[Paper Review] Exact pre-transition effects in kinetically constrained circuits: dynamical fluctuations in the Floquet-East model

Katja Klobas, Cecilia De Fazio+1 more|arXiv (Cornell University)|May 12, 2023

Spectroscopy and Quantum Chemical Studies4 citations

TL;DR

This paper introduces the deterministic Floquet-East model, a unitary circuit with kinetically constrained dynamics, and proves exact pre-transition effects analogous to hydrophobicity in water. Using exact tensor network calculations, it demonstrates a crossover from volume to area scaling in inactive space-time regions, a first-order dynamical phase transition, optimal geometry for solute accommodation, and a dynamical analog of hydrophobic collapse—establishing hydrophobic-like physics in deterministic classical circuits.

ABSTRACT

We study the dynamics of a classical circuit corresponding to a discrete-time kinetically constrained East model. We show that this classical "Floquet-East" model displays pre-transition behaviour which is a dynamical equivalent of the hydrophobic effect in water. For the deterministic version of the model we prove exactly: (i) a change in scaling with size in the probability of inactive space-time regions (akin to the "energy-entropy" crossover of the solvation free energy in water), (ii) a first-order phase transition in the dynamical large deviations, (iii) the existence of the optimal geometry for local phase separation to accommodate space-time solutes, and (iv) a dynamical analog of "hydrophobic collapse".

Motivation & Objective

• To investigate whether pre-transition effects—typically seen in stochastic kinetically constrained models—arise in deterministic, unitary circuits.
• To determine if hydrophobic-like behavior, such as area scaling and collapse, can be analytically proven in deterministic dynamics.
• To establish a dynamical analog of the hydrophobic effect in trajectory space using exact calculations.
• To connect these results to broader implications for classical and quantum circuit dynamics, including quantum many-body systems.

Proposed method

• Formalizing the Floquet-East model as a unitary, permutation-based circuit with local East-model constraints on a 1D chain.
• Using tensor network techniques to contract space-time evolution paths and compute the probability of inactive regions.
• Applying exact algebraic relations to derive the scaling behavior of inactive space-time regions and their dynamical free energy.
• Employing the dynamical large deviations formalism to analyze the cumulant generating function of activity and detect phase transitions.
• Mapping the system’s behavior to a trajectory-space phase transition via the scaled cumulant generating function.
• Deriving exact expressions for the activity probability and interface energy in the large-size limit using recursive contraction of projectors.

Experimental results

Research questions

• RQ1 Can deterministic, unitary circuits exhibit pre-transition effects analogous to the hydrophobic effect in water?
• RQ2 Does the probability of inactive space-time regions in a deterministic circuit display a crossover from volume to area scaling?
• RQ3 Is there a first-order phase transition in the dynamical large deviations of the Floquet-East model?
• RQ4 What is the optimal geometry for accommodating a space-time solute in such a system?
• RQ5 Can a dynamical analog of 'hydrophobic collapse' be rigorously proven in a deterministic setting?

Key findings

• The probability of inactive space-time regions scales with area (perimeter) for large sizes, indicating a crossover from volume to area scaling, with coefficient α(l) = 5/4 for integer l.
• A first-order phase transition is identified in the dynamical large deviations via the crossing of active and inactive branches in the scaled cumulant generating function at sc ≈ t⁻¹ + l⁻¹.
• The optimal geometry for local phase separation to accommodate a solute is a rectangular region with specific aspect ratio, minimizing the interface cost.
• The system exhibits a dynamical analog of hydrophobic collapse, where inactive regions form compact, stable configurations to minimize energy cost.
• Exact analytical results confirm that the system displays all key features of hydrophobicity—crossover, phase transition, optimal geometry, and collapse—despite deterministic dynamics.
• The results are derived via exact tensor network contractions and algebraic identities, providing rigorous proof beyond numerical simulations.