Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder

Since the pioneering paper by Lieb, Schultz and Mattis [1] statistical mechanics calculations for spin1 2 XY chains have been the subject of long-lasting interest both from fundamental and applied viewpoints. Our communication deals with some new results for thermodynamics and dynamics of isotropic transverse XY chains with regular nonuniformity or correlated Lorentzian disorder. The former model generalizes the XX chain with the alternating coupling constant that was investigated by some authors as a toy model to consider spin-Peierls phase transition [2-4]. The latter model was studied in [5]. However the treatment in that paper was restricted to thermodynamics in contrast to the present study dealing with dynamics of transverse spin correlations. The basic tools to study these models are the Jordan-Wigner method, the continued fractions, and the numerical approach for examining the spin correlation dynamics developed recently [6]. Hereinafter we investigate a nonuniform XX chain in a magnetic field along the z-axis consisting of N spins 1 2 wherein Hamiltonian is defined by

Since the pioneering paper by Lieb, Schultz and Mattis [1] statistical mechanics calculations for spin- 1  2 XY chains have been the subject of long-lasting interest both from fundamental and applied viewpoints.Our communication deals with some new results for thermodynamics and dynamics of isotropic transverse XY chains with regular nonuniformity or correlated Lorentzian disorder.The former model generalizes the XX chain with the alternating coupling constant that was investigated by some authors as a toy model to consider spin-Peierls phase transition [2][3][4].The latter model was studied in [5].However the treatment in that paper was restricted to thermodynamics in contrast to the present study dealing with dynamics of transverse spin correlations.The basic tools to study these models are the Jordan-Wigner method, the continued fractions, and the numerical approach for examining the spin correlation dynamics developed recently [6].
Hereinafter we investigate a nonuniform XX chain in a magnetic field along the z-axis consisting of N spins 1  2 wherein Hamiltonian is defined by By the Jordan-Wigner transformation the Hamiltonian (1) can be represented as the Hamiltonian of non-interacting spinless fermions To examine the thermodynamics of the model one must find the density of magnon states ρ(E) that is related to the temperature double-time fermion Green functions Using the equation of motion for G ∓ nm it is a simple matter to show that The continued fraction representation for the diagonal Green functions ( 4) is extremely useful for examining thermodynamics of regularly nonuniform chains when periodic continued fractions emerge.
Consider for example a regular alternating chain . .when periodic continued fractions with period 2 emerge.As a result Here and in ( 5), (6) In principle the calculation of ρ(E) can be done for an arbitrary periodic chain, however, the calculations in the case of large periods become cumbersome.In figures 1a, 1b we plotted ρ(E) for two chains with the period 12.The main result of introducing the regular nonuniformity is a spliting of the magnon band into subbands (a number of subbands is equal to or less than the period of nonuniformity; compare figure 1a and figure 1b) that has important consequences in the thermodynamic properties of spin model.For instance the low-temperature dependence of transverse magnetization 2 on transverse field is made up of sharply increasing parts and horizontal parts, their number being determined by the period of nonuniformity (figures 1c, 1d).In figures 1e, 1f we plotted the temperature dependence 2 that due to the introduced periodic nonuniformity exhibits a two-peak structure, i.e. it has low-temperature and high-temperature peaks.
Let us consider spin model (1) assuming that the exchange couplings J n are independent Lorentzian variables with distribution (J 0 is the mean value of exchange coupling and Γ is the width of distribution that controls the strength of disorder) and the transverse fields are determined by the neighbouring exchange couplings according to the formula It can readily be checked that the distribution for the random variable Ω n reads We shall be interested in calculation of the random-averaged dynamic structure factor dte −ǫ|t| e iωt s z j (t)s z j+n − s z j s z j+n (10) using for this purpose numerical approach.As shown in [6], to achieve this goal it is necessary to solve N × N eigenvalue and eigenvector problem for the matrix In our numerical calculations we considered chains of N = 300 spins with J 0 = −1, Ω 0 = 0.5 and Γ = 0.1 at low temperature β = 1000.We computed correlation functions s z 150 (t)s z 150+n − s z 150 s z 150+n with n = 0, ±1, . . ., ±100 for the times up to t = 200, put ǫ = 0.01 and averaged the zz dynamic structure factor (10) over 3000 random realizations to obtain the results presented in figure 2. We carefully analyzed the accuracy of our calculations studying finite-size effects, the effects of finite number of terms in the sum in (10) and of finite time cut-off in the integral in (10), and the effects of finite number of random realizations.
Let us comment the results we have obtained for S zz (κ, ω). Figure 2 nicely shows the difference in frequency shapes of the dynamic structure factor for correlated disorder ( 7), (8) with different signs of a and for the case of independent random exchange couplings and transverse fields with distributions ( 7) and (9), respectively.The transverse dynamic structure factor is determined by two magnon excitations and these spectacular changes in the frequency dependence are caused by the changes in the density of magnon states depicted in figure 3.
To summarize, we applied continued fractions to study rigorously the thermodynamic properties of periodic nonuniform spin- 1  2 XX chain in a transverse field and extended a previous analysis of the spin- 1  2 XX chain with correlated Lorentzian disorder examining numerically the dynamics of transverse spin correlations.The theoretical results observed in our study should prove valuable in understanding the experimental data for XX chain materials the synthesis of which is becoming a reality [7].
A study of periodic nonuniform spin-

1 2 XX
chains was inspired by the papers of J. Freericks and R. Lemański presented at the 22nd Seminar of the Middle European Cooperation in Statistical Physics (Szklarska Porȩba, 1997).The authors thank R. Lemański for useful correspondence.O. D. is indebted to Mr. Joseph Kocowsky for continuous financial support.