From local to critical fluctuations in lattice models: a nonperturbative renormalization-group approach

q-state Potts model from the nonperturbative renormalization group

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 Collapse

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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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Multiparameter universality and intrinsic diversity of critical phenomena in weakly anisotropic systems

Recently a unified hypothesis of multiparameter universality for the critical behavior of bulk and confined anisotropic systems has been formulated [V. Dohm, Phys. Rev. E 97, 062128 (2018)2470-004510.1103/PhysRevE.97.062128]. We prove the validity of this hypothesis in d≥2 dimensions on the basis of the principle of two-scale-factor universality for isotropic systems at vanishing external field. We introduce an angular-dependent correlation vector and a generalized shear transformation that transforms weakly anisotropic systems to isotropic systems. As examples we consider the O(n)-symmetric φ^{4} model, Gaussian model, and n-vector model. By means of the inverse of the shear transformation we determine the general structure of the bulk order-parameter correlation function, of the singular bulk part of the critical free energy, and of critical bulk amplitude relations of anisotropic systems at and away from T_{c}. It is shown that weakly anisotropic systems exhibit a high degree of intrinsic diversity due to d(d+1)/2-1 independent parameters that cannot be determined by thermodynamic measurements. Exact results are derived for the d=2 Ising universality class and for the spherical and Gaussian universality classes in d≥2 dimensions. For the d=3 Ising universality class we identify the universal scaling function of the isotropic bulk correlation function from the nonuniversal result of the functional renormalization group. A proof is presented for the validity of multiparameter universality of the exact critical free energy and critical Casimir amplitude in a finite rectangular geometry of weakly anisotropic systems with periodic boundary conditions in the Ising universality class. This confirms the validity of recent predictions of self-similar structures of finite-size effects in the (d=2,n=1) universality class at T=T_{c} derived from conformal field theory [V. Dohm and S. Wessel, Phys. Rev. Lett. 126, 060601 (2021)PRLTAO0031-900710.1103/PhysRevLett.126.060601]. This also substantiates the previous notion of an effective shear transformation for anisotropic two-dimensional Ising models. Our theory paves the way for a quantitative theory of nonuniversal critical Casimir forces in anisotropic superconductors for which experiments have been proposed by G. A. Williams [Phys. Rev. Lett. 92, 197003 (2004)PRLTAO0031-900710.1103/PhysRevLett.92.197003].

Weakly-Interacting Bose–Bose Mixtures from the Functional Renormalisation Group

We provide a detailed presentation of the functional renormalisation group (FRG) approach for weakly-interacting Bose–Bose mixtures, including a complete discussion on the RG equations. To test this approach, we examine thermodynamic properties of balanced three-dimensional Bose–Bose gases at zero and finite temperatures and find a good agreement with related works. We also study ground-state energies of repulsive Bose polarons by examining mixtures in the limit of infinite population imbalance. Finally, we discuss future applications of the FRG to novel problems in Bose–Bose mixtures and related systems.

Precision calculation of universal amplitude ratios in O(N) universality classes: Derivative expansion results at order O(∂^{4})

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.

Functional-renormalization-group approach to classical liquids with short-range repulsion: A scheme without repulsive reference system

The renormalization-group approaches for classical liquids in previous works required a repulsive reference such as a hard-core one when applied to systems with short-range repulsion. The need for the reference is circumvented here by using a functional-renormalization-group approach for integrating the hierarchical flow of correlation functions along a path of variable interatomic coupling. We introduce the cavity distribution functions to avoid the appearance of divergent terms and choose a path to reduce the error caused by the decomposition of higher order correlation functions. We demonstrate using exactly solvable one-dimensional models that the resulting scheme yields accurate thermodynamic properties and interatomic distribution at various densities when compared to integral-equation methods such as the hypernetted chain and the Percus-Yevick equation, even in the case where our hierarchical equations are truncated with the Kirkwood superposition approximation, which is valid for low-density cases.

Dimensional reduction breakdown and correction to scaling in the random-field Ising model

We provide a theoretical analysis by means of the nonperturbative functional renormalization group (NP-FRG) of the corrections to scaling in the critical behavior of the random-field Ising model (RFIM) near the dimension d_{DR}≈5.1 that separates a region where the renormalized theory at the fixed point is supersymmetric and critical scaling satisfies the d→d-2 dimensional reduction property (d>d_{DR}) from a region where both supersymmetry and dimensional reduction break down at criticality (d<d_{DR}). We show that the NP-FRG results are in very good agreement with recent large-scale lattice simulations of the RFIM in d=5 and we detail the consequences for the leading correction-to-scaling exponent of the peculiar boundary-layer mechanism by which the dimensional-reduction fixed point disappears and the dimensional-reduction-broken fixed point emerges in d_{DR}.

Conformal invariance in the nonperturbative renormalization group: A rationale for choosing the regulator

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.

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

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.

Berezinskii-Kosterlitz-Thouless Paired Phase in Coupled XY Models

We study the effect of a linear tunneling coupling between two-dimensional systems, each separately exhibiting the topological Berezinskii-Kosterlitz-Thouless (BKT) transition. In the uncoupled limit, there are two phases: one where the one-body correlation functions are algebraically decaying and the other with exponential decay. When the linear coupling is turned on, a third BKT-paired phase emerges, in which one-body correlations are exponentially decaying, while two-body correlation functions exhibit power-law decay. We perform numerical simulations in the paradigmatic case of two coupled XY models at finite temperature, finding evidences that for any finite value of the interlayer coupling, the BKT-paired phase is present. We provide a picture of the phase diagram using a renormalization group approach.

Anti-Newtonian Expansions and the Functional Renormalization Group

Anti-Newtonian expansions are introduced for scalar quantum field theories and classical gravity. They expand around a limiting theory that evolves only in time while the spatial points are dynamically decoupled. Higher orders of the expansion re-introduce spatial interactions and produce overlapping lightcones from the limiting isolated world line evolution. In scalar quantum field theories, the limiting system consists of copies of a self-interacting quantum mechanical system. In a spatially discretized setting, a nonlinear “graph transform” arises that produces an in principle exact solution of the Functional Renormalization Group for the Legendre effective action. The quantum mechanical input data can be prepared from its 1 + 0 dimensional counterpart. In Einstein gravity, the anti-Newtonian limit has no dynamical spatial gradients, yet remains fully diffeomorphism invariant and propagates the original number of degrees of freedom. A canonical transformation (trivialization map) is constructed, in powers of a fractional inverse of Newton’s constant, that maps the ADM action into its anti-Newtonian limit. We outline the prospects of an associated trivializing flow in the quantum theory.

Dual lattice functional renormalization group for the Berezinskii-Kosterlitz-Thouless transition: Irrelevance of amplitude and out-of-plane fluctuations

We develop a functional renormalization group (FRG) approach for the two-dimensional XY model by combining the lattice FRG proposed by Machado and Dupuis [Phys. Rev. E 82, 041128 (2010)PLEEE81539-375510.1103/PhysRevE.82.041128] with a duality transformation that explicitly introduces vortices via an integer-valued field. We show that the hierarchy of FRG flow equations for the infinite set of relevant and marginal couplings of the model can be reduced to the well-known Kosterlitz-Thouless renormalization group equations for the renormalized temperature and the vortex fugacity. Within our approach it is straightforward to include weak amplitude as well as out-of-plane fluctuations of the spins, which lead to additional interactions between the vortices that do not spoil the Berezinskii-Kosterlitz-Thouless transition. This demonstrates that previous failures to obtain a line of true fixed points within the FRG are a mathematical artifact of insufficient truncation schemes.

Resummed mean-field inference for strongly coupled data

We present a resummed mean-field approximation for inferring the parameters of an Ising or a Potts model from empirical, noisy, one- and two-point correlation functions. Based on a resummation of a class of diagrams of the small correlation expansion of the log-likelihood, the method outperforms standard mean-field inference methods, even when they are regularized. The inference is stable with respect to sampling noise, contrarily to previous works based either on the small correlation expansion, on the Bethe free energy, or on the mean-field and Gaussian models. Because it is mostly analytic, its complexity is still very low, requiring an iterative algorithm to solve for N auxiliary variables, that resorts only to matrix inversions and multiplications. We test our algorithm on the Sherrington-Kirkpatrick model submitted to a random external field and large random couplings, and demonstrate that even without regularization, the inference is stable across the whole phase diagram. In addition, the calculation leads to a consistent estimation of the entropy of the data and allows us to sample form the inferred distribution to obtain artificial data that are consistent with the empirical distribution.

Ordered phase of the O(N) model within the nonperturbative renormalization group

We analyze nonperturbative renormalization group flow equations for the ordered phase of Z_{2} and O(N) invariant scalar models. This is done within the well-known derivative expansion scheme. For its leading order [local potential approximation (LPA)], we show that not every regulator yields a smooth flow with a convex free energy and discuss for which regulators the flow becomes singular. Then we generalize the known exact solutions of smooth flows in the "internal" region of the potential and exploit these solutions to implement an improved numerical algorithm, which is much more stable than previous ones for N>1. After that, we study the flow equations at second order of the derivative expansion and analyze how and when the LPA results change. We also discuss the evolution of the field renormalization factors.

Thermodynamics of the two-dimensional XY model from functional renormalization

We solve the nonperturbative renormalization-group flow equations for the two-dimensional XY model at the truncation level of the (complete) second-order derivative expansion. We compute the thermodynamic properties in the high-temperature phase and compare the nonuniversal features specific to the XY model with results from Monte Carlo simulations. In particular, we study the position and magnitude of the specific-heat peak as a function of temperature. The obtained results compare well with Monte Carlo simulations. We additionally gauge the accuracy of simplified nonperturbative renormalization-group treatments relying on ϕ^{4}-type truncations. Our computation indicates that such an approximation is insufficient in the high-T phase and a correct analysis of the specific-heat profile requires account of an infinite number of interaction vertices.

Scale invariance implies conformal invariance for the three-dimensional Ising model

Using the Wilson renormalization group, we show that if no integrated vector operator of scaling dimension -1 exists, then scale invariance implies conformal invariance. By using the Lebowitz inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising universality class. This shows, in particular, that scale invariance implies conformal invariance for the three-dimensional Ising model.

Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition

We reexamine the two-dimensional linear O(2) model (φ4 theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension η(T(KT))≃0.24 and the stiffness jump ρ(s)(T(KT)(-))≃0.64 at the transition temperature T(KT), in very good agreement with the exact results η(T(KT))=1/4 and ρ(s)(T(KT)(-))=2/π, as well as an essential singularity of the correlation length in the high-temperature phase as T→T(KT).

Thermodynamics in the vicinity of a relativistic quantum critical point in 2+1 dimensions

We study the thermodynamics of the relativistic quantum O(N) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form P(T)=P(0)+N(T(3)/c(2))F(N)(Δ/T), where c is the velocity of the excitations at the QCP and |Δ| a characteristic zero-temperature energy scale. Using both a large-N approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function F(N). For small values of N (N</~10) we find that F(N)(x) is nonmonotonic in the quantum critical regime (|x|</~1) with a maximum near x=0. The large-N approach-if properly interpreted-is a good approximation both in the renormalized classical (x</~-1) and quantum disordered (x>/~1) regimes, but fails to describe the nonmonotonic behavior of F(N) in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio T(BKT)/ρ(s)(0) is very close to π/2, implying that the stiffness ρ(s)(T(BKT)(-)) at the transition is only slightly reduced with respect to the zero-temperature stiffness ρ(s)(0). Finally, we briefly discuss the experimental determination of the universal function F(2) from the pressure of a Bose gas in an optical lattice near the superfluid-Mott-insulator transition.

Long-range and many-body effects in coagulation processes

We study the problem of diffusing particles which coalesce upon contact. With the aid of a nonperturbative renormalization group, we first analyze the dynamics emerging below the critical dimension two, where strong fluctuations imply anomalously slow decay. Above two dimensions, the long-time, low-density behavior is known to conform with the law of mass action. For this case, we establish an exact mapping between the physics at the microscopic scale (lattice structure, particle shape and size) and the macroscopic decay rate in the law of mass action. In addition, we identify a term violating this classical law. It originates in long-range and many-particle fluctuations and is a simple, universal function of the macroscopic decay rate.

Nonperturbative renormalization group preserving full-momentum dependence: implementation and quantitative evaluation

We present the implementation of the Blaizot-Méndez-Wschebor approximation scheme of the nonperturbative renormalization group we present in detail, which allows for the computation of the full-momentum dependence of correlation functions. We discuss its significance and its relation with other schemes, in particular, the derivative expansion. Quantitative results are presented for the test ground of scalar O(N) theories. Besides critical exponents, which are zero-momentum quantities, we compute the two-point function at criticality in the whole momentum range in three dimensions and, in the high-temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.

Renormalization group flow equations with full momentum dependence

After a short elementary introduction to the exact renormalization group for the effective action, I discuss a particular truncation of the hierarchy of flow equations that allows for the determination of the full momentum of the n-point functions. Applications are then briefly presented, to critical O(N) models, to Bose-Einstein condensation and to finite-temperature field theory.