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arXiv:cond-mat/0408033 · Physical Review E

Regularly alternating spin- 12 anisotropic XY chains: The ground-state and thermodynamic properties

Oleg Derzhko, Johannes Richter, Taras Krokhmalskii, O. Zaburannyi

Published Jun 4, 2004• 19 citations
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Abstract

Using the Jordan-Wigner transformation and continued fractions we calculate rigorously the thermodynamic quantities for the spin-1/2 transverse Ising chain with periodically varying intersite interactions and/or on-site fields. We consider in detail the properties of the chains having a period of the transverse field modulation equal to 3. The regularly alternating transverse Ising chain exhibits several quantum phase transition points, where the number of transition points for a given period of alternation strongly depends on the specific set of the Hamiltonian parameters. The critical behavior in most cases is the same as for the uniform chain. However, for certain sets of the Hamiltonian parameters the critical behavior may be changed and weak singularities in the ground-state quantities appear. Due to the regular alternation of the Hamiltonian parameters the transverse Ising chain may exhibit plateaulike steps in the zero-temperature dependence of the transverse magnetization vs transverse field and many-peak temperature profiles of the specific heat. We compare the ground-state properties of regularly alternating transverse Ising and transverse XX chains and of regularly alternating quantum and classical chains. Making use of the corresponding unitary transformations we extend the elaborated approach to the study of thermodynamics of regularly alternating spin-1/2 anisotropic XY chains without field. We use the exact expression for the ground-state energy of such a chain of period 2 to discuss how the exchange interaction anisotropy destroys the spin-Peierls dimerized phase.

DOI
10.1103/physreve.69.066112

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Using the Jordan-Wigner transformation and continued fractions we calculate rigorously the thermodynamic quantities for the spin-1/2 transverse Ising chain with periodically varying intersite interactions and/or on-site fields. We consider in detail the properties of the chains having a period of the transverse field modulation equal to 3. The regularly alternating transverse Ising chain exhibits several quantum phase transition points, where the number of transition points for a given period of alternation strongly depends on the specific set of the Hamiltonian parameters. The critical behavior in most cases is the same as for the uniform chain. However, for certain sets of the Hamiltonian parameters the critical behavior may be changed and weak singularities in the ground-state quantities appear. Due to the regular alternation of the Hamiltonian parameters the transverse Ising chain may exhibit plateaulike steps in the zero-temperature dependence of the transverse magnetization vs transverse field and many-peak temperature profiles of the specific heat. We compare the ground-state properties of regularly alternating transverse Ising and transverse XX chains and of regularly alternating quantum and classical chains. Making use of the corresponding unitary transformations we extend the elaborated approach to the study of thermodynamics of regularly alternating spin-1/2 anisotropic XY chains without field. We use the exact expression for the ground-state energy of such a chain of period 2 to discuss how the exchange interaction anisotropy destroys the spin-Peierls dimerized phase.
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.
Using continued fractions we study the ground state properties of the spin-½ Ising chain in a transverse field with periodically varying interaction strengths and external fields. We consider in detail the chain having a period-2 modulation of the interaction strengths and compare the results obtained with those corresponding to the spin-½ isotropic XY chain in a transverse field. In contrast to the behaviour of the transverse XY chain, the transverse Ising chain does not exhibit a step-like magnetization versus field dependence caused by the alternation of bonds. Its susceptibility exhibits a logarithmic singularity at a field determined by interaction strengths, and it is stable with respect to spin-Peierls dimerization.
We examine the ground state properties of the s=12s=\frac{1}{2}s=21​ transverse Ising chain with regularly alternating bonds and fields, using exact analytical results and exact numerical data for long (up to N=900)N=900)N=900) and short (N=20)(N=20)(N=20) chains. For a given period of alternation the system may exhibit a series of quantum phase transitions, where the number of transitions depends on the concrete set of the parameters of the Hamiltonian. The critical behavior for the nonuniform and uniform chains is the same.
We consider the spin-12\frac{1}{2}21​ isotropic XY chain in an external magnetic field directed along the zzz axis with periodically varying ggg factors. To reveal the effects of regularly alternating ggg factors, we calculate various static and dynamic equilibrium quantities in the ground state and at finite temperatures. We demonstrate that because of the regularly alternating ggg factors the saturation field may disappear and the field dependence of the susceptibility in the ground state has additional logarithmic singularity at zero field. Moreover, the zero-field susceptibility has a logarithmic singularity as $T\ensuremath{\rightarrow}0$. Furthermore, the dynamic structure factors exhibit much more structure in the ``wave vector--frequency'' plane that can be traced out to modifications of the two-fermion excitation continua which exclusively determine ${S}_{zz}(\ensuremath{\kappa},\ensuremath{\omega})$ and dominate the properties of ${S}_{xx}(\ensuremath{\kappa},\ensuremath{\omega})$. We discuss what changes can be observed in dynamic experiments on the corresponding substances.
We study a finite spin-1/2 Ising chain with a spatially alternating transverse field of period 2. By means of a Jordan-Wigner transformation for even and odd sites, we are able to map it into a one-dimensional model of free fermions. We determine the ground-state energies in the positive- and negative-parity subspaces (subspaces with an even or odd total number of down spins, respectively) and compare them in order to establish the ground-state energy for the entire Hamiltonian. We derive closed-form expressions for this energy gap between the different parity subspaces and analyze its behavior and dependence on the system size in the various regimes of the applied field. Finally, we suggest an expression for the correlation length of such a model that is consistent with the various values found in the literature for its behavior in the vicinity of critical points.
We consider the quantum XY spin chain with quasi-periodic two-valued exchange couplings and a uniform transverse field. The equivalence of the system with a free fermionic model permits a detailed analysis of its thermodynamics. The zero-temperature magnetization is a Cantor function of the applied field. The zero-field specific heat and susceptibility have a power-law behaviour at low temperature, modulated by periodic amplitudes. The critical exponents interpolate continuously between those of the periodic and of the random model.