Low temperature thermodynamics of finite spin-1/2 XX Chain with some distorted links

We present the results of our analytical and numerical study of quantum stationary states and the thermodynamics of exactly solvable finite linear spin-1/2 XX chain with one or two distorted links. The exact dispersion equations for the stationary states with one inverted spin for both cases have been derived. The corresponding spectrum consists of one quasi-continuous band and some localized impurity levels. The analytical inequalities for the values of the critical model parameters describing the appearance of local energy impurity levels above and below quasi-continuous band are obtained and analyzed. The field and the temperature dependences of the main thermodynamic characteristics of these models are studied. It is shown, that the appearance of the localized levels near the distorted links may effect significantly on the thermodynamic properties at low temperatures, leading to the additional singularities in the field and the temperature dependences of the principal thermodynamic characteristics. For example, the field dependence of z-projection of average total spin and the field dependence of the magnetization at zero temperature have the finite jumps associated with the quasi-continuous spectrum, and the impurity levels. The remnants of these jumps at rather low temperature are clearly visible. The average z-projections of the spins of distorted link(s) may decrease with the increasing of the magnetic field for some values of model parameters. The temperature dependence of the specific heat may demonstrate additional maxima at very low temperatures. The complex oscillatory pattern of the time dependence of the dynamical longitudinal pair correlation function and the autocorrelation function is associated with the finiteness of the and the appearance of


Introduction
Last decades the discovery of a big number of new compounds with quasi-one-dimensional and quasi-twodimensional magnetic structures has given further impetus to the theoretical studies of low-dimensional systems [1][2][3]. In this context the exactly solvable models like spin-1/2 XX chain attract much attention of theoreticians [1][2][3][4][5][6][7][8][9][10][11]. For many years, above chain model was one of the theoretical instruments for checking the numerical results for more general spin systems, and also applied for adequate simulation of the magnetic properties of real compounds [12,13]. Any real magnetic material has different types of structure defects. The theoretical study of the influence of these defects on the energy spectrum and the thermodynamics of spin chain structures is of interest.

The Hamiltonians
This work is devoted to the theoretical treatment of the low temperature thermodynamics of some exactly solvable quantum models based on spin-1/2 XX-chain. We study analytically and numerically the finite spin-1/2 XX chains with defects of different nature: finite linear XX-chain with one and two distorted XX bond.
The corresponding Hamiltonians for these models have the following form 1  N n n n    for the second one (two distorted links).
The spin-1/2 XX chain is a well-known example of an exactly solvable spin system (see, for example, [2][3][4]). Jordan-Wigner transformation [14] reduces (1) to the Hamiltonian of the ideal gas of spinless fermions, so both models are exactly solvable.

Stationary states with one inverted spin
We diagonalized the Hamiltonian (1) by solving the one particle Schrödinger equation in the lattice site representation and derived the following dispersion relations for the stationary states with one inverted spin for both models and     for the chain with two distorted links.

Thermodynamics
Using the dispersion relations (2)(3)(4), one can investigate the thermodynamic properties of the models considered. We need only the total number of stationary states and the degeneracy rate of each state for the calculating of the partition function and a several thermodynamic quantities.
We derived the analytical formulas for some principal and local thermodynamic characteristics and studied numerically their field and temperature dependence as the functions of the model parameters.
In particular, we found a complex dependence of the heat capacity on the applied magnetic field with numerous minima and maxima (see fig. 1(a, b)).
We associate the maxima and minima in the weak fields with the quasi-continuous spectra, and the additional extrema in strong magnetic field with the localized levels.
The Вісник ХНУ імені В.Н. Каразіна, серія «Фізика», вип. 31, 2019 The remnants of these jumps at rather low temperature are clearly visible (see fig. 2). We associate the maxima and minima in weak fields with quasi-continuous spectra, and additional extrema in strong magnetic field correspond to the localized levels.
In the case of weak interaction with the main XX chain the field dependence of average value of the spins of distorted link may be decreased and even becomes negative with the increasing of magnetic field, and jumps in the critical field along the direction of the magnetic field (see fig. 3a). Pair correlation function ( fig. 3b) is negative for nearest neighbors at zero and weak fields. We associate this behavior with antiferromagnetic character of XX exchange interaction. The temperature dependence of specific heat per spin for both models with the same number of lattice spins and the same parameters for the first "distortion" at 11 ,1 nn  positions in very weak magnetic field is presented at Fig.  4.
The temperature dependence of the specific heat per spin for the model with one distorted bond (dashed line at fig. 4) has no additional low temperature peaks of XX chains with two distorted links (solid line at fig. 4). The low temperature peaks for the model with two distorted links can be associated with the second "distortion" at 22 ,1 nn  for the model with two links.
The absolute values of longitudinal pair dynamical correlation function and autocorrelation function for the spins of distorted link are pictured at fig. 5. These characteristics also affected noticeably by the local levels, which leads to the complex oscillations pattern, including some kind of beats.

Summary and conclusions
Two exactly solvable spin-1/2 models with XX exchange interactions with one and two distorted links and finite total number of spins are investigated analytically and numerically.
In particular, we obtained the dispersion equations for the stationary states with one inverted spin are derived, and the analytical relations for the critical values of model parameters which correspond to the appearance of the localized impurity levels. The low temperature thermodynamics of above models are studied numerically.
We found a complex character of the dependence of the heat capacity on applied longitudinal magnetic field with numerous minima and maxima.
The field dependence of z-projection of the average total spin and the field dependence of magnetization at zero temperatures have the finite jumps associated with the quasi-continuous spectrum, and the impurity levels. At rather low temperatures the remnants of these jumps are clearly visible. The average z-projection of the spins of distorted link(s) may decrease with the increasing of the magnetic field for some values of model parameters.
The temperature dependence of specific heat may demonstrate additional maxima at very low temperatures.
The pair dynamical correlation function and autocorrelation function also affected noticeably by the local levels, which results to the complex oscillations patterns. So, the finite nature of the models, and the appearance of the localized levels may effect noticeably on the local thermodynamic characteristics.