Transport in out-of-equilibrium XXZ chains: Nonballistic behavior and correlation functions

We consider the nonequilibrium protocol where two semi-infinite gapped XXZ chains, initially prepared in different equilibrium states, are suddenly joined together. At large times, a generalized hydrodynamic description applies, according to which the system can locally be represented by space- and time-dependent stationary states. The magnetization displays an unusual behavior: depending on the initial state, its profile may exhibit abrupt jumps that can not be predicted directly from the standard hydrodynamic equations and which signal nonballistic spin transport. We ascribe this phenomenon to the structure of the local conservation laws and make a prediction for the exact location of the jumps. We find that the jumps propagate at the velocities of the heaviest quasiparticles. By means of time-dependent density matrix renormalization group simulations we show that our theory yields a complete description of the long-time steady profiles of conserved charges, currents, and local correlations.

Figure 1

Pictorial representation of the quench protocol. After joining together two semi-infinite XXZ chains, quasiparticle excitations are created. Different types of quasiparticles move with different maximum and minimum velocities ${\zeta }_n^{\pm }$. The heaviest quasiparticles move with velocity ${\zeta }_{\infty }$ (cf. Sec. 3b1).

Figure 2

Space-time profiles of densities and currents of spin and energy. Solid black lines display the theoretical predictions, while points correspond to the exact time evolution computed by tDRMG simulations up to times $t=20$. Vertical dashed lines represent positive and negative light cones of the different $n$-quasiparticle bound states, see Sec. 3b1. The corresponding rays $\zeta ={\zeta }_n^{\pm }$, with $n=1,2,3$, are displayed as gray dashed lines, while the black dashed line corresponds to the largest string $\zeta ={\zeta }_{\infty }$.

Figure 3

Space-time profiles of local correlators, same notations as in Fig. 2. Note that the absolute value of correlators along the $x$ direction is two orders of magnitude smaller than that along the $z$ direction. In the former case, the visible small ripples on the theoretical curves are numerical artifacts.

Figure 4

NESS energy current ${\langle {\mathit{J}}_e\rangle }_{\zeta =0}$ as a function of the anisotropy $\mathrm{\Delta }$. The initial state is prepared by joining together two semi-infinite chains with vanishing magnetic field and different temperatures.

Figure 5

Space-time profiles of energy and magnetization (top) and spin-spin correlation functions (bottom), same notations as in Fig. 2. The function ${\mathcal{R}}_1^{{\mathbf{\sigma }}^z{\mathbf{\sigma }}^z}$ is computed as the ratio between two different profiles: the first is the profile of the correlator $\langle {\mathbf{\sigma }}_i^z{\mathbf{\sigma }}_{i+1}^z\rangle$ obtained by joining thermal states with ${\mu }_L=1$, ${\beta }_L=0$ and ${\mu }_R=-2$, ${\beta }_R=0$; the second is the profile for ${\mu }_L=1$, ${\beta }_L=0$ and ${\mu }_R=+2$, ${\beta }_R=0$. The plot for ${\mathcal{R}}_1^{{\mathbf{\sigma }}^z{\mathbf{\sigma }}^z{\mathbf{\sigma }}^z}$ is obtained analogously from the correlator $\langle {\mathbf{\sigma }}_i^z{\mathbf{\sigma }}_{i+1}^z{\mathbf{\sigma }}_{i+2}^z\rangle$. Note that the odd operators show a genuine discontinuity at ${\zeta }_{\infty }\sim -0.086$ (black vertical dashed line).

Figure 6

Space-time profiles of densities and currents of spin and energy. Different plots correspond to different values of $\mathrm{\Delta }$, in the gapless regime $\mathrm{\Delta }=cos\pi /\ell$, with $\ell =3,7$ and in the gapped regime $\mathrm{\Delta }=1.2$. The small circles on top of the profiles indicate the positions of the light cones ${\zeta }_n^{\pm }$ for each values of $\mathrm{\Delta }$. Note that the number of light cones in the gapless regime is finite as the number of species is also finite. In the gapped regime instead, there is an infinite number of light cones converging to the ray ${\zeta }_{\infty }$, where the magnetization density and the spin current change sign.

Figure 7

Space-time profiles of densities and currents of spin and energy, same notations as in Fig. 2. Remarkably, we see that the leftmost light cones of the magnetization and of the energy profiles do not coincide. This is due to the special properties of the initial state, as explained in detail in the main text.

Figure 8

Space-time profiles of densities and currents of spin and energy, same notations as in Fig. 2. The magnetization exhibits a nonmonotonic behavior, which is naturally interpreted as a thermoelectric effect.