2D Ising Model Research Articles

Spontaneous Symmetry Breaking in Systems Obeying the Dynamics of On–Off Intermittency and Presenting Bimodal Amplitude Distributions

In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the symbolic dynamics of two symbols produced by the 3D Ising model in its critical state. After this numerical verification of the reliability of PNA-STSM, three examples of how PNA-STSM can be applied to the category of systems that obey the dynamics of the on–off intermittency are presented. Usually, such time series, with on–off intermittency, present bimodal amplitude distributions (i.e., with two lobes). As has recently been found, the phenomenon of on–off intermittency is associated with the spontaneous symmetry breaking (SSB) of the second-order phase transition. Thus, the revelation that a system is close to SSB supports a deeper understanding of its dynamics in terms of criticality, which is quite useful in applications such as the analysis of pre-earthquake fracture-induced electromagnetic emission (also known as fracture-induced electromagnetic radiation) (FEME/FEMR) signals. Beyond the case of on–off intermittency, PNA-STSM can provide credible results for the dynamics of any two-symbol symbolic dynamics, even in cases in which there is an imbalance in the probability of the appearance of the two respective symbols since the two symbols are not considered separately but, instead, simultaneously, considering the information from both branches of the symbolic dynamics.

Magnetocaloric effect in the Ising model with RKKY interaction on the Shastry–Sutherland lattice

The classical Monte Carlo method is used to examine effects of electron and hole doping on the magnetocaloric effect in the generalized 2D Ising model with the long-range RKKY interaction on the Shastry–Sutherland lattice. The electron and hole doping in this model is controlled by deviations of the Fermi wave vector kF from the initial value kF=2π/1.24 which models the most realistically situation in the undoped metallic Shastry–Sutherland rare-earth tetraboride TmB4. For the undoped case we have found a heating hill (the entropy change ΔS>0) occurring in the vicinity of metamagnetic transitions and a cooling depression in the paramagnetic phase which is in a very good qualitative agreement with experimental measurements in TmB4. In the case of small electron doping (kF=2π/1.26), a significant decrease of the maximum of −ΔS is observed, while in the strongly hole (electron) doped case kF=2π/1.17 (kF=2π/1.43) the maximum of −ΔS is considerably enhanced against the undoped system, which points to the fact that some of doped systems could have better cooling properties than the undoped one. This is confirmed independently by calculations of relative cooling power for a wide range of parameter kF, where we have found two intervals kF∈〈2π/1.15,2π/1.21〉 and kF∈〈2π/1.39,2π/1.5〉 with much larger relative cooling power in comparison to the undoped case.

Magnetization dynamics and Peierls instability in topological Josephson structures

We study a long topological Josephson junction with a ferromagnetic strip between two superconductors. The low-energy theory exhibits a non-local in time and space interaction between chiral Majorana fermions, mediated by the magnonic excitations in the ferromagnet. While short ranged interactions turn out to be irrelevant by power counting, we show that sufficiently strong and long-ranged interactions may induce a $\mathbb{Z}_2$-symmetry breaking. This spontaneous breaking leads to a tilting of the magnetization perpendicular to the Majorana propagation direction and the opening of a fermionic gap (Majorana mass). It is analogous to the Peierls instability in the commensurate Fr\"ohlich model and reflects the nontrivial interplay between Majorana modes and magnetization dynamics. Within a Gaussian fluctuation analysis, we estimate critical values for the temporal and spatial non-locality of the interaction, beyond which the symmetry breaking is stable at zero temperature -- despite the effective one-dimensionality of the model. We conclude that non-locality, i.e., the stiffness of the magnetization in space and time, stabilizes the symmetry breaking. In the stabilized regime, we expect the current-phase relation to exhibit an experimentally accessible discontinuous jump. At nonzero temperatures, as usual in the 1D Ising model, the long-range order is destroyed by solitonic excitations, which in our case carry each a Majorana zero mode. In order to estimate the correlation length, we investigate the solitons within a self-consistent mean-field approach.

Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism

Inspired by the 1995 paper of Baake–Grimm–Pisani, we aim to explain the empirical observation that the distribution of Lee–Yang zeros corresponding to a one-dimensional Ising model appears to follow the gap labelling theorem. This follows by combining two main ingredients: first, the relation between the transfer matrix formalism for the 1D Ising model and an ostensibly unrelated matrix formalism generating the Szegő recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices.

Eigenvalue Crossing as a Phase Transition in Relaxation Dynamics.

When a system's parameter is abruptly changed, a relaxation toward the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation operator results in a singularity in the dynamics analogous to a first-order equilibrium phase transition. While dynamical phase transitions are intrinsically hard to detect in nature, here we show how this kind of transition can be observed in an experimentally feasible four-state colloidal system. Finally, analytical proof of survival in the thermodynamic limit of a many body (1D Ising) model is provided.

Excitations of Ising strings on a lattice

The 3d Ising model in the low temperature (ferromagnetic) phase describes dynamics of two-dimensional surfaces — domain walls between clusters of parallel spins. The Kramers-Wannier duality maps these surfaces into worldsheets of confining strings in the Wegner’s ℤ2 gauge theory. We study the excitation spectrum of long Ising strings by simulating the ℤ2 gauge theory on a lattice. We observe a strong mixing between string excitations and the lightest glueball state and do not find indications for light massive resonances on the string worldsheet.

Critical behavior and room temperature magnetocaloric effect of La-doped Pr0.7Sr0.3MnO3 compounds

In this work, we present the critical and the magnetocaloric properties of Pr0.7-xLaxSr0.3MnO3 compounds around room temperature. The samples exhibit a single phase with an orthorhombic structure. However, the lattice parameters increase with an increase in La content (x). The Curie temperature (TC) of the ferromagnetic (FM)-paramagnetic (PM) phase transition of compounds increases from 256 K for x = 0–319 K for x = 0.3. We observed a large magnetocaloric effect with high maximum magnetic entropy change (|ΔSmax|) values occurring around room temperature, corresponding to |ΔSmax| = 6.65 and 5.21 J/kgK under a magnetic field change (ΔH) of 50 kOe for x = 0.2 and 0.3, respectively. We also pointed out that the change in |ΔSmax| obeys an exponential function of ΔH (|ΔSmax| ∼ ΔHn), where the value of field exponent (n) is approximately 0.52. Furthermore, the critical behavior around TC for compounds with x = 0.2 and 0.3 has been investigated. The results indicate that these compounds undergo a second-order phase transition with the critical parameters (β = 0.311, γ = 1.137, and δ = 4.656 at TC = 298.4 K for x = 0.2; and β = 0.319, γ = 1.098, and δ = 4.442 at TC = 318.7 K for x = 0.3) that are quite close to those expected for the 3D-Ising model. This suggests the existence of the short-range magnetic interactions in these compounds.

Positivity, low twist dominance and CSDR for CFTs

We consider a crossing symmetric dispersion relation (CSDR) for CFT four point correlation with identical scalar operators, which is manifestly symmetric under the cross-ratios u,vu,v interchange. This representation has several features in common with the CSDR for quantum field theories. It enables a study of the expansion of the correlation function around u=v=1/4u=v=1/4, which is used in the numerical conformal bootstrap program. We elucidate several remarkable features of the dispersive representation using the four point correlation function of \Phi_{1,2}Φ1,2 operators in 2d minimal models as a test-bed. When the dimension of the external scalar operator (\Delta_\sigmaΔσ) is less than \frac{1}{2}12, the CSDR gets contribution from only a single tower of global primary operators with the second tower being projected out. We find that there is a notion of low twist dominance (LTD) which, as a function of \Delta_\sigmaΔσ, is maximized near the 2d Ising model as well as the non-unitary Yang-Lee model. The CSDR and LTD further explain positivity of the Taylor expansion coefficients of the correlation function around the crossing symmetric point and lead to universal predictions for specific ratios of these coefficients. These results carry over to the epsilon expansion in 4-\epsilon4−ϵ dimensions. We also conduct a preliminary investigation of geometric function theory ideas, namely the Bieberbach-Rogosinski bounds.

Possible new phase transition in the 3D Ising model associated with boundary percolation

In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.

Anomalous Dynamical Phase Transitions of the Ising Model Studied by the Nonequilibrium Ensemble Method: Freezing-by-Heating and Permanent Liquid.

We investigate the nonequilibrium, dynamical phase transition behavior of the one-dimensional Ising model, using the trajectory ensemble approach in the context of the large deviation theory. We introduce a double-biased ensemble, named the s, g-ensemble, based on nonequilibrium steady-state trajectories. The ensemble invokes the time-integrated, trajectory energy as an order parameter, coupled to its conjugate g-field in addition to the dynamical activity and its conjugate field s in the trajectory space. Using the dynamical free energy obtained from the large deviation formalism, we explore the rich behaviors of the dynamical phase transition of the 1D Ising model in the (s, g, T) parameter space, with T being temperature. Among other features, we discover that novel, anomalous dynamical phase transitions are possible due to the decoupling between the dynamical activity and trajectory energy under specific conditions. In particular, we observe that the system exhibits a freezing-by-heating phenomenon as the dynamical activity decreases with temperature under a specific condition. We also find a permanent liquid phase when the equilibrium temperature and the nonequilibrium g-field are exactly balanced against each other. Our results provide a useful tool for exploring the dynamical phase transition phenomena to be investigated in various systems.

Non-reciprocal interactions spatially propagate fluctuations in a 2D Ising model

Motivated by the anisotropic interactions between fish, we implement spatially anisotropic and therefore non-reciprocal interactions in the 2D Ising model. First, we show that the model with non-reciprocal interactions alters the system critical temperature away from that of the traditional 2D Ising model. Further, local perturbations to the magnetization in this out-of-equilibrium system manifest themselves as traveling waves of spin states along the lattice, also seen in a mean-field model of our system. The speed and directionality of these traveling waves are controllable by the orientation and magnitude of the non-reciprocal interaction kernel as well as the proximity of the system to the critical temperature.

Holographic cameras: an eye for the bulk

We consider four-point correlators in an excited quantum state of a field theory. We show that, when the theory and state are holographic, a judiciously applied Fourier transform produces high-quality images of point-like bulk particles, revealing the geometry in which they move. For translation-invariant states, the bulk Einstein’s equations amount to local differential equations on correlator data. In theories or states that are not holographic, images are too blurry to extract a bulk geometry. We verify this for gauge theories at various couplings and the 3D Ising model by adapting formulas from conformal Regge theory.

3D quantum gravity from holomorphic blocks

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the ${3nj}$ symbols of the recoupling of the SU(2) representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as the gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the ${6j}$ symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity and offers a new twist on the construction of a duality between 3D quantum gravity and a 2D conformal theory.

Topological correlations in three-dimensional classical Ising models: An exact solution with a continuous phase transition

We study a three-dimensional (3D) classical Ising model that is exactly solvable when some coupling constants take certain imaginary values. The solution combines and generalizes the Onsager-Kaufman solution [L. Onsager, Phys. Rev. 65, 117 (1944); B. Kaufman, Phys. Rev. 76, 1232 (1949)] of the 2D Ising model and the solution of Kitaev's honeycomb model [A. Kitaev, Ann. Phys, 321, 2 (2006)], leading to a three-parameter phase diagram with a third-order phase transition between two distinct phases. Interestingly, the phases of this model are distinguished by topological features: the expectation value of a certain family of loop observables depend only on the topology of the loop (whether the loop is contractible), and are quantized at rational values that differ in the two phases. We show that a related exactly solvable 3D classical statistical model with real coupling constants also shows the topological features of one of these phases. Furthermore, even in the model with complex parameters, the partition function has some physical relevance, as it can be interpreted as the transition amplitude of a quantum dynamical process and may shed light on dynamical quantum phase transitions.

Vaccination dilemma in the thermodynamic limit.

The vaccination game is a social dilemma that refers to the conundrum individuals face (to get immunized or not) when the population is exposed to an infectious disease. The model has recently gained much traction due to the COVID-19 pandemic since the public perception of vaccines plays a significant role in disease dynamics. This paper studies the vaccination game in the thermodynamic limit with an analytical method derived from the 1D Ising model called Nash equilibrium mapping. The individual dilemma regarding vaccination comes from an internal conflict wherein one tries to balance the perceived advantages of immunizing with the apparent risks associated with vaccination, which they hear through different news media. We compare the results of Nash equilibrium (NE) mapping from other 1D Ising-based models, namely, Darwinian evolution (DE) and agent-based simulation. This study aims to analyze the behavior of an infinite population regarding what fraction of people choose to vaccinate or not vaccinate. While Nash equilibrium mapping and agent-based simulation agree mostly, DE strays far from the two models. DE fails to predict the equilibrium behavior of players in the population reasonably. We apply the results of our study to analyze the AstraZeneca (AZ) COVID-19 vaccine risk vs disease deaths debate, both via NE mapping and the agent-based method. Both predict nearly 100% AZ vaccine coverage for people aged above 40, notwithstanding the risk. At the same time, younger people show a slight reluctance. We predict that while government intervention via vaccination mandates and/or advertisement campaigns are unnecessary for the older population, for the younger population (ages: 20-39), some encouragement from the government via media campaigns and/or vaccine mandates may be necessary.

Two-dimensional van der Waals ferromagnetic thin film CrTe2 with high Curie temperature and metallic conductivity

Two-dimensional van der Waals (2D vdW) materials have opened up an opportunity to explore an innovative spin-based magnetic nanodevice. However, controllable fabrication of 2D vdW ferromagnets with high Curie temperature remains challenging. In this paper, we reported the growth of 2D CrTe2 single-crystal films epitaxially on Al2O3 substrates using pulsed laser deposition. We find that it shows a typical paramagnetic–ferromagnetic (PM–FM) phase transition around 200 K. The precise Curie temperature and Weiss temperature are 189 and 206.7 K, respectively. The saturation magnetization reaches 73.64 emu/g for the film thickness of 30 nm. The critical exponent β = 0.329 indicates that the magnetic interactions obey the 3D-Ising model. Electronic transport measurement confirms that a CrTe2 film always remains a metallic behavior at 5 K ≤ T ≤ 320 K and the resistivity of room temperature is 1.5 mΩ/cm. The first-principles calculation uncovers that the FM ordering state mainly stems from an exchange coupling of the adjacent Cr-spin t2g polarized electrons and the metallic conductivity is due to p–d orbital hybridization between Cr and Te atoms. This work would shed new light on studying large-scale growth of 2D magnets and developing 2D magnet-based nanodevices of room temperature.

The Airy2 process and the 3D Ising model

The Ferrari–Spohn diffusion process arises as limit process for the 2D Ising model as well as random walks with area penalty. Motivated by the 3D Ising model, we consider M such diffusions conditioned not to intersect. We show that the top process converges to the Airy2 process as . We then explain the relation with the 3D Ising model and present some conjectures about it.

Mapping between Spin-Glass Three-Dimensional (3D) Ising Model and Boolean Satisfiability Problem

The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean satisfiability (K-SAT) problems for K ≥ 3 MSATK≥3 are nontrivial, due to the existence of non-planarity graphs, nonlocalities, and the randomness. In this work, the relation between a spin-glass three-dimensional (3D) Ising model MSGI3D with the lattice size N = mnl and the K-SAT problems is investigated in detail. With the Clifford algebra representation, it is easy to reveal the existence of the long-range entanglements between Ising spins in the spin-glass 3D Ising lattice. The internal factors in the transfer matrices of the spin-glass 3D Ising model lead to the nontrivial topological structures and the nonlocalities. At first, we prove that the absolute minimum core (AMC) model MAMC3D exists in the spin-glass 3D Ising model, which is defined as a spin-glass 2D Ising model interacting with its nearest neighboring plane. Any algorithms, which use any approximations and/or break the long-range spin entanglements of the AMC model, cannot result in the exact solution of the spin-glass 3D Ising model. Second, we prove that the dual transformation between the spin-glass 3D Ising model and the spin-glass 3D Z2 lattice gauge model shows that it can be mapped to a K-SAT problem for K ≥ 4 also in the consideration of random interactions and frustrations. Third, we prove that the AMC model is equivalent to the K-SAT problem for K = 3. Because the lower bound of the computational complexity of the spin-glass 3D Ising model CLMSGI3D is the computational complexity by brute force search of the AMC model CUMAMC3D, the lower bound of the computational complexity of the K-SAT problem for K ≥ 4 CLMSATK≥4 is the computational complexity by brute force search of the K-SAT problem for K = 3 CUMSATK=3. Namely, CLMSATK≥4=CLMSGI3D≥CUMAMC3D=CUMSATK=3. All of them are in subexponential and superpolynomial. Therefore, the computational complexity of the K-SAT problem for K ≥ 4 cannot be reduced to that of the K-SAT problem for K < 3.

Critical fluctuations in liquid-liquid extraction organic phases controlled by extractant and diluent molecular structure.

Extractant aggregation in liquid-liquid extraction organic phases impacts extraction energetics and is related to the deleterious efficiency-limiting liquid-liquid phase transition known as third phase formation. Using small angle X-ray scattering, we find that structural heterogeneities across a wide range of compositions in binary mixtures of malonamide extractants and alkane diluents are well described by Ornstein-Zernike scattering. This suggests that structure in these simplified organic phases originates from the critical point associated with the liquid-liquid phase transition. To confirm this, we measure the temperature dependence of the organic phase structure, finding critical exponents consistent with the 3D Ising model. Molecular dynamics simulations were also consistent with this mechanism for extractant aggregation. Due to the absence of water or any other polar solutes required to form reverse-micellar-like nanostructures, these fluctuations are inherent to the binary extractant/diluent mixture. We also show how the molecular structure of the extractant and diluent modulate these critical concentration fluctuations by shifting the critical temperature: critical fluctuations are suppressed by increasing extractant alkyl tail lengths or decreasing diluent alkyl chain lengths. This is consistent with how extractant and diluent molecular structure are known to impact metal and acid loading capacity in many-component LLE organic phases, suggesting phase behavior of practical systems may be effectively studied in simplified organic phases. Overall, the explicit connection between molecular structure, aggregation and phase behavior demonstrated here will enable the design of more efficient separations processes.

Connections between efficient control and spontaneous transitions in an Ising model.

A system can be driven between metastable configurations by a time-dependent driving protocol, which uses external control parameters to change the potential energy of the system. Here we investigate the correspondence between driving protocols that are designed to minimize work and the spontaneous transition paths of the system in the absence of driving. We study the spin-inversion reaction in a 2D Ising model, quantifying the timing of each spin flip and heat flow to the system during both a minimum-work protocol and a spontaneous transition. The general order of spin flips during the transition mechanism is preserved between the processes, despite the coarseness of control parameters that are unable to reproduce more detailed features of the spontaneous mechanism. Additionally, external control parameters provide energy to each system component to compensate changes in internal energy, showing how control parameters are tuned during a minimum-work protocol to counteract underlying energetic features. This paper supports a correspondence between minimum-work protocols and spontaneous transition mechanisms.