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

TL;DR

The paper demonstrates evidence for a two-dimensional quantum glass state at high temperatures by combining real-space dynamics and Hilbert-space probes in a disordered spin array of superconducting qubits. It identifies an intermediate non-ergodic extended (NEE) regime where certain degrees of freedom freeze while others remain dynamic, evidenced by a power-law decay of the Hilbert-space return probability and a nonzero Edwards-Anderson order parameter alongside suppressed diffusion. The authors develop a qualitative and quantitative framework, including a Cayley-tree treatment and Schrieffer–Wolff analysis, to separate dephasing and relaxation processes and to locate thresholds for ergodicity breaking in 2D; they also observe 1/f spin noise consistent with glassy dynamics. Overall, the work provides finite-size, finite-temperature experimental confirmation of a non-ergodic, glass-like phase in a two-dimensional quantum many-body system and outlines a mechanism where dephasing dominates over relaxation in a broad disorder range, implying energy transport without full ergodicity.

Abstract

Disorder in quantum many-body systems can drive transitions between ergodic and non-ergodic phases, yet the nature--and even the existence--of these transitions remains intensely debated. Using a two-dimensional array of superconducting qubits, we study an interacting spin model at finite temperature in a disordered landscape, tracking dynamics both in real space and in Hilbert space. Over a broad disorder range, we observe an intermediate non-ergodic regime with glass-like characteristics: physical observables become broadly distributed and some, but not all, degrees of freedom are effectively frozen. The Hilbert-space return probability shows slow power-law decay, consistent with finite-temperature quantum glassiness. In the same regime, we detect the onset of a finite Edwards-Anderson order parameter and the disappearance of spin diffusion. By contrast, at lower disorder, spin transport persists with a nonzero diffusion coefficient. Our results show that there is a transition out of the ergodic phase in two-dimensional systems.

Figures (6)

• Figure 1: Phase Diagram. At low temperatures and weak disorder, the system exhibits long-range order, while strong disorder localizes the bosons. At high temperatures, weak disorder yields an ergodic diffusive phase; increasing disorder drives a transition to a glassy non-ergodic state with finite Edwards-Anderson order parameter, and ultimately to a fully frozen MBL phase.
• Figure 2: Evidence of glass formation in the frozen magnetization. a, Temporal evolution of magnetization for disorders of $w=9$ and $w=15$ (corresponding disorder patterns showed on top), revealing slower relaxation at the higher disorder. b, Time dependence of the spin glass correlation function $C(t)$, showing power law relaxation of $C(t)$ in the ergodic phase. c, Disorder-dependence of the dynamic exponent $\alpha$ that describes the power-law decay of $C(t)$. d, Same as b, but with linear axis, revealing the absence of complete relaxation at higher disorders. e, The Edwards-Anderson order parameter, $Q_\text{EA} = C(\infty)$, as a function of $w$ shows an abrupt change in slope near $w=w_c\simeq 10$. $Q_\text{EA}$ is extracted from fitting $C(t)$ with a decaying functional form (exponential with or without a power-law prefactor, see SI) with a constant offset.
• Figure 3: Signatures of glassy behavior in the return probability. a, Real-space measurement instances show that fewer spins are flipped when $w$ is higher. The return probability is defined as the fraction of the measurements in which no spins are flipped. b, Time dependence of measured return probability for $n=25$ (circles), which shows a local minimum ("correlation hole") at low disorder, and power-law decay for larger $w$. Solid curves show results from exact numerical modeling, averaged over $\sim 2500$ disorder instances (100 for $w=1$), while straight lines indicate power-law fits. c, Return probability on logarithmic scales shows a wide distribution, for $n=42$ and at three different evolution times ($w=30$). Solid lines are Gaussian fits. d, Time dependence of return probability $R(t)$ for a range of system sizes, displaying power-law dependence on time for $w=25$. Solid straight lines show power-law fits. e, The power-law exponents extracted in d scale as $\eta\propto n^{2.4}$. f, Exponent $\eta$ as a function of disorder for different system sizes. Continuous lines in panel d,$n=20,25$, as well as grey symbols in panel e,$n=16, 20$ show results of numerical simulations; for $n=25$ numerical results coincide with experimental ones.
• Figure 4: Disappearance of diffusion at strong disorder. a, The measured relaxation of magnetization $\langle m_k(t)m_k(0)\rangle$ for the smallest eigenmode ($k=1$) at several values of disorder ($n=70$). b, Extracted relaxation rates for $n=42$, $59$ and $70$. Solid lines are fits using $\Gamma = \mathcal{D} \lambda$. c, The extracted diffusion constant $\mathcal{D}$ (left axis) and non-zero frozen amplitude $M_\infty$ (right axis). Background colors denote $\beta$, resulting from fitting data in b with $\Gamma = \mathcal{D} \lambda^\beta$. Diffusion applies for $\beta \approx 1$ and fails as $\beta$ departs from unity and hence $\mathcal{D}$ cannot be extracted above $w>10$.
• Figure S5: Comparison of remanent magnetization fits by different time dependencies. Left panel: data for $n=59$$w=14$ and their fits with different laws, with and without offset. The data shown here were filtered with degree $r=2$ filter. Right panel: data deviation from different fits as a function of $w$ for $n=59$. Below phase transition power law is clearly a winner, while above the transition exponential fit with offset quickly wins over the power law. One cannot distinguish pure exponential and exponential with a power law prefactor.