The two-particle density matrix of a Luttinger liquid
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 $\gamma^2=(K+K^{-1}-2)/2$ which governs the spatial decay of matrix elements, the result exposes a second exponent, $\lambda=(K^{-1}-K)/2$, 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.