卢廷格液体的两粒子密度矩阵
The two-particle density matrix of a Luttinger liquid
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中文总结 AI 辅助
研究卢廷格液体的两粒子密度矩阵，通过构造性玻色化推导其封闭有限尺寸表达式，揭示新指数，该结果再现密度矩阵重整化群计算，将通用标度与微观系统可观测量联系起来。
中文摘要 AI 辅助
两粒子相干性是约化密度矩阵层次结构的第一级，包含单粒子可观测量无法获取的相关性，但即使在一维中解析两粒子密度矩阵也很少见。我们使用具有显式紫外截断的构造性玻色化，推导了无自旋费米子在Tomonaga - Luttinger液体中的等时两粒子约化密度矩阵的封闭有限尺寸表达式。除了熟悉的依赖卢廷格参数\(K\)的指数\(\gamma^2=(K + K^{-1}-2)/2\)控制矩阵元的空间衰减外，结果还揭示了第二个指数\(\lambda=(K^{-1}-K)/2\)，它编码相反手征之间的相关性并控制非对角结构。两粒子约化密度矩阵的对角极限产生密度相关性和静态结构因子，其相干性解决了排斥时的代数\(2k_F\)电荷密度波相关性和吸引时的奇偶性p波配对相关性。在从单粒子密度矩阵确定截断后，解析结果定量地再现了卢廷格液体区域内相互作用J - V链的密度矩阵重整化群计算。该结果将通用卢廷格液体标度与有限微观系统中的可观测量联系起来。
英文摘要
Two-particle coherence is the first level of the reduced-density-matrix hierarchy that contains correlations inaccessible to single-particle observables, yet analytic two-particle density matrices are rare even in one dimension. We derive a closed, finite-size expression for the equal-time two-particle reduced density matrix of spinless fermions in a Tomonaga-Luttinger liquid using constructive bosonization with an explicit ultraviolet cutoff. In addition to the familiar Luttinger parameter $K$-dependent exponent $γ^2=(K+K^{-1}-2)/2$ which governs the spatial decay of matrix elements, the result exposes a second exponent, $λ=(K^{-1}-K)/2$, which encodes correlations between opposite chiralities and controls the off-diagonal structure. The diagonal limit of the two-particle reduced density matrix yields density correlations and the static structure factor, while its coherences resolve algebraic $2k_F$ charge-density-wave correlations for repulsion and odd-parity p-wave pairing correlations for attraction. After fixing the cutoff from the one-particle density matrix, the analytic result quantitatively reproduces density matrix renormalization group calculations of the interacting J-V chain within the Luttinger liquid regime. The result connects universal Luttinger liquid scaling with observables in finite microscopic systems.
