Charge- and spin-gap formation in exactly solvable Hubbard chains with long-range hopping

Abstract

We discuss the transition from a metal to charge or spin insulating phases characterized by the opening of a gap in the charge or spin excitation spectra, respectively. These transitions are addressed within the context of two exactly solvable Hubbard and tJ chains with long range, 1/r hopping. We discuss the specific heat, compressibility, and magnetic susceptibility of these models as a function of temperature, band filling, and interaction strength.We then use conformal field theory techniques to extract ground state correlation functions. Finally, by employing the g-ology analysis we show that the charge insulator transition is accompanied by an infinite discontinuity in the Drude weight of the electrical conductivity. While the magnetic properties of these models reflect the genuine features of strongly correlated electron systems, the charge transport properties, especially near the Mott-Hubbard transition, display a non-generic behavior.(3) which describes spin ( s Kσ ; S z K = ±1/2, C K = 0) and charge ( d K , e K ; S K = 0, C z K = ±1/2) degrees of freedom in an occupation number representation with a hard core constraint,The chemical potential in the presence of an external magnetic field is given by µ σ = µ − σ(gµ B H 0 )/2. In the following we will set µ B ≡ 1, and g = 2. To be precise, we also restrict ourselves to t ≥ 0 and U ≥ −2πt in which case we were allowed to identify K min − ∆ ≡ K max because J K vanishes for K = K min = −∆(L − 1)/2. Note that, formally, the entire spectrum appears to display spin-charge separation at all energies in the sense that the Hamiltonian splits up into independent spin and charge contributions ("strong" spin-charge separation). In reality, however, spin and charge excitations are coupled by the constraint, σ n s K,σ + n d K + n e K = 1; and spin-charge separation only occurs at sufficiently low energies/temperatures, where spin and charge excitations contribute independently to various physical properties.As already discussed above, we will concentrate on the physics of two special limits:(i) the 1/r-Hubbard model in the vicinity of half filling, n < ∼ 1, and (ii) the 1/r-tJ model for n < 1. The latter is obtained by taking the limit U → ∞ of (3) which projects out all double occupancies (h d K → ∞). With the help of the completeness constraint one arrives at the tJ effective Hamiltonian,where, for J = 4t 2 /U ≪ 1, the exchange coupling, J K (U/t), reduces to J K = (J/4) π 2 − (K − ∆/2) 2 . Below, we will adopt the usual standpoint for the tJ model 7 , and treat J as an independent parameter. A. 1/r-Hubbard ModelFrom eq. (3) we can immediately extract the form of eigenstates in an occupation number representation in terms of the effective spin ( s K,σ ) and charge ( e K , d K ) degrees of freedom.

The Hubbard chain at finite temperatures: ab initio calculations of Tomonaga-Luttinger liquid properties

“…For the numerical treatment of equations (47,49) we rewrite them in terms of usual convolutions of functions of a real variable…”

“…Finally, we want to comment on the structure of the equations determining the thermodynamical properties of the Hubbard model. In contrast to long-range interaction systems [46,47] we have to solve a set of subsidiary equations (47) for the "distribution functions" b, c, and c before evaluating the free energy (49). Obviously, the dynamics of the elementary excitations of the nearest-neighbor systems is more involved than those of [46,47] which may be viewed as "free particles with exclusion statistics".…”

Relaxation of a one-dimensional Mott insulator after an interaction quench

“…These expressions involve several energy scales, apart from the interaction U and the bandwidth W (= 1), namely the Mott gap ∆ = U − 1, the total bandwidth of the spectrum Ω = 2 + ∆ = U + 1, and ω = √ Ω 2 − 4U n, a characteristic density-dependent energy scale appearing in the holon and spinon excitation energies [22,33]. As functions of U the constants c ± have the remarkable symmetry that both are invariant under the replacement U → 1/U (for all n).…”

“…1, together with the long-time limit (dotted blue lines) and the thermodynamic prediction (solid red lines). The latter is determined from the exact grandcanonical potential f (T, µ, [22,32,33], using the temperature T and chemical potential µ that correspond to the same internal energy and density as the final state. The density is given by N gcan /L = ∂f /∂µ, which yields the chemical potential µ(T, n, U ) by inversion, and the internal energy per site is e(T, µ, U ) = H gcan /L = −T 2 ∂(f /T )/∂T − ∂f /∂µ.…”

“…We consider only n ≤ 1; larger densities can be treated by means of a particle-hole transformation, c kσ → c † −kσ [32]. For U ≥ −W and any number of lattice sites L, the model (4) is represented by an effective noninteracting bosonic Hamiltonian, from which ground-state and thermodynamic properties can be obtained analytically [22,32,33]. For U = 0 the ground state of (4) is of course the Fermi sea,…”