Interplay of magnetic order, pairing, and phase separation in a one-dimensional spin-fermion model

We consider a lattice model of itinerant electrons coupled to an array of localized classical Heisenberg spins. The nature of the ground-state-ordered magnetic phases that result from the indirect spin-spin coupling mediated by the electrons is determined as a function of density and the spin-fermion coupling $J$. At a fixed chemical potential, spiral phases exist only up to values of $J$ which are less than roughly half the electronic bandwidth. At a fixed electron density and near half filling, the system phase-separates into a half-filled antiferromagnetic phase and a spiral phase. The ferromagnetic phases are shown to be fully polarized, while the spiral phases have equal admixture of up and down spins. Phase separation survives in the presence of weak pairing field $\mathrm{\Delta }$ but disappears when $\mathrm{\Delta }$ exceeds a critical value ${\mathrm{\Delta }}_c$. If pairing fields are large enough, an additional spiral state arises at strong coupling $J$. The relevance of this study, especially the phase separation, to artificially engineered systems of adjacent itinerant electrons and localized spins is discussed. In particular, we propose a method which might allow for the braiding of Majorana fermions by changing the density and moving their location as they are pulled along by a phase separation boundary.

Figure 1

Classical spin wave vector $q_{*}$ which minimizes the ground state energy, as a function of chemical potential $\mu$ for fixed $J=1.0$ and $J=2.5$. Inset: Red circles denote results with lattice size $L=1000$ sites. Data for $L=500$ (black squares) and $L=5000$ (blue triangles) indicate finite-size effects are small.

Figure 2

Phase diagram in the chemical potential $\left(\mu \right)$, exchange constant $\left(J\right)$ plane. At low and high densities (the Hamiltonian is particle-hole symmetric) ferromagnetic order $q_{*}=0$ minimizes the energy. For $J\lesssim W/2=2t$, F order gives way first to spiral (incommensurate $q_{*}$) and then AF ($q_{*}=\pi$) order. The red lines mark the boundary between full and partial polarization in a F state. See text.

Figure 3

Top: Populations of the individual species with spin in the $\pm \widehat{x}$ directions, as functions of chemical potential. The F phase at low (and high) density is fully polarized, while the spiral and AF phases have balanced populations. Bottom: The corresponding ground state (F, S, or AF). Discontinuities in $N_{x\sigma }$ occur precisely at the magnetic phase boundaries.

Figure 4

Density as a function of chemical potential at $J=1$. The F-spiral phase boundary is signaled by a jump in the compressibility $\kappa =d\rho /d\mu$. The density jumps abruptly at the spiral-AF boundary. Inset A: Compressibility $\kappa$ vs $\mu$. Inset B: Density as a function of $\mu$ at $J=2.5$.

Figure 5

Density is shown as a function of chemical potential in the graph. Black squares denote $\rho$ vs $\mu$ in the GCE. Red circles denote $\rho$ vs $\mu =d{E}_0/d\rho$ in the CE. Inset A: Enlarged $\mu$ vs $\rho$ graph. The two areas $A_1\approx A_2$ satisfy the Maxwell construction. Here the critical $\mu$ is 0.851. Inset B: $\rho$ vs $\mu$ in GCE and $\rho$ vs $dE/d\rho$ in CE at $J=2.5$, which shows a direct phase transition from F to AF.

Figure 6

Top: The mixing of $k$ with $k-q$, for $q=0.506\pi$, gives rise to a pair of overlapping energy bands. Bottom: Occupation numbers in the $k$ space. The jumps of $n_{\uparrow }\left(k\right)$ or $n_{\downarrow }\left(k\right)$ curves show Fermi wave vectors $k_F$.

Figure 7

Optimal wave vector $q_{*}$ which minimizes the ground state energy $E_0$, as a function of chemical potential $\mu$ for fixed $J=1.0$ but different $\mathrm{\Delta }$ values. Finite-size effects are verified to be small. Inset: Enlarged $q_{*}/\pi$ vs $\mu$ graph. It has an abrupt jump at $\mathrm{\Delta }=0.4$ but grows continuously at $\mathrm{\Delta }=0.6$. $\mathrm{\Delta }=0.5$ is around the critical value ${\mathrm{\Delta }}_c$.

Figure 8

Optimal wave vector $q_{*}$ as a function of chemical potential $\mu$ for fixed $J=2.5$ but different $\mathrm{\Delta }$ values. There are two critical values ${\mathrm{\Delta }}_c$ for $J=2.5$. Left: Wave vector $q_{*}$ vs $\mu$ for $\mu \approx {\mathrm{\Delta }}_c^{-}$. This lower ${\mathrm{\Delta }}_c^{-}$ distinguishes F-AF phase separation and spiral-AF phase separation. Right: Wave vector $q_{*}$ vs $\mu$ for $\mu \approx {\mathrm{\Delta }}_c^{+}$. The upper ${\mathrm{\Delta }}_c^{+}$ distinguishes spiral-AF phase separation and no phase separation.

Figure 9

Critical value ${\mathrm{\Delta }}_c$ vs exchange constant $J$. There are two critical values, upper ${\mathrm{\Delta }}_c^{+}$ and lower ${\mathrm{\Delta }}_c^{-}$, for fixed $J$. Below the critical value $J_c=2.10\pm 0.02$, only ${\mathrm{\Delta }}_c^{+}$ is nonzero. There are three regions distinguished by second-order spiral-AF phase transition (PT), first-order spiral-AF PT, and first-order F-AF PT.

Figure 10

Top: Phase diagram in the chemical potential $\left(\mu \right)$, pairing field $\left(\mathrm{\Delta }\right)$ plane, with fixed exchange constants $\left(J=2.5\right)$. The upper red line corresponds to upper ${\mathrm{\Delta }}_c^{+}$. The lower red line corresponds to lower ${\mathrm{\Delta }}_c^{-}$. Bottom: Phase diagram in $\mu ,\mathrm{\Delta }$ plane, with fixed $J=1.0$. The red line corresponds to ${\mathrm{\Delta }}_c^{+}$. ${\mathrm{\Delta }}_c^{-}$ is zero at $J=1.0$.