Chebyshev matrix product state approach for spectral functions

Holzner, Andreas, Weichselbaum, Arnold, McCulloch, Ian P., Schollwock, Ulrich and von Delft, Jan (2011) Chebyshev matrix product state approach for spectral functions. Physical Review B, 83 19 Article No.195115: . doi:10.1103/PhysRevB.83.195115

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Author Holzner, Andreas
Weichselbaum, Arnold
McCulloch, Ian P.
Schollwock, Ulrich
von Delft, Jan
Title Chebyshev matrix product state approach for spectral functions
Journal name Physical Review B   Check publisher's open access policy
ISSN 1098-0121
1550-235X
Publication date 2011-05-01
Year available 2015
Sub-type Article (original research)
DOI 10.1103/PhysRevB.83.195115
Open Access Status File (Publisher version)
Volume 83
Issue 19 Article No.195115
Total pages 20
Place of publication College Park, MD, United States
Publisher American Physical Society
Language eng
Abstract 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 vertical bar t(n)>, (v) the entanglement entropies of which were found to remain bounded with increasing recursion order n for all cases analyzed here; and (vi) it distributes the total entanglement entropy that accumulates with increasing n over the set of Chebyshev vectors vertical bar t(n)>, 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-1/2 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.
Formatted abstract
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 |tn⟩, (v) the entanglement entropies of which were found to remain bounded with increasing recursion order n for all cases analyzed here; and (vi) it distributes the total entanglement entropy that accumulates with increasing n over the set of Chebyshev vectors |tn⟩, 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-1/2 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.
Keyword Dependent Schrodinger-Equation
Quantum Renormalization-Groups
Moments
Systems
Q-Index Code C1
Q-Index Status Confirmed Code
Grant ID 321918
FOR 801
CE110001013
FT100100515
Institutional Status UQ
Additional Notes Received 29 January 2011; revised 24 March 2011; published 10 May 2011; ©2011 American Physical Society

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2012 Collection
 
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