Analytic two-body density-matrix spectrum of the Fermionic Tonks-Girardeau gas
Abstract
The two-body density matrix (TBDM) of a quantum many-body system provides information about particle correlations, e.g., pairing. In this work, we derive the complete spectrum of the TBDM of the fermionic Tonks-Girardeau gas using its exact ground-state wavefunction in second quantization. We obtain analytical expressions for all eigenvalues and eigenvectors for an arbitrary number of particles and identify two different types of eigenstates: collective states, formed by superpositions of paired momentum modes, and single-determinant states. We provide a physical interpretation of this structure, explaining the origin of the different eigenstates and their degeneracies. We further derive an asymptotic expansion for the largest eigenvalue, improving on previous estimates and analytically confirming its linear scaling with the particle number, thereby indicating off-diagonal long-range order. For the high-index collective states, we obtain simpler expressions for the eigenvalues and eigenvectors, showing that the eigenvalues decay as n−4 and that the eigenvectors become independent of the number of particles. Finally, we discuss the connection between the collective states and an effective description in terms of composite bosons, whose spectral properties suggest a behavior consistent with Tan's contact for interacting bosonic systems.
Related
- Computable fermionic non-Gaussianity from the covariance matrix
- Entanglement spectrum and magic in higher-dimensional free fermionic systems
- Exact fermionic matrix elements from the Bethe-ansatz solution of the Richardson-Gaudin model
- Pauli spectrum and nonstabilizerness of random fermionic Gaussian states
- Hydrodynamic description of correlation matrix eigenvalues in a finite temperature expanding free fermionic gas
BibTeX
@misc{rojofrancas2026analytic,
title = {{Analytic two-body density-matrix spectrum of the Fermionic Tonks-Girardeau gas}},
author = {Abel Rojo-Francàs and Joan Martorell and Francesc Sabater and Bruno Juliá-Díaz},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05820},
eprint = {2611.05820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05820},
}
RIS
TY - UNPB
TI - Analytic two-body density-matrix spectrum of the Fermionic Tonks-Girardeau gas
AU - Abel Rojo-Francàs
AU - Joan Martorell
AU - Francesc Sabater
AU - Bruno Juliá-Díaz
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05820
UR - https://arxiv.org/abs/2611.05820
AB - The two-body density matrix (TBDM) of a quantum many-body system provides information about particle correlations, e.g., pairing. In this work, we derive the complete spectrum of the TBDM of the fermionic Tonks-Girardeau gas using its exact ground-state wavefunction in second quantization. We obtain analytical expressions for all eigenvalues and eigenvectors for an arbitrary number of particles and identify two different types of eigenstates: collective states, formed by superpositions of paired momentum modes, and single-determinant states. We provide a physical interpretation of this structure, explaining the origin of the different eigenstates and their degeneracies. We further derive an asymptotic expansion for the largest eigenvalue, improving on previous estimates and analytically confirming its linear scaling with the particle number, thereby indicating off-diagonal long-range order. For the high-index collective states, we obtain simpler expressions for the eigenvalues and eigenvectors, showing that the eigenvalues decay as n^−4 and that the eigenvectors become independent of the number of particles. Finally, we discuss the connection between the collective states and an effective description in terms of composite bosons, whose spectral properties suggest a behavior consistent with Tan's contact for interacting bosonic systems.
ER -