⚛️ quantum physics

Free fermion models with an operator entanglement growing as $O(\sqrt{t})$

This paper demonstrates that in one-dimensional number-conserving free fermion systems subject to space- and time-dependent randomness, the Operator Space Entanglement Entropy of local spin operators and the evolution operator grows as $\mathcal{O}(\sqrt{t})$ , a behavior that is analytically established by mapping the quantum models to the symmetric simple exclusion process and contrasts with the logarithmic or linear growth observed in non-random or different disorder scenarios.

Original authors: Federico Tonetto, Jérôme Dubail

Published 2026-10-06

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CC BY 4.0

Original authors: Federico Tonetto, Jérôme Dubail

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the rules of how information spreads are fundamentally different from our everyday experience. When physicists study a chain of tiny particles, like electrons or atoms, they often look at a quantity called entanglement entropy. This measures how deeply two parts of a system are connected, such that knowing the state of one part instantly reveals something about the other. For decades, a clear pattern seemed to emerge: in systems that are "integrable"—meaning they are simple, predictable, and lack the chaotic chaos of most real-world materials—this entanglement grows very slowly, only increasing as the logarithm of time. It is a gentle, sluggish rise that suggests the system remains orderly and easy to describe. In contrast, in chaotic systems where particles interact wildly, this entanglement explodes linearly, doubling its growth rate with every tick of the clock. This distinction has long been used as a litmus test to tell the difference between simple, predictable quantum motion and the complex, messy behavior of true chaos.

However, a new study by Federico Tonetto and Jérôme Dubail challenges this neat division. They investigated a specific type of quantum system: a one-dimensional chain of free fermions, which are particles that do not interact with each other but can still become entangled. In previous studies of these "clean" systems, the entanglement of local operators—mathematical tools used to describe the state of the particles—was known to grow logarithmically, fitting the slow, orderly pattern. But the researchers asked a different question: what happens if you introduce randomness into the system? They did not just add static disorder, like a frozen impurity, but allowed the environment to fluctuate randomly in both space and time, mimicking a noisy, unpredictable world. They focused on two specific things: the evolution of a local spin operator (which involves a mathematical "string" that connects it to the rest of the chain) and the evolution operator itself, which describes how the entire system changes over time.

The results were surprising and fell into a category that had not been predicted. When the researchers simulated these systems with full randomness, they found that the entanglement did not grow slowly as a logarithm, nor did it explode linearly like a chaotic system. Instead, it grew at a rate proportional to the square root of time. This is a middle ground: faster than the slow, orderly growth of the clean systems, but much slower than the rapid, chaotic growth of interacting systems. The researchers confirmed this behavior using two different models: one that evolves continuously like a noisy Hamiltonian, and another that evolves in discrete steps like a random quantum circuit. In both cases, the growth followed the same square-root pattern, suggesting this is a robust feature of random, non-interacting quantum systems.

The study also explored what happens when the randomness is different. When the disorder was "quenched," meaning it was frozen in place and did not change over time, the behavior split depending on the model. In the discrete circuit model, the entanglement stopped growing entirely and saturated at a fixed value, a sign of localization where particles get stuck. In the continuous Hamiltonian model, however, the entanglement grew incredibly slowly, following a double-logarithmic pattern, which is even slower than the standard logarithmic growth. This distinction highlights that the nature of the randomness matters deeply. But when the randomness was active in both space and time, the square-root growth appeared consistently, regardless of whether the system was a continuous chain or a discrete circuit.

To understand why this happens, the authors mapped their complex quantum problem onto a simpler, classical model known as the symmetric simple exclusion process. Imagine a line of particles that can hop to neighboring spots but cannot occupy the same spot at the same time; this is a classical model of diffusion. The researchers showed that the quantum entanglement in their system behaves, on average, just like the flow of particles in this classical model. By assuming that the quantum fluctuations average out in a predictable way—a property they verified with extensive computer simulations—they were able to derive exact formulas for the growth rate. These formulas matched their numerical data perfectly, confirming that the square-root growth is a genuine physical phenomenon driven by the interplay of randomness and the non-interacting nature of the particles.

This finding is significant because it reshapes our understanding of quantum complexity. For a long time, it was believed that if a system was "integrable" or non-interacting, its complexity would always remain low, growing only logarithmically. The discovery of this square-root growth suggests that even in systems without direct particle interactions, the introduction of randomness can significantly increase the complexity of the quantum state, making it harder to simulate on a classical computer than previously thought. Yet, it is not as hard as a fully chaotic system. The work provides a new, precise benchmark for how quantum information spreads in noisy environments, showing that the universe of quantum dynamics contains a rich spectrum of behaviors between the extremes of perfect order and total chaos. The researchers' conclusion is that the square-root growth is the generic behavior for these random free-fermion models, a result that holds true across different types of randomness and different ways of modeling time.

Technical Summary: Free Fermion Models with Operator Entanglement Growing as $O(\sqrt{t})$

Problem Statement
The Operator Space Entanglement Entropy (OSEE) serves as a quantifier of the complexity of operators acting on the Hilbert space of a one-dimensional quantum system, specifically characterizing the bond dimension required to approximate operators via Matrix Product Operators (MPOs). It is widely believed that in integrable systems (such as clean free fermion chains), the OSEE of local operators grows at most logarithmically with time ( $O(\log t)$ ), whereas in chaotic systems, it grows linearly ( $O(t)$ ).

This paper investigates the behavior of OSEE in random free fermion models. The authors ask whether the introduction of randomness (in space, time, or both) preserves the "integrable" logarithmic scaling or induces chaotic linear scaling. Specifically, they focus on:

Local spin operators involving a Jordan-Wigner (JW) string (e.g., $X_j, Y_j$ in XX chains), which map to non-local fermionic operators.
The time-evolution operator $U(t)$ itself.

Methodology
The authors analyze two distinct models of random free fermions:

Continuous-time: A Brownian Hamiltonian known as the Quantum Symmetric Simple Exclusion Process (QSSEP), featuring space- and time-dependent noisy hopping.
Discrete-time: A brick-wall quantum circuit composed of random Gaussian gates (matchgates) drawn from the Haar measure on $U(2)$ .

The study considers three types of disorder:

Quenched disorder: Randomness is static in time (frozen in space).
Purely temporal disorder: Randomness is redrawn at every time step but maintains spatial translation invariance (e.g., $\nu=2$ sites).
Fully random disorder: Independent randomness in both space and time.

Technical Framework
The core technical contribution lies in mapping the vectorized operators to free-fermionic Gaussian states.

Vectorization: The operator $O$ is mapped to a state $|O\rangle\rangle$ in a doubled Hilbert space. For Gaussian unitaries $U$ and the Jordan-Wigner string $JW$ , these vectorized states are Gaussian.
Correlation Matrices: The OSEE is expressed entirely in terms of the spectrum $\{\tau_k\}$ ${τ_{k}}$ of a reduced correlation matrix $G_{BB}$ $G_{B B}$ (restricted to subsystem $B$ $B$ ).
- For the evolution operator $U(t)$ , the OSEE is $S_\alpha(U) = 2 \sum_k f_\alpha(\tau_k)$ .
- For the conjugated JW string $U^\dagger JW U$ , the OSEE is $S_\alpha(JW) = 2 \sum_k h_\alpha(\tau_k)$ .
Analytical Derivation (Fully Random Case): To compute the average OSEE for fully random disorder, the authors map the quantum dynamics to the classical Symmetric Simple Exclusion Process (SSEP).
- They utilize the Full Counting Statistics (FCS) of the particle number in the subsystem.
- Assuming the FCS is self-averaging (supported numerically), they relate the moments of the correlation matrix to the large deviation function of the SSEP current.
- This allows the derivation of closed-form integral expressions for the coefficients of the $\sqrt{t}$ growth.

Key Results

Fully Random Disorder (Space and Time):
- Both the OSEE of the Jordan-Wigner string (and thus local spin operators with strings) and the OSEE of the evolution operator grow as $O(\sqrt{t})$ .
- This growth holds for all Rényi indices $\alpha$ .
- The authors derive analytical prefactors $2s_\alpha$ and $c_\alpha$ for the JW string and evolution operator, respectively, which match numerical simulations for both the Brownian Hamiltonian and the matchgate circuit.
- Contrast: This is distinct from the $O(\log t)$ growth in clean integrable systems and the $O(t)$ growth in generic chaotic systems.
Purely Temporal Disorder:
- Numerical results show that the OSEE for both operators also grows as $O(\sqrt{t})$ .
- The system is not self-averaging in this regime, preventing a simple analytical derivation via the SSEP mapping used for the fully random case.
Quenched Disorder (Static in Time):
- Matchgate Circuits: The OSEE saturates to a constant value ( $O(1)$ ), consistent with Anderson localization.
- Disordered Hamiltonian: The OSEE grows as $O(\log \log t)$ . The authors attribute this to the chiral symmetry of the Hamiltonian (which lacks on-site terms), placing the system at an infinite-randomness critical point, similar to the random transverse-field Ising chain.

Significance and Claims
The paper claims that the discovery of $O(\sqrt{t})$ growth challenges the dichotomy that OSEE must be either logarithmic (integrable) or linear (chaotic).

The random free fermion models investigated are "integrable" in the sense that they are non-interacting (free fermions), yet they exhibit a growth rate ( $O(\sqrt{t})$ ) that is strictly faster than the logarithmic scaling expected for integrable dynamics.
This suggests that the OSEE of local operators in the Heisenberg picture may not be a perfect quantifier of dynamical complexity or integrability, as it fails to distinguish between these specific random free fermion models and truly chaotic systems (which scale as $O(t)$ ) as sharply as previously thought.
The work establishes a concrete link between the operator entanglement of random free fermions and the classical diffusion of the Symmetric Simple Exclusion Process.

Limitations and Future Directions
The authors note that the analytical derivation relies on the assumption of self-averaging for the FCS, which is supported numerically but not rigorously proven. They identify the purely temporal disorder case as an open problem for analytical treatment due to the lack of a classical underlying model and the absence of self-averaging. Future work is suggested to explore models without $U(1)$ particle number conservation and the effects of weak interactions on this diffusion behavior.

Technical Summary: Free Fermion Models with Operator Entanglement Growing as O(t)O(\sqrt{t})O(t​)

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Technical Summary: Free Fermion Models with Operator Entanglement Growing as $O(\sqrt{t})$