Study of multi-particle states with tensor renormalization group method

This paper investigates the multi-particle states of the (1+1)-dimensional Ising model using a tensor renormalization group-based spectroscopy scheme to identify energy eigenstates, determine particle numbers, and compute two-particle scattering phase shifts that align with exact predictions.

The Gist

Imagine you are trying to listen to a complex symphony, but you are standing inside a tiny, echoey room. You can hear the music, but it's distorted by the walls. Your goal is to figure out exactly which instruments are playing (the particles) and how they interact, even though you can't step outside the room to hear the "real" infinite concert hall.

This paper is about a new, clever way to solve that problem using a method called Tensor Renormalization Group (TRG). Here is the breakdown in simple terms:

1. The Problem: The "Noisy Room"

In physics, scientists use computers to simulate how particles behave. Usually, they use a method called "Monte Carlo," which is like trying to guess the shape of a cloud by throwing darts at it millions of times. It works, but it gets very messy and noisy when you try to look at excited states (like the 2nd or 3rd note in a chord) or multiple particles interacting at once. The signal gets lost in the static.

2. The Solution: The "Digital Origami"

The authors use a different approach based on Tensor Networks. Think of the entire universe of the simulation as a giant, complex piece of paper covered in a grid of numbers.

• The Old Way: To understand the music, they used to fold this paper into a perfect square. This worked great for the main, loud notes (the ground state), but when they tried to hear the faint, high-pitched notes (higher excited states), the paper got too crumpled, and the details were lost.
• The New Trick: The authors realized that if they folded the paper into a long, thin rectangle (specifically, 8 units tall and 64 units wide) instead of a square, they could preserve the details of those faint notes. It's like unfolding a map just enough to see the small towns without losing the big cities.

3. The Experiment: Listening to the Ising Model

They tested this on a famous, simplified physics model called the Ising Model (think of it as a grid of tiny magnets that can point up or down).

• The Goal: They wanted to find out how many "magnets" were dancing together. Are they dancing alone (1 particle)? In a pair (2 particles)? Or in a trio (3 particles)?
• The Method: They used a "quantum number" filter. Imagine a bouncer at a club. The bouncer checks your ID (quantum number). If you have the right ID, you get in. By checking which "IDs" the energy levels had, they could sort the particles into different groups.

4. The Results: Finding the Dancers

By using their new "long rectangle" folding technique, they successfully identified:

• Solo dancers: Single particles moving through the grid.
• Pairs: Two particles interacting (scattering off each other).
• Trios: Three particles moving together.

They didn't just find them; they could also "see" the wave function. If the energy levels are the notes being played, the wave function is the shape of the sound wave. They mapped out exactly how these particles looked as they moved around the grid.

5. The Grand Finale: The Phase Shift

The ultimate test was to see how two particles bounce off each other. In physics, this is called the scattering phase shift.

• Method A (The Math Formula): They used a famous formula (Lüscher's formula) that translates the "echoes" in their small room into the behavior of the infinite world.
• Method B (The Wave Fit): They looked at the actual shape of the wave function they calculated and tried to fit a smooth curve to it, like matching a puzzle piece.

The Result: Both methods gave the exact same answer, and that answer matched the "perfect" theoretical prediction perfectly. It was like tuning two different radios to the same station and hearing the exact same song.

Summary

This paper is a success story of better folding. By changing how they compressed the data (folding the tensor network into a rectangle instead of a square), the authors managed to hear the "quiet notes" of the quantum symphony that were previously drowned out by noise. They successfully identified single, double, and triple particle states and proved their new method works perfectly by comparing it to known mathematical truths.

In a nutshell: They built a better pair of ears to listen to the quantum world, allowing them to count and describe how particles dance together with unprecedented clarity.

Technical Summary

1. Problem Statement

The investigation of multi-particle interactions in lattice field theory is crucial for understanding scattering processes and bound states. While the Monte Carlo method is standard, it suffers from significant technical limitations when studying excited states:

• Temporal Extent: A very large temporal extent is required to suppress contamination from higher excited states.
• Statistical Noise: Extracting higher excited states often leads to large statistical errors.

Tensor network methods, specifically the Tensor Renormalization Group (TRG), offer a deterministic alternative with no statistical noise. However, previous implementations (using square tensor networks where time and space extents are equal, $L_t = L_s$) faced a different challenge:

• Degeneracy Errors: While effective for low-lying energy states, the errors in estimating higher excited states (which correspond to multi-particle states) increase drastically due to the close degeneracy of eigenvalues in the transfer matrix.

Objective: The authors aim to develop a modified spectroscopy scheme using the Tensor Renormalization Group (TRG) to accurately extract the energy spectrum, quantum numbers, and momenta of higher excited states (specifically one-, two-, and three-particle states) in the (1+1)-dimensional Ising model.

2. Methodology

The study employs a spectroscopy scheme based on the Tensor Renormalization Group (TRG) and the Higher-Order Tensor Renormalization Group (HOTRG) algorithm.

A. Transfer Matrix and Spectroscopy

• The partition function $Z$ is expressed as the trace of the transfer matrix $T$ raised to the power of the temporal extent $L_t$: $Z = \text{Tr}[T^{L_t}]$.
• The energy spectrum is derived from the eigenvalues ($\lambda_a$) of the transfer matrix: $E_a = -\ln(\lambda_a)$. The energy gap is $\omega_a = \ln(\lambda_0 / \lambda_a)$.
• Quantum Number Identification: Since eigenstate quantum numbers are not known a priori, the authors use interpolating operators ($\hat{O}_q$) and their matrix elements $\langle \Omega | \hat{O}_q | a \rangle$.
  • The system has $Z_2$ symmetry. By choosing the ground state $|\Omega\rangle$ with $q=+1$, the selection rule $\langle b | \hat{O}_q | a \rangle \neq 0 \implies q_b q_q q_a = 1$ allows the classification of states into $q = \pm 1$ sectors.
  • Matrix elements are computed using an impurity tensor network, where the operator is inserted into the tensor network structure.

B. Modified Coarse-Graining Strategy

To address the error accumulation in higher excited states, the authors modify the standard coarse-graining procedure:

• Standard Approach: Uses a square network ($L_t = L_s$), leading to high errors for excited states.
• Proposed Approach: Uses a rectangular network where the temporal extent is smaller than the spatial extent ($1 < L_t < L_s$).
  • Procedure:
    1. Apply 2D HOTRG iterations to reduce both time and space dimensions.
    2. Apply 1D HOTRG iterations only in the spatial direction to further reduce the spatial size while keeping the temporal structure intact.
    3. Perform exact contraction of the final tensors to obtain the effective transfer matrix.
  • Rationale: This strategy reduces the number of coarse-graining iterations in the time direction, mitigating the accumulation of truncation errors that cause eigenvalue degeneracy issues.

C. Momentum and Wave Function Extraction

• Momentum: Identified by computing matrix elements of momentum-projected operators, e.g., $\hat{O}_1(P) = \frac{1}{L_s} \sum \hat{s}_x e^{-iPx}$. Non-zero matrix elements indicate the momentum $P$ of the state.
• Wave Function: The two-particle wave function $\psi_a(x)$ is extracted from matrix elements of a two-point operator $\hat{O}_2(x) = \frac{1}{L_s} \sum \hat{s}_{x'} \hat{s}_{x'+x}$.
• Scattering Phase Shift: Computed via two independent methods:
  1. Lüscher's Formula: Relates finite-volume energy levels to the scattering phase shift $\delta(k)$.
  2. Wave Function Fitting: Fits the extracted wave function to a free wave form $\psi(x) \propto \cos(k(x + L_s/2))$ outside the effective interaction range $R$ to extract $\delta(k)$.

3. Key Contributions

1. Optimization of TRG for Excited States: Demonstrated that using a rectangular tensor network ($L_t < L_s$) significantly reduces numerical errors for higher excited states compared to the standard square network approach.
2. Multi-Particle State Identification: Successfully identified and classified one-, two-, and three-particle states in the (1+1)d Ising model by analyzing the system-size dependence of energy levels and their quantum numbers.
3. Dual Verification of Phase Shifts: Developed a method to extract the two-particle scattering phase shift directly from the wave function and showed its consistency with the traditional Lüscher's formula method and exact analytical predictions.
4. High-Resolution Spectrum: Achieved reliable energy spectrum calculations up to the 42nd excited state ($a=42$), a significant improvement over previous works limited to lower states.

4. Results

• Optimal Parameters: For the Ising model at $T=2.44$ (symmetric phase) with spatial size $L_s=64$, a temporal extent of $L_t = 8$ yielded the smallest relative error. This balance minimized eigenvalue degeneracy issues while limiting the number of coarse-graining steps.
• Quantum Number Classification:
  • Using the selection rule with the spin operator $\hat{s}_0$, states were successfully classified into $q=-1$ and $q=+1$ sectors.
  • The numerical results matched exact quantum numbers for states up to $a=42$.
• Particle Identification:
  • $q=-1, P=0$ sector: The lowest energy level converged to the rest mass $m$ (1-particle). Higher levels converged to $3m$ (3-particle states).
  • $q=+1, P=0$ sector: The lowest three levels converged to $2m$, identifying them as the ground, first, and second excited two-particle states.
• Wave Function Analysis:
  • Extracted wave functions for $L_s=56$ and $L_s=112$.
  • The wave function of an ambiguous state (violet triangles in Fig 4b) was found to have a significantly different shape and smaller amplitude compared to the two-particle states, aiding in its identification.
• Scattering Phase Shift:
  • The effective interaction range was determined to be $R \approx 40$.
  • Phase shifts calculated via Lüscher's formula and wave function fitting (for $L_s > 2R$) showed excellent agreement with each other and the exact prediction $\delta(k) = -\pi/2$.

5. Significance

This work establishes a robust, deterministic framework for studying multi-particle physics in lattice field theories without the statistical noise inherent in Monte Carlo simulations.

• Overcoming Limitations: It solves the specific problem of error accumulation in higher excited states within TRG methods by optimizing the geometry of the tensor network.
• Methodological Advancement: The ability to extract wave functions and scattering phase shifts directly from tensor networks opens new avenues for studying few-body and many-body dynamics in quantum field theories.
• Future Applicability: The authors propose extending this scheme to analyze three-particle states in detail and applying it to more complex quantum field theories beyond the Ising model.

In summary, the paper provides a validated, high-precision spectroscopy tool that bridges the gap between lattice field theory and experimental scattering data, specifically for multi-particle systems.