Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

For phase transitions in dissipative quantum impurity models, the existence of a quantum-to-classical correspondence has been discussed extensively. We introduce a variational matrix product state approach involving an optimized boson basis, rendering possible high-accuracy numerical studies across the entire phase diagram. For the sub-Ohmic spin-boson model with a power-law bath spectrum $\propto {\omega }^s$, we confirm classical mean-field behavior for $s<1/2$, correcting earlier numerical renormalization-group results. We also provide the first results for an $XY$-symmetric model of a spin coupled to two competing bosonic baths, where we find a rich phase diagram, including both critical and strong-coupling phases for $s<1$, different from that of classical spin chains. This illustrates that symmetries are decisive for whether or not a quantum-to-classical correspondence exists.

Figure 1

Depiction of the MPS Eq. (4), with each $A$-matrix expressed in an optimal boson basis via $A=\tilde{A}V$ [Eq. (5)].

Figure 2

VMPS results for the order parameter of SBM1 near criticality. (a) $⟨{\sigma }_x⟩$ vs ($\alpha -{\alpha }_c$) at $h_x=0$, and (b) $⟨{\sigma }_x⟩$ vs $h_x$ at $\alpha ={\alpha }_c$, on linear plots (insets) or log-log plots (main panels). Dashed lines are power-law fits in the ranges between the vertical marks. (c),(d) Comparison of the exponents $\beta$ and $\delta$ for different $s$ obtained from the VMPS method, NRG studies [4], mean-field theory, and, in (d), the exact hyperscaling result $\delta =(1+s)/(1-s)$ which applies for $s>1/2$. (See also 18, Fig. S7).

Figure 3

(a) Phase diagram of SBM2 in the $s\mathrm{\text{−}}\alpha$ plane for $\overrightarrow{h}=0$, with dissipation strength $\alpha \equiv {\alpha }_x={\alpha }_y$. The critical phase only exists for $s^{*}<s<1$, and its boundary ${\alpha }_c\to \infty$ for $s\to 1^{-}$. (Ref. 18 describes the determination of the phase boundary and gives a 3D sketch of the $s\mathrm{\text{−}}\alpha \mathrm{\text{−}}{h}_z$ phase diagram, see Fig. S8.) (b) Tranverse-field response of SBM2, $⟨{\sigma }_z⟩\propto h_z^{1/{\delta }^{\prime }}$, for four choices of $s$ and $\alpha$, showing free (diamonds), critical (squares) and localized (triangles, circles) behavior.

Figure 4

Schematic RG flow for SBM2 in the $\alpha \mathrm{\text{−}}{h}_z$ plane ($h_x=h_y=0$). The thick lines correspond to a continuous QPT; the full (open) circles are stable (unstable) fixed points, for labels see text. (a) $s^{*}<s<1$: CR is reached for small $\alpha$ and $h_z=0$, it is separated from $L$ by a QPT controlled by the multicritical QC1 fixed point. Equation (6) implies that CR is located at ${\alpha }^{*}=1-s+\mathcal{O}[(1-s{)}^2]$. For finite $h_z$, a QPT between $D$ and $L$ occurs, controlled by QC2. (b) $0<s<s^{*}$: both CR and QC1 have disappeared, such that the only transition is between $D$ and $L$.