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:1507.02909 · Physical Review B

Critical phenomena in hyperbolic space

Karim Mnasri, Bhilahari Jeevanesan, Jörg Schmalian

Published Oct 30, 2015• 8 citations
arXivHTMLPDF

Abstract

How does the presence of a finite length scale in a negatively curved spatial geometry, namely the spatial curvature, modify statistical physics properties of many-particle systems such as critical exponents or the texture of ordered states? Here the authors show that phenomena such as unusual magnetization textures and a strong curvature fixed point can emerge, and that this is a promising theoretical approach for investigating uniform frustration and nontrivial critical behavior.

DOI
10.1103/physrevb.92.134423

Related papers

Geometry of bounded critical phenomena

Giacomo Gori, Andrea Trombettoni

Abstract The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d . This is achieved by altering the flat metric with a space dependent scale factor γ ( x ), x belonging to a bounded domain Ω. γ ( x ) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q -curvature. The equation for γ ( x ) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ ( x ) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q -hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.

arXiv:1904.08919•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2020•4 citations

Critical behavior in spherical and hyperbolic spaces

Dario Benedetti

We study the effects of curved background geometries on the critical behavior of scalar field theory. In particular we concentrate on two maximally symmetric spaces: d -dimensional spheres and hyperboloids. In the first part of the paper, by applying the Ginzburg criterion, we find that for a large correlation length the Gaussian approximation is valid on the hyperboloid for any dimension d ⩾ 2, while it is not trustable on the sphere for any dimension. This is understood in terms of various notions of the effective dimension, such as the spectral and Hausdorff dimension. In the second part of the paper, we apply functional renormalization group methods to develop a different perspective on such phenomena and to deduce them from a renormalization group analysis. By making use of the local potential approximation, we discuss the consequences of having a fixed scale in the renormalization group equations. In particular, we show that in the case of spheres there is no true phase transition, as symmetry restoration always occurs at large scales. In the case of hyperboloids, the phase transition is still present, but as the only true fixed point is the Gaussian one, mean field exponents are valid also in dimensions lower than four.

arXiv:1403.6712•Journal of Statistical Mechanics Theory and Experiment•Jan 6, 2015•62 citations

Effects of curvature in two - dimensional condensed matter systems

Silva, Ygor Pará

The influence of curved background on the properties of Condensed Matter systems is a significant question for a broad class of physical applications, catching the curiosity because of their interesting features in a rich interface between many research areas. In this thesis, we introduce an analytically solvable model of Dirac fermions with imaginary mass on the sphere. We show the existence of an in nite sequence of exceptional points (EP), which depend on the radius (curvature) of the sphere. We employ quench dynamics to characterize curvaturedependent Non-Hermitian phase transitions. We demonstrate that the existence of singular points of the Loschmidt echo and the fidelity are an unambiguous signature of geometric EPs that distinguish between different phases of the model. Also, we use numerical techniques to solve the Bose-Hubbard hamiltonian that simulates the hyperbolic geometry. In this system, we attempt to verify whether we can observe a bound-states comprising a pair of atoms in the presence of repulsive interactions in a hyperbolic lattice. This system is a playground to simulate a space with negative curvature, that is difficult to simulate in a euclidean space without distortions.

•Aug 23, 2022•0 citations

Classical spin liquids from frustrated Ising models in hyperbolic space

Fabian Köhler, Johanna Erdmenger B, Roderich Moessner +1 more

Antiferromagnetic Ising models on frustrated lattices can realize classical spin liquids, with highly degenerate ground states and, possibly, fractionalized excitations and emergent gauge fields. Motivated by the recent interest in many-body systems in negatively curved space, we study hyperbolic frustrated Ising models. Specifically, we consider nearest-neighbor Ising models on tesselations with odd-length loops in two-dimensional hyperbolic space. For finite systems with open boundaries we determine the ground-state degeneracy exactly, and we perform extensive finite-temperature Monte Carlo simulations to obtain thermodynamic data as well as correlation functions. We show that the shape of the boundary, constituting an extensive part of the system, can be used to control low-energy states: Depending on the boundary, we find ordered or disordered ground states. Our results demonstrate how geometric frustration acts in curved space to produce classical spin liquids.

arXiv:2506.02113•Physical review. E•Mar 2, 2026•0 citations

Geometric effects on critical behaviours of the Ising model

Hiroyuki Shima, Yasunori Sakaniwa

We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.

•May 5, 2006•29 citations

XY frustrated systems: Continuous exponents in discontinuous phase transitions

Matthieu Tissier, Bertrand Delamotte, D. Mouhanna

$\mathrm{XY}$ frustrated magnets exhibit an unusual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first-order phase transitions. We show that all perturbative approaches that have been used to investigate $\mathrm{XY}$ frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.

arXiv:cond-mat/0107183•Physical review. B, Condensed matter•Apr 24, 2003•55 citations
How does the presence of a finite length scale in a negatively curved spatial geometry, namely the spatial curvature, modify statistical physics properties of many-particle systems such as critical exponents or the texture of ordered states? Here the authors show that phenomena such as unusual magnetization textures and a strong curvature fixed point can emerge, and that this is a promising theoretical approach for investigating uniform frustration and nontrivial critical behavior.
Abstract The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d . This is achieved by altering the flat metric with a space dependent scale factor γ ( x ), x belonging to a bounded domain Ω. γ ( x ) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q -curvature. The equation for γ ( x ) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ ( x ) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q -hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.
We study the effects of curved background geometries on the critical behavior of scalar field theory. In particular we concentrate on two maximally symmetric spaces: d -dimensional spheres and hyperboloids. In the first part of the paper, by applying the Ginzburg criterion, we find that for a large correlation length the Gaussian approximation is valid on the hyperboloid for any dimension d ⩾ 2, while it is not trustable on the sphere for any dimension. This is understood in terms of various notions of the effective dimension, such as the spectral and Hausdorff dimension. In the second part of the paper, we apply functional renormalization group methods to develop a different perspective on such phenomena and to deduce them from a renormalization group analysis. By making use of the local potential approximation, we discuss the consequences of having a fixed scale in the renormalization group equations. In particular, we show that in the case of spheres there is no true phase transition, as symmetry restoration always occurs at large scales. In the case of hyperboloids, the phase transition is still present, but as the only true fixed point is the Gaussian one, mean field exponents are valid also in dimensions lower than four.
The influence of curved background on the properties of Condensed Matter systems is a significant question for a broad class of physical applications, catching the curiosity because of their interesting features in a rich interface between many research areas. In this thesis, we introduce an analytically solvable model of Dirac fermions with imaginary mass on the sphere. We show the existence of an in nite sequence of exceptional points (EP), which depend on the radius (curvature) of the sphere. We employ quench dynamics to characterize curvaturedependent Non-Hermitian phase transitions. We demonstrate that the existence of singular points of the Loschmidt echo and the fidelity are an unambiguous signature of geometric EPs that distinguish between different phases of the model. Also, we use numerical techniques to solve the Bose-Hubbard hamiltonian that simulates the hyperbolic geometry. In this system, we attempt to verify whether we can observe a bound-states comprising a pair of atoms in the presence of repulsive interactions in a hyperbolic lattice. This system is a playground to simulate a space with negative curvature, that is difficult to simulate in a euclidean space without distortions.
Antiferromagnetic Ising models on frustrated lattices can realize classical spin liquids, with highly degenerate ground states and, possibly, fractionalized excitations and emergent gauge fields. Motivated by the recent interest in many-body systems in negatively curved space, we study hyperbolic frustrated Ising models. Specifically, we consider nearest-neighbor Ising models on tesselations with odd-length loops in two-dimensional hyperbolic space. For finite systems with open boundaries we determine the ground-state degeneracy exactly, and we perform extensive finite-temperature Monte Carlo simulations to obtain thermodynamic data as well as correlation functions. We show that the shape of the boundary, constituting an extensive part of the system, can be used to control low-energy states: Depending on the boundary, we find ordered or disordered ground states. Our results demonstrate how geometric frustration acts in curved space to produce classical spin liquids.
We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.
XY\mathrm{XY}XY frustrated magnets exhibit an unusual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first-order phase transitions. We show that all perturbative approaches that have been used to investigate XY\mathrm{XY}XY frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.