One variation on Lloyd's theme

One random spin-1/2 XY chain that after Jordan-Wigner fermionization reduces to the extended Lloyd's model is considered. The random-averaged one-fermion Green functions have been calculated exactly that yields thermodynamics of the spin model.Comment: 15 pages, Latex, 5 figures (in TeX format

Quantum spin chains with regularly alternating bonds and fields

We consider the spin-1/2 XY chain in a transverse field with regularly varying exchange interactions and on-site fields. In two limiting cases of the isotropic XX and extremely anisotropic (Ising) exchange interaction the thermodynamic quantities are calculated rigorously with the help of continued fractions. We discuss peculiarities of the low-temperature magnetic properties and a possibility of the spin-Peierls instability.Comment: Presented at 11-th Czech and Slovak Conference on Magnetism, Ko\v{s}ice, 20-23 August 200

Quantum phase transitions in alternating transverse Ising chains

This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.Comment: 37 pages, 7 figures, talk at the "Ising lectures" (ICMP, L'viv, March 2002

Jordan-Wigner Fermionization and the Theory of Low-Dimensional Quantum Spin Models. Dynamic Properties

The Jordan-Wigner transformation is known as a powerful tool in condensed matter theory, especially in the theory of low-dimensional quantum spin systems. The aim of this chapter is to review the application of the Jordan-Wigner fermionization technique for calculating dynamic quantities of low-dimensional quantum spin models. After a brief introduction of the Jordan-Wigner transformation for one-dimensional spin one-half systems and some of its extensions for higher dimensions and higher spin values we focus on the dynamic properties of several low-dimensional quantum spin models. We start from the famous s=1/2 XX chain. As a first step we recall well-known results for dynamics of the z-spin-component fluctuation operator and then turn to the dynamics of the dimer and trimer fluctuation operators. The dynamics of the trimer fluctuations involves both the two-fermion (one particle and one hole) and the four-fermion (two particles and two holes) excitations. We discuss some properties of the two-fermion and four-fermion excitation continua. The four-fermion dynamic quantities are of intermediate complexity between simple two-fermion (like the zz dynamic structure factor) and enormously complex multi-fermion (like the xx or xy dynamic structure factors) dynamic quantities. Further we discuss the effects of dimerization, anisotropy of XY interaction, and additional Dzyaloshinskii-Moriya interaction on various dynamic quantities. Finally we consider the dynamic transverse spin structure factor $S_{zz}({\bf{k}},\omega)$ for the s=1/2 XX model on a spatially anisotropic square lattice which allows one to trace a one-to-two-dimensional crossover in dynamic quantities.Comment: 53 pages, 22 figure

There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains

We review some recent results on statistical mechanics of the one-dimensional spin-1/2 XY systems paying special attention to the dynamic and thermodynamic properties of the models with Dzyaloshinskii-Moriya interaction, correlated disorder, and regularly alternating Hamiltonian parameters.Comment: 21 pages, 4 figure