Scaling up the transcorrelated density matrix renormalization group

TL;DR

The paper tackles the challenge of accurately solving strongly correlated electron systems by scaling transcorrelated DMRG to large 2D lattices. It combines a highly optimized MPO construction, entanglement-informed mappings from momentum-space modes to MPS tensors, and self-consistent optimization of the Gutzwiller correlator J to produce a non-Hermitian transcorrelated Hamiltonian whose ground state is more amenable to MPS representation. The approach yields substantial improvements in ground-state energy accuracy over conventional DMRG, achieving up to $12\times12$ lattices with relative errors significantly reduced, particularly in dilute, closed-shell regimes, while identifying limits posed by entanglement near the Fermi surface at half-filling. These advances broaden the practical applicability of TC-DMRG to larger, more complex systems and lay groundwork for further refinements in correlator design and MPO efficiency. Overall, the work demonstrates that carefully engineered MPOs and entanglement-aware tensor mappings can unlock scalable, high-precision TC-DMRG calculations for strongly correlated electron models.

Abstract

Explicitly correlated methods, such as the transcorrelated method which shifts a Jastrow or Gutzwiller correlator from the wave function to the Hamiltonian, are designed for high-accuracy calculations of electronic structures, but their application to larger systems has been hampered by the computational cost. We develop improved techniques for the transcorrelated density matrix renormalization group (DMRG), in which the ground state of the transcorrelated Hamiltonian is represented as a matrix product state (MPS), and demonstrate large-scale calculations of the ground-state energy of the two-dimensional Fermi-Hubbard model. Our developments stem from three technical inventions: (i) constructing matrix product operators (MPO) of transcorrelated Hamiltonians with low bond dimension and high sparsity, (ii) exploiting the entanglement structure of the ground states to increase the accuracy of the MPS representation, and (iii) optimizing the non-linear parameter of the Gutzwiller correlator to mitigate the non-variational nature of the transcorrelated method. We examine systems of size up to $12 \times 12$ lattice sites, four times larger than previous transcorrelated DMRG studies, and demonstrate that transcorrelated DMRG yields significant improvements over standard non-transcorrelated DMRG for equivalent computational effort. Transcorrelated DMRG reduces the error of the ground state energy by $3\times$-$17 \times$, with the smallest improvement seen for a small system at half-filling and the largest improvement in a dilute closed-shell system.

Figures (9)

• Figure 1: The MPO bond dimension of the (transcorrelated) momentum space Fermi-Hubbard Hamiltonian $\hat{H}_\text{ks} (\widetilde{H}_\text{ks})$ for square periodic lattices. MPOs were constructed with either the bipartite algorithm from 'Renormalizer' Ren-Shuai-2018 or the hybrid algorithm from 'ITensorMPOConstruction' ITensorMPOConstruction. We discuss the anomaly for the $12 \times 12$ transcorrelated system in Appendix \ref{['sec:Transcorrelated bond dimension']}.
• Figure 2: The bipartite entanglement entropy with the $\epsilon$ mapping. Displayed is the entanglement entropy of each bipartition of the MPS approximation at $m = 1024$ to the ground state of the transcorrelated Hamiltonian for the $8 \times 8$ system with $N_e \in \{ 26, 28, 44 \}$ and $J \in \{ 0, J^*_r \}$, where $J^*_r$ is given in Table \ref{['tab:j values']}. The vertical lines correspond to the shell structure of $\epsilon(\bm{k})$.
• Figure 3: The relative error in the DMRG energy of the $6 \times 6$ system with $N_e \in \{ 24, 36 \}$ and the row-major (R), Fiedler (F) and $\epsilon$ mappings. For $N_e = 36$ we include the bipartite (B) mapping. We also show the three data points, one at $N_e = 24$ and two at $N_e = 36$, from Ref. Baiardi-Reiher-2020 with the Fiedler mapping.
• Figure 4: The bipartite entanglement entropy with the bipartite mapping. Displayed is the entanglement entropy for each bipartition of the MPS approximation at $m = 512$ to the ground state of the transcorrelated Hamiltonian for the $8 \times 8$ system with $N_e = 64$ and $J \in \{ 0, J^*_r \}$, where $J^*_r$ is given in Table \ref{['tab:j values']}. The vertical lines correspond to the shell structure of $|\epsilon(\bm{k})|$.
• Figure 5: The energy and variance obtained for the $6 \times 6$ transcorrelated system with $N_e = 36$ and $J \in \{ 0, -0.4, -0.5, -0.6 \}$ as a function of the MPS bond dimension. The horizontal line is the AFQMC reference energy given in Table \ref{['tab:energies']}, with a width proportional to the reported uncertainty.