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arXiv:2311.04607 · Journal of High Energy Physics

Finite-size versus finite-temperature effects in the critical long-range O(N) model

Dario Benedetti, Razvan Gurău, Sabine Harribey, Davide Lettera

Published Feb 13, 2024• 9 citations

Abstract

A bstract In this paper we consider classical and quantum versions of the critical long-range O ( N ) model, for which we study finite-size and finite-temperature effects, respectively, at large N . First, we consider the classical (isotropic) model, which is conformally invariant at criticality, and we introduce one compact spatial direction. We show that the finite size dynamically induces an effective mass and we compute the one-point functions for bilinear primary operators with arbitrary spin and twist. Second, we study the quantum model, mapped to a Euclidean anisotropic field theory, local in Euclidean time and long-range in space, which we dub fractional Lifshitz field theory . We show that this model admits a fixed point at zero temperature, where it displays anisotropic Lifshitz scaling, and show that at finite temperature a thermal mass is induced. We then compute the one-point functions for an infinite family of bilinear scaling operators. In both the classical and quantum model, we find that, as previously noted for the short-range O ( N ) model in [1], the large- N two-point function contains information about the one-point functions, not only of the bilinear operators, but also of operators that appear in the operator product expansion of two fundamental fields only at subleading order in 1/ N , namely powers of the Hubbard-Stratonovich intermediate field.

DOI: 10.1007/jhep02(2024)078

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A bstract In this paper we consider classical and quantum versions of the critical long-range O ( N ) model, for which we study finite-size and finite-temperature effects, respectively, at large N . First, we consider the classical (isotropic) model, which is conformally invariant at criticality, and we introduce one compact spatial direction. We show that the finite size dynamically induces an effective mass and we compute the one-point functions for bilinear primary operators with arbitrary spin and twist. Second, we study the quantum model, mapped to a Euclidean anisotropic field theory, local in Euclidean time and long-range in space, which we dub fractional Lifshitz field theory . We show that this model admits a fixed point at zero temperature, where it displays anisotropic Lifshitz scaling, and show that at finite temperature a thermal mass is induced. We then compute the one-point functions for an infinite family of bilinear scaling operators. In both the classical and quantum model, we find that, as previously noted for the short-range O ( N ) model in [1], the large- N two-point function contains information about the one-point functions, not only of the bilinear operators, but also of operators that appear in the operator product expansion of two fundamental fields only at subleading order in 1/ N , namely powers of the Hubbard-Stratonovich intermediate field.

This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!

We present analytical results for the finite-size scaling in

d

-dimensional

O(N)

systems with strong anisotropy where the critical exponents (e.g., ${\ensuremath{\nu}}_{\ensuremath{\Vert}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance

r

as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ with different exponents $\ensuremath{\sigma}$ in corresponding spatial directions, systems with space-``time'' anisotropy near a quantum critical point, and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider

O(N)

systems in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, confined to a

d

-dimensional layer with geometry ${L}^{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{n};m+n=d$ and periodic boundary conditions across the finite

m

dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.

We numerically exhibit two strange phenomena of finite-size fluctuation in thermal equilibrium of a paradigmatic long-range interacting system having a second-order phase transition. One is a nonclassical finite-size scaling at the critical point, which differs from the prediction by statistical mechanics. With the aid of this strange scaling, the scaling theory for infinite-range models conjectures the nonclassical values of critical exponents for the correlation length. The other is relaxation of the fluctuation strength from one level to another in spite of being in thermal equilibrium. A scenario is proposed to explain these phenomena from the viewpoint of the Casimir invariants and their nonexactness in finite-size systems, where the Casimir invariants are conserved in the Vlasov dynamics describing the long-range interacting systems in the limit of large population. This scenario suggests appearance of the reported phenomena in a wide class of isolated long-range interacting systems.

We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.

Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.