We develop a GPU-accelerated hybrid quantum Monte Carlo (QMC) algorithm to solve the fundamental yet difficult problem of $U(1)$ gauge field coupled to fermions, which gives rise to a $U(1)$ Dirac spin liquid state under the description of (2+1)d quantum electrodynamics QED$_3$. The algorithm renders a good acceptance rate and, more importantly, nearly linear space-time volume scaling in computational complexity $O(N_τ V_s)$, where $N_τ$ is the imaginary time dimension and $V_s$ is spatial volume, which is much more efficient than determinant QMC with scaling behavior of $O(N_τV_s^3)$. Such acceleration is achieved via a collection of technical improvements, including (i) the design of the efficient problem-specific preconditioner, (ii) customized CUDA kernel for matrix-vector multiplication, and (iii) CUDA Graph implementation on the GPU. These advances allow us to simulate the $U(1)$ Dirac spin liquid state with unprecedentedly large system sizes, which is up to $N_τ\times L\times L = 660\times66\times66$, and reveal its novel properties. With these technical improvements, we see the asymptotic convergence in the scaling dimensions of various fermion bilinear operators and the conserved current operator when approaching the thermodynamic limit. The scaling dimensions find good agreement with field-theoretical expectation, which provides supporting evidence for the conformal nature of the $U(1)$ Dirac spin liquid state in the QED$_3$. Our technical advancements open an avenue to study the Dirac spin liquid state and its transition towards symmetry-breaking phases at larger system sizes and with less computational burden.

翻译：我们开发了一种GPU加速的混合量子蒙特卡罗（QMC）算法，用于求解U(1)规范场耦合费米子这一基础且困难的物理问题，该问题在(2+1)维量子电动力学QED$_3$描述下会产生U(1)狄拉克自旋液体态。该算法具有较高的接受率，更重要的是其计算复杂度呈现近乎线性的时空体积标度行为$O(N_τ V_s)$，其中$N_τ$为虚时间维度，$V_s$为空间体积，这比具有$O(N_τV_s^3)$标度行为的行列式QMC方法效率显著提升。这种加速通过一系列技术改进实现，包括：（i）设计针对特定问题的高效预处理器，（ii）定制矩阵-向量乘法的CUDA内核，（iii）GPU上的CUDA Graph实现。这些进展使我们能够以前所未有的大系统尺寸（高达$N_τ\times L\times L = 660\times66\times66$）模拟U(1)狄拉克自旋液体态，并揭示其新颖特性。借助这些技术改进，我们在趋近热力学极限时观察到各种费米子双线性算符及守恒流算符标度维度的渐近收敛行为。所得标度维度与场论预期高度吻合，为QED$_3$中U(1)狄拉克自旋液体态的共形性质提供了支持性证据。我们的技术进步为在更大系统尺寸和更低计算负担下研究狄拉克自旋液体态及其向对称破缺相的转变开辟了新途径。