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6.4. THREE-SITE RING 139
6.4 A pair of particles in a three-site ring
Let us first consider a simple model describing a pair of up and down spin particles
on a simple three-site ring. The sites are connected by a hopping integral t, and
we consider an on-site repulsion U>0 and a nearest neighbor Coulomb repulsion
V :
H 3 = t ∑
n i↑ n i↓ + V ∑ n i n j (6.1)
〈i,j〉
c + iσ c jσ + U ∑
〈i,j〉σ i
The Hilbert space contains 9 states, and when we consider translational symmetries,
it is reduced to 3 states :
P 1 = |↑↓ ◦〉 + e ik |◦ ↑↓〉 − e −ik |↓ ◦ ↑〉
P 2 = |↓↑ ◦〉 + e ik |◦ ↓↑〉 − e −ik |↑ ◦ ↓〉
P 3 = |d ◦◦〉+ e ik |◦d◦〉 + e −ik |◦ ◦ d〉
where d is a doubly occupied site, and k takes the value k =0, 2π/3, 4π/3. The
ground state of free particles (U = V =0)|ψ 0 〉 is in the sector k =0fort0. The ground-state
energy for t0withenergyE = −2|t|. In the latter case, two of the
eigenvectors lie in the sector k = 0 and the two other eigenvectors lie respectively
in the sectors k =2π/3 andk = −2π/3. This is a trivial result understood in
terms of the free dispersion of a one dimensional chain with periodic boundary
conditions. However, it is worth noting that the circulation of the current around
the ring is finite for the ground state component lying in the k = ±2π/3 sectors
when t>0. In the latter case we get, in each of the k = ±2π/3 sectors of the
Hilbert space, the circulation of the current: F =1.1547|t|, where:
F = 〈J 12 〉 + 〈J 23 〉 + 〈J 32 〉 (6.2)
and the definition of the current is obtained by the conservation of the density:
δn i
δt =0=e c [H, n i]= ∑ J i,j (6.3)
〈i,j〉
which leads to the definition of the current operator on a link 1 :
J ij = ∑ (
)
it ij c † iσ c jσ + c.c.
σ
(6.4)
We emphasize that this definition of the current-operator is gauge invariant.
When U, V > 0, the states k = ±2π/3 have higher energies, but the degeneracy
1 This leads to the same definition as the derivative of the Hamiltonian with respect to the
δH
gauge field A ij :
δA ij
= J ij .