Learning the Renyi entropy of multiple disjoint intervals in transverse-field quantum Ising models with restricted Boltzmann machine

TL;DR

The paper develops an improved swapping operation to compute Renyi entropies for multiple disjoint intervals and demonstrates it in the 1D TFQIM using two state-preparation methods: exact diagonalization and neural-network quantum states based on restricted Boltzmann machines. It shows that the RBM-based approach yields results for $S_2$ with two, three, and four disjoint intervals that closely match SVD benchmarks, across ferromagnetic, critical, and paramagnetic regimes. The work highlights the method's ability to explore multi-interval entanglement in many-body systems and points to future extensions to higher Renyi orders, higher dimensions, and non-ground states, while noting current limitations due to potential underfitting near criticality. Overall, the approach provides a versatile, accurate framework for accessing complex entanglement structures in quantum spin chains.

Abstract

Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: one is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one-dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.

Figures (5)

• Figure 1: Schematic picture to illustrate the alternate arrangements of the intervals. The subsystem $A$ consists of the blue disjoint intervals $a_i$, i.e., $A=\cup_ia_i$ in which $(i=1,2,\cdots n)$ while the complement part $B=\bar{A}$ contains the red parts, i.e., $B=\cup_ib_i$.
• Figure 2: Schematic figure of the swapping operation on the $m$ replica states $\otimes_{j=1}^m|\Psi^j\rangle$. In the upper plot, we have separated the whole states into multiple parts, i.e., $A_1B_1A_2B_2\cdots$ which are enclosed in the dashed rectangles. The arrows appear only in the $A$'s parts, indicating that the swapping operation will exchange the $(j+1)$-st states to the $j$-th states, and the first states will be moved to the $m$-th states. The states in the $B$'s parts stay unchanged. After the swapping operation, the final states turn into $S_{\rm wap}^{(m)}\otimes_{j=1}^m|\Psi^{j}\rangle$ as the lower plot shows.
• Figure 3: Schematic figure of the structure of the RBM. The visible layer (in green) consists of $N$ visible neurons $s_j$ where $(j=1, 2, \cdots N)$, each of which corresponds to the spin direction $s_j=\{-1,1\}$ at every site of the system, while the hidden layer (in red) is composed of $M$ hidden neurons $h_i$ where $(i=1, 2, \cdots M)$, which take values as $h_i=\{-1,1\}$.
• Figure 4: (Left) Renyi entropy $S_2$ with two disjoint intervals ($A=\alpha\cup\gamma$) against $l_1$, the length of $\beta$ interval; (Right) Renyi entropy $S_2$ with two disjoint intervals ($A=\alpha\cup\gamma$) against $l_2$, the length of $\gamma$ interval. In both panels, the circles in the top-left corners are the schematic pictures to show the arrangements of the disjoint intervals. Meanwhile, different styles of the points represent the Renyi entropy obtained from the machine learning methods with neural network for various magnetic fields, while the dashed lines correspond to the diagonalization of the Hamiltonian from the SVD method. They match each other very well.
• Figure 5: (Left) Renyi entropy $S_2$ with three disjoint intervals $A=a_1\cup a_2\cup a_3$ against $l_1$, the size of each $b_i$$(i=1, 2, 3)$. (Right) Renyi entropy $S_2$ with four disjoint intervals $A=a_1\cup a_2\cup a_3\cup a_4$ against $l_1$, the size of each $b_i$$(i=1, 2, 3, 4)$. In both panels, the circles in the top-left corners are the schematic pictures to show the arrangements of the disjoint intervals. Meanwhile, different styles of the points represent the Renyi entropy obtained from the machine learning methods with neural network for various magnetic fields, while the dashed lines correspond to the diagonalization of the Hamiltonian from the SVD method. They match each other very well within errors.