Fermionic magic resources in disordered quantum spin chains

TL;DR

This work uses fermionic antiflatness (FAF) to quantify fermionic non-Gaussianity as a resource in disordered spin chains, linking it to the ergodic–MBL transition. Through exact diagonalization and covariance-matrix analysis, FAF is shown to be near maximal in ergodic regimes and suppressed in deep MBL, with the scaling governed by interaction extent and disorder: $\mathcal{F}_1 \propto \Delta N_{\rm int}/W^{2}$. Rare cat-like resonances can dramatically boost FAF, tying nonperturbative MBL instability mechanisms to increased non-Gaussianity. Time evolution from product states reveals slow, power-law growth of FAF in MBL, saturating to an extensive plateau, consistent with LIOM dynamics. Overall, FAF provides a robust diagnostic of fermionic complexity beyond free-fermion descriptions and motivates exploring FAF in other nonergodic phenomena.

Abstract

Fermionic non-Gaussianity quantifies a quantum state's deviation from a classically tractable free-fermionic description, constituting a necessary resource for computational quantum advantage. Here we use fermionic antiflatness (FAF) to measure this deviation across ergodic and many-body localized (MBL) regimes. We focus on the paradigmatic disordered spin-$1\!/2$ XXZ chain and its impurity variant with local interactions. Across highly excited eigenstates, FAF evolves from typical-state behavior at weak disorder to strongly suppressed values deep in the MBL regime, with volume-law scaling in the XXZ chain and an area-law bound in the impurity setting. Rare long range catlike eigenstates exhibit a pronounced enhancement of FAF, making it a sensitive diagnostic of mechanisms proposed to destabilize MBL. Starting from product states, we find that in the MBL regime FAF grows slowly in time, approaching saturation via a power-law relaxation. Overall, our results show that MBL suppresses fermionic non-Gaussianity, and the associated complexity beyond free fermions, while ergodicity restores it, motivating explorations of fermionic non-Gaussianity in other ergodicity-breaking phenomena.

Figures (10)

• Figure 1: (a) The disordered XX chain is Anderson localized and fermionic Gaussian, hence $\mathcal{F}_k=0$. Adding interactions $\Delta\ne 0$ gives (b) an impurity model and (c) the XXZ chain, featuring ergodic and MBL regimes with non-Gaussian eigenstates and $\mathcal{F}_k>0$, decaying with disorder strength $W$ as $\mathcal{F}_k\propto W^{-2}$.
• Figure 2: The FAF density $\langle f_1\rangle$ across the ergodic--MBL crossover in (a) the XXZ chain and (b) the impurity model ($\Delta=1$). At weak disorder $W$, $\langle f_1\rangle$ approaches its typical-state value; upon entering MBL it decreases as $\langle f_1\rangle \propto N_{\mathrm{int} } W^{-2}$ [Eq. \ref{['Eq:F1_Scaling']}], with a subleading $W^{-4}$ correction (red dashed lines). Inset: $\langle \mathcal{F}_1\rangle$ as a function of $L$ in the impurity model for fixed $W$, highlighting the volume-law and area-law scalings of FAF.
• Figure 3: (a) The FAF density $\langle f_1\rangle$ as a function of interaction strength $\Delta$ in XXZ model for different disorder strengths $W$ and system sizes $L$. The dashed lines show fits $\braket{f_1}=a\Delta + b\Delta^2$, with $b\Delta^2$ being a small, sub-leading correction; (b) Expectation values $\langle \hat{\sigma}^{z}_i\rangle$ in two long-range cat-like eigenstates of $\hat{\mathcal{H}}_{\rm xxz}$ for $L=18$ and $W=14$, and the corresponding $\mathcal{F}_1$ values.
• Figure 4: Time dynamics of $\langle \mathcal{F}_1\rangle$ for the (a) XXZ and (b) impurity model in the MBL regime, with interaction strength set as $\Delta=1$. The evolution of $\langle \mathcal{F}_1\rangle$ follows the prediction in Eq. \ref{['TD_MBL']}, as shown by the red-dashed lines; in the impurity model the second-order term $t^{-2\beta}$ is non-negligible.
• Figure 5: (a-b) Time-evolution of Majorana correlators $|\langle \gamma_m \gamma_n\rangle|^2$ in the MBL regime for the XXZ Hamiltonian (blue) and the $\ell$-bit model (red) for $L=16$ and $W=12$ (for the XXZ model). In the noninteracting case ($\Delta=\lambda=0$, light colors), the correlators rapidly saturate to a constant, reflecting trivial dynamics. In contrast, in the interacting case ($\Delta=\lambda=1$, dark colors), spin dephasing induces a slow power-law decay prior to saturation. (c) Dynamics of $\langle f_1\rangle$ for the $\ell$-bit Hamiltonian at $\lambda=1$ for several system sizes $L$. The dynamics display a clear power-law behavior (red-dashed line), consistent with Eq. \ref{['Eq:Proof_TD']}. Inset: Time-evolution of $\delta f_1={f}_1^{\mathrm{MBL}} -\langle f_1\rangle$, highlighting the same power-law behavior.