Dissipation effects in random transverse-field Ising chains

We study the effects of Ohmic, super-Ohmic, and sub-Ohmic dissipation on the zero-temperature quantum phase transition in the random transverse-field Ising chain by means of an (asymptotically exact) analytical strong-disorder renormalization-group approach. We find that Ohmic damping destabilizes the infinite-randomness critical point and the associated quantum Griffiths singularities of the dissipationless system. The quantum dynamics of large magnetic clusters freezes completely, which destroys the sharp phase transition by smearing. The effects of sub-Ohmic dissipation are similar and also lead to a smeared transition. In contrast, super-Ohmic damping is an irrelevant perturbation; the critical behavior is thus identical to that of the dissipationless system. We discuss the resulting phase diagrams, the behavior of various observables, and the implications to higher dimensions and experiments.

Figure 1

Decimation scheme. The magnitudes of the transverse fields, interactions, and magnetic moments are depicted by the length of the arrows, the thickness of the lines, and the size of the dots, respectively. The bosonic baths are depicted by the array of springs and oscillators in which lighter (small) oscillators are faster. Finally, the magnitude of the coupling to the bath (or the local dissipation strength) is depicted by the thickness of the spring connecting the spin to the bath. (a) Global adiabatic renormalization of the fast oscillators, (b.i) decimation of a strong local transverse field, and (b.ii) decimation of a strong local interaction.

Figure 2

Schematic of the possible transverse-field decimations in the $h$ vs $\mu$ plane. (a) Before the integration of the fast oscillators, the hatched area represents all the potential fields to be decimated. (b) After integrating out the oscillators, some of these fields have been shifted below $\Omega -d\Omega$ and are not eligible to be decimated anymore, only those in the triangular shaded area still are. The dashed line $h=\Omega$ maps into the dotted line $\mu =(\Omega -h^{\prime })/\left[\alpha {(\Omega /{\Omega }_I)}^{s-1}d\Omega \right]$ after the adiabatic-renormalization step (11).

Figure 3

Schematic phase diagrams in the transverse-field ($h_{\mathrm{typ}}$) temperature ($T$) plane. (a) Infinite-randomness critical-point scenario of the dissipationless and super-Ohmic chains. SO and SD denote the strongly ordered and disordered conventional phases, while WO and WD denote the ordered and disordered quantum Griffiths phases. (b) Smeared-transition scenario for Ohmic dissipation ($0<\alpha \ll 1)$, with the inhomogeneously ordered (IO) ferromagnet replacing the WD Griffiths phase. The open circle marks the end of the tail of the smeared transition. The thin dashed lines represent finite-temperature crossovers between a quantum critical (QC) regime and a quantum paramagnet (PM) or ferromagnet (FM). The crossover temperatures $T^{*}$ and $T^{**}$ are discussed in the text.