Study of multi-particle states with tensor renormalization group method

TL;DR

This work develops a tensor-network–based spectroscopy scheme to extract finite-volume spectra and multi-particle content in the (1+1)-dimensional Ising model. By computing the transfer-matrix spectrum from coarse-grained tensor networks and using impurity tensors to assign quantum numbers and momenta, the authors identify one-, two-, and three-particle states and compute two-particle scattering phase shifts using both Lüscher's formula and wave-function fits. The approach achieves reliable high-lying states (with $L_t=8$) and demonstrates consistency with exact predictions, including a matching of phase shifts to $\delta(k)=-\pi/2$. This method offers a deterministic, low-time-extent alternative to Monte Carlo, with potential applicability to other quantum field theories and multi-particle scattering analyses.

Abstract

We investigate the multi-particle states of the (1+1)-dimensional Ising model using a spectroscopy scheme based on the tensor renormalization group method. We start by computing the finite-volume energy spectrum of the model from the transfer matrix, which is numerically estimated using the coarse-grained tensor network. We then identify the quantum number and momentum of the eigenstates by using the symmetries of the system and the matrix elements of an appropriate interpolating operator. Next, we plot the energy for a particular quantum number and momentum as a function of system size to identify the number of particles in the corresponding energy eigenstates. With this method, we obtain one-, two-, and three-particle states. We also compute the two-particle scattering phase shift using Lüscher's formula as well as the wave function approach, and compare the results with the exact prediction.

Figures (6)

• Figure 1: A graphical image of (a) transfer matrix ${\cal T}_{S'S}$, (b) one time slice tensor network ${\cal A}_{kj}$.
• Figure 2: The coarse-graining procedure for the (a) pure tensor network, (b) impurity tensor network. Note that $\hat{0}(\hat{1})$ is the unit vector showing time(space) direction and we show $L_{\rm t}=2$ for readability.
• Figure 3: (a) The relative error of energy spectrum for $L_{\rm s}=64$ computed with $L_{\rm t}=2,4,8,16,64$ using $\chi=80$. (b) The numerical energy spectrum, matrix elements for quantum number classification, and exact quantum number for $L_{\rm s}=64$ computed with $\chi=80$ and $L_{\rm t}=8$.
• Figure 4: The energy spectrum as function of system size computed with $\chi=80$ and $L_{\rm t}=8$ for (a) $q=-1,P=0$ channel, (b) $q=+1, P=0$ channel.
• Figure 5: The wave function for $q=+1, P=0$ sector, computed with $\chi=80, L_{\rm t}=8$ for (a) $L_{\rm s}=56$, (b) $L_{\rm s}=112$.