Loop Algorithm for Quantum Transverse Ising Model in a Longitudinal Field

TL;DR

The paper presents a novel merge-unmerge loop update for quantum Monte Carlo simulations of the quantum transverse Ising model in a longitudinal field, implemented within the stochastic series expansion framework. This loop algorithm enables the off-diagonal updates to migrate spatially, significantly reducing autocorrelation times relative to traditional line and loop methods, especially at large longitudinal fields. Demonstrations on a Rydberg atom chain and Kagome qubit ice show improved convergence, accurate reproduction of phase transitions and structure factors, and substantial computational gains. The method is described as open-source, adaptable to other quantum-many-body systems and models beyond QTIM, with potential extensions to more complex lattices and entanglement measurements.

Abstract

The quantum transverse Ising model and its extensions play a critical role in various fields, such as statistical physics, quantum magnetism, quantum simulations, and mathematical physics. Although it does not suffer from the sign problem in most cases, the corresponding quantum Monte Carlo algorithm performs inefficiently, especially at a large longitudinal field. The main hindrance is the lack of loop update method which can strongly decrease the auto-correlation between Monte Carlo steps. Here, we successfully develop a loop algorithm with a novel merge-unmerge process. It demonstrates a great advantage over the state-of-the-art algorithm when implementing it to simulate the Rydberg atom chain and Kagome qubit ice. This advanced algorithm suits various systems such as Rydberg atom arrays, trapped ions, quantum materials, and quantum annealers.

Figures (6)

• Figure 1: Schematic diagram of the loop algorithm. (a) During the loop update process, the QMC sample is depicted in the diagram with the diagonal operator $H_d$ (black bond), constant operator $\sigma^{(0)}$ (red bond), and off-diagonal operator $\sigma^{(1)}$ (blue bond) locating at each integer imaginary time. The black (white) circles denote the spin up (down). (b) The merge-unmerge processes can move the off-diagonal operator to the other site, in conjunction with the loop update process which flips the spins along the path.
• Figure 2: Schematic picture of the off-diagonal update process: (a) The initial configuration. (b) The loop path after the merge step; the dashed boxes indicate the merged sites. (c) The resulting configuration after flipping all spins along the loop path. (d) The final configuration after the unmerge process; the dashed boxes now indicate the unmerged sites.
• Figure 3: Rydberg atom chain.(a) Staggered magnetization per site $\mathrm{|M_s|/N}$ and (b) the ratio of its auto-correlation time by line and loop algorithm (Inset: by $\mathrm{loop}_c$ and loop algorithm).
• Figure 4: Kagome qubit ice.(a) Structure factor and (b) the ratio of its auto-correlation time by line and loop algorithm. The inset of (a) shows the magnetization per site and the dashed black line highlights the 1/3 corresponding to the ice rule filling. The inset of (b) shows the ratio of its auto-correlation time by $\mathrm{loop_c}$ and loop algorithm. The parameters are $J_z=0.25$, $\Gamma=0.15$, $\beta=20$, and $N=24\times24\times3$ with periodical boundary condition.
• Figure 5: The loop length of (a,b) the Rydberg atom chain and (c,d) Kagome qubit ice. (a) The Rabi frequency $\Omega$ is changing at $\Delta =0$ and (b) the detuning $\Delta$ is changing at $\Omega=1.0$. The other parameters are identical to those in Fig. \ref{['fig3']}: $R_b=1.2$, $\beta=20$, and $L=51$. (c) The transverse field $\Gamma$ is changing at $B$=0.5 and (d) the longitudinal field $B$ is changing at $\Gamma$=0.15. The other parameters are identical to those in Fig. \ref{['fig4']}: $J_z = 0.25$, $\beta = 20$ and $N=6\times6\times3$.