Evidence for a two-dimensional quantum glass state at high temperatures

1. Problem Statement

The nature of the transition between ergodic (thermalizing) and non-ergodic phases in quantum many-body systems is a central open question in condensed matter physics.

• The Debate: While Many-Body Localization (MBL) describes a phase where all degrees of freedom are frozen and transport ceases, its existence in dimensions $D > 1$ is controversial. Conversely, classical glasses exhibit "glassy" behavior (slow relaxation, history dependence) without full localization.
• The Gap: There is a theoretical proposal for an intermediate phase called Non-Ergodic Extended (NEE) states. These states possess a support in Hilbert space that is exponentially large but smaller than the full dimension, allowing for energy transport (dephasing) while preventing full thermalization (particle transport).
• The Challenge: Distinguishing between NEE states, classical quantum glasses, and fully localized MBL states is difficult due to the exponential growth of the Hilbert space, which makes classical numerical simulations (like exact diagonalization) intractable for large 2D systems. Furthermore, most theoretical work focuses on the infinite-temperature limit ($T \to \infty$) or zero temperature, leaving the finite-temperature regime in 2D largely unexplored experimentally.

2. Methodology

The authors utilized a 59-qubit (and up to 70-qubit) superconducting quantum processor (Willow architecture) to simulate a 2D disordered interacting spin model.

• Hamiltonian: The system models a 2D square lattice of spin-1/2 particles with nearest-neighbor XY coupling ($J$) and random on-site magnetic fields ($h_i$) drawn from a box distribution of width $W$:
  $$H = -J \sum_{\langle i,j \rangle} (S^+_i S^-_j + S^-_i S^+_j) + \sum_i h_i S^z_i$$
  The dimensionless disorder strength is defined as $w = W/J$.
• Experimental Protocol:
  1. Initialization: Prepare a product state (bit-string) corresponding to a specific energy near the middle of the spectrum.
  2. Evolution: Evolve the system under the analog Hamiltonian for variable times $t$.
  3. Measurement: Measure in the $Z$-basis.
  4. Averaging: Repeat the process $10^5 - 10^6$ times per initial state and average over $\sim 10^3$ disorder realizations to build statistical distributions.
• Dual-Space Analysis: The study uniquely probes dynamics in two domains:
  • Real Space: Measuring local magnetization, spin diffusion, and Edwards-Anderson order parameters.
  • Hilbert Space: Measuring the Return Probability ($R(t) = |\langle \Psi(0)|\Psi(t) \rangle|^2$) and Wavefunction Statistics (distribution of bit-string probabilities).

3. Key Contributions

1. Experimental Realization of 2D Quantum Glass: The first experimental evidence of a glassy, non-ergodic phase in a 2D disordered system at finite temperature ($T \approx \infty$).
2. Identification of the NEE Phase: Demonstration that the "quantum glass" phase observed is physically equivalent to the theoretical Non-Ergodic Extended (NEE) state, bridging the gap between classical glass theory and many-body localization physics.
3. Decoupling of Transport Channels: Evidence that energy transport (dephasing) can persist even when spin diffusion (particle transport) is suppressed, distinguishing this phase from fully localized MBL.
4. Scaling Laws: Discovery of specific scaling behaviors for the return probability exponent ($\eta \propto n^{2.4}$) and wavefunction statistics that characterize the glassy phase.

4. Key Results

A. Real-Space Signatures: Glassy Magnetization

• Freezing: At low disorder ($w < w_c \approx 10$), magnetization relaxes completely (ergodic). At high disorder ($w > w_c$), a significant fraction of magnetization remains frozen indefinitely.
• Edwards-Anderson Order Parameter ($Q_{EA}$): The dynamic spin-glass correlation function $C(t)$ decays as a power law in the ergodic phase but saturates to a non-zero value ($Q_{EA} = C(t \to \infty)$) in the glassy phase. $Q_{EA}$ shows a sharp onset near $w_c \approx 10$.
• Breakdown of Diffusion: In the ergodic phase, spin diffusion follows normal diffusion ($\Gamma \propto \lambda$, where $\lambda$ is the Laplacian eigenvalue). In the glassy phase, diffusion breaks down; the diffusion coefficient $D$ drops to near zero, and the relaxation becomes anomalous (slower than exponential).

B. Hilbert-Space Signatures: Return Probability

• Power-Law Decay: In the ergodic phase, the return probability $R(t)$ decays exponentially to the inverse Hilbert space dimension. In the glassy regime ($10 \le w \le 35$), $R(t)$ exhibits a slow power-law decay:
  $$R_{typ}(t) \sim (Jt)^{-\eta}$$
• System Size Scaling: The decay exponent $\eta$ scales superlinearly with system size $n$:
  $$\eta \approx \kappa(w) \cdot n^{2.4}$$
  This superlinear scaling indicates that dephasing arises from internal interactions between many spin pairs (two-level systems), rather than external noise.
• Distribution Width: The distribution of $\ln R(t)$ becomes extremely broad (log-normal to power-law tails), indicating that the system explores a "cluster" of states in Hilbert space rather than the entire space.

C. Wavefunction Statistics

• Porter-Thomas to Power-Law: In the ergodic phase, wavefunction amplitudes follow the Porter-Thomas distribution (characteristic of random matrices). As disorder increases into the glassy phase, the distribution evolves into a power-law form $P(|\psi|^2) \sim |\psi|^{-\zeta}$.
• Critical Exponent: The exponent $\zeta$ decreases with increasing disorder, approaching $\zeta \to 1$ at very strong disorder. This differs from the MBL transition on random regular graphs (where $\zeta = 1/2$), suggesting a distinct universality class for 2D quantum glasses.

D. Theoretical Consistency

• Dephasing vs. Relaxation: Theoretical modeling on a Cayley tree (Bethe lattice) demonstrates that in a specific range of disorder, the dephasing rate ($\Gamma_\phi$) is much larger than the relaxation rate ($\Gamma_r$).
  • $\Gamma_\phi > 0$: Energy can propagate (allowing power-law decay and 1/f noise).
  • $\Gamma_r \approx 0$: Particles/spins cannot relax to equilibrium (preventing ergodicity).
• 1/f Noise: The observed power-law decay of $R(t)$ implies a spin noise spectrum $S(\omega) \propto 1/\omega$, a hallmark of glassy systems.

5. Significance

• Resolution of the "Glass vs. MBL" Debate: The paper provides strong evidence that the "quantum glass" and the "Non-Ergodic Extended (NEE)" state are likely the same physical phenomenon. It shows that a system can be non-ergodic without being fully localized (MBL).
• High-Temperature Physics: It establishes that glassy behavior is not limited to low temperatures or specific integrable models but is a robust feature of 2D disordered quantum systems at high temperatures.
• Quantum Simulation Advantage: The study highlights the unique capability of quantum simulators to access Hilbert-space diagnostics (like return probability and wavefunction statistics) that are impossible to compute classically for these system sizes.
• Implications for Quantum Information: The existence of a phase with suppressed spin diffusion but active dephasing has implications for quantum memory (protection against bit-flips) and the generation of 1/f noise, which is a major source of decoherence in superconducting qubits.

In conclusion, the authors have experimentally mapped a phase diagram for 2D disordered matter, identifying a distinct Quantum Glass/NEE phase characterized by frozen spin transport, power-law relaxation, and specific wavefunction statistics, fundamentally distinguishing it from both the ergodic fluid and the fully frozen MBL phase.