Simulating strongly correlated quantum systems with tree tensor networks

By V. Murg, Ors Legeza, R. Noack, Frank Verstraete. 2010

204 citations recorded by the source corpus.

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 z, where the z=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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b32180e9234e8e9ce8171b7cb8887bacec752823

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Citations in the corpus

Changing the topology of Tensor Networks (2012, background): “Tensor networks are of interest especially in quantum chemistry (see [1, 2]) but also for general large data sets, they can become of practical use.”
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors (2013, methodology): “Formats of this type have been used successfully in the quantum physics and chemistry communities in the last decade [21, 23, 27, 32].”
Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, background): “Tree tensor networks are also interesting quite apart from the connection to the all-to-all circuit [56–65].”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, methodology): “using the numbering rules (169) for each orbitals.”
The density matrix renormalization group for ab initio quantum chemistry (2014, methodology): “For a given desired accuracy, the optimization algorithms for TTNS and CGTNS are currently less efficient than QC-DMRG.”