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arXiv:2411.08036

Superpositions of tree-tensor networks for single-reference ground states in the strong correlation regime

Dominic Bergeron

Published Nov 12, 2024• 0 citations
arXivHTMLPDF

Abstract

The fermionic many-body problem in the strong correlation regime is notoriously difficult to tackle. In a previous work (Phys. Rev. B 101, 045109 (2020)), we have proposed to extend the single-reference coupled-cluster (SRCC) method to the strong correlation regime using low-rank tensor decompositions (LRTD) to express the cluster operator, without truncating it with respect to the number of excitations. For that purpose, we have proposed a new type of LRTD called ``superpositions of tree-tensor networks'' (STTN), which use the same set of building blocs to define all the tensors involved in the CC equations, and combine different ``channels'', i.e. different types of pairing among excited particles and holes, in the decomposition of a given tensor. Those two principles are aimed at globally minimizing the total number of free parameters required to accurately represent the ground state. In this work, we show that STTN can indeed be compact and accurate representations of strongly correlated ground states by using them to express the CC cluster operator amplitudes and wave function coefficients of exact grounds states of small two-dimensional Hubbard clusters, at half-filling, up to three particle-hole excitations. We show the compactness of STTN by using a number of free parameters smaller than the number of equations in the CCSD approximation, i.e. much smaller than the number of fitted tensor elements. We find that, for the systems considered, the STTN are more accurate as the size of the system increases and that combining different channels in the decompositions of the most strongly correlated tensors is crucial to obtain good accuracy.

DOI
10.48550/arxiv.2411.08036

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The fermionic many-body problem in the strong correlation regime is notoriously difficult to tackle. In a previous work (Phys. Rev. B 101, 045109 (2020)), we have proposed to extend the single-reference coupled-cluster (SRCC) method to the strong correlation regime using low-rank tensor decompositions (LRTD) to express the cluster operator, without truncating it with respect to the number of excitations. For that purpose, we have proposed a new type of LRTD called superpositions of tree-tensor networks'' (STTN), which use the same set of building blocs to define all the tensors involved in the CC equations, and combine different channels'', i.e. different types of pairing among excited particles and holes, in the decomposition of a given tensor. Those two principles are aimed at globally minimizing the total number of free parameters required to accurately represent the ground state. In this work, we show that STTN can indeed be compact and accurate representations of strongly correlated ground states by using them to express the CC cluster operator amplitudes and wave function coefficients of exact grounds states of small two-dimensional Hubbard clusters, at half-filling, up to three particle-hole excitations. We show the compactness of STTN by using a number of free parameters smaller than the number of equations in the CCSD approximation, i.e. much smaller than the number of fitted tensor elements. We find that, for the systems considered, the STTN are more accurate as the size of the system increases and that combining different channels in the decompositions of the most strongly correlated tensors is crucial to obtain good accuracy.
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Abstract. Compactly representing and efficently applying linear operators are fundamental ingredients in tensor network methods for simulating quantum many-body problems and solving high-dimensional problems in scientific computing. In this work, we study such representations for tree tensor networks, the so-called tree tensor network operators (TTNOs), paying particular attention to Hamiltonian operators that involve long-range pairwise interactions between particles. Generalizing the work by Lin, Tong, and others on matrix product operators, we establish a direct connection between the hierarchical low-rank structure of the interaction matrix and the TTNO property. This connection allows us to arrive at very compact TTNO representations by compressing the interaction matrix into a hierarchically semiseparable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.
We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-NNN tensor from exponential to polynomial in NNN, and this has become a popular approach for machine learning. We introduce the idea of imposing low-rank constraints on the tensors that compose the tensor network. With this modification, the time and space complexities for the network optimization can be substantially reduced while maintaining high accuracy. We detail this idea for tree tensor network states (TTNS) and projected entangled-pair states. Simulations of spin models on Cayley trees with low-rank TTNS exemplify the effect of rank constraints on the expressive power. We find that choosing the tensor rank rrr to be on the order of the bond dimension mmm, is sufficient to obtain high-accuracy groundstate approximations and to substantially outperform standard TTNS computations. Thus low-rank tensor networks are a promising route for the simulation of quantum matter and machine learning on large data sets.
A constant goal of quantum chemistry is devising accurate and computationally effective methods for molecular simulations. In this work an application of tensor decompositions in the context of highly accurate coupled cluster theory, which is often considered a "gold standard", is investigated. The scheme we develop is aimed to mitigate the steep growth of the computational cost with the system size, and hence to overcome the "curse of dimensionality" common in many potent methods in electronic structure. We show how to reduce the computational effort of the restricted coupled cluster with singles and doubles (RCCSD) by two orders of magnitude by introducing alternative parameterizations of the method using "canonical polyad decomposition" (CPD) and "tensor hypercontraction" (THC) formats. After describing CPD and THC formats in detail, we demonstrate how to cast regular index based tensors into a decomposed form. The number of parameters and the accuracy of these representations depend on the expansion length (rank) of the approximation. We investigate the dependence of rank upon the size of the system and a target accuracy and show it to be low for typical tensors in electronic structure. We then provide a generic procedure to reformulate any coupled cluster method using tensor decompositions. Two specific approximate methods, THC-RCCSD and CPD-RCCSD, are derived. We demonstrate the accuracy of these new approaches by calculating energies of a large set of organic molecules, as well as by simulations of Hubbard models. Finally, it is shown how the restriction of the number of parameters in approximate coupled cluster can improve the accuracy in the challenging strong correlation regime. We conclude by discussing a connection of our findings to other new developments in the coupled cluster theory and propose possible extensions of our approach.
We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination number zzz, where the z=2z=2z=2 case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach.