The Ising model describes collective behaviors such as phase transitions and critical phenomena in various physical, biological, economical, and social systems. It is well known that spontaneous phase transition at finite temperature does not exist in the Ising model with short-range interactions in one dimension. Yet, little is known about whether this forbidden phase transition can be approached arbitrarily closely—at fixed finite temperature. Here I use symmetry analysis of the transfer matrix to reveal the existence of spontaneous ultranarrow phase crossover (UNPC) at finite temperature in one class of one-dimensional Ising models on decorated two-leg ladders, in which the crossover temperature T0 is determined solely by on-rung interactions and decorations, while the crossover width 2δT is independently, exponentially reduced ( δT=0 means a genuine phase transition) by on-leg interactions and decorations. These findings establish a simple ideal paradigm for realizing an infinite number of one-dimensional Ising systems with spontaneous UNPC at desirable T0 , which would be characterized in routine laboratory measurements as a genuine first-order phase transition with large latent heat thanks to the ultranarrow δT (say less than one nanokelvin), paving a way to push the limit in our understanding of phase transitions and the dynamical actions of frustration arbitrarily close to the forbidden regime. Published by the American Physical Society 2024
The Ising model describes collective behaviors such as phase transitions and critical phenomena in various physical, biological, economical, and social systems. It is well known that spontaneous phase transition at finite temperature does not exist in the Ising model with short-range interactions in one dimension. Yet, little is known about whether this forbidden phase transition can be approached arbitrarily closely—at fixed finite temperature. Here I use symmetry analysis of the transfer matrix to reveal the existence of spontaneous ultranarrow phase crossover (UNPC) at finite temperature in one class of one-dimensional Ising models on decorated two-leg ladders, in which the crossover temperature T0 is determined solely by on-rung interactions and decorations, while the crossover width 2δT is independently, exponentially reduced ( δT=0 means a genuine phase transition) by on-leg interactions and decorations. These findings establish a simple ideal paradigm for realizing an infinite number of one-dimensional Ising systems with spontaneous UNPC at desirable T0 , which would be characterized in routine laboratory measurements as a genuine first-order phase transition with large latent heat thanks to the ultranarrow δT (say less than one nanokelvin), paving a way to push the limit in our understanding of phase transitions and the dynamical actions of frustration arbitrarily close to the forbidden regime. Published by the American Physical Society 2024
In a previous paper [Weiguo Yin, Phys. Rev. Res. 6, 013331 (2024)], the forbidden spontaneous phase transition in the one-dimensional Ising model was found to be approachable arbitrarily closely in decorated ladders by ultranarrow phase crossover (UNPC) at a given finite temperature T0 with the crossover width $2\ensuremath{\delta}T$ reduced exponentially, which resembles a genuine first-order transition with large latent heat. Here, I reveal that the forbidden phase transition can be approached at fixed T0 as well in decorated single-chain Ising models in the presence of a magnetic field, in which T0 is determined by the interactions involving only the decorated parts and the magnetic field, while $2\ensuremath{\delta}T$ is independently, exponentially reduced ($\ensuremath{\delta}T=0$ means a genuine transition) by restoring the ferromagnetic interaction between the ordinary spins on the chain backbone---which was neglected in the previous studies of pseudotransition---thus manifesting that this asymptoticity to the forbidden transition is essentially the buildup of coherence in preformed crossover of local states. Furthermore, I show that the UNPC can be realized even in the absence of the conventional geometric frustration because the magnetic field itself can induce previously unnoticed hidden spin frustration. These findings make the doors wide open to the engineering and utilization of UNPC as a new paradigm for exploring exotic phenomena and 1D device applications.
The prohibition of finite-temperature phase transition in one-dimensional (1D) Ising models and 1D/2D quantum Heisenberg models with short-range interactions fundamentally constrains the application potentials of low-dimensional magnetic materials. Recently, ultranarrow phase crossover (UNPC), which can approach a transition at a desirable finite temperature T0 arbitrarily closely, was discovered in 1D decorated Ising chains and ladders. Here we present a theoretical study of similarly decorated, yet much more challenging, quantum Heisenberg ferrimagnets in a magnetic field, which features ferromagnetic backbone exchange J, antiferromagnetic site-decoration coupling JAF, and different magnetic moments for the backbone and decorating spins μaSa<μbSb. We exactly solved the model in the large J limit -- as a central-macrospin model -- and found two finite-temperature second-order transitions; just above Tc2 a ``half-ice, half-fire'' regime appears. Finite-J weak-field results follow from an effective-field mapping, suggesting the emergence of UNPC at finite T0 in 2D square lattices thanks to its exponentially strong initial magnetic susceptibility χ0∝e4πSa2J/T0, though less likely in 1D chains where χ0∝J/T0. These results may shed light on new technological applications of low-dimensional quantum spin systems and attract experimental and computational tests.
As is well known, one-dimensional spin systems with short-range interactions cannot undergo a phase transition at finite temperature. However, in many 1D Ising spin models with frustration, at a certain crossover point, the thermodynamic quantities show anomalies that resemble such a transition W. Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331]. Recently, Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331] has shown, using decorated ladders as an example, that a one-dimensional Ising model can be arbitrarily close to the genuine phase transition. Using the transfer matrix method and real space renormalization group technique, we find that crossover resembling a phase transition accompanied by gigantic specific heat can also occur in two-dimensional systems. We consider coupled Ising spin chains with a zig-zag structure and a four-spin j_{4} interaction, which allows us to control the temperature at which the spins become unbound in the model with short-range interactions. The crossover point is defined by two parameters: the crossover temperature t=t^{*} and the crossover interaction j_{4}=j_{4}^{*}, and occurs when the temperature at which the effective interaction between the spin subsystems zeros out coincides with the temperature at which the spins become unbound. In systems in which a genuine phase transition occurs at the critical temperature t=t_{c}, there are two possibilities: if t_{c}>t^{*} the system undergoes a sequence of three phase transitions with a reentrant transition to disordered phase, while for t_{c}<t^{*} the phase transition is preceded by the considered crossover. We study the key role in the appearance of anomalies in the thermodynamic quantities of the four-spin interactions which are suspected to have a significant impact on the properties of magnetic two-dimensional lattices, but possible also on the construction of social systems modeling the phenomenon of social validation or peer pressure.
Very recently, the discovery of a marginal, or practically perfect, phase transition (MPT or PPPT) at finite temperature in the 2-leg ladder Ising model with trimer rungs was reported [1]. The MPT or PPPT stems from a new mathematical structure that has not appeared before in phase-transition problems. This opens the door to simulations and utilizations of the spontaneous phase-transition phenomena in one-dimensional systems, despite the proof that phase transitions do not exist in the one-dimensional Ising models with short-range interactions back to year 1924 [2]. Naturally, the urgent fundamental and practical question is how we, now guided by the new light on phase transitions, can find the next cases of MPT or PPPT in one-dimensional systems. Here, I present a generalization of the original idea to a new parents-children Ising model with ice-cream-cone rungs, and exactly prove that the model possesses the same mathematical structure and thus MPT or PPPT. Yet, the ice-cream-cone structure features internal degrees of freedom, making the MPT or PPPT cases countless and highly tunable---with interesting behaviors like phase reentrance, Tc domes, pairing etc. These discoveries open the door wide to new interdisciplinary researches in studying, engineering, and utilizing the rich phenomenology of MPT or PPPT in various one-dimensional physical, biological, economical, industrial, and social systems, ranging from building the first-generation phase-transition-ready one-dimensional circuits to developing advanced theories that contain the new mathematical structure for frustration-driven phase transitions.
OpenAI’s reasoning model was used to carry out an exact analytic study of one-dimensional ferrimagnetic site- and bond-decorated q -state Potts models. We demonstrate that the finite-temperature ultranarrow phase crossover (UNPC), driven by a hidden “half-ice, half-fire” state recently discovered in the q = 2 case (Ising model), persists for q > 2 . Moreover, we identify unique features for q > 2 , including the dome structure in the field-temperature phase diagram, and for large q a secondary high-temperature UNPC to the fully disordered paramagnetic state. As the UNPC quickly approaches a genuine transition by enhancing J , the interaction between the backbone spins, two distinct behaviors emerge: In the site-decorated Potts model, T 0 is independent of J and thus remains unchanged (Type-I UNPC), and in the bond-decorated Potts model with q > 2 , T 0 depends on J and quickly shifts toward a finite temperature as J increases (Type-II UNPC). These results establish a versatile framework for engineering controlled fast state-flipping switches in low-dimensional systems. Our nine-dan artificial intelligence (AI)-contribution framework assigns AI the meritorious status of AI-co-led discovery in this work.
In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width m connected by sequences of vertical strings of length n mutually separated by distance N, with N=1,2 and 3. We find that the critical temperature Tc(N,m,n), arising from the collective effects, decreases as n and N increase, and increases as m increases, as it should be. The amplitude A(N,m,n) of the logarithmic divergence at the bulk critical temperature Tc(N,m,n) becomes smaller as n and m increase. A rounded peak, with size of order lnm and signifying the one-dimensional behavior of strips of finite width m, appears when n is large enough. The appearance of these rounded peaks does not depend on m as much, but depends rather more on N and n, which is rather perplexing. Moreover, for fixed m and n, the specific heats are not much different for different N. This is a most surprising result. For N=1, the spin-spin correlation in the center row of each strip can be written as a Toeplitz determinant with a generating function which is much more complicated than in Onsager's Ising model. The spontaneous magnetization in that row can be calculated numerically and the spin-spin correlation is shown to have two-dimensional Ising behavior.