Abstract: We present a spectroscopy scheme for the lattice field theory by using tensor
renormalization group method combining with the transfer matrix formalism. By
using the scheme, we can not only compute the energy spectrum for the lattice
theory but also determine quantum numbers of the energy eigenstates.
Furthermore, 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. Secondly, 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 an appropriate
matrix elements represented by tensor network. As a demonstration of the new
scheme, we show the spectroscopy of (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.
概要: シータ項とはゲージ理論に許される冗長な項であり、系に純粋に量子的な効果を与える。さらに、$\theta = \pi$の特殊な場合を除いて、CP対称性を明示的に破る。理論的な興味にもかかわらず、シータ項の効果をモンテカルロ法によって解析することは符号問題のため困難であり、これまで主にテンソルネットワーク法や量子アルゴリズムがその解析に用いられてきた。本発表では、シータ項つきのシュウィンガー模型（1＋1次元時空の量子電気力学）が、ボソン化法と呼ばれるフェルミオン系を等価なボソン系に変換する手法を用いることで、極めて単純なモンテカルロ計算によって第一原理計算可能であることを指摘する[1]。この手法の正当性と有効性を、他の手法から得られた計算結果との比較より実証した後、有限温度・有限シータ領域における閉じ込め力の精密計算を行い、この領域における閉じ込めの様相を定量的に明らかにする[1]。さらに、作用レベルではCP不変な$\theta = \pi$でのシュウィンガー模型の、有限温度・有限フェルミオン質量領域での相構造を、量子イジング鎖とのユニバーサリティという観点と本手法を組み合わせることで決定する[2]。
References
[1] H. Ohata, “Monte Carlo study of Schwinger model without the sign problem”, JHEP 12, 007 (2023), arXiv:2303.05481 [hep-lat].
[2] H. Ohata, “Phase diagram near the quantum critical point in Schwinger model at θ = π: analogy with quantum Ising chain”, PTEP 2024, 013B02 (2024), arXiv:2311.04738 [hep-lat].
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