Entanglement Membrane Dynamics

Updated 19 December 2025

Entanglement membrane is a coarse-grained, geometric framework that models entanglement growth in quantum chaotic systems by minimizing a codimension-one surface with an orientation-dependent tension.
It unifies diverse approaches from random unitary circuits, field theories, and holographic duals to explain thermalization, operator spreading, and black hole information dynamics.
The framework enables extraction of key parameters like entanglement velocity and butterfly velocity, linking quantum chaos to hydrodynamic and gravitational phenomena.

An entanglement membrane is a coarse-grained, geometric construct that encapsulates the leading dynamics of entanglement growth in quantum chaotic many-body systems, both in lattice models and in systems with holographic duals. It is characterized by a minimal (codimension-one) surface in the emergent spacetime of the quantum evolution, whose orientation-dependent tension function encodes the full local “equation of state” for entanglement production. The entanglement membrane framework unifies and extends the scaling theory of entanglement propagation derived in random circuit models, operator growth, field theories, and holographic duals, playing a central role in the current understanding of thermalization, information propagation, and black hole information dynamics.

1. Definition and Variational Principle

The core of entanglement membrane theory is an effective variational principle for the von Neumann or Rényi entanglement entropy $S(A,t)$ of a spatial subregion $A$ at time $t$ in a chaotic many-body system. In the scaling limit of large $A$ and late times, this entropy is given by

$S(A, t) = \min_{\Sigma:\,\partial\Sigma=\partial A(t)} \int_\Sigma d^{d-1}\xi\, \sqrt{\gamma}\, \frac{s_{\mathrm{th}}\,\mathcal{E}(v)}{\sqrt{1-v^2}}$

where $\Sigma$ is a codimension-one membrane anchored on the boundary of $A$ at $t$ , parameterized by local transverse velocity $v$ , and $\mathcal{E}(v)$ is the membrane tension, a convex, model-dependent function that encapsulates the entanglement dynamics. $s_{\mathrm{th}}$ is the thermal (or equilibrium) entropy density, and $\gamma$ is the induced metric on the membrane's worldvolume (Jonay et al., 2018, Mezei, 2018, Zhou et al., 2019, Mezei et al., 2019).

In one spatial dimension (1+1D), the membrane reduces to a directed curve $x(\tau)$ in spacetime, and the variational problem simplifies to

$S(x,t) = \min_{x_0} \left\{ s_{\mathrm{eq}} \int_0^t d\tau\, \mathcal{E}(\dot{x}(\tau)) + S(x_0, 0) \right\}$

with $x(t)$ fixed and $x(0)$ free (Blake et al., 2023). The entanglement velocity $v_E = \mathcal{E}(0)$ governs the linear growth regime, while the butterfly velocity $v_B$ is determined by $\mathcal{E}(v_B) = v_B$ .

2. Membrane Tension Function and Universal Constraints

The tension function $\mathcal{E}(v)$ encodes the local entanglement production rate for a membrane segment with local velocity $v$ (i.e., slope in spacetime). Its general properties are determined by physical consistency:

Convexity: $\mathcal{E}''(v)\geq 0$ , otherwise the minimum surface would fragment.
Evenness/Symmetry: $\mathcal{E}(v) = \mathcal{E}(-v)$ in systems with inversion symmetry.
Causality: For $|v| > v_B$ , where $v_B$ is the butterfly velocity (speed limit for information/entanglement propagation), $\mathcal{E}(v) = |v|$ and $\mathcal{E}'(v_B) = 1$ (Jonay et al., 2018).
Model dependence: While the structure is universal, the explicit form of $\mathcal{E}(v)$ varies with the microscopic Hamiltonian, gate statistics in circuits, and the dynamical universality class.

In higher dimensions, the membrane is a surface in spacetime with a local orientation vector $\vec{v}$ and an orientation-dependent tension $\mathcal{E}(\vec{v})$ (Jonay et al., 2018). Lattice anisotropy and multiple light-cone directions can give rise to piecewise linear $\mathcal{E}(\vec{v})$ (e.g., in Clifford circuits) (Sommers et al., 3 Apr 2024).

3. Emergence in Random Circuits, Floquet Chains, and Holography

In random unitary circuits, the membrane appears as the minimal energy domain wall in a classical statistical mechanics mapping of the replicated partition function. The tension function can be calculated analytically in various limits (large- $q$ , Clifford circuits) and is observed to be convex, universal, and intimately related to the equilibrium entropy density and gate statistics (Sierant et al., 2023, Zhou et al., 2019).

In exactly solvable Floquet circuits, e.g., generalized dual-unitary models, the entanglement membrane tension is linear for dual-unitary gates and generically convex for more generic gates. Analytical expressions can be constructed, confirming the hierarchy $v_E < v_B$ and the emergent universality in entanglement growth (Rampp et al., 2023).

In field-theoretic and holographic systems, the minimal membrane action arises directly from the extremal area prescription of the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surfaces in the black brane spacetime. The tension function is given by

$\mathcal{E}(v) = \sqrt{\left. \frac{-a'(z)}{2(d-1) z^{2d-3}} \right|_{z=c^{-1}(v^2)}}$

with $v^2 = a(z) - z a'(z)/[2(d-1)]$ and $a(z)$ the emblackening factor of the black brane metric (Mezei, 2018, Mezei et al., 2019). This provides a quantitative link between quantum information and black hole thermodynamics, yielding linear growth ("entanglement tsunami"), saturation, and the Page curve in evaporating scenarios (Blake et al., 2023, Jiang et al., 19 Dec 2024).

4. Operator Growth, Out-of-Time-Order Correlators, and the Membrane

The entanglement membrane not only accounts for entropy growth but also for operator spreading. The out-of-time-order commutator (OTOC) $C(x,t)$ decays as

$C(x,t) \sim \exp[ - \mu(v) t ] \,, \quad v = x/t, \quad \mu(v) = s_{\mathrm{eq}} \mathcal{E}(v) - v$

Defining $v_B$ as the unique $v$ for which $\mathcal{E}(v_B)=v_B$ yields the butterfly velocity, marking the leading edge of operator and entanglement fronts (Mezei et al., 3 Dec 2025, Zhou et al., 2019).

The tension function is typically extracted from either entanglement measurements or operator entanglement spectra, with connected Legendre transformations relating the tension and the local entropy production rate $\Gamma(s)$ . Numerical algorithms compute $\mathcal{E}(v)$ efficiently even in complex spin chains (Zhou et al., 2019).

5. Hydrodynamic Generalizations and Coupling to Other Modes

The membrane theory admits systematic hydrodynamic generalizations, including spatially inhomogeneous quenches, local fluid velocity fields (with the tilt $v$ measured relative to $u^a$ ), and coupling to hydrodynamic modes such as conserved charges or energy in the presence of dissipation (Mezei et al., 2019). First-order subleading corrections and higher-derivative terms are classified according to membrane worldvolume geometry and hydrodynamic invariants, forming a derivative expansion in analogy with fluctuating hydrodynamics.

Joining quenches (sudden connections of two or more halves) are described in this framework by permitting the membrane to end on branes at joining interfaces, with tensionless boundary conditions imprinted (Mezei et al., 2019).

6. Quantum Information Theory, Page Curve, and Black Hole Dynamics

The entanglement membrane formalism provides a comprehensive account of black hole information dynamics within both semiclassical gravity and quantum circuits. The competition between membrane configurations at different slopes encodes the transition from linear entropy growth (Hartman-Maldacena surface, no island) to saturation (minimal sloped membranes, island saddle), thereby yielding the full Page curve in toy models, double holography, and directly in quantum chaotic systems (Blake et al., 2023, Jiang et al., 19 Dec 2024). Scrambling time and information retrieval windows (Hayden-Preskill protocol) both follow from membrane geometry.

In AdS/CFT, the tension function is determined by the bulk metric, and the membrane minimization reduces to an extremization problem on the boundary, matching the projection of the bulk HRT or quantum extremal surface (Mezei, 2018, Jiang et al., 19 Dec 2024). In matrix theory constructions, entanglement between supergravity modes and local probes is mapped to local curvature and gravitational potential via an area/entropy dictionary paralleling but distinct from the Ryu-Takayanagi formula (Sahakian et al., 2017, Sahakian, 2016).

7. Fluctuations, Bit Threads, and Extensions

The minimal membrane picture extends to fluctuations, with entanglement entropy statistics following the roughening exponents of stochastic elastic manifolds (e.g., KPZ scaling in 1D, elastic membrane roughness in higher dimensions) (Sierant et al., 2023). In quantum circuits, the effective statistical mechanics is that of a zero-temperature directed polymer in a random medium (Sommers et al., 3 Apr 2024).

A convex optimization duality links the minimal membrane cut to a bit-thread max-flow problem, generalizing holographic bit threads to chaotic systems out of equilibrium (Agón et al., 2019). The dual variable is a divergence-free flow with an anisotropic norm constraint dictated by the Legendre transform of the membrane tension.

Generalizations include descriptions in monitored circuits, systems with multiple velocity scales, higher-dimensional and lattice-anisotropic systems, and calculations of finer quantum information measures such as reflected entropy, information velocity, and mutual information (Jiang et al., 25 Nov 2024).