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arXiv:cond-mat/0302211 · Physical review. B, Condensed matter

Finite-temperature dynamical correlations using the microcanonical ensemble and the Lanczos algorithm

M W Long, Peter Prelovšek, Samir El Shawish, J. Karadamoglou, X. Zotos

Published Dec 15, 2003• 94 citations
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Abstract

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.

DOI
10.1103/physrevb.68.235106

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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.
Based on the microcanonical ensemble theory in statistical mechanics, we devise a method that can be used to enhance the capability of numerical calculation on statistical models with large lattice sizes. In our method, we take the expectation value of the energy, as defined in quantum mechanics, instead of the eigenvalue as the energy of a physical system. We show mathematically that the relevant physical quantities obtained in this way are unchanged in the thermodynamical limit and we apply this method to numerical calculations. In this paper, we present our numerical results with the one-dimensional spinless fermionic model as a first test of our method. The numerical calculations are done to a 4096 lattice size using a computer with a speed of about 40 mflops. Our numerical data agree quite well with the exact values. Also, the fluctuation of data is small, in contrast to that obtained using the quantum Monte Carlo method. \textcopyright{} 1996 The American Physical Society.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.
We present a numerical method for the evaluation of dynamical response functions at finite temperatures in one-dimensional strongly correlated systems. The approach is based on the density-matrix renormalization group method, combined with the finite-temperature Lanczos diagonalization. The feasibility of the method is tested on the example of dynamical spin correlations in the anisotropic Heisenberg chain, in particular, it yields nontrivial results for the critical behavior in the isotropic case.
A numerical algorithm to calculate exact finite-temperature spectra of many-body lattice Hamiltonians is formulated by combining the typicality approach and the shifted Krylov subspace method. The combined algorithm, which we name finite-temperature shifted Krylov subspace method for simulating spectra (FTKωωω), efficiently reproduces the canonical-ensemble probability distribution at finite temperatures with the computational cost proportional to the Fock space dimension. The present FTKωωω enables us to exactly calculate finite-temperature spectra of many-body systems whose system sizes are twice larger than those handled by the canonical ensemble average and allows us to access the frequency domain without sequential real-time evolution often used in previous studies. By employing the reweighting method with the present algorithm, we obtain significant reduction of the numerical costs for temperature sweeps. Application to the Kiteav-Heisenberg model (KHM) on a honeycomb lattice demonstrates the capability of the FTKωωω. The KHM shows quantum phase transitions from the quantum spin liquid (QSL) phase to magnetically ordered phases when the finite Heisenberg exchange coupling is introduced. We examine temperature dependence of dynamical spin structure factors of the KHM in proximity to the QSL. It is clarified that the crossover from a spin-excitation continuum, which is a characteristic of the QSL, to a damped high-energy magnon mode occurs at temperatures higher than the energy scale of the Heisenberg couplings or the spin gap that is a signature of the QSL at zero temperature. The crossover and the closeness to the Kitaev's QSL are quantitatively measured by the width of the excitation continuum or the magnon spectrum. The present results shed new light on analysis of neutron scattering and other spectroscopy measurements on QSL candidates.