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

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 nonredundant parameterization of the matrix product state (MPS) tangent space [J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization 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.

Linear response theory for the density matrix renormalization group: Efficient algorithms for strongly correlated excited states

“…Formulating the determination of the poles as an eigenvalue problem yields the Tamm-Dancoff and random phase approximations to excited states in DMRG. An explicit route to derive the DMRG-TDA and DMRG-RPA eigenvalue equations is to use a linearization of the time-dependent variational principle, 22,26,27 from which the TDA can be understood as a variational approximation to RPA. Our objective here is to formulate an efficient sweep algorithm to solve the DMRG-TDA and DMRG-RPA equations.…”

“…In this study, we focus on a so-called non-redundant parameterization in terms of the projectorsQ (n) , while previous studies focused on explicit expressions for the tangent space vectors in MPS terminology. 23,26,27 The projectors for the first order space were already introduced during the discussion of DMRG-LRT in Eqs. (12)- (26), i.e., the representation ofQ (1) i is chosen from…”

“…The connection between the projector and non-redundant tangent space parameterization of the MPS is discussed in detail in Ref. 26. Briefly, an explicit parameterization of the non-redundant tangent space vectors is given by…”

“…One way to relieve this drawback of the SA-DMRG algorithm is to use the stateaveraged harmonic Davidson (SA-HD) DMRG algorithm to target higher excited states directly. 20 Recently, excitations have been constructed on top of a reference MPS wavefunction, [21][22][23][24][25][26][27] analogous to the concept of particle-hole excitations on top of a reference Slater determinant. In this post-MPS or post-DMRG theory, the reference MPS wavefunction provides a site-based meanfield ansatz, 22,28 and excitations consist of "local" changes in this mean-field ansatz.…”

“…In this post-MPS or post-DMRG theory, the reference MPS wavefunction provides a site-based meanfield ansatz, 22,28 and excitations consist of "local" changes in this mean-field ansatz. 26 In this way, analogues of the Tamm-Dancoff approximation (TDA), 24-27, 29, 30 the random phase approximation (RPA), 22,26,27,31 and configuration interaction with singles and doubles (CISD) 26 were derived for an MPS reference wavefunction. The main advantage of the post-DMRG theory is that it allows to derive excited state information from a ground state calculation, without the need to update or augment the renormalized basis states.…”