Equations of two-fluid hydrodynamics in the Hubbard model
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Equations of TwoFluid Hydrodynamics in the Hubbard Model R. O. Zaіtsev Moscow Institute of Physics and Technology, Institutskiі per. 9, Dolgoprudnyі, Moscow oblast, 141700 Russia *email: [email protected] Received October 13, 2009
Abstract—A set of equations is obtained using stationary and nonstationary superconductivity theories in terms of the Hubbard model. The equations near Tc and for a weak magnetic field have the general form of superconductivity equations with a strongly anisotropic Fermi surface. Calculations are performed using a kinetic equation for quasiparticles. The lowtemperature collective excitations in a superconductor are stud ied. An explicit relationship for the temperature dependence of second sound is obtained. DOI: 10.1134/S1063776110040060
1. INTRODUCTION To date, the study of superconductivity in the Hub bard model has been restricted to finding the Cooper instability possibility. In the Hubbard model with repulsion, superconductivity was shown to occur in a 1
limited electron concentration range (2/3 < ne < 1). In this work, we obtain superconductivity equations for this range, derive kinetic equations, and determine the temperature dependence of second sound. We also obtain equations to determine the spatial and time dependences of a function that relates the density of particles to the density of quasiparticle states. We con sider the possibility of finding the corresponding dependences (end factors) in both the normal and superconductivity parts of the phase diagram in the Hubbard model.
n, m From this point on, Xˆ r are the Hubbard X operators of the Fermi type that satisfy the permutation relation ships n, m n, q p, m p, q (2) { Xˆ , Xˆ } = δ ( δ Xˆ + δ Xˆ ), r
σ, 0 0, σ Xˆ r Xˆ r' ϕ ( r – r' )
σ, r, r' ( r ≠ r' )
–μ
∑
(1) σ, 0 0 , σ Xˆ r Xˆ r .
σ, r
1 This
r 1, r 2
3+
1
r1
2
agrees
qualitatively
with
experiments
r
σ
R r1, r2 ( τ 1, τ 2 ) =
r2
1
2
ˆ ( Xˆ 0, σ ( τ )Xˆ 0, –σ ( τ ) )〉 , –〈T r1 r2 1 2
ˆ ( Xˆ –σ, 0 ( τ )Xˆ σ, 0 ( τ ) )〉 , ˜ rσ , r ( τ , τ ) = – 〈 T R r1 r2 1 2 1 2 1 2
–σ
= – D r2, r1 ( τ 2, τ 1 ).
(2')
Here, the fourth Green’s function is expressed ˆ is the operator of order through the first function, T ing in parameter τ, and the X operators are written in the Matsubara representation with Hamiltonian (1). The equations for the Green’s functions written in the oneloop approximation represent the generaliza tion of Gor’kov’s equations for the X Hubbard opera tors. Allowing for permutation relationships (2), we obtain ˆ + ( τ , r )D + ( τ , τ ) U 1
finding
n, q
r
ˆ ( Xˆ –σ, 0 ( τ )Xˆ 0, –σ ( τ ) )〉 ˜ rσ , r ( τ , τ ) = – 〈 T D r1 r2 1 2 1 2 1 2
Assume that Hubbard energy U is the maximum energy parameter, which corresponds to the consider ation of transitions inside the lower Hubbard subband with the Hamiltonian
∑
m, p
where μ is the chemical potential and ϕ(r) are the hop ping integrals. If a system is unstable to the appearance of a con densate of Cooper pairs, then it passes to the super c
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