Competing interlayer charge order and quantum monopole reorganisation in bilayer kagome spin ice via quantum annealing

Abstract

Magnetic monopoles in frustrated magnets are paradigmatic fractionalised quasiparticles, yet no experimental platform simultaneously tunes their confinement and preserves ice-rule physics. Here we exploit the native bilayer architecture of a D-Wave Advantage2 quantum annealer to realise the first programmable two-plane kagome spin ice, spanning $1{,}536$ logical spins across a $4\times13\times14$ grid of system size, interlayer coupling, and quantum drive. We find that interlayer exchange drives a sharp transition from ferroelectric to antiferroelectric staggered charge order, an Ice-II phase with no classical or single-layer analogue, with a critical onset at $(J_{\perp}/J_1)^{*} \approx 0.044$ that is stable across five decades of annealing time. Restricting the charge structure factor to ice-rule plaquettes reveals an order-of-magnitude enhancement over conventional all-plaquette estimators, demonstrating that quantum-selected charge order is invisible to defect-diluted probes and establishing a methodological standard for future quantum spin ice experiments. The quantum renormalisation of the monopole chemical potential sets a concrete engineering target for the transmon circuit-QED kagome ice required to enter the monopole deconfinement regime. Three falsifiable predictions follow for existing Ni$_{81}$Fe$_{19}$ nanowire bilayer architectures: a critical interlayer separation, an elevated monopole activation temperature, and an order-of-magnitude enhancement of the Ice-II signal in published X-ray datasets, all testable without new fabrication.

Figures (11)

• Figure 1: Local patch of bilayer kagome spin ice as implemented on the D-Wave Advantage2 (Zephyr Z15) processor. Layer 1 (navy, Zephyr qubit orientation group 1) and Layer 2 (green, group 2) form two coupled kagome planes with intralayer nearest-neighbour coupling $J_1>0$ (solid bonds) and interlayer coupling $J_\perp$ (dashed bonds). Each triangular plaquette carries a monopole charge $Q_m(\triangle)=\frac{1}{2}\sum_{i\in\triangle}\sigma_i^z$; plaquettes satisfying the kagome ice rule ($|Q_m|=\frac{1}{2}$, blue shading) correspond to two-in-one-out or one-in-two-out configurations, while all-in or all-out configurations are monopole defects ($|Q_m|=\frac{3}{2}$, red shading). Sublattice labels A, B, C identify the three-site basis of the front unit cell. Red (blue) arrows denote spin-up (spin-down) states $\sigma^z=\pm1$. The interlayer staggered correlator $C_s^\perp<0$ (Eq. \ref{['eq:csperp']}) signals the antiferroelectric Ice-II phase reported in this work, in which the two layers develop opposite staggered charge order, a configuration with no single-layer analogue.
• Figure 2: Composite bilayer kagome results at $N=768$/layer. (a) Monopole density $\rho_m$ vs. quantum-drive proxy $\Gamma_{\mathrm{eff}}$ for all $J_{\perp}/J_1$ values; error bars are binomial standard errors. (b) Effective chemical-potential proxy $\mu_{\rm mon}^{\mathrm{eff}}/J_1$ [Eq. \ref{['eq:mueff_exp']}]; classical reference $2J_1$ (dotted) and deconfinement threshold $\mu_c=0.808\,J_1$ (dashed red) are indicated. The persistent gap above $\mu_c$ is a quantitative measure of the hardware distance from deconfinement. (c) Monopole pair correlation $G(r)$ at three representative annealing times ($J_{\perp}/J_1=0.5$); exponential decay confirms the confined phase at all conditions. (d) All-plaquette charge structure factor $S_Q(\mathbf{k}^{*})$ vs. system size $N$ at $t_a=5\,\mathrm{ns}$; the absence of $N$-dependence contrasts with the strong ice-manifold signal in Fig. \ref{['fig:sqcomp']} and is explained by defect dilution. (e) Interlayer monopole correlator $C_m^\perp$ vs. $J_{\perp}/J_1$; the sign change is a bilayer-specific result. (f) Phase diagram of $\rho_m$ in $(\log_{10}\Gamma_{\mathrm{eff}},\,J_{\perp}/J_1)$ space at $N=768$, confirming orthogonal control of the two experimental axes.
• Figure 3: Finite-size study of monopole density $\rho_m(\Gamma_{\mathrm{eff}})$ and phase-boundary trend. Left: Raw $\rho_m(\Gamma_{\mathrm{eff}})$ for all four system sizes at $J_{\perp}/J_1=0.5$; larger systems exhibit sharper crossovers. Centre: Scaling collapse with 2D Coulomb gas exponents ($\nu=0.5$, $\beta=0.125$), locating the crossover region at $\Gamma_{\mathrm{eff}}\sim1$--$10$ and quantifying the additional system sizes required for a definitive collapse. Right: Phase-boundary trend; crossover location $\Gamma_{\mathrm{eff}}^*$ vs. $J_{\perp}/J_1$, extracted by inflection point and variance peak. The monotone shift provides a quantitative design rule for the transverse field required for equivalent monopole activation as a function of interlayer coupling.
• Figure 4: Interlayer staggered correlator $C_s^\perp$ [Eq. \ref{['eq:csperp']}] at $N=768$. Left:$C_s^\perp$ vs. $J_{\perp}/J_1$ at four representative annealing times. The reversal from weakly ferroelectric ($C_s^\perp>0$) to strongly antiferroelectric ($C_s^\perp\approx-0.40$ to $-0.44$) is the primary result of this work and has no single-layer analogue. Right:$C_s^\perp$ vs. $\Gamma_{\mathrm{eff}}$ for all $J_{\perp}$ values, showing that the sign transition is stable over five decades of annealing time and is a ground-state property of the bilayer exchange; only the fastest $5\,\mathrm{ns}$ point shows a noticeable non-adiabatic deviation.
• Figure 5: Staggered charge structure factor $S_Q^{\mathrm{stag}}(\mathbf{k}^{*})$ [Eq. \ref{['eq:SQstag']}]. Left:$S_Q^{\mathrm{stag}}$ vs. system size $N$ at $J_{\perp}/J_1=0.5$, showing $N$-dependent growth at fast anneals (quantum regime) absent at slow anneals (classical regime), a finite-size signature of quantum-driven Ice-II order selection. Right:$S_Q^{\mathrm{stag}}(\mathbf{k}^{*})$ (ice-rule manifold) vs. all-plaquette $S_Q(\mathbf{k}^{*})$ for $N=768$, $J_{\perp}/J_1=0.5$, as a function of annealing time $t_a$. The order-of-magnitude enhancement ($9.8$--$13.9\times$, typical value $\approx12$) demonstrates that conventional all-plaquette probes systematically underestimate Ice-II charge order when monopole density is non-negligible.