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arXiv:1607.01696 · Physical review. B./Physical review. B

Matrix product purifications for canonical ensembles and quantum number distributions

Thomas Barthel

Published Sep 26, 2016• 30 citations
arXivHTMLPDF

Abstract

Matrix product purifications (MPPs) are a very efficient tool for the simulation of strongly correlated quantum many-body systems at finite temperatures. When a system features symmetries, these can be used to reduce computation costs substantially. It is straightforward to compute an MPP of a grand-canonical ensemble, also when symmetries are exploited. This paper provides and demonstrates methods for the efficient computation of MPPs of canonical ensembles under utilization of symmetries. Furthermore, we present a scheme for the evaluation of global quantum number distributions using matrix product density operators (MPDOs). We provide exact matrix product representations for canonical infinite-temperature states, and discuss how they can be constructed alternatively by applying matrix product operators to vacuum-type states or by using entangler Hamiltonians. A demonstration of the techniques for Heisenberg spin-$1/2$ chains explains why the difference in the energy densities of canonical and grand-canonical ensembles decays as $1/L$.

DOI
10.1103/physrevb.94.115157

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Matrix product purifications (MPPs) are a very efficient tool for the simulation of strongly correlated quantum many-body systems at finite temperatures. When a system features symmetries, these can be used to reduce computation costs substantially. It is straightforward to compute an MPP of a grand-canonical ensemble, also when symmetries are exploited. This paper provides and demonstrates methods for the efficient computation of MPPs of canonical ensembles under utilization of symmetries. Furthermore, we present a scheme for the evaluation of global quantum number distributions using matrix product density operators (MPDOs). We provide exact matrix product representations for canonical infinite-temperature states, and discuss how they can be constructed alternatively by applying matrix product operators to vacuum-type states or by using entangler Hamiltonians. A demonstration of the techniques for Heisenberg spin-1/21/21/2 chains explains why the difference in the energy densities of canonical and grand-canonical ensembles decays as 1/L1/L1/L.
For the simulation of equilibrium states and finite-temperature response functions of strongly correlated quantum many-body systems, we compare the efficiencies of two different approaches in the framework of the density matrix renormalization group (DMRG). The first is based on matrix product purifications. The second, more recent one, is based on so-called minimally entangled typical thermal states (METTS). For the latter, we highlight the interplay of statistical and DMRG truncation errors, discuss the use of self-averaging effects, and describe schemes for the computation of response functions. For critical as well as gapped phases of the $\text{spin}\ensuremath{-}1/2$ XXZ chain and the one-dimensional Bose-Hubbard model, we assess the computation costs and accuracies of the two methods at different temperatures. For almost all considered cases, we find that, for the same computation cost, purifications yield more accurate results than METTS---often by orders of magnitude. The METTS algorithm becomes more efficient only for temperatures well below the system's energy gap. The exponential growth of the computation cost in the evaluation of response functions limits the attainable time scales in both methods and we find that in this regard, METTS do not outperform purifications.
We develop a numerical method based on matrix product states for simulating quantum many-body systems at finite temperatures without importance sampling and evaluate its performance in spin-1/2 systems. Our method is an extension of the random phase product state (RPPS) approach introduced recently (Iitaka, arXiv:2006.14459). We show that the original RPPS approach often gives unphysical values for thermodynamic quantities even in the Heisenberg chain. We find that by adding the operation of Trotter gates to the RPPS, the sampling efficiency of the approach significantly increases, and its results are consistent with those of the purification approach. We also apply our method to a frustrated spin-1/2 system to exemplify that it can simulate a system in which the purification approach fails.
Based on the density matrix renormalization group (DMRG), strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). When a system features symmetries, these can be utilized to substantially reduce MPS computation costs. It is conceptually straightforward to simulate canonical ensembles using symmetric METTS. In practice, it is important to alternate between different symmetric collapse bases to decrease autocorrelations in the Markov chain of METTS. To this purpose, we introduce symmetric Fourier and Haar-random block bases that are efficiently mixing. We also show how grand-canonical ensembles can be simulated efficiently with symmetric METTS. We demonstrate these approaches for spin-1/21/21/2 XXZXXZXXZ chains and discuss how the choice of the collapse bases influences autocorrelations as well as the distribution of measurement values and, hence, convergence speeds.
We present a new wavefunction ansatz that combines the technique of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as |Ψ^(N,S,M)_SP−MPS⟩=P_S|Ψ(N,M)_MPS⟩, where P_S is the spin projector for total spin S and |Ψ(N,M)MPS⟩ is an MPS wavefunction with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of Ŝ^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |Ψ(N,M)MPS⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) SP-MPS are closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector P_S naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories. Unlike usual spin projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |Ψ(N,M)MPS⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.
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.
An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.