Quantum criticality in spin chains with non-Ohmic dissipation

Abstract

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 ${d}_{\mathrm{eff}}=d+z$ 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 $\mathrm{U}(1)$-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 ${d}_{\mathrm{eff}}=2$. 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 $\mathrm{U}(1)$ 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 $z=\mathrm{max}{(2\ensuremath{-}\ensuremath{\eta})/s,1}$. We also connect our results to quantum criticality in nondissipative spin chains with long-range spatial interactions.