Circuit Quantum Simulation of a Tomonaga-Luttinger Liquid with an Impurity

The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures. A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity. However, this transition impedes experimental explorations of real-world TLLs. Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths. Quantum simulation can provide a powerful workaround. Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity. Measurements reveal the renormalization group “beta function” for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling. Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored. With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.

Popular Summary

Strong correlations among a large ensemble of quantum particles can give rise to intriguing states of matter with unconventional behaviors. Quantum phase transitions at zero temperature are believed to underpin many of these phenomena, yet the complexity of real-world strongly correlated materials and the theoretical challenge posed by even simplified models impedes a microscopic understanding. The realization of simple, well-characterized systems for studying these behaviors is therefore highly desirable. To that end, we investigate one-dimensional collective physics—and the resulting metal-insulator quantum phase transition—using quantum simulation with a nanoengineered circuit.

In one-dimensional systems, the enhanced interactions result in phases of strongly correlated matter described by the “Tomonaga-Luttinger liquid,” a concept describing interacting electrons. However, the metal-insulator quantum phase transition induced by even a single impurity impedes experimental exploration. Furthermore, this hallmark signature still eludes a full theoretical treatment. Quantum simulation can provide a powerful workaround, as we show here with a quantum circuit implementing an analog of a Tomonaga-Luttinger liquid with adjustable electronic interactions and a fully tunable scattering impurity.

Measurements reveal the universal scaling flows to an insulating state, establish a quantitative relation with the circuit parameters, and explore the out-of-equilibrium regime. With the quantum simulator benchmarked by the precise agreement with untested and novel predictions, we then achieve quantum simulation in its strongest sense by elucidating theoretically unsolved regimes.

Our approach opens the path to in-depth investigations of various facets of correlated physics and sheds a new light on the modified transport properties of quantum components when embedded into a circuit.

Figure 1

TLL quantum simulator and quantum phase transition between metallic and insulating states. (a) The device’s schematic circuit (left) and colorized $e$-beam micrograph (right). A QPC set to a single electronic channel of bare transmission probability $\tau$ is formed in a 2DEG with the split gates colored yellow. The gates colored in green control the series resistance $R$ and, thereby, the Luttinger interaction parameter $K=1/(1+R{e}^2/h)$. A bright metallic island on top of $\mathsf{Y}$-shaped trenches separates the electronic channel and resistance. (b) Colored continuous lines display the device’s conductance $G=dI/dV$ measured at zero bias ($V=0$) for different temperatures $T$ in the presence of a series resistance adjusted to $R=h/e^2$ ($K=1/2$). The data are plotted versus bare (unrenormalized) $\tau$, which is determined from $G(V=60\mu \mathrm{V},T=7.9\mathrm{mK})=(h/\tau e^2+R{)}^{-1}$ (black dashed line). The conductance vanishes upon cooling, except at the ballistic quantum-critical point $\tau =1$.

Figure 2

Universal conductor-insulator crossover. In both panels, data points for a fixed device tuning of $\tau$ at different temperatures are shown with identical symbols. For clarity, vertical shifts of 0.1 are applied between $K\in \{1/2,2/3,3/4,4/5\}$ ($R\in \{1,1/2,1/3,1/4\}h/e^2$, respectively). Colored continuous lines represent the experimental quantum-simulated solutions, obtained by averaging the data ensemble. Exact TLL predictions ($K\in \{1/2,2/3\}$ only) are shown as black dashed lines. (a) displays the renormalization-group beta functions ${\beta }_K(g)$. Individual data points are the discrete temperature differentiation, at fixed $\tau$, of conductance measurements $\delta g/\delta \ln T$, with $g\equiv G/(K{e}^2/h)$. Black dash-dotted lines show the asymptotic ${\beta }_K(g)$ slopes predicted near the $g=0$ and $g=1$ fixed points. (b) displays the universal conductance flows $G_K(T/T_I)$. Quantum-simulated curves are obtained by integrating the experimental ${\beta }_K(g)$ shown in (a). Direct conductance measurements are also displayed for representative settings of $\tau$, with $T_I(\tau ,K)$ adjusted at temperatures well below the high-energy $RC$ cutoff. Full symbols indicate that the universality criteria $T\lesssim h/25k_{B}RC$ is verified. Nonuniversal deviations can develop at higher temperatures (open symbols).

Figure 3

Scaling temperature versus impurity strength. Symbols display the experimentally extracted TLL scaling temperature $T_I$ versus unrenormalized transmission probability $\tau \in [0,1]$. Each set of identical symbols corresponds to the same tuning of the Luttinger interaction parameter $K$. Continuous lines represent the asymptotic predictions at $\tau \ll 1$ and $1-\tau \ll 1$. The fully quantitative $K=1/2$ prediction is compared to the data without any adjustable parameter. At other $K\in \{2/3,3/4,4/5\}$, the unknown multiplicative theoretical factor is freely adjusted. The same data and predictions are also shown in a log-log scale for $\tau <0.15$ (bottom-left inset) and $\tau >0.85$ (top-right inset).

Figure 4

Out-of-equilibrium conductance renormalization. Symbols in all panels display the measured differential conductance versus several normalizations of the dc bias voltage $V$. For each device setting of $K$ and $\tau$ [color code in (c)], $T_I$ is separately determined from $G(V=0)$ (Fig. 3). Theoretical predictions are shown for $T=0$ [continuous lines; see Refs. [28, 29] and Eq. (b6)] and for values of $T/T_I$ fixed by the corresponding data [dashed lines; see Refs. [28, 29] and Eqs. (b1) and (b3)]. In (b),(c), we apply 0.15 vertical shifts between different $K\in \{1/2,2/3,4/5\}$. (a) illustrates the $eV/k_{B}T$ universality. Data obtained at $K=2/3$ for several settings of $\{\tau ,T\}$, each corresponding to the same $G(V=0)$ (two values shown), are plotted versus $V$ (left side) and $eV/k_{B}T$ (right side). (b) shows the nontrivial thermal-nonequilibrium crossover as $V$ is increased, for the representative settings $\tau =0.62$ and 0.94 at $T=7.7\mathrm{mK}$. We show only $|eV|<h/6RC$ data points, sufficiently below the high-energy cutoff. Large $eV/k_{B}T>12$ are signaled by full symbols. (c) provides a parameter-free comparison between theoretical $G_K^{T/T_I=0}(eV/k_B{T}_I)$ and data points within $|eV|\in [12k_{B}T,h/6RC]$. The distinct $G_K^{V/V_I=0}(T/T_I)$ universal scalings at equilibrium are also plotted, versus $T/T_I$, as gray dash-dotted lines.

Figure 5

Quantum phase transition between metallic and insulating states. (a),(b),(c) Colored continuous lines display the device’s conductance $G=dI/dV$ measured at $T=7.7\mathrm{mK}$ and different dc bias voltage $V$ for a series resistance set to $R=\{1,1/2,1/4\}\times h/e^2$ ($K=\{1/2,2/3,4/5\}$). The data are plotted versus bare transmission probability $\tau$. (d) Colored continuous lines display $G=dI/dV$ measured at $V=0$ for different $T$ and a series resistance set to $R=h/4e^2$ ($K=4/5$) versus bare transmission probability $\tau$. Whatever $K$ ($R$), the conductance progressively vanishes as $V$ or $T$ is reduced except at the ballistic quantum critical point $\tau =1$.

Figure 6

Reproducibility in equivalent circuit configurations. (a) Renormalization beta function for two physical implementations of the QPC in series with a linear resistance $R\in \{1,1/2\}\times h/e^2$. The continuous lines show the same experimental quantum simulations displayed Fig. 2, while the dash-dotted lines are obtained with another QPC (at a lower signal-to-noise level). (b) The experimental scaling temperature $T_I$ (symbols) does not depend on the impurity realization by different QPCs at our experimental accuracy (different superimposed symbols correspond to different physical realizations of the QPC). Inset: Equivalence of the simulated renormalization flows $G_K(T/T_I)$ (continuous lines, main article data; dash-dotted lines, additional data).