Ground State and Finite Temperature Lanczos Methods

Abstract

The present review will focus on recent development of exact-diagonalization (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 finitetemperature 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.

Photoinduced in-gap excitations in the one-dimensional extended Hubbard model

“…The average incoming number of photons per lattice site during the pump is estimated to be ∝ A 2 0 ω 0 t d . In our calculation, the system's temporal evolution is evaluated by the time-dependent Lanczos method, where, within the reach of the exact diagonalization, the time-dependent wave function ψ(t) can be obtained exactly 19 . Based on the information of ψ(t), we calculate the time-dependent current density j(t) = ψ(t)|ĵ|ψ(t) , whereĵ is the current operator.…”

Absence of full many-body localization in the disordered Hubbard chain

“…That is, the (full) MBL requires that both, C 0 and S 0 , are finite. For the calculation of imbalance correlations we employ the microcanonical Lanczos method (MCLM) [42,43] on finite systems of maximum length L = 14 forn = 1 (forn = 1/2 see the Supplement [41] ). The high frequency resolution is achieved by large number of Lanczos steps N L = 10…”

Krylov-Projected Quantum Monte Carlo Method

“…This is achieved by extending the recently developed Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method [2,15,16], by combining it with ideas from the dynamical and finite-temperature Lanczos (FTLM) methods. [17][18][19] The key advantage of the approach is that it avoids any explicit storage over the full Hilbert space, instead only storing occupied states in the discretized wavefunction at each snapshot. This allows for sparsity in the wavefunction to be exploited to minimize memory bottlenecks, which are a primary limitation in conventional approaches which require explicit storage over the space [17,18,20,21].…”

“…[17][18][19] The key advantage of the approach is that it avoids any explicit storage over the full Hilbert space, instead only storing occupied states in the discretized wavefunction at each snapshot. This allows for sparsity in the wavefunction to be exploited to minimize memory bottlenecks, which are a primary limitation in conventional approaches which require explicit storage over the space [17,18,20,21]. The result is a QMC method which although weakly exponentially scaling, in common with the ground state FCIQMC approach, can allow for systems to be treated well outside that possible by conventional means, and retains many of the important features of the parent method [15,22].…”

“…N can be very large for systems of interest and so in practice one starts from a much smaller number of states, R ≪ N , chosen as a random linear combination of all basis states, |r = n η rn |n . This turns out to converge quickly with R, particularly at high temperatures [17,18,[29][30][31]. In our stochastic approach the initial random vectors are created by distributing a given number of walkers randomly throughout the Hilbert space with coefficients ±1.…”