Spin-Lattice Order in One-Dimensional Conductors: Beyond the RKKY Effect

We investigate magnetic order in a lattice of classical spins coupled to an isotropic gas of one-dimensional conduction electrons via local exchange interactions. The frequently discussed Ruderman-Kittel-Kasuya-Yosida effective exchange model for this system predicts that spiral order is always preferred. Here we consider the problem nonperturbatively, and find that such order vanishes above a critical value of the exchange coupling that depends strongly on the lattice spacing. The critical coupling tends to zero as the lattice spacing becomes commensurate with the Fermi wave vector, signaling the breakdown of the perturbative Ruderman-Kittel-Kasuya-Yosida picture, and spiral order, even at weak coupling. We provide the exact phase diagram for arbitrary exchange coupling and lattice spacing, and discuss its stability. Our results shed new light on the problem of utilizing a spiral spin-lattice state to drive a one-dimensional superconductor into a topological phase.

Figure 1

Electronic band structure for the spin lattice in the antiferromagnetic state (left panel) and spiral phase (right panel). The corresponding points in the phase diagram are shown by the spade and clover symbols in Fig. 2. Close to the commensurate points, $k_{F}a=(2n+1)\pi /2$, the AF state fully gaps the electronic system. When the lattice spacing is increased the gaps in the AF state become smaller and at a critical value of the lattice spacing the partially gapped spiral state becomes the lowest energy state.

Figure 2

Ground state phase diagram of the model in Eq. (2) as a function of lattice spacing $a$ and exchange coupling $J$. Shading schematically represents the wave vector $q$ of the spiral, lying between $q=0$ in the ferromagnetic ($F$) phase and $q=\pi /a$ in the antiferromagnetic (AF) phase. Solid phase boundary lines correspond to first order transitions, double lines to second order transitions and dots to triple points. The spade and clover symbols correspond to the parameters used for the band structures shown in Fig. 1.

Figure 3

Optimal wave vector of the spin spiral $q$ as a function of exchange coupling at $k_{F}a=0.9\pi$ (red) and $k_{F}a=0.6\pi$ (blue), shown as vertical cuts of the inset.

Figure 4

Optimal wave vector $q$ and energy per impurity spin ${\overline{E}}_0=E_{0}a/L$ as a function of $k_{F}a$ at $J\nu =0.2$, shown as a horizontal cut of the inset in Fig. 3. The energy is measured relative to the $q=0$ state.