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We propose the Hirota bilinearization of the Fokas–Lenells derivative nonlinear Schrödinger equation (FLE) with a non-vanishing background. In the proposed method, we have introduced an auxiliary function to transform the equation into bilinear form. The use of an auxiliary function makes the method simpler than the ones reported earlier. Using the proposed method we have obtained the dark (and bright) one soliton solution. We have also discussed the properties of the obtained soliton and mentioned the criteria upon which the nature of the soliton being dark or bright depends upon. Then we have obtained the dark (and bright) two soliton solution and discussed the respective properties and also through asymptotic analysis showed how the phase between the two individual solitons changes before and after interaction. Eventually we have proposed the scheme for obtaining N soliton solutions. The proposed method can be extended to other nonlinear equations where straightforward bilinearization is not feasible. Later, we have introduced a gauge transformation which transforms the spectral problem of FLE into a spectral problem for the Landau–Lifshitz (LL) spin system. Soliton act as an information carrier and LL system exhibits a variety of nonlinear structures so the study is worth doing. [ABSTRACT FROM AUTHOR]

In this work, a simple modification of the Monte Carlo algorithm is introduced, which is called step Monte Carlo (sMC). The sMC approach allows to simulate processes far from equilibrium and obtain information about the dynamic properties of the system under investigation. In the approach proposed, here the probability of accepting the final (trial) state depends on the activation energy, not on the relative energy between the final and initial state. This barrier height is probed on an ongoing basis, by generating intermediate states along the path connecting the initial and trial positions. Importantly, to calculate the activation energy, the model only requires knowledge of the Hamiltonian without having to introduce additional input parameters such as transition rates etc. The details of sMC are explained in the case of a simple spin model. The comparison of its results with the ones obtained within the frame of stochastic Landau‐Lifshitz‐Gilbert indicates the correctness of sMC. In this study's opinion, the proposed method can be applied to simulate other processes, for example, dynamics of classical atoms and complex fluids, diffusion, nucleation, surface adsorption, and crystal growth processes. [ABSTRACT FROM AUTHOR]

We generalize the theory of extending breather solutions in the anticontinuous limit to the case of discrete spin systems. We formulate necessary conditions and determine the upper bound for the intersite coupling constant for which the extension procedure is possible. Using a numerical algorithm, we obtain breather modes of a discrete spin chain related to single-site and two-site excitations of the anticontinuous limit and show their linear stability. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the energy landscape of the two-component spin systems, known as the Curie–Weiss–Potts model, which is a generalization of the Curie–Weiss model consisting of q ≥ 3 spins. In the energy landscape of a multi-component model, the most important element is the relative strength between the inter-component interaction strength and the component-wise interaction strength. If the inter-component interaction is stronger than the component-wise interaction, we can expect all the components to be synchronized in the course of metastable transition. However, if the inter-component interaction is relatively weaker, then the components will be desynchronized in the course of metastable transition. For the two-component Curie–Weiss model, the phase transition from synchronization to desynchronization has been precisely characterized in studies owing to its mean-field nature. The purpose of this paper is to extend this result to the Curie–Weiss–Potts model with three spins. We observe that the nature of the phase transition for the three-spin case is entirely different from the two-spin case of the Curie–Weiss model, and the proof as well as the resulting phase diagram is fundamentally different and exceedingly complicated. [ABSTRACT FROM AUTHOR]

In this study, we investigate the metastable behavior of Metropolis-type Glauber dynamics associated with the Blume–Capel model with zero chemical potential and zero external field at very low temperatures. The corresponding analyses for the same model with zero chemical potential and positive small external field were performed in Cirillo and Nardi (J Stat Phys 150:1080–1114, 2013) and Landim and Lemire (J Stat Phys 164:346–376, 2016). We obtain both large deviation-type and potential-theoretic results on the metastable behavior in our setting. To this end, we perform highly thorough investigation on the energy landscape, where it is revealed that no critical configurations exist and alternatively a massive flat plateau of saddle configurations resides therein. [ABSTRACT FROM AUTHOR]

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from spin systems , or spin unitaries , which are finite sets S of selfadjoint unitary matrices such that any two unitaries in S anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all m -tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C ⁎ -envelope of any operator system generated by a spin system of cardinality 2 k or 2 k + 1 is the simple matrix algebra M 2 k (C). As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [13] that the spin ball B m spin and max ball B m max coincide as matrix convex sets in the cases m = 1 , 2. We also derive analogous results for countable spin systems and their C ⁎ -envelopes. [ABSTRACT FROM AUTHOR]

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk. [ABSTRACT FROM AUTHOR]

In mathematics an inverse scattering transformation is a method for solving some nonlinear equations with partial derivatives. Discovering of the method became one of the crucial events in mathematical physics in the last 40 years [1]–[6]. The method presents a nonlinear analogue, in a sense generalized Fourier transformation, which is applied to solve a lot of linear equations with particular derivatives. Title "inverse scattering problem"is originated from key idea of recovery time evolution of the potential from time evolution its scattering data: inverse scattering is related to the problem about recovery of the potential from its scattering matrix, in difference from direct scattering the problem of finding a scattering matrix of potential. The inverse scattering transformation can be applied to many co-called exactly decided models, i.e. completely integrable infinite systems. Firstly it was presented by Clifford S. Gardner, John M. Greene and Martin D. Kruskal and others (1967, 1974) for Korteweg de Vries equation, and soon spread to nonlinear Scrodinger equation, sine-Gordon equation and Toda chain equation. Later the method was used to solve many another equations, such that Kadomtsev-Petviashvili equation, Ishimori equation, Dime equation etc. Characteristic property of solutions obtained by inverse scattering method is existence of solitons, solutions reminding as particles as waves, which do not have any analogues for linear equations with particular derivatives and are applied to nonlinear optics and plasma physics, and its quantum version describes many-particle system with б-shaped interaction.

We study the evolution of a spin- s system described by the long-range z z -type Ising interaction. The Fubini–Study metric of the quantum state manifold defined by this evolution is obtained. We explore the topology of this manifold and show that it corresponds to a sphere. Exploration of the Riemannian curvature allows us to determine the manifold geometry. Also we calculate the speed of evolution of the system and represent the curvature by means of this speed. This is important for an experimental measurement of the curvature. The conditions for achieving the minimal and maximal values of the speed of evolution are obtained. Also we examine the geometry of state manifold and speed of evolution of spin system in the thermodynamic limit. We propose the physical system of methane molecule for application of our considerations. Finally, we study the influence of an external magnetic field on the metric of state manifold and on the speed of evolution. In this case we obtain the conditions for achieving the minimal possible speed of evolution. For some predefined initial states the orientations of magnetic fields to reach the minimal and maximal values of the speed are found. [ABSTRACT FROM AUTHOR]

We prove exponential convergence to equilibrium for the Fredrickson-Andersen one-spin facilitated model on bounded degree graphs satisfying a subexponential, but larger than polynomial, growth condition. This was a classical conjecture related to non-attractive spin systems. Our proof relies on coupling techniques based on Harris graphical construction for interacting particle systems. [ABSTRACT FROM AUTHOR]

The two-time correlation function for probe spin interacting with spin system (bath) is studied. We show that zeros of this function correspond to zeros of partition function of spin system in complex magnetic field. The obtained relation gives new possibility to observe the Lee–Yang zeros experimentally. Namely, we show that measuring of the time dependence of correlation function allows direct experimental observation of the Lee–Yang zeros. [ABSTRACT FROM AUTHOR]

The superheating effect, in which a solid is heated well above its melting temperature and remains in an ordered, metastable phase, is a well-known phenomenon in materials science. Superheating can be observed experimentally under carefully controlled conditions, and more routinely in atomistic computer simulations. In the context of simulations of superheating, the so-called Z-method is a recently developed technique which allows a precise characterization of the metastable solid state and the melting temperature. However, metastable states are also present in other first-order phase transitions such as the order–disorder transition in spin systems. In spite of all the available work on the behavior of superheated solids, there have been few attempts to characterize the metastable ordered phase in contexts other than melting. In this work we study the ordered metastable state in a two-dimensional Potts lattice in the microcanonical ensemble under the conditions for first-order transitions. We connect our results with the melting transition in solids, and in particular with the empirical findings of the Z-method. In contrast with the sharp, discontinuous slope of the isochoric curve observed in the melting of solids, an S-shape curve is observed for T ( E ) in the case of spins. We discuss the implications of our results to extend the Z-method and its applicability for general first-order phase transitions. [ABSTRACT FROM AUTHOR]

We present a protocol for accelerating adiabatic quantum state transfer through a spin chain by adding an effective control pulse. Using Feshbach P-Q partitioning technique, we obtain the one-component dynamical equation which guides us to set the amplitude and period of the control field. This field can be a sequence of alternatively negative and positive (zero-energy change) pulses which can be realized easily in experiment. As an example, we discuss a three particles spin chain and the numerical calculation shows that the accelerated quantum state transfer can be obtained. [ABSTRACT FROM AUTHOR]

We present a geometric description of the collective dynamics in the system of spins with the Hamiltonian consisting of exchange interactions and an external magnetic field. This Hamiltonian yields the classical Landau—Lifshitz equation for each individual spin. However, magnetic fields of all spins evolve in a complicated way, due to interactions between particles. Passing to stereographic coordinates, evolution of spins and their magnetic fields can be represented by actions of one-parameter families of Möbius transformations. On the other side, the corresponding evolution on the group of Möbius transformations is determined by the configuration of all local fields in the system. On the whole, local magnetic fields are identified with Möbius transformations acting on them. [ABSTRACT FROM AUTHOR]

We prove a complexity dichotomy theorem for the six-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit. [ABSTRACT FROM AUTHOR]

In this paper, we study a spin system gauge equivalent to the integrable Fokas–Lenells equation, the so-called Myrzakulov-LXXII equation. Moreover, the (2+1)-dimensional Myrzakulov-LXXII equation is obtained and a new Lax representation for it is constructed. A connection between the differential geometry of the curves and the spin system under study is established. Also, a one-soliton solution of the Myrzakulov-LXXII equation is found. [ABSTRACT FROM AUTHOR]