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.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
The static and dynamic properties of the anisotropic
XY model
(s=1∕2) on the inhomogeneous periodic chain, composed of
N cells with
n different exchange interactions and magnetic moments, in a transverse field
h, are determined exactly at arbitrary temperatures. The properties are obtained by introducing the Jordan-Wigner fermionization and by reducing the problem to a diagonalization of a finite matrix of
nth order. The quantum transitions are determined exactly by analyzing, as a function of the field, the induced magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}{\ensuremath{\mu}}_{m}⟨{S}_{j,m}^{z}⟩$ (
j denotes the cell,
m the site within the cell, ${\ensuremath{\mu}}_{m}$ the magnetic moment at site
m within the cell) and the spontaneous magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}⟨{S}_{j,{m}^{\ensuremath{'}}}^{x}⟩$ which is obtained from the correlations
⟨Sj,mxSj+r,mx⟩ for large spin separations. These results, which are obtained for infinite chains, correspond to an extension of the ones obtained by Tong and Zhong [Physica B 304, 91 (2001)]. The dynamic correlations, $⟨{S}_{j,m}^{z}(t){S}_{{j}^{\ensuremath{'}},{m}^{\ensuremath{'}}}^{z}(0)⟩$, and the dynamic susceptibility, ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$, are also obtained at arbitrary temperatures. Explicit results are presented in the limit
T=0, where the critical behavior occurs, for the static susceptibility ${\ensuremath{\chi}}_{q}^{zz}(0)$ as a function of the transverse field
h, and for the frequency dependency of dynamic susceptibility ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$.
We introduce a quantum antiferromagnet model, having exactly soluble thermodynamic properties. It is an infinite range antiferromagnetic Ising model put in a transverse field. The free energy gives the ground state energy in the zero temperature limit and it also gives the low temperature behaviour of the specific heat, the exponential variation of which gives the precise gap magnitude in the excitation spectrum of the system. The detailed behaviour of the (random sublattice) staggard magnetisation and susceptibilities are obtained and studied near the Néel temperature and the zero temperature quantum critical point.
Thermodynamic properties of one-dimensional Ising chains with random nearest neighbor exchange interaction J are studied exactly in the case of distributions of the type P(J) = A|J|ν for Jmin ≤J ≤Jmax and P(J) = 0, otherwise. The results are compared with those for a periodic chain. We find that the specific heat C ∼Tν+1 for Jmin = 0 and kT ≪Jmax. The dependence of spin-spin correlation function on distance and temperature is discussed. The absence of the peak in the zero field χ⊥ as a function of temperature when ν = 0 is noted in contrast to the case of the periodic chain. The effect of randomness on the low temperature behaviour of these thermodynamic quantities is different. For instance whereas C becomes analytic at T = 0, χ|| either remains singular (ferromagnetic) or becomes analytic (antiferromagnetic). The effect of random xy interaction (∑iJi⊥(SixSi + 1x + SiySi + 1y)) on the high temperature behaviour of χ|| and χ⊥ is discussed.