Entanglement of Disjoint Intervals in Dual-Unitary Circuits: Exact Results

TL;DR

This work delivers an exact microscopic confirmation that entanglement growth for two disjoint intervals after a quantum quench in generic dual-unitary circuits follows the entanglement-membrane picture, while charge-bearing dual-unitary circuits exhibit a quasiparticle-like entanglement drop due to conserved charges. The authors develop a diagrammatic, transfer-matrix framework and prove that the leading transfer-matrix eigenstructure drives the reduced state toward a maximal-entropy form, yielding $S_A(t)=\min(8t,4\ell)\log(d)$ (up to small corrections) for generic circuits. When charges are present, a bound involving the classical mutual information of left/right-moving charges bounds and can saturate, producing a dip consistent with the quasiparticle picture; these charged circuits are not generally Yang–Baxter integrable. The results provide the first rigorous microscopic validation of membrane-like entanglement dynamics in interacting lattice systems and clarify how conservation laws alter the late-time entanglement, with implications for related probes such as OTOCs and operator entanglement.

Abstract

The growth of the entanglement between two disjoint intervals and its complement after a quantum quench is regarded as a dynamical chaos indicator. Namely, it is expected to show qualitatively different behaviours depending on whether the underlying microscopic dynamics is chaotic or integrable. So far, however, this could only be verified in the context of conformal field theories. Here we present an exact confirmation of this expectation in a class of interacting microscopic Floquet systems on the lattice, i.e., dual-unitary circuits. These systems can either have zero or a super extensive number of conserved charges: the latter case is achieved via fine-tuning. We show that, for almost all dual unitary circuits on qubits and for a large family of dual-unitary circuits on qudits the asymptotic entanglement dynamics agrees with what is expected for chaotic systems. On the other hand, if we require the systems to have conserved charges, we find that the entanglement displays the qualitatively different behaviour expected for integrable systems. Interestingly, despite having many conserved charges, charge-conserving dual-unitary circuits are in general not Yang-Baxter integrable.

Figures (4)

• Figure 1: The entanglement of a subsystem $A$ composed by two disjoint intervals of size $\ell$ separated by a distance $x>\ell$, i.e. $A=[0,\ell]\cup[\ell+x, 2\ell+x]$, according to the membrane picture (dashed blue) and the quasiparticle one (dashed red). The continuous black line refers to the points on which both predictions agree. The quasiparticles are taken to move at speed $v_{\rm qp}=1$.
• Figure 2: Reduced density matrix of the subsystem $A=[0,\ell]\cup[\ell+x, 2\ell+x]$ according to the diagrammatic representation described in Eqs. \ref{['eq:foldedgatepicture']}--\ref{['eq:DUgraph']}. The case depicted refers to the early-time scenario $t\le x$ when the two intervals are not causally connected, and their contribution to the entanglement factorises. We took $t=1$, $x=4$, and $\ell=3$.
• Figure 3: Reduced density matrix of the subsystem $A=[0,\ell]\cup[\ell+x, 2\ell+x]$ according to the diagrammatic representation described in Eqs. \ref{['eq:foldedgatepicture']}--\ref{['eq:DUgraph']}. The case depicted refers to the time regime where the two intervals are connected by some light-cones, i.e. at times $x/2<t<\ell+x/2$, which cannot be simplified using only dual unitarity. We took $t=4$, $x=5$, and $\ell=3$.
• Figure 4: Numerical simulation of the evolution of the entanglement entropy in the time interval not fixed by dual unitarity, i.e. $t\in [x/2,\ell+x/2]$, for chaotic, dual unitary gates acting on qubits ($d=2$). The plot considers different values $\ell$ keeping the ratio ${x}/{\ell}$ constant (thus providing a scaling limit). The circle marks correspond to the choice ${x}/{\ell}=1$, while the squares correspond to the choice ${x}/{\ell}=4$. In both cases for increasing $x$ and $\ell$ we approach the straight line (black dashed) describing the $x\rightarrow\infty$ limit.

Theorems & Definitions (5)

• Definition 1
• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3