Self-stabilizing temperature-driven crossover between topological and nontopological ordered phases in one-dimensional conductors

We present a self-consistent analysis of the topological superconductivity arising from the interaction between self-ordered localized magnetic moments and electrons in one-dimensional conductors in contact with a superconductor. We show that, due to a gain in entropy, there exists a magnetically ordered yet nontopological phase at finite temperatures that is relevant for systems of magnetic adatom chains on a superconductor. The spin-orbit interaction is taken into account, and we show that it causes a modification of the magnetic order, yet without affecting the topological properties.

Figure 1

Zoom on a 1D conductor with embedded magnetic moments on top of a superconductor. The magnetic moments self-order in the form a spiral order with spatial period $\pi /k_m$.

Figure 2

Example of dispersion relations in the transformed basis obtained by the spin-dependent shift of momenta $k\to k+\sigma k_m$ for which the spiral effective magnetic field becomes ferromagnetic and the Zeeman-like gap $J$ opens at $k=0$ [33, 38]. In the plots, the chemical potential $\mu$ remains constant, but $k_m$ varies. Top panels for the normal state $\left({\epsilon }_k\right)$, with red and blue colors corresponding to opposite spin projections perpendicular to the spiral field. Bottom panels for the induced superconducting state $(E_k$, with ${\mathrm{\Delta }}_s<J$). For $k_m=k_F$, the chemical potential $\mu$ (dashed line) lies in the middle of the $J$ gap (b) and the superconducting system is fully gapped (e). For smaller or larger $k_m$, the gap lies at lower or higher energies and $\mu$ eventually touches the upper (a) or lower gap edge (c). At both touching points the superconducting gap closes [(d), (f)] and the state becomes nontopological.

Figure 3

Dispersion relations as in Fig. 2 for the tight binding model associated with Eq. (1) and corresponding to Fig. 4. (a) and (d) represent the low-temperature topological phase, (b) and (e) the crossover to the nontopological phase where the gap ${\mathrm{\Delta }}^{*}$ closes, and (c) and (f) the final high-temperature phase where $k_m=\pi /2a$ and the system is magnetically ordered yet nontopological.

Figure 4

(a) Free energy $F$ for an adatom chain conductor of length $L=1 \mu \mathrm{m}$ as a function of total spiral momentum $k_m$ (including SOI contributions) and temperature $T$ [38]. Contour lines complement the color coding. The minima $k_m\left(T\right)$ are marked by the red line and correspond to the ground state configuration. The values $k_F,\pi /a-k_F$, and $\pi /2a$ are marked by horizontal dashed lines. The parameters for the tight binding model are $t=10$ meV, $a=3$ Å, ${\mathrm{\Delta }}_s=2$ meV, $J=3$ meV, $L=1\mu \mathrm{m}$, and $\mu =-0.3t$ (7/20 filling). Stability of the order is ensured up to $k_B{T}^{*}=0.37{\mathrm{\Delta }}_s$ (at the right plot limits). The gap ${\mathrm{\Delta }}^{*}$ closes at the phase boundary between the topological (clear) and nontopological (hatched) regions. The system becomes nontopological at $k_{B}T\approx 0.22{\mathrm{\Delta }}_s$ and $k_m$ then stabilizes at $\pi /2a$, corresponding to an antiferromagnetic order. (b) Values of $k_m$ minimizing the free energy $F$ [same as in (a)], the entropy $S$, and the energy $E$. While $E$ alone favors a topological phase, the entropy $S$ favors a gapless phase, but is at higher temperatures further enhanced by tuning $k_m\to \pi /a$, and eventually dominates the minimum of $F$ at $k_{B}T>0.22{\mathrm{\Delta }}_s$. For $k_m=\pi /2a$, the two identical gaps indicated in Fig. 3 are ${\mathrm{\Delta }}^{*}=0.30{\mathrm{\Delta }}_s$.