons in the lowest Landau levels and the rest of the many-body system of particles. This definition came as an entropy of the probability distribution {\it à la} Shannon. Fractons are charge-flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or Hausdorff dimension defined within the interval 1<h<21 < h < 21<h<2 and associated with the fractal quantum curves of such objects. They carry rational or irrational values of spin and the spin-statistics connection takes place in this fractal approach to the fractional spin particles. We take into account the fractal von Neumann entropy associated with the fractal distribution function which each universal class of fractons satisfies. We consider the fractional quantum Hall effect-FQHE given that fractons can model Hall states. According to our formulation entanglement between occupaton-numbers in this context increases with the universality classes of the quantum Hall transitions considered as fractal sets of dual topological quantum numbers filling factors. We verify that the Hall states have stronger entanglement between ocuppation-numbers and so we can consider this resource for fracton quantum computing.
Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.
We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter = ν + 1 which is identified with the controlling parameter e -2φ of the Luttinger model. In this way, a bosonic system characterized by a fractal index i f [ h ] = i f [2] = 1 is considered in a conformal field theory with central charge c [ν = 0] = 1 = i f [2] with a compactified radius R = 1/ 1/2 = 1. Thus, we have a mapping of a bosonic theory to a fermionic one and vice versa, i.e. the duality symmetry = 3- h of the universal class h of fractons defined in the interval 1< h <2 is satisfied.
Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with qqq-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising model for 0<q≤20 < q \leq 20<q≤2 (q=2q = 2q=2 is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as ννν, δδδ, and ηηη, but also define the Hausdorff dimension HDH_DHD​ of the system tied geometrically to the anomalous dimension ηηη. We discover that for classical systems, HDH_DHD​ is precisely equal to the fractional order qqq. In contrast, for quantum systems, HDH_DHD​ deviates from this direct equivalence, scaling more gradually, driven by additional degrees of freedom introduced by quantum fluctuations. These results reveal how fractional derivatives fundamentally modify the fractal geometry of many-body interactions, directly influencing the universal symmetries of the system and overcoming traditional dimensional restrictions on phase transitions. Specifically, we find that for q<1q < 1q<1 in the classical regime and q<2q < 2q<2 in the quantum regime, fractional interactions allow phase transitions in one dimension. This work establishes fractional derivatives as a powerful tool for engineering critical behavior, offering new insights into the geometry of multiscale systems and opening avenues for exploring tunable quantum materials on NISQ devices.
We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is log⁡3/log⁡2≈1.585\log 3/ \log 2 \approx 1.585log3/log2≈1.585, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum 333-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from εεε-expansion that the transition becomes first-order for D>1.3D > 1.3D>1.3. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for D∈[1,3]D\in[1,3]D∈[1,3]. We find that only the correlation length exponent ννν interpolates the values of integer-dimensional models. This implies that, contrary to a generally held belief, the universality class of quantum phase transition may not be uniquely determined by symmetry and spatial dimension of the system. This work initiates studies on quantum critical phenomena on graphs and networks which may be of significant importance in the context of quantum networks and communication.
In non-interacting systems, disorder can drive a trivial phase into a topological one. However little is known how to construct a fractional quantum Hall ground-state, a paradigmatic topologically ordered state, that exists both in crystalline and disordered lattices and is qualitatively different to known topological phases. Here, we propose a general method for building such a phase. This is done by coupling quantum wires placed aperiodically in real-space, where the spatial positioning allows us to tune the inter-wire couplings. We call the emergent phase the Fractonic Fractional Quantum Hall Effect as it displays a rich interplay of fractional quantum Hall physics with fractonic constraints, formed by coupling differently-fractionalised wires into a globally gapped phase. The ground state has an exponential degeneracy in system size, a signature of the emergence of fractons. It displays a rich phenomenology of excitations, which can either behave like anyons confined to move in one dimension (lineons), multiples of which can then hop between two wires (s-lineons) or be free to travel across the system (C-anyons), depending on the multiplicity. Both the ground state degeneracy and mutual statistics are directly determined by the real-space positions of the wires, which can be disordered. Our method provides an analytically solvable pathway to non-crystalline fractional quantum Hall effects and fractonic theories in two-dimensions, examples of which were lacking.