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ON THE STATE OF A FERMI-SYSTEM WITH CORRELATION OF PAIRS OF PARTICLES WITH PARALLEL SPINS. I. GROUND--STATE

Zygmunt M. Galasiewicz

Published Jan 1, 1960• 0 citations
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Abstract

S>The solution of a compensation equation of dangerous graphs, for a system of interacting Fermi-particles, as an odd function of wave vectors is investigated. It is shown that this solution leads to the anomalous state, the energy of which is smaller than the energy of the normal state. The energy difference is equal to the binding energy of the pairs of fermions with parallel spin-moments. This anomalous state can exist in the presence of attractive Coulomb interactions. The elementary excitations are separated from the ground state by a gap the width of which depends on the direction of the wave vector. For the direction perpendicular to the spin direction, energy gap is zero and therefore the state cannot be superconducting. The binding energy of the pairs and the width of the energy gap are of the same order as in the theory based on the creation of pairs with antiparallel spin-moments, where in the expansion of the interaction term in spherical harmonics, the first coefficient is more than three times greater than the zero coefficient. In the absence of the magnetic field the total spin of a system is zero. In the presence of the magnetic field the total spinmore » differs from zero. The paramagnetic susceptibility is calculated. (auth)« less

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ON THE STATE OF A FERMI-SYSTEM WITH CORRELATION OF PAIRS OF PARTICLES WITH PARALLEL SPINS. II. THERMODYNAMICS

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The thermodynamics of the anomalous state of a Fermi-system (in which the correlation of pairs of particles with parallel spins is taken into account) were investigated. The transition temperature T/sub c/ to the anomalous state was found, as well as the temperature dependence the specific heat for T approx equal i O and T approx equal T/sub c/. For T = T/sub c/ the specific heat has a jump. In addition, formulas for the paramagnetic susceptibilities for T approx equal O and T approx equal T/sub c/ were obtained. (auth)

Acta Phys. Polon.•Jan 1, 1960•0 citations

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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.

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Collective excitations are considered for a system of Fermi particles for which attraction in a state with nonzero angular momentum is predominant. In such a system, beside sound excitations, other collective excitations without a gap in the energy spectrum will exist. The velocity of these excitations can be expressed in terms of the gap in the single-particle excitation spectrum. It is pointed out that the solution for the ground state function with DELTA (n)~Y/sub 22/ (n) is unstable. (auth)

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The properties of a Fermi system with a non-zero angular momentum pair interaction are investigated. Uncoupling of the three-particle Green function that yields a solution with isotropic and anisotropic gaps is suggested. It is shown that the solution with the anisotropic gap is asymptotically exact. The difficulties that appear in deriving the solutions with the isotropic gap are discussed. (auth)

•Apr 1, 1963•0 citations
S>The solution of a compensation equation of dangerous graphs, for a system of interacting Fermi-particles, as an odd function of wave vectors is investigated. It is shown that this solution leads to the anomalous state, the energy of which is smaller than the energy of the normal state. The energy difference is equal to the binding energy of the pairs of fermions with parallel spin-moments. This anomalous state can exist in the presence of attractive Coulomb interactions. The elementary excitations are separated from the ground state by a gap the width of which depends on the direction of the wave vector. For the direction perpendicular to the spin direction, energy gap is zero and therefore the state cannot be superconducting. The binding energy of the pairs and the width of the energy gap are of the same order as in the theory based on the creation of pairs with antiparallel spin-moments, where in the expansion of the interaction term in spherical harmonics, the first coefficient is more than three times greater than the zero coefficient. In the absence of the magnetic field the total spin of a system is zero. In the presence of the magnetic field the total spinmore » differs from zero. The paramagnetic susceptibility is calculated. (auth)« less
The thermodynamics of the anomalous state of a Fermi-system (in which the correlation of pairs of particles with parallel spins is taken into account) were investigated. The transition temperature T/sub c/ to the anomalous state was found, as well as the temperature dependence the specific heat for T approx equal i O and T approx equal T/sub c/. For T = T/sub c/ the specific heat has a jump. In addition, formulas for the paramagnetic susceptibilities for T approx equal O and T approx equal T/sub c/ were obtained. (auth)
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.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.
The pair susceptibility and superconducting gap equation are obtained when a branch-cut spectrum, with either spin-charge separation or an anomalous Fermi surface exponent $\ensuremath{\alpha}>0$ is introduced for the normal state. For k\mathbf{k}k-nondiagonal pairing interactions, spin-charge separation leads to an enhancement of Tc{T}_{c}Tc​ and ${\ensuremath{\Delta}}_{\mathbf{k}}(0)$ compared to a Fermi liquid. For $\ensuremath{\alpha}>0$, a critical coupling is required for a solution. For spin-charge separation or $0<\ensuremath{\alpha}<\frac{1}{2}$, an arbitrarily small k\mathbf{k}k-diagonal contribution to the pairing interaction still leads to a solution, as in a Fermi liquid.