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Chinese Physics B

Gauss quadrature based finite temperature Lanczos method

Jian Li, Hai‐Qing Lin

Published Mar 2, 2022• 2 citations
Publisher

Abstract

The finite temperature Lanczos method (FTLM), which is an exact diagonalization method intensively used in quantum many-body calculations, is formulated in the framework of orthogonal polynomials and Gauss quadrature. The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one- and two-dimensional Gauss quadratures. Then lower order Gauss quadrature, which is generated from Lanczos iteration, can be applied to approximate the initial weighted summation. This framework fills the conceptual gap between FTLM and kernel polynomial method, and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.

DOI
10.1088/1674-1056/ac5986

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The finite temperature Lanczos method (FTLM), which is an exact diagonalization method intensively used in quantum many-body calculations, is formulated in the framework of orthogonal polynomials and Gauss quadrature. The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one- and two-dimensional Gauss quadratures. Then lower order Gauss quadrature, which is generated from Lanczos iteration, can be applied to approximate the initial weighted summation. This framework fills the conceptual gap between FTLM and kernel polynomial method, and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.
Magnetic molecules promise progress in a wide variety of fields ranging from high density memory storage to quantum computing as well as magnetic cooling. In order to gain better insight this work aims at the study of theoretical models and evaluation of thermodynamic observables by implementing and testing a new numerical approximation method - the Finite-Temperature Lanczos-Method (FTLM). Magnetic centres of molecules are modelled as spins of fixed quantum number, position and interactions. The model is specified in a spin Hamiltonian consisting of three parts: Heisenberg exchange, single ion anisotropy and Zeeman interaction. Although finite, the dimension of the Hilbert space grows exponentially with the number of spin sites, so full diagonalisation of the Hamiltonian is rendered infeasible: Even for a small number of spins the Hilbert space dimension exceeds several millions, which is beyond exact diagonalisation techniques. In order to calculate observables yet, it is necessary to implement approximation methods. Following previous work, where the Hilbert space could be partitioned into invariant subspaces with respect to magnetic quantum numbers, now systems with anisotropy are studied. These anisotropic terms generally do not commute with total spin operators, therefore partitioning no longer helps reducing the problem size. There are several techniques for approximate solutions to eigenvalue equations, among these adaptations of the original Lanczos method. As a Krylov subspace method, its main advantage is that the most complex operations required are matrix-vector multiplications. When matrix elements are calculated on-the-fly, the most economical version requires storage of only two vectors. The second ingredient for FTLM is a typicality based Monte Carlo approach: Traces are estimated as an average over a tiny set of initial vectors compared to the full Hilbert space. First tests revealed a major flaw due to numerical instability of traces that normally are zero. By partitioning with respect to magnetic quantum numbers, especially exploiting their inversion symmetry, this problem was unknowingly dodged in previous calculations. It was solved while preserving the advantage of minimal computational effort: An arbitrary set of initial vectors would regain this time-reversal symmetry only, if a substantial number of initial vectors where used. Instead, a doubling of the initial vectors by inverting them with respect to magnetic quantum numbers yields the desired effect. Time-consuming evaluations of matrix elements were also avoided by calculating the magnetisation as a difference quotient. Finally FTLM is implemented and tested for several magnetic molecules. Analogues of the hour glass molecules synthesised by Glaser et al. were first model systems, showing good agreement of exact diagonalisation results and approximations. The method can be applied to any Hamiltonian, given that matrix-vector products in the underlying Hilbert space can be calculated. Here limiting factors are memory usage and computing time. As a proof of principle observables in a Hilbert space consisting of d=100,000,000 states were calculated using the parameter set suggested by Mazurenko et al. for the Mn12-acetate molecule. Single crystal as well as powder averaged results are compared to experimental data. So this previously impossible calculation is now executed within hours by use of FTLM.
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.
We present a GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians that targets workstation hardware rather than distributed-memory clusters. The Hamiltonian action is evaluated matrix-free in a row-wise gather formulation. We introduce and compare two state-to-index strategies: a compressed lookup table (CLT), which reduces lookup memory by a factor of 16 relative to a full table while retaining a fixed, branch-light access pattern, and a GPU-adapted combinatorial-ranking scheme that removes the lookup table altogether. Numerical tests against FP64 CPU references show that FP32 GPU arithmetic changes heat capacities and magnetic susceptibilities by amounts several orders of magnitude below the stochastic uncertainty of the FTLM trace estimator at typical sample sizes. Benchmarks show speedups of up to about one order of magnitude over optimized multicore CPU calculations and enable Hilbert-space sectors of dimension ~10^8 on a single workstation GPU. The MATLAB/CUDA implementation, including example input files and benchmark scripts, is openly available at https://github.com/ghasdeke/ftlm-gpu (archived at DOI: 10.5281/zenodo.20378647) under the Apache-2.0 license.
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.
The computation of thermal properties of quantum many-body systems is a central challenge in our understanding of quantum mechanics. We introduce the Quantum Finite Temperature Lanczos Method (QFTLM), which extends the finite-temperature Lanczos method to quantum computers by combining real-time quantum Krylov methods with efficient preparation of typical states for trace estimation. This approach enables the computation of thermal expectation values while avoiding the exponential scaling inherent to classical exact simulation techniques. Numerical experiments on the transverse-field Ising model show that QFTLM can reproduce thermal observables over a wide temperature range. We further analyze the influence of Krylov dimension, number of trace-estimator states, and Trotter error, and show that suitable regularization is essential for robustness in noisy settings. These results establish QFTLM as a promising framework for finite-temperature quantum simulation.