自旋链模型中 bootstrap 方法的数值探索
Numerical exploration of the bootstrap in spin chain models
AI总结：
本文利用半定规划研究一维自旋链的 bootstrap 方法，探讨了在多种模型中提取共形中心荷等物理数据的可靠性，并揭示了该方法在热力学极限和收敛性方面面临的指数复杂性挑战。
摘要：
我们分析了应用于一维自旋链的 bootstrap 方法（一种与变分方法对偶的优化方法），利用半定规划来提取数值结果。我们研究了基态中的关联函数在临界点、非临界点以及松弛最优性下如何收敛到其真实值。我们考虑了横向 Ising 模型、三态 Potts 模型以及其他非可积自旋链，并研究了半定方法在多大程度上能够可靠地提取数值化的涌现物理数据，包括共形中心荷、关联长度和标度维数。我们展示了提取这些数据的流程，并给出了所考虑的各种模型的初步结果。我们将结果与精确解析解以及在系统体积足够小时的精确对角化结果进行了比较。当我们尝试趋向热力学极限时，将平移不变性作为约束条件施加的半定数值方法找到了带有周期性边界条件的解，即使这些边界条件并未被明确指定。这意味着关联函数中所有共形数据的确定都必须在有限体积下处理。我们的研究揭示了该方法面临实际挑战。具体而言，从最优解中提取的关联函数（在优化中作为松弛变量）存在收敛问题，这表明系统大小具有潜在的指数复杂性。
英文摘要：
We analyze the bootstrap approach (a dual optimization method to the variational approach) to one-dimensional spin chains, leveraging semidefinite programming to extract numerical results. We study how correlation functions in the ground state converge to their true values at and away from criticality and at relaxed optimality. We consider the transverse Ising model, the three state Potts model, and other non-integrable spin chains and investigate to what extent semidefinite methods can reliably extract numerical emergent physical data, including conformal central charges, correlation lengths and scaling dimensions. We demonstrate procedures to extract these data and show preliminary results in the various models considered. We compare to exact analytical results and to exact diagonalization when the system volume is small enough. When we attempt to go to the thermodynamic limit, the semidefinite numerical method with translation invariance imposed as a constraint finds the solution with periodic boundary conditions even if these have not been specified. This implies that the determination of all conformal data in correlators has to be handled at finite volume. Our investigation reveals that the approach has practical challenges. In particular, the correlation functions extracted from the optimal solution, which function as slack variables in the optimization, have convergence issues that suggest an underlying exponential complexity in the system size.
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