Single-pole ladder at quarter filling

We study the ground state and excitation spectrum of a quasi-one-dimensional system consisting of a pole and rungs oriented in opposite directions (“centipede ladder,” CL) at quarter filling. The spin and charge excitation spectra are found in the limits of small and large longitudinal hoppings $t_{\Vert }$ compared to the on-rung hopping rate $t_{\perp }$ and exchange coupling $I_{\perp }$. At small $t_{\Vert }$, the system with ferromagnetic on-rung exchange demonstrates instability against dimerization. Coherent propagation of charge-transfer excitons is possible in this limit. At large $t_{\Vert }$, CL behaves like two-orbital Hubbard chain, but the gap opens in the charge excitation spectrum, thus reducing the symmetry from SU(4) to SU(2). The spin excitations are always gapless and their dispersion changes from quadratic magnonlike for ferromagnetic on-rung exchange to linear spinonlike for antiferromagnetic on-rung exchange in weak longitudinal hopping limit.

Figure 1

(Color online) Centipede (a) and strongly asymmetric two-leg ladder (b). Several possible rung states (c) corresponding to vacancy, single occupied states, singlet and triplet states. Excited states with double on-site occupancy are not shown.

Figure 2

(Color online) Two-electron level structure for two neighboring rungs. (a) Ferromagnetic on-rung exchange. The ground state is dimerized. (b) Antiferromagnetic on-rung exchange. The ground state is array of bonding orbitals. (c) Occupation scheme for two different ground states: filled, empty, and half-filled circles correspond to occupied (triplet), empty (vacancy), and bonding electron states, respectively. Transitions within a given charge spin or charge sector and transitions between different sectors are marked by vertical and slanting dashed arrows, respectively. The energies of these transitions are also shown. The two-electron energy levels for singlet $\left(E_S\right)$, triplet $\left(E_T\right)$ bonding-bonding $\left(E_{++}\right)$, and bonding-antibonding $\left(E_{+-}\right)$ states are given in Eq. (11).

Figure 3

Scheme of superexchange induced in fourth-order hopping processes in dimerized CL. Arrows indicate sequential electron hops resulting in the intermediate states (16). Notations for the occupation scheme are the same as for Fig. 2c.

Figure 4

(Color online) Tight-binding band structure for quarter-filled centipede model at $t_{\Vert }⪡t_{\perp }$. The occupied and empty orbital subbands ${ϵ}_{\pm }\left(k\right)$ are shown. The chemical potential is chosen as a reference energy level.

Figure 5

Coherent motion of the vacancy-singlet exciton due to the fourth-order electron hopping $(1\to 2;2\to 3;3\to 2;4\to 3)$. Diagrams illustrate electron configurations in the initial (a), subsequent intermediate [(b)–(d)], and finite (e) states of exciton translation. Filled, empty, and half-filled (black down) and half-filled (black up) circles correspond to singlet, vacancy bonding, and antibonding electron states, respectively.

Figure 6

Incoherent motion of the vacancy and singlet due to the fourth-order electron hopping $(2\to 3;2\to 3;1\to 2;3\to 4)$. Notations for the occupation scheme are the same as in Fig. 5.

Figure 7

(Color online) Tight-binding band structure of two lowest Hubbard subbands for quarter-filled centipede model at $t_{\perp }⪡t_{\Vert }$. The electrochemical potential ${ϵ}_{\alpha }={ϵ}_{\beta }$. The direct gap corresponds old Fermi surface point $k_F=\pi ∕2$. The indirect gap ${\Delta }_{\mathit{ind}}$ is between the bottom of conduction band and the top of the valence band is characterized by the “flat” parts of the spectra. The inset shows the energy dependence of the density of states (DoS). The character of the DoS singularities and its influence on the response functions are discussed in the Sec. 4.