Quench dynamics of entanglement entropy under projective charge measurements: the free fermion case

TL;DR

This paper analyzes how periodic projective measurements of a subsystem particle number alter the quench-induced entanglement growth in a 1D free-fermion chain. Using an operatorial quasiparticle picture, it derives analytic expressions showing two types of measurement-induced corrections: a classical, outcome-independent term that scales with the charge-variance variance and a quantum, outcome-dependent term that can be sizable for specific outcomes but vanishes upon averaging. The framework recovers known results on symmetry-resolved entanglement and full counting statistics in relevant limits and is validated against exact results for the Néel state. Across single and multiple measurements and for both symmetric and symmetry-breaking initial states, the findings highlight that meaningful deviations from unitary dynamics require conditioning on measurement outcomes; averaging over outcomes largely washes out quantum corrections, leaving a dominant classical correction and suggesting routes to extend the approach to interacting regimes.

Abstract

We consider the effect of projective measurements on the quench dynamics of the bipartite entanglement entropy in one dimensional free fermionic systems. In our protocol, we consider projective measurements of a $U(1)$ conserved charge, the particle number, on some large subsystem, and study the entanglement entropies between the same subsystem and its complement. We compare the dynamics emanating from two classes of initial states, one which is an eigenstate of the charge and another which is not. Moreover, we consider the effects of a single measurement as well as multiple which are periodically performed. Using the quasiparticle picture, we obtain analytic expressions for the behaviour of the entanglement which admit a transparent physical interpretation. In general, we find that measurements introduce two distinct types of corrections to the entanglement, which can be interpreted separately as classical and quantum contributions. The classical contribution is independent of the measurement outcome and scales logarithmically with variance of the charge distribution. In contrast, the quantum contribution depends on the specific measurement outcome and can be significant for individual realizations; however, it becomes negligible when averaged over all possible outcomes. Our expressions reduce to previously known results for symmetry resolved entanglement and full counting statistics in some relevant limits, and are confirmed by an exact calculation performed on the Néel initial state.

Figures (5)

• Figure 1: Quasiparticle picture of quench dynamics in free fermions. In the picture, the quench acts on the initial state by producing pairs of entangled quasiparticles, relating, therefore, entanglement propagation to transport. At each instant of time, dashed lines correspond to pairs which are not shared between $A$ and $\overline{A}$ and therefore do not contribute to the entropy. Solid lines in contrast correspond to shared pairs, which contribute to the entanglement.
• Figure 2: Measurement protocol \ref{['eq:full_state_multiple']}. The system undergoes a non-equilibrium free evolution, from a non-equilibrium initial state. After a time step $\tau$, it is projected within the system $A$ to an eigenstate of the $U(1)$ charge. The unitary evolution-projection is then repeated an arbitrary number of times; after the $m$-th step, at some time t, the entanglement entropy of the interval $A$ is then computed.
• Figure 3: $S_A(t|\tau,q)$ in the Néel state for several values of $\Delta q/\tau$, at fixed $\tau=\ell/3$, and varying time. Immediately after the measurement, the entropy is decreased and the difference with respect to $S_A(t)$ remains constant up to $t=\ell/2$. Successively, all curves tend to the same stationary value. Since in \ref{['eq:neelcorrection']} the entropic contributions are momentum-independent, the approach to the stationary value is dictated for any $q$ by the integral of the counting function, implying that the ratios of the distances between the curves remains constant in time.
• Figure 4: Comparison between the exact solution for the average correction to the entanglement entropy, $\langle\delta S_A(t|\tau,q)\rangle =\langle S_A(t|\tau,q)\rangle-S_A(t)$ of the Néel state, obtained from the exact solution discussed in appendix \ref{['appB']}, and the average of the quasiparticle/saddle point prediction \ref{['eq:correctionneel']}. The saddle point result is accurate, exhibiting a logarithmic correction in the time of the measurement.
• Figure 5: Behaviour of $S_A(t|\tau,q)$ for several values of $\Delta q/\tau$, at fixed $\tau=\ell/4$, and varying time. We see that immediately after the measurement the entropy is decreased and the difference with respect to $S_A(t)$ remains constant up to $t=\ell/2$. At this point the structure of the counting function becomes more complex, and this leads to a faster increase in the entropy of the measured state. In particular, this surpasses both the unitary value and the asymptotic $t\to \infty$ value, which is then approached from above, as shown in the inset. Note that the deviations from the unitary value appear quite significant for $t<\ell/2$; it is however necessary to consider that the variance of the probability distribution is only $\sigma^2 =\frac{2\tau}{3\pi}$, therefore all values shown in the plot are highly unlikely. In particular, the logarithmic correction is truly negligible for such values.