Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1110.2821 · Physical Review A

One-dimensional multicomponent fermions with δ -function interaction in strong- and weak-coupling limits: Two-component Fermi gas

Xi‐Wen Guan, Zhong-Qi Ma

Published Mar 26, 2012• 34 citations
arXivHTMLPDF

Abstract

The Fredholm equations for one-dimensional two-component fermions with repulsive and with attractive $\ensuremath{\delta}$-function interactions are solved by an asymptotic expansion for strong repulsion, weak repulsion, weak attraction, and strong attraction. Consequently, we obtain the first few terms of the expansion of the ground-state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two types. Thus the asymptotic expansions of the energies of repulsive and attractive fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy connects analytically.

DOI
10.1103/physreva.85.033632

Related papers

One-dimensional multicomponent fermions with δ -function interaction in strong- and weak-coupling limits: κ -component Fermi gas

Xi‐Wen Guan, Zhong-Qi Ma, Brendan Wilson

We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional $\ensuremath{\kappa}$-component fermions with repulsive and attractive $\ensuremath{\delta}$-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the $\ensuremath{\kappa}$-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You's result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as $\ensuremath{\kappa}\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as for spinless Bosons.

Physical Review A•Mar 26, 2012•30 citations

1D multicomponent Fermions with delta function interaction in strong and weak coupling limits: $κ$-component Fermi gas

Xi-Wen Guan, Zhong-Qi Ma, Brendan Wilson

We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional $κ$-component fermions with repulsive and with attractive delta-function interaction in strong and weak coupling regimes. We thus obtain a highly accurate result for the ground state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair correlation functions shed light on the quantum statistic effects of the $κ$-component interacting fermions. For the balanced spin case with repulsive interaction, the ground state energy obtained confirms Yang and You's result [Chin. Phys. Lett. {\bf 28}, 020503 (2011)] that the energy per particle as $κ\to \infty$ is the same as for spinless Bosons.

arXiv:1201.4629•Jan 23, 2012•0 citations

Exact Analysis of δ-Function Attractive Fermions and Repulsive Bosons in One-Dimension

Toshiaki Iida, Miki Wadati

The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.

arXiv:cond-mat/0503033•Journal of the Physical Society of Japan•Jun 1, 2005•45 citations

Ground-state and elemental excitations of the one-dimensional multicomponent Fermi gas with delta -function interaction

P. Schlottmann

We consider a gas of fermions with parabolic dispersion and N spin components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. The ground-state Bethe ansatz integral equations are solved numerically for both repulsive and attractive interactions to obtain the energy, the chemical potential, and the magnetic susceptibility as a function of the band filling and the interaction strength. For the repulsive interaction the Fermi gas has the properties of a Luttinger liquid. In the attractive case, on the other hand, the fermions in the ground-state form bound states of up to N fermions of different spin components. The spectrum of elemental charge and spin excitations is derived for the repulsive and attractive situations. The spectrum is discussed in the limits of vanishing interaction strength and very strong coupling. For the repulsive interaction the low-lying charge excitations can be characterized by the Fermi momentum and the Fermi velocity. The range of the spin-wave excitations is correlated with the Fermi momentum of the charges. The spin wave velocity is inversely proportional to the magnetic susceptibility. The spin-wave excitations become soft in the infinite-repulsive-coupling limit. In the attractive case in zero field all excitation branches except that of bound states of N fermions have an energy gap. It requires a finite energy to break these bound states and hence there is no response to a field smaller than a critical field. The low-T specific heat is proportional to the temperature.

Journal of Physics Condensed Matter•Feb 14, 1994•47 citations

Thermodynamics of the one-dimensional multicomponent Fermi gas with a delta -function interaction

P. Schlottmann

The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.

Journal of Physics Condensed Matter•Aug 9, 1993•33 citations

One-dimensional multicomponent Fermi gas in a trap: quantum Monte Carlo study

Matveeva, N., Astrakharchik, Grigori

A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.

•Jan 1, 2016•30 citations
The Fredholm equations for one-dimensional two-component fermions with repulsive and with attractive \ensuremathδ\ensuremath{\delta}\ensuremathδ-function interactions are solved by an asymptotic expansion for strong repulsion, weak repulsion, weak attraction, and strong attraction. Consequently, we obtain the first few terms of the expansion of the ground-state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two types. Thus the asymptotic expansions of the energies of repulsive and attractive fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy connects analytically.
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional $\ensuremath{\kappa}$-component fermions with repulsive and attractive $\ensuremath{\delta}$-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the $\ensuremath{\kappa}$-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You's result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as $\ensuremath{\kappa}\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as for spinless Bosons.
1D multicomponent Fermions with delta function interaction in strong and weak coupling limits: κκκ-component Fermi gas
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional κκκ-component fermions with repulsive and with attractive delta-function interaction in strong and weak coupling regimes. We thus obtain a highly accurate result for the ground state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair correlation functions shed light on the quantum statistic effects of the κκκ-component interacting fermions. For the balanced spin case with repulsive interaction, the ground state energy obtained confirms Yang and You's result [Chin. Phys. Lett. {\bf 28}, 020503 (2011)] that the energy per particle as κ→∞κ\to \inftyκ→∞ is the same as for spinless Bosons.
The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.
We consider a gas of fermions with parabolic dispersion and N spin components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. The ground-state Bethe ansatz integral equations are solved numerically for both repulsive and attractive interactions to obtain the energy, the chemical potential, and the magnetic susceptibility as a function of the band filling and the interaction strength. For the repulsive interaction the Fermi gas has the properties of a Luttinger liquid. In the attractive case, on the other hand, the fermions in the ground-state form bound states of up to N fermions of different spin components. The spectrum of elemental charge and spin excitations is derived for the repulsive and attractive situations. The spectrum is discussed in the limits of vanishing interaction strength and very strong coupling. For the repulsive interaction the low-lying charge excitations can be characterized by the Fermi momentum and the Fermi velocity. The range of the spin-wave excitations is correlated with the Fermi momentum of the charges. The spin wave velocity is inversely proportional to the magnetic susceptibility. The spin-wave excitations become soft in the infinite-repulsive-coupling limit. In the attractive case in zero field all excitation branches except that of bound states of N fermions have an energy gap. It requires a finite energy to break these bound states and hence there is no response to a field smaller than a critical field. The low-T specific heat is proportional to the temperature.
The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.
A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.