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arXiv:2606.22321 · Physical review. D/Physical review. D.

Multiparticle states investigation with the tensor renormalization group method

Fathiyya Izzatun Az-zahra, Shinji Takeda, Takeshi Yamazaki

Published Aug 21, 2026• 0 citations
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Abstract

We investigate multiparticle states of the ( 1 + 1 ) d Ising Model using a spectroscopy scheme based on the 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 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 Lüscher’s formula and the 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.

DOI
10.1103/zhq7-swns

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We investigate multiparticle states of the ( 1 + 1 ) d Ising Model using a spectroscopy scheme based on the 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 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 Lüscher’s formula and the 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.
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.
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 L\"uscher's formula as well as the wave function approach, and compare the results with the exact prediction.
We present a spectroscopy scheme using transfer matrix and tensor network. With this method, the energy spectrum is obtained from the eigenvalues of the transfer matrix which is estimated by coarse grained tensor network of a lattice model, and the quantum number is classified from the matrix elements of a proper operator that can be represented as an impurity tensor network. Additionally, the momentum of one-particle state and two-particle state whose total momentum is zero are classified using matrix elements of proper momentum operators. Furthermore, using Lüscher's formula, the scattering phase shift is also computed from the energy of two-particle state. As a demonstration, the method is applied to (1+1)(1+1)(1+1)d Ising model.
We present a spectroscopy scheme for the lattice field theory by using the tensor renormalization group method combining with the transfer matrix formalism. By using the scheme, we cannot only compute the energy spectrum for the lattice theory but also determine quantum numbers of the energy eigenstates. Furthermore, the wave function of the corresponding eigenstate can also be computed. The first step of the scheme is to coarse grain the tensor network of a given lattice model by using the higher order tensor renormalization group, and then after making a matrix corresponding to a transfer matrix from the coarse-grained tensors, its eigenvalues are evaluated to extract the energy spectrum. Second, the quantum number of the eigenstates can be identified by a selection rule that requires to compute matrix elements of an associated insertion operator. The matrix elements can be represented by an impurity tensor network and computed by the coarse-graining scheme. Moreover, we can compute the wave function of the energy eigenstate by putting the impurity tensor at each point in space direction of the network. Additionally, the momentum of the eigenstate can also be identified by computing appropriate matrix elements represented by the tensor network. As a demonstration of the new scheme, we show the spectroscopy of the (1 + 1)d Ising model and compare it with exact results. We also present a scattering phase shift obtained from two-particle state energy using Luscher's formula.
We present a spectroscopy scheme for the lattice field theory by using the tensor renormalization group method combining with the transfer matrix formalism. By using the scheme, we cannot only compute the energy spectrum for the lattice theory but also determine quantum numbers of the energy eigenstates. Furthermore, the wave function of the corresponding eigenstate can also be computed. The first step of the scheme is to coarse grain the tensor network of a given lattice model by using the higher order tensor renormalization group, and then after making a matrix corresponding to a transfer matrix from the coarse-grained tensors, its eigenvalues are evaluated to extract the energy spectrum. Second, the quantum number of the eigenstates can be identified by a selection rule that requires to compute matrix elements of an associated insertion operator. The matrix elements can be represented by an impurity tensor network and computed by the coarse-graining scheme. Moreover, we can compute the wave function of the energy eigenstate by putting the impurity tensor at each point in space direction of the network. Additionally, the momentum of the eigenstate can also be identified by computing appropriate matrix elements represented by the tensor network. As a demonstration of the new scheme, we show the spectroscopy of the ( 1 + 1 ) d Ising model and compare it with exact results. We also present a scattering phase shift obtained from two-particle state energy using Lüscher’s formula. Published by the American Physical Society 2024
We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to 102441024^410244, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.