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.
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.
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 notion of "half fire, half ice" was recently introduced to describe an exotic macroscopic ground-state degeneracy emerging in a ferrimagnet under the critical magnetic field, in which the "hot" spins are fully disordered on the sublattice with smaller magnetic moments and the "cold" spins are fully ordered on the sublattice with larger magnetic moments. Here, we further point out that this state has a twin named "half ice, half fire" in which the hot and cold spins switch positions. The new state is an excited state-thus hidden in the ground-state phase diagram-and is robust with respect to the interactions that destroy the half-fire, half-ice state. We demonstrate with exact results how this hidden state can drive phase switching at desirable finite temperature, even for the one-dimensional Ising model where phase transition at finite temperature is forbidden. We suggest that our findings may open a new door to the understanding and controlling of phase competition and transition in unconventional frustrated systems.
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 one-dimensional (1D) ${J}_{1}\text{\ensuremath{-}}{J}_{2}$
q-state Potts model is solved exactly for arbitrary
q by analytically block-diagonalizing the original ${q}^{2}\ifmmode\times\else\texttimes\fi{}{q}^{2}$ transfer matrix into a simple $2\ifmmode\times\else\texttimes\fi{}2$ maximally symmetric subspace, based on using OpenAI's reasoning model o3-mini-high to exactly solve the
q=3 case. Furthermore, by matching relevant subspaces, we map the Potts model onto a simpler effective 1D
q-state Potts model, where
J2 acts as the nearest-neighbor interaction and
J1 as an effective magnetic field, nontrivially generalizing a 56-year-old theorem previously limited to the simplest case (
q=2, the Ising model). Our exact results provide insights to phenomena such as atomic or electronic order stacking in layered materials and the emergence of dome-shaped phases in complex phase diagrams. This work is anticipated to fuel both research in 1D frustrated magnets for recently discovered finite-temperature application potentials and the fast moving topic area of AI in science.
We investigate the ground-state and finite-temperature properties of a mixed spin-1/2 and spin-1 Ising model on a decorated square (Lieb) lattice incorporating a uniaxial single-ion anisotropy and magnetic field. By employing the generalized decoration-iteration transformation, the model is mapped exactly onto an effective spin-1/2 Ising model on the square lattice characterized by an effective nearest-neighbor interaction and an effective field. The studied model consequently becomes exactly solvable even for finite values of the applied magnetic field whenever the effective field vanishes. The ground-state analysis reveals three distinct phases: ferrimagnetic phase (FRI), disordered phase (DP), and ferromagnetic (FM) phase. The ground-state boundary between FRI and DP phases gives rise to a dome-shaped surface of discontinuous thermal phase transitions, which is terminated by a line of Ising-type critical points associated with continuous thermal phase transitions. Both continuous and discontinuous thermal phase transitions belong to the exactly solvable parameter regime defined by a vanishing effective field in spite of the fact that the applied magnetic field is finite. Two consecutive discontinuous thermally-induced reentrant phase transitions DP-FRI-DP are identified in a narrow parameter region. The exact analytical predictions including reentrance, field- and thermally-driven phase transitions are independently verified by classical Monte Carlo simulations.