Nature of a single doped hole in two-leg Hubbard and $t-J$ ladders

In this paper, we have systematically studied the single-hole problem in two-leg Hubbard and $t-J$ ladders by large-scale density-matrix renormalization-group calculations. We found that the doped holes in both models behave similarly, while the three-site correlated hopping term is not important in determining the ground-state properties. For more insights, we have also calculated the elementary excitations, i.e., the energy gaps to the excited states of the system. In the strong-rung limit, we found that the doped hole behaves as a Bloch quasiparticle in both systems where the spin and charge of the doped hole are tightly bound together. In the isotropic limit, while the hole still behaves like a quasiparticle in the long-wavelength limit, our results show that its spin and charge components are only loosely bound together inside the quasiparticle, whose internal structure can lead to a visible residual effect which dramatically changes the local structure of the ground-state wave function.

Figure 1

The parameters of the (a) anisotropic Hubbard model and (b) $t-J$ model on a two-leg square ladder. Here $t_{ij}=t$ ($t_{ij}=\alpha t$) and $J_{ij}=J$ ($J_{ij}=\alpha J$) describe the interchain (intrachain) hopping and spin superexchange couplings, respectively. At $\alpha =1$, it reduces to the isotropic limit.

Figure 2

Ground-state phase diagram of the two-leg Hubbard ladder. The blue line is the phase boundary labeled by the critical value ${\alpha }_c$ as a function of $U/t$. In the left region ($\alpha <{\alpha }_c$), there is no charge modulation in the hole density profile, while the right region ($\alpha >{\alpha }_c$) has clear charge modulation.

Figure 3

Second derivative of single-hole kinetic energy $E_k^{\prime \prime }\left(h\right)=d^2{E}_k^h/d{\alpha }^2$ vs $\alpha$ with different sample sizes at $U/t=20$. The peak position, labeled by the blue dashed line, determines the phase boundary between the two distinct phases.

Figure 4

Hole density distribution $〈n_x^h〉$ of the Hubbard model at $U/t=20$ and the $t-J$ model at $t/J=5$ for (a) $\alpha =0.5$ and (b) $\alpha =1.0$. Here the system size is $100\times 2$, and $x$ is the rung index.

Figure 5

Hole-spin correlation functions $|〈n_{i_0}^h{s}_i^z〉|$ for (a) the $t-J$ model at $t/J=5$ and (b) the Hubbard model at $U/t=20$. Here $i_0=(50,2)$ is the hole site index, $i=(x,y)$ is the spin site index, and $d=|x-50|$ is the distance between the hole and spin along the ladder. The exponentially decaying correlation functions show that the spin degrees of freedom are exponentially localized close to the dynamic hole for both $\alpha =0.5$ and $\alpha =1.0$. The solid lines show the linear fit to the data, and $\xi$ is the spin-charge correlation length.

Figure 6

The effective mass $m$ (in units of $t$) of the single hole (or spin-charge object) of the two-leg Hubbard ladder at $U/t=20$. The effective mass diverges at ${\alpha }_c=0.79$ but remains finite in both $\alpha <{\alpha }_c$ and $\alpha >{\alpha }_c$ parameter regions. The inset shows the inverse of effective mass $1/m$ as a function of $\alpha$.

Figure 7

Excited-state energy gaps and density profiles. (a) Scaling of excitation energy $E_1-E_0$ with lattice size at $\alpha =0.5$. (b) Density profiles for the ground state and first excited state at $\alpha =0.5$. (c) Excitation energies $E_1-E_0$ and $E_2-{\tilde{E}}_0$ at $\alpha =1.0$, where ${\tilde{E}}_0=(E_1+E_0)/2$. (d) Density profiles for the ground state, first excited state, and second excited state at $\alpha =1.0$.

Figure 8

Ground-state hole density profile $n_x^h$ for the (a) free- fermion model in Fig. 9 and (b) isotropic two-leg $t-J$ ladder. It is clear that nodes are present when the ground state has momentum $k\ne 0$ for the free-fermion system, while they are absent for the isotropic $t-J$ ladder.

Figure 9

A noninteracting fermion model with the top of the valence band at nonzero momentum at half filling under the open boundary condition. (a) Coupling parameters and (b) energy dispersion obtained directly by Bloch theorem. With $t_{\parallel }=1, t_{\parallel }^{\prime }=0.632$, and $t_{\perp }=50$, the top of valence band is at $k=\pm k_0=\pm 0.629\pi$.

Figure 10

Effective mass calculated in two different ways. (a) $E_{\mathrm{OBC}}=\mathrm{const}+{\pi }^2/2m{L}^2$, with $L=400–1200$. (b) $|E_{\mathrm{PBC}}-E_{\mathrm{ABC}}|={\pi }^2/2m{L}^2$, whose decaying behavior is quite noisy, and it is hard to make a clear conclusion for small lattice size. However, with large enough system size (e.g., $l=200–1200$), the overall envelop clearly decays as $1/L^2$. The effective mass determined by both ways is consistent with the exact value.

Figure 11

(a) Excitation energy and (b) hole density profile for the noninteracting model in Ref. [19]. There are two quasidegenerate ground states with energy splitting scaling as $1/L^3$ and similar hole density profiles with a single wave packet, whereas the second excited state has a much higher energy, where the energy difference between the excited state and the ground state is much larger and decays as $1/L^2$. Moreover, the hole density distribution has double wave packets.