SciPost Phys. 10, 058 (2021) ·
published 9 March 2021
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Quantum lattice models with large local Hilbert spaces emerge across various fields in quantum many-body physics. Problems such as the interplay between fermions and phonons, the BCS-BEC crossover of interacting bosons, or decoherence in quantum simulators have been extensively studied both theoretically and experimentally. In recent years, tensor network methods have become one of the most successful tools to treat such lattice systems numerically. Nevertheless, systems with large local Hilbert spaces remain challenging. Here, we introduce a mapping that allows to construct artificial $U(1)$ symmetries for any type of lattice model. Exploiting the generated symmetries, numerical expenses that are related to the local degrees of freedom decrease significantly. This allows for an efficient treatment of systems with large local dimensions. Further exploring this mapping, we reveal an intimate connection between the Schmidt values of the corresponding matrix\hyp product\hyp state representation and the single\hyp site reduced density matrix. Our findings motivate an intuitive physical picture of the truncations occurring in typical algorithms and we give bounds on the numerical complexity in comparison to standard methods that do not exploit such artificial symmetries. We demonstrate this new mapping, provide an implementation recipe for an existing code, and perform example calculations for the Holstein model at half filling. We studied systems with a very large number of lattice sites up to $L=501$ while accounting for $N_{\rm ph}=63$ phonons per site with high precision in the CDW phase.
Sebastian Paeckel, Thomas Köhler, Salvatore R. Manmana
SciPost Phys. 3, 035 (2017) ·
published 17 November 2017
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We present an algorithmic construction scheme for matrix-product-operator
(MPO) representations of arbitrary $U(1)$-invariant operators whenever there is
an expression of the local structure in terms of a finite-states machine (FSM).
Given a set of local operators as building blocks, the method automatizes two
major steps when constructing a $U(1)$-invariant MPO representation: (i) the
bookkeeping of auxiliary bond-index shifts arising from the application of
operators changing the local quantum numbers and (ii) the appearance of phase
factors due to particular commutation rules. The automatization is achieved by
post-processing the operator strings generated by the FSM. Consequently, MPO
representations of various types of $U(1)$-invariant operators can be
constructed generically in MPS algorithms reducing the necessity of expensive
MPO arithmetics. This is demonstrated by generating arbitrary products of
operators in terms of FSM, from which we obtain exact MPO representations for
the variance of the Hamiltonian of a $S=1$ Heisenberg chain.
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