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 $d_{\mathrm{eff}}=d+z$ 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 $\mathrm{U}\left(1\right)$-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 $d_{\mathrm{eff}}\approx 3$ 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 $\eta$ 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 $\mathrm{U}\left(1\right)$ 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 $z=\max \left\{\right(2-\eta )/s,1\}$. We also connect our results to quantum criticality in nondissipative spin chains with long-range spatial interactions.

Figure 1

Dynamical critical exponent $z$ as a function of $s$ for the $Z_4$ and the $XY$ model. The naive scaling estimate $z_0$ (the solid curve) does not coincide with the calculated $z$ for values of $s$ other than the integer-valued end points of our span of $s$ values.

Figure 2

Anomalous scaling dimension $\eta$ as a function of $s$ for the $Z_4$ and the $XY$ model. ${\eta }_0$ indicates the discrepancy between the naive scaling estimate $z_0=2/s$ and the actually calculated value of the dynamical critical exponent $z$ (based on the mean for the $Z_4$ and $XY$ model).

Figure 3

Critical exponent ratio $\beta /\nu$ as a function of $s$. This ratio appears to be independent of order parameter symmetry and is also well defined in the limit of large $s$. The dashed line represents $\beta /\nu =1/8$.

Figure 4

Correlation length exponent $\nu$ as a function of $s$. As the dissipation term becomes more short ranged with larger $s$, the exponent for the $Z_4$ model approaches the 2D Ising value $\nu =1$. In the $XY$ case, $\nu$ is expected to diverge in the limit $d_{\mathrm{eff}}\to 2$. The results (obtained for finite $L$) presented for the largest values of $s$ should therefore be regarded only as effective exponents.

Figure 5

Critical exponent $\beta$ as a function of $s$. For the $Z_4$ model, $\beta$ evolves smoothly from the 3D Ising to the 2D Ising limit as $s$ is increased from 1 to 2. The $XY$ result, on the other hand, starts out near the 3D $XY$ value for $s=1$ and features a nonmonotic evolution of $\beta$ with $s$, with a divergent $\beta$ in the limit of large $s$; cf. Fig. 4.

Figure 6

The dynamical critical exponent $z$ as in Fig. 1, but compared with the scaling estimate $z^{\prime }=(2-\eta )/s$. The values of $\eta$ used in the scaling estimates are the same as reported in Fig. 2, with the left panel corresponding to the $Z_4$ model and the right panel to the $XY$ model.