Mott-Glass Phase of a One-Dimensional Quantum Fluid with Long-Range Interactions

We investigate the ground-state properties of quantum particles interacting via a long-range repulsive potential ${\mathcal{V}}_{\sigma }(x)\sim 1/|x{|}^{1+\sigma }$ ($-1<\sigma$) or ${\mathcal{V}}_{\sigma }(x)\sim -|x{|}^{-1-\sigma }$ ($-2\le \sigma <-1$) that interpolates between the Coulomb potential ${\mathcal{V}}_0(x)$ and the linearly confining potential ${\mathcal{V}}_{-2}(x)$ of the Schwinger model. In the absence of disorder the ground state is a Wigner crystal when $\sigma \le 0$. Using bosonization and the nonperturbative functional renormalization group we show that any amount of disorder suppresses the Wigner crystallization when $-3/2<\sigma \le 0$; the ground state is then a Mott glass, i.e., a state that has a vanishing compressibility and a gapless optical conductivity. For $\sigma <-3/2$ the ground state remains a Wigner crystal.

Figure 1

Phase diagram of a pure (a) or disordered (b) one-dimensional Bose fluid with short-range interactions and a long-range interaction potential ${\mathcal{V}}_{\sigma }(x)$ [Eq. (1)]. $K_{\mathrm{LL}}$ denotes the Luttinger parameter associated with short-range interactions; it determines the ground state when $\sigma >0$ (in that case ${\mathcal{V}}_{\sigma }$ is effectively short range) but plays no important role for long-range interactions.

Figure 2

Flow trajectories $[K_k,{\delta }_k(0)]$ for various values of $\sigma$. The solid and dash-dotted lines are obtained for different values of the disorder strength. The red solid line for $-3/2<\sigma \le 0$ corresponds to the Mott-glass fixed point defined by $K^{*}=0$ and ${\delta }^{*}(u)$ [Eq. (13)]. Disorder is irrelevant for $\sigma <-3/2$ and the solid green line corresponds to the Wigner-crystal fixed point.

Figure 3

Disorder correlator ${\delta }_k(u)$ for various values of $k$ and $\sigma =0$ (left), $\sigma =-0.5$ (right). The green dash-dotted curve shows the initial condition ${\delta }_{\mathrm{\Lambda }}(u)\propto \cos (2u)$ and the red dashed one the fixed-point solution (13).