Uniform Long-range Interactions Research Articles

Entanglement and quantum-classical crossover in the extended XX model with long-range interactions

In this work we considered the one-dimensional extended isotropic XY model (s=1/2) in a transverse field with uniform long-range interactions among the z components of the spins. We studied the classical critical behaviour of the model through the behaviour of the magnetization, isothermal susceptibility, internal energy and specific heat. We have obtained exact expressions for these functions and evaluated the critical exponents. The phase diagrams for the classical critical behaviour were built for three cases of the multiplicity p of the multiple spin interaction, namely p=2, p=3 and p→∞. We have also shown that the quantum phase transitions can also be characterized through two quantifiers of entanglement, namely, the concurrence and the von Neumann entropy. We have also verified through the von Neumann entropy how the central charge of the model is affected by the multiplicity p, the coupling exchange J2 and the uniform long-range interaction I.

Quantum phase transitions of the extended isotropic XY model with long-range interactions

The one-dimensional extended isotropic XY model ( s=1/2) in a transverse field with uniform long-range interactions among the z components of the spin is considered. The model is exactly solved by introducing the Gaussian and Jordan–Wigner transformations, which map it in a non-interacting fermion system. The partition function can be determined in closed form at arbitrary temperature and for arbitrary multiplicity of the multiple spin interaction. From this result, all relevant thermodynamic functions are obtained and, due to the long-range interactions, the model can present classical and quantum transitions of first and second orders. The study of its critical behavior is restricted to the quantum transitions, which are induced by the transverse field at T=0. The phase diagram is explicitly obtained for multiplicities p=2,3,4 and ∞ , as a function of the interaction parameters, and, in these cases, the critical behavior of the model is studied in detail. Explicit results are also presented for the induced magnetization and isothermal susceptibility χ T zz , and a detailed analysis is also carried out for the static longitudinal 〈 S j z S l z 〉 and transversal 〈 S j x S l x 〉 correlation functions. The different phases presented by the model can be characterized by the spatial decay of these correlations, and from these results some of these can be classified as quantum spin liquid phases. The static critical exponents and the dynamic one, z, have also been determined, and it is shown that, besides inducing first order phase transition, the long-range interaction also changes the universality class of the model.

Thermodynamic properties of the XXZ double chain with long-range interactions

Using the Jordan–Wigner transformation and cut-off approximation, the Letter studies the one-dimensional spin-1/2 XXZ double chain with z-directional uniform long-range interactions in the external magnetic field. Thermodynamic properties as the Helmholtz free energy, internal energy, specific heat and magnetization of the system are obtained. The first and second-order phase transition of the system at a finite temperature are also given. Our results agree with those of the other literatures.

Application of Mean-Field Jordan–Wigner Transformation to Antiferromagnet System

By using the mean-field Jordan–Wigner transformation analysis, this paper studies the one-dimensional spin-1/2 XYZ antiferromagnetic chain in the transverse field with uniform long-range interactions among the z-components of the spins. The thermodynamic quantities, such as Helmholtz free energy, the internal energy, the specific heat, and the isothermal susceptibility, are obtained. Under degenerating condition, our results agree with numerical results of the other literatures

A study of the spin- [formula omitted] XZ chain with long-range interactions in the external magnetic field

By using the Jordan–Wigner transformation and the integral Gaussian transformation, this paper studies the thermodynamic property of the one-dimensional spin- 1 2 XZ chain in the external magnetic field, with uniform long-range interactions among the z components of the spins. Analytical formulas and numerical solutions of the Helmholtz free energy, internal energy, specific heat, magnetization and isothermal susceptibility per site are obtained. The phase diagram of the system is also given at finite temperature. Under degenerate condition, our results agree with numerical results of the other literatures.

Quantum transitions of the isotropicXYmodel with long-range interactions on the inhomogeneous periodic chain

The isotropic $XY$ model $(s=1/2)$ in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of $N$ segments with $n$ different exchange interactions and magnetic moments, is exactly solved by introducing the integral Gaussian transformation and the generalized Jordan--Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of $n\text{th}$ order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at $T=0$, for the induced magnetization and isothermal susceptibility ${\ensuremath{\chi}}_{T}^{zz}$ as a function of the transverse field.

Thermodynamic properties of the spin-1/2 ferromagnetic Heisenberg chain with long-range interactions

The one-dimensional spin-1/2 XYZ ferromagnetic model in a transverse field and uniform long-range interactions among the z components of the spins is studied using the mean-field Jordan–Wigner transformation. The thermodynamic quantities results like Helmholtz free energy, the internal energy, the specific heat and the isothermal susceptibility are obtained both analytically and numerically. The phase transition of the system at a finite temperature is also discussed. Our results agree with numerical results of the XYZ spin chain by others.

Study of the XY chain with long-range interactions

Using the mean-field Jordan-Wigner transformation analysis, we have studied the one-dimensional spin-1/2 XY chain with Z-directional uniform long-range interactions in the external magnetic field. We have obtained such results as the Helmholtz free energy, internal energy, specific heat, magnetization and isothermal susceptibility. We also discuss the first and second-order phase transitions of the system at a finite temperature. Our results agree with results published in literature.

Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions

The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin- 1 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions. By using the emptiness formation probability P(n), the block-block entanglement Sn and the magnetization M z , the properties of the system are also studied. It is worthy of note that when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation Probability P(n) of the system decreases, but the block-block entanglement Sn increases.

Quantum criticality of a d-dimensional spin-S XXZ ferromagnetic model with longitudinal infinite-range interactions

A d-dimensional spin-S XXZ ferromagnetic model with uniform long-range interactions in the z-direction is mapped into an effective XY model in a transverse field. Then, using the two-time Green function equation of motion method, the zero-temperature properties for a periodic cubic lattice are exactly obtained close to the quantum critical point signaling the field-induced quantum phase transition from the fully polarized state into a phase with easy-plane spin ordering. We determine the relevant critical exponents which have a mean-field character for any d and S.

Block-block entanglement and quantum phase transition in the spin-12XXchain

The block-block entanglement in the ground state of spin-$\frac{1}{2}\phantom{\rule{0.3em}{0ex}}XX$ chain with three-spin and uniform long-range interactions among the $z$ components of the spins is investigated. Our results show that the block-block entanglement manifests quantum phase transitions. Besides the logarithmic scaling behavior of the block-block entanglement with the block size, that is, ${S}_{L}\ensuremath{\sim}{\mathrm{log}}_{2}(L)$, in phase I region, it is found that there is the power scaling behavior, ${S}_{L}\ensuremath{\sim}L$, in the quantum frustration regions. It is also found that the quantum frustration gives rise to an increase of block-block entanglement.

Quantum phase transition in a solvable spin model

The spin-1/2XX chain with three-spin and uniform long-range interactions among the z components of the spins is investigated. Using the Jordan-Wigner transformation, the thermodynamic quantities and spin transports of the model are evaluated exactly. The phase diagram together with the key parameters characterizing quantum phase transitions is presented.

Static and dynamic properties of the XXZ chain with long-range interactions

The one-dimensional XXZ model ( s = 1 2 ) in a transverse field, with uniform long-range interactions among the transverse components of the spins, is studied. The model is exactly solved by introducing the Jordan–Wigner transformation and the integral Gaussian transformation. The complete critical behaviour and the critical surface for the quantum and classical transitions, in the space generated by the transverse field and the interaction parameters, are presented. The crossover lines for the various classical/quantum regimes are also determined exactly. It is shown that, besides the tricritical point associated with the classical transition, there are also two quantum critical points: a bicritical point where the classical second-order critical line meets the quantum critical line, and a first-order transition point at zero field. It is also shown that the phase diagram for the first-order classical/quantum transitions presents the same structure as for the second-order classical/quantum transitions. The critical classical and quantum exponents are determined, and the internal energy, the specific heat and the isothermal susceptibility, χ T zz , are presented for the different critical regimes. The two-spin static and dynamic correlation functions, 〈 S j z S l z 〉 , are also presented, and the dynamic susceptibility, χ q zz ( ω ) , is obtained in closed form. Explicit results are presented at T = 0 , and it is shown that the isothermal susceptibility, χ T zz , is different from the static one, χ q zz ( 0 ) . Finally, it is shown that, at T = 0 , the internal energy close to the first-order quantum transition satisfies the scaling form recently proposed by Continentino and Ferreira.

The one-dimensional XXZ model with long-range interactions

The one-dimensional XXZ model ( s= 1 2 , N sites) with uniform long-range interactions among the transverse components of the spins is considered. The Hamiltonian of the model is explicitly given by H=J∑ j=1 N s j xs j+1 x+s j ys j+1 y −(I/N) ∑ j,k=1 N s j zs k z−h∑ j=1 Ns j z, where the s x, y, z are half the Pauli spin matrices. The model is exactly solved by applying the Jordan–Wigner fermionization, followed by a Gaussian transformation. In the absence of the long-range interactions ( I=0), the model, which reduces to the isotropic XY model, is known to exhibit a second-order quantum-phase transition driven by the field at zero temperature. It is shown that in the presence of the long-range interactions ( I≠0) the nature of the transition is strongly affected. For I>0, which favours the ordering of the transverse components of the spins, the transition is changed from second to first order, due to the competition between transverse and xy couplings. On the other hand, for I>0, which induces complete frustration of the spins, a second-order transition is still present, although the system is driven out of its usual universality class, and its critical exponents assume typical mean-field values.

One-dimensional Ising model with long-range and random short-range interactions

The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4 (1994) 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a `ferrimagnetic’ phase with magnetization 0< σ<1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points.

Phase diagram of random lattice gases in the annealed limit

An analysis of the random lattice gas in the annealed limit is presented. The statistical mechanics of disordered lattice systems is briefly reviewed. For the case of the lattice gas with an arbitrary uniform interaction potential and random short-range interactions the annealed limit is discussed in detail. By identifying and extracting an entropy of mixing term, a correct physical expression for the pressure is explicitly given. The one-dimensional lattice gas with uniform long-range interactions and random short-range interactions satisfying a bimodal annealed probability distribution is discussed. The model is exactly solved and is shown to present interesting behavior in the presence of competition between interactions, such as the presence of three phase transitions with different critical temperatures and the occurrence of triple and quadruple points.