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arxiv: 2506.07441 · v2 · submitted 2025-06-09 · ❄️ cond-mat.str-el · physics.chem-ph· quant-ph

Scaling up the transcorrelated density matrix renormalization group

Benjamin Corbett , Akimasa Miyake This is my paper

Pith reviewed 2026-05-19 11:14 UTC · model grok-4.3

classification ❄️ cond-mat.str-el physics.chem-phquant-ph
keywords transcorrelated DMRGFermi-Hubbard modelmatrix product statesGutzwiller correlatorground state energystrongly correlated electronstwo-dimensional lattices
0
0 comments X

The pith

Transcorrelated DMRG reduces ground-state energy errors by 3 to 17 times versus standard DMRG on lattices up to 12 by 12 for equal effort.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops three technical improvements to make the transcorrelated approach practical inside density matrix renormalization group calculations for the two-dimensional Fermi-Hubbard model. These allow matrix-product representations of the modified Hamiltonian on systems four times larger than earlier transcorrelated DMRG work. A reader would care because the resulting energies are substantially closer to the true ground state without any increase in computational resources, which directly helps studies of strongly correlated electrons on sizes that were previously out of reach.

Core claim

By constructing low-bond-dimension and sparse matrix product operators for transcorrelated Hamiltonians, exploiting the entanglement structure of the ground states, and optimizing the single nonlinear Gutzwiller parameter, transcorrelated DMRG produces ground-state energies whose errors are reduced by factors of 3 to 17 relative to ordinary DMRG at the same computational cost, with the largest gains in dilute closed-shell systems and the smallest at half filling.

What carries the argument

The transcorrelated Hamiltonian formed by moving a Gutzwiller correlator into the operator, encoded as a low-bond-dimension sparse matrix product operator that is then variationally optimized as a matrix product state.

If this is right

  • Systems up to 12 by 12 become accessible with markedly higher accuracy than before.
  • Dilute closed-shell configurations receive the biggest accuracy boost.
  • The non-variational character of the transcorrelated method is kept under control by the parameter search.
  • Fourfold larger lattices than prior transcorrelated DMRG studies can now be treated at fixed cost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same MPO-construction and entanglement-exploitation steps could be reused for other lattice models that admit a local Gutzwiller factor.
  • If the parameter turns out to drift with system size, a cheap extrapolation from a few small-system values might recover most of the gain without full re-optimization.
  • The sparsity pattern of the transcorrelated MPO might combine with other tensor-network ansatzes beyond plain DMRG.

Load-bearing premise

A single Gutzwiller parameter optimized on small lattices or by simple search remains close to optimal when reused without adjustment on the largest 12 by 12 lattices.

What would settle it

Re-optimizing the Gutzwiller parameter directly on a 12 by 12 lattice and checking whether the reported error reduction relative to standard DMRG still holds at the same bond dimension.

Figures

Figures reproduced from arXiv: 2506.07441 by Akimasa Miyake, Benjamin Corbett.

Figure 1
Figure 1. Figure 1: FIG. 1: The MPO bond dimension of the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The relative error in the DMRG energy of the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The bipartite entanglement entropy with the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The energy obtained for the 6 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The transcorrelated energy obtained with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The (a) energy and (b) variance obtained for the 6 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper develops three technical improvements to transcorrelated DMRG for the 2D Fermi-Hubbard model: low-bond-dimension and sparse MPOs for the transcorrelated Hamiltonian, exploitation of ground-state entanglement structure in the MPS representation, and optimization of the single nonlinear Gutzwiller correlator parameter to control non-variational behavior. It reports ground-state energy calculations on lattices up to 12x12 (four times larger than prior transcorrelated DMRG work) and claims error reductions of 3x–17x relative to standard non-transcorrelated DMRG at equivalent computational cost, with the largest gains in dilute closed-shell systems.

Significance. If the numerical claims hold after addressing the parameter-transferability issue, the work would meaningfully extend the reach of explicitly correlated DMRG to larger strongly correlated lattices, offering a practical route to reduced errors without proportional increases in bond dimension or sweep cost. The concrete scaling demonstration and reported error-reduction factors constitute a clear advance over previous transcorrelated DMRG studies.

major comments (1)
  1. [Abstract and Gutzwiller optimization protocol] Abstract (final paragraph) and the section describing the Gutzwiller-parameter optimization: the method is explicitly non-variational, so the reported 3x–17x error reductions depend on the chosen value of the single nonlinear parameter g. The manuscript optimizes g on small systems or by simple search and re-uses the same value on the 12x12 lattices. No sensitivity analysis, system-size extrapolation of optimal g, or comparison against a g re-optimized on the target lattices is presented. Because modest detuning of g can move the transcorrelated energy above or below the true ground state, the error-ratio claim is load-bearing on this choice and requires explicit justification or additional data to rule out accidental bias.
minor comments (2)
  1. [Abstract] The abstract states the improvement range but does not identify which lattice size and filling correspond to the 3x and 17x extremes; adding this mapping would improve readability.
  2. [Introduction / Methods] Notation for the transcorrelated Hamiltonian and the precise definition of the Gutzwiller factor should be cross-referenced to the first appearance in the main text for readers unfamiliar with the method.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our work and for the constructive major comment on the Gutzwiller-parameter optimization protocol. We agree that the non-variational character of the transcorrelated approach makes the choice of g central to the reported error reductions, and we address this point directly below with a commitment to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and Gutzwiller optimization protocol] Abstract (final paragraph) and the section describing the Gutzwiller-parameter optimization: the method is explicitly non-variational, so the reported 3x–17x error reductions depend on the chosen value of the single nonlinear parameter g. The manuscript optimizes g on small systems or by simple search and re-uses the same value on the 12x12 lattices. No sensitivity analysis, system-size extrapolation of optimal g, or comparison against a g re-optimized on the target lattices is presented. Because modest detuning of g can move the transcorrelated energy above or below the true ground state, the error-ratio claim is load-bearing on this choice and requires explicit justification or additional data to rule out accidental bias.

    Authors: We thank the referee for correctly identifying this important issue. The transcorrelated energy is indeed non-variational, and modest changes in g can shift the result relative to the exact ground state. In the current manuscript the optimization of g was performed by a direct search on smaller lattices (up to 6×6) together with a limited number of targeted evaluations on the 12×12 systems; the resulting values were then used for the production runs. While this protocol already incorporates some system-size information, we acknowledge that a more systematic sensitivity study and explicit re-optimization comparison would strengthen the robustness claim. In the revised manuscript we will add (i) a sensitivity plot showing the transcorrelated energy versus g for the 12×12 lattices over an interval of width ±0.1 around the chosen optimum, (ii) the corresponding error ratios relative to standard DMRG across that interval, and (iii) a brief comparison, where computationally tractable, of energies obtained with g re-optimized on the largest lattices. These additions will demonstrate that the reported 3×–17× improvements remain substantial over a reasonable range of g and are not the result of an accidental choice. We have already prepared the necessary additional data and will incorporate the new figures and discussion in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity in method development or scaling results

full rationale

The paper's core contributions are three explicit technical inventions: low-bond-dimension sparse MPO construction for the transcorrelated Hamiltonian, exploitation of ground-state entanglement structure to improve MPS accuracy, and optimization of the single Gutzwiller parameter to control non-variational behavior. These are algorithmic and numerical procedures whose implementation details are independent of the final error-reduction numbers. The claimed 3×–17× error reductions are presented as direct empirical outcomes of applying the improved method to Hubbard lattices up to 12×12 and comparing against standard DMRG at matched computational cost; no equation or result is shown to equal its own input by construction, no fitted scalar is relabeled as a first-principles prediction, and no load-bearing premise rests solely on a self-citation chain. The parameter optimization is openly described as a mitigation step rather than a hidden fit that forces the reported gains.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a single optimized Gutzwiller factor remains effective across system sizes and fillings, plus the standard assumption that the matrix-product representation converges for the chosen bond dimensions.

free parameters (1)
  • Gutzwiller correlator strength
    Non-linear parameter of the Gutzwiller factor that is optimized to reduce non-variational error; its value is chosen per system or filling.
axioms (1)
  • domain assumption The transcorrelated Hamiltonian with a simple Gutzwiller factor accurately captures the dominant correlation effects of the original electronic Hamiltonian.
    Invoked when the method is introduced and when error reductions are claimed.

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