We demonstrate ballistic spin transport of an integrable unitary quantum circuit, which can be understood either as a paradigm of an integrable periodically driven (Floquet) spin chain, or as a Trotterized anisotropic (XXZ) Heisenberg spin-1/2 model. We construct an analytic family of quasilocal conservation laws that break the spin-reversal symmetry and compute a lower bound on the spin Drude weight, which is found to be a fractal function of the anisotropy parameter. Extensive numerical simulations of spin transport suggest that this fractal lower bound is in fact tight.
We demonstrate ballistic spin transport of an integrable unitary quantum circuit, which can be understood either as a paradigm of an integrable periodically driven (Floquet) spin chain, or as a Trotterized anisotropic (XXZ) Heisenberg spin-1/2 model. We construct an analytic family of quasilocal conservation laws that break the spin-reversal symmetry and compute a lower bound on the spin Drude weight, which is found to be a fractal function of the anisotropy parameter. Extensive numerical simulations of spin transport suggest that this fractal lower bound is in fact tight.
Abstract We consider the integrable one-dimensional spin-1 chain defined by the Zamolodchikov–Fateev (ZF) Hamiltonian. The latter is parametrized, analogously to the XXZ spin-1/2 model, by a continuous anisotropy parameter and at the isotropic point coincides with the well-known spin-1 Babujian–Takhtajan Hamiltonian. Following a procedure recently developed for the XXZ model, we explicitly construct a continuous family of quasi-local conserved operators for the periodic spin-1 ZF chain. Our construction is valid for a dense set of commensurate values of the anisotropy parameter in the gapless regime where the isotropic point is excluded. Using the Mazur inequality, we show that, as for the XXZ model, these quasi-local charges are enough to prove that the high-temperature spin Drude weight is non-vanishing in the thermodynamic limit, thus establishing ballistic spin transport at high temperature.
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.
An explicit matrix product ansatz is presented, in the first two orders in the (weak) coupling parameter, for the nonequilibrium steady state of the homogeneous, nearest neighbor Heisenberg XXZ spin 1/2 chain driven by Lindblad operators which act only at the edges of the chain. The first order of the density operator becomes, in the thermodynamic limit, an exact pseudolocal conservation law and yields-via the Mazur inequality-a rigorous lower bound on the high-temperature spin Drude weight. Such a Mazur bound is a nonvanishing fractal function of the anisotropy parameter Δ for |Δ|<1.
Purely ballistic transport is a rare feature even for integrable models. By numerically studying the Heisenberg chain with the power-law exchange, $J\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$, where r is a distance, we show that for spin anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}exp(\ensuremath{-}\ensuremath{\alpha}+2)$ the system exhibits a quasiballistic spin transport and the presence of fermionic excitation, which do not decay up to extremely long times $\ensuremath{\sim}{10}^{3}/J$. This conclusion is reached on the basis of the dynamics of spin domains, the dynamical spin conductivity, inspection of the matrix elements of the spin-current operator, and analysis of most conserved operators. Our results smoothly connect two models in which fully ballistic transport is present: free particles with nearest-neighbor hopping and the isotropic Haldane-Shastry model.
This thesis studies transport properties of low-dimensional quantum spin systems with a particular focus on the transport of heat and spin. While the integrable anisotropic spin-1/2 chain exhibits ballistic transport (finite Drude weights), the numerical results of this work show that heat and spin transport of non-integrable models (spin ladder, frustrated and dimerized chain) is not ballistic, characterized by vanishing Drude weights in the thermodynamic limit. The results for the thermal conductivity of spin ladders are compared to experimental data. For the case of the spin-1/2 Heisenberg chain, this work contains detailed results for the temperature and magnetic field dependence of the Drude weights. Finally, bond-disordered spin systems are considered, namely the spin-1/2 chain and the spin ladder. In the first case, the transport coefficients are computed by means of exact diagonalization and discussed as a function of frequency, temperature, and magnetic field. In the second case, spectral properties of the elementary excitation of disordered spin ladders are analyzed using diagrammatic perturbation theory and numerical techniques.
A continuous family of quasilocal exact conservation laws is constructed in the anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted) boundary conditions and for a set of commensurate anisotropies densely covering the entire easy plane interaction regime. All local conserved operators follow from the standard (Hermitian) transfer operator in fundamental representation (with auxiliary spin s=1/2), and are all even with respect to a spin flip operation. However, the quasilocal family is generated by differentiation of a non-Hermitian highest weight transfer operator with respect to a complex auxiliary spin representation parameter s and includes also operators of odd parity. For a finite chain with open boundaries the time derivatives of quasilocal operators are not strictly vanishing but result in operators localized near the boundaries of the chain. We show that a simple modification of the non-Hermitian transfer operator results in exactly conserved, but still quasilocal operators for periodic or generally twisted boundary conditions. As an application, we demonstrate that implementing the new exactly conserved operator family for estimating the high-temperature spin Drude weight results, in the thermodynamic limit, in exactly the same lower bound as for almost conserved family and open boundaries. Under the assumption that the bound is saturating (suggested by agreement with previous thermodynamic Bethe ansatz calculations) we propose a simple explicit construction of infinite time averages of local operators such as the spin current.