Nuclear magnetism and electron order in interacting one-dimensional conductors

The interaction between localized magnetic moments and the electrons of a one-dimensional conductor can lead to an ordered phase in which the magnetic moments and the electrons are tightly bound to each other. We show here that this occurs when a lattice of nuclear spins is embedded in a Luttinger liquid. Experimentally available examples of such a system are single wall carbon nanotubes grown entirely from ${}^{13}\mathrm{C}$ and GaAs-based quantum wires. In these systems the hyperfine interaction between the nuclear spin and the conduction electron spin is very weak; yet it triggers a strong feedback reaction that results in an ordered phase consisting of a nuclear helimagnet that is inseparably bound to an electronic density wave combining charge and spin degrees of freedom. This effect can be interpreted as a strong renormalization of the nuclear Overhauser field and is a unique signature of Luttinger liquid physics. Through the feedback the order persists up into the millikelvin range. A particular signature is the reduction in the electric conductance by the universal factor of 2.

Figure 1

(Color online) Illustration of the helical nuclear magnetism of the SWNT, which is triggered by the RKKY interaction over the electron system (not shown). The nuclear spins (red arrows) order ferromagnetically on a cross-section of the SWNT and rotate along the SWNT axis with a period $\pi ∕k_F={\lambda }_F∕2$ in the spin $xy$ plane (chosen here arbitrarily orthogonal to the SWNT axis). The blue ribbon is a guide to the eye for the helix. The feedback of this nuclear magnetic field strongly renormalizes the electron system through the opening of a partial gap due to a strongly renormalized Overhauser field and so modifies the RKKY interaction. Through this strong coupling of electron and nuclear systems the combined order persists up into the millikelvin range. As a particular consequence the electric conductance of the SWNT is reduced by a factor of precisely 2.

Figure 2

(Color online) Illustration of the helical nuclear magnetism for a GaAs-based quantum wire. Compared with the SWNT (Fig. 1) the number of ferromagnetically locked spins on a cross section through the wire is much larger, and the wavelength $\pi ∕k_F$ of the helical rotation along the wire is longer. The feedback effect remains otherwise the same and the combined electron and nuclear order persists into the millikelvin range as well.

Figure 3

Magnetization $m_{2k_F}$ as a function of $T$ for ${}^{13}\mathrm{C}$ single wall nanotubes. Dashed line: without the feedback [Eq. (38)]; solid line: including the feedback [Eq. (70)]. The vertical lines mark the cross-over temperatures written next to them [Eqs. (39, 71)]. $T_0^{*}$ is evaluated with the two-band model of Sec. 8, while we use the quantitatively correct one-band description for $T^{*}$. Note that compared with Ref. 9 we use a larger $A_0$ (according to the results of Ref. 28) and so obtain a slightly larger $T^{*}$. On the other hand $T_0^{*}$ is smaller because the exponent $g$ is larger the two-band model.

Figure 4

Magnetization $m_{2k_F}$ as a function of $T$ for GaAs quantum wires. Dashed line: without the feedback [Eq. (38)]; solid line: including the feedback [Eq. (70)]. The vertical lines mark the crossover temperatures written next to them [Eqs. (39, 71)]. Since the exponent $g$ changes only little through the renormalization $(g\to g^{\prime })$, the values of $T_0^{*}$ and $T^{*}$ are of the same order of magnitude.

Figure 5

Crossover temperature $T^{*}$ [Eq. (71)] as a function of the hyperfine coupling constant $A_0$ for ${}^{13}\mathrm{C}$ single wall nanotubes. $T^{*}$ follows a power-law $T^{*}\propto A_0^{2∕(3-g_{x,y}^{\prime })}=A_0^{0.86}$ and is plotted up to the self-consistency limit $T^{*}\approx v_F∕L{k}_B=3\mathrm{K}$. The values about $A_0\sim {10}^{-4}\mathrm{eV}$ correspond to those deduced in Ref. 12. Note that through the whole range we have $k_B{T}^{*}\sim B^{*}$ [Eq. (74)].

Figure 6

Crossover temperatures $T_0^{*}$ [dashed line, Eq. (39)], $T^{*}$ [full line, Eq. (71)], and the bound $B^{*}$ [dash-dotted line, Eq. (74)] for GaAs quantum wires as functions of the interaction strength, expressed by the Luttinger liquid parameter $K_c$ (keeping $K_s=1$). The noninteracting limit is $K_c=1$, smaller $K_c>0$ correspond to increasingly stronger repulsive interactions. The dotted line is the continuation of $T^{*}$ beyond the energy scale set by $B^{*}$, for which the validity of the theory becomes uncertain. The curves of $T^{*}$ and $T_0^{*}$ cross at $K_c\approx 0.6$ because the RKKY interaction $J_{2k_F}^{\prime }$ defining $T^{*}$ has a prefactor that is by $1∕2$ smaller than in the case without the feedback. Since nuclear spin order unavoidably leads to the feedback, $T^{*}$ defines the crossover temperature for the order even when $T_0^{*}>T^{*}$. Note that close to $K_c=1$, Eqs. (39, 71) diverge because the cutoff $\delta$ in Eq. (A12) was neglected in the further evaluation of the RKKY interaction. This is valid for $K_c$ smaller than $\sim 0.8$. Reintroducing the cutoff self-consistently close to $K_c=1$ regularizes the divergence and leads to the displayed curves (see Appendix ).

Figure 7

Sketch of the RKKY interaction $J_q^{\alpha }$ [Eq. (23)] or equivalently the spin susceptibility ${\chi }_{\alpha }\left(q\right)$ [Eq. (A16)].

Figure 8

(Color online) Illustration of the cross section through the 1D conductor. The $N_{\perp }⪢1$ nuclear spins within the support of the transverse confinement of the electron wave function (central colored region) are ferromagnetically locked and behave like a single large spin $\tilde{I}=N_{\perp }I$. The nuclear spins outside this support do not interact with the electron system and are generally disordered because their direct dipolar interaction is very weak.

Figure 9

(Color online) Illustration of the RKKY coupling between two large spins composed of $N_{\perp }$ individual spins at sites $i$ and $j$. The hyperfine interaction is reduced by distributing a single electron over $N_{\perp }$ nuclear spins, $A=A_0∕N_{\perp }$, resulting in an RKKY interaction $J_{ij}∕N_{\perp }^2$. This reduction by $1∕N_{\perp }^2$ is compensated because $N_{\perp }^2$ nuclear spins are mutually coupled through the same RKKY interaction $J_{ij}$.

Figure 10

(Color online) Illustration of the nuclear spin configuration minimizing the energy of Eq. (80) for the case of a complete electron polarization $S$ pointing upwards in the figure. The figure shows how the sign of the hyperfine constant changes when going around the nanotube cross section. The overall magnetization of this configuration is $m=-0.17$ (along $x$, normalized to $-1<m<1$). See also Ref. 28.