Numerically exploring the 1D-2D dimensional crossover on spin dynamics in the doped Hubbard model

Using determinant quantum Monte Carlo (DQMC) simulations, we systematically study the doping dependence of the crossover from one to two dimensions and its impact on the magnetic properties of the Hubbard model. A square lattice of chains is used, in which the dimensionality can be tuned by varying the interchain coupling $t_{\perp }$. The dynamical spin structure factor and static quantities, such as the static spin susceptibility and nearest-neighbor spin correlation function, are characterized in the one- and two-dimensional limits as a benchmark. When the dimensionality is tuned between these limits, the magnetic properties, while evolving smoothly from one to two dimensions, drastically change regardless of the doping level. This suggests that the spin excitations in the two-dimensional Hubbard model, even in the heavily doped case, cannot be explained using the spinon picture known from one dimension. The DQMC calculations are complemented by cluster perturbation theory studies to form a more complete picture of how the crossover occurs as a function of doping and how doped holes impact magnetic order.

Figure 1

The doping dependence of the average fermion sign is shown for different values of interchain coupling at $\beta =3/t$. A $10\times 10$ system is used, such that in the 1D limit it consists of 10-site chains. The $t_{\perp }=0t$ values are essentially hidden under the $t_{\perp }=0.1t$ values.

Figure 2

False-color plots of the dynamical spin structure factor are shown at (a) $n=1.0$, (b) $n=0.8$, and (c) $n=0.6$. The solid lines in panel (a) correspond to the upper and lower boundaries of the two-spinon continuum. Calculations are performed on a 48-site Hubbard chain with $U=8t$ and $\beta =15/t$ to access the expected low-temperature behavior.

Figure 3

False-color plots of the dynamical spin structure factor $S(\mathbf{q},\omega )$ along high-symmetry directions are shown at (a) $n=1.0$, (b) $n=0.8$, and (c) $n=0.6$ electron density on a $10\times 10$ cluster with $U=8t$ and $\beta =3/t$. The spectra at $n=0.9$ and $n=0.7$ have also been obtained and interpolate smoothly between the spectra shown here. The dots correspond to the maximum intensity for a given momentum point, and the solid line at half filling is the linear spin-wave dispersion calculated for a nearest-neighbor Heisenberg model with $J=4t^2/U=0.5t$.

Figure 4

Static spin susceptibility $\chi \left(\mathbf{q}\right)$ for different values of $t_{\perp }$ at (a) half filling and (b) $40\%$ hole doping. The magnetic order is transferred from one to two dimensions. The crossover only occurs for $t_{\perp }\ge 0.4t$ because of the high temperature.

Figure 5

NN spin correlation as a function of electron density $n$ for different values of the interchain hopping $t_{\perp }$, shown both along (top row) and perpendicular to (bottom row) the chain. Perpendicular to the chain, the spin correlation function for $t_{\perp }=0.1t$ is shown instead of that for $t_{\perp }=0t$, which is identically zero. The values are very small close to one dimension, leading to larger error bars. The solid lines show the $(1-p^2)$ curve predicted by the local static picture.

Figure 6

False-color plot of the dynamical spin structure factor, $S(\mathbf{q},\omega )$, along the main symmetry directions at half filling, for different values of $t_{\perp }$. In one dimension, the solid line corresponds to the two-spinon continuum; for $t_{\perp }\ne 0$, it represents the linear spin-wave dispersion. The dots indicate the maximum intensity at each momentum point (in units of $\pi$).

Figure 7

False-color plot of the dynamical spin structure factor $S(\mathbf{q},\omega )$ along the main symmetry directions, for different doping levels and values of the interchain coupling, $t_{\perp }$. At half filling in one dimension, the solid line corresponds to the two-spinon continuum; for $t_{\perp }\ne 0$, it represents the linear spin-wave dispersion. The dots indicate the maximum intensity for each momentum point, which is given in units of $\pi$.

Figure 8

Energy of the spin excitations at different momenta as a function of $t_{\perp }$ for different doping levels. Panel (a) focuses on the detailed doping dependence at $\mathbf{k}=(0,\pi )$, while panel (b) compares the evolution of the energy of spin excitations at $\mathbf{k}=(\pi ,0)$ and $\mathbf{k}=(\pi ,\pi )$ at two different doping levels.

Figure 9

False-color plot of the dynamical spin structure factor $S(\mathbf{q},\omega )$ computed by CPT along the main symmetry directions, at half filling (left column), $16.7\%$ (center column), and $33.3\%$ (right column) hole doping, for different values of the transverse hopping integral $t_{\perp }$, as calculated by CPT. The first row corresponds to $t_{\perp }=0$, the second to $t_{\perp }=0.4t$, and the third to $t_{\perp }=t$. The color scale is the same for all plots. At half filling, the solid line corresponds to the two-spinon continuum and the dashed line to the linear spin-wave dispersion.