Ground State and Finite Temperature Lanczos Methods

Oliver Hanebaum, Jürgen Schnack

Peter Prelovšek, J. Bonča

The present review will focus on recent development of exact-diagonali- zation (ED) methods that use Lanczos algorithm to transform large sparse matrices onto the tridiagonal form. We begin with a review of basic principles of the Lanczos method for computing ground-state static as well as dynamical properties. Next, generalization to finite-temperatures in the form of well established finite-temperature Lanczos method is described. The latter allows for the evaluation of temperatures T>0 static and dynamic quantities within various correlated models. Several extensions and modification of the latter method introduced more recently are analysed. In particular, the low-temperature Lanczos method and the microcanonical Lanczos method, especially applicable within the high-T regime. In order to overcome the problems of exponentially growing Hilbert spaces that prevent ED calculations on larger lattices, different approaches based on Lanczos diagonalization within the reduced basis have been developed. In this context, recently developed method based on ED within a limited functional space is reviewed. Finally, we briefly discuss the real-time evolution of correlated systems far from equilibrium, which can be simulated using the ED and Lanczos-based methods, as well as approaches based on the diagonalization in a reduced basis.

Markus Aichhorn, Maria Daghofer, Hans Gerd Evertz +1 more

We present a modified finite-temperature Lanczos method for the evaluation of dynamical and static quantities of strongly correlated electron systems that complements the finite-temperature method introduced by Jakli\ifmmode \check{c}\else \v{c}\fi{} and Prelov\ifmmode \check{s}\else \v{s}\fi{}ek for low temperatures. Together they allow accurate calculations at any temperature with moderate effort. As an example we calculate the static spin-correlation function and the regular part of the optical conductivity ${\ensuremath{\sigma}}^{\mathrm{reg}}(\ensuremath{\omega})$ of the one-dimensional Hubbard model at half filling and show in detail the connection between the ground-state and finite-temperature method. By using cluster perturbation theory, the finite-temperature spectral function is extended to the infinite system, clearly exhibiting the effects of spin-charge separation.

Massimo Capone, Luca de’ Medici, Antoine Georges

We present an efficient method to solve the impurity Hamiltonians involved in dynamical mean-field theory at low but finite temperature based on the extension of the Lanczos algorithm from ground state properties alone to excited states. We test the approach on the prototypical Hubbard model and find extremely accurate results from $T=0$ up to relatively high temperatures up to the scale of the critical temperature for the Mott transition. The algorithm substantially decreases the computational effort involved in finite temperature calculations.