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arXiv:2507.15095 · Physical review. E

Criticality in one-dimensional field theories with mesoscopic, infinite-range interactions

Kurt Langfeld, Amanda Turner

Published Dec 23, 2025• 0 citations
arXivHTMLPDF

Abstract

This research investigates a class of one-dimensional theories characterized by a distinctly defined infinite interaction range. We propose that such theories emerge naturally through a mesoscopic feedback mechanism. In this proof-of-concept study, we examine Ising-type models and a model with continuous O(3) symmetry, and demonstrate that the natural emergence of phase transitions, criticality, spontaneous symmetry breaking and previously unidentified universality classes is evident. The framework introduced here holds particular relevance for monolayer spintronics research, where the ultimate goal is to achieve a strong ferromagnetic order at room temperature.

DOI
10.1103/5mjz-d6tx

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This research investigates a class of one-dimensional theories characterized by a distinctly defined infinite interaction range. We propose that such theories emerge naturally through a mesoscopic feedback mechanism. In this proof-of-concept study, we examine Ising-type models and a model with continuous O(3) symmetry, and demonstrate that the natural emergence of phase transitions, criticality, spontaneous symmetry breaking and previously unidentified universality classes is evident. The framework introduced here holds particular relevance for monolayer spintronics research, where the ultimate goal is to achieve a strong ferromagnetic order at room temperature.
Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-Tc{T}_{c}Tc​ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of qqq-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(nnn) symmetry with nnn=qqq-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(nnn=qqq-1) universality classes.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interaction emerges from the interplay between the Kitaev ( K ) and Heisenberg ( J ) exchanges, and is accompanied by an additional bond-dependent interaction (Γ), leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J - K -Γ model with four-fold screw symmetry in a spin-orbit-entangled chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic ordered structures, the transition remains in the Ising universality class with central charge c = 1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to Ising materials such as chains of Co 2+ ions is also discussed.
Quantum criticality of metal-insulator transitions in correlated electron systems is shown to belong to an unconventional universality class with violation of the Ginzburg-Landau-Wilson (GLW) scheme formulated for symmetry breaking transitions. This unconventionality arises from an emergent character of the quantum critical point, which appears at the marginal point between the Ising-type symmetry breaking at nonzero temperatures and the topological transition of the Fermi surface at zero temperature. We show that Hartree-Fock approximations of an extended Hubbard model on square lattices are capable of such metal-insulator transitions with unusual criticality under a preexisting symmetry breaking. The obtained universality is consistent with the scaling theory formulated for Mott transitions and with a number of numerical results beyond the mean-field level, implying that preexisting symmetry breaking is not necessarily required for the emergence of this unconventional universality. Examinations of fluctuation effects indicate that the obtained critical exponents remain essentially exact beyond the mean-field level. It further clarifies the whole structure of singularities by a unified treatment of the bandwidth-control and filling-control transitions. Detailed analyses of the criticality, containing diverging carrier density fluctuations around the marginal quantum critical point, are presented from microscopic calculations and reveal the nature as quantum critical ``opalescence.'' The mechanism of emerging marginal quantum critical point is ascribed to a positive feedback and interplay between the preexisting gap formation present even in metals and kinetic energy gain (loss) of the metallic carrier. Analyses of crossovers between GLW type at nonzero temperature and topological type at zero temperature show that the critical exponents observed in (V,Cr)2O3{(\mathrm{V},\mathrm{Cr})}_{2}{\mathrm{O}}_{3}(V,Cr)2​O3​ and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.
Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground-state disorder-to-order phase transitions. We consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick model describing the collective interaction of N spins and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, we focus on critical quenches, where the initial state and/or the postquench Hamiltonian are critical. Depending on the type of quench, we identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely nonequilibrium, characterized by distinct universality classes and static and dynamical critical exponents. These behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as N^{1/3}), respectively. The quench dynamics is studied through a combination of exact numerics and analytical calculations utilizing the nonequilibrium Keldysh field theory. Our results are amenable to realization in experiments with trapped-ion experiments where long-range interactions naturally arise.
In this study, we present theoretical investigations of phase transitions and critical phenomena in materials through the lens of second-order Ginzburg-Landau theory, in conjunction with considerations of symmetry groups and thermal fluctuations. By addressing the residual effects after a renormalization process, a small number of macroscopic degrees of freedom can effectively replace the infinite number of microscopic degrees of freedom, emphasizing the significant role of dimensionality and the intrinsic characteristics of the system in understanding and analyzing transitions. We highlight several non-universal characteristics of continuous phase transitions near the transition temperature, including the non-monotonic relationship between the critical temperature and dimensionality, as well as the enhancement or disappearance of the specific heat jump in complex superconductors. While the resulting expressions for thermodynamic quantities are complex for one-dimensional systems, obeying Mermin-Wagner's theorem, they are considerably simplified for two-dimensional and three-dimensional systems.