Hierarchy of Modes in an Interacting One-Dimensional System

Studying interacting fermions in one dimension at high energy, we find a hierarchy in the spectral weights of the excitations theoretically, and we observe evidence for second-level excitations experimentally. Diagonalizing a model of fermions (without spin), we show that levels of the hierarchy are separated by powers of ${\mathcal{R}}^2/L^2$, where $\mathcal{R}$ is a length scale related to interactions and $L$ is the system length. The first-level (strongest) excitations form a mode with parabolic dispersion, like that of a renormalized single particle. The second-level excitations produce a singular power-law line shape to the first-level mode and multiple power laws at the spectral edge. We measure momentum-resolved tunneling of electrons (fermions with spin) from or to a wire formed within a GaAs heterostructure, which shows parabolic dispersion of the first-level mode and well-resolved spin-charge separation at low energy with appreciable interaction strength. We find structure resembling the second-level excitations, which dies away quite rapidly at high momentum.

Figure 1

The main features of spectral function for spinless fermions in the region $-k_F<k<k_F$ ($k_F<k<3k_F$) labeled by 0(1). The gray areas mark nonzero values, $p(h)$ shows the particle(hole) sector, $k_F$ is the Fermi momentum, $a,b,c$, respectively, identify the level in the hierarchy in powers 0,1,2 of ${\mathcal{R}}^2/L^2$, and $(r,l)$ specifies the origin in the range—modes on the edge have no such label.

Figure 2

The local density of states for spinless fermions: red and green lines show the contribution of $a$ and $b$ excitations and the blue line indicates the Luttinger-liquid regime. Inset is a log-log plot around the Fermi energy: the blue points are numerical data for $N=71$, $L=700$, $mV=6$ giving $K=0.843$, and the dashed line is $\rho (ϵ)=\text{const}{|ϵ-\mu |}^{(K+K^{-1})/2-1}$.

Figure 3

Measurement of the tunneling differential conductance $G=dI/dV$ for two samples, each consisting of a set of identical wires of length $L=10$ (a)–(d) and $L=18\mu \mathrm{m}$ (e), sketched in inset (f). (a) Intensity plot of $dG/d{V}_{\text{dc}}$ to $2\mu$. The line is the 2D dispersion. The green solid line marks the $a$ modes, dashed green lines the $b$ modes, and dashed blue the $c$ modes (as in Fig. 1); dotted magenta and blue lines are parasitic 2D dispersions. Spin ($S$) and charge ($C$) modes are indicated with black dashed lines. $T=300\mathrm{mK}$. (b) Enlargement of the replica feature in (a) just above $k_{\mathrm{F}}$. It appears as a pale band (slowly varying $G$) between the two green curves, after a red region (sharp rise in $G$). (c) The same as in (b), but with the gate voltage over most of the parasitic ($p$’) region changed to move the $p$’ parabolas. (d) $G$ vs $V_{\text{dc}}$ at various fields $B$ from 3 to 4.8 T [from (a)]; $+$ and $\times$ symbols on each curve indicate, respectively, the voltages corresponding to the dashed and solid [$p1b$ and $p1a(l)$] green lines in (a) and (b), showing the enhanced conductance between the two. (e) $dG/d{V}_{\text{dc}}$ for a second device, at $T<100\mathrm{mK}$. The replica feature is similar to that shown in (b) and (c) for the other sample.