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arxiv: 2606.22321 · v1 · pith:EFPANMSInew · submitted 2026-06-21 · ✦ hep-lat

Multi-particle states investigation with tensor renormalization group method

Fathiyya Izzatun Az-zahra , Shinji Takeda , Takeshi Yamazaki This is my paper

Pith reviewed 2026-06-26 09:42 UTC · model grok-4.3

classification ✦ hep-lat
keywords Ising modeltensor renormalization groupmulti-particle statesscattering phase shiftLüscher formulafinite volume spectrumtransfer matrix
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The pith

Tensor renormalization group identifies multi-particle states and phase shifts in the Ising model.

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

This paper presents a spectroscopy method for the (1+1)d Ising model that combines tensor renormalization group for estimating the transfer matrix with impurity tensor networks for computing operator matrix elements. The approach extracts the energy spectrum and assigns quantum numbers and momenta to eigenstates. Dependence of energies on system size distinguishes one-, two-, and three-particle states for given quantum numbers. Scattering phase shifts extracted from two-particle states using Lüscher's formula and wave functions match theoretical expectations, enabling further study of state degeneracies and three-particle energies.

Core claim

Using a transfer matrix estimated by coarse-grained tensor networks and matrix elements from impurity tensor networks, the method identifies energy eigenstates' quantum numbers and momenta, distinguishes particle numbers by finite-size scaling, and computes two-particle scattering phase shifts that agree with theory in the (1+1)d Ising model.

What carries the argument

Spectroscopy scheme using transfer matrix from tensor renormalization group and impurity tensor network for matrix elements to assign quantum numbers and momenta.

Load-bearing premise

The coarse-grained tensor network sufficiently approximates the transfer matrix eigenvalues and the impurity tensor network accurately computes the matrix elements for quantum number and momentum assignment.

What would settle it

Observing a significant mismatch between the extracted scattering phase shift and the known theoretical value for the Ising model at the studied volumes would indicate the method fails to capture the states correctly.

Figures

Figures reproduced from arXiv: 2606.22321 by Fathiyya Izzatun Az-zahra, Shinji Takeda, Takeshi Yamazaki.

Figure 1
Figure 1. Figure 1: FIG. 1: A graphical image of (a) transfer matrix, (b) single-time slice tensor network, and [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a-b) The coarse-graining procedure for constructing the initial pure and impurity [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The coarse-graining procedure of the main tensor network using HOTRG. The [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The coarse-graining procedure for the impurity tensor network [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) The relative error of the energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Relative errors of energy spectrum for [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: (a) The initial tensor network representation with size [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: One-particle state energy spectrum for [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: (a) The pair of light green and gray circle represents the contracted initial impurity [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: , the networks with Lt = 4 and Lt = 8 yield almost the same energy level in the sector for each system size. The three energy levels in this sector shown by orange, green, and red markers, which are classified by looking at the shape of the corresponding wave function, see Sec. III E, are the two-particle state energy as they approach ω = 2m in the large system size. On the other hand, the number of parti… view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Energy spectrum in the [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: The tensor network representation for the computation of two-particle state wave [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: (a) The wave function [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: A diagram of the two-particle state analysis performed in this work. The black [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Scattering phase shift of two-particle state on CM and moving frame, i.e. [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: (a) The effective potential from the two-particle ground state wave function for [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Comparison of the phase shift in CM frame, computed at [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Effective phase shift [PITH_FULL_IMAGE:figures/full_fig_p038_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Finite volume energy of two-particle states in the CM and moving frames, for [PITH_FULL_IMAGE:figures/full_fig_p039_22.png] view at source ↗
Figure 12
Figure 12. Figure 12: The same argument can be applied for the [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Finite-volume energy of three-particle states for [PITH_FULL_IMAGE:figures/full_fig_p041_23.png] view at source ↗
read the original abstract

We investigate multi-particle states of the (1+1)d Ising Model using a spectroscopy scheme based on transfer matrix and tensor renormalization group method. The scheme begins with computing the energy spectrum of the system from the transfer matrix estimated by the coarse-grained tensor network. The quantum number and momentum of these energy eigenstates are not a priori known, thus we identify them using matrix elements of an interpolating operator that is numerically computed with an impurity tensor network. Furthermore, by observing the dependence of the energy as a function of system size, we identify the number of particles of the eigenstates and obtain one-, two-, and three-particle states for a specific quantum number and momentum. From the two-particle state sector, we compute the scattering phase shift using L\"uscher's formula and wave function approach, and observe their consistency with theoretical prediction. Using the information of the two-particle scattering phase shift, we investigate the degeneracy of the two-particle states, the theoretical prediction of the three-particle finite volume energy and also the degeneracy in the three-particle states.

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

0 major / 3 minor

Summary. The manuscript introduces a spectroscopy scheme using the transfer matrix and tensor renormalization group (TRG) method to investigate multi-particle states in the (1+1)-dimensional Ising model. The approach computes the energy spectrum from the coarse-grained tensor network, identifies quantum numbers and momenta via matrix elements from an impurity tensor network, determines particle numbers from system-size dependence of energies, extracts one- to three-particle states, and computes scattering phase shifts from two-particle states using Lüscher's formula and wave-function methods, showing consistency with theoretical predictions. It further analyzes degeneracies in two- and three-particle sectors and three-particle finite-volume energies.

Significance. If the TRG approximation accuracy and state identification hold as described, the work demonstrates a viable tensor-network route to multi-particle spectroscopy and phase-shift extraction in a solvable lattice model. The direct comparison to exact Ising-model predictions supplies a falsifiable validation that strengthens the case for applying similar methods to theories where Monte Carlo or exact diagonalization become prohibitive.

minor comments (3)
  1. [Method description] The abstract states that quantum numbers and momenta are identified from matrix elements of an interpolating operator computed with the impurity tensor network, but the main text should include an explicit example (e.g., a table of operator matrix elements for the lowest few states) showing how the assignment is performed unambiguously.
  2. [Results on particle-number identification] Finite-volume scaling is used to assign particle number; the manuscript should report the range of volumes studied and the functional form assumed for the extrapolation (e.g., exponential corrections) so that readers can judge the robustness of the one-, two-, and three-particle classifications.
  3. [Scattering phase shift section] The phase-shift extraction via both Lüscher's formula and the wave-function approach is reported to agree with theory; a quantitative table or plot of the difference (with bond-dimension dependence) would make the level of numerical agreement explicit rather than qualitative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on multi-particle spectroscopy using TRG in the 1+1d Ising model and for recommending minor revision. No specific major comments appear in the provided report, so we offer no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard TRG numerics validated against external Ising theory

full rationale

The paper computes the transfer-matrix spectrum via coarse-grained TRG tensors, assigns quantum numbers/momenta via impurity-tensor matrix elements, identifies particle number from finite-volume energy scaling, and extracts two-particle phase shifts via Lüscher's formula and wave-function methods before comparing the results to independent theoretical predictions for the (1+1)d Ising model. None of these steps reduce by construction to fitted parameters or self-citations; the central claims are externally falsifiable against known exact results and the numerical implementation is self-contained with stated bond-dimension choices and consistency checks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the scheme implicitly assumes standard properties of the Ising transfer matrix and Lüscher quantization condition.

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Reference graph

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