Calculus of continuous matrix product states

Abstract

We discuss various properties of the variational class of continuous matrix product states, a class of Ansatz states for one-dimensional quantum fields that was recently introduced as the direct continuum limit of the highly successful class of matrix product states. We discuss both attributes of the physical states, e.g., by showing in detail how to compute expectation values, as well as properties intrinsic to the representation itself, such as the gauge freedom. We consider general translation noninvariant systems made of several particle species and derive certain regularity properties that need to be satisfied by the variational parameters. We also devote a section to the translation invariant setting in the thermodynamic limit and show how continuous matrix product states possess an intrinsic ultraviolet cutoff. Finally, we introduce a new set of states, which are tangent to the original set of continuous matrix product states. For the case of matrix product states, this construction has recently proven relevant in the development of new algorithms for studying time evolution and elementary excitations of quantum spin chains. We thus lay the foundation for similar developments for one-dimensional quantum fields.

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

“…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…”

“…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 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.…”

“…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.…”

An Efficient Variational Principle for the Direct Optimization of Excited States

“…Instead, existing excited state methods typically require an ansatz to use its variational freedom to satisfy the needs of many eigenstates simultaneously, the difficulty of which has limited our predictive power over the doubly-excited states in light harvesting systems, the spectra of excited state absorption experiments, and the band gaps of transition metal oxides. For example, linear response (LR) methods such as time de- * Electronic mail: eneuscamman@berkeley.edu pendent HF and DFT [8], CI singles (CIS) [8], equation of motion CC with singles and doubles (EOM-CCSD) [9], and LR DMRG [10][11][12] are limited by the requirement that all excited states of interest must be found in the ground state's LR space, which for a nonlinear ansatz is typically much less flexible than its full variational space. In many other cases, such as state-averaged complete active space methods [13,14], some VMC approaches [15], and directly targeted DMRG [16], crucial ansatz components (often the one particle basis) are required to be the same for the ground and all excited states.…”