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arXiv:2511.06706

Dynamics of number entropy for free fermionic systems in presence of defects and stochastic processes

Atharva Naik, Bijay Kumar Agarwalla, Manas Kulkarni

quant-phcond-mat.quant-gascond-mat.stat-mechSubmitted Nov 10, 2025• Announced Nov 11, 2025• 0 citations
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Abstract

We investigate the dynamics of number entropy in a chain of free fermions subjected to both defects and stochastic processes. For a special class of defects, namely conformal defects, we present analytical and numerical results for the temporal growth of number entropy, the time evolution of the number distribution, and the eigenvalue profile of the associated correlation matrix within a subsystem. We show that the number entropy exhibits logarithmic growth in time, originating from the Gaussian structure of the number distribution. We find that the eigenvalue dynamics reveal a profound connection to the reflection and transmission coefficients of the associated scattering problem for a broad range of defects. When stochastic processes are introduced, specifically Stochastic Unitary Processes (SUP) and Quantum State Diffusion (QSD), the number entropy scales as $\ln(t)$ in the SUP case and shows strong hints of $\ln [\ln(t)]$ scaling in the QSD case. These findings establish compelling evidence that number entropy grows logarithmically slower than the corresponding von Neumann entanglement entropy across a wide class of systems.

Comments
7 pages , 4 figures + Supplemental Material

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We investigate the dynamics of number entropy in a chain of free fermions subjected to both defects and stochastic processes. For a special class of defects, namely conformal defects, we present analytical and numerical results for the temporal growth of number entropy, the time evolution of the number distribution, and the eigenvalue profile of the associated correlation matrix within a subsystem. We show that the number entropy exhibits logarithmic growth in time, originating from the Gaussian structure of the number distribution. We find that the eigenvalue dynamics reveal a profound connection to the reflection and transmission coefficients of the associated scattering problem for a broad range of defects. When stochastic processes are introduced, specifically Stochastic Unitary Processes (SUP) and Quantum State Diffusion (QSD), the number entropy scales as ln⁡(t)\ln(t)ln(t) in the SUP case and shows strong hints of ln⁡[ln⁡(t)]\ln [\ln(t)]ln[ln(t)] scaling in the QSD case. These findings establish compelling evidence that number entropy grows logarithmically slower than the corresponding von Neumann entanglement entropy across a wide class of systems.
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
Free fermion models with an operator entanglement growing as O(t)O(\sqrt{t})O(t​)
We study the time evolution of the Operator Space Entanglement Entropy (OSEE) of one-dimensional number-conserving free fermions subject to a random space- and time-dependent evolution. It is known that, in free-fermion systems without randomness, the OSEE of \emph{local} operators grows at most \emph{logarithmically} in time. Here we show that, in the presence of randomness, the OSEE of local spin operators involving a Jordan-Wigner string grows as O(t)\mathcal O(\sqrt t)O(t​), for all values of the Rényi index. We establish this in two models: a Brownian Hamiltonian and a brick-wall matchgate circuit. We also consider the OSEE of the evolution operator, and we show that it also grows as O(t)\mathcal O(\sqrt{t})O(t​), in contrast with the linear growth generically expected without randomness. We analytically compute the average entropies of the two operators by mapping the quantum models to the symmetric simple exclusion process, relying on an assumption of self-averaging which we support numerically. We also consider two other types of randomness: random evolution in time that keeps spatial translation invariance, and quenched disorder. In the former case, we find that the OSEE also grows as t\sqrt{t}t​. In the latter case, we find that the entropy grows as O(log⁡log⁡t)\mathcal O( \log \log t)O(loglogt) for Hamiltonian evolution both for local operators with a Jordan-Wigner string and for the evolution operator, whereas it remains bounded at all times in the brick-wall matchgate circuit.
Abstract We investigate the dynamics of the fermionic logarithmic negativity in a free-fermion chain with a localized loss, which acts as a dissipative impurity. The chain is initially prepared in a generic Fermi sea. In the standard hydrodynamic limit of large subsystems and long times, with their ratio fixed, the negativity between two subsystems is described by a simple formula, which depends only on the effective absorption coefficient of the impurity. The negativity grows linearly at short times, then saturating to a volume-law scaling. Physically, this reflects the continuous production with time of entangling pairs of excitations at the impurity site. Interestingly, the negativity is not the same as the Rényi mutual information with Rényi index 1 / 2 , in contrast with the case of unitary dynamics. This reflects the interplay between dissipative and unitary processes. The negativity content of the entangling pairs is obtained in terms of an effective two-state mixed density matrix for the subsystems. Criticality in the initial Fermi sea is reflected in the presence of logarithmic corrections. The prefactor of the logarithmic scaling depends on the loss rate, suggesting a nontrivial interplay between dissipation and criticality.
Abstract Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.
Abstract The study of the entanglement dynamics plays a fundamental role in understanding the behaviour of many-body quantum systems out of equilibrium. In the presence of a globally conserved charge, further insights are provided by the knowledge of the resolution of entanglement in the various symmetry sectors. Here, we carry on the program we initiated in Parez et al (2021 Phys. Rev. B 103 L041104), for the study of the time evolution of the symmetry resolved entanglement in free fermion systems. We complete and extend our derivations also by defining and quantifying a symmetry resolved mutual information. The entanglement entropies display a time delay that depends on the charge sector that we characterise exactly. Both entanglement entropies and mutual information show effective equipartition in the scaling limit of large time and subsystem size. Furthermore, we argue that the behaviour of the charged entropies can be quantitatively understood in the framework of the quasiparticle picture for the spreading of entanglement, and hence we expect that a proper adaptation of our results should apply to a large class of integrable systems. We also find that the number entropy grows logarithmically with time before saturating to a value proportional to the logarithm of the subsystem size.