Helical nuclear spin order in two-subband quantum wires

In quantum wires, the hyperfine coupling between conduction electrons and nuclear spins can lead to a (partial) ordering of both of them at low temperatures. By an interaction-enhanced mechanism, the nuclear spin order, caused by Ruderman-Kittel-Kasuya-Yosida exchange, acts back onto the electrons and gaps out part of their spectrum. In wires with two subbands characterized by distinct Fermi momenta $k_{F1}$ and $k_{F2}$, the nuclear spins form a superposition of two helices with pitches $\pi /k_{F1}$ and $\pi /k_{F2}$, thus exhibiting a beating pattern. This order results in a reduction of the electronic conductance in two steps upon lowering the temperature.

Figure 1

Panel (a) depicts a quantum wire with two subbands defined in a two-dimensional electron gas by a harmonic confining potential. The subbands have transversal wave functions ${\Psi }_j(x,y)={\psi }_j\left(x\right)\otimes {\psi }_1\left(y\right)$ and therefore exhibit different $x$-inversion symmetry with respect to the wire axis. A sketch of the nuclear spin lattice in the wire is given on the left-hand side of panel (b) (for visibility reasons, not all spins are shown). Within a given cross section, the nuclear spins align ferromagnetically, while they form helical orders along the wire direction (see Sec. 4). The right-hand side of panel (b) shows the effective description of the nuclear spins as helical magnetic fields.

Figure 2

Fermionic diagrams contributing to the ($x$ and $y$ components of the) spin susceptibilities ${\chi }_{ij}$. Solid lines denote a particle in band 1 at, e.g., momentum $k+q$, frequency $\omega$, and spin up; dashed lines denote a particle in band 2 (note that the Luttinger liquid approach yields the density-density interacting versions of these diagrams). The dotted lines denote the momentum $q$ emitted/absorbed by the nuclear spins.

Figure 3

RKKY exchange interactions $J_{\mathrm{s}}$ and $J_{\mathrm{a}}$ for $g_{22}<g_{12}<g_{11}$ and at finite temperature.

Figure 4

Opening of gaps around the chemical potential $\mu$ in the dispersion $E\left(q\right)$ due to the magnetic Overhauser field in a two-subband quantum wire. Panel (a) depicts the effect of a helical field of positive helicity and momentum $2k_{F1}$ in the symmetric channel, which allows for scattering between right-moving spin-down particles and left-moving spin-up particles. The scattering gaps out these two modes. Panel (b) shows how a helical field in the antisymmetric channel of negative helicity and momentum $k_{F1}+k_{F2}$ gaps out right-moving spin-up particles and left-moving spin-down particles in both bands. As explained in the main text, the intraband order depicted in panel (a) and the interband order shown in panel (b) are mutually exclusive because their combination would gap out the entire lower subband.

Figure 5

Exponents $2g_{ij}$ of the RKKY exchange couplings [see Eq. (9)] as a function of $v_{F2}/v_{F1}$ for fixed $v_{F1}$ and interaction strength $U$. These exponents describe the regime of disordered nuclear spins, e.g., at sufficiently high temperatures. Consequently, they do not contain any feedback effect between electrons and nuclear spins; see Sec. 4.

Figure 6

Exponents $2g_{ij}$ of the RKKY exchange couplings [see Eq. (9)] as a function of $v_{F2}/v_{F1}$ for fixed $v_{F1}$ and interaction strength $U$. On this larger scale logarithmic plot, we also show the exponents $2g_{ii,0}$ for decoupled bands; see main text. Like for the smaller scale plot in Fig. 5, the nuclear spins are assumed to be disordered.

Figure 7

RG flow of the two-subband quantum wire system. Panel (a) shows the flow for density-density coupled subbands in the absence of hyperfine coupling. For any value of the interband interaction $U_{12}$, the system can be described by two decoupled spinful Luttinger liquids (LLs), characterized, however, by different Luttinger liquid parameters and velocities. Panel (b) shows how the flow is modified due to the presence of the hyperfine coupling and a resulting ordering of the nuclear spins in the symmetric channel. Here, the interband interaction would correspond to the out-of-plane axis (not shown), and panel (b) is the projection of the full flow into the given plane.

Figure 8

Exponents $2g_{ij}^{\prime }$ of the residual RKKY exchange couplings after the formation of nuclear spin helices at momenta $2k_{F1}$ and $2k_{F2}$, as a function of $v_{F2}/v_{F1}$ for fixed $v_{F1}$ and interaction strength $U$ such that $K_{1c}=0.5$.

Figure 9

Experimental signatures of the nuclear spin order. Panel (a) shows the two superimposed helices of different pitch lengths ${\lambda }_1=\pi /k_{F1}$ and ${\lambda }_2=\pi /k_{F2}$ formed by the nuclear spins, leading to a beating pattern of the nuclear magnetization in real space. Panel (b) depicts the conductance of the two-subband quantum wire as a function of temperature. At the higher critical temperature $T_{ci}>T_{cj}$, the helix at momentum $2k_{Fi}$ forms, and the conductance drops from $4e^2/h$ to (approximately) $3e^2/h$. At the second ordering temperature $T_{cj}$, the second helix forms and further reduces the conductance to (approximately) $2e^2/h$.