Long-lived binary tunneling spectrum in the quantum Hall Tomonaga-Luttinger liquid

The existence of long-lived nonequilibrium states without showing thermalization, which has previously been demonstrated in time evolution of ultracold atoms, suggests the possibility of their spatial analog in transport behavior of interacting electrons in solid-state systems. Here we report long-lived nonequilibrium states in one-dimensional edge channels in the integer quantum Hall regime. An indirect heating scheme in a counterpropagating configuration is employed to generate a nontrivial binary spectrum consisting of high- and low-temperature components. This unusual spectrum is sustained even after traveling 5–$10\mu \mathrm{m}$, much longer than the length for electronic relaxation (about 0.$1\mu \mathrm{m})$, without showing significant thermalization. This observation is consistent with the integrable model of Tomonaga-Luttinger liquid. The long-lived spectrum implies that the system is well described by noninteracting plasmons, which are attractive for carrying information for a long distance.

Figure 1

(a) Schematic diagram of interacting quantum Hall edge channels $\left(\mathrm{r}/\ell ,\uparrow /\downarrow \right)$ with nearest neighbor coupling capacitances $C_{\mathrm{X}}$ and $C_{\mathrm{Z}}$ . Charge wave packets for plasmon eigenmodes $\left[\mathrm{L}/\mathrm{R},\mathrm{C}/\mathrm{S}\right]$ are illustrated. (b) The indirect heating scheme for interacting quantum-Hall edge channels $\left(\mathrm{r}/\ell ,\uparrow /\downarrow \right)$. Each time a single-electron charge wave packet is injected from a PC to $\left(\ell ,\uparrow \right)$ as shown by “inj,” four plasmon charge wave packets in the eigenmodes $\left[\mathrm{L}/\mathrm{R},\mathrm{C}/\mathrm{S}\right]$ are generated. Carriers are drained to ohmic contacts marked by $⊠$. The energy distribution function $f\left(E\right)$ shown at the bottom insets exhibits a double step structure for the initial state (the central inset), close to Fermi distribution at $T_{\mathrm{H}}$ in the DH region (the left inset), and an unusual binary spectrum with two temperatures $T_{\mathrm{H}}$ and $T_{\mathrm{base}}$ in the IH region (the right inset).

Figure 2

(a) The measurement setup with three magnified scanning-electron micrographs with false color. The QD spectrometer and one of the heat injectors (PC1–PC4) are activated with the surface gates $({\mathrm{SG}}_{\mathrm{iso}},{\mathrm{SG}}_{1-4}$, etc.), while inactivated PCs are fully opened. The double lines indicate edge channels that form when PC1 and PC3 are activated, while only one of the PCs are activated in the actual measurements. The outer and inner channels serve spin-up and -down transport, respectively. The QD with a few electrons shows a charging energy of about 2 meV. Terminals labeled “fl” are floating ohmic contacts. (b) A schematic energy diagram of the transport through a QD level $\varepsilon$. Electronic spectra $f_{\mathrm{a},\uparrow }\left(E\right)$ and $f_{\mathrm{b},\uparrow }\left(E\right)$ can be determined from the current $I\left(\varepsilon \right)$. (c) Current spectra of the QD for various effective bias voltages $V_{\mathrm{D}}-V_{\mathrm{E}1}$ taken with all PCs (including PC1) fully opened $\left(G_i/G_{\mathrm{q}}=2\right)$. Dashed lines indicate the energy alignment of the ground state with the chemical potential of the channels $\left(\varepsilon ={\mu }_{\mathrm{a}/\mathrm{b},\uparrow }\right)$, while dotted lines indicate that of the excited states. Each trace is offset for clarity.

Figure 3

(a) A schematic channel layout for measuring the spectrum in the DH region formed downstream of PC1 in $\left(\mathrm{a},\uparrow /\downarrow \right)$. (b) Schematic channel geometries near PC1 for (i) $G_1=0$, (ii) $G_1=0.5G_{\mathrm{q}}$, (iii) $G_1=G_{\mathrm{q}}$, (iv) $G_1=1.5G_{\mathrm{q}}$, and (v) $G_1=2G_{\mathrm{q}}$. Channels $\left(\mathrm{a},\uparrow \right)$ and $\left(\mathrm{a},\downarrow \right)$ can be heated in (ii) and (iv), respectively. (c) $V_{\mathrm{PC}1}$ dependence of $G_1/G_{\mathrm{q}}$, measured by the current $I_{\mathrm{E}1}$ at the excitation voltage $V_{\mathrm{E}1}=-100$ $\mu \mathrm{V}$. (d) Current spectra with a flat-top region of the width associated with the effective bias. Measured current (circles) is fitted well with the Fermi distribution function (solid lines). The broadened edges measure the electron temperatures $T_{\mathrm{a}/\mathrm{b},\uparrow }$ in the channels. The $I_{\mathrm{D}}$ and $V_{\mathrm{C}}$ scales apply to the trace (v), while other traces are offset for clarity. (e) $k_{\mathrm{B}}{T}_{\mathrm{a}/\mathrm{b},\uparrow }$ as a function of $V_{\mathrm{PC}1}$. $T_{\mathrm{H}}^{\left(0.1\right)}$ is the expected temperature considering the coupling $p=0.1$. Small difference between $T_{\mathrm{a},\uparrow }$ and $T_{\mathrm{b},\uparrow }$ at unheated conditions (iii) and (v) might come from external noise in the ammeter for $I_{\mathrm{E}1\text{.}}$

Figure 4

Binary tunneling spectra in the IH regions with heat current injected from (a) PC2, (b) PC3, and (c) PC4. Measured current (circles) is fitted well with the binary spectrum (solid lines) with two temperatures, $T_{\mathrm{a},\uparrow }^{\left(\mathrm{H}\right)}$ and $T_{\mathrm{a},\uparrow }^{\left(\mathrm{L}\right)}$, coexisting in the channel. Each inset shows the schematic channel geometry with the location of IH regions of interest.

Figure 5

Current spectrum in a linear scale (i) with heat injection from PC2 and (ii) without heat injection. The positive and negative current tails in the red and blue hatched regions in (i) represent the excitation of hot electrons and hot holes, respectively, consistent with the binary spectrum (the solid lines). The deduced distribution functions $f_{\mathrm{a},\uparrow }\left(E\right)$ and $1-f_{\mathrm{b},\uparrow }\left(E\right)$ are shown in the inset. The binary spectrum in $f_{\mathrm{a},\uparrow }\left(E\right)$ contrasts with a single Fermi distribution function $f_{\mathrm{F}}\left(E\right)$ (dashed line).

Figure 6

(a) Conductance of PC2, $G_2=I_{\mathrm{E}2}/V_{\mathrm{E}2}$, as a function of the gate voltage, $V_{\mathrm{PC}2}$. (b and c) $V_{\mathrm{PC}2}$ dependence of the fraction $p$ of hot minority carriers in (b) and thermal energies for $T_{\mathrm{a},\uparrow }^{\left(\mathrm{H}\right)}$ (circles), $T_{\mathrm{a},\uparrow }^{\left(\mathrm{L}\right)}$ (black solid line), and $T_{\mathrm{b},\uparrow }$ (black dashed line) in (c) deduced from the fitting. The parameter $T_{\mathrm{a},\uparrow }^{\left(\mathrm{H}\right)}$ is smaller than but comparable to $T_{\mathrm{PER}}^{\left(0.1\right)}$, which is expected in the assumption of reaching an identical temperature. The blue solid, dashed, and dotted lines are $T_{\mathrm{PER}}^{\left(0.1\right)}$ by assuming $D_{2,\uparrow }=D_{2,\downarrow },D_{2,\downarrow }=0$, and $D_{2,\uparrow }=0$, respectively.

Figure 7

(a) Tunneling spectra measured in the DH region induced by PC1 at $G_1=1.5G_{\mathrm{q}}$. Its $V_{\mathrm{E}1}$ dependence is taken by changing both $V_{\mathrm{E}1}$ and $V_{\mathrm{D}}$ to keep $V_{\mathrm{E}1}-V_{\mathrm{D}}=200$ $\mu \mathrm{V}$. Excited-state features are indicated by arrows (ES and ${\mathrm{ES}}^{\prime })$. (b) Tunneling spectra measured in the IH region induced by PC2 at $G_2=0.5G_{\mathrm{q}}$. The binary spectrum develops with increasing $V_{\mathrm{E}2}$.

Figure 8

(a) The bias voltage $V_{\mathrm{E}1}$ dependence of $k_{\mathrm{B}}{T}_{\mathrm{a},\uparrow }$ obtained at $G_2/G_{\mathrm{q}}\sim 1.5$ (circles) and $\sim 0.5$ (squares). Solid and dashed lines show $T_{\mathrm{H}}^{\left(0.1\right)}$ and $T_{\mathrm{H}}^{\left(0\right)}$, respectively, considering the variation of $k_{\mathrm{B}}{T}_{\mathrm{base}}=10$ (lower bounds) and $13\mu \mathrm{eV}$ (upper bounds). (b and c) The bias voltage $V_{\mathrm{E}2}$ dependence of $p$ in (b) and $k_{\mathrm{B}}{T}_{\mathrm{a},\uparrow }^{\left(\mathrm{H}\right)},k_{\mathrm{B}}{T}_{\mathrm{a},\uparrow }^{\left(\mathrm{L}\right)}$, and $k_{\mathrm{B}}{T}_{\mathrm{b},\uparrow }$ in (c). The error bars show typical ambiguity in fitting the binary spectrum. Blue solid lines show $T_{\mathrm{H}}^{\left(0.1\right)}$, considering the ambiguity of tunneling probabilities between $D_{2,\uparrow }=D_{2,\downarrow }=G_2/2G_{\mathrm{q}}$ (upper bound) and $D_{2,\uparrow }=G_2/G_{\mathrm{q}}$ and $D_{2,\downarrow }=0$ (lower bound).

Figure 9

Momentum distribution functions for (a) initial states in noninteracting channels and (b) steady state at $t=\infty$ after the interaction is turned on at $t=0$. Each trace is offset for clarity. Initially, channels are in thermal equilibrium at $T_1=T_3=T_4=T_{\mathrm{base}}$ and $T_2=10T_{\mathrm{base}}$ in (a). Nonequilibrium steady states are reached at $t=\infty$ in (b), where quasibinary spectrum emerges in counterpropagating channels 3 and 4. The dashed and dash-dotted lines show single and binary Fermi distribution functions, respectively, adjusted to the spectra based on the TL theory (solid lines). The schematic channel geometries without and with interaction are shown in the inset of (a) and (b), respectively.

Figure 10

Spectrum after traveling $10\mu \mathrm{m}$. (a) The energy diagram at a negative bias ${\mu }_{\mathrm{a},\uparrow }-{\mu }_{\mathrm{b},\uparrow }=e{V}_{\mathrm{D}}=-80$ $\mu \mathrm{eV}$. (b) The location of DH and IH regions in the schematic channel layout. (c) Current spectrum of the QD measured with PC4 in the reversed magnetic field. Binary spectrum is seen on the left side of the peak only when the PC4 conductance is set in the tunneling regime $\left(G_4\simeq 0.5G_{\mathrm{q}}\right)$.