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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Microcanonical Lanczos method

X. Zotos

Published Mar 10, 2006• 3 citations
Publisher

Abstract

In this article, the development of a novel numerical simulation method based on the Lanczos technique and the microcanonical statistical ensemble is discussed. It allows the evaluation of finite temperature dynamic correlations for quantum many-body systems on lattices larger than hitherto possible. To demonstrate the applicability of the method, a series of recently obtained results for one-dimensional spin and electronic models are reviewed.

DOI
10.1080/14786430500227830

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In this article, the development of a novel numerical simulation method based on the Lanczos technique and the microcanonical statistical ensemble is discussed. It allows the evaluation of finite temperature dynamic correlations for quantum many-body systems on lattices larger than hitherto possible. To demonstrate the applicability of the method, a series of recently obtained results for one-dimensional spin and electronic models are reviewed.
We show how to generalize the zero-temperature Lanczos method for calculating dynamical correlation functions to finite temperatures. The key is the microcanonical ensemble which allows us to replace the involved canonical ensemble with a single appropriately chosen state; in the thermodynamic limit it provides the same physics as the canonical ensemble but with the evaluation of a single expectation value. We can employ the same system sizes as for zero temperature but, whereas the statistical fluctuations present in small systems are prohibitive, the spectra of the largest system sizes are surprisingly smooth. We investigate, as a test case, the spin conductivity of the spin-1/2 anisotropic Heisenberg model and in particular we present a comparison of spectra obtained by the canonical and microcanonical ensemble methods.
We examine the accuracy of the microcanonical Lanczos method (MCLM) developed by Long et al. [Phys. Rev. B 68, 235106 (2003)PRBMDO0163-182910.1103/PhysRevB.68.235106] to compute dynamical spectral functions of interacting quantum models at finite temperatures. The MCLM is based on the microcanonical ensemble, which becomes exact in the thermodynamic limit. To apply the microcanonical ensemble at a fixed temperature, one has to find energy eigenstates with the energy eigenvalue corresponding to the internal energy in the canonical ensemble. Here, we propose to use thermal pure quantum state methods by Sugiura and Shimizu [Phys. Rev. Lett. 111, 010401 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.010401] to obtain the internal energy. After obtaining the energy eigenstates using the Lanczos diagonalization method, dynamical quantities are computed via a continued fraction expansion, a standard procedure for Lanczos-based numerical methods. Using one-dimensional antiferromagnetic Heisenberg chains with S=1/2, we demonstrate that the proposed procedure is reasonably accurate, even for relatively small 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.
In our earlier work, we proposed a new microcanonical ensemble method and showed that the numerical results of this method are correct in 1D models. In this paper, the numerical data in 2D models are presented. One of our calculations is concentrated on fermionic models, because fermionic models behave quite differently from bosonic models when the dimension is larger than two (there may be boson-fermion duality in 1D). Our results come out to be consistent with exact values. To our knowledge, there is no other numerical methods which can present reliable calculation on fermionic models with large lattice, especially when the temperature is low. In addition to the test on fermionic models, we also show our results on quantum 2D XY model. We calculate longitudinal spin-spin correlation, specific heat, vortex density and vortex pair density. These results agree with the calculations by other methods. The data of vortex density and vortex pair density seem to display directly the unbinding of vortices- and antivortices. PACS. 05.30.Ch- Quantum ensemble theory. PACS. 65.50.+m- Thermodynamic properties and entropy.
We present a method to compute dynamical correlation functions in quantum many-body systems at zero temperature. The ground state of a finite system is evaluated exactly by a modified Lanczos method and from it the real-frequency correlation functions are obtained by a projective technique for the memory-function formalism. We apply the method to the density-density correlation function of a one-dimensional spinless fermion system. Exact results are recovered in the noninteracting limit; in the strong-coupling limit our results are compared with Monte Carlo simulation and analytical approximations.