量子自旋链中弦序的Bootstrap认证
Bootstrap certification of string order in quantum spin chains
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- Cornell University（康奈尔大学）
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中文总结 AI 辅助
本文提出一种基于弦算符代数闭合性的bootstrap方法，以多项式成本认证量子自旋链中的非局域弦序，并直接验证了团簇相和Haldane相。
中文摘要 AI 辅助
多体bootstrap通过仅受正定性约束的关联函数上最小化能量来认证基态性质，无需任何变分波函数。然而，其成本随所涉及算符的尺寸呈指数增长，使得用于诊断拓扑相的长程弦关联函数难以触及。我们通过将弦算符视为基本对象来克服这一困难：裸弦和端点修饰弦与彼此及局域词满足闭合的算符代数，从而得到一个半定规划，其成本仅随弦长呈多项式增长。单次计算即可获得所有长度的弦关联函数的估计。一旦基态能量被限定在一个窗口内，每个目标弦都能获得严格的双侧界限。在与团簇Ising链和自旋-1 Heisenberg链上的密度矩阵重正化群计算进行基准测试时，该方法直接从哈密顿量认证了团簇相和Haldane相的非局域弦序。我们对弦序参数的严格认证揭示出，界限的尖锐程度由进入计算的算符决定，而非仅由其长度决定。通过利用弦和局域词的代数闭合性，我们的框架将空间扩展的可观测量提升为多项式可处理的bootstrap变量，为量子多体系统中非局域序的无波函数认证开辟了道路。
英文摘要
The many-body bootstrap certifies ground-state properties by minimizing energy over correlators constrained only by positivity, without any variational wavefunction. Its cost, however, grows exponentially with the size of the operators involved, placing the long-range string correlators that diagnose topological phases out of reach. We overcome this by treating string operators as primary objects: bare and endpoint-dressed strings satisfy a closed operator algebra with one another and with local words, yielding a semidefinite program whose cost grows only polynomially with string length. A single computation then yields estimates of string correlators of all lengths. Rigorous two-sided bounds are obtained for each target string once the ground-state energy is pinned within a window. When benchmarked against density-matrix renormalization group computations on the cluster Ising and spin$-1$ Heisenberg chains, the method certifies the nonlocal string order of the cluster and Haldane phases directly from the Hamiltonian. Our rigorous certification of string order parameters reveal that the sharpness of the bounds is set by which operators enter the calculation, not by their length alone. By exploiting algebraic closure of strings and local words, our framework lifts spatially extended observables into polynomially tractable bootstrap variables, opening up a route to wavefunction-free certification of nonlocal order in quantum many-body systems.
