The two particle density matrix of a Luttinger Liquid

Harini Radhakrishnan, Matthias Thamm, Hatem Barghathi, Bernd Rosenow, and Adrian Del Maestro

arXiv:XXXX.YYYYY

Abstract

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. We outline the derivation of the exact $2$-RDM from bosonization and map it to the $J$-$V$ model of interacting spinless fermions in one dimension, where the low-energy sector can be described 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 the two-body lattice energy and static structure factor.

Description

This repository includes links, code, scripts, and data to generate the figures in a paper.

Requirements

To reproduce the plots of the paper, run the code in the jupyter notebook generate_figures.ipynb. This notebook has the following python packages as dependencies:

• numpy
• scipy
• matplotlib

The DMRG simulations were performed using the code in the following repository: DelMaestroGroup/tV2RDMdmrg_julia

Support

The creation of these materials was supported in part by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0024333.

Figures

Figure 02: Interaction exponents

Figure 03: Density-density correlations

Figure 04: Two-RDM

Figure 05: Second cumulant

Figure 06: Difference between interacting and free fermion two-RDM

Figure 07: Unormalized background-subtracted coherence

Figure 08: Pairing wave function

Figure 10: One-RDM and two-RDM cut comparison to DMRG

Figure 11: Two-RDM cuts comparison to DMRG

Figure 12: Structure factor and pair correlation function

Figure 13: Energies comparison to DMRG