Ground-state and spectral properties of an asymmetric Hubbard ladder

We investigate a ladder system with two inequivalent legs, namely, a Hubbard chain and a one-dimensional electron gas. Analytical approximations, the density-matrix renormalization group method, and continuous-time quantum Monte Carlo simulations are used to determine ground-state properties, gaps, and spectral functions of this system at half-filling. Evidence for the existence of four different phases as a function of the Hubbard interaction and the rung hopping is presented. First, a Luttinger liquid exists at very weak interchain hopping. Second, a Kondo-Mott insulator with spin and charge gaps induced by an effective rung exchange coupling is found at moderate interchain hopping or strong Hubbard interaction. Third, a spin-gapped paramagnetic Mott insulator with incommensurate excitations and pairing of doped charges is observed at intermediate values of the rung hopping and the interaction. Fourth, the usual correlated band insulator is recovered for large rung hopping. We show that the wave numbers of the lowest single-particle excitations are different in each insulating phase. In particular, the three gapped phases exhibit markedly different spectral functions. We discuss the relevance of asymmetric two-leg ladder systems as models for atomic wires deposited on a substrate.

Figure 1

The asymmetric Hubbard ladder described by Hamiltonian (1), with intrachain hopping $t_{\parallel }$ and interchain hopping $t_{\perp }$. On the lower (Fermi, $y=\text{F}$) leg, electrons do not interact, whereas on the upper (Hubbard, $y=\text{H}$) leg, they experience an onsite repulsion $U$.

Figure 2

Single-particle dispersions [Eqs. (2) and (3)] of the noninteracting ladder for (a) $t_{\perp }=2.5t_{\parallel }$, (b) $t_{\perp }=t_{\parallel }$; (b) also shows the four Fermi points $\pm k_{\text{ab}}$ and $\pm k_{\text{b}}$ defined by Eq. (5). (c) Single-particle dispersion of the tight-binding chain (solid blue line) and single holon-spinon continuum (shaded area) of the half-filled Hubbard chain with $U=4t_{\parallel }$ from the Bethe ansatz solution. A horizontal dashed line shows the Fermi energy at half-filling in all three figures.

Figure 3

Staggered magnetization of the Hubbard leg ($m_{\text{H}}$) and Fermi leg ($m_{\text{F}}$) for $U=4t_{\parallel }$ in the Hartree-Fock approximation as a function of the rung hopping $t_{\perp }$.

Figure 4

Hartree-Fock gap $E_{\text{HF}}$ for $U/t_{\parallel }=2,4$, and 8 as a function of the rung hopping $t_{\perp }$.

Figure 5

The four Hartree-Fock bands ${\epsilon }_{n\sigma }\left(k\right)$ for $U=5t_{\parallel }$ with (a) weak ($t_{\perp }=0.5t_{\parallel }$), (b) intermediate ($t_{\perp }=1.5t_{\parallel }$), and (c) strong ($t_{\perp }=3t_{\parallel }$) rung hopping.

Figure 6

Hartree-Fock “phase diagram” in the $\left(U,t_{\perp }\right)$ plane with three different regions. The lowest single-particle excitations have wave numbers $k_{\text{HF}}$ at the edges of the reduced Brillouin zone $(k_{\text{HF}}=\pm \frac{\pi }{2})$, at its center $\left(k_{\text{HF}}=0\right)$, and at incommensurate values $0<\left|k_{\text{HF}}\right|<\frac{\pi }{2}$, respectively.

Figure 7

(a) Charge gap $E_{\text{c}}$ and (b) spin gap $E_{\text{s}}$ calculated with the DMRG as a function of the rung hopping $t_{\perp }$ in a half-filled asymmetric two-leg Hubbard ladder with $L=128$ rungs. Finite-size corrections are of the order of the symbol size in (a) and of the order of $0.01t_{\parallel }$ in (b). Error bars indicate DMRG truncation errors larger than the symbol size.

Figure 8

Pair binding energy $E_{\text{pb}}$ calculated with DMRG in an asymmetric two-leg Hubbard ladder with $L=128$ rungs (a) as a function of the rung hopping $t_{\perp }$ at half-filling for several values of the Hubbard interaction $U$ and (b) as a function of the band filling $n=N/2L$ for $U=5t_{\parallel }$ and two values of $t_{\perp }$. Finite-size corrections are smaller than $0.05t_{\parallel }$. Error bars indicate DMRG truncation errors larger than the symbol size.

Figure 9

Ground-state charge density distribution on the Fermi leg for two electrons added to a half-filled ladder with $U=8t_{\parallel }$ and three values of $t_{\perp }$.

Figure 10

Fourier transform of the ground-state charge density on the Fermi leg with two electrons added to a half-filled ladder with $U=5t_{\parallel }$ and four values of $t_{\perp }$.

Figure 11

Spectral functions $A(k,y,\omega )$ on the Hubbard leg (left column) and the Fermi leg (right column) calculated using the CT-INT method with $\beta t_{\parallel }=30$ on a periodic ladder with $L=30$ rungs and $U=5t_{\perp }$. (a), (b) Luttinger liquid phase ($t_{\perp }=0.1t_{\parallel }$), (c), (d) Kondo-Mott insulator ($t_{\perp }=0.3t_{\parallel }$), (e), (f) incommensurate spin-gapped Mott insulator ($t_{\perp }=t_{\parallel }$), (g), (h) correlated band insulator ($t_{\perp }=3t_{\parallel }$).

Figure 12

Schematic phase diagram of the half-filled asymmetric Hubbard ladder.