Experimental observation of exact quantum critical states

Abstract

Anderson localization physics features three fundamental types of eigenstates: extended, localized, and critical, with the third one exhibiting the exotic properties in-between the former two. Confirming the presence of critical states is challenging, as it typically necessitates either advancing the analysis to the thermodynamic limit or identifying a universal mechanism which can rigorously determine these states. Here we report the unambiguous experimental realization of critical states, governed by a rigorous mechanism for exact quantum critical states, and further observe a generalized mechanism that quasiperiodic zeros in hopping couplings protect the critical states. We implement a programmable quasiperiodic mosaic model with tunable couplings and on-site potentials through a multiple superconducting qubit quantum system. By measuring the time-evolving observables, we identify the coexisting delocalized dynamics and incommensurately distributed zeros in the couplings, which are the defining features of the critical states. We map the localized-to-critical phase transition and demonstrate that critical states persist until quasiperiodic zeros are removed by strong long-range couplings, highlighting a novel generalized mechanism discovered in this experiment and shown with rigorous theory. Finally, we resolve the energy-dependent transition between localized and critical states, revealing the presence of anomalous mobility edges.

Figures (4)

• Figure 1: Schematic for the mechanism of quantum critical states.a-b Density profile of the eigenstates in extended, localized (a) and critical phases (b), which are delocalized in both position and momentum space. Blue circles denote the nodes caused by incommensurately distributed zeros (IDZs), which are key ingredients for the critical states. c Visualization of the self-similar structure characteristic of critical states. d The quantum spin model comprising a 2D array of spin qubits. Blue spheres represent lattice sites, and blue (orange) bonds mark the nearest-neighbour (long-range) couplings controlled by tunable couplers in the experiment. e Illustration of one-dimensional spin chain model incorporating long-range couplings. The 2D geometry facilitates long-range couplings. By activating orange bonds in 2D array and relabeling the system as a 1D chain, nearest-neighbour couplings effectively realize long-range interactions within the redefined 1D configuration.
• Figure 2: Characteristic dynamics of localized, critical and extended phases.a-b Dynamics in the localized phase with $\lambda/J=0.25$. c-d Dynamics in the critical phase with $\lambda/J=2.5$. e-f Dynamics in the extended phase with the long-range coupling $J^L_{m,n}=\lambda$ and $\lambda/J=2.5$. In a,c,e, the top panels show illustrations for the density profiles of each phases, and bottom panels show the measured dynamics of on-site population $n_j(t)$, where the system is initialized in $\vert \psi (t=0)\rangle = (\prod \sigma^{+}_{6k-5} )\vert 0^{\otimes N}\rangle$, with $k$ indexing every sixth site starting from site 1. b,d,f show the measured dynamics of $n_j(t)$ with the initial state prepared at the left and right to the zeros in coupling coefficients marked by the stars, highlighting the distinct behaviors of different phases. The parameters of systems are $\lambda/(2\pi)=1~\mathrm{MHz}$, $J^L_{m,n}=0$ (localized phase); $\lambda/(2\pi)=10~\mathrm{MHz}$, $J^L_{m,n}=0$ (critical phase); $\lambda/(2\pi)=J^L_{m,n}/(2\pi)=10~\mathrm{MHz}$ (extended phase); with $J/(2\pi)=4~\mathrm{MHz}$, $V_0=0$ and $\theta = \pi/5$.
• Figure 3: Localized-critical phase transition and its breakdown. a Measured time-averaged observable $\overline{\mathcal{D}}$ within the time range from $0$ to $t_f$ for long-ranged coupling $J_{m,n}^L=0$, where $\lambda/J=1/4$ ($\lambda/J=2$) are chosen from a deeply localized (critical) phase. b-d Characterization of the localized-critical phase transition for $J_{m,n}^L=0$ in (b) and its breakdown in the presence of long-ranged coupling in (c-d), where (c) shows $\overline{\mathcal{D}}$ and time-averaged $\overline{n}_{\mathrm{r}}=\sum_{j>j_0} n_j$ for different $J_{m,n}^L/J$ with $\lambda/(2\pi)=10~\mathrm{MHz}$, and (d) gives $\overline{\mathcal{D}}$ and $M(t_f)$ for different $\lambda/J$ with $J_{m,n}^L=2.5 J$. The system is initialized in $\vert \psi (t=0)\rangle = \vert 1\rangle_{14}$, $J/(2\pi)=4~\mathrm{MHz}$, and the size of system is $N=24$. The markers represent experimental data, and the solid lines correspond to simulations.
• Figure 4: Mobility edges in the quasiperiodic mosaic lattice.a Measured time-averaged observable $\overline{\mathcal{D}}$, where the system is quenched from the localized state $\vert \psi_g^+\rangle$ and zero-energy state $\vert \psi_g^-\rangle$ of the Hamiltonian $H$ as $\lambda \to 0$ with $\vert \psi_g^{\pm}\rangle$$= (\vert 1\rangle_{12}\pm\vert 1 \rangle_{13})/\sqrt{2}$. The site $j$ ranging from 1 to $N=24$ and $\theta=\pi/5$. b-c Measured $\overline{\mathcal{D}}$ and $M(t_f)$ for the system quenched from the superposition of localized and critical state $\vert \psi^{\phi}_n \rangle = (\vert 1\rangle_{12} + e^{i\phi}\vert 1 \rangle_{13})/\sqrt{2}$ (b), as well as for states $\vert \psi^{\pm}_n \rangle$ located near the edge (inside the novel mobility edges) and center (outside the novel mobility edges) of the spectrum (c), where $t_f=\mathrm{300~ns}$ and $\vert \psi^{\pm}_n \rangle = (\vert 1\rangle_{n}\pm\vert 1 \rangle_{n+1})/\sqrt{2}$. The dashed lines mark the mobility edges separating the localized and critical states. The markers represent experimental data, and the solid lines correspond to numerical simulations.