The toric code under antiferromagnetic isotropic Heisenberg interactions

Abstract

We investigate the impact of an isotropic antiferromagnetic Heisenberg perturbation on the toric code, focusing on the resulting quantum phase transition and the nature of the phase that emerges beyond topological order. Using neural-network quantum states (NQS), we compute ground states over a wide range of Heisenberg couplings while fully respecting the exact symmetries of the model. In the weak-coupling regime, the numerical results are in excellent agreement with an effective low-energy description derived from a Schrieffer-Wolff (SW) transformation, providing analytic control over the perturbative breakdown of topological order. We show that the Heisenberg perturbation only renormalizes local operators at low orders, whereas mixing between topological sectors occurs only at a perturbative order proportional to the system size. At intermediate values of the Heisenberg interaction, the topological phase breaks down. We estimate the critical point through a combination of the fidelity susceptibility and the logarithmic susceptibility of non-contractible Wilson loops for various system sizes. Furthermore, we utilize the topological entanglement entropy to provide a comprehensive characterization of the phase transition. Beyond the transition, an antiferromagnetic $\pm X/\pm Z$ Néel phase emerges, characterized by a fourfold-degenerate symmetry-broken manifold, which is explicitly probed using staggered-magnetization-based diagnostics. Our results show how local two-spin interactions, which naturally arise in realistic implementations of the toric code, drive the breakdown of topological order. Moreover, we establish the SW approach as a systematic framework for analyzing such perturbations in combination with variational many-body methods.

Figures (14)

• Figure 1: Structure of the toric-code lattice. (a) The four red sites indicate the star operator $A_v=\prod_{e\in +_v}\sigma^x_e$; the four blue sites indicate the plaquette operator $B_p=\prod_{e\in \partial p}\sigma^z_e$. The four purple sites show the nearest neighbors of the green site used in the Heisenberg interaction. Moreover, the gray dashed lines describe the dual lattice. We show examples of non-contractible Wilson loops under Periodic Boundary Condition, where $W_z^{(v)}$ and $W_z^{(h)}$ are products of $\sigma^z$ along the light-blue vertical solid and horizontal dashed lines. In the same manner, the pink solid vertical and horizontal dashed lines represent the $\sigma^x$ loops $W_x^{(v)}$ and $W_x^{(h)}$kitaev_toric_code. (b) The Marshall-gauge pattern used in this work, where gray sites belong to the $A$ sublattice (flipped by the Marshall transformation) and black sites belong to the $B$ sublattice. We specifically use $L_x=L_y=L$
• Figure 2: Ground state energy per spin $E_0/N$ (with $N=2L^2$) versus the AFM Heisenberg coupling $J$ for $L=3,4,5,6$. Symbols show NQS results and error bars represent stochastic uncertainty; for $L=3$ we also plot the exact-diagonalization (ED) benchmark. The dashed curve is the SW prediction truncated at $\mathcal{O}(J^2)$, see Eq. \ref{['energyperspin']}. Note that this approximation coincides with the result up to third order for $L \geq 4$; see Appendix \ref{['sec:methods_sw_third']}. In the perturbative regime $J\lesssim 0.1$, SW is in quantitative agreement with NQS/ED. For $J\gtrsim 0.13$--$0.16$, systematic deviations from the $\mathcal{O}(J^2)$ curve appear and the $L$-dependence increases, consistent with the breakdown of low-order perturbation theory and the approach to the critical region.
• Figure 3: Fidelity susceptibility $\chi_F$ obtained from NQS for $L=3,4,5,6$ and from ED for $L=3$. Prominent peaks indicate a phase crossover in this range of $J$. The peak height increases with system size $L$, while the peak position exhibits a slight finite-size drift.
• Figure 4: Finite-size scaling analysis of the fidelity-susceptibility peak. (a) Log--log plot of the peak height, $\chi_F^{\mathrm{peak}}(L)$, versus $L$. From a linear fit, we obtain the critical exponent from $2/\nu = 2.85 \pm 0.54$. (b) Extrapolated critical coupling $J_c(\infty)$, obtained from the drift of the peak position by fitting $J_{\mathrm{peak}}(L)=J_c(\infty)+a/L^{1/\nu}$, shown as a function of the assumed shift exponent $1/\nu$. we obtained $a=-0.086$ in our fitting. The vertical dashed lines indicate the value of $1/\nu$ from (a), and the horizontal dashed line and band marks the corresponding $J_c = 0.1641^{+0.0034}_{-0.0024}$. The blue vertical band indicates the uncertainty range of $1/\nu$ from (a).
• Figure 5: Logical-qubit diagnostics from non-contractible Wilson loops for odd $L$. Shown are the parity-even double-winding correlators $\langle W_x^{(h)}W_x^{(v)}\rangle_{\rm phys}$ and $\langle W_z^{(h)}W_z^{(v)}\rangle_{\rm phys}$ for (a) $L=3$ and (b) $L=5$. Their deviation from the ideal toric-code values directly quantifies correlated logical errors induced by the isotropic Heisenberg interaction. Markers show NQS results for $L=3,5$, including the sampling/optimization uncertainty, as well as ED results for $L=3$. Dashed curves correspond to the $\mathcal{O}(J^2)$ SW dressing of the logical correlators; good agreement at small $J$ validates the perturbative error model, while the rapid change and strong deviations near the transition region indicate the onset of the breakdown of the topological protection as the system crosses into the Néel-ordered phase.