The three-dimensional Ising model on the m$\times$n$\times$l cubic lattice with the screw boundary condition along the ${\bf X}$ direction and the periodic boundary conditions along both ${\bf Y}$ and ${\bf Z}$ directions is exactly solved by using the $2^{mn}$-dimensional representation of the rotation group in $2mn$-dimensions, similar to the Kaufman's spinor approach in two dimensions. The exact partition function is obtained from two sets of $2^{mn-1}$ eigenvalues of the $2^{mn}$-dimensional transfer matrix {\bf V} corresponding to the even and odd eigenvectors, respectively. Such the eigenvalues are determined by the angles of the 2mn-dimensional rotation associated with {\bf V}.

The Ising model, a cornerstone in the understanding of critical points, phase transitions, and magnetic systems, has been pivotal in advancing our knowledge of statistical physics. Although analytical solutions exist for the 1D and 2D Ising models, complexities rise significantly with the inclusion of external magnetic fields and in higher dimensions. Here, we demonstrate a novel quantum computational approach to simulate the 1D Ising model with basic quantum gates in the QISKIT(Python) that is both accessible to beginners and executable on personal computers, bypassing the need for advanced computational infrastructure. By initializing the system in a comprehensive superposition state, we effectively utilize quantum gates to replicate the intricate interactions between spins and external fields. This method offers a more intuitive and direct observational means of the system's evolution, distinguishing it from previous methodologies. Our results not only repeat all the physics behavior in the 1D Ising model but also showcase the expanding capabilities of quantum computing to tackle complex physical systems, promising advancements in both theoretical and applied physics.

In the framework of mean field approximation, we consider a spin system consisting of two interacting sub-ensembles: spins interactions with in each sub-ensemble are ferromagnetic, while the inter-ensemble interactions are antiferromagnetic. We define the effective number of the nearest neighbors, and show that if the two sub-ensembles have the same effective number of the nearest neighbors, the classical form of critical exponents (, , , ) gives way to the non-classical form (, , , ), and the scaling function changes simultaneously. We demonstrate that this system allows for two second-order phase transitions and two first-order phase transitions. We observe that an external magnetic field does not destroy the phase transitions, but only shifts their critical points, allowing for control of the system’s parameters. We discuss the regime when the magnetization as a function of the magnetic field develops a low-magnetization plateau, and show that the height of this plateau abruptly rises to the value of one when the magnetic field reaches a critical value. Our analytical results are supported by a Monte Carlo simulation of a three-dimensional layered model.

In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-(s,(2t−1)/2) with nearest neighbors and the external magnetic field. We establish the partition function of the model by means of the transfer matrix. Under a null boundary condition, we compute certain thermodynamic quantities for the 1D-MSIM. We find some precise formulas to determine the model’s free energy, entropy, magnetization, and susceptibility. By examining the model’s associated iterative equations, we use the cavity approach to investigate the phase transition problem. We numerically determine the model’s periodicity.

Social insects, such as honey bees exhibit complex behavioral patterns and their inconspicuous coordination enables decision-making on the colony level. It is thus proposed, that a high-level description of their collective behavior might share commonalities with neural processes in the brains. At the same time, recent research concerning overarching features of neural activity implies that brains are poised at the edge of the critical phase transition and that such a state is enabling maximal computational power and adaptability. In our research, we applied some tools developed in the computational neuroscience field to the dataset of bee trajectories recorded within the hive, during the course of many days. Our results imply that certain characteristics of the system are remarkably similar to the Ising model when it operates at critical temperature and also shares some of the features with the human brain at the resting state

The exact dynamics of emergence remains one of the most prominent outstanding questions for the field of complexity science. I first discuss various perspectives on emergence in various contexts, then offer a different perspective on understanding emergence in a graph-theoretic representation. From the discussion, an observer’s choice in state space seems to have an effect for that observer to detect emergent behavior. To test these ideas, I analyze the dynamics of all possible spatial state spaces near the critical temperature in an Ising model. As a result, state space topologies that appear more deterministic flip more bits than topologies that appear more random, which is contrary to our intuitions about randomness. In addition, the size of different state spaces constrain a system’s ability to explore various states within the same time frame. These results are important to understanding emergent phenomena in biological systems, which are layered with various state spaces and observational perspectives.

Phase transitions- both in the classical and in the quantum version- are the perfect playground for appreciating universality at work. Indeed, the fine details become unimportant and a classification in very few universality classes is possible. Very recently, a striking deviation from this picture has been discovered: some antiferromagnetic spin chains with competing interactions show a different set of phase transitions depending on the parity of number of spins in the chain. The aim of this article is to demonstrate that the same behavior also characterizes the most simple quantum spin chain: the Ising model in a transverse field. By means of an exact solution based on a Wigner-Jordan transformation, we show that a first order quantum phase transition appears at zero applied field in the odd spin case, while it is not present in the even case. A hint of a possible physical interpretation is given by the combination of two fact: at the point of the phase transition, the degeneracy of the ground state in the even and the odd case substantially differ, being respectively 2 and 2N, with N the number of spins; the spin of the most favorable kink states changes at that point.

The dynamics of many complex systems can be classified as ordered, chaotic, or critical. Order offers stability and robustness, while chaos allows for change and adaptability. Criticality, then, is often seen as a balance required by living systems at different scales. In classical models, however, criticality is only found near phase transitions, restricting the parameter space (and thus the likelihood) of critical dynamics, as most parameters yield ``undesirable'' solutions. Here we show that this limitation is due to the homogeneity built-in these models, i.e., all elements sharing parameter values. By exploring heterogeneous versions of archetypal models in physics and computer science, we observe critical dynamics in a broader range of parameters, and thus could be more common than previously thought.

Soft-matter systems when driven out-of-equilibrium often give rise to structures that usually lie in-between the macroscopic scale of the material and microscopic scale of its constituents. In this paper we review three such systems, the two-dimensional square-lattice Ising model, the Kuramoto model and the Rayeligh-Bénard convection system which when driven out-of-equilibrium give rise to emergent spatio-temporal order through self-organization. A common feature of these systems is that the entities that self-assemble are coupled to one another in some way, either through local interactions or through a continuous media. Therefore, the general nature of non-equilibrium fluctuations of the intrinsic variables in these systems are found to follow similar trends as order emerges. Through this paper, we attempt to look for connections between among these systems and systems in general which give rise to emergent order when driven out-of-equilibrium.

The out-of-equilibrium dynamics of many body systems has recently received a burst of interest, also due to experimental implementations. The dynamics of both observables, such as magnetization and susceptibilities, and quantum information related quantities, such as concurrence and entanglement entropy, have been investigated under different protocols bringing the system out of equilibrium. In this paper we focus on the entanglement entropy dynamics under a sinusoidal drive of the tranverse magnetic field in the 1D quantum Ising model. We find that the area and the volume law of the entanglement entropy coexist under periodic drive for an initial non-critical ground state. Furthermore, starting from a critical ground state, the entanglement entropy exhibits finite size scaling even under such a periodic drive. This critical-like behaviour of the out-of-equilibrium driven state can persist for arbitrarily long time, provided that the entanglement entropy is evaluated on increasingly subsytem sizes, whereas for smaller sizes a volume law holds. Finally, we give an interpretation of the simultaneous occurrence of critical and non-critical behaviour in terms of the propagation of Floquet quasi-particles.

^{We consider the collective pricing orders in a minimum supply chain that is composed of a monopolistic manufacturer and many retailers that belong to the same chain store firm. The retailers have the freedom to raise or lower the local price. The chain store firm sets up the commercial rules for local retail stores to maximize its total payoff. The monopolistic manufacturer firm controls the total quantity supplied for the market to achieve maximum benefits. We applied the two dimensional Ising model in statistical physics to map the collective distribution of microscopic strategy of local retailers into the macroscopic total payoff of the chain store firm. The local stores choose to raise the price or lower the price based their own mind when the supply in market surpasses the demand. When the supply in market is far less than the demand, the stores synchronously raise prices, even though a local store only have the incomplete information of their nearest neighboring supermarket. We find the critical equation for the balance point between the action of supplier and the action of chain store management based on game theory and statistical physics. The critical equation can identify the Nash equilibrium point of the non-cooperative game between the manufacturer and the chain-store seller, and reveal different levels of collective operations. This statistical physics method also holds for more complicate supply chains and economic systems.}

The total enthalpies of the 16 different spin configurations that can be realized in the unit cell of the archetype spin crossover complex [Fe(phen)₂(NCS)₂] (phen=1,2-phenanthroline) were calculated applying periodic density functional theory combined with the Hubbard model and the Grimme-D2 dispersion correction (DFT+U/D2). The obtained enthalpy differences between the individual spin configurations were used to determine spin couplings of an Ising-like model and subsequent Monte Carlo simulations for this model allowed to estimate the phenomenological interaction parameter Г of the Slichter-Drickamer model, which is commonly used to describe the cooperativity of the spin transition. The calculational procedure described here, which led to an estimate of about 3 kJ/mol⁻¹ for Г, in good agreement with experiment, may be used to predict from first principles how modifications of spin crossover complexes can change the character of the spin transition from gradual to abrupt and vice versa.

Here we investigate how the local properties of particles in a thermal bath may influence the thermodynamics of the bath, and consequently alter the statistical mechanics of subsystems that comprise the bath. We are guided by the theory of small-system thermodynamics, which is based on two primary postulates: that small systems can be treated self-consistently by coupling them to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt this “nanothermodynamics” to investigate how a large system may subdivide into an ensemble of smaller subsystems, causing internal heterogeneity across multiple size scales. For the semi-classical ideal gas, maximum entropy favors subdividing a large system of “atoms” into an ensemble of “regions” of variable size. The mechanism of region formation could come from quantum exchange symmetry that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by their distinct locations. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by a theorem in quantum mechanics for sub-additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for atoms that may be meters apart. Other models presented here are based on Ising-like spins, which are solved analytically in one dimension. Focusing on the bonds between the Ising-like spins we find similarity in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for fully-accurate predictions. Another result using Ising-like spins involves simulations that include a nonlinear correction to Boltzmann’s factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin exchange on intermediate lengths. These simulations exhibit 1/f-like noise at low frequencies (f), and white noise at higher f, similar to equilibrium thermal fluctuations found in many materials.

The classical Rotational Isomeric State (RIS) model, originally proposed by Flory, has been used to rationalize a wide range of physicochemical properties of neutral polymers. However, many weak polyelectrolytes of interest are able to regulate their charge depending on the conformational state of the bonds. Recently, it has been shown that the RIS model can be coupled with the Site Binding (SB) model, for which the ionizable sites can adopt two states: protonated or deprotonated. The resulting combined scheme, the SBRIS model, allows to analyse ionization and conformational equilibria on the same foot. In the present work this approach is extended to include pH-dependent electrostatic Long Range (LR) interactions, ubiquitous in weak polyelectrolytes at moderate and low ionic strengths. With this aim the original LR interactions are taken into account by defining effective Short Range (SR) and pH-dependent parameters, such as effective microscopic protonation constants and rotational bond energies. The new parameters are systematically calculated using variational methods. The machinery of statistical mechanics for SR interactions, including the powerful and fast transfer matrix methods, can then be applied. The resulting technique, to which we will refer as Local Effective Interaction Parameters (LEIP) method, is illustrated with a minimal model of a flexible linear polyelectrolyte containing only one type of rotating bonds. LEIP reproduces very well the pH dependence of the degree of protonation and bond probabilities obtained by semi-grand canonical Monte Carlo simulations, where LR interactions are taken explicitly into account. The reduction in the computational time in several orders of magnitude suggests that the LEIP technique could be useful in a range of areas involving linear weak polyelectrolytes, allowing direct fitting of the relevant physical parameters to the experimental quantities.

Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights in quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows to extract high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo which enables calculations on large finite systems. Finite-size scaling can then be used to determine physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.