Quantum Spin Glass in the Two-Dimensional Disordered Heisenberg Model via Foundation Neural-Network Quantum States

TL;DR

The paper addresses the existence of a quantum spin-glass phase in a two-dimensional disordered Heisenberg model by employing Foundation Neural-Network Quantum States (FNQS) to efficiently compute disorder-averaged ground-state properties across many realizations, mitigating the sign problem and boundary-condition issues. It identifies an intermediate quantum spin-glass region between ferromagnetic and antiferromagnetic phases, evidenced by a finite overlap order parameter $Q$ when magnetic orders vanish, with phase boundaries near $p_{\mathrm{FM}}\approx0.2$ and $p_{\mathrm{AFM}}\approx0.8$. A complementary semiclassical Holstein-Primakoff spin-wave analysis shows that quantum fluctuations do not destroy the spin-glass order at leading order in $1/S$, with $Q_{\text{SW}}^2=Q_0^2+\Delta Q^2$ agreeing with the fully quantum FNQS results at $S=1/2$. The work demonstrates FNQS as a scalable method for disordered quantum magnets and provides a coherent 2D phase diagram, highlighting the robustness of QSG phases against quantum fluctuations and motivating future studies of low-energy excitations and potential spin-liquid regimes near phase boundaries.

Abstract

We investigate the two-dimensional frustrated quantum Heisenberg model with bond disorder on nearest-neighbor couplings using the recently introduced Foundation Neural-Network Quantum States framework, which enables accurate and efficient computation of disorder-averaged observables with a single variational optimization. Simulations on large lattices reveal an extended region of the phase diagram where long-range magnetic order vanishes in the thermodynamic limit, while the overlap order parameter, which characterizes quantum spin glass states, remains finite. These findings, supported by a semiclassical analysis based on a large-spin expansion, provide compelling evidence that the spin glass phase is stable against quantum fluctuations, unlike the classical case where it disappears at any finite temperature.

Figures (7)

• Figure 1: Ground-state phase diagram of the disordered Heisenberg model [see Eq. \ref{['eq:ham_model']}] in the thermodynamic limit, as a function of the probability $p \in [0,1]$ [see Eq. \ref{['eq:prob_couplings']}]. Three distinct phases are identified: a ferromagnetic phase for ${p \le p_{\mathrm{FM}} \approx 0.2}$, an antiferromagnetic phase for ${p \ge p_{\mathrm{AFM}} \approx 0.8}$, and an intermediate quantum spin glass phase characterized by the absence of magnetic order and a finite overlap parameter $Q$ (see Order parameters for details). The phase diagram is obtained using the variational approach based on Foundation Neural-Network Quantum States (refer to Methods).
• Figure 2: Left panel: Ferromagnetic order parameter $\mathcal{M}^2_{\text{Ferro}}$ (blue circles, dashed lines) and Néel antiferromagnetic order parameter $\mathcal{M}^2_{\text{N\'eel}}$ (red squares, dashed lines) are shown as a function of the probability $p$ for system sizes ranging from $L=4$ to $L=14$. Extrapolated values in the thermodynamic limit (TL), obtained via finite-size scaling in $1/L$ (see \ref{['sec:extrapolations']} of the Methods for details), are indicated by solid lines connecting the corresponding symbols. Right panel: Overlap order parameter $Q$ (green circles, dashed lines) as a function of $p$ for the same system sizes and averaging procedure. Extrapolated TL values are shown as green circles connected by a solid line. Results in both panels are averaged over $\mathcal{R} = 600$ disorder realizations. The error on the extrapolated values in the thermodynamic limit are estimated via a resampling technique with Gaussian noise. The semiclassical value of the overlap obtained in the TL of the non-interacting spin-wave theory is shown for comparison (orange rhombi), in great agreement with the FNQS values (see Semiclassical analysis for the details).
• Figure 3: Finite-size scaling of the overlap order parameter $Q$ [see Eq. \ref{['eq:overlap_order_par']}] as a function of the inverse system size $1/L$ for $L = 4$ to $L = 14$ at $p = 0.5$. Quadratic (solid line and yellow circle) and power-law (dashed line and green circle) fits are performed over all data points, while a linear fit (dotted line and red circle) is applied to the last three points. Inset: Zoom of the extrapolations close to $1/L \to 0$. The error bars of the extrapolated values in the thermodynamic limit are estimated via a resampling technique with Gaussian noise. The red star indicating the extrapolated value at $p = 1.0$ is showed for comparison.
• Figure 4: Semiclassical prediction for the overlap order parameter in the thermodynamic limit as a function of the inverse spin representation $1/S$ at $p=0.5$. On the vertical axes we plot $Q_{\text{SW}}$ [see Eq. \ref{['eq:Qlsw']}] divided by $4S^2$ to have a comparison with the FNQS result at $S=\tfrac{1}{2}$ and at the same time with the classical value $Q_0 / (4S^2) = 0.1443$ (see main text for the definition). The semiclassical correction is obtained through a non-interacting spin wave-theory computation (see Semiclassical analysis and the SI Appendix for details). For $S=\tfrac{1}{2}$ the value of the overlap is compatible with the value obtained for the fully quantum model with the FNQS, within the error bars.
• Figure 5: Spin-spin correlations for a disorder realization with $p = 0.7$ on a $6 \times 6$ cluster. The two-dimensional lattice is unrolled using the row-major convention for visualization. Exact diagonalization results (blue empty circles) are compared with variational calculations: an NQS optimized on the specific realization (green circles), and two FNQS models optimized over $\mathcal{R} = 600$ independent disorder realizations. A smaller FNQS (orange triangles) with $P \approx 8\times10^5$ parameters and a larger FNQS-L (red triangles) with $P \approx 1.2\times10^6$ parameters. The inset shows the corresponding spin structure factor along the path connecting $\mathbf{k} = (0,0)$ and $\mathbf{k} = (\pi,\pi)$.