Entanglement Line Tension in Quantum Systems

• Entanglement line tension is defined as the effective free-energy cost per unit length of the entanglement membrane, capturing key velocity bounds and scaling behaviors in quantum circuits.
• Exact analytic forms in level‑2 dual‑unitary circuits reveal a linear dependence on velocity, distinguishing critical quantities like the entanglement and butterfly velocities.
• The membrane framework links entanglement growth, operator spreading, and causal bounds, offering a unified approach for both solvable and generic chaotic quantum many-body systems.

Entanglement line tension is a central concept in the coarse-grained description of entanglement dynamics and operator spreading in quantum many-body systems, particularly in unitary circuit models. It characterizes the effective free-energy cost per unit length of the so-called "entanglement membrane" that traverses spacetime in replicated circuit or path-integral representations. The form and properties of the entanglement line tension encode key dynamical velocities, capture universal aspects of operator growth, and serve as a central organizing principle linking entanglement growth, correlation functions, and causal bounds in both solvable and generic chaotic systems (Rampp et al., 2023, Zhou et al., 2019).

1. Definition of Entanglement Line Tension

In a 1+1D quantum circuit, such as a Floquet chain of qudits with local Hilbert space dimension $q$, the $t$-step evolution operator $U(t)$ admits an operator-to-state mapping. The Rényi-$\alpha$ operator entanglement across a spatial cut at position $x$ is defined as

$S_\alpha(x,t) = \frac{1}{1-\alpha} \ln \Tr \left[ \left( \Tr_A|U(t)\rangle\langle U(t)| \right)^\alpha \right],$

where $A$ denotes the subset of input $\oplus$ output legs to the left of the cut. In the scaling limit $x, t \to \infty$ with fixed velocity $v = x/t$, the membrane theory predicts a "membrane" law: $$S_\alpha(x, t) \approx s_{\rm eq}\, \mathcal{E}_\alpha(v)\, t,\qquad s_{\rm eq} = \ln q,$$ with the (Rényi-$\alpha$) entanglement line tension defined as

$$\mathcal{E}_\alpha(v) = \lim_{\substack{x,t\to\infty \ x/t=v}} \frac{1}{s_{\rm eq}\, t} S_\alpha(x,t).$$

For $\alpha=1$, this yields the von Neumann entanglement line tension $\mathcal{E}(v) = \mathcal{E}_1(v)$ (Rampp et al., 2023). The same object can be analyzed via the free-energy density in associated tensor-network partition functions or replicated spin models (Zhou et al., 2019).

2. Structure and Computation in Solvable Lattice Models

Entanglement line tension calculations for generic microscopic models are exponentially difficult, but recent results provide exact solutions in certain classes of circuits, notably generalized dual-unitary circuits ("level-2" or $\bar{\mathcal{L}}_2$ circuits). For these, the partition function for the second Rényi entropy is

$Z_2(m,n) = \frac{B_1^{\min(m,n)}}{q^{m+n}},$

where $B_1 = \frac{1}{q^2} \sum_i \lambda_i^4$ is the two-site operator purity, and $\{\lambda_i\}$ are the Schmidt coefficients of the two-site gate. The entanglement line tension in level-2 circuits has the closed analytic form

$$\mathcal{E}(v) = 1 - (1-v)\frac{\ln B_1}{\ln q^2}, \qquad -1 \leq v \leq 1.$$

This linear form, arising from flat nonzero Schmidt spectra, holds for all $\alpha \geq 2$ (Rampp et al., 2023).

3. Physical Velocities and Bounds

The tension $\mathcal{E}(v)$ encodes two physically significant velocities:

• Entanglement velocity $v_E = \mathcal{E}(0) = 1 - \frac{\ln B_1}{\ln q^2}$, governing the half-chain entanglement growth rate.
• Butterfly velocity $v_B$ as the solution to $\mathcal{E}(v_B) = v_B = 1$, delineating the maximal front velocity for operator spreading.

For entangling circuits ($B_1<q^2$), $v_E < v_B = 1$. Convexity of $\mathcal{E}(v)$ enforces general bounds, including:

• The "tsunami-volume" bound:

$S_A(t) \le s_{\rm eq} |\partial A| v_B t,$

• The "surface-growth" bound:

$$\frac{dS_A(t)}{dt} \le v_E |\partial A| s_{\rm eq}.$$

Level-2 circuits saturate both bounds, mirroring saturation found in holographic models (Rampp et al., 2023).

4. Connection to Entanglement Membrane and Effective Spin Models

The entanglement membrane framework arises from coarse-graining pairing configurations in replicated circuit path integrals, generalizing the early random-circuit results. The membrane is a codimension-1 object whose free-energy per unit length is the entanglement line tension $E(v)$ (Zhou et al., 2019). In deterministic circuits, one can represent the system via an effective $S_N \cup \{\perp\}$ "spin" model: paired (permutation) blocks cost no energy, while orthogonal blocks contribute local weights. The membrane manifests as a domain wall in the effective spin model.

The tension $E(v)$ can be extracted via generating functions and a Legendre duality: $$E(v) = (1/s_{\rm eq}) \max_s \left[ \ln b_0(s) + v s \right],$$ where $b_0(s)$ is the leading root of a characteristic function derived from irreducible step weights. In generic chaotic models, $E(v)$ is convex with $E(0) > 0$, rising quadratically near $v = 0$, and tangent to $|v|$ at $v_B$ (Zhou et al., 2019).

5. Exact Results, Temporal Entanglement, and Correlation Functions

In exactly solvable level-2 generalized dual-unitary circuits, repeated application of folded-gate identities yields a factorized influence-matrix, resulting in area-law temporal entanglement along arbitrary rays in spacetime. This leads to the exact linear form for $\mathcal{E}(v)$. Connected two-point functions (at infinite temperature) vanish for $|v| < 1$, except at the edges $v=0,\pm 1$, linking information propagation and operator growth directly to the line tension (Rampp et al., 2023).

6. Extensions to Higher Hierarchy Levels and Other Models

For higher-level generalized dual-unitary circuits ($k \geq 3$), similar analytic reduction is possible only in specific spacetime sectors ($v \ge v_* = (k-2)/k$). There, the line tension takes the form: $$\mathcal{E}(v) = 1 - (1-v)\frac{\ln B_{k-1}}{\ln q^2},$$ where $B_{k-1}$ is the operator purity of a stack of $k-1$ folded gates. The butterfly velocity remains $v_B=1$ unless $B_{k-1}=q^2$, but $v_E$ no longer admits an explicit solution; instead, convexity provides upper and lower bounds: $$v_E \ge 1-\frac{\ln B_{k-1}}{\ln q^2},\qquad v_E \le 1 - \frac{2}{k}\frac{\ln B_{k-1}}{\ln q^2} \ \text{(for parity-symmetric circuits)}.$$ Thus, partial solvability still constrains entanglement growth, extending the membrane and line-tension formalism to broader settings (Rampp et al., 2023).

7. Generic vs. Dual-Unitary Circuits and Algorithmic Extraction

The shape of $E(v)$ distinguishes universality classes:

• In generic chaotic systems, $E(v)$ is smooth, convex, with a quadratic minimum and linear growth outside the light-cone.
• In dual-unitary circuits, $E(v)$ is exactly flat within $|v|<1$ and equals $|v|$ otherwise. Efficient numerical algorithms compute $E(v)$ from operator purities and recursion over pinched wall configurations, allowing verification of convexity, velocity tangency, and diffusive/wavelike properties (Zhou et al., 2019).

The entanglement line tension thus provides a unifying quantitative framework for understanding entanglement propagation, operator spreading, and causal bounds in both exactly solvable and generic chaotic quantum many-body systems.