Quantum Phase Transition in the Sub-Ohmic Spin-Boson Model: Quantum Monte Carlo Study with a Continuous Imaginary Time Cluster Algorithm

A continuous time cluster algorithm for two-level systems coupled to a dissipative bosonic bath is presented and applied to the sub-Ohmic spin-boson model. When the power $s$ of the spectral function $J(\omega )\propto {\omega }^s$ is smaller than $1/2$, the critical exponents are found to be classical, mean-field like. Potential sources for the discrepancy with recent renormalization group predictions are traced back to the effect of a dangerously irrelevant variable.

Figure 1

(a) Realization of an imaginary time world line of a spin-$1/2$ in a transverse field. (b) Sketch of the continuous time cluster update: (i) Starting configuration. (ii) Random insertion of new potential spin flips (red dots) with Poissonian statistics. (iii) Connection of segments with probabilities given by Eq. (7). Different colors indicate the resulting clusters. (iv) Each cluster is flipped with probability $1/2$ (the blue one was not flipped). (v) Resulting new imaginary time world line.

Figure 2

Results for the spin-boson model for $s=0.2$ and $\Delta =0.1$. (a) Moment-ratio $Q$ as a function of the coupling constant $\alpha$ for different values of $\beta$. The critical coupling is at ${\alpha }_c=0.0175\pm 0.0002$. (b)–(c) Finite $\beta$-scaling for the moment-ratio $Q$, magnetization $m$ and susceptibility $\chi$ according to (8). The values for the critical exponents are $y_t^{*}=0.5$, $y_h^{*}=0.75$. For large positive values of the scaling variable corrections to scaling are stronger.

Figure 3

(a)–(c) Data for the susceptibility $\chi$ as a function of the distance from the critical point $\delta >0$ (i.e., in the delocalized phase) for $s=0.1$ (a), $s=0.2$ (b) and $s=0.3$ for different values of $\beta$ ($\Delta$ and ${\omega }_c$ as in Fig. 2). For increasing inverse temperature $\beta$ the data points approach the straight line, which is the zero temperature behavior $\chi \propto {\delta }^{-1}$. (d) Magnetization $m$ as a function of $\delta >0$ (i.e., in the localized phase) for $\beta =2^{16}$ for different values of $s$ (multiplied with 2, 4 and 8 for $s=0.2$, $s=0.3$ and $s=0.4$, respectively, for better visibility). The straight lines are guides for the eye proportional to the zero temperature behavior $(\alpha -{\alpha }_c{)}^{-1/2}$. Shown are only the data that are free from finite-$\beta$ corrections.

Figure 4

(a)–(b) Numerical estimates of the critical exponents $1/\nu$ and ${\beta }_m$ as a function of $s$. Triangles: QMC result (from finite temperature scaling of the QMC data as described in the text); squares: RG results (from 13, cf. also 7); straight lines: mean-field values for $s<1/2$. (c) Critical coupling strength ${\alpha }_c$ as a function of $s$ for the SB model (1, 2) with ${\omega }_c=1$ and $\Delta =0.1$. Triangles: QMC result; squares: NRG results for fixed NRG discretization parameter $\Lambda =2$ (from 12). Performing the limit $\Lambda \to 1$ moves the NRG estimates for ${\alpha }_c$ slightly downward.