Critical states and anomalous wave transport in an aperiodic polariton monotile

Technical Summary: Critical States and Anomalous Wave Transport in an Aperiodic Polariton Monotile

Problem Statement
Quasicrystals and aperiodic tilings occupy a physical regime between ordered periodic systems and disordered amorphous materials. While they lack translational symmetry and unit cells, they possess long-range aperiodic order, self-similar eigenmodes, and fractal spectra, leading to critical states and anomalous transport. Despite extensive study in platforms ranging from ultracold gases to photonics, research on exciton-polaritons in aperiodic systems has been predominantly limited to one-dimensional (1D) models or vertex (tight-binding) approximations. Recently, the "Hat" monotile (or "einstein") was introduced as a new aperiodic tiling capable of covering the plane without gaps or overlaps. This paper addresses the gap in understanding the two-dimensional (2D) single-particle physics and mean-field dynamics of polaritons within this specific "Monotile" quasilattice, specifically investigating the existence of critical states and their impact on wave transport.

Methodology
The authors model the system using 2D exciton-polaritons in a planar microcavity, described initially by the Hermitian Schrödinger equation with an optically induced potential. The potential is constructed from repulsive Gaussian pump spots located at the vertices of the Monotile tiling (derived from a deltoidal trihexagonal "tetrille" lattice).

The study employs a multi-stage numerical approach:

1. Spectral Analysis: Direct diagonalization of the 2D Schrödinger equation for various system sizes ($L = 20a \to 40a$) to obtain eigenenergies and eigenstates.
2. Scaling Analysis: Calculation of the Inverse Participation Ratio (IPR) and generalized dimensions ($D_q$) to characterize the localization properties of eigenstates.
3. Wavepacket Dynamics: Direct numerical integration of the 2D Schrödinger equation (using split-step fast Fourier transform) to simulate the spreading of an initial Gaussian wavepacket. This yields the diffusion exponent ($\nu$) and temporal correlation exponent ($\delta$).
4. Non-Hermitian Extension: Incorporation of gain and loss terms ($\kappa$ and $\gamma$) to model the dissipative nature of polaritons and analyze transport below the condensation threshold.
5. Mean-Field Modeling: Application of the generalized Gross-Pitaevskii equation (2DGPE) coupled with a reservoir density equation to simulate nonlinear polariton fluids under both non-resonant (ballistic condensation) and resonant (pulsed) driving conditions.

Key Contributions and Results

• Existence of Critical States: Through direct diagonalization, the authors confirm that the Monotile spectrum hosts a coexistence of localized, extended, and critical states. Critical states are predominantly found at lower energies and near spectral pseudogaps.
• Multifractal Hilbert Space: Scaling analysis of the wavefunction moments reveals a multifractal structure in the Hilbert space. The generalized dimension $D_2$ is found to be less than the spatial dimension ($D_2 < 2$) in critical regions, with the scaling exponents $D_q$ following a power law that approaches an asymptotic value $D_\infty \approx 1.01$, consistent with critical system theory.
• Anomalous Transport Regimes: The study identifies distinct transport regimes for polariton wavepackets in the Monotile:
  • Super-diffusion: Regimes where the diffusion exponent $1/2 < \nu < 1$.
  • Near Sub-diffusion: Regimes where $0 < \nu < 1/2$.
    These regimes are directly linked to the fractal nature of the spectrum and the specific parameters of the optical lattice (potential amplitude $V_0$ and lattice constant $a$).
• Non-Hermitian Dynamics: In the presence of gain ($\kappa$) and loss ($\gamma$), the authors observe that increasing $\kappa$ drives the system toward ballistic transport ($\nu \to 1$). This occurs because low-energy critical states have poor overlap with the gain regions (pump spots) and are suppressed relative to high-energy extended states, which survive and expand ballistically.
• Nonlinear Mean-Field Behavior:
  • Non-resonant Driving: When driven above the condensation threshold, the system forms ballistic condensates characterized by extended modes and long-range coherence, inheriting the $C_6$ rotational symmetry of the underlying tetrille lattice.
  • Resonant Driving: Using a resonant picosecond pulse, the authors demonstrate that anomalous transport (super- and sub-diffusive) can be probed in the nonlinear regime. The shape of the expanding wavefront follows a stretched exponential profile, with the exponent $b$ correlating with the diffusion exponent $\nu$ (e.g., $b > 2$ for super-diffusion, $b < 2$ for sub-diffusion), validating the single-particle findings in a many-body context.

Significance and Claims
The paper claims to provide the first comprehensive investigation of 2D wave transport and criticality in the "Hat" monotile using a full Schrödinger picture for polariton scatterer potentials, moving beyond 1D or tight-binding approximations. The primary significance lies in demonstrating that the Monotile quasilattice supports critical states with multifractal properties that lead to tunable anomalous transport regimes (super-diffusive and near sub-diffusive).

The authors propose that these findings can be experimentally verified using reconfigurable optical lattices for cavity polaritons. Specifically, they suggest that by tuning the optical potential parameters ($a$ and $V_0$) and utilizing resonant pulsed excitation, researchers can observe these anomalous transport signatures in macroscopic polariton quantum fluids. The work establishes a foundation for studying critical phenomena and topological properties in all-optically versatile aperiodic systems, noting that while similar results were found in Penrose tilings, the unique advantages of the Monotile remain a subject for future investigation. The paper concludes by outlining future directions, including stochastic discrete nonlinear networks and the exploration of chiral aperiodic monotiles for spin-related phenomena.