Anomalous spin transport in integrable random quantum circuits

TL;DR

We construct genuinely interacting, time-translation–breaking quantum circuits from a three-site inhomogeneous transfer matrix built on the XXZ $R$-matrix, yielding two commuting but inequivalent layers $U_0$ and $U_1$ that preserve a common set of conserved charges. By exploring quasiperiodic (Fibonacci, Thue–Morse) and random protocols, we map a spin-transport phase diagram with ballistic, superdiffusive, and diffusive regimes governed by the underlying $R$-matrix (trigonometric, rational, hyperbolic). Generalized hydrodynamics, augmented with protocol-weighted dressing, predicts Euler-scale correlation functions, dressed velocities, and diffusion bounds, with TBA data providing microscopic insight into peak structures and velocity degeneracies observed in $C(x,t)$. Large-scale tDMRG confirms the GHD predictions across regimes, demonstrating that exact Yang–Baxter integrability coexists with stochastic time dependence and establishing GHD as a predictive tool for transport in time-dependent integrable quantum circuits.

Abstract

High-temperature spin transport in integrable quantum spin chains exhibits a rich dynamical phase diagram, including ballistic, superdiffusive, and diffusive regimes. While integrability is known to survive in static and periodically driven systems, its fate in the complete absence of time-translation symmetry, particularly in interacting random quantum circuits, has remained unclear. Here we construct integrable random quantum circuits built from inhomogeneous XXZ R-matrices. Remarkably, integrability is preserved for arbitrary sequences of gate layers, ranging from quasiperiodic to fully random, thereby explicitly breaking both continuous and discrete time-translation symmetry. Using large-scale time-dependent density-matrix renormalization group simulations at infinite temperature and half filling, we map out the resulting spin-transport phase diagram and identify ballistic, superdiffusive, and diffusive regimes controlled by the spectral parameters of the R-matrices. The spatiotemporal structure of spin correlations within each regime depends sensitively on the inhomogeneity, exhibiting spatial asymmetry and sharp peak structures tied to near-degenerate quasiparticle velocities. To account for these findings, we develop a generalized hydrodynamics framework adapted to time-dependent integrable circuits, yielding Euler-scale predictions for correlation functions, Drude weights, and diffusion bounds. This approach identifies the quasiparticles governing transport and quantitatively captures both the scaling exponents and fine structures of the correlation profiles observed numerically. Our results demonstrate that exact Yang-Baxter integrability is compatible with stochastic quantum dynamics and establish generalized hydrodynamics as a predictive framework for transport in time-dependent integrable systems.

Figures (16)

• Figure 1: Graphical interpretation of Eq. \ref{['YBE']}. The red (blue) circle denotes $R(u-\tau_1)$ [$R(u-\tau_2)$], while the rounded rectangle denotes the local gate $(PR)(\tau_2-\tau_1)$, which swaps the two $R$-matrices.
• Figure 2: (a) Schematic illustration of the monodromy and transfer matrices with two-site translational symmetry. (b) Action of the circuit $U$ on the staggered monodromy matrix. A single layer of $U$ exchanges the $R$ - matrices on odd and even bonds. In the thermodynamic limit this corresponds to a one-site translation of the inhomogeneity pattern depicted by the blue and red lines. Two successive layers restore the original configuration. As a result, $U$ commutes with $T_{12}(u)$ for all $u$, confirming the integrability of the circuit.
• Figure 3: Schematic circuit construction from a three-site inhomogeneous transfer matrix. Red, blue and green circles represent $R(u-\tau_1)$, $R(u-\tau_2)$, and $R(u-\tau_3)$, respectively. Local gates $(PR)(\tau_i-\tau_j)$ are color-coded using the same RGB convention: when a gate exchanges two $R$-matrices associated with $\tau_i$ and $\tau_j$, the resulting block is assigned the remaining third color. We construct two unitarily inequivalent circuit elements $U_0$ and $U_1$ that both commute with the transfer matrix $T(u)$.
• Figure 4: Quasiperiodic circuits generated from Thue-Morse (left) and Fibonacci (right) words. Blue (red) blocks represent applications of $U_0$ ($U_1$). Both sequences are aperiodic but have well-defined densities of $U_0$ and $U_1$ in the long time limit.
• Figure 5: Estimated values of the dynamical exponent $z$ in the gapless phase. Results for Fibonacci (Thue--Morse) sequence are shown in red (blue). All dynamical exponents converge to $1$, independent of both the sequence and spectral parameters. These results are obtained from simulations of a system of size $L= 1500$ with bond dimension $\chi= 256$, evolved for $400$ time steps (each step consists of applying either $U_0$ or $U_1$).