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Cambridge University Press eBooks

Dynamical Correlation Functions

Tao Xiang

Published Aug 31, 2023• 0 citations
Publisher

Abstract

This chapter presents the methods of calculating dynamical correlation functions, including the continued-fraction expansion, Lanczos-DMRG, Lanczos-MPS, Chebyshev-MPS, correction vector, conjugate gradient, and dynamical DMRG methods. In the practical application of the Lanczos-MPS or Chebyshev-MPS method, a reorthogonalization scheme is introduced to optimize all the MPS generated with these methods. As an example of application, we investigate the dynamical spectra of the spin-1/2 Heisenberg model using the Chebyshev-MPS method.

DOI
10.1017/9781009398671.019

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This chapter presents the methods of calculating dynamical correlation functions, including the continued-fraction expansion, Lanczos-DMRG, Lanczos-MPS, Chebyshev-MPS, correction vector, conjugate gradient, and dynamical DMRG methods. In the practical application of the Lanczos-MPS or Chebyshev-MPS method, a reorthogonalization scheme is introduced to optimize all the MPS generated with these methods. As an example of application, we investigate the dynamical spectra of the spin-1/2 Heisenberg model using the Chebyshev-MPS method.
The density-matrix renormalization group (DMRG) algorithm can be adapted to the calculation of dynamical correlation functions in various ways which all represent compromises between computational efficiency and physical accuracy. In this paper we reconsider the oldest approach based on a suitable Lanczos-generated approximate basis and implement it using matrix product states (MPS) for the representation of the basis states. The direct use of matrix product states combined with an ex post reorthogonalization method allows us to avoid several shortcomings of the original approach, namely the multitargeting and the approximate representation of the Hamiltonian inherent in earlier Lanczos-method implementations in the DMRG framework, and to deal with the ghost problem of Lanczos methods, leading to a much better convergence of the spectral weights and poles. We present results for the dynamic spin structure factor of the spin-1/2 antiferromagnetic Heisenberg chain. A comparison to Bethe ansatz results in the thermodynamic limit reveals that the MPS-based Lanczos approach is much more accurate than earlier approaches at minor additional numerical cost.
The density matrix renormalization group (DMRG) method allows for very precise calculations of ground state properties in low-dimensional strongly correlated systems. We investigate two methods to expand the DMRG to calculations of dynamical properties. In the Lanczos vector method the DMRG basis is optimized to represent Lanczos vectors, which are then used to calculate the spectra. This method is fast and relatively easy to implement, but the accuracy at higher frequencies is limited. Alternatively, one can optimize the basis to represent a correction vector for a particular frequency. The correction vectors can be used to calculate the dynamical correlation functions at these frequencies with high accuracy. By separately calculating correction vectors at different frequencies, the dynamical correlation functions can be interpolated and pieced together from these results. For systems with open boundaries we discuss how to construct operators for specific wave vectors using filter functions.
The Chebyshev expansion offers a numerically efficient and easy-implement algorithm for evaluating dynamic correlation functions using matrix product states (MPS). In this approach, each recursively generated Chebyshev vector is approximately represented by an MPS. However, the recurrence relations of Chebyshev polynomials are broken by the approximation, leading to an error which is accumulated with the increase of the order of expansion. Here we propose a reorthonormalization approach to remove this error introduced in the loss of orthogonality of the Chebyshev polynomials. Our approach, as illustrated by comparison with the exact results for the one-dimensional XY and Heisenberg models, improves significantly the accuracy in the calculation of dynamical correlation functions.
Current widely-used approaches to calculate spectral functions using the density-matrix renormalization group in frequency space either necessarily include an artificial broadening (correction-vector method), have limited resolution (time-domain density-matrix renormalization group with Fourier transform method), or are limited to low-energy properties or single dominant modes (original continued fraction method). Here we propose an adaptive Lanczos-vector method to calculate the coefficients of a continued fraction expansion of the spectral function iteratively. We show that one can obtain a very accurate representation of the spectral function very efficiently, and that one can also directly extract the spectral weights and poles for the discrete system.
Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help understand condensed matter experiments. Within the DMRG framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction-vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that Krylov-space approach can be more accurate and efficient than conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.
Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help illustrate condensed matter experiments. Within the density matrix renormalization group (DMRG) framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that the Krylov-space approach can be more accurate and efficient than the conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.