Quantum criticality in spin chains with non-ohmic dissipation

We investigate the critical behavior of a spin chain coupled to bosonic baths characterized by a spectral density proportional to $\omega^s$, with $s>1$. Varying $s$ changes the effective dimension $d_\text{eff} = d + z$ 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 $\eta$ are independent of the order parameter symmetry for all values of $s$. The dynamical critical exponent varies continuously from $z \approx 2$ for $s=1$ to $z=1$ for $s=2$, and the anomalous scaling dimension evolves correspondingly from $\eta \gtrsim 0$ to $\eta = 1/4$. The latter exponent values are readily understood from the effective dimensionality of the system being $d_\text{eff} \approx 3$ for $s=1$, while for $s=2$ the anomalous dimension takes the well-known exact value for the 2D Ising and XY models, since then $d_{\rm{eff}}=2$. A noteworthy feature is, however, that $z$ approaches unity and $\eta$ 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 $z=1,\eta=1/4$ for $s \approx 1.75$ for both Ising-like and U(1) order parameter symmetry. These results lead us to conjecture that for all site-dissipative $Z_q$ chains, these two exponents are related by the scaling relation $z = \text{max} {(2-\eta)/s, 1}$. We also connect our results to quantum criticality in nondissipative spin chains with long-range spatial interactions.