Magnetic phase diagram of the infinite-$U$ Hubbard model with nearest- and next-nearest-neighbor hoppings

We study the infinite-$U$ Hubbard model on ladders of two, four, and six legs with nearest- $\left(t\right)$ and next-nearest- $\left(t^{\prime }\right)$ neighbor hoppings by means of the density-matrix renormalization group algorithm. In particular, we analyze the stability of the Nagaoka state for several values of $t^{\prime }$ when we vary the electron density $\rho$ from half filling to the low-density limit. We build the two-dimensional phase diagram, where the fully spin polarized and paramagnetic states prevail. We find that the inclusion of a nonfrustrating next-nearest-neighbor hopping stabilizes the fully spin polarized phase up until $|t^{\prime }/t|=0.5$. Surprisingly, for this value of $t^{\prime }$, the ground state is fully spin polarized for almost any electron density $1\gtrsim \rho \gtrsim 0$, connecting the Nagaoka state to itinerant ferromagnetism at low density. Also, we find that the previously found checkerboard insulator phase at $t^{\prime }=0$ and $\rho =0.75$ is unstable against $t^{\prime }$.

Figure 1

A $4\times 6$ lattice with open boundary conditions. Shown by solid black lines $t$ is the nearest-neighbor hopping integral, and shown by dashed blue lines, $t^{\prime }$ is the next-nearest-neighbor one.

Figure 2

Magnetization value in $2\times 20$ ladders with open boundary conditions for three different values of $t^{\prime }$. $M=1$ is the fully spin polarized state, and $M=0$ is the paramagnetic state. The purple squares show the limit $t^{\prime }=0$ (square lattice), blue circles show an intermediate case, $t^{\prime }=0.3$, and green triangles illustrate the limit case of $t^{\prime }=0.5$.

Figure 3

Nearest- and next-nearest-neighbor correlations in the central part of the $2\times 20$ lattice for several values of $t^{\prime }$. The thickness of the lines varies as the third power of the bond density $B_{ij}={\sum }_{\sigma }\langle [{\tilde{c}}_{i\sigma }^{†}{\tilde{c}}_{j\sigma }+\text{H.c.}]\rangle$, while the color indicates the sign of the total spin correlations. The size of the arrows indicates the magnitude of the local value of $S_i^z$.

Figure 4

Magnetization value at $t^{\prime }=0.2$ (top panel) and $t^{\prime }=0.4$ (bottom panel) as a function of $\rho$ for several lattice sizes: $2\times 20$ (purple squares), $2\times 30$ (yellow diamonds), $4\times 10$ (blue circles), $4\times 12$ (green upward triangles), and $6\times 8$ (orange downward triangles).

Figure 5

Phase diagram for the $2\times 20$ lattice with open boundary conditions as a function of the next-nearest-neighbor hopping $t^{\prime }$ and electron density $\rho$. Green squares show the boundary of the fully spin polarized ferromagnetic state, and blue circles indicate the boundary of the paramagnetic state. The dark blue triangles represent the points where the checkerboard insulator is found. Points are connected by smooth spline fits.

Figure 6

Spin structure factor of the $4\times 10$ lattice for $t^{\prime }=0.0$ and several electron densities $\rho$: 0.8 (top left), 0.775 (top right), 0.75 (bottom left), and 0.725 (bottom right). The white square represents the first Brillouin zone, and $k_x$ and $k_y$ are in units of $\pi$.

Figure 7

Two-dimensional phase diagram of the system based upon results for four- and six-leg ladders. In green we show the FSP phase, in blue we give the paramagnetic one, and in yellow we show the intermediate region. The green squares and blue circles signal the end points of the FSP and paramagnetic phases, respectively. Points are connected by smooth spline fits.