Topological frustration and quantum resources

TL;DR

This work investigates topological frustration (TF) in quantum spin chains by analyzing how frustrated boundary conditions induce a delocalized topological excitation that yields a $2N$-fold classical ground-state degeneracy lifting into a gapless low-energy band. By examining entanglement entropy (EE) and disconnected entropy (DEE), as well as non-stabilizerness through Stabiliser Rényi entropies (SRE), the authors demonstrate a universal two-term decomposition of quantum resources: the TF contribution adds a distinct, phase-robust term atop the non-frustrated baseline, anchored to the delocalized kink-like excitation that resembles a $W$-state. They further show that TF manifests a detectable jump in SRE at a critical field $h^{*}$ in the XYZ chain, revealing a quantum phase transition not captured by conventional order parameters, while the half-chain entanglement remains continuous. The findings highlight TF as a practical diagnostic for nonlocal quantum correlations and resourcefulness, with implications for quantum technologies (e.g., quantum batteries) and potential extensions to higher dimensions and experimental platforms.

Abstract

Although in general boundary conditions do not affect the bulk properties of a system, some of them are special and defy such expectation. This is the case, for instance, of those inducing geometrical frustration in a classical magnet. Recently, the study of such settings in quantum systems (dubbed topological frustration) has uncovered peculiar features, interesting both from a fundamental and technological point of view. In this work, we present and discuss the behavior of several quantum resources in presence of TF, namely the (disconnected) entanglement entropy and the non-stabilizerness Renyi entropy. We will show that, compared to their non-frustrated counterparts, TF adds a distinct contribution to these resources, due to a stable, delocalized, topological excitation. Remarkably, this contribution can be calculated analytically, due to its similarities with that of a W-state.

Figures (7)

• Figure 1: (Left) Illustration of geometric frustration through simple geometry. The square lattice admits a Néel-ordered configuration in which all antiferromagnetic bonds are simultaneously satisfied and is therefore unfrustrated. In contrast, the triangular plaquette does not allow all AFM interactions to be minimized at the same time: assigning alternating spins on two vertices inevitably forces the third one to violate one AFM bond. (Right) Example of frustration induced by competing interactions. Here, a next-to-nearest AFM coupling favours arrangements incompatible with the nearest-neighbor interaction, regardless of whether it is FM or AFM. Since the two interactions cannot be simultaneously satisfied, the system becomes frustrated.
• Figure 2: The mechanism underlying topological frustration can be understood through the following steps: (a) impose AFM couplings, which naturally favour a two-site periodic Néel configuration; (b) disrupt this periodic pattern by selecting an odd number of spins, thereby preventing the Néel order; (c) finally, close the system into a ring by enforcing PBCs, which forces the incompatibility introduced in (b) to extend throughout the entire system and gives rise to topological frustration.
• Figure 3: Schematic representation of two different entanglement partitions. (Left) Bipartition of the system for the entanglement entropy. The system is divided into two complementary subsystems $A$ made of $M=m N$ spins and $B$ made of $N-M=N(1-m)$ spins. (Right) Disconnected entanglement entropy, defined between two disjoint subsystems $A$ and $B$ embedded in a larger system, which captures non-local quantum correlations beyond the area law contribution. The lengths $n$, $m$, and $l$ are normalized with respect to the length of the system.
• Figure 4: Disconnected $2$-Rényi entanglement entropy in the TF phase. (Left) Normalized $S^{D}_{2}(m,l)$ with respect to the value in Eq. \ref{['DEE_near_zero']}, plotted as a function of the magnetic field $h$ for different system sizes. The partition lengths are fixed to $m = (N-1)/2$, $l = (N-1)/8$, and $r =(N-1)/4$ (see Fig. \ref{['fig:EE_partitions']}). (Right) Scaling of $\Delta S^D_2 (N) = \tilde{S}^D_2 (1/2,1/8) - S^D_2 (N, 1/2, 1/8, 1/4)$ as a function of the chain length $N$. In the thermodynamic limit, $\Delta S_2^D$ vanishes as a power law, $\Delta S^{(2)}_{\mathrm{D}} = a N^{b}$, with exponent $b \simeq -0.935 \pm 0.006$, independently of $h$.
• Figure 5: Pictorial representation of the Clifford circuit $\hat{\mathcal{S}}$ in Eq. \ref{['cliffordcircuit']} for $5$ qubits. The boxes labeled $H$ and $Z$ denote the Hadamard gate and the $\sigma^{z}$ operator acting on the corresponding qubit. CNOT gates are depicted as lines connecting a filled black dot, indicating the control qubit, to a circle marking the target qubit.