REVIEW
Free fermion models with an operator entanglement growing as $O(\sqrt{t})$
Quantum Physics · Submitted 5 Oct 2026 · 86 references
Pith's verdict: Genuinely new: random free fermions give sqrt(t) OSEE growth with parameter-free prefactors; the two unproven analytic steps are honestly flagged and backed by numerics strong enough to carry the paper through refereeing.
T0 · AI review
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Referee report
The paper
The paper claims
Randomness turns free-fermion operator entanglement into a square-root growth law
The reading
What carries the argument
The computation rests on vectorizing the two operators of interest into Gaussian states: the Jordan-Wigner string becomes a tensor product of two domain-wall states, and the evolution operator becomes a BCS-like state. Both entropies then reduce, via Peschel's formula, to sums over the spectrum $\{\tau_k\}$ of the reduced correlation matrix $G^U_{BB} = (u_{AB})^\dagger u_{AB}$, giving factors of two for different reasons in the two cases (Eq. 49). The analytic step replaces the quantum expectation by a classical one through the identity $E[\mathrm{Tr}(\rho e^{\sum_i a_i \hat n_i})] = E[e^{\sum_j a_j n_j}]_{\mathrm{SSEP}}$, mapping the full counting statistics to the integrated current of the symmetric simple exclusion process, whose large-deviation function (Derrida–Gerschenfeld, Eq. 66) fixes the moments $E[\mathrm{Tr}(G^U_{BB})^n] = \sqrt{t/(n\pi)}$ at leading order; a self-averaging assumption on the full counting statistics (Eq. 68) converts the annealed average into the quenched one that the entropy requires, and a Laplace-trick resummation yields the closed-form prefactors $2s_\alpha$ and $c_\alpha$ in Eqs. (71)–(73).
Extended reading notes
Core claim
For one-dimensional number-conserving free fermions with fully random spatio-temporal disorder, the operator-space entanglement entropy of a local spin operator containing a Jordan-Wigner string grows on average as $S_\alpha^A(O) = 2s_\alpha \sqrt{t} + o(\sqrt{t})$ for every Rényi index, with the closed-form prefactor $2s_\alpha$ given in Eq. (71), replacing the logarithmic growth of the clean chain; the OSEE of the evolution operator grows as $S_\alpha^A(U) = c_\alpha \sqrt{t} + o(\sqrt{t})$ with $c_\alpha$ in Eq. (73), against the linear growth generic for evolution operators. Both a Brownian hopping Hamiltonian (the QSSEP) and a brick-wall matchgate circuit with Haar-random $U(2)$ gates give the same asymptotic behavior, and the $\sqrt{t}$ growth holds for each single realization, not only on average. With purely temporal (translation-invariant) randomness the $\sqrt{t}$ growth persists, while quenched disorder yields $\log\log t$ growth for the Hamiltonian (protected by chiral symmetry) and saturation for the circuit.
Load-bearing premise
The derivation assumes the full counting statistics is self-averaging — that the average of its logarithm equals the logarithm of its average up to $o(\sqrt{t})$ corrections (Eq. 68) — which the authors state they cannot prove and support only numerically; this step, together with the unproved SSEP identity (64) for the circuit, is what turns the solvable classical average into the quenched entropy, and if it fails the closed-form prefactors (71)–(73) do not follow from the stated argument.
Editorial extensions
If this is right
- The growth rate of operator entanglement in free-fermion systems is not fixed by freeness: adding randomness moves the OSEE of Jordan-Wigner-string operators from $O(\log t)$ to $O(\sqrt{t})$ (Eqs. 3 versus 4).
- The OSEE of the evolution operator, normally linear in time, is reduced to $\sqrt{t}$ by randomness (Eq. 6), so local noise acts as an operator-space entanglement bottleneck.
- The same $\sqrt{t}$ growth appears with purely temporal, translation-invariant randomness, so spatial randomness is not required for the effect.
- Quenched disorder splits the two models: the chiral-symmetric Hamiltonian escapes Anderson localization and grows as $\log\log t$, while the matchgate circuit saturates (Eq. 58).
- For translation-invariant single-bond noise ($\nu=1$) the quasiparticles remain coherent and the clean logarithmic law survives with the light-cone radius randomized (Eq. B1).
Reading between the lines
- If the $\sqrt{t}$ growth survives weak interactions beyond free fermions — a question the authors leave open — the OSEE growth rate would cease to be a reliable integrability-versus-chaos diagnostic, and the natural test is a noisy interacting chain where the SSEP mapping fails.
- The fact that the whole Rényi spectrum is fixed by the single functions $2s_\alpha$ and $c_\alpha$ suggests the SSEP current fluctuations determine the full entanglement spectrum, so one could probe whether the leading quantum correction $\mathrm{Var}[\log\chi]$ obeys a similar universal form.
- The SSEP identity (64) for the matchgate circuit is asserted without proof; a direct proof would make the circuit results fully analytic, rather than numerically consistent, and would likely extend the mapping to other Gaussian noise ensembles.
Original abstract
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 $\mathcal O(\sqrt t)$, for all values of the R\'enyi 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 $\mathcal O(\sqrt{t})$, 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 $\sqrt{t}$. In the latter case, we find that the entropy grows as $\mathcal O( \log \log t)$ 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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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 $\mathcal O(\sqrt t)$, for all values of the R\'enyi 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 $\mathcal O(\sqrt{t})$, 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 $\sqrt{t}$. In the latter case, we find that the entropy grows as $\mathcal O( \log \log t)$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.97 | 0.89 | 0.04 | asserted 0.66 |
| s1 | It is known that, in free-fermion systems without randomness, the OSEE of \emph{local} operators grows at most \emph{logarithmically} in time. | background 1.00 | 0.04 | 0.03 | asserted 0.98 |
| s2 | Here we show that, in the presence of randomness, the OSEE of local spin operators involving a Jordan-Wigner string grows as $\mathcal O(\sqrt t)$, for all values of the R\'enyi index. | result 0.98 | 0.94 | 0.77 | asserted 1.00 |
| s3 | We establish this in two models: a Brownian Hamiltonian and a brick-wall matchgate circuit. | method 0.59 | 0.92 | 0.08 | asserted 0.99 |
| s4 | We also consider the OSEE of the evolution operator, and we show that it also grows as $\mathcal O(\sqrt{t})$, in contrast with the linear growth generically expected without randomness. | result 0.97 | 0.96 | 0.20 | asserted 1.00 |
| s5 | 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. | method 0.98 | 0.96 | 0.07 | asserted 0.99 |
| s6 | We also consider two other types of randomness: random evolution in time that keeps spatial translation invariance, and quenched disorder. | contribution 0.97 | 0.89 | 0.03 | asserted 0.75 |
| s7 | In the former case, we find that the OSEE also grows as $\sqrt{t}$. | result 1.00 | 0.96 | 0.10 | asserted 1.00 |
| s8 | In the latter case, we find that the entropy grows as $\mathcal O( \log \log t)$ 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. | result 1.00 | 0.96 | 0.09 | asserted 1.00 |
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Figures
Figures reproduced from arXiv:2610.05995 by the authors.
References
Discussion
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