Ising Model Research Articles

Exotic phases induced by RKKY interaction in the square lattice

We study a model of Ising spins in which direct exchange interactions compete with the Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling, which may arise effectively from itinerant electrons. We consider the model in the two-dimensional square lattice and focus on values of the RKKY coupling constant (JRKKY) and the Fermi momentum (kF) that induce strong frustration. We study the low-temperature magnetic field phase diagram using Monte Carlo simulations, considering several nearest neighbors in the RKKY interaction. Magnetic frustration yields a rich phase diagram with a variety of exotic phases. We compute the structure factors and define appropriate order parameters to describe the transitions. We also discuss the validity of the approximation made for long-range couplings and analytically demonstrate the change in the coexistence lines in the phase diagram by including up to the tenth nearest neighbors.

The economophysical model of the propagation of innovation: the Ising model

The process of technical and economic justification of investment, analysis and evaluation of the effectiveness of innovation requires a tool for describing and modeling the process of distribution of technology in the industry. The work presents a model of propagation of innovation involving physical approaches, describing the market saturation point, i.e. the point at which the exponential growth of the innovation propagation speed is replaced by the logarithmic growth. The object of the study is the propagation of innovation, and the subject is the development of the market saturation point functional. The authors justified the implementation and described the approach to modeling of the process of saturation of the market with innovation by the physical Ising model. The value of the Ising model’s toolkit is presented by the Curie point in ferromagnets which characterizes the second order phase transition. The article presents the mathematical model of the compliance of physical parameters with economic ones: the amount of inter-company influence, barriers to implementation and breakthrough of innovation. The authors adduce the discussion of the limitations and applicability of this model as well as further potential directions of study of economophysical models. The tools developed by the authors can be used in all sectors of economics to improve their innovation activity level.

Magnetic properties, critical behavior, and magnetocaloric effect of Nd1−xSrxMnO3 (0.2 ≤ x ≤ 0.5): The role of Sr doping concentration

Magnetic characteristics, magnetocaloric effect, and critical behavior of Nd1−xSrxMnO3 compounds by Sr doping (x = 0.2, 0.3, 0.4, 0.5) were studied. All samples maintained orthorhombic structures, but the space group changed from Pnma (No. 62) for x = 0.2, 0.3 to Imma (No. 74) for x = 0.4, 0.5. As Sr doping increased, the Curie temperature (TC), Curie–Weiss temperature (TCW), and magnetization increased, attributed to the double exchange (DE) interaction. A discrepancy between TCW and TC was observed due to the competition between polarons and DE interaction. The critical behavior was investigated systematically using the self-consistent (modified Arrott plots, MAP) method and the Kouvel–Fisher (KF) relation. The KF relation was suitable for the samples with x = 0.2 and 0.5, while the MAP method was suitable for the samples with x = 0.3 and 0.4. Among the Ising, XY, Heisenberg, and mean-field models, the samples with x = 0.2, 0.3, and 0.4 aligned more closely with the mean-field model, except for the x = 0.5 sample. Entropy change (−ΔSM) of Nd1−xSrxMnO3 (0.2 ≤ x ≤ 0.5) increased with the applied field, with the maximum value observed around TC. For the sample with x = 0.3, (−ΔSM) reached 4.315 J/kg K at μ0ΔH = 50 kOe, corresponding to a relative cooling power (RCP) of 280.48 J/kg. Remarkably, the x = 0.4 sample displayed (−ΔSM) of 3.298 J/kg K at μ0ΔH = 50 kOe near room temperature, with the RCP of 283.64 J/kg. These findings underscore the role of Sr doping in tuning the magnetic properties, critical behavior, and magnetocaloric effect of NdMnO3.

Holographic Analysis of Boundary Correlation Functions for the Hyperbolic-Lattice Ising Model

Abstract We analyze boundary spin correlation functions of the hyperbolic-lattice Ising model from the holographic point of view. Using the corner-transfer-matrix renormalization-group (CTMRG) method, we demonstrate that the boundary correlation function exhibits power-law decay with quasiperiodic oscillation, while the bulk correlation function always decays exponentially. On the basis of the geometric relation between the bulk correlation path and distance along the outer edge boundary, we find that scaling dimensions for the boundary correlation function can be well explained by a combination of the bulk correlation length and background curvatures inherent to the hyperbolic lattice. We also investigate the cutoff effect of the bond dimension in CTMRG, revealing that the long-distance behavior of the boundary spin correlation is accurately described even with a small bond dimension. In contrast, the short-distance behavior rapidly loses its accuracy.

Hysteretic response to different modes of ramping an external field in sparse and dense Ising spin glasses

We consider the hysteretic behavior of Ising spin glasses at T=0 for various modes of driving. Previous studies mostly focused on an infinitely slow speed Ḣ by which the external field H was ramped to trigger avalanches of spin flips by starting with destabilizing a single spin while few have focused on the effect of different driving methods. First, we show that this conventional protocol imposes a system size dependence. Then, we numerically analyze the response of Ising spin glasses at rates Ḣ that are fixed as well, to elucidate the differences in the response. Specifically, we compare three different modes of ramping (Ḣ=c/N, Ḣ=c/N, and Ḣ=c for constant c) for two types of spin glass systems of size N, representing dense networks by the Sherrington–Kirkpatrick model and sparse networks by the lattice spin glass in d=3 dimensions known as the Edwards Anderson model. Depending on the mode of ramping, we find that the response of each system, in form of spin-flip avalanches and other observables, can vary considerably. In particular, in the N-independent mode applied to the lattice spin glass, which is closest to experimental reality, we observe a percolation transition with a broad avalanche distribution between phases of localized and system-spanning responses. We explore implications for combinatorial optimization problems pertaining to sparse systems.

Dynamic magnetic characteristics of the kinetic Ising model under the influence of randomness.

In this paper, we propose to solve the issues of long-range or next-neighbor interactions by introducing randomness. This approach is applied to the square lattice Ising model. The Monte Carlo method with the Metropolis algorithm is utilized to calculate the critical temperature T_{C}^{*} under equilibrium thermodynamic phase transition conditions and to investigate the characterization of randomness in terms of magnetization. In order to further characterize the effect of this randomness on the magnetic system, clustering coefficients C_{p} are introduced. Furthermore, we investigate a number of dynamic magnetic behaviors, including dynamic hysteresis behaviors and metamagnetic anomalies. The results indicate that noise has the effect of destabilizing the system and promoting the dynamic phase transition. When the system is subjected to noise, the effect of this noise can be mitigated by the addition of a time-oscillating magnetic field. Finally, the evolution of anomalous metamagnetic fluctuations under the influence of white noise is examined. The relationship between the bias field corresponding to the peak of the curve h_{b}^{peak} and the noise parameter σ satisfies the exponential growth equation, which is consistent with other results.

Finite Temperature Magnetism in the Triangular Lattice Antiferromagnet KErTe2

Abstract After the discovery of the ARECh2 (A = alkali or monovalent ions, RE = rare-earth, Ch = chalcogen) triangular lattice quantum spin liquid (QSL) family, a series of its oxide, sulfide, and selenide counterparts has been consistently reported and extensively investigated. While KErTe2 represents the initial synthesized telluride member, preserving its triangular spin lattice, it was anticipated that the substantial tellurium ions could impart more pronounced magnetic attributes and electronic structures to this material class. This study delves into the magnetism of KErTe2 at finite temperatures through magnetization and electron spin resonance (ESR) measurements. Based on the angular momentum J ^ after spin-orbit coupling (SOC) and symmetry analysis, we obtain the magnetic effective Hamiltonian to describe the magnetism of Er3+ in R 3 ¯ m space group. Applying the mean-field approximation to the Hamiltonian, we can simulate the magnetization and magnetic heat capacity of KErTe2 in paramagnetic state and determine the crystalline electric field (CEF) parameters and partial exchange interactions. The relatively narrow energy gaps between the CEF ground state and excited states exert a significant influence on the magnetism. For example, small CEF excitations can result in a significant broadening of the ESR linewidth at 2 K. For the fitted exchange interactions, although the values are small, given a large angular momentum J = 15/2 after SOC, they still have a noticeable effect at finite temperatures. Notably, the heat capacity data under different magnetic fields along the c axis direction also roughly match our calculated results, further validating the reliability of our analytical approach. These derived parameters serve as crucial tools for future investigations into the ground state magnetism of KErTe2. The findings presented herein lay a foundation for exploration of the intricate magnetism within the triangular-lattice delafossite family.

Quantum subspace expansion in the presence of hardware noise

Finding ground state energies on current quantum processing units (QPUs) using algorithms such as the variational quantum eigensolver (VQE) continues to pose challenges. Hardware noise severely affects both the expressivity and trainability of parameterized quantum circuits, limiting them to shallow depths in practice. Here, we demonstrate that both issues can be addressed by synergistically integrating VQE with a quantum subspace expansion, allowing for an optimal balance between quantum and classical computing capabilities and costs. We perform a systematic benchmark analysis of the iterative quantum-assisted eigensolver in the presence of hardware noise. We determine ground state energies of 1D and 2D mixed-field Ising spin models on noisy simulators and the IBM QPUs ibmq_quito (5 qubits) and ibmq_guadalupe (16 qubits). To maximize accuracy, we propose a suitable criterion to select the subspace basis vectors according to the trace of the noisy overlap matrix. Finally, we show how to systematically approach the exact solution by performing controlled quantum error mitigation based on probabilistic error reduction on the noisy backend fake_guadalupe.

Bipartite OTOC in open quantum systems: information scrambling and irreversibility

Abstract The field of information scrambling has seen significant growth over the last decade, where the out-of-time-ordered correlator (OTOC) has emerged as a prominent tool to probe it. In this work, we use bipartite OTOC, a particular form of OTOC, to study information scrambling in the atom–field interaction models and the model of the Ising spin chain interacting with a tilted magnetic field. This is done considering the effects of open quantum systems. A relationship between information scrambling, using bipartite OTOC, and irreversibility, using entropy production, is probed under unitary dynamics. The equivalence of bipartite OTOC with operator entanglement is explicitly shown for the Ising model.

Dynamical critical behavior on the Nishimori point of frustrated Ising models.

By considering the quench dynamics of two-dimensional frustrated Ising models through numerical simulations, we investigate the dynamical critical behavior on the multicritical Nishimori point (NP). We calculate several dynamical critical exponents, namely, the relaxation exponent z_{c}, the autocorrelation exponent λ_{c}, and the persistence exponent θ_{c}, after a quench from the high temperature phase to the NP. We confirm their universality with respect to the lattice geometry and bond distribution. For a quench from a power-law correlated initial state to the NP, the aging dynamics are much slower. We also look up the issue of multifractality during the critical dynamics by investigating different moments of the spatial correlation function. We observe a single growth law for all the length scales extracted from different moments, indicating that the equilibrium multifractality at the NP does not affect the dynamics.

A comprehensive neural network study of the non-equilibrium phase transition of the two-dimensional Ising model on the square lattice

A comprehensive neural network study of the non-equilibrium phase transition of the two-dimensional Ising model on the square lattice

Possible application of quantum computing in the field of ocean engineering: optimization of an offshore wind farm layout with the Ising model

AbstractAs a possible application of quantum computing in the field of ocean engineering, an attempt is made to identify the optimum layout of an offshore wind farm with the quasi-quantum annealing method while assuming a wake model that conforms to the standard Jensen–Katic wake model but can be implemented in the Ising model. The optimum layouts obtained in the present study are compared with those identified with other conventional methods such as the genetic algorithm. It is confirmed that the quasi-quantum annealing method works as well as other methods in identifying the optimum layouts, while the optimum layout of an offshore wind farm composed of even more than 100 wind turbines can be identified with reasonable CPU time.

Frequency tunable CMOS ring oscillator‐based Ising machine

SummaryOscillator‐based Ising machines (OIMs) particularly those realized in complementary metal oxide semiconductor (CMOS) have gained popularity for solving combinatorial optimization problems (COPs) in recent years due to its scalability, low‐power consumption, and room temperature operation. The implemented OIMs have thus far focused on solving optimization problems with a single global minima. However, real‐life optimization problems often have multiple solutions. In this paper, we propose a generalized approach to solve COPs with single (without contention), as well as multiple (with contention) solutions using frequency tunable CMOS ring oscillator (ROSC)‐based Ising machine. A capacitive frequency tunable CMOS ring‐oscillator coupled with an internal subharmonic injection locking (SHIL) generator realized using 14‐nm FinFET models works as Ising spin in the proposed approach. We demonstrate how frequency tuning can help in attaining good quality results and also determine all possible solutions of COP with contention. We also propose a generalized algorithm for monitoring the states of the oscillator network to indicate tuning necessity and extract solutions from the oscillator's output irrespective of the type of COP.

Quantum wetting transition in the cluster Ising model

The wetting transition in the one-dimensional cluster Ising model with opposite boundary fields is studied. Tuning one boundary field while fixing another leads to the occurrence of phase transitions, where the transition points depend on the cluster coupling. Furthermore, the phase diagram is divided into three regions with different phase transition. For weak and strong cluster coupling, the phase transition is continuous and belongs to the same universality of transverse Ising model with boundary fields. For intermediate cluster coupling, the phase transition is first order. In the strong cluster coupling region, the critical region becomes exponentially small and the asymptotic behavior is absent even for lattice size up to 104. A numerical method to solve the energy gap and the correlation length is proposed on an infinite long spin chain. With this method, one can get the critical behavior as close to the critical point as possible provided that the numerical accuracy is high enough. In the light of this method, we clearly show that there is a preasymptotic regime in which the apparent critical exponents depend on the cluster coupling. Moreover, we obtain the accurate energy gap exponent zν and the correlation length exponent ν in the asymptotic critical region.

Spread and spectral complexity in quantum spin chains: from integrability to chaos

We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.

Complementarity between quantum entanglement and geometric and dynamical appearances in N spin-1/2 system under all-range Ising model

Complementarity between quantum entanglement and geometric and dynamical appearances in N spin-1/2 system under all-range Ising model

Metastable states in the Ising model with Glauber-Kawasaki competing dynamics.

Metastable states are identified in the Ising model with competition between the Glauber and Kawasaki dynamics. The model of interaction between magnetic moments was implemented on a network where the degree distribution follows a power law of the form P(k)∼k^{-α}. The evolution toward the stationary state occurred through the competition between two dynamics, driving the system out of equilibrium. In this competition, with probability q, the system was simulated in contact with a heat bath at temperature T by the Glauber dynamics, while with probability 1-q, the system experienced an external energy influx governed by the Kawasaki dynamics. The phase diagrams of T as a function of q were obtained, which are dependent on the initial state of the system, and exhibit first- and second-order phase transitions. In all diagrams, for intermediate values of T, the phenomenon of self-organization between the ordered phases was observed. In the regions of second-order phase transitions, we obtained the critical exponents of the order parameter β, susceptibility γ, and correlation length ν. Furthermore, in the regions of first-order phase transitions, we have demonstrated the instability due to transitions between ordered phases through hysteresislike curves of the order parameter, in addition to the existence of absorbing states. We also estimated the value of tricritical points when the discontinuity in the order parameter in the phase transitions was no longer observed.

Quantum neural network approach to Markovian dissipative dynamics of many-body open quantum systems.

Numerous variational methods have been proposed for solving quantum many-body systems, but they often face exponentially increasing computational complexity as the Hilbert space dimension grows. To address this, we introduce a novel approach using quantum neural networks to simulate the dissipative dynamics of many-body open quantum systems. This method combines neural-network quantum state representation with the time-dependent variational principle, both implemented via quantum algorithms. This results in accurate open quantum dynamics described by the Lindblad quantum master equation, exemplified by the spin-boson and transverse field Ising models. Our approach avoids the computational expense of classical algorithms and demonstrates the potential advantages of quantum computing for many-body simulations. To reduce measurement errors, we introduce a projection reset procedure, which could benefit other quantum simulations. In addition, our approach can be extended to simulate non-Markovian quantum dynamics.

Insight into compensation behaviors and hysteresis characteristics of an Ising-type borophene monolayer

Abstract Boron is another novel two-dimensional material after graphene with splendid chemical and physical properties and potential applications in magnetism, electronics, energy materials and other fields. Studying the magnetic properties of borophene is important to improve the relevant theories and experiments. In this paper, we extend a single spin-3/2 Ising model to analyze the magnetic characteristics of the borophene monolayer by using Monte Carlo simulation. The influence of crystal field, exchange coupling and external magnetic field on the magnetization, magnetic susceptibility and internal energy are studied separately and the phase diagrams are presented. The calculation indicates that the system presents the compensation temperature and triple-loop hysteresis behaviors.

Evidence and Structural Insights into a Ligand-Mediated Phase Transition in the Solvated Ligand Shell of Quantum Dots.

As synthesized, nanocrystal surfaces are typically covered in coordinating organic ligands, and the degree of packing and order of these ligands are ongoing questions in the field of colloidal nanocrystals, particularly in the solution state. Recently, isothermal titration calorimetry coupled with 1H NMR has been used to probe ligand exchanges on colloidal quantum dots, revealing the importance of the composition of the ligand shell on exchange thermodynamics. Previous work has shown that the geometry and length of a ligand's aliphatic chain can influence the thermodynamics of exchange. This has been attributed to interligand interactions, and the use of a modified Ising model simulation to account for these collective effects has been critical in describing these reactions. In this report, we explore the reaction between indium phosphide quantum dots and zinc chloride on a size series of nanocrystals capped with two different lengths of aliphatic, straight-chain carboxylate ligands to investigate the effect that nanocrystal size has on these interligand interactions. We demonstrate that interligand interactions increase as the nanocrystal size increases, changing the thermodynamics of the ligand exchange reaction. Critically, we show that a self-consistent model of these ligand exchanges does not fit the data without the use of a phase transition term in the model and that the strength of this phase transition depends on the nanocrystal size. Combined with solution state X-ray diffraction, these results provide indirect evidence that ligands are ordered on nanocrystals in the solution state.