Over the past few years we have embarked upon a program to investigate the physics of low-dimensional quantum spin systems from the viewpoint of exactly solved (integrable) models in statistical mechanics.
A highlight of this work is our recent publication in the 21 November 2003 issue of Physical Review Letters, in which we investigated the thermal and magnetic properties of the integrable su(4) ladder model by means of the quantum transfer matrix method. We evaluated the magnetic susceptibility, specific heat, magnetic entropy, and high field magnetization from the free energy derived via the recently proposed method of high temperature expansion for exactly solved models. We showed that the integrable model can be used to describe the physics of the strong coupling ladder compounds. We saw excellent agreement between the theoretical results and the experimental data for a sample of known ladder compounds.
This work represents direct contact with experiments in condensed matter physics.
We are currently extending this approach in a number of directions.
Link to the paper (M.T. Batchelor, X.W. Guan, N.Oelkers, K. Sakai, Z. Tsuboi and A. Foerster)
Summary for qantum spin ladders
The ultimate challenge for theoretical physics is comparison with experiment. Yet often the development of theory is a fascinating and worthwhile endeavor in its own right. Over the past 40 years the theory of integrable models in statistical mechanics has developed to an extraordinary level of mathematical sophistication. By integrable it is meant that the eigenspectrum of the relevant Hamiltonian can be derived exactly. An essential ingredient is the Yang-Baxter equation, the so-called master-key to integrability. Recent progress has seen the development of a new approach to calculate the free energy of integrable models in terms of nonlinear integral equations. In 2002 Shiroishi and Takahashi solved the nonlinear integral equations using an exact high temperature expansion method to obtain the free energy of the spin-1/2 Heisenberg chain. Building on this approach, Batchelor, Guan, Oelkers and collaborators obtained the free energy of a spin-1/2 Heisenberg ladder model. This has led to comparison with experimental results for the strong coupling ladder compounds.
The experimental realisation of compounds with a ladderlike structure has contributed to the intense interest in low-dimensional quantum systems. The existence of a spin gap, magnetisation plateaux, quantum critical points and superconductivity under hole doping are examples of key physical properties observed in the ladder compounds. The calculation of properties such as the full temperature phase diagram, the high field magnetisation curve and the specific heat have provided a significant challenge. These quantities have been obtained for the integrable ladder model using the exact high temperatute expansion (HTE) method. The figures show a comparison between the experimental (EXP) and theoretical results for the susceptibility, magnetisation and the specific heat as functions of magnetic field and temperature for a typical compound. The critical magnetic fields H_c1 ~ 7.8 Tesla and H_c2 ~ 13.0 Telsa are also in good agreement with the experimental results. The excellent agreement between theory and experiment is also seen for other known ladder compounds.
The research reported here brings over 40 years of mathematical development into direct contact with experiment. Indeed, there is a strong expectation that integrable models will play a crucial role in understanding low-dimensional quantum effects, where they are most pronounced.