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arxiv: 2509.15483 · v2 · submitted 2025-09-18 · 🪐 quant-ph · cond-mat.str-el

Dispersion Relations in Two- and Three-Dimensional Quantum Systems

Valeriia Bilokon , Elvira Bilokon , Illya Lukin , Andrii Sotnikov , Denys Bondar This is my paper

Pith reviewed 2026-05-18 15:14 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords dispersion relationstensor networksiPEPStransverse-field Ising modelexcitation spectraimaginary-time evolutionquantum lattice modelsthree-dimensional systems
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The pith

A tensor-network approach using infinite projected entangled-pair states computes dispersion relations in two- and three-dimensional quantum lattice models.

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

The paper develops a method to extract momentum-resolved excitation spectra in strongly correlated systems by performing imaginary-time evolution inside the iPEPS tensor-network framework. It tests the technique on the transverse-field Ising model and recovers the expected dispersion curves in both the paramagnetic and ferromagnetic phases for square and cubic lattices. The same procedure works in three dimensions, where previous methods had been unable to deliver reliable momentum-resolved data. Because the calculations use only modest bond dimensions and standard computational resources, the approach makes previously inaccessible spectra available across wide ranges of parameters.

Core claim

Imaginary-time evolution within the iPEPS ansatz yields dispersion relations for the transverse-field Ising model on two- and three-dimensional lattices that agree with series-expansion results wherever the latter exist, thereby providing the first such spectra for three-dimensional quantum lattice models.

What carries the argument

Imaginary-time evolution of iPEPS states to extract momentum-resolved excitation spectra by analyzing the decay of correlations in imaginary time.

If this is right

  • Dispersion relations become computable in both paramagnetic and ferromagnetic regimes of the model.
  • Results match independent series-expansion data in regimes where such expansions are reliable.
  • Only modest bond dimensions and computational effort are required to maintain accuracy across a broad range of parameters.
  • The same procedure applies to other two- and three-dimensional lattice models that admit an iPEPS description.

Where Pith is reading between the lines

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

  • The technique could be applied to models with longer-range interactions or different symmetries to map out previously unknown phase diagrams in three dimensions.
  • Momentum-resolved spectra obtained this way could serve as input for designing photonic or quantum-information devices that rely on precise band structures.
  • Extending the evolution to finite temperature or to real-time dynamics would test whether the same framework yields dynamical structure factors in three dimensions.

Load-bearing premise

The iPEPS representation with modest bond dimensions remains accurate enough to capture the dispersion relations in three dimensions without large truncation errors over wide parameter ranges.

What would settle it

Direct numerical comparison of the iPEPS-derived dispersion curves against exact diagonalization or higher-bond-dimension results on small three-dimensional clusters at the same parameter values.

Figures

Figures reproduced from arXiv: 2509.15483 by Andrii Sotnikov, Denys Bondar, Elvira Bilokon, Illya Lukin, Valeriia Bilokon.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Expectation value [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dispersion relation for the 3D TFIM calculated using [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Convergence of the dispersion relation as a function [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Extracting momentum-resolved excitation spectra in strongly correlated quantum systems remains a major challenge, especially beyond one spatial dimension. We present an efficient tensor-network approach to compute dispersion relations via imaginary-time evolution within the infinite projected entangled-pair states (iPEPS) framework. Benchmarking on the transverse-field Ising model, the method successfully captures dispersion relations in both paramagnetic and ferromagnetic phases for two- and three-dimensional lattices, achieving strong agreement with series expansion methods, where these are applicable. Crucially, this work presents the first demonstration of dispersion relation calculations for three-dimensional quantum lattice models - a long-standing computational challenge that opens entirely new research frontiers. The method demonstrates remarkable efficiency, requiring only modest computational resources while maintaining high accuracy across wide parameter ranges. Its broad applicability makes it a powerful tool for quantum simulation, photonic material design, and quantum information platforms requiring precise momentum-resolved spectra.

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 manuscript introduces a tensor-network method using infinite projected entangled-pair states (iPEPS) and imaginary-time evolution to extract momentum-resolved dispersion relations in 2D and 3D quantum lattice models. Benchmarking is performed on the transverse-field Ising model in both paramagnetic and ferromagnetic phases, with reported strong agreement to series-expansion results where available; the work claims to provide the first such calculations for three-dimensional systems.

Significance. If the accuracy and convergence claims hold under systematic checks, the result would be significant as the first demonstration of dispersion-relation extraction for 3D quantum lattice models via tensor networks, addressing a long-standing computational challenge and enabling new studies in quantum simulation and photonic materials. The reported efficiency with modest resources strengthens the potential impact if truncation errors are shown to be controlled.

major comments (1)
  1. [Abstract and 3D results] Abstract and 3D results section: the central claim that the iPEPS method with modest bond dimensions accurately captures dispersion relations in 3D rests on 'strong agreement' with series expansions, yet no error bars, bond-dimension scaling plots, or truncation-error estimates are provided; this is load-bearing because the stricter entanglement area law in 3D makes long-range correlations (which enter the excitation energies) vulnerable to systematic truncation precisely in regimes where series data are unavailable.
minor comments (2)
  1. [Methods] The methods section would benefit from an explicit statement of the bond-dimension values employed for the 2D and 3D benchmarks and the convergence criterion used in the imaginary-time evolution.
  2. [Figures] Figure captions should specify the momentum grid resolution and any fitting procedure used to extract the dispersion curves from the imaginary-time data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point about the presentation of the 3D results. We address the concern directly below and outline the revisions we will make to strengthen the supporting evidence.

read point-by-point responses

  1. Referee: [Abstract and 3D results] Abstract and 3D results section: the central claim that the iPEPS method with modest bond dimensions accurately captures dispersion relations in 3D rests on 'strong agreement' with series expansions, yet no error bars, bond-dimension scaling plots, or truncation-error estimates are provided; this is load-bearing because the stricter entanglement area law in 3D makes long-range correlations (which enter the excitation energies) vulnerable to systematic truncation precisely in regimes where series data are unavailable.

    Authors: We agree that additional quantitative checks would strengthen the central claim for the 3D calculations. The current manuscript shows direct comparisons to series-expansion data in the regimes where those benchmarks exist and reports good visual agreement for both paramagnetic and ferromagnetic phases. However, we acknowledge that explicit bond-dimension scaling plots, error bars on the extracted dispersion points, and truncation-error estimates derived from the iPEPS singular-value spectrum are not included. In the revised manuscript we will add these elements: (i) a figure showing the dispersion relation for several bond dimensions D in the 3D transverse-field Ising model, (ii) a quantitative assessment of the change in excitation energies with D, and (iii) a brief discussion of how the area-law truncation affects long-range correlations and the resulting dispersion. For parameter regions without series-expansion data we will explicitly note that the accuracy is inferred from the controlled convergence observed in the benchmarked regimes and from the 2D results, rather than claiming direct validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent benchmarks

full rationale

The paper introduces an iPEPS-based imaginary-time evolution method to extract momentum-resolved spectra. It validates the approach by direct comparison to series-expansion results on the transverse-field Ising model in both 2D and 3D, explicitly noting agreement “where these are applicable.” The central claim of a first 3D demonstration therefore rests on numerical output cross-checked against an external, non-self-referential technique rather than on any fitted parameter, self-definition, or load-bearing self-citation. No equation or step reduces the reported dispersion relations to the input ansatz by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard tensor-network approximations whose accuracy depends on chosen bond dimension and truncation; the abstract provides no explicit free parameters but implies domain assumptions about iPEPS representability.

free parameters (1)
  • bond dimension
    Typical for iPEPS; controls approximation quality and must be increased until convergence for accurate 3D dispersion.
axioms (1)
  • domain assumption The infinite projected entangled-pair states ansatz can faithfully represent ground states and low-lying excitations of the transverse-field Ising model in 2D and 3D lattices.
    Invoked implicitly to justify that imaginary-time evolution within iPEPS yields reliable dispersion relations.

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Reference graph

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