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Progress of Theoretical Physics

One-Dimensional Anisotropic Heisenberg Model at Finite Temperatures

Minoru Takahashi, Masuo Suzuki

Published Dec 1, 1972• 403 citations
Publisher

Abstract

The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.

DOI
10.1143/ptp.48.2187

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Progress of Theoretical Physics•Mar 1, 1972•246 citations
The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.
In this thesis we take a look at the one-dimensional Heisenberg model. The Heisenberg model is the most used model to study the magnetic properties of materials and is of paramount importance for understanding magnets. First we sketch a brief derivation of the model and discuss its strengths and weaknesses. Afterwards we develop a transformation of the Heisenberg model to a model of spinless fermions, which allows for an easy solution of the Jz = 0 model. After solving the model for Jz = 0, we solve the model for general Jz. The results we find are very complicated and it is hard to calculate anything explicitly. So we move to to the thermodynamic limit to calculate the ground state energy for Jz = 1. After that we have found a solution for big system sizes, we use exact diagonalisation to solve the Heisenberg model. We do this for system sizes up too 12 particles. With our results we check how accurate some expansions for smaller systems sizes are. We find some accurate expansions for small ∆ and ∆ = 1. We also look at a energy gap of the system for ∆ > 1. We find how this gap behaves as a function of ∆.
Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the Finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, we confirm that the Euler–Maclaurin formula gives the correct low–temperature expansion of the free energy for the XXZ Heisenberg model.
A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in T/∣J∣\sqrt{T/|J|}T/∣J∣​ about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.
Part I. Thermodynamics of Non-Interacting Systems and Ground States on Interacting Systems: 1. Free energy and correlation functions of XY models 2. Systems with delta-function potential 3. Isotropic Heisenberg model 4. XXZ model 5. XYZ and eight-vertex models 6. Hubbard model Part II. Finite Temperature Integral Equations for Un-Nested Systems: 7. Repulsive delta-function bosons 8. Thermodynamics of the XXX chain 9. Thermodynamics of the XXZ model 10. Thermodynamics of the XYZ model 11. Low-temperature thermodynamics Part III. Finite Temperature Integral Equations for Nested Systems: 12. S=1/2 fermions with repulsive potential in the continuum 13. S=1/2 fermions with an attractive potential 14. Thermodynamics of the Hubbard model Part IV. Quantum Transfer Matrix and Recent Developments: 15. Transfer matrix and correlation length 16. The spin 1/2 XXZ model in a magnetic field 17. The XYZ model with no magnetic field 18. Recent developments and related topics.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.