Universal Quantum Suppression in Frustrated Ising Magnets across the Quasi-1D to 2D Crossover via Quantum Annealing

Abstract

Quantum magnets in the $M\mathrm{Nb_2O_6}$ and BaCo$_2$V$_2$O$_8$ families realise frustrated transverse-field Ising models whose competing ferromagnetic and antiferromagnetic couplings generate a sign problem provably intractable for quantum Monte Carlo at any system size, leaving their quantum phase boundaries numerically Inaccessible. Using a D-Wave Advantage2 quantum annealer at $L\leq27$ (729 spins), we obtain the large-$L$ critical points for this model family, measuring quantum-driven transitions at ${g_c^{\mathrm{QPU}}}\in\{0.286,\,0.210,\,0.156,\,0.093\}$ for $α\in\{1.0,\,0.7,\,0.5,\,0.3\}$, where the analytically exact classical threshold is ${g_c^{\mathrm{class}}}(α)=2α/3$. The suppression ratio $r(α)$ exhibits a sharp two-regime structure: the three quasi-1D geometries ($α\leq0.7$) are mutually consistent with a universal plateau $\bar{r}=0.450$ ($χ^2/\mathrm{dof}=1.10$, $p=0.33$), demonstrating that quantum fluctuations destroy approximately $55\%$ of the classical FM stability window independently of coupling anisotropy, while $r$ steps down to the 2D limit above the empirical crossover scale $α^*\approx0.7$. Inner Binder cumulant pairs, which converge fastest to the thermodynamic limit, resolve $r(1.0)\approx0.412$ and a step $Δr=0.038\pm0.015$ from the quasi-1D plateau. A four-point linear fit $r(α)=0.494-0.063\,α$ summarises both regimes; its $α\to0$ intercept recovers the exact 1D result of Pfeuty within 1.7 standard deviations, and its slope is a lower bound on the true crossover amplitude concentrated in $α\in[α^*,1]$. Two sequential blind predictions, confirmed at $0.2σ$ and $0.7σ$ before each measurement, validate the crossover law. All four geometries show a direct ferromagnet-to-paramagnet transition, complete quantum ergodicity ($f_{\rm uniq}=1.000$), and null valence-bond solid order.

Figures (6)

• Figure 1: Classical spinwave spectra and quantum-only windows. Each row shows $J(\mathbf{k})-J(\mathbf{0})$ along $\Gamma\to K\to X\to\Gamma$ (left) and the instability diagnostic $J(\mathbf{k}_*)-J(\mathbf{0})=6\alpha-9g$ vs $g$ (right). Top ($\alpha=1$): $g_c^{\mathrm{class}}=2/3$, $\Delta g=0.381$. Middle ($\alpha=0.7$, NiNb$_2$O$_6$): $g_c^{\mathrm{class}}=0.467$, $\Delta g=0.257$. Third ($\alpha=0.5$, BaCo$_2$V$_2$O$_8$): $g_c^{\mathrm{class}}=1/3$, $\Delta g=0.177$. Bottom ($\alpha=0.3$, FeNb$_2$O$_6$): $g_c^{\mathrm{class}}=0.200$, $\Delta g=0.107$. In all four cases the measured $g_c^{\mathrm{QPU}}$ (solid vertical line) lies within the classically stable FM regime, demonstrating that the transitions are quantum-driven.
• Figure 2: Dimensional crossover of the quantum suppression ratio.Left: $r(\alpha)=g_c^{\mathrm{QPU}}(\alpha)/g_c^{\mathrm{class}}(\alpha)$ vs lattice anisotropy $\alpha=J_1'/J_1$. Filled circles: QPU measurements; error bars from $g_c^{\mathrm{QPU}}$ uncertainties. Solid line: weighted linear fit $r(\alpha)=0.494-0.063\,\alpha$ ($1.9\sigma$); shaded band: $\pm1\sigma$. Horizontal dashed line: universal plateau $\bar{r}=0.450\pm0.002$ ($\chi^2/\mathrm{dof}=1.10$) confirmed across three quasi-1D geometries. Vertical dotted line: empirical crossover scale $\alpha^*\approx0.7$. Open circle at $\alpha=1$: best estimate from inner Binder pairs ($r\approx0.412$, $2.5\sigma$ below plateau); filled circle: conservative weighted mean ($r=0.428\pm0.018$). Filled star at $\alpha=0$: exact 1D TFIM critical point ($r^{\rm 1D}=1/2$, Pfeuty Pfeuty1970), consistent with the $\alpha\to0$ extrapolation within 1.7 standard deviations. Boxed points: blind predictions confirmed at 0.2 and 0.7 standard deviations (red: $\alpha=0.5$; orange: $\alpha=0.3$). Right: quantum-only windows $\Delta g(\alpha)=g_c^{\mathrm{class}}(\alpha)-g_c^{\mathrm{QPU}}(\alpha)$. Solid curve: Eq. \ref{['eq:prediction']}. Dashed curve: $g_c^{\mathrm{class}}(\alpha)=2\alpha/3$. The ratio $\Delta g/g_c^{\mathrm{class}}=1-r(\alpha)$ increases as $\alpha$ decreases, quantifying the growing role of quantum fluctuations as the lattice becomes quasi-one-dimensional.
• Figure 3: Three independent estimates of $g_c$ at $t_a=500\,\mu\mathrm{s}$. Each row: (left) pairwise $U_{4,\sqrt{3}}$ crossings vs $L_a\times L_b$; (centre) $U_{4,\sqrt{3}}$ vs $1/L$ with $L\to\infty$ extrapolation; (right) zoom of the critical region. Top ($\alpha=1.0$): inner pairs $(18,21)$ and $(21,24)$ both cross at $g\approx0.275$; outer $(24,27)$ pair crosses at $0.300$ due to non-monotonic finite-size drift; weighted mean $g_c^{\mathrm{QPU}}=0.286\pm0.012$. Middle ($\alpha=0.7$): three inner pairs span $\delta g_c=0.003$; weighted mean $0.210\pm0.001$. Third ($\alpha=0.5$): inner-pair crossing at $0.156$; sign change between $g=0.154$ and $0.158$. Bottom ($\alpha=0.3$): pairwise $U_4$ crossings give weighted mean $0.093\pm0.005$; $\chi_{\rm FM}$-peak extrapolation $0.095$--$0.098$. All three methods agree within $\pm0.006$ for $\alpha\geq0.5$ and $\pm0.008$ for $\alpha=0.3$.
• Figure 4: Quantum phase boundary $g_c^{\mathrm{QPU}}(t_a)$. Each row: (left) pairwise Binder cumulant crossings vs annealing time $t_a\propto1/\Gamma_{\mathrm{eff}}$ ($U_{4,\sqrt{3}}$ for $\alpha\geq0.5$; $U_4$ for $\alpha=0.3$); (centre) Binder cumulant for $L\in\{24,27\}$ across all 14 $t_a$ values; (right) $(g,\Gamma_{\mathrm{eff}})$ phase diagram. In all four geometries $g_c^{\mathrm{QPU}}(t_a)$ converges to a $t_a$-independent value for $t_a\gtrsim20\,\mu\mathrm{s}$, confirming the critical points are intrinsic to the model Hamiltonian and not artefacts of the annealing protocol. Top: $\Delta g=0.381$. Middle: $\Delta g=0.257$. Third: $\Delta g=0.177$. Bottom ($\alpha=0.3$): $\Delta g=0.107$. The ratio $\Delta g/g_c^{\mathrm{class}}=1-r(\alpha)$ increases with decreasing $\alpha$, confirming the dimensional crossover.
• Figure 5: Direct FM-to-paramagnet transition across all four geometries. Each row: (left) sublattice asymmetry $||m_A|-|m_B||$ vs $g$ at $L=18$, confirming three-sublattice symmetry throughout; (centre) sublattice magnetisations $m_A$, $m_B$, $m_C$ vs $g$ at $L=27$, $t_a=500\,\mu\mathrm{s}$, all three vanishing simultaneously at $g_c^{\mathrm{QPU}}(\alpha)$; (right) sublattice magnetisations vs $1/L$ at $g=g_c^{\mathrm{QPU}}(\alpha)$, $t_a=5\,\mathrm{ns}$, consistent with zero in the thermodynamic limit. No relative sublattice splitting at any $g$ or $t_a$ rules out an intermediate $\sqrt{3}\!\times\!\sqrt{3}$ phase. Top: $\alpha=1.0$. Second: $\alpha=0.7$. Third: $\alpha=0.5$. Bottom: $\alpha=0.3$.