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Acta Physica Polonica A

Spin-1/2 XX Chains with Three-Spin Interactions

Taras Krokhmalskii, Oleg Derzhko, Taras Verkholyak, Joachim Stolze

Published Jan 1, 2008• 3 citations
Publisher

Abstract

We consider a spin-1/2 XX chain with three-spin interactions which is equivalent to a system of noninteracting spinless fermions.We examine some dynamic quantities of the spin model.In particular,we calculate analytically the dynamic transverse (zz) structure factor which is governed by a two-fermion excitation continuum.Moreover,we compute numerically the dynamic xx structure factor which is a many-fermion dynamic quantity.We illustrate how the three-spin interactions manifest themselves in the dynamic probes.

DOI
10.12693/aphyspola.113.437

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We consider a spin-1/2 XX chain with three-spin interactions which is equivalent to a system of noninteracting spinless fermions.We examine some dynamic quantities of the spin model.In particular,we calculate analytically the dynamic transverse (zz) structure factor which is governed by a two-fermion excitation continuum.Moreover,we compute numerically the dynamic xx structure factor which is a many-fermion dynamic quantity.We illustrate how the three-spin interactions manifest themselves in the dynamic probes.
We consider the spin-12\frac{1}{2}21​ anisotropic XYXYXY chain in a transverse (z)(z)(z) field with the Dzyaloshinskii-Moriya interaction directed along the zzz axis in spin space to examine the effect of the Dzyaloshinskii-Moriya interaction on the zzzzzz, xxxxxx, and yyyyyy dynamic structure factors. Using the Jordan-Wigner fermionization approach we analytically calculate the dynamic transverse spin structure factor. It is governed by a two-fermion excitation continuum. We analyze the effect of the Dzyaloshinskii-Moriya interaction on the two-fermion excitation continuum. Other dynamic structure factors which are governed by many-fermion excitations are calculated numerically. We discuss how the Dzyaloshinskii-Moriya interaction manifests itself in the dynamic properties of the quantum spin chain at various fields and temperatures.
We consider a spin-12\frac{1}{2}21​ XYXYXY chain in a transverse (z)(z)(z) field with multisite interactions. The additional terms introduced into the Hamiltonian involve products of spin components related to three adjacent sites. A Jordan--Wigner transformation leads to a simple bilinear Fermi form for the resulting Hamiltonian and, hence, the spin model admits a rigorous analysis. We point out the close relationships between several variants of the model, which were discussed separately in previous studies. The ground-state phases (ferromagnet and two kinds of spin liquid) of the model are reflected in the dynamic structure factors of the spin chains, which are the main focus in this study. First, we consider the zzzzzz dynamic structure factor, reporting for this quantity a closed-form expression and analyzing the properties of the two-fermion (particle-hole) excitation continuum, which governs the dynamics of transverse spin component fluctuations and of some other local operator fluctuations. Then we examine the xxxxxx dynamic structure factor, which is governed by many-fermion excitations, reporting both analytical and numerical results. We discuss some easily recognized features of the dynamic structure factors, which are signatures of the presence of the three-site interactions.
One-dimensional quantum spin-1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors).This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions.A number of dynamic quantities (e.g.related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.We consider the spin-1/2 XY chain in a transverse ( z) magnetic field with the Hamiltonianand calculate the dynamic structure factorsis the dimer operator.The results for the dynamic transverse structure factor S zz (κ, ω) and for the dynamic dimer structure factor S DD (κ, ω) are known, whereas the analysis of the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been reported so far.We compare different two-fermion dynamic quantities contrasting their generic and specific features.
We have considered a numerical scheme for the calculation of the equilibrium properties of spin-1/2 XY chains. Within its frames it is necessary to solve in the last resort only the 2N\times 2N eigenvalue and eigenvector problem but not the 2^N\times 2^N one as for an arbitrary system consisting of N spins 1/2. To illustrate the approach we have presented some new results. Namely, the xx dynamic structure factor for the Ising model in transverse field, the density of states for the isotropic chain with random intersite couplings and transverse fields that linearly depend on the surrounding couplings, and the zz dynamic structure factor for the Ising model in random transverse field. The results obtained are hopped to be useful for an interpretation of observable data for one-dimensional spin-1/2 XY substances.
We have considered a numerical scheme for the calculation of the equilibrium properties of spin{ 1 2 XY chains.Within its frames it is necessary to solve in the last resort only the 2N 2N eigenvalue and eigenvector problem but not the 2 N 2 N one as for an arbitrary system consisting of N spins 1 2 .To illustrate the approach we have presented some new results.Namely, the xx dynamic structure factor for the Ising model in transverse eld, the density of states for the isotropic chain with random intersite couplings and transverse elds that linearly depend on the surrounding couplings, and the zz dynamic structure factor for the Ising model in the random transverse eld.The results obtained are hoped to be useful for an interpretation of observable data for one{dimensional spin{ 1 2 XY substances.