Quantum Entanglement’s Secrets Unlocked with New Model of Particle Interactions

Researchers are increasingly focused on understanding how information spreads within complex quantum systems. Tobias Swann from the Rudolf Peierls Centre for Theoretical Physics and Adam Nahum from Laboratoire de Physique de l’ Ecole Normale Sup erieure, along with et al., present a new continuum mechanics approach to describe entanglement in noisy, interacting chains of fermions. Their work develops a semiclassical treatment of information scrambling, yielding exact results for entanglement and operator spreading under weak interactions. This research is significant because it provides a controlled theoretical framework, distinct from that used in random unitary circuits, to explore the crossover between free and interacting quantum behaviours and reveals a fascinating unbinding phenomenon within the entanglement membrane itself.

Weak interaction dynamics and the continuum limit of fermion scrambling are deeply connected concepts

Scientists have developed a continuum description of information scrambling within a chain of randomly interacting fermions. This work centres on a semiclassical treatment of the path integral for an effective spin chain, designed to model “two-replica” observables including entanglement purity and the out-of-time-ordered correlator.
The formalism yields exact results for the entanglement membrane and operator spreading specifically in the limit of weak interactions. A significant feature of this research is the identification of a large crossover lengthscale separating free and interacting behaviours, enabling a continuum limit and controlled saddle-point calculation.

The resulting framework differs from those previously established for random unitary circuits, offering a novel approach to understanding chaotic many-body dynamics. The entanglement membrane emerges as a bound state of two travelling waves, exhibiting an unbinding phenomenon as its velocity converges towards the butterfly velocity.

This unbinding reveals crucial insights into the dynamics of quantum information and operator propagation within the system. Researchers mapped the replicated fermion model onto a Heisenberg spin chain, allowing for semiclassical treatment following established methods. Solving the equations of motion governing two spacetime fields, z(x, t) and z(x, t), provides a continuum description of the entanglement membrane as a composite of two smooth domain walls.

Predictions for the entanglement membrane tension E(v), dependent on the velocity v defining the membrane’s orientation in spacetime, are derived through solutions of differential equations. Furthermore, the study identifies a critical velocity, vc, at which the domain walls in z and z become unbound.

As the velocity approaches vc from below, the characteristic size of the bound state diverges, indicating a transition towards ballistic spreading of operators within the Majorana chain. This critical velocity is identified with the butterfly velocity, signifying a fundamental limit on the speed of information transfer. The work extends previous analyses of entanglement after a quench and offers a complementary approach to studies employing nonlinear sigma models.

Semiclassical Path Integrals and Weak Interaction Limits for Entanglement Propagation reveal novel dynamics

A semiclassical treatment of the path integral underpins the work, focusing on an effective spin chain describing two-replica observables like entanglement purity and the out-of-time-ordered correlation function. This formalism yields exact results for the entanglement membrane and operator spreading in the weak interaction limit, where a significant lengthscale separates free and interacting behaviours.

The researchers leveraged this large lengthscale to perform a continuum limit and a controlled saddle-point calculation, differing from approaches used in random unitary circuits. The study begins with the classical Hamiltonian, defined as H = H0 + HI, where H0 = 4∆2 0 Z dx z′z′ (1 + zz)2 and HI = 16∆2 I Z dx zz(1 − z2)(1 −z2) (1 + zz)4.

A Berry phase contribution, SBerry = −1 2 Z dxdt zz − z z 1 + zz, is also incorporated into the action, alongside a boundary term dependent on the fields z and z. Solving the equations of motion derived from varying this action, z = −(1 + zz)2 δH δ z and z = +(1 + zz)2 δH δz, requires treating z and z as independent functions of both space and time at the saddle point.

Subsequently, the researchers re-parameterised the fields using z = tan θ 2 and z = tan φ 2 to simplify the coupled equations of motion for θ and φ, which become θ = +4∆2 0 θ′′ + (θ′)2 tan θ −φ 2 + ∆2 IF(θ, φ) and φ = −4∆2 0 φ′′ + (φ′)2 tan φ −θ 2 −∆2 IF(φ, θ). The function F(θ, φ) is defined as 8 sec2 θ −φ 2 sin 2θ tan θ −φ 2 sin 2φ −cos 2φ.

Dimensional analysis revealed characteristic length and timescales of l ≡∆0 ∆I and τ ≡1 ∆2 I, respectively, demonstrating the absence of dimensionless parameters in the continuum problem. The entanglement purity was then calculated using a saddle-point approximation, achieved by applying specific boundary conditions to zI and zF and solving for z(x, t) and z(x, t).

These boundary conditions impose initial and final domain walls in z and z at spatial positions determined by the velocity v. The researchers observed that in the interacting system, these domain walls persist as relatively sharp structures, forming a bound state when the velocity v falls below a certain threshold. This bound state represents the continuum form of the entanglement membrane, previously identified in other studies of entanglement growth, and its action cost per-unit-time is a function of velocity, E(v).

Entanglement membrane dynamics and operator spreading in Majorana fermion chains reveal novel transport phenomena

Researchers developed a continuum description for information scrambling within a chain of randomly interacting Majorana fermions. The approach utilises a semiclassical treatment of the path integral for an effective spin chain, examining two-replica observables including entanglement purity and the out-of-time-ordered correlator.

This formalism yields exact results for the entanglement membrane and operator spreading in the limit of weak interactions. A significant crossover lengthscale exists between free and interacting behaviour, enabling a continuum limit and controlled saddle-point calculation. The entanglement membrane emerges as a bound state of two travelling waves, demonstrating an unbinding phenomenon as the velocity of the membrane approaches the butterfly velocity.

Analysis of the entanglement membrane tension E(v) is achieved through solutions of differential equations, relating the tension to the velocity v which defines the membrane’s orientation in spacetime. A specific case of a static membrane, with v = 0, simplifies calculations due to symmetries, reducing the problem to a classical energy minimisation for z(x).

Dynamical considerations reveal a critical velocity, vc, at which domain walls in z and z become unbound. As v approaches vc from below, the characteristic size of the bound state diverges, exhibiting a behaviour proportional to 1/√vc −v. In the unbound regime, two domain walls propagate separately as travelling waves at a fixed velocity, vc.

This critical velocity is identified as the butterfly velocity, indicating ballistic spreading of operators within the Majorana chain. The emergent entanglement membrane shares similarities with those observed in Haar-random circuits, particularly in the velocity-dependent effective line tension E(v) and universal features of entanglement dynamics. However, differences arise in the treatment of degrees of freedom, with z and z remaining coupled in the path integral, unlike the independent summation in the circuit model.

Entanglement membrane dynamics and universal scrambling in weakly interacting fermionic chains reveal fundamental connections to quantum chaos

Scientists have developed a continuum description of information scrambling within a chain of interacting fermions. This approach utilises a semiclassical treatment of the path integral, focusing on observables related to entanglement purity and operator spreading. The resulting formalism yields precise results for entanglement membranes and operator propagation when interactions are weak.

The analysis reveals that the entanglement membrane arises as a bound state of two travelling waves, exhibiting an unbinding phenomenon as its velocity nears the butterfly velocity. A binary search algorithm was employed to accurately determine critical values within the model, demonstrating logarithmic time efficiency.

Furthermore, the research indicates a degree of universality, suggesting similar scrambling properties across various noisy Majorana lattice models with weak interactions, differing only in scaling parameters related to length, time, and entropy density. Limitations acknowledged by the researchers include a focus on only two observables and a restriction to leading order in the semiclassical expansion.

Future work could extend the coherent-states formalism to a broader range of observables, such as correlation function moments and cross-entropies. Investigating higher-order terms in the semiclassical expansion also represents a potential avenue for further research, potentially refining the understanding of information scrambling in these systems.