Non-equilibrium dynamics of charged dual-unitary circuits

TL;DR

This work introduces $ G$-symmetric dual-unitary circuits with arbitrary numbers of $U(1)$ charges and derives exact results for both solvable and non-solvable initial states. Charged solvable states admit exact quench dynamics that relax to non-trivial generalized Gibbs ensembles while preserving maximal entanglement velocity, yielding linear entanglement growth and an extensive number-entropy contribution. In contrast, non-solvable states exhibit a two-stage entanglement growth with a second, slower phase dictated by charge-conservation constraints and the entanglement-membrane framework, and they can display quantum Mpemba effects in symmetry restoration. The results reveal how conservation laws shape non-equilibrium dynamics in minimal, highly structured quantum circuits and open avenues for exploring operator spreading, simulation complexity, and extensions to non-abelian symmetries.

Abstract

The interplay between symmetries and entanglement in out-of-equilibrium quantum systems is currently at the centre of an intense multidisciplinary research effort. Here we introduce a setting where these questions can be characterised exactly by considering dual-unitary circuits with an arbitrary number of $U(1)$ charges. After providing a complete characterisation of these systems we show that one can introduce a class of solvable states, which extends that of generic dual unitary circuits, for which the non-equilibrium dynamics can be solved exactly. In contrast to the known class of solvable states, which relax to the infinite temperature state, these states relax to a family of non-trivial generalised Gibbs ensembles. The relaxation process of these states can be simply described by a linear growth of the entanglement entropy followed by saturation to a non-maximal value but with maximal entanglement velocity. We then move on to consider the dynamics from non-solvable states, combining exact results with the entanglement membrane picture we argue that the entanglement dynamics from these states is qualitatively different from that of the solvable ones. It shows two different growth regimes characterised by two distinct slopes, both corresponding to sub-maximal entanglement velocities. Moreover, we show that non-solvable initial states can give rise to the quantum Mpemba effect, where less symmetric initial states restore the symmetry faster than more symmetric ones.

Figures (8)

• Figure 1: Graphical representation of the state of a $\mathcal{G}$-symmetric dual-unitary circuit in a given charge sector. Left-moving (blue shades) and right-moving (red shades) legs have generically different local dimensions. See Sec. \ref{['sec:diagrams']} for a detailed explanation of the diagrammatic notation.
• Figure 2: Schematic representation of entanglement growth of a finite, connected interval of size $L_A$ from a charged solvable state (orange) and a generic one (blue). Solvable charged states behave as those studied in piroli2020exact, meaning they thermalise at the fastest possible rate and the entanglement saturates at $t=L_A/2$. In the generic case, instead, there is a secondary growth phase which is always slower than the first at times between the thermalization time $t_{th}$ and $L_A/2$.
• Figure 3: Entanglement growth at a single edge (meaning that we evaluate $\lim_{n\rightarrow 1} \frac{1}{1-n}\log{(\sqrt{\tr_A{\rho_A^n(t)}})}$) for randomly generated initial states (coloured dots) divided by the analytic prediction \ref{['eq:entgrowthmembrane']} obtained measuring the values of $c^{({r})}/c^{({\ell})}$ randomly chosen. More details on the numerics is presented in App. \ref{['app:numericsdetails']}.
• Figure 4: Pictorial representation of the domain-wall configuration giving the leading contribution to Eq. \ref{['eq:rdm2']}. The domain of squares does not reach the bottom part of the diagram (where the initial states are located) as the contribution would otherwise be suppressed as $a^t$, for some $a<1$. Instead, if the square-state domain is confined to the top part as in this picture, then the suppression does not scale with $t$. We stress that this domain wall approach works once the gates are projected in a sector where they are "chaotic" and then one looks for the dominant domain configuration in each of these sectors.
• Figure 5: Pictorial representation of the dominant domain contributions to evaluate Eq. \ref{['eq:6']} for small values of $t'=t-L_A/2$. We used the fact that, since $s^{{{({r})}}}_n<s^{{{({\ell})}}}_n$, the dominant domain at the left/right borders is always the one on the left for small values of $t'$, and we sum over all possible choices of cuts in between the triangles (dashed lines in Fig. \ref{['fig:domainconfig2ndphase']}), which we label as $x=1,\ldots 2t'$.

Theorems & Definitions (4)

• Theorem 1
• Definition 1: Charged Solvable States
• Theorem 2: Equivalent MPS
• Theorem 3: Compatibility Condition