Ballistic Spin Transport in a Periodically Driven Integrable Quantum System

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.

Figure 1

Schematics of the time evolution. The red gates represent ${\mathcal{U}}_{\mathrm{even}}$ and the blue ones ${\mathcal{U}}_{\mathrm{odd}}$. The direction of the time is upwards. The schematic shows two full time steps in the bulk of the system.

Figure 2

The colored area corresponds to real $\eta$ and imaginary $\lambda$. The blue and yellow lines are constant $\lambda$ and constant $\eta$ contours, respectively. The continuous-time limit corresponds to $\mathcal{J}$, ${\mathcal{J}}^{\prime }\to 0$. Note, $\eta =\pi /2$ corresponds to the free model (${\mathcal{J}}^{\prime }=0$). In the limit $\lambda \to \infty$ ($\mathcal{J}\to \pi /4$) the local propagator (4) reduces to a swaplike gate with some ${\mathcal{J}}^{\prime }$-dependent phase.

Figure 3

A schematic phase diagram of the model based on TEBD simulations. The three circles mark the values of $\mathcal{J}$ and ${\mathcal{J}}^{\prime }$ used in the right plot, which in turn depicts the time dependence of the exponent $\alpha$, defined through the transport of magnetization between two half-chains [32] $\alpha (t)=(d/d\log t)\log ({\sum }_{\tau =0}^t⟨j_{N/2+1}(\tau )⟩)$. We can recognize ballistic, superdiffusive and diffusive regimes (red, yellow, and blue, respectively). Here we have used bond dimension 64 on a chain of length $N=3600$.

Figure 4

Drude weight at $|\lambda |=1$ as computed using the Mazur inequality (blue) and TEBD (yellow-red). The color scales from yellow to red as the simulation time increases from $t=50$ to $t=1000$. The TEBD simulations were performed using a bond dimension of 64 and a system size $N=3600$. The inset in the center of the lower panel shows a more precise set of simulations using a bond dimension of 128 for a small section of $\cos \eta$. The top-left inset of the lower panel shows convergence towards the fractal peak at $\eta =3\pi /4$ for bond dimension 256. To demonstrate fractality, the upper panel only shows the Mazur bound without the rescaling.

Figure 5

A comparison between the analytic and numerical results. Additionally we show the $m=3$ case as an example of more generic behavior with respect to $|\lambda |$. For all $m$, the $|\lambda |$ dependence lies between the $m=2$ and $m\to \infty$ curves.