Abstract The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d . This is achieved by altering the flat metric with a space dependent scale factor γ ( x ), x belonging to a bounded domain Ω. γ ( x ) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q -curvature. The equation for γ ( x ) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ ( x ) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q -hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.
Abstract The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d . This is achieved by altering the flat metric with a space dependent scale factor γ ( x ), x belonging to a bounded domain Ω. γ ( x ) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q -curvature. The equation for γ ( x ) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ ( x ) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q -hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.
Abstract We study the critical properties of the 3d O(2) universality class in bounded domains through Monte Carlo simulations of the clock model. We use an improved version of the latter, chosen to minimize finite-size corrections at criticality, with 8 orientations of the spins and in the presence of vacancies. The domain chosen for the simulations is the slab configuration with fixed spins at the boundaries. We obtain the universal critical magnetization profile and two-point correlations, which favorably compare with the predictions of the critical geometry approach based on the Yamabe equation. The main result is that the correlations, once the dimensionful contributions are factored out with the critical magnetization profile, are shown to depend only on the distance between the points computed using a metric found solving the corresponding fractional Yamabe equation. The quantitative comparison with the corresponding results for the Ising model at criticality is shown and discussed. Moreover, from the magnetization profiles the critical exponent η is extracted and found to be in reasonable agreement with up-to-date results.
We show how the apparent power-law scaling of classical two-dimenional structure factors in bounded domains at criticality is related to fixed and free boundary conditions on the order parameter. This enables us to predict the behavior of the corresponding apparent scaling dimensions based on a simple symmetry analysis. With the use of correlation functions from conformal field theory, our predictions are confirmed with exact calculations for multicritical Ising models. Effects of random boundary fields are also discussed.
We investigate two-dimensional spin models and begin with an introduction to critical phenomena with emphasis on classical systems with boundaries and conformal invariance. Then we introduce the antiferromagnetic quantum Heisenberg model with frustrating interactions. Conformal invariance of a statistical mechanical system at criticality has powerful implications in two dimensions. With the use of conformal field theory, we investigate finite-size and boundary effects for a class of N-critical models that describe the simplest type of multicritical behaviour. In Papers I and II, correlation functions for the order parameter and other operators of these multicritical Ising models are derived for restricted domains (semi-infinite plane, infinite strip, disc and rectangle) with free or fixed boundary conditions. These yield the corresponding universal surface exponents and allow for a detailed analysis of correlations lengths, structure factors and response functions. The interplay between duality and boundary conditions is also made explicit. In Paper III, we analyse structure factors at criticality and show how apparent power- law scaling at intermediate momenta - induced by finite size and boundaries - depends on the boundary conditions applied. The predictions are confirmed by using correlation functions calculated in Papers I and II. Two-dimensional Heisenberg antiferromagnets have attracted much attention because of their relation to high-Tc superconductors. The second part of the thesis (Papers IV and V) treats spin-1/2 antiferromagnets defined on square- and honeycomb lattices with frustrating interactions. By using a Schwinger-boson mean-field theory, we calculate low-temperature quantum corrections to thermodynamic parameters, such as spin-wave velocity and transverse susceptibility. Their dependence on frustration is explicated and used to estimate the stability of the Néel state via a mapping to the nonlinear .sigma. model that describes the low-energy limit of the antiferromagnet.
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.
We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a d-dimensional lattice with periodic boundary conditions and n=Ld sites, the Fisher manifold has m=d⋅n dimensions (one per bond), and we find ∣R(Jc)∣∼ndR with dR=(dν+2η)/(dν+η), where ν and η are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (ν=1, η=1/4), this predicts dR=10/9, confirmed by exact transfer-matrix computations (L=6--9: dR=1.1115±0.0002) and multi-seed MCMC through L=24. For 3D Ising (ν=0.630, η=0.0363), the prediction dR=1.019 is consistent with MCMC on L3 tori up to L=10 (power-law fit: dR=1.040). For 2D Potts q=3 (predicted 33/29≈1.138), FFT-MCMC through L=40 shows deff oscillating non-monotonically around ∼1.20, consistent with O(1/(lnL)2) logarithmic corrections. For q=4 (predicted 22/19), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity R3=−R1/2, R4=−R2/2 holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.
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!