We investigate the critical behavior of a spin chain coupled to bosonic baths characterized by a spectral density proportional to ${\ensuremath{\omega}}^{s}$, with
s>1. Varying
s changes the effective dimension
deff=d+z of the system, where
z is the dynamical critical exponent and the number of spatial dimensions
d is set to one. We consider two extreme cases of clock models, namely Ising-like and
U(1)-symmetric ones, and find the critical exponents using Monte Carlo methods. The dynamical critical exponent and the anomalous scaling dimension $\ensuremath{\eta}$ are independent of the order parameter symmetry for all values of
s. The dynamical critical exponent varies continuously from $z\ensuremath{\approx}2$ for
s=1 to
z=1 for
s=2, and the anomalous scaling dimension evolves correspondingly from $\ensuremath{\eta}\ensuremath{\gtrsim}0$ to $\ensuremath{\eta}=1/4$. The latter exponent values are readily understood from the effective dimensionality of the system, being ${d}_{\mathrm{eff}}\ensuremath{\approx}3$ for
s=1, while for
s=2 the anomalous dimension takes the well-known exact value for the two-dimensional Ising and
XY models, since then
deff=2. However, a noteworthy feature is that
z approaches unity and $\ensuremath{\eta}$ approaches
1/4 for values of
s<2, while naive scaling would predict the dissipation to become irrelevant for
s=2. Instead, we find that $z=1,\ensuremath{\eta}=1/4$ for $s\ensuremath{\approx}1.75$ for both Ising-like and
U(1) order parameter symmetry. These results lead us to conjecture that for all site-dissipative
Zq chains, these two exponents are related by the scaling relation $z=\mathrm{max}{(2\ensuremath{-}\ensuremath{\eta})/s,1}$. We also connect our results to quantum criticality in nondissipative spin chains with long-range spatial interactions.
We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time
t and by Monte Carlo simulation. Both the magnetization
(m) and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent
z is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to
m. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension: the dynamical exponent is infinite and, at the critical point, the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point, there may be different exponents for the divergence of the average and typical correlation lengths, again as in one-dimension, but the evidence for this is less strong.