Breakdown of the Nagaoka phase at finite doping

The Nagaoka $\left(U=\infty \right)$ limit of the Hubbard model on a square lattice is mapped onto the itinerant-localized Kondo model at infinitely strong coupling. Such a model is well suited to perform quantum Monte Carlo (QMC) simulations to compute spin correlation functions. For periodic boundary conditions, this model is shown to exhibit no short-range ferromagnetic (FM) spin correlations at any doping $\delta \ge 0.01$ and at finite temperature $T=0.1t.$ Our simulations give no indication that there is a tendency towards ferromagnetic ordering in the ground state, with more than one hole. Employing on the other hand the open boundary conditions (or mixed boundary conditions) may result in the qualitatively different results for the thermodynamic limit depending on the way one chooses to approach this limit. These observations imply that the relevant thermodynamic limit remains unclear.

Figure 1

The figure shows the energy difference $\mathrm{\Delta }E=E(Q_{\text{max}}-1)-E\left(Q_{\text{max}}\right)$ between the states with $Q=Q_{\text{max}}$ and $Q=Q_{\text{max}}-1$ for the different numbers of holes $n_h$. The lattice size $N=L\times L$ and the doping level $\delta =n_h/L^2$.

Figure 2

The spin value in the ground state for the different numbers of electrons $n_e=N-n_h$. Periodic boundary conditions are imposed.

Figure 3

The $\langle Q_i^z{Q}_{i+1}^z\rangle$ correlation between nearest sites for the different numbers of electrons $n_e=N-n_h$. Periodic boundary conditions are imposed.

Figure 4

(a) Correlation functions for $L=10$ for different doping levels for the hard-core boson case at $T=0.1t$. (b) The same for fermion case. Solid (dashed) lines show results obtained for $L=20$ $\left(L=10\right)$, respectively. Number of lattice size is $N=L\times L$. (c) The highlighted fragments of (b) with the doping level being separately presented.

Figure 5

The figure shows the energy difference between the fully polarized (FP) state $\left(Q=Q_{\text{max}}\right)$ and the one spin flipped (SF) state $\left(Q=Q_{\text{max}}-1\right)$ in case of two holes for the different lattice sizes $N=L_x\times L_y$. The SF areas correspond to the case $E(Q_{\text{max}}-1)<E\left(Q_{\text{max}}\right)$, the FP areas correspond to the case $E(Q_{\text{max}}-1)>E\left(Q_{\text{max}}\right)$. (a) The lattice with the open boundary conditions. (b) The lattice with the periodic boundary condition for the axis $x$ and the open boundary condition for the axis $y$. (c) The lattice with the periodic boundary conditions.