量子转子网络的规范符号规则
A Gauge Sign Rule for Quantum Rotor Networks
- National Laboratory of the Rockies（落基山国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对量子转子网络，证明符号问题由回路通量控制，提出规范符号规则，并展示其统一了Mott转变、格点规范理论和阻挫优化中的符号问题。
AI中文摘要：
符号问题是路径积分权重非正定导致量子系统难以经典模拟的问题。它出现在看似无关的问题中，从费米子反对称性和几何阻挫到实时演化、拓扑角和有限密度。我们在这里针对紧致连续变量研究它，将其建模为量子转子网络，$\hat H=\sum_i 4E_C(\hat n_i-n_{g,i})^2-\sum_{\langle ij\rangle} E_{ij}\cos(\hat\phi_i-\hat\phi_j-\theta_{ij})$。对于这些系统，一个单一的规范不变量控制着阻碍，即相互作用图独立回路中的阻挫通量。我们证明了一个符号规则，即Marshall规则的转子类比。电荷基哈密顿量无符号当且仅当每个回路通量模$2\pi$为零。精确对角化和密度矩阵重整化随后表明，符号代价是通量的规范不变函数。对于单个回路，它在零通量时消失，在$\pi$处达到峰值，并且随回路周长呈指数小。对许多回路求和后，它是广延的，随系统尺寸增长。相同的回路通量产生了通常分开处理的三个问题的符号，即Mott转变（通过奴隶转子映射）、紧致$U(1)$格点规范理论和阻挫连续优化。超导转子阵列天然实现了该模型及其符号。
英文摘要：
The sign problem is the non-positive path-integral weight that makes a quantum system hard to simulate classically. It shows up in problems that look unrelated, from fermion antisymmetry and geometric frustration to real-time evolution, topological angles, and finite density. We study it here for compact continuous variables, modeled as networks of quantum rotors, $\hat H=\sum_i 4E_C(\hat n_i-n_{g,i})^2-\sum_{\langle ij\rangle} E_{ij}\cos(\hatϕ_i-\hatϕ_j-θ_{ij})$. For these, a single gauge-invariant quantity controls the obstruction, the frustration flux through the independent loops of the interaction graph. We prove a sign rule, the rotor analogue of Marshall's. The charge-basis Hamiltonian is sign-free if and only if every loop flux vanishes modulo $2π$. Exact diagonalization and density-matrix renormalization then show that the sign cost is a gauge-invariant function of the flux. For a single loop it vanishes at zero flux, peaks at $π$, and is exponentially small in the loop's perimeter. Summed over many loops it is extensive, growing with system size. The same loop flux generates the sign of three problems usually treated apart, namely the Mott transition (through the slave-rotor mapping), compact $U(1)$ lattice gauge theory, and frustrated continuous optimization. A superconducting rotor array realizes both the model and its sign natively.
