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arXiv:0803.4495 · Journal of Physics Condensed Matter

Heat capacity of Schottky type in low-dimensional spin systems

S. Mahdavifar, Alireza Akbari

Published Apr 22, 2008• 16 citations
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Abstract

The heat capacity of low-dimensional spin systems is studied using theoretical and numerical techniques. Keeping only two energy states, the system is mapped onto the two-level-system (TLS) model. Using the low-temperature Lanczos method, it is confirmed that the behavior of T M and the energy gap as functions of the control parameter is the same in the two models studied; a conclusion that can probably be extrapolated to the general case of any system that possesses an energy gap.

DOI
10.1088/0953-8984/20/21/215213

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The heat capacity of low-dimensional spin systems is studied using theoretical and numerical techniques. Keeping only two energy states, the system is mapped onto the two-level-system (TLS) model. Using the low-temperature Lanczos method, it is confirmed that the behavior of T M and the energy gap as functions of the control parameter is the same in the two models studied; a conclusion that can probably be extrapolated to the general case of any system that possesses an energy gap.
A formulation is given to map the one-dimensional S = 1/2 quantum spin systems onto the two-dimensional Ising model with many-spin interactions on the basis of the n-spin cluster decomposition (n CLD). The cluster transfer matrix is defined which characterizes the thermodynamical properties of the equivalent system. A prescription is presented to construct the cluster transfar matrix and to evaluate its maximum eigenvalue numerically. The method has been applied to the calculation of the energy and specific heat of the XY model for various values of the cluster size n and th Trotter size m. The convergence properties of the energy as a function of n and m have been studied in detail. The main results obtained are: (1) The low-temperature behavior is improved much more by considering larger spin clusters; (2) the energy depends on the spin cluster through the effective cluster size ne defined by ne =(n+1)/2; and (3) the energy converges linearly with 1/n2em2 except at very low temperatures.
We have developed a method for evaluating the specific heat of lattice spin systems. It is based on the knowledge of high-temperature series expansions, the total entropy of the system, and the low-temperature expected behavior of the specific heat as well as the ground-state energy. By the choice of an appropriate variable (entropy as a function of energy), a stable interpolation scheme between low and high temperature is performed. Contrary to previous methods, the constraint that the total entropy is log(2S+1)\mathrm{log}(2S+1)log(2S+1) for a spin S on each site is automatically satisfied. We present some applications to quantum spin models on one- and two-dimensional lattices. Remarkably, in most cases, a good accuracy is obtained down to zero temperature.
Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method which takes into account not only eigenvalues of ϵn\epsilon_nϵn​ for 0≤n≤Nm0 \leq n \leq N_m0≤n≤Nm​ obtained by the energy-matrix diagonalization but also their extrapolated ones for Nm+1≤n<∞N_m+1 \leq n < \inftyNm​+1≤n<∞ (Nm=20N_m=20Nm​=20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.
Abstract In this work, we present a comparative study of thermodynamic quantum equilibrium observables of spin-1 Heisenberg chain. As a theoretical approach, the modified spin-wave theory is chosen. From the numerical side, we consider finite-temperature Lanczos, kernel polynomial, and density matrix renormalization techniques. The results show general consistency of thermodynamic quantities, such as heat capacity and magnetic susceptibility, calculated within the approaches. For the modified spin-wave, the results show some inaccuracy especially at low temperature, failure in capturing the gapped features of spin-1 Heisenberg chain, while the numerical approaches illustrate a good achievement at T→0T\rightarrow 0T→0 T → 0 within the proper parameters chosen.
The thermodynamic properties of the two-dimensional Ising model with random (+or-J) nearest-neighbour interactions are determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The central result is that the spin-glass correlation length diverges as T- nu at low temperature T with nu =2.59+or-0.13. The heat capacity is similar to the Schottky anomaly. At zero temperature the entropy per spin is 0.0701+or-0.0005 and the internal energy per spin is -(1.4024+or-0.0012)J.
Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method, which takes into account not only eigenvalues of ε(n) for 0 ≤ n ≤ N(m) obtained by the energy-matrix diagonalization but also their extrapolated ones for N(m) + 1 ≤ n <∞ (N(m) = 20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov, and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.