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Multi-particle states investigation with tensor renormalization group method

This paper investigates multi-particle states in the (1+1)d Ising model by combining the tensor renormalization group method with transfer matrix spectroscopy and impurity tensor networks to identify quantum numbers, extract one- to three-particle energy levels, and validate two-particle scattering phase shifts and three-particle degeneracies against theoretical predictions.

Original authors: Fathiyya Izzatun Az-zahra, Shinji Takeda, Takeshi Yamazaki

Published 2026-06-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Fathiyya Izzatun Az-zahra, Shinji Takeda, Takeshi Yamazaki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the music of a tiny, invisible orchestra. In the world of physics, this orchestra is made of particles, and the "music" they play is their energy levels. Usually, figuring out the exact notes (energy) and who is playing them (particle count) is like trying to hear a single violin in a hurricane; the noise is too loud, and the signal is too faint.

This paper is about a team of physicists who built a new, ultra-precise "noise-canceling headphone" to listen to this orchestra. They used a mathematical tool called a Tensor Renormalization Group (TRG) method to study a simple model called the Ising Model (think of it as a grid of tiny magnets that can point up or down).

Here is what they did, explained simply:

1. The Problem: The "Square" Bottleneck

Previously, to hear the orchestra, scientists used a method that required a perfectly square grid. It was like trying to take a photo of a long hallway using only a square camera sensor; you could only see a small part of the hallway clearly before the image got blurry and distorted. This meant they could only study the "easy" notes (low energy) and missed the complex, high-pitched ones (multi-particle states).

2. The Solution: A New "Coarse-Graining" Strategy

The authors invented a new way to process the data. Instead of forcing everything into a square, they used a flexible strategy to "zoom out" (coarse-grain) the information.

The Analogy: Imagine looking at a high-resolution photo of a forest. If you zoom out too fast, the trees blur together. Their new method is like zooming out slowly and carefully, preserving the details of individual trees even when you are looking at the whole forest.
The Result: They could now see much further down the "hallway" of energy levels. They successfully identified not just single particles, but groups of two and three particles playing together.

3. Identifying the Players (Quantum Numbers)

Once they heard the notes, they needed to know who was playing.

The Method: They used "interpolating operators," which act like specific musical filters. If you play a filter tuned to "spin up," and the orchestra responds, you know a "spin up" player is there.
The Discovery: They mapped out the energy levels and assigned them "quantum numbers" (like ID cards) and "momentum" (how fast they are moving). They found single particles, pairs, and triplets.

4. The Two-Particle Dance (Scattering)

The most exciting part was studying how two particles interact.

The Phase Shift: When two particles bump into each other, they change their rhythm slightly. This change is called a "phase shift."
The Discovery: They measured this shift in three different ways (using energy levels, looking at the wave outside the interaction, and looking inside). All three methods agreed perfectly.
The Surprise: They found that in this specific model, the "dance" is always the same, regardless of how fast the particles are moving. The phase shift is always exactly -90 degrees (or $-\pi/2$ ).
The Consequence: Because the dance is so predictable, it causes a "degeneracy." In physics, this means different groups of particles end up singing the exact same note. Specifically, they found that some states with different total momenta become four-fold degenerate (four different setups sound exactly the same).

5. The Three-Particle Trio

They also looked at groups of three particles.

The Prediction: They calculated what the energy should be if three particles were just bumping into each other in pairs (like a game of tag where everyone only touches two people at a time).
The Match: The numbers they calculated matched their computer simulation perfectly.
The Complexity: Unlike the two-particle group, the three-particle group had a more complex "choir" structure. They found states that were unique, some that were doubled, some quadrupled, and even some that were eight-fold degenerate (eight different setups sounding identical).

Summary

In short, the authors built a better mathematical microscope. They used it to listen to a simple magnetic model and successfully identified the "notes" played by one, two, and three particles. They confirmed that the two-particle interaction is perfectly predictable (always a -90 degree shift), which causes strange "echoes" where different groups of particles sound identical. They also showed that three-particle groups follow the rules of pairwise interactions, creating a complex pattern of echoes (degeneracies) that matches theoretical predictions.

They did not apply this to real-world medical devices or new engines; they simply proved that their new mathematical "headphones" work incredibly well for understanding the fundamental rules of how particles group together in this specific model.

Technical Summary: Multi-particle states investigation with tensor renormalization group method

Problem and Motivation
The study of multi-particle states is critical for understanding systems with strong interactions, such as those in nuclear and high-energy physics (e.g., hadron decays). While Lattice Quantum Chromodynamics (LQCD) utilizes Monte Carlo (MC) algorithms for non-perturbative spectroscopy, these methods face significant challenges: they require large lattice extents in the time direction to suppress contamination from higher eigenstates and demand extremely high statistics to reduce noise in excited state extraction. Tensor network methods have emerged as a deterministic alternative that avoids statistical errors. However, previous applications of the Lagrangian approach using the Transfer Matrix (TM) and Higher Order Tensor Renormalization Group (HOTRG) on single-time slice networks suffered from large systematic errors due to the near-degeneracy of eigenvalues, particularly for higher excited states and larger system sizes. This limited the reliable extraction of energy spectra to low-lying states.

Methodology
This work proposes an improved spectroscopy scheme for the (1+1)d Ising Model based on the transfer matrix formalism and tensor network representation. The core of the methodology involves:

Transfer Matrix and Tensor Network Formulation: The partition function is reformulated as the trace of a transfer matrix $T$ . The eigenvalues of $T$ correspond to the energy spectrum ( $E_a = -\ln \lambda_a$ ). The matrix elements required for quantum number identification are represented as impurity tensor networks.
Novel Coarse-Graining Strategy: To address the accuracy issues of previous single-time slice approaches, the authors introduce a new coarse-graining strategy. Instead of using a square tensor network (where time and space dimensions are equal), they employ an initial tensor network with a small spatial size $L_s^{(init)}$ $L_{s}^{(ini t)}$ (specifically $L_s^{(init)} \in \{1, 3, 5, 7\}$ $L_{s}^{(ini t)} \in {1, 3, 5, 7}$ ) embedded into a larger main network. The total spatial size is $L_s = L_s^{(init)} \times 2^n$ $L_{s} = L_{s}^{(ini t)} \times 2^{n}$ .
- The initial small network is coarse-grained using a "HOTRG-like" algorithm to produce a single effective tensor.
- This tensor is then embedded and further coarse-grained using standard 2D and 1D HOTRG steps.
- The authors systematically vary the time direction size $L_t$ (testing $L_t = 2, 4, 8, 16, 64$ ) to optimize accuracy, finding that $L_t = 4$ and $L_t = 8$ yield the smallest relative errors by balancing the accumulation of coarse-graining errors against the degeneracy of transfer matrix eigenvalues.
Quantum Number and Momentum Identification: Since quantum numbers are not a priori known, the authors identify them using selection rules derived from system symmetries. They compute matrix elements of interpolating operators (spin fields for $Z_2$ symmetry and momentum operators) represented as impurity tensor networks.
Scattering Analysis: The study computes the two-particle scattering phase shift using three distinct approaches to ensure consistency:
- Finite Volume Energy: Using Lüscher's formula applied to the energy spectrum in both the Center of Mass (CM) frame ( $P=0$ ) and moving frames ( $P \neq 0$ ).
- Wave Function (Outside Interaction Range): Fitting the Bethe-Salpeter (BS) wave function in the free region to extract the phase shift.
- Wave Function (Inside Interaction Range): Using the reduced BS wave function method within the interaction range to extract the phase shift.

Key Results

Improved Accuracy: The new coarse-graining strategy with $L_t = 4$ and $L_t = 8$ significantly reduces relative errors in the energy spectrum to the order of $O(10^{-3})$ . This allows for the reliable identification of higher excited states (up to $a \approx 57$ for $L_s=64$ ) and larger system sizes ( $L_s$ up to 112) compared to previous square-network approaches.
State Classification:
- One-Particle States: Identified in the $q=-1$ sector by their dispersion relation and energy approaching the rest mass $m$ as $L_s \to \infty$ .
- Two-Particle States: Identified in the $q=+1$ sector. Their energies approach $2m$ in the large volume limit.
- Three-Particle States: Identified in the $q=-1$ sector. Their energies approach $3m$ in the large volume limit.
Scattering Phase Shift: All three methods (Lüscher's formula, wave function fitting outside $R$ , and BS wave function inside $R$ ) yield consistent results. The scattering phase shift for the (1+1)d Ising model is found to be constant, $\delta = -\pi/2$ , for all relative momenta within the elastic region ( $0 \le k/m < \sqrt{3}$ ).
Degeneracy and Dispersion:
- The constant phase shift $\delta = -\pi/2$ leads to specific quantization conditions for relative momentum, resulting in four-fold degeneracies for two-particle states in moving frames ( $P \neq 0$ ).
- The authors derive and verify the theoretical dispersion relations for two- and three-particle states.
- In the three-particle sector, the energy spectrum exhibits varied degeneracies (non-degenerate, two-fold, four-fold, and eight-fold) depending on the total momentum and the specific combinations of individual particle momenta allowed by the quantization conditions.
Wave Functions: The effective potential derived from the two-particle ground state wave function shows a strong repulsive interaction at the origin and an interaction range of approximately $R \approx 40$ at $T=2.44$ .

Significance and Claims
The paper claims that the proposed coarse-graining strategy successfully overcomes the limitations of previous tensor network spectroscopy methods, enabling the reliable extraction of higher excited states and multi-particle dynamics. The work demonstrates that:

Tensor networks can deterministically identify multi-particle states (up to three particles in this study) and their quantum numbers without statistical noise.
The consistency of phase shift extraction across three different methods (energy-based and wave-function-based) validates the robustness of the approach.
The observed degeneracies in two- and three-particle sectors are direct consequences of the specific scattering phase shift ( $\delta = -\pi/2$ ) of the Ising model, providing a clear link between scattering properties and finite-volume spectral degeneracies.

The authors conclude that this framework is ready for future investigation of four-particle states and application to other quantum field theories.