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arXiv:1501.07216 · Physical Review B

Spectral functions and time evolution from the Chebyshev recursion

F. Alexander Wolf, Jorge A. Justiniano, Ian P. McCulloch, Ulrich Schollwöck

Published Mar 31, 2015• 65 citations
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Abstract

We link linear prediction of Chebyshev and Fourier expansions to analytic continuation. We push the resolution in the Chebyshev-based computation of $T=0$ many-body spectral functions to a much higher precision by deriving a modified Chebyshev series expansion that allows to reduce the expansion order by a factor $\ensuremath{\sim}\frac{1}{6}$. We show that in a certain limit the Chebyshev technique becomes equivalent to computing spectral functions via time evolution and subsequent Fourier transform. This introduces a novel recursive time-evolution algorithm that instead of the group operator ${e}^{\ensuremath{-}iHt}$ only involves the action of the generator $H$. For quantum impurity problems, we introduce an adapted discretization scheme for the bath spectral function. We discuss the relevance of these results for matrix product state (MPS) based DMRG-type algorithms, and their use within the dynamical mean-field theory (DMFT). We present strong evidence that the Chebyshev recursion extracts less spectral information from $H$ than time evolution algorithms when fixing a given amount of created entanglement.

DOI
10.1103/physrevb.91.115144

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We link linear prediction of Chebyshev and Fourier expansions to analytic continuation. We push the resolution in the Chebyshev-based computation of T=0T=0T=0 many-body spectral functions to a much higher precision by deriving a modified Chebyshev series expansion that allows to reduce the expansion order by a factor \ensuremath∼16\ensuremath{\sim}\frac{1}{6}\ensuremath∼61​. We show that in a certain limit the Chebyshev technique becomes equivalent to computing spectral functions via time evolution and subsequent Fourier transform. This introduces a novel recursive time-evolution algorithm that instead of the group operator e\ensuremath−iHt{e}^{\ensuremath{-}iHt}e\ensuremath−iHt only involves the action of the generator HHH. For quantum impurity problems, we introduce an adapted discretization scheme for the bath spectral function. We discuss the relevance of these results for matrix product state (MPS) based DMRG-type algorithms, and their use within the dynamical mean-field theory (DMFT). We present strong evidence that the Chebyshev recursion extracts less spectral information from HHH than time evolution algorithms when fixing a given amount of created entanglement.
We improve a recently developed expansion technique for calculating real-frequency spectral functions of any one-dimensional model with short-range interactions, by postprocessing computed Chebyshev moments with linear prediction. This can be achieved at virtually no cost, and in sharp contrast to existing methods based on the dampening of the moments, improves the spectral resolution rather than lowering it. We validate the method for the exactly solvable resonating level model and the single impurity Anderson model. It is capable of resolving sharp Kondo resonances, as well as peaks within the Hubbard bands when employed as an impurity solver for dynamical mean-field theory. Our method works at zero temperature and allows for arbitrary discretization of the bath spectrum. It achieves similar precision as the dynamical density matrix renormalization group, at lower cost. We also propose an alternative expansion, of $\mathbb{1}\ensuremath{-}exp(\ensuremath{-}\ensuremath{\tau}H)$ instead of the usual HHH, which opens the possibility of using established methods for the time evolution of matrix product states to calculate the spectral functions directly.
We compute the spectral functions for the two-site dynamical cluster theory and for the two-orbital dynamical mean-field theory in the density-matrix renormalization group (DMRG) framework using Chebyshev expansions represented with matrix product states (MPS). We obtain quantitatively precise results at modest computational effort through technical improvements regarding the truncation scheme and the Chebyshev rescaling procedure. We furthermore establish the relation of the Chebyshev iteration to real-time evolution and discuss technical aspects as computation time and implementation in detail.
We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main features of this Chebyshev matrix product state (CheMPS) approach are as follows: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it can exploit the fact that the latter can be much smaller than the model's many-body bandwidth; (iii) it offers a well-controlled broadening scheme that allows finite-size effects to be either resolved or smeared out, as desired; (iv) it is based on using MPS tools to recursively calculate a succession of Chebyshev vectors $|{t}_{n}\ensuremath{\rangle}$, (v) the entanglement entropies of which were found to remain bounded with increasing recursion order nnn for all cases analyzed here; and (vi) it distributes the total entanglement entropy that accumulates with increasing nnn over the set of Chebyshev vectors $|{t}_{n}\ensuremath{\rangle}$, which need not be combined into a single vector. In this way, the growth in entanglement entropy that usually limits density matrix renormalization group (DMRG) approaches is packaged into conveniently manageable units. We present zero-temperature CheMPS results for the structure factor of spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to three benchmark methods, we find that CheMPS (a) yields results comparable in quality to those of correction-vector DMRG, at dramatically reduced numerical cost; (b) agrees well with Bethe ansatz results for an infinite system, within the limitations expected for numerics on finite systems; and (c) can also be applied in the time domain, where it has potential to serve as a viable alternative to time-dependent DMRG (in particular, at finite temperatures). Finally, we present a detailed error analysis of CheMPS for the case of the noninteracting resonant level model.
We present and test a new algorithm for time-evolving quantum many-body systems initially proposed by Holzner et al. [Phys. Rev. B 83, 195115 (2011)]. The approach is based on merging the matrix product state (MPS) formalism with the method of expanding the time-evolution operator in Chebyshev polynomials. We calculate time-dependent observables of a system of hardcore bosons quenched under the Bose-Hubbard Hamiltonian on a one-dimensional lattice. We compare the new algorithm to more standard methods using the MPS architecture. We find that the Chebyshev method gives numerically exact results for small times. However, the reachable times are smaller than the ones obtained with the other state-of-the-art methods. We further extend the new method using a spectral-decomposition-based projective scheme that utilizes an effective bandwidth significantly smaller than the full bandwidth, leading to longer evolution times than the nonprojective method and more efficient information storage, data compression, and less computational effort.
This thesis is concerned with two main topics: first, the advancement of the density matrix renormalization group (DMRG) and, second, its applications. In the first project of this thesis we exploit the common mathematical structure of the numerical renormalization group and the DMRG, namely, matrix product states (MPS), to implement an efficient numerical treatment of a two-lead, multi-level Anderson impurity model. By adopting a star-like geometry, where each species (spin and lead) of conduction electrons is described by its own so-called Wilson chain, instead of a single Wilson chain we achieve a very significant reduction in the numerical resources required to obtain reliable results. Moreover, we show that it is possible to find an "optimal" chain basis, in which chain degrees of freedom of different Wilson chains become effectively decoupled from each other further out on the Wilson chains. This basis turns out to also diagonalize the model's chain-to-chain scattering matrix. In the second project we show that Chebychev expansions offer numerically efficient representations for calculating spectral functions of one-dimensional lattice models using MPS methods. The main features of this Chebychev matrix product state (CheMPS) approach are: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it offers a well-controlled broadening scheme; (iii) it is based on using MPS tools to recursively calculate a succession of Chebychev vectors, (iv) whose entanglement entropies were found to remain bounded with increasing recursion order for all cases analyzed here. We present CheMPS results for the structure factor of spin-1/2 antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to benchmark methods, we find that CheMPS yields results comparable in quality to those of correction vector DMRG, at dramatically reduced numerical cost and agrees well with Bethe Ansatz results for an infinite system, within the limitations expected for numerics on finite systems. Following these technologically focused projects we study the so-called Kondo cloud by means of the DMRG in the third project. The Kondo cloud describes the effect of spatially extended spin-spin correlations of a magnetic moment and the conduction electrons which screen the magnetic moment through the Kondo effect at low temperatures. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. We investigate how perturbations which destroy the Kondo effect, like an applied gate potential or a magnetic field, affect the formation of the screening cloud. In a forth project we address the impact of Quantum (anti-)Zeno physics resulting from repeated single-site resolved observations on the many-body dynamics. We use time-dependent DMRG to obtain the time evolution of the full many-body wave function that is then periodically projected in order to simulate realizations of stroboscopic measurements. For the example of a 1-D lattice of spin-polarized fermions with nearest-neighbor interactions, we find regimes for which many-particle configurations are stabilized and destabilized depending on the interaction strength and the time between observations.
We present a method for extrapolation of real-time dynamical correlation functions which can improve the capability of matrix product state methods to compute spectral functions. Unlike the widely used linear prediction method, which ignores the origin of the data being extrapolated, our recursion methods utilize a representation of the wavefunction in terms of an expansion of the same wavefunction and its translations at earlier times. This recursion method is exact for a noninteracting Fermi system. Surprisingly, the recursion method is also more robust than linear prediction at large interaction strength. We test this method on the Hubbard two-leg ladder, and present more accurate results for the spectral function than previous studies.