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arXiv:1702.00487 · Physical review. E

Interpolation between low and high temperatures of the specific heat for spin systems

Heinz–Jürgen Schmidt, Andreas Hauser, Andre Lohmann, Johannes Richter

Published Apr 6, 2017• 28 citations
arXivHTMLPDF

Abstract

The high temperature expansion (HTE) of the specific heat of a spin system fails at low temperatures, even if it is combined with a Padé approximation. On the other hand, we often have information about the low-temperature asymptotics (LTA) of the system. Interpolation methods combine both kind of information, HTE and LTA, in order to obtain an approximation of the specific heat that holds for the whole temperature range. Here we revisit the entropy method that has been previously published and propose two variants that better cope with problems of the entropy method for gapped systems. We compare all three methods applied to the antiferromagnetic Haldane spin-one chain and especially apply the second variant, called log Z method, to the cuboctahedron for different spin quantum numbers. In particular, we demonstrate that the interpolation method is able to detect an extra low-temperature maximum in the specific heat that may appear if a separation of two energy scales is present in the considered system. Finally, we illustrate how interpolation also works for classical spin systems.

DOI
10.1103/physreve.95.042110

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The high temperature expansion (HTE) of the specific heat of a spin system fails at low temperatures, even if it is combined with a Padé approximation. On the other hand, we often have information about the low-temperature asymptotics (LTA) of the system. Interpolation methods combine both kind of information, HTE and LTA, in order to obtain an approximation of the specific heat that holds for the whole temperature range. Here we revisit the entropy method that has been previously published and propose two variants that better cope with problems of the entropy method for gapped systems. We compare all three methods applied to the antiferromagnetic Haldane spin-one chain and especially apply the second variant, called log Z method, to the cuboctahedron for different spin quantum numbers. In particular, we demonstrate that the interpolation method is able to detect an extra low-temperature maximum in the specific heat that may appear if a separation of two energy scales is present in the considered system. Finally, we illustrate how interpolation also works for classical spin systems.
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.
We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.
An improved Padé method is applied to the entropy and the reduced susceptibility χ R (=χ· k B T / N ( g µ) 2 ) expanded in powers of the internal energy in order to calculate the specific heat c ( T ) for S ≤2 and the zero-field susceptibility for S =1/2 of a quantum Heisenberg spin chain, respectively. The exponents α( c ( T )∼ T α ) obtained for an antiferromagnetic exchange and 1≤ S ≤2 are 1.6, 1.4 and 1.3, which are much different from the value (α=1) given by the spin-wave theory (SW). It is also shown that the polynomials with exponents given by SW cannot fit in with the Padé approximants for these cases. This disagreement with the behaviors of SW is ascribed, at least for S =1, to the nonzero gap found by Botet and Jullien.
A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the S =1/2 XY model and is then applied to the Heisenberg model of S =1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S =1/2 antiferromagnet. In antiferromagnets of S ≧1 the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.
The ground state and thermodynamic properties of spin-1 and spin- chains are investigated via exactly solved su (3) and su (4) models with physically motivated chemical potential terms. The analysis involves the thermodynamic Bethe ansatz and the high temperature expansion (HTE) methods. For the spin-1 chain with large single-ion anisotropy, a gapped phase occurs which is significantly different from the valence-bond-solid Haldane phase. The theoretical curves for the magnetization, susceptibility and specific heat are favourably compared with experimental data for a number of spin-1 chain compounds. For the spin- chain a degenerate gapped phase exists starting at zero external magnetic field. A middle magnetization plateau can be triggered by the single-ion anisotropy term. Overall, our results lend further weight to the applicability of integrable models to the physics of low-dimensional quantum spin systems. They also highlight the utility of the exact HTE method.
Many frustrated spin models on three-dimensional (3D) lattices are currently being investigated, both experimentally and theoretically, and develop new types of long-range orders in their respective phase diagrams. They present finite-temperature phase transitions, most likely in the Heisenberg 3D universality class. However, the combination between the 3D character and frustration makes them hard to study. We present here several methods derived from high-temperature series expansions (HTSEs), which give exact coefficients directly in the thermodynamic limit up to a certain order; for several 3D lattices, supplementary orders than in previous literature are reported for the HTSEs. We introduce an interpolation method able to describe thermodynamic quantities at T>TcT>{T}_{c}T>Tc​, which we use here to reconstruct the magnetic susceptibility and the specific heat and to extract universal and nonuniversal quantities (for example, critical exponents, temperature, energy, entropy, and other parameters related to the phase transition). While the susceptibility associated with the order parameter is not usually known for more exotic long-range orders, the specific heat is indicative of a phase transition for any kind of symmetry breaking. We present examples of applications on ferromagnetic and antiferromagnetic models on various 3D lattices and benchmark our results whenever possible.