Finite-energy spectral function of an anisotropic two-dimensional system of coupled Hubbard chains

We study the crossover from the one-dimensional to the two-dimensional Hubbard model in the photoemission spectra of weakly coupled chains. The chains with on-site repulsion are treated using the spin-charge factorized wave function, known to provide an accurate description of the dynamics of the model in the strong coupling limit, while the hoppings between the chains are considered as a perturbation. We calculate dynamical spectral functions at all energies in the random-phase approximation by resuming an infinite set of diagrams. Even though the hoppings drive the system from a fractionalized Luttinger-liquid-like system to a Fermi-liquid-like system at low energies, significant characteristics of the one-dimensional system remain in the two-dimensional system. Furthermore, we find that, of introducing (frustrated) hoppings beyond the nearest-neighbor one, the interference effects increase the energy and momentum range of the one-dimensional character.

Figure 1

Direction of the hopping terms in the 2D Hubbard model.

Figure 2

Left panel: Values of the charge exponents ${\beta }_c^{+}$and ${\beta }_c^{-}$ as functions of the LL parameter $\theta$. Central and right panels: Spectral function obtained using RPA expression (1) for $t^{\prime }=0.2$ and $t^{\prime }=-0.2$ and for $\theta =1/8$, $v_s=1$, and $v_c=2.718$. The gray lines signal the boundaries of spin and charge continua of the 1D spectral function: $\omega =\pm u_{s}k$ and $\omega =\pm u_{c}k$. The bound state (green line) is obtained by solving Eq. (3). Red (large) and blue (small) dots displayed at $\omega =0$ correspond to values of $k_x$ for which $\text{sign}\hspace{0.5em}\text{Re}\left[G(\omega =0,k_x)\right]$ is respectively positive or negative.

Figure 3

Spectral function obtained by RPA formula (1) for different values of the LL parameter $\theta$ and interchain coupling $t^{\prime }$. Axes and labels are the same as for Fig. 2, central and right panels. For $\theta =0$ we set $v_c=v_s$ in order to obtain the exact free-particle result; for all other values of $\theta$ $v_s=1$ and $v_c=2.718$. The gray lines signal the boundaries of spin and charge continuum. Red (large) and blue (small) dots displayed at $\omega =0$ correspond to values of $k_x$ for which $\text{sign}\hspace{0.5em}\text{Re}\left[G(\omega =0,k_x)\right]$ is, respectively, positive or negative. For the cases where $Z\ne 0$ this criterion corresponds to $k_x$ being inside or outside the FS. The green line corresponds to the bound states obtained by solving Eq. (3).

Figure 4

QP residue as a function of $t^{\prime }$ for different values of $\theta$: $\theta =1/16$ (orange), $\theta =1/8$ (blue), $\theta =1/2$ (gray), $\theta =3/4$ (red), and $\theta =1$ (black). For each value of $t^{\prime }$, $k_x^{+}$ and $k_x^{-}$ were determined on each side of the FS. For these values the bound-state equation was solved in order to find ${\omega }_{+/-}\simeq 0$; $Z_{+/-}$ was computed using Eq. (4).

Figure 5

SF for the 1D Hubbard model within the large-$U$ approximation. Upper panel: quarter-filling $n=0.5$, computed for $U/t=7.5$. Lower panel: $n=0.7$, computed for $U/t=11.5$

Figure 6

Top three rows: SF of the Hubbard model at quarter-filling ($n=0.5$) and $U=7.5$ in an anisotropic square lattice obtained by weakly coupling Hubbard chains within RPA (1) for different values of the interchain hopping $t_1^{\prime }$ and transverse momentum $k_y$. The axes labels and the scale are the same as for Fig. 5. As $t_1^{\prime }$ increases the bound states, corresponding to a coherent excitation, start on the boundaries of the continuous region, changing the shape of the FS. Lower row: Spectral function of the Hubbard model for $n=0.7$ and $t_1^{\prime }=0.25$ and $U/t=11.5$.

Figure 7

Left panel: The gray and white regions correspond, respectively, to $\text{sign}\hspace{0.5em}\text{Re}\left[G(\omega =0,k_x)\right]<0$ and $\text{sign}\hspace{0.5em}\text{Re}\left[G(\omega =0,k_x)\right]>0$, i.e., to the exterior and interior of the FS obtained within the RPA. The FS of noninteracting fermions is given by the orange curves. Right panel: QP residue along the FS as function of $k_y$. For each value of $k_y$, $k_x^{+}$ and $k_x^{-}$ were determined on each side of the FS. For these values, the bound-state equation was solved in order to find ${\omega }_{+/-}\simeq 0$; $Z_{+/-}$ (black lines) was computed using Eq. (4). The RPA $t^{\prime }\left(\mathbf{k}\right)$ along the FS is plotted as a function of $k_y$ (orange curve). As implied by the RPA expression, when the self-energy vanishes the coherent excitations disappear ($Z=0$). Upper, central, and lower panels correspond, respectively, to $t_1^{\prime }=0.1,0.2,0.5$, and to $n=0.5$.

Figure 8

SF of the Hubbard model for $n=0.5$ and $t_1^{\prime }=0.25$ for $U=\infty$, where $u_s=0$.

Figure 9

SF of the Hubbard model for several anisotropic lattices: $t_1^{\prime }=0.5,t_2^{\prime }=t_3^{\prime }=0$ (top row); $t_1^{\prime }=0.25,t_2^{\prime }=0.25,t_3^{\prime }=0$ (central row); $t_1^{\prime }=0.05,t_2^{\prime }=-0.2,t_3^{\prime }=0.25$ (bottom row). Note the decrease of coherent modes and the increase of continuum-like features as frustration increases.

Figure 10

Coherent modes computed for the anisotropic square lattice with $t_1^{\prime }=0.2$ and $U/t=7.5$. Left panels: Spectral function computed for $k_y=0$ and $k_y=\pi$. Regions with strictly zero spectral weight are shaded in blue, the bound state (blue line) corresponds to a pole of the 2D Green’s function. The high-energy regions, shaded in red, have a low but nonvanishing spectral weight; therefore the antibound states (red line) arising in this region have a small but finite width. Right panel: Dispersion relation of bound and antibound states.