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Journal of Physical Studies

Critical behavior of structurally disordered systems with long-range interaction

M. Dudka, D. Shapoval, Yurij V. Holovatch

Published Jan 1, 2024• 1 citation
Publisher

Abstract

Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.

DOI
10.30970/jps.28.2601

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Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.
We study a stability boundary of a region where nontrivial critical behavior of an n-vector model with long-range power-law decaying interactions is induced by the presence of a structural disorder (e.g., weak quenched dilution). This boundary is given by the marginal dimension of the order parameter nc dependent on space dimension, d, and a control parameter of the interaction decay, σ, below which the model belongs to the new dilution-induced universality class. Exploiting the Harris criterion and recent field theoretical renormalization group results for the pure model with long-range interactions, we get nc as a three loop ɛ = 2σ – d-expansion. We provide numerical values for nc applying series resummation methods. Our results show that not only the Ising systems (n = 1) can belong to the new disorder-induced long-range universality class at d = 2 and 3.
We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance x as ∼ x - d - σ in a d-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter σ. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent ν characterising the new universality class for d = 3 and for those values of σ for which long-range interactions are relevant for the critical behaviour.
Changes in magnetic critical behaviour of quenched structurally-disordered magnets are usually exemplified in experiments and in MC simulations by diluted systems consisting of magnetic and non-magnetic components. In our study we aim to show that similar effects can be observed not only for diluted magnets with non-magnetic impurities but may be implemented, e.g., by the presence of two (and more) chemically different magnetic components as well. Therefore we consider a model of the structurally-disordered quenched magnet where all lattice sites are occupied by Ising-like spins of different lengths L. In such a random spin length Ising model, the length L of each spin is a random variable governed by the distribution function p(L). We demonstrate that this model belongs to the universality class of the site-diluted Ising model. This proves that both models are described by the same values of asymptotic critical exponents. However, their effective critical behaviour differs. As a case study, we consider a quenched mixture of two different magnets with values of elementary magnetic moments L1=1 and L2=s, and of concentration c and 1−c, correspondingly. We apply field-theoretical renormalization group approach to analyse the renormalization group flow for different initial conditions, triggered by s and c, and to calculate effective critical exponents further away from the fixed points of the renormalization group transformation. We show how the effective exponents are governed by difference in properties of the magnetic components.
Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^{d+σ} to a local system, i.e., with finite-range interactions, in an effective fractal dimension d_{eff}. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97%. To do so, we review perturbative renormalization-group (RG) results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.