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/9612164 · Europhysics Letters (EPL)

Critical limit and anisotropy in the two-point correlation function of three-dimensional O(N) models

Massimo Campostrini B, A Pelissetto, Paolo Rossi, Ettore Vicari

Published Jun 10, 1997• 19 citations
arXivHTMLPDF

Abstract

In three-dimensional O( N ) models, we investigate the low-momentum behavior of the two-point Green's function G ( x ) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. In non-rotational invariant physical systems with O( N )-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent ρ, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N = ∞ one finds ρ = 2. We show that, for all values of N ⩾ 0, ρ ≃ 2. Non-Gaussian corrections to the universal low-momentum behavior of G ( x ) are evaluated, and found to be very small.

DOI
10.1209/epl/i1997-00286-8

Related papers

Two-point correlation function of three-dimensional O(N) models: The critical limit and anisotropy

Massimo Campostrini B, Andrea Pelissetto C, Paolo Rossi +1 more

In three-dimensional $O(N)$ models, we investigate the low-momentum behavior of the two-point Green's function $G(x)$ in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotationally invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice $N$-vector model, and $1/N$ expansion, field-theoretical methods within the ${\ensuremath{\varphi}}^{4}$ continuum formulation. Non-Gaussian corrections to the universal low-momentum behavior of $G(x)$ are evaluated, and found to be very small. In nonrotationally invariant physical systems with $O(N)$-invariant interactions, the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent $\ensuremath{\rho}$, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At $N=\ensuremath{\infty}$ one finds $\ensuremath{\rho}=2$. We show that, for all values of $N>~0$, $\ensuremath{\rho}\ensuremath{\simeq}2$.

arXiv:cond-mat/9705086•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 1, 1998•74 citations

Criticality of the O(2) model with cubic anisotropies from nonperturbative renormalization

Andrzej Chlebicki, Paweł Jakubczyk

We study the O(2) model with Z_{4}-symmetric perturbations within the framework of the nonperturbative renormalization group (RG) for spatial dimensionality d=2 and 3. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In d=3 the system is controlled by the XY, Ising, and low-T fixed points in the presence of a dangerously irrelevant anisotropy coupling λ. In d=2 the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed points, giving rise to nonuniversal critical behavior, and an isolated Ising fixed point. In addition to inducing crossover behavior in universal properties, the presence of the Ising fixed point yields a generic, abrupt change of critical temperature at a specific value of the anisotropy field.

arXiv:1909.10600•Physical review. E•Nov 7, 2019•6 citations

Critical behavior of a 3D Ising-like system in the higher non-Gaussian approximation: Inclusion of the critical exponent of the correlation function

I. R. Yukhnovskii, Mykhailo P. Kozlovskii, Ihor V. Pylyuk

A microscopic description of the critical behavior of systems belonging to the universality class of the three-dimensional (3D) Ising model is developed within the collective variables (CV) approach. The higher non-Gaussian approximation (the sextic distribution for the modes of spin-moment density oscillations or the ρ6model) is used. A specific feature of the partition function calculation for an Ising-like system is the inclusion of the correction for the potential averaging. This correction leads to the modified recurrence relations (RR) for the ρ6model and a nonzero critical exponent of the correlation function η. The RR between the coefficients of the effective sextic distributions are written and analyzed. A technique for determining the small critical exponent η is elaborated in the higher non-Gaussian approximation. It is shown that the renormalized critical exponent of the correlation length has a tendency to a reduction in the case when the exponent η is taken into account.

International Journal of Modern Physics B•Jun 30, 2014•1 citation

Critical behaviour of a 3D Ising-like system in the ρ^6 model approximation: Role of the correction for the potential averaging

Ihor V. Pylyuk, Ulyak

The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.

arXiv:1212.6139•Condensed Matter Physics•Dec 1, 2012•1 citation

Breakdown of conformal invariance at strongly random critical points

M. B. Hastings, Shivaji L. Sondhi

We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system that has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the renormalization group flow.

arXiv:cond-mat/0011124•Physical review. B, Condensed matter•Aug 14, 2001•9 citations

Strong-coupling analysis of two-dimensional O(N) σ models withN⩽2 on square, triangular, and honeycomb lattices

Massimo Campostrini B, Andrea Pelissetto C, Paolo Rossi +1 more

The critical behavior of two-dimensional (2D) O(N) \ensuremath{\sigma} models with N\ensuremath{\le}2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N2 the critical behavior is of power-law type, and the exponents \ensuremath{\gamma} and \ensuremath{\nu} extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent \ensuremath{\sigma}=1/2, and with universality. The value \ensuremath{\sigma}=1/2 is confirmed within an uncertainty of few percent. The prediction \ensuremath{\eta}=1/4 is also roughly verified. For various values of N\ensuremath{\le}2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N\ensuremath{\le}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. \textcopyright{} 1996 The American Physical Society.

arXiv:hep-lat/9603002•Physical review. B, Condensed matter•Sep 1, 1996•53 citations
In three-dimensional O( N ) models, we investigate the low-momentum behavior of the two-point Green's function G ( x ) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. In non-rotational invariant physical systems with O( N )-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent ρ, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N = ∞ one finds ρ = 2. We show that, for all values of N ⩾ 0, ρ ≃ 2. Non-Gaussian corrections to the universal low-momentum behavior of G ( x ) are evaluated, and found to be very small.
In three-dimensional O(N)O(N)O(N) models, we investigate the low-momentum behavior of the two-point Green's function G(x)G(x)G(x) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotationally invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice NNN-vector model, and 1/N1/N1/N expansion, field-theoretical methods within the ${\ensuremath{\varphi}}^{4}$ continuum formulation. Non-Gaussian corrections to the universal low-momentum behavior of G(x)G(x)G(x) are evaluated, and found to be very small. In nonrotationally invariant physical systems with O(N)O(N)O(N)-invariant interactions, the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent $\ensuremath{\rho}$, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At $N=\ensuremath{\infty}$ one finds $\ensuremath{\rho}=2$. We show that, for all values of N> 0N>~0N> 0, $\ensuremath{\rho}\ensuremath{\simeq}2$.
We study the O(2) model with Z_{4}-symmetric perturbations within the framework of the nonperturbative renormalization group (RG) for spatial dimensionality d=2 and 3. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In d=3 the system is controlled by the XY, Ising, and low-T fixed points in the presence of a dangerously irrelevant anisotropy coupling λ. In d=2 the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed points, giving rise to nonuniversal critical behavior, and an isolated Ising fixed point. In addition to inducing crossover behavior in universal properties, the presence of the Ising fixed point yields a generic, abrupt change of critical temperature at a specific value of the anisotropy field.
A microscopic description of the critical behavior of systems belonging to the universality class of the three-dimensional (3D) Ising model is developed within the collective variables (CV) approach. The higher non-Gaussian approximation (the sextic distribution for the modes of spin-moment density oscillations or the ρ6model) is used. A specific feature of the partition function calculation for an Ising-like system is the inclusion of the correction for the potential averaging. This correction leads to the modified recurrence relations (RR) for the ρ6model and a nonzero critical exponent of the correlation function η. The RR between the coefficients of the effective sextic distributions are written and analyzed. A technique for determining the small critical exponent η is elaborated in the higher non-Gaussian approximation. It is shown that the renormalized critical exponent of the correlation length has a tendency to a reduction in the case when the exponent η is taken into account.
The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.
We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system that has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the renormalization group flow.
The critical behavior of two-dimensional (2D) O(N) \ensuremath{\sigma} models with N\ensuremath{\le}2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N2 the critical behavior is of power-law type, and the exponents \ensuremath{\gamma} and \ensuremath{\nu} extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent \ensuremath{\sigma}=1/2, and with universality. The value \ensuremath{\sigma}=1/2 is confirmed within an uncertainty of few percent. The prediction \ensuremath{\eta}=1/4 is also roughly verified. For various values of N\ensuremath{\le}2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N\ensuremath{\le}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. \textcopyright{} 1996 The American Physical Society.