Itinerant ferromagnetism in the repulsive Hubbard chain with spin-anisotropic odd-wave attraction

The ground-state properties of the Hubbard chain with on-site repulsion and anisotropic nearest-neighbor attraction are investigated by means of density matrix renormalization group calculations. The nonlocal attraction acts between fermions of one spin component only, mimicking the effect of $p$-wave Feshbach resonances in cold-atom systems. We analyze the onset of itinerant ferromagnetism, pinpointing the critical attraction strength where partially and fully ferromagnetic states occur. In the cold-atom setup, where the two (pseudo)spin populations are separately conserved, ferromagnetism occurs with the nucleation of a fully imbalanced band-insulating domain hosting the attractive component only. The size of this domain grows with the attraction strength, therefore increasing the (opposite) imbalance of the other domain, until the two spin components are fully separated. In the presence of a harmonic trap, the ferromagnetic state hosts a partially imbalanced domain in the center with an excess of the attractive component and filling lower than one. This central region is surrounded by fully imbalanced domains, located in the trap tails, hosting only fermions belonging to the other component.

Figure 1

Ground-state energy $E$ as a function of the spin polarization $P=(N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })$, plotted for different values of the interaction strength $V$, ranging from $V=0$ (upper curve) to $V=-1$ (bottom curve). The total number of fermions is fixed to $N=N_{\uparrow }+N_{\downarrow }=40$, the length of the chain is $L=60$. Here and in all other figures the on-site Hubbard-repulsion strength is $U=5$; furthermore, the connecting lines are a guide to the eye, unless otherwise specified.

Figure 2

Ground-state energy $E$ as a function of the nearest-neighbor attraction strength $V$, for three different sets of population numbers. The red squares correspond to a fully polarized gas of spin-up fermions, with $N_{\uparrow }=40$ and $N_{\downarrow }=0$. The other two data sets represent the energy of the same system upon addition of, respectively, a further spin-up fermion (green circles) or a spin-down fermion (blue triangles). The dashed vertical segment indicates the position of the crossing point, $V=-0.849\left(1\right)$. This corresponds to the critical value of the attraction strength for the onset of full ferrromagnetism. The inset shows an enlargment of the energy difference $\mathrm{\Delta }E$ between the green circle and blue triangle data sets. The length of the chain is $L=60$.

Figure 3

Density profiles of spin-up fermions (upward triangles) and spin-down fermions (downward triangles) in a chain with $L=120$ sites, labeled by the index $i$. To improve visibility, only every other point is shown. The four panels (a)–(d) refer to different values of the nearest-neighbor attraction strength $V$ between spin-up fermions: $V=-0.5,-1.2,-1.3,-1.4$ (from top to bottom). The spin populations are $N_{\uparrow }=N_{\downarrow }=40$. For large and negative $V$, a growing fully imbalanced domain hosting a band insulator of spin-up particles (i.e., the attractive component) coexists with a partially ferromagnetic domain with an excess of spin-down fermions.

Figure 4

Double occupancy $d$ as a function of the nearest-neighbor attraction strength $V$. The different curves correspond to calculations for different system sizes. For each value of $L$, we choose the spin populations so that the overall densities are kept constant to $N_{\uparrow }/L=N_{\downarrow }/L=1/3$, so that the global spin polarization is $P=0$. Notice the sudden variation around $V=-1.25$, corresponding to the nucleation of the band-insulator domain hosting spin-up fermions only, leading to an essentially vertical drop of the double occupancy. The inset shows the double occupancy as a function of $1/L$ for $V=-1.34$, showing that $d$ vanishes for infinite system sizes as the ground state becomes fully ferromagnetic.

Figure 5

Extrapolation to the infinite-size limit of the critical strength $V=V_c$ for the nucleation of the band-insulating domain, corresponding to the phase transition to a partially ferromagnetic state. For each system size $L$, we extract the value $V=V_L$ at which the double occupancy in Fig. 4 displays the sudden variation and plot it as a function of $1/L$. The position of the critical point is inferred by fitting the data with a low-order polynomial $p\left(x\right)$, with $x=1/L$, and setting $V_c=p\left(0\right)$. Specifically, a linear fitting function (continuous red line), obtained by retaining only the three largest system sizes, and a quadratic fitting function (dashed green curve) are shown, yielding $V_c=-1.2385\left(3\right)$ and $V_c=-1.233\left(2\right)$, respectively.

Figure 6

Ground-state energy per particle $E/N$, extrapolated to the thermodynamic limit, plotted as a function of the nearest-neighbor attraction strength $V$. The asymptotic behavior for large and negative $V$, see Eq. (2), is shown as a dashed line. The dot-dashed line is the prediction from first-order perturbation theory, holding for small and negative $V$. The inset shows the dependence of the ground-state energy per particle on the inverse system size (data symbol) for $V=-1.3$; the spin densities are fixed at $N_{\uparrow }/L=N_{\downarrow }/L=1/3$. The infinite-lattice result corresponds to the intercept of the straight line fitting the data (dashed curve).

Figure 7

Density profiles of the two spin components in a harmonic trap of intensity $K=0.002$. The nearest-neighbor attraction strength is $V=-1.35$, which in the flat trap with open boundaries corresponds to a fully ferromagnetic ground state. The spin populations are $N_{\uparrow }=N_{\downarrow }=40$.