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Finding and classifying an infinite number of cases of the practically perfect phase transition in an Ising model in one dimension

Wei‐Guo Yin

Published Jun 26, 2020• 3 citations
OpenAlex

Abstract

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, $T_c$ 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.

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Finding and classifying an infinite number of cases of the marginal phase transition in one-dimensional Ising models

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One-dimensional systems---ranging from travelling light to circuit cables and from DNA to superstrings---are ubiquitous and critically important to the human knowledge of the universe. However, our engagement with one-dimensional systems in the research and education of spontaneous phase transitions, the phenomena wherein materials can change rapidly between different phases (e.g., gas, liquid, solid, etc.) on their own, has not been largely exercised, since it was proven that one-dimensional systems do not contain phase transitions in the textbook Ising model almost 100 years ago [1] and its quantum counterpart, the Heisenberg model, over 50 years ago [2]. Recently, a spontaneous marginal phase transition (MPT) was discovered in a one-dimensional Ising model containing strong geometrical frustration [3]. Here, by exploring the symmetry of the new mathematical structure underlying the MPT, I report the finding and classification of an infinite number of MPT cases---with highly tunable intriguing behaviors like phase reentrance, the dome shape of transition temperature, pairing, and gauge freedom. These discoveries reveal the possibility of building the MPT-based one-dimensional Ising Machine that can be used to simulate the complex phenomena of phase competition in strongly correlated systems and provide insights with its unambiguous exact solutions. They also form a rich playground for exploring unconventional phase transitions in one-dimensional Heisenberg models.

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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

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From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.

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Paradigm for approaching the forbidden phase transition in the one-dimensional Ising model at fixed finite temperature: Single chain in a magnetic field

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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 ${T}_{0}$ 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 ${T}_{0}$ as well in decorated single-chain Ising models in the presence of a magnetic field, in which ${T}_{0}$ 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.

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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, TcT_cTc​ 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.
One-dimensional systems---ranging from travelling light to circuit cables and from DNA to superstrings---are ubiquitous and critically important to the human knowledge of the universe. However, our engagement with one-dimensional systems in the research and education of spontaneous phase transitions, the phenomena wherein materials can change rapidly between different phases (e.g., gas, liquid, solid, etc.) on their own, has not been largely exercised, since it was proven that one-dimensional systems do not contain phase transitions in the textbook Ising model almost 100 years ago [1] and its quantum counterpart, the Heisenberg model, over 50 years ago [2]. Recently, a spontaneous marginal phase transition (MPT) was discovered in a one-dimensional Ising model containing strong geometrical frustration [3]. Here, by exploring the symmetry of the new mathematical structure underlying the MPT, I report the finding and classification of an infinite number of MPT cases---with highly tunable intriguing behaviors like phase reentrance, the dome shape of transition temperature, pairing, and gauge freedom. These discoveries reveal the possibility of building the MPT-based one-dimensional Ising Machine that can be used to simulate the complex phenomena of phase competition in strongly correlated systems and provide insights with its unambiguous exact solutions. They also form a rich playground for exploring unconventional phase transitions in one-dimensional Heisenberg models.
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
As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.
From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.
Abstract Phase transitions, as one of the most intriguing phenomena in nature, are divided into first-order phase transitions (FOPTs) and continuous ones in current classification. While the latter shows striking phenomena of scaling and universality, the former has recently also been demonstrated to exhibit scaling and universal behavior within a mesoscopic, coarse-grained Landau–Ginzburg theory. Here we apply this theory to a microscopic model—the paradigmatic Ising model, which undergoes FOPTs between two ordered phases below its critical temperature—and unambiguously demonstrate universal scaling behavior in such FOPTs. These results open the door for extending the theory to other microscopic FOPT systems and experimentally testing them to systematically uncover their scaling and universal behavior.
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{T}_{0}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{T}_{0}T0​ as well in decorated single-chain Ising models in the presence of a magnetic field, in which T0{T}_{0}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.