Transfer matrix simulation technique: Effectiveness and applicability to the low-dimensional magnetic spin systems

The deterministic quantum transfer-matrix (QTM) technique, its mathematical background, effectiveness and applicability to the real physical low-dimensional magnetic systems are presented. Modelling is based on the Heisenberg Hamiltonian which describes the macroscopic Haldane-gap and molecular-based spin S = 1 chains, small size magnetic clusters embedded in some supramolecules and other interesting compounds. Using QTM, the spin degrees of freedom are accurately taken into account, yielding the thermodynamical functions at finite temperatures. The finite-temperature results for some isotropic and anisotropic systems as well as for systems with uniform and non-uniform interactions are reviewed. For Yb4As3 with antisymmetric interactions new results are presented—the field-dependent specific heat for finite chains and the field dependence of the energy gap. In order to demonstrate the effectiveness and applicability of the method to modelling of non-magnetic impurity effects, variation of the QTM approximants is shown for finite segments. computational complexity of our problems is exponential, the efficiency of parallelization using the Message Passing Interface (MPI) system library was analysed and estimated as close to 100 % for our platform SGI Origin 3800 with 64 processor units. For the quantum chain simulations, both the memory and CPU bound for k B T / J ⩽ 0.1 was established. For the finite ring simulations, the CPU time resources imposed the limits.