发表机构
National Tsing Hua University（国立清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义通道集中度以量化量子几何中激发通道的响应分配，推导出临界伊辛模型和XX模型的精确值，并在共形场论中给出普适公式，揭示总度量标度遗漏的响应差异。
AI 中文摘要
量子度量量化了沿参数方向的基态总响应，但并未解析激发如何分担该响应。我们将通道集中度（CC）定义为在指定激发通道（如动量块）上归一化响应权重的平方和。一个精确的有限尺寸定理得出，对于临界横场伊辛模型的场响应，热力学极限下的CC为2/3；对于半填充XX配对响应，则为1/3，尽管两者具有相同的主导度量标度。在XY Lifshitz点的有限尺寸方法中，场和各向异性扰动产生不同的集中度，尽管它们具有共同的具有二次色散的限制哈密顿量。在圆上的一维1+1维共形场论（CFT）中，我们考虑固定扇区中的非简并真空受到一个空间积分的标量初级场的扰动。我们推导出完整的零动量能级响应权重，包括后裔态。对于标度维度$0<\Delta<3/2$，这些权重决定了归一化响应分布和一个精确的普适集中度函数。该表达式重现了精确的伊辛晶格极限2/3，并对三态Potts热场给出约0.8515的值，而仅基于指数的近似给出约0.800。有限尺寸相互作用计算比较了多体能级上的集中度和排序响应权重。这些精确基准显示了总度量标度所遗漏的响应差异，并指导了与有限尺寸相互作用谱的比较。
英文摘要
The quantum metric quantifies the total ground-state response along a parameter direction, but does not resolve how excitations share that response. We define channel concentration (CC) as the sum of squared normalized response weights over specified excitation channels, such as momentum blocks. An exact finite-size theorem yields thermodynamic CCs of 2/3 for the field response of the critical transverse-field Ising model and 1/3 for the half-filled XX pairing response, despite the same leading metric scaling. In finite-size approaches to the XY Lifshitz point, field and anisotropy perturbations yield different concentrations despite a common limiting Hamiltonian with quadratic dispersion. In a unitary 1+1-dimensional conformal field theory (CFT) on a circle, we consider a nondegenerate vacuum in a fixed sector perturbed by one spatially integrated scalar primary. We derive complete zero-momentum energy-level response weights, including descendants. For scaling dimension $0<Δ<3/2$, these weights determine the normalized response distribution and an exact universal concentration function. The expression reproduces the exact Ising lattice limit 2/3 and gives approximately 0.8515 for the three-state Potts thermal field, compared with approximately 0.800 from an exponent-only approximation. Finite-size interacting calculations compare concentrations and ranked response weights over many-body energy levels. These exact benchmarks show which response distinctions total metric scaling misses and guide comparisons with finite-size interacting spectra.
Comments44 pages, 7 figures (including Supplemental Material); code and data available at https://github.com/ToelUl/channel-concentration/tree/companion-2026-09-21-rc1
