Effects of dissipation on disordered quantum spin models

We study the effects of the coupling to an Ohmic quantum environment on the static and dynamical properties of a class of disordered spin models in a transverse magnetic field using a method of direct spin summation. We discuss the influence of the environment on various features of the phase diagram of the models as well as on the stability of the possible phases.

Figure 1

Disorder-averaged free-energy density, $\overline{f}$, of the quantum $S=1∕2$ $p=3$ model at temperature $T=0.3$ as a function of the transverse magnetic field $\Gamma$. The coupling to the bath is $\eta =1.0$. The three phases of the model are represented: a physical paramagnet labeled PM1, an unphysical paramagnet that one discard on physical grounds labeled PM2 and the spin glass. The values of $\overline{f}$ obtained for finite $M$ are shown with thin lines [$M=8$ (bottom), $M=9$ (middle), and $M=10$ (top)]. The result of the extrapolation to $M\to \infty$ is displayed with bold lines.

Figure 2

The critical averaged free-energy density, $\overline{f}$, at inverse temperature $\beta =3.3$, as a function of the inverse number of imaginary-time slices, $1∕M$. The coupling to the bath is $\eta =1.0$. The dashed and solid lines are the results of fits linear in $1∕M$ and in $1∕M^2$, respectively. Circles: results of DSS for $M=8,9,10,11,12$, and 13. Stars: results of DSS for $M=14,15$, and 16.

Figure 3

Upper panel: The diagonal order parameter $q_d\left(\tau \right)$ for the $p=3$ model as a function of $\tau ∕\left(\beta \hslash \right)=t∕M$ where $t$ is the Trotter index, $t=0,1,\cdots ,M$. The temperature is $T=0.3$ and the transverse magnetic field is $\Gamma =0.8$. The coupling to the bath is $\eta =1$. The six upper curves are the solutions for $M=8,9,10,11,12,13$, from top to bottom. The symbols are the actual data and the lines represent the results of the spline interpolation. The lowest curve is the result of the extrapolation to $M\to \infty$. Lower panel: The limiting curve ${\lim }_{M\to \infty }{q}_d^M\left(\tau \right)$ for three couplings to the environment: $\eta =0$ (bottom curve), $\eta =0.5$ (middle curve), and $\eta =1$ (top curve), The other parameters are the same as in the upper panel. (See text for the details of the method of extrapolation used.)

Figure 4

Upper panel: Static phase diagram for the $p=2$ model as obtained using the DSS technique for finite number of time slices $M$ and extrapolating the data to $M\to \infty$. The three lines correspond to $\eta =0,0.5,1$, from bottom to top. Lower panel: Static phase diagram for the $p=3$ model obtained using the same numerical method. The continuous line (dashed line) indicates a second order (first order) phase transition. The critical lines continue below the lowest value of $T$ for which we trust the algorithm, $T\approx 0.25$, to reach a quantum critical point at $T=0$.

Figure 5

Free energy density, $\overline{f}$, entropy, $\overline{s}$, and susceptibility, $\chi$, as functions of the transverse field, $\Gamma$, for the $p=3$ model at $T=0.5>T_s^{*}$ for three values of the coupling to the bath, $\eta =0,0.5,1$. The continuous (dashed) line corresponds to the paramagnetic (glassy) phase. The entropy and susceptibility are continuous at the transition indicating a second order phase transition.

Figure 6

Free energy, $\overline{f}$, entropy, $\overline{s}$, and susceptibility, $\chi$, as function of the transverse field, $\Gamma$, for the $p=3$ model at $T=0.3<T_s^{*}$ for three values of the coupling to the bath, $\eta =0,0.5,1$. The continuous (dashed) line corresponds to the paramagnetic (glassy) phase. The entropy and susceptibility are discontinuous at the transition indicating a first order phase transition.

Figure 7

Dependence of $m$ and $q_{\mathrm{ea}}$ on the critical temperature $T_s(\Gamma ,\eta )$ for the $p=3$ model. Three values of $\eta$ are considered, $\eta =0,0.5,1$. For increasing values of $\eta$ the interval in which $m=1$ increases and, hence, the region where a thermodynamic first order transition occurs decreases.

Figure 8

Comparison between the static critical line $(T_s,{\Gamma }_s)$ and Gardner’s instability line $(T_G,{\Gamma }_G)$ for the $p=3$ model with $\eta =0$.

Figure 9

Comparison between the static and marginal critical lines for the $p=10$ model. The solid lines represent second order transitions and the dashed lines first order ones. The set of curves with $T_s^{\text{class}}\approx 0.6$ (thin lines) are the static transition lines for $\eta =0$ (below) and $\eta =1$ (above); the set of curves with $T_s^{\text{class}}\approx 0.82$ (bold lines) correspond to the dynamic transition for $\eta =0$ (below) and $\eta =1$ (above).