Percolation transition in quantum Ising and rotor models with sub-Ohmic dissipation

We investigate the influence of sub-Ohmic dissipation on randomly diluted quantum Ising and rotor models. The dissipation causes the quantum dynamics of sufficiently large percolation clusters to freeze completely. As a result, the zero-temperature quantum phase transition across the lattice percolation threshold separates an unusual super-paramagnetic cluster phase from an inhomogeneous ferromagnetic phase. We determine the low-temperature thermodynamic behavior in both phases, which is dominated by large frozen and slowly fluctuating percolation clusters. We relate our results to the smeared transition scenario for disordered quantum phase transitions, and we compare the cases of sub-Ohmic, Ohmic, and super-Ohmic dissipation.

Figure 1

Schematic ground state phase diagram of the diluted dissipative quantum Ising model [Eq. (2)] for fixed values of $\zeta <1$, ${\omega }_c$, and $J$. The three panels show three cuts through the three-dimensional parameter space of dilution $p$, transverse field $h_x$, and dissipation strength $\alpha$. (a) $\alpha$-$p$ phase diagram at a fixed transverse field $h_x$ with $h_x>h_{\infty }(\alpha =0)$ such that the dissipationless system is in the paramagnetic phase. This phase diagram also applies to the rotor model Eq. (4). (b) $h_x$-$p$ phase diagram at a fixed dissipation strength $\alpha$. (c) $h_x$-$\alpha$ phase diagram at fixed dilution $p<p_c$. CSPM refers to the cluster super-paramagnetic phase, transition (i) denotes the smeared generic (field or dissipation-driven) quantum phase transition, and (ii) and (iii) denote the percolation quantum phase transitions in the two regimes with or without dynamic clusters, respectively.

Figure 2

The magnetization as a function of dilution $p$ for different ordering fields $H_z$ at absolute zero. The solid line is the magnetization at $H_z=0$ (the contribution of the infinite cluster only). The dashed line is for an infinitesimal field, and the remaining ones represent stronger fields. Insets display the hysterisis curves in the (i) ordered and (ii) disordered phases.