True Bilayer Exciton Condensate of One-Dimensional Electrons

We theoretically predict that a true bilayer exciton condensate, characterized by off-diagonal long-range order and global phase coherence, can be created in one-dimensional solid state electron systems. The mechanism by which this happens is to introduce a single particle hybridization of electron and hole populations, which locks the phase of the relevant mode and hence invalidates the Mermin-Wagner theorem. Electron-hole interactions then amplify this tendency towards off-diagonal long-range order, enhancing the condensate properties by more than an order of magnitude over the noninteracting limit. We show that the temperatures below which a substantial condensate fraction would form could reach hundreds of Kelvin, a benefit of the weak screening in one-dimensional systems.

Figure 1

(a) Sketch of the proposed system. (b) Band structure of noninteracting electrons in parallel nanowires with weak interwire tunneling $t_{\perp }$. The tunneling forces a gap ${\delta }_{\mathrm{sp}}=2t_{\perp }$ (solid shaded lines) to open at the Fermi level. (c) Spatial dependence of the exciton-exciton correlator, showing the strong enhancement of excitonic off-diagonal long-range order in the ideal model at zero temperature when $U_{\perp }=2t$, for $t_{\perp }=0.001t$ (blue dashed line), $t_{\perp }=0.0025t$ (orange dotted line), $t_{\perp }=0.005t$ (green dash-dotted line), $t_{\perp }=0.01t$ (purple solid line), $t_{\perp }=0$ (grey solid line). Free electrons ($U_{\perp }=0$) with $t_{\perp }=0.01$ (black dashed line) shown for comparison. (d) Ratio of the order parameter $A$ of the exciton condensate with interactions ($U_{\perp }\ne 0$) to the noninteracting case (i.e., free fermions, $U_{\perp }=0$). In all cases, we see that sufficient $U_{\perp }$ can enhance the excitonic order by an order of magnitude or more. The line styles match (c). Results in (c) and (d) are for $M=300$.

Figure 2

(a) Spectral function of $G_u^R(x,\omega )$ for the model system with $t_{\perp }=0.001t$ and $U_{\perp }=2t$, exhibiting the large gap $\mathrm{\Delta }$. (b) Scaling of $\mathrm{\Delta }$ with $t_{\perp }$ for $U_{\perp }=0.25t$ (dark blue line), $U_{\perp }=0.5t$ (bright red line), $U_{\perp }=0.75t$ (yellow line), $U_{\perp }=t$ (violet line) $U_{\perp }=1.25t$ (green line), $U_{\perp }=1.5t$ (light blue line), $U_{\perp }=2t$ (dark red line) for the model system. (c) Spectral function of ${\chi }_{J_{\perp }}(\omega )$ for the model system with $t_{\perp }=0.001t$ and $\eta =0.001t$, for $U_{\perp }=2t$ (blue line), $U_{\perp }=1.5t$ (green dotted line), and $U_{\perp }=t$ (red dash-dotted line). Weight below $\delta$ is entirely due to finite $\eta$. (d) Scaling of $\delta$ with $U_{\perp }$, for $t_{\perp }=0.001t$ (dark blue line), $t_{\perp }=0.0025t$ (bright red line), $t_{\perp }=0.005t$ (yellow line), $t_{\perp }=0.01t$ (violet line) $t_{\perp }=0.025t$ (green line), $t_{\perp }=0.05t$ (light blue line), $t_{\perp }=0.1t$ (dark red line) for the model system. (e) Order parameter $A$ as a fraction of its ground state value, against inverse temperature $\beta$ for the model system with $U_{\perp }=2t$, $t_{\perp }=0.01t$. Once $\beta >1/\delta$, the system approaches ground state properties exponentially fast in $\beta$. (f) dc I-V characteristic of the model system with $t=1\mathrm{eV}$, $U_{\perp }=2t$, $t_{\perp }=0.01t$ (blue), and $t_{\perp }=0.001t$ (red), showing both dissipationless and dissipative regimes. All results are for $M=96$.

Figure 3

Achievable gap $\delta /k_B$ in the strong screening case as a function of $t_{\perp }$ for $t=1\mathrm{eV}$ (green line), $t=0.5\mathrm{eV}$ (red line), and $t=0.4,\mathrm{eV}$ (blue line). Inset: Range of achievable $A/A_{\mathrm{IWI}=0}$, indicated by showing high and low values as a function of $t_{\perp }$, with colors matching the main figure.