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 ωˢ, 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 η are independent of the order parameter symmetry for all values of s. The dynamical critical exponent varies continuously from z≈2 for $s=1$ to $z=1$ for $s=2$, and the anomalous scaling dimension evolves correspondingly from η0 to η=1/4. The latter exponent values are readily understood from the effective dimensionality of the system, being deff≈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 η 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,η=1/4 for s≈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=max(2-η)/s,1. We also connect our results to quantum criticality in nondissipative spin chains with long-range spatial interactions.