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:1612.08739 · Physical review. D/Physical review. D.

Lee-Yang model from the functional renormalization group

Luca Zambelli, Omar Zanusso

Published Apr 5, 2017• 49 citations
arXivHTMLPDF

Abstract

We investigate the critical properties of the Lee-Yang model in less than six spacetime dimensions using truncations of the functional renormalization group flow. We give estimates for the critical exponents, study the dependence on the regularization scheme, and show the convergence of our results for increasing size of the truncations in four and five dimensions. While with our truncations it is numerically challenging to approach the three-dimensional case, we provide a simple approximation which allows us to qualitatively study the Lee-Yang model in two and three dimensions and use it to argue the existence of further nonunitary multicritical theories including one which is relevant for the universality class of the Blume-Capel model.

DOI
10.1103/physrevd.95.085001

Related papers

New universality class in three dimensions: The critical Blume-Capel model

Alessandro Codello, Mahmoud Safari, Gian Paolo Vacca +1 more

We study the Blume-Capel universality class in $d=\frac{10}{3}\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The renormalization group flow is extracted by looking at poles in fractional dimension of three loop diagrams using $\overline{\mathrm{MS}}$. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $\ensuremath{\epsilon}=\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the operator product expansion coefficients to the leading order. Our findings agree with and complement conformal field theory results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as special cases.

arXiv:1706.06887•Physical review. D/Physical review. D.•Oct 2, 2017•27 citations

Rethinking Dimensional Regularization in Critical Phenomena

P. Beretta, Alessandro Codello

We show that it is possible to use dimensional regularization (DR) beyond the usual $\varepsilon$-expansion in the context of renormalization group (RG) calculations in Critical Phenomena. Based on this fact, we propose a new functional RG scheme - Functional Dimensional Regularization (FDR) - and apply it to a scalar theory in three dimensions. We compute the critical exponents of the Ising universality class directly in $d=3$ under various typical approximations. The method that emerges combines the agility typical of DR with the generality proper of functional RG. Moreover, at a given order of approximation, FDR seems to provide faster convergence and better estimates than other functional RGs.

arXiv:2604.25103•Apr 28, 2026•0 citations

Continuous field theory and the renormalization group

C. Itzykson, Jean-Michel Drouffe

Continuous field models were originally systematically elaborated (and still are) in the context of particle physics. Most relevant to the study of critical phenomena is the derivation of a renormalization group flow, characterized by a set of Callan–Symanzik equations for Green functions, with regular coefficients derived from perturbation theory. This is supplemented by the idea due to Fisher and Wilson of an expansion in powers of the deviation from the strict renormalizability dimension, i.e. four in the case of the ϕ 4 model, but can also be presumed to work directly in the physical dimension three (and even possibly two). We devote this chapter to a general presentation and a survey of some applications. An appendix gives a short introduction to multicritical phenomena. The Lagrangian and dimensional analysis Introduction We want to investigate universal critical properties. A discrete lattice model appears as a regularizing intermediate stage, which allows a precise meaning to be given to the functional integrals of the field theory. Following the scheme suggested by the mean field approximation, one is tempted to start directly from a continuous model, in which the lattice is replaced by a d -dimensional continuous Euclidean space, and also, even for models with a discrete symmetry, the dynamical variables are replaced by continuous fields. From the original formulation, we just retain the idea of a cutoff factor ∧ large with respect to all momenta, which ensures that some possibly divergent expressions remain finite.

Cambridge University Press eBooks•Sep 29, 1989•0 citations

Revisiting the Lee-Yang singularities in the four-dimensional Ising model: a tribute to the memory of Ralph Kenna

Juan Jesus Ruiz-Lorenzo

We have studied numerically the Lee-Yang singularities of the four dimensional Ising model at criticality, which is believed to be in the same universality class as the φ44 scalar field theory. We have focused in the numerical characterization of the logarithmic corrections to the scaling of the zeros of the partition function and its cumulative probability distribution, finding a very good agreement with the predictions of the renormalization group computation on the φ44 scalar field theory. To obtain these results, we have extended a previous study [R. Kenna, C. B. Lang, Nucl. Phys., 1993, B393, 461] in which there were computed numerically the first two zeros for L≤24 lattices, to the computation of the first four zeros for L≤64 lattices.

arXiv:2402.03932•Condensed Matter Physics•Sep 24, 2024•3 citations

Functional Renormalization Group Methods for Multicritical Scalar Field Theories

ISMAELE BARTOLUCCIO

Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.

•Sep 16, 2026•0 citations

Renormalization group for the φ4 theory with long-range interaction and the critical exponent η of the Ising model

Raphael Goll, Peter Kopietz

We calculate the critical exponent η of the D-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long range to short range; the numerical value of η can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result η=0.03651 is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions D and obtain the resulting critical exponent η(D) between two and four dimensions. For dimensions 2≤D≤3 our result for η is consistent with previous calculations.

arXiv:1804.04150•Physical review. E•Aug 28, 2018•6 citations
We investigate the critical properties of the Lee-Yang model in less than six spacetime dimensions using truncations of the functional renormalization group flow. We give estimates for the critical exponents, study the dependence on the regularization scheme, and show the convergence of our results for increasing size of the truncations in four and five dimensions. While with our truncations it is numerically challenging to approach the three-dimensional case, we provide a simple approximation which allows us to qualitatively study the Lee-Yang model in two and three dimensions and use it to argue the existence of further nonunitary multicritical theories including one which is relevant for the universality class of the Blume-Capel model.
We study the Blume-Capel universality class in $d=\frac{10}{3}\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The renormalization group flow is extracted by looking at poles in fractional dimension of three loop diagrams using MS‾\overline{\mathrm{MS}}MS. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $\ensuremath{\epsilon}=\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the operator product expansion coefficients to the leading order. Our findings agree with and complement conformal field theory results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as special cases.
We show that it is possible to use dimensional regularization (DR) beyond the usual ε\varepsilonε-expansion in the context of renormalization group (RG) calculations in Critical Phenomena. Based on this fact, we propose a new functional RG scheme - Functional Dimensional Regularization (FDR) - and apply it to a scalar theory in three dimensions. We compute the critical exponents of the Ising universality class directly in d=3d=3d=3 under various typical approximations. The method that emerges combines the agility typical of DR with the generality proper of functional RG. Moreover, at a given order of approximation, FDR seems to provide faster convergence and better estimates than other functional RGs.
Continuous field models were originally systematically elaborated (and still are) in the context of particle physics. Most relevant to the study of critical phenomena is the derivation of a renormalization group flow, characterized by a set of Callan–Symanzik equations for Green functions, with regular coefficients derived from perturbation theory. This is supplemented by the idea due to Fisher and Wilson of an expansion in powers of the deviation from the strict renormalizability dimension, i.e. four in the case of the ϕ 4 model, but can also be presumed to work directly in the physical dimension three (and even possibly two). We devote this chapter to a general presentation and a survey of some applications. An appendix gives a short introduction to multicritical phenomena. The Lagrangian and dimensional analysis Introduction We want to investigate universal critical properties. A discrete lattice model appears as a regularizing intermediate stage, which allows a precise meaning to be given to the functional integrals of the field theory. Following the scheme suggested by the mean field approximation, one is tempted to start directly from a continuous model, in which the lattice is replaced by a d -dimensional continuous Euclidean space, and also, even for models with a discrete symmetry, the dynamical variables are replaced by continuous fields. From the original formulation, we just retain the idea of a cutoff factor ∧ large with respect to all momenta, which ensures that some possibly divergent expressions remain finite.
We have studied numerically the Lee-Yang singularities of the four dimensional Ising model at criticality, which is believed to be in the same universality class as the φ44 scalar field theory. We have focused in the numerical characterization of the logarithmic corrections to the scaling of the zeros of the partition function and its cumulative probability distribution, finding a very good agreement with the predictions of the renormalization group computation on the φ44 scalar field theory. To obtain these results, we have extended a previous study [R. Kenna, C. B. Lang, Nucl. Phys., 1993, B393, 461] in which there were computed numerically the first two zeros for L≤24 lattices, to the computation of the first four zeros for L≤64 lattices.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
We calculate the critical exponent η of the D-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long range to short range; the numerical value of η can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result η=0.03651 is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions D and obtain the resulting critical exponent η(D) between two and four dimensions. For dimensions 2≤D≤3 our result for η is consistent with previous calculations.