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arxiv: 2509.06241 · v2 · submitted 2025-09-07 · ❄️ cond-mat.str-el · quant-ph

Site Basis Excitation Ansatz for Matrix Product States

Steven R. White This is my paper

Pith reviewed 2026-05-18 17:32 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords matrix product statesexcitation ansatzHeisenberg chainmagnon dispersionquantum spin modelsDMRGone-dimensional systems
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The pith

Site basis excitation ansatz computes high-accuracy one-magnon dispersion for the S=1 Heisenberg chain from a single diagonalization step.

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

The paper presents the site basis excitation ansatz as a simplified way to calculate elementary excitations in one-dimensional quantum lattice models described by matrix product states. It forms a small basis of excitation tensors through one diagonalization step similar to a DMRG update on the infinite ground state. This basis then allows the momentum-dependent energies to be found by diagonalizing small matrices for each wavevector. The method is demonstrated on the spin-one Heisenberg chain where it reproduces the magnon dispersion curve accurately. Sympathetic readers would see this as a practical advance that reduces the computational effort needed to study spin-wave spectra compared to full tangent-space optimizations.

Core claim

The author claims that an infinite matrix product state ground state, prepared with a simple finite-system DMRG procedure, combined with a site-basis excitation ansatz formed by a single multi-state diagonalization, produces the full one-magnon dispersion relation through repeated diagonalization of small non-orthogonal overlap and Hamiltonian matrices. Leaving the basis non-orthogonal proves essential for convergence, while imposing a left-orthonormal gauge degrades performance. The construction further permits exact recovery of all momentum modes from a single localized Wannier excitation translated across sites.

What carries the argument

The site basis excitation ansatz, which generates a compact non-orthogonal set of excitation tensors from one DMRG-like diagonalization to enable efficient momentum-space spectrum calculation.

If this is right

  • The one-magnon dispersion of the S=1 Heisenberg chain is obtained efficiently and with high accuracy.
  • Not imposing a gauge condition on the excitation basis is crucial for the method to converge well.
  • Wannier excitations can be built that, when translated, exactly reconstruct the plane-wave magnon modes at every momentum.
  • The simple finite DMRG approach serves as an effective alternative for obtaining the infinite MPS ground state.

Where Pith is reading between the lines

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

  • If the approach generalizes, it could reduce the cost of computing excitations in other strongly correlated 1D systems.
  • The non-orthogonal basis choice may inspire similar simplifications in higher-dimensional tensor network methods.
  • Localized Wannier excitations open a path to studying real-space properties of magnons more directly.

Load-bearing premise

The infinite matrix product state must accurately represent the true ground state, and the single diagonalization must generate an excitation basis adequate for the desired accuracy without additional refinement.

What would settle it

A direct comparison showing that the SBEA-computed dispersion curve for the S=1 Heisenberg chain differs substantially from well-established numerical benchmarks at key momenta would indicate the method falls short.

Figures

Figures reproduced from arXiv: 2509.06241 by Steven R. White.

Figure 2
Figure 2. Figure 2: FIG. 2. Spin profile of single site excitations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. One-magnon dispersion of the spin-1 Heisenberg [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spin profile of a Wannier excitation for the Heisen [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

We introduce a simple and efficient variation of the tangent-space excitation ansatz used to compute elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). A small basis for the excitation tensors is formed based on a single diagonalization analogous to a single site DMRG step but for multiple states. Once overlap and Hamiltonian matrix elements are found, obtaining the excitation for any momentum only requires diagonalization of a tiny matrix, akin to a non-orthogonal band-theory diagonalization. The approach is based on an infinite MPS description of the ground state, and we introduce an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite system DMRG. For the $S=1$ Heisenberg chain, our method -- site basis excitation ansatz (SBEA) -- efficiently produces the one-magnon dispersion with high accuracy. We also examine the role of MPS gauge choices, finding that not imposing a gauge condition -- leaving the basis nonorthogonal -- is crucial for the approach, whereas imposing a left-orthonormal gauge (as in prior work) severely hampers convergence. We also show how one can construct Wannier excitations, analogous to the Wannier functions of band theory, where one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta.

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

2 major / 2 minor

Summary. The paper introduces the Site Basis Excitation Ansatz (SBEA), a variation of the tangent-space excitation ansatz for elementary excitations in 1D quantum lattice systems with matrix product states. A small non-orthogonal basis for excitation tensors is constructed via a single diagonalization step analogous to a multi-state DMRG sweep; momentum-dependent excitations then follow from a small generalized eigenvalue problem. The ground state is represented by an infinite MPS obtained from a simple finite-system DMRG procedure presented as an alternative to VUMPS. For the S=1 Heisenberg antiferromagnet the method is shown to reproduce the one-magnon dispersion with high accuracy; the authors further demonstrate that omitting any left-orthonormal gauge condition is essential for convergence and that the resulting modes admit an exact Wannier reconstruction.

Significance. If the reported accuracy is confirmed by systematic benchmarks, SBEA would constitute a lightweight, easily implementable route to excitation spectra that avoids both full tangent-space optimization and the more involved VUMPS machinery. The explicit demonstration that a non-orthogonal basis is required, together with the Wannier-function construction, supplies concrete technical insight that could be reused in other MPS-based excitation calculations.

major comments (2)
  1. [Numerical results] Numerical results section: the central claim of “high accuracy” reproduction of the one-magnon dispersion is stated without quantitative error tables, maximum deviations from literature values, or convergence data versus bond dimension and basis size; such tables are required to substantiate the accuracy assertion and to rule out post-hoc parameter adjustment.
  2. [Ground-state construction] Ground-state construction paragraph: the finite-system DMRG procedure used to generate the iMPS is described only qualitatively; a direct comparison of its energy or correlation length against a converged VUMPS reference for the same S=1 Heisenberg chain is needed, because any systematic error in the ground state directly propagates into the excitation spectrum.
minor comments (2)
  1. [Abstract] The abstract and introduction should state the bond dimension and the dimension of the excitation basis explicitly rather than referring only to “small basis.”
  2. [Figures] Figure captions for the dispersion plots should include the precise literature reference curve against which the SBEA data are compared.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific suggestions that will help strengthen the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Numerical results] Numerical results section: the central claim of “high accuracy” reproduction of the one-magnon dispersion is stated without quantitative error tables, maximum deviations from literature values, or convergence data versus bond dimension and basis size; such tables are required to substantiate the accuracy assertion and to rule out post-hoc parameter adjustment.

    Authors: We agree that quantitative benchmarks are needed to support the accuracy claim. In the revised manuscript we will add tables reporting the maximum deviation of our one-magnon energies from established literature values over the Brillouin zone, together with convergence data for the excitation energies as functions of MPS bond dimension and site-basis size. These additions will allow readers to assess the precision systematically. revision: yes

  2. Referee: [Ground-state construction] Ground-state construction paragraph: the finite-system DMRG procedure used to generate the iMPS is described only qualitatively; a direct comparison of its energy or correlation length against a converged VUMPS reference for the same S=1 Heisenberg chain is needed, because any systematic error in the ground state directly propagates into the excitation spectrum.

    Authors: We acknowledge that a quantitative comparison would improve the presentation. We will compute and include a direct comparison of the ground-state energy per site and correlation length obtained from our finite-system DMRG procedure against a converged VUMPS reference for the S=1 Heisenberg chain. This will confirm that the ground state is sufficiently accurate for the subsequent excitation calculations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper presents SBEA as a variation on the standard tangent-space excitation ansatz for infinite MPS, with the ground state obtained from a simple finite-system DMRG procedure explicitly positioned as an alternative to VUMPS and excitations generated by a single diagonalization to form a non-orthogonal basis followed by a small generalized eigenvalue problem. Accuracy for the S=1 Heisenberg one-magnon dispersion is reported by direct comparison to established external literature benchmarks rather than by fitting or self-referential prediction on the same data. Gauge-choice tests and Wannier reconstruction are performed internally with explicit convergence demonstrations. No load-bearing step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the method remains falsifiable against independent numerical or experimental results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard MPS representational assumptions and the new SBEA construction; no new physical particles or forces are introduced.

free parameters (1)
  • MPS bond dimension
    Numerical accuracy depends on the chosen bond dimension, a standard tunable parameter in MPS calculations.
axioms (1)
  • domain assumption An infinite MPS ground state can be obtained via the paper's finite-system DMRG procedure as a practical alternative to VUMPS.
    The method is built on an infinite MPS description of the ground state obtained this way.

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Forward citations

Cited by 2 Pith papers

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    A method using dressed creation operators from MLWFs enables selective preparation and detection of quasiparticles in lattice theories, tested via MPS on hardcore QCD ladders to separate known excitations from resonances.

  2. Checkerboard Bose Hubbard Ladders using Transmon Arrays

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    Sublattice bias in the checkerboard Bose-Hubbard model makes the commensurate superfluid phase experimentally accessible via transmon arrays and supplies new probes for superfluid and insulating phases.

Reference graph

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