Diffusion and Ballistic Transport in One-Dimensional Quantum Systems

It has been conjectured that transport in integrable one-dimensional systems is necessarily ballistic. The large diffusive response seen experimentally in nearly ideal realizations of the $S=1/2$ 1D Heisenberg model is therefore puzzling and has not been explained so far. Here, we show that, contrary to common belief, diffusion is universally present in interacting 1D systems subject to a periodic lattice potential. We present a parameter-free formula for the spin-lattice relaxation rate which is in excellent agreement with experiment. Furthermore, we calculate the current decay directly in the thermodynamic limit using a time-dependent density matrix renormalization group algorithm and show that an anomalously large time scale exists even at high temperatures.

Figure 1

In a diffusive channel, the conductivity is limited by the dominant of the various scattering processes pictured as a serial arrangement of resistors. If part of the current is, however, protected by a conservation law, a parallel ballistic channel for charge transport is opened.

Figure 2

Experimental data for $1/T_1$ of the spin chain compound ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$ at $h=9\mathrm{T}$ taken from Ref. 16 (dots) compared to our theory (blue solid line). Without diffusion, $\gamma =0$, $1/(T_{1}T)$ would be almost constant (red dashed line).

Figure 3

(a) $C(t)=⟨\mathcal{J}(t)\mathcal{J}(0)⟩/L$ at $T=\infty$ for various $\Delta$ as indicated on the plot. The solid (dashed) lines correspond to $\Delta$ in the critical (gapped) regime, respectively. (b) $\mathrm{Re}[C(t)]/2JT$ at $T=0.2J$ for $\Delta =0.6$, $\Delta =0.8$, and $\Delta =1.0$ (solid lines). The dashed lines are linear fits $\mathrm{Re}[C(t)]/2JT=A-BJt$ for $Jt\in [3.5,7]$.