Quantum creep and variable-range hopping of one-dimensional interacting electrons

The variable range hopping results for noninteracting electrons of Mott and Shklovskii are generalized to one-dimensional disordered charge density waves and Luttinger liquids using an instanton approach. Following a recent paper by Nattermann, Giamarchi, and Le Doussal [Nattermann, Giamarchi, and Le Doussal [Phys. Rev. Lett. 91, 56603 (2003)] we calculate the quantum creep of charges at zero temperature and the linear conductivity at finite temperatures for these systems. The hopping conductivity for the short range interacting electrons acquires the same form as for noninteracting particles if the one-particle density of states is replaced by the compressibility. In the present paper we extend the calculation to dissipative systems and give a discussion of the physics after the particles materialize behind the tunneling barrier. It turns out that dissipation is crucial for tunneling to happen. Contrary to pure systems the new metastable state does not propagate through the system but is restricted to a region of the size of the tunneling region. This corresponds to the hopping of an integer number of charges over a finite distance. A global current results only if tunneling events fill the whole sample. We argue that rare events of extra low tunneling probability are not relevant for realistic systems of finite length. Finally we show that an additional Coulomb interaction only leads to small logarithmic corrections.

Figure 1

Flow diagram in $K\text{‐}u$ plane. RG is stopped at scale $l=l*$ such that $p^{2}u(l*)=\mathcal{M}⪢1$.

Figure 2

Upper figure, two equivalent ground states with two bifurcation sites which facilitate the formation of an instanton in nonzero external field. Middle, top view of an almost rectangular instanton, the width of the wall corresponds to $\xi$, i.e., the distance between the grid points of the upper figure. Lower figure, hopping process corresponding to the instanton formation, see also Fig. 3 in Ref. 36.

Figure 3

Phase profile $\phi \left(x\right)$ (bold line) after several tunneling events have happened, starting from the zero field ground state characterized by the dashed line. The thin lines show different zero field ground states which follow from each other by phase shifts of $2\pi ∕p$. A current flows only if the instantons form a percolating path through the sample.