Continuum mechanics of entanglement in noisy interacting fermion chains

We develop an effective continuum description for information scrambling in a chain of randomly interacting Majorana fermions. The approach is based on the semiclassical treatment of the path integral for an effective spin chain that describes “two-replica” observables such as the entanglement purity and the OTOC. This formalism gives exact results for the entanglement membrane and for operator spreading in the limit of weak interactions. In this limit there is a large crossover lengthscale between free and interacting behavior, and this large lengthscale allows for a continuum limit and a controlled saddle-point calculation. The formalism is also somewhat different from that known from random unitary circuits. The entanglement membrane emerges as a kind of bound state of two travelling waves, and shows an interesting unbinding phenomenon as the velocity of the entanglement membrane approaches the butterfly velocity.

💡 Research Summary

In this work the authors develop a continuum description of information scrambling in a one‑dimensional chain of Majorana fermions with both random quadratic (free‑fermion) couplings and weak random quartic interactions. By averaging over the Gaussian white‑noise disorder they obtain an effective Hamiltonian acting on four replicated copies of the system. This Hamiltonian can be rewritten in terms of SO(4) spin operators, revealing that the replicated dynamics is equivalent to a 1+1‑dimensional field theory of two complex fields, denoted z(x,t) and \bar z(x,t).

The key technical step is a semiclassical (saddle‑point) treatment of the path integral for this effective spin chain. The resulting equations of motion for z and \bar z are nonlinear wave‑type equations; in the static limit ((v=0)) the two fields are complex conjugates and the problem reduces to a simple energy‑minimisation. For a moving “entanglement membrane” with velocity (v) the fields become independent, each obeying a Fisher‑Kolmogorov‑Petrovsky‑Piskunov (FKPP)‑like equation. The membrane therefore appears as a bound state of two travelling waves—one in each field.

The authors compute the line tension (E(v)) of the membrane, which determines the scaling of the averaged entanglement purity (or second Rényi entropy) as (S_2\sim s_{\rm eq}E(v)t) with (v=x/t). They find that (E(v)=v_B,g(v/v_B)), where (v_B) emerges from the saddle‑point equations and is identified with the butterfly velocity governing the spread of out‑of‑time‑ordered correlators (OTOCs). Importantly, as (v) approaches a critical value (v_c) (which coincides with (v_B)), the bound state “unbinds”: its size diverges as (\sim1/\sqrt{v_c-v}) and the two domain walls separate, each propagating with the fixed velocity (v_c). This unbinding marks a transition from a collective membrane dynamics to independent ballistic operator spreading.

The paper emphasizes that the weak‑interaction regime introduces a large crossover length (\ell_{\rm int}=\Delta_0/\Delta_I) (ratio of noise strengths). When (\ell_{\rm int}\gg1) the system behaves almost like a free fermion chain at short scales, but beyond (\ell_{\rm int}) the interacting physics dominates, allowing a controlled continuum limit and a systematic saddle‑point expansion. Unlike Haar‑random unitary circuits, where the two replica fields are summed over separately and time‑reversal symmetry is not manifest, here the fields (z) and (\bar z) are treated symmetrically, and the time‑reversal relation only emerges at the saddle point.

Both the entanglement purity and the OTOC are expressed in terms of the same effective Hamiltonian; they differ only in boundary conditions (the purity fixes both initial and final “input‑output” cuts, while the OTOC inserts an operator at one end). This unified treatment enables the authors to extract the same line tension from both observables, confirming the universality of the entanglement‑membrane picture beyond random circuits and holographic models.

In summary, the work provides an analytically tractable continuum framework for scrambling in noisy, weakly interacting fermion chains. It demonstrates how a large interaction‑induced crossover length permits a semiclassical description, derives the exact velocity‑dependent membrane tension, and uncovers a novel unbinding transition that ties the membrane dynamics to the butterfly velocity. These results broaden the applicability of the entanglement‑membrane paradigm and offer new insights into the universal features of quantum information spreading in disordered many‑body systems.