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Odd-Frequency Superfluidity in the Fermionic Tonks-Girardeau Gas

Matthew Wong

Published Sep 9, 2026• 0 citations
Publisher

Abstract

The fermionic Tonks-Girardeau (FTG) gas is known to exhibit conventional superfluidity, but may also be a vessel for unconventional odd-frequency superfluidity that is not revealed when studying only equal-time pairing correlations. In this thesis, we investigate odd-frequency superfluidity in the FTG gas. Specifically, this system is explored as a possible example of situations where odd-frequency pairing arises alongside conventional, even-frequency superfluidity. Previous works have extensively studied macroscopic quantum phenomena in the FTG gas via reduced density matrices, but this work is the first to investigate the hidden dynamical pairing in this system. We search for even fermion-exchange symmetry in the time-dependent two-body correlation matrix (T2bCM) that was formalised and shown to be synonymous with odd-frequency pair correlations by previous work. The eigenfunctions and eigenvalues of this matrix are studied for the homogeneous and harmonically trapped FTG gas. In the homogeneous case, it is found that the T2bCM eigenfunction is entirely antisymmetric under exchange of fermion coordinates and therefore no symmetric pairing occurs and no odd-frequency superfluidity exists within the system. Furthermore, the conventional macroscopic order present at zero time continues to exist at finite values of the pairing-time difference, but the magnitude of the macroscopic eigenvalue decreases monotonically with this time parameter. In the harmonically trapped case, the T2bCM eigenfunction contains a finite symmetric component that becomes concentrated at the boundary of the system for a large number of particles. It is found that, like the untrapped scenario, the macroscopic eigenvalue is finite and increases with particle number. We then consider a symmetric pairing order parameter and find that it decays with the particle number according to a power law, indicating that the harmonically trapped FTG gas exhibits non-macroscopic interface-localised odd-frequency order.

DOI
10.26686/kcv2-wb8d

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The fermionic Tonks-Girardeau (FTG) gas is known to exhibit conventional superfluidity, but may also be a vessel for unconventional odd-frequency superfluidity that is not revealed when studying only equal-time pairing correlations. In this thesis, we investigate odd-frequency superfluidity in the FTG gas. Specifically, this system is explored as a possible example of situations where odd-frequency pairing arises alongside conventional, even-frequency superfluidity. Previous works have extensively studied macroscopic quantum phenomena in the FTG gas via reduced density matrices, but this work is the first to investigate the hidden dynamical pairing in this system. We search for even fermion-exchange symmetry in the time-dependent two-body correlation matrix (T2bCM) that was formalised and shown to be synonymous with odd-frequency pair correlations by previous work. The eigenfunctions and eigenvalues of this matrix are studied for the homogeneous and harmonically trapped FTG gas. In the homogeneous case, it is found that the T2bCM eigenfunction is entirely antisymmetric under exchange of fermion coordinates and therefore no symmetric pairing occurs and no odd-frequency superfluidity exists within the system. Furthermore, the conventional macroscopic order present at zero time continues to exist at finite values of the pairing-time difference, but the magnitude of the macroscopic eigenvalue decreases monotonically with this time parameter. In the harmonically trapped case, the T2bCM eigenfunction contains a finite symmetric component that becomes concentrated at the boundary of the system for a large number of particles. It is found that, like the untrapped scenario, the macroscopic eigenvalue is finite and increases with particle number. We then consider a symmetric pairing order parameter and find that it decays with the particle number according to a power law, indicating that the harmonically trapped FTG gas exhibits non-macroscopic interface-localised odd-frequency order.
The fermionic Tonks-Girardeau gas is an exactly solvable model realizing a conventional macroscopic condensate of p-wave Cooper pairs. Here we demonstrate that, when confined by a trapping potential, it also hosts unconventional odd-frequency pairing, i.e., pair correlations with s-wave symmetry that are only present with a finite time delay t between fermions forming a pair. Quantitative results for fixed particle number are obtained using a generalization of Yang's theory of off-diagonal long-range order to define a t-dependent order parameter that is antisymmetric under combined t inversion and exchange of fermion-pair indices. Odd-frequency pairing in the fermionic Tonks-Girardeau gas is found to be a mesoscopic effect, localized at the system's boundary and therefore nonextensive thermodynamically. Our results show that this elusive hidden order is accessible in systems having fixed particle number, suggesting new avenues towards its experimental realization and further detailed study.
We determine some exact static and time-dependent properties of the fermionic Tonks-Girardeau (FTG) gas, a spin-aligned one-dimensional Fermi gas with infinitely strongly attractive zero-range odd-wave interactions. We show that its two-particle reduced density matrix exhibits superconductive off-diagonal long-range order, and on a ring an FTG gas with an even number of atoms has a highly degenerate ground state with quantization of Coriolis rotational flux and high sensitivity to rotation and to external fields and accelerations. For a gas initially under harmonic confinement, we show that during an expansion the momentum distribution undergoes a "dynamical bosonization," approaching that of an ideal Bose gas without violating the Pauli exclusion principle.
The fermionic Tonks-Girardeau (FTG) gas is a one-dimensional spin-polarized Fermi gas with infinitely strong attractive zero-range odd-wave interactions, arising from a confinement-induced resonance reachable via a three-dimensional ppp-wave Feshbach resonance. We investigate the off-diagonal long-range order (ODLRO) of the FTG gas subjected to a longitudinal harmonic confinement by analyzing the two-particle reduced density matrix for which we derive a closed-form expression. Using a variational approach and numerical diagonalization we find that the largest eigenvalue of the two-body density matrix is of order N∕2N∕2N∕2, where NNN is the total particle number, and hence a partial ODLRO is present for a FTG gas in the trap.
We investigate odd-in-time—or —pairing of fermions in equilibrium systems within the particle-number-conserving framework of Penrose, Onsager, and Yang, where superfluid order is defined by macroscopic eigenvalues of reduced density matrices. We show that odd-frequency pair correlations are synonymous with even fermion-exchange symmetry in a time-dependent correlation function that generalises the two-body reduced density matrix. Macroscopic even-under-fermion-exchange pairing is found to emerge from conventional Penrose-Onsager-Yang condensation in two-body or higher-order reduced density matrices through the symmetry-mixing properties of the Hamiltonian. We identify and characterize a matrix responsible for producing macroscopic even fermion-exchange correlations that coexist with a conventional Cooper-pair condensate, while a matrix is shown to be responsible for creating macroscopic even fermion-exchange correlations from hidden orders such as a multiparticle condensate. The transformer scenario is illustrated using the spin-balanced s -wave superfluid with Zeeman splitting as an example. The generator scenario is demonstrated by the composite-boson condensate arising for itinerant electrons coupled to magnetic excitations. Structural analysis of the transformer and generator matrices is shown to provide general conditions for odd-frequency pairing order to arise in a given system. Our formalism facilitates a fully general derivation of the Meissner effect for odd-frequency superconductors that holds also beyond the regime of validity for mean-field theory. Published by the American Physical Society 2024
We theoretically investigate an ultracold spin-polarized atomic Fermi gas with resonant odd-channel (p(p(p-wave) interactions trapped in one-dimensional harmonic traps. We solve the Yang-Yang thermodynamic equations based on the exact Bethe ansatz solution, and predict the finite-temperature density profile and breathing mode frequency by using a local density approximation to take into account the harmonic trapping potential. The system features an exotic super fermionic Tonks-Girardeau (super-fTG) phase, due to the large effective range of the interatomic interactions. We explore the parameter space for such a fascinating super-fTG phase at finite temperature and provide smoking-gun signatures of its existence in both breathing mode frequencies and density profiles. Our results suggest that the super-fTG phase can be readily probed at temperatures of about 0.1TF0.1{T}_{F}0.1TF​, where TF{T}_{F}TF​ is the Fermi temperature. These results are to be confronted with future cold-atom experiments with 6Li^{6}\mathrm{Li}6Li and 40K^{40}\mathrm{K}40K atoms.
We demonstrate that the ground state of the fermionic Tonks-Girardeau gas is a number-conserving Bardeen-Cooper-Schrieffer (BCS) state. This offers a compelling new perspective on describing the ground-state wave function of the fermionic Tonks-Girardeau gas, enabling a complete description in second quantization. The proposed wave function is constructed based on the occupation numbers and natural orbitals of the one-body density matrix and is valid under any external potential. Specifically, we provide the exact coefficients that allow one to describe the fermionic Tonks-Girardeau gas as a number-conserving BCS state. Furthermore, the proposed wave function's form in second quantization allows us to derive the necessary expectation values to detect pairing in the fermionic Tonks-Girardeau gas. Through this approach, we demonstrate and show how to detect that the fermionic Tonks-Girardeau gas not only exhibits nontrivial quantum correlations but also is a paired state.