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A SIMPLIFIED FORMULATION OF THE LANCZOS TECHNIQUE FOR STRONGLY CORRELATED SYSTEMS

S. Ehika

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The basic idea of the Lanczos method is to construct a special basis where the Hamiltonian has a tridiagonal representation. Once in this form, the matrix can be diagonalized easily using standard library subroutines. This approach is known as the standard Lanczos technique (SLT) which still has the problem of diagonalizing large matrix sizes emanating from increase of the size of the Hilbert space with the size of the system. In this current presentation, we develop a simplified formulation of the SLT and then use it to study the rapid convergence to the ground state energy and wavefunction of some finite systems.

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The basic idea of the Lanczos method is to construct a special basis where the Hamiltonian has a tridiagonal representation. Once in this form, the matrix can be diagonalized easily using standard library subroutines. This approach is known as the standard Lanczos technique (SLT) which still has the problem of diagonalizing large matrix sizes emanating from increase of the size of the Hilbert space with the size of the system. In this current presentation, we develop a simplified formulation of the SLT and then use it to study the rapid convergence to the ground state energy and wavefunction of some finite systems.
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.
A closed form expression for the ground-state energy density of the general extensive many-body problem is given in terms of the Lanczos tridiagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, ${\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}$(N) and ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}}$(N), asymptotic forms \ensuremath{\alpha}(z) and \ensuremath{\beta}(z) can be defined in terms of a parameter z\ensuremath{\equiv}n/N (n is the Lanczos iteration and N is the size of the system). By application of theorems on the zeros of orthogonal polynomials we find the ground-state energy density in the bulk limit to be given, in general, by E0{\mathit{E}}_{0}E0​=inf[\ensuremath{\alpha}(z)-2\ensuremath{\beta}(z)]. \textcopyright{} 1996 The American Physical Society.
The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size sis_{i}si​ to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
We review a recent approach for the simulation of many-body interacting systems based on an efficient generalization of the Lanczos method for Quantum Monte Carlo simulations. This technique allows to perform systematic corrections to a given variational wavefunction, that allow to estimate exact energies and correlation functions, whenever the starting variational wavefunction is a qualitatively correct description of the ground state. The stability of the variational wavefunction against possible phases, not described at the variational level can be tested by using the ''effective Hamiltonian'' approach. In fact Monte Carlo methods, such as the ''fixed node approximation'' and the ''generalized Lanczos technique'' (Phys. Rev. B 64,024512, 2001) allow to obtain exact ground state properties of an effective Hamiltonian, chosen to be as close as possible to the exact Hamiltonian, thus yielding the most reasonable estimates of correlation functions. We also describe a simplified one-parameter scheme that improve substantially the efficiency of the generalized Lanczos method. This is tested on the t-J model, with a special effort to obtain accurate pairing correlations, and provide a possible non-phonon mechanism for High temperature superconductivity.
In the eigen-functional theoretical framework, exact/accurate treatment of high-dimensional strongly correlated electron systems may be realized. The two key steps of this method are that under the path integral formulation we first change a D-dimensional quantum many-particle system into an (D+1)-dimensional (time-dependent) effective "single-particle" problem, then by solving the eigen-equation of the propagator operator of the particles, we can obtain the ground state energy functional and action, respectively. Under the eigen-functional theory, the problems of quantum many-particle systems end in to solve the equation of phase fields that are completely determined by the electron interaction. In practice this equation of the phase fields can be numerically solved for large lattice sites. After replacing the real time by the imaginary time, this method can also be applied for finite temperature cases.
We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.