The Thouless theorem for matrix product states and subsequent post-density matrix renormalization group methods

TL;DR

This paper extends the Thouless theorem to matrix product states (MPS), establishing a non-redundant parametrization of the MPS manifold, and applies these concepts to improve excitation energy calculations in quantum many-body systems.

Contribution

It introduces the Thouless theorem for MPS, linking MPS tangent space parametrization to excitation theories like TDA, CI, and RPA, and demonstrates practical applications in quantum chemistry and condensed matter.

Findings

Derived the Thouless theorem for MPS.

Extended the analogy between Hartree-Fock and DMRG to excitation theories.

Applied RPA-MPS and TDA-MPS to compute excitation energies in Hubbard and polyene models.

Abstract

The similarities between Hartree-Fock (HF) theory and the density-matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function ansatz. Linearization of the time-dependent variational principle near a variational minimum allows to derive the random phase approximation (RPA). We show that the non-redundant parametrization of the matrix product state (MPS) tangent space [J. Haegeman et al., Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e. an explicit non-redundant parametrization of the entire MPS manifold, starting from a specific MPS reference. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation (TDA), the configuration interaction (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS ansatz with single and double excitations to improve on the ground state and to calculate low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calculations of the RPA-MPS correlation energy. With the long-range quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.