1
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Delamotte B, De Polsi G, Tissier M, Wschebor N. Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group. Phys Rev E 2024; 109:064152. [PMID: 39020923 DOI: 10.1103/physreve.109.064152] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2024] [Accepted: 05/21/2024] [Indexed: 07/20/2024]
Abstract
It is expected that conformal symmetry is an emergent property of many systems at their critical point. This imposes strong constraints on the critical behavior of a given system. Taking them into account in theoretical approaches can lead to a better understanding of the critical physics or improve approximation schemes. However, within the framework of the nonperturbative or functional renormalization group and, in particular, of one of its most used approximation schemes, the derivative expansion (DE), nontrivial constraints apply only from third order [usually denoted O(∂^{4})], at least in the usual formulation of the DE that includes correlation functions involving only the order parameter. In this work we implement conformal constraints on a generalized DE including composite operators and show that new constraints already appear at second order of the DE [or O(∂^{2})]. We show how these constraints can be used to fix nonphysical regulator parameters.
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2
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Sánchez-Villalobos CA, Delamotte B, Wschebor N. q-state Potts model from the nonperturbative renormalization group. Phys Rev E 2023; 108:064120. [PMID: 38243545 DOI: 10.1103/physreve.108.064120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2023] [Accepted: 11/05/2023] [Indexed: 01/21/2024]
Abstract
We study the q-state Potts model for q and the space dimension d arbitrary real numbers using the derivative expansion of the nonperturbative renormalization group at its leading order, the local potential approximation (LPA and LPA^{'}). We determine the curve q_{c}(d) separating the first [q>q_{c}(d)] and second [q
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Affiliation(s)
- Carlos A Sánchez-Villalobos
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, 75005 Paris, France
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J. H. y Reissig 565, 11300 Montevideo, Uruguay
| | - Bertrand Delamotte
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, 75005 Paris, France
| | - Nicolás Wschebor
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J. H. y Reissig 565, 11300 Montevideo, Uruguay
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3
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Chlebicki A, Jakubczyk P. Low-temperature behavior of the O(N) models below two dimensions. Phys Rev E 2023; 107:014121. [PMID: 36797959 DOI: 10.1103/physreve.107.014121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Accepted: 12/22/2022] [Indexed: 06/18/2023]
Abstract
We investigate the critical behavior and the nature of the low-temperature phase of the O(N) models treating the number of field components N and the dimension d as continuous variables with a focus on the d≤2 and N≤2 quadrant of the (d,N) plane. We precisely chart a region of the (d,N) plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension η. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the O(N) models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in d<2. In particular, we discuss how this framework leads to destabilization of the long-range order in favor of the quasi-long-range order in systems with d<2 and N<2. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi-long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents η(d,N) and ν^{-1}(d,N) and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in d<2.
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Affiliation(s)
- Andrzej Chlebicki
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Pawel Jakubczyk
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
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4
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Chlebicki A, Sánchez-Villalobos CA, Jakubczyk P, Wschebor N. Z_{4}-symmetric perturbations to the XY model from functional renormalization. Phys Rev E 2022; 106:064135. [PMID: 36671161 DOI: 10.1103/physreve.106.064135] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 12/07/2022] [Indexed: 06/17/2023]
Abstract
We employ the second order of the derivative expansion of the nonperturbative renormalization group to study cubic (Z_{4}-symmetric) perturbations to the classical XY model in dimensionality d∈[2,4]. In d=3 we provide accurate estimates of the eigenvalue y_{4} corresponding to the leading irrelevant perturbation and follow the evolution of the physical picture upon reducing spatial dimensionality from d=3 towards d=2, where we approximately recover the onset of the Kosterlitz-Thouless physics. We analyze the interplay between the leading irrelevant eigenvalues related to O(2)-symmetric and Z_{4}-symmetric perturbations and their approximate collapse for d→2. We compare and discuss different implementations of the derivative expansion in cases involving one and two invariants of the corresponding symmetry group.
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Affiliation(s)
- Andrzej Chlebicki
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Carlos A Sánchez-Villalobos
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J.H.y Reissig 565, 11300 Montevideo, Uruguay
| | - Pawel Jakubczyk
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Nicolás Wschebor
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J.H.y Reissig 565, 11300 Montevideo, Uruguay
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5
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Balog I, Rançon A, Delamotte B. Critical Probability Distributions of the Order Parameter from the Functional Renormalization Group. PHYSICAL REVIEW LETTERS 2022; 129:210602. [PMID: 36461981 DOI: 10.1103/physrevlett.129.210602] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Accepted: 11/02/2022] [Indexed: 06/17/2023]
Abstract
We show that the functional renormalization group (FRG) allows for the calculation of the probability distribution function of the sum of strongly correlated random variables. On the example of the three-dimensional Ising model at criticality and using the simplest implementation of the FRG, we compute the probability distribution functions of the order parameter or, equivalently, its logarithm, called the rate functions in large deviation theory. We compute the entire family of universal scaling functions, obtained in the limit where the system size L and the correlation length of the infinite system ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} held fixed. It compares very accurately with numerical simulations.
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Affiliation(s)
- I Balog
- Institute of Physics, Bijenička cesta 46, HR-10001 Zagreb, Croatia
| | - A Rançon
- Univ. Lille, CNRS, UMR 8523-PhLAM-Laboratoire de Physique des Lasers, Atomes et Molécules, F-59000 Lille, France
| | - B Delamotte
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensé, F-75005 Paris, France
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6
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Yabunaka S, Fleming C, Delamotte B. Incompleteness of the large-N analysis of the O(N) models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite N. Phys Rev E 2022; 106:054105. [PMID: 36559345 DOI: 10.1103/physreve.106.054105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Accepted: 09/21/2022] [Indexed: 11/06/2022]
Abstract
We summarize the usual implementations of the large-N limit of O(N) models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit N→∞. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as N increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of N. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all N>0 and show that they play an important role for the tricritical physics of O(N) models. Finally, we show that some of these fixed points are bivalued when they are considered as functions of d and N thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at N=∞ and d=3 is shown to play a crucial role for the internal consistency of all our results.
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Affiliation(s)
- S Yabunaka
- Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan
| | - C Fleming
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, F-75005, Paris, France
| | - B Delamotte
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, F-75005, Paris, France
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7
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Wilkins A, Rigopoulos G, Masoero E. Coarse graining in time with the functional renormalization group: Relaxation in Brownian motion. Phys Rev E 2022; 106:054109. [PMID: 36559426 DOI: 10.1103/physreve.106.054109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Accepted: 10/13/2022] [Indexed: 06/17/2023]
Abstract
We apply the functional renormalization group (fRG) to study relaxation in a stochastic process governed by an overdamped Langevin equation with one degree of freedom, exploiting the connection with supersymmetric quantum mechanics in imaginary time. After reviewing the functional integral formulation of the system and its underlying symmetries, including the resulting Ward-Takahashi identities for arbitrary initial conditions, we compute the effective action Γ from the fRG, approximated in terms of the leading and subleading terms in the gradient expansion: the local potential approximation and wave-function renormalization, respectively. This is achieved by coarse graining the thermal fluctuations in time resulting in, e.g., an effective potential incorporating fluctuations at all timescales. We then use the resulting effective equations of motion to describe the decay of the covariance and the relaxation of the average position and variance toward their equilibrium values at different temperatures. We use as examples a simple polynomial potential, an unequal Lennard-Jones type potential, and a more complex potential with multiple trapping wells and barriers. We find that these are all handled well, with the accuracy of the approximations improving as the relaxation's spectral representation shifts to lower eigenvalues, in line with expectations about the validity of the gradient expansion. The spectral representation's range also correlates with temperature, leading to the conclusion that the gradient expansion works better for higher temperatures than lower ones. This paper demonstrates the ability of the fRG to expedite the computation of statistical objects in otherwise long-timescale simulations, acting as a first step to more complicated systems.
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Affiliation(s)
- Ashley Wilkins
- School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
| | - Gerasimos Rigopoulos
- School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
| | - Enrico Masoero
- Cardiff School of Engineering, Cardiff University Queens Buildings, The Parade, Cardiff CF24 3AA, United Kingdom
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8
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De Polsi G, Wschebor N. Regulator dependence in the functional renormalization group: A quantitative explanation. Phys Rev E 2022; 106:024111. [PMID: 36109989 DOI: 10.1103/physreve.106.024111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2022] [Accepted: 07/22/2022] [Indexed: 06/15/2023]
Abstract
The search for controlled approximations to study strongly coupled systems remains a very general open problem. Wilson's renormalization group has shown to be an ideal framework to implement approximations going beyond perturbation theory. In particular, the most employed approximation scheme in this context, the derivative expansion, was recently shown to converge and yield accurate and very precise results. However, this convergence strongly depends on the shape of the employed regulator. In this paper we clarify the reason for this dependence and justify, simultaneously, the most commonly employed procedure to fix this dependence, the principle of minimal sensitivity.
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Affiliation(s)
- Gonzalo De Polsi
- Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
| | - Nicolás Wschebor
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J.H.y Reissig 565, 11000 Montevideo, Uruguay
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9
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De Polsi G, Hernández-Chifflet G, Wschebor N. Precision calculation of universal amplitude ratios in O(N) universality classes: Derivative expansion results at order O(∂^{4}). Phys Rev E 2021; 104:064101. [PMID: 35030839 DOI: 10.1103/physreve.104.064101] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2021] [Accepted: 11/11/2021] [Indexed: 06/14/2023]
Abstract
In the last few years the derivative expansion of the nonperturbative renormalization group has proven to be a very efficient tool for the precise computation of critical quantities. In particular, recent progress in the understanding of its convergence properties allowed for an estimate of the error bars as well as the precise computation of many critical quantities. In this work we extend previous studies to the computation of several universal amplitude ratios for the critical regime of O(N) models using the derivative expansion of the nonperturbative renormalization group at order O(∂^{4}) for three-dimensional systems.
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Affiliation(s)
- Gonzalo De Polsi
- Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J. H. y Reissig 565, 11300 Montevideo, Uruguay
| | - Guzmán Hernández-Chifflet
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J. H. y Reissig 565, 11300 Montevideo, Uruguay
| | - Nicolás Wschebor
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J. H. y Reissig 565, 11300 Montevideo, Uruguay
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10
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Baldazzi A, Percacci R, Zambelli L. Limit of vanishing regulator in the functional renormalization group. Int J Clin Exp Med 2021. [DOI: 10.1103/physrevd.104.076026] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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11
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Péli Z. Derivative expansion for computing critical exponents of O(N) symmetric models at next-to-next-to-leading order. Phys Rev E 2021; 103:032135. [PMID: 33862715 DOI: 10.1103/physreve.103.032135] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2020] [Accepted: 03/04/2021] [Indexed: 11/07/2022]
Abstract
We apply the derivative expansion of the effective action in the exact renormalization group equation up to fourth order to the Z_{2} and O(N) symmetric scalar models in d=3 Euclidean dimensions. We compute the critical exponents ν, η, and ω using polynomial expansion in the field. We obtain our predictions for the exponents employing two regulators widely used in exact renormalization group computations. We apply Wynn's epsilon algorithm to improve the predictions for the critical exponents, extrapolating beyond the next-to-next-to-leading order prediction of the derivative expansion.
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Affiliation(s)
- Zoltán Péli
- MTA-DE Particle Physics Research Group, H-4010 Debrecen, P.O. Box 105, Hungary
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12
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Shalaby AM. λ
-point anomaly in view of the seven-loop hypergeometric resummation for the critical exponent
ν
of the
O(2)
ϕ4
model. Int J Clin Exp Med 2020. [DOI: 10.1103/physrevd.102.105017] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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13
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Connelly A, Johnson G, Rennecke F, Skokov VV. Universal Location of the Yang-Lee Edge Singularity in O(N) Theories. PHYSICAL REVIEW LETTERS 2020; 125:191602. [PMID: 33216565 DOI: 10.1103/physrevlett.125.191602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2020] [Revised: 09/15/2020] [Accepted: 09/28/2020] [Indexed: 06/11/2023]
Abstract
We determine a previously unknown universal quantity, the location of the Yang-Lee edge singularity for the O(N) theories in a wide range of N and various dimensions. At large N, we reproduce the N→∞ analytical result on the location of the singularity and, additionally, we obtain the mean-field result for the location in d=4 dimensions. In order to capture the nonperturbative physics for arbitrary N, d and complex-valued external fields, we use the functional renormalization group approach.
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Affiliation(s)
- Andrew Connelly
- Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
| | - Gregory Johnson
- Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
| | - Fabian Rennecke
- Department of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA
| | - Vladimir V Skokov
- Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
- Riken-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
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14
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Balog I, De Polsi G, Tissier M, Wschebor N. Conformal invariance in the nonperturbative renormalization group: A rationale for choosing the regulator. Phys Rev E 2020; 101:062146. [PMID: 32688494 DOI: 10.1103/physreve.101.062146] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2020] [Accepted: 06/12/2020] [Indexed: 11/07/2022]
Abstract
Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.
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Affiliation(s)
- Ivan Balog
- Institute of Physics, Bijenička cesta 46, HR-10001 Zagreb, Croatia
| | - Gonzalo De Polsi
- Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
| | - Matthieu Tissier
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France
| | - Nicolás Wschebor
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J.H. y Reissig 565, 11000 Montevideo, Uruguay
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15
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Coquand O, Mouhanna D, Teber S. Flat phase of polymerized membranes at two-loop order. Phys Rev E 2020; 101:062104. [PMID: 32688533 DOI: 10.1103/physreve.101.062104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2020] [Accepted: 04/27/2020] [Indexed: 06/11/2023]
Abstract
We investigate two complementary field-theoretical models describing the flat phase of polymerized-phantom-membranes by means of a two-loop, weak-coupling, perturbative approach performed near the upper critical dimension D_{uc}=4, extending the one-loop computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)PRLTAO0031-900710.1103/PhysRevLett.60.2634]. We derive the renormalization group equations within the modified minimal substraction scheme, then analyze the corrections coming from two-loop with a particular attention paid to the anomalous dimension and the asymptotic infrared properties of the renormalization group flow. We finally compare our results to those provided by nonperturbative techniques used to investigate these two models.
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Affiliation(s)
- O Coquand
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt, Linder Höhe, 51147 Köln, Germany
| | - D Mouhanna
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France
| | - S Teber
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, 75005 Paris, France
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