The two body density matrix of a Luttinger liquid

The $n$-body reduced density matrix ($n$-RDM) characterizes higher order correlations in a many-body system. This quantity can be used to compute any $n$-body observable without direct access to the full wavefunction, and is experimentally measurable. Analytically, the problem of computing higher order density matrices becomes increasingly challenging as the number of coordinates grows. However, within the Luttinger liquid regime, bosonization provides access to correlation functions by representing them as exponentials of bosonic field operators. In this talk, we outline the derivation of the exact $2$-RDM from bosonization and map to the $J$-$V$ model of interacting, spinless fermions in one dimension, where the low-energy sector is describable by Luttinger liquid theory. We demonstrate that the non-interacting limit reproduces the result predicted by Wick's theorem and present an analytical result for density-density correlations, allowing for an investigation of the effects of a finite size lattice. Our results demonstrate agreement with those obtained from density matrix renormalization group calculations. Finally, we discuss the application of our expression for computing two-body observables such as the energy and the particle entanglement.