Magic phase transitions in monitored gaussian fermions

TL;DR

This work investigates how continuous monitoring impacts non-stabilizerness (magic) in Gaussian fermionic systems by computing Stabilizer Rényi Entropies (SREs) with scalable Majorana sampling. It demonstrates that the leading volume-law scaling of magic, $M_\alpha\sim a_\alpha L$, is robust under monitoring, while the subleading logarithmic corrections, $-b_\alpha\log L$, undergo sharp transitions at critical measurement rates in both hopping fermions and the Ising chain. These monitor-induced transitions in complexity are invisible to standard entanglement diagnostics and highlight how integrability, initial conditions, and measurement strength shape quantum resource generation. The results point to the power of magic-based diagnostics for uncovering hidden dynamical features in monitored many-body systems and suggest avenues for analytical understanding and extensions to interacting or non-Gaussian settings.

Abstract

Monitored quantum systems, where unitary dynamics compete with continuous measurements, exhibit dynamical transitions as the measurement rate is varied. These reflect abrupt changes in the structure of the evolving wavefunction, captured by complementary complexity diagnostics that include and go beyond entanglement aspects. Here, we investigate how monitoring affects magic state resources, the nonstabilizerness, of Gaussian fermionic systems. Using scalable Majorana sampling techniques, we track the evolution of stabilizer Rényi entropies in large systems under projective measurements. While the leading extensive (volume-law) scaling of magic remains robust across all measurement rates, we uncover a sharp transition in the subleading logarithmic corrections. This measurement-induced complexity transition, invisible to standard entanglement probes, highlights the power of magic-based diagnostics in revealing hidden features of monitored many-body dynamics.

Figures (16)

• Figure 1: Left panels. Plain time evolution (no measurement, i.e. $\gamma=0$) under the hopping Hamiltonian in Eq. (\ref{['eq:H_XX']}) of the SRE $M_1$ after having initialized the system into the Néel state with particle density $n = N/L = 1/2$ (top) and $n = N/L = 1/4$ (bottom). Each point has been obtained by averaging over $\mathcal{S}=1000$ Pauli strings. Gray horizontal lines correspond to $L\log(2)$. Right panels. Extensive behavior of the stationary stabilizer entropies averaged over a time window $\Delta t = [L/8,L/4]$ for $n=1/2$, and $\Delta t = [L/4,L/2]$ for $n=1/4$. The dashed line corresponds to $L \log(2)$ while, for $n=1/4$ also $\log\binom{L}{L/4}$ is drawn (dot-dashed line).
• Figure 2: Top panels. Late-time stabilizer Rényi entropies after a plain dynamics in the hopping fermions for $n = 1/2$ and $n = 1/4$, after subtracting the (non-symmetric) Haar average $L \log(2)$. Bottom panels. Logarithmic corrections to the non-stabilizerness, extracted via finite-size analysis (see main text for details).
• Figure 3: Log-linear plot of the relaxation dynamics of the SREs after having subtracted the best-fit finite-size stationary values $\overline{M_{\alpha}(L)} \sim a_{\alpha}L - b_{\alpha} \log L - c_{\alpha}$; dashed lines are size-dependent exponential decay drown as guide for the eyes.
• Figure 4: Time evolution of the stabilizer Rényi entropy density approaching its thermodynamic limit for quenches with the Hopping Hamiltonian. The log-log plot highlights the thermodynamic scaling regime for times $t < t^*(L)$, where finite-size effects are negligible. The dashed line indicates the algebraic decay observed in this regime, as discussed in the main text.
• Figure 5: Time evolution of the stabilizer Rényi entropy densities $M_1/L$ (left panel) and $M_2/L$ (right panel) under unitary free-fermion dynamics interspersed with local projective measurements performed at various rates $\gamma$. The system is initialized in the Néel state at half filling ($n = 1/2$), and averages are taken over $N_{\mathrm{traj}} = 500$ independent quantum trajectories. The rescaled entropies exhibit an overall extensive behavior, with finite-size corrections becoming increasingly suppressed as the measurement rate $\gamma$ increases.