SU(2)-Symmetric Spin-Boson Model: Quantum Criticality, Fixed-Point Annihilation, and Duality

The annihilation of two intermediate-coupling renormalization-group (RG) fixed points is of interest in diverse fields from statistical mechanics to high-energy physics, but has so far only been studied using perturbative techniques. Here we present high-accuracy quantum Monte Carlo results for the SU(2)-symmetric $S=1/2$ spin-boson (or Bose-Kondo) model. We study the model with a power-law bath spectrum $\propto {\omega }^s$ where, in addition to a critical phase predicted by perturbative RG, a stable strong-coupling phase is present. Using a detailed scaling analysis, we provide direct numerical evidence for the collision and annihilation of two RG fixed points at $s^{*}=0.6540(2)$, causing the critical phase to disappear for $s<s^{*}$. In particular, we uncover a surprising duality between the two fixed points, corresponding to a reflection symmetry of the RG beta function, which we utilize to make analytical predictions at strong coupling which are in excellent agreement with numerics. Our work makes phenomena of fixed-point annihilation accessible to large-scale simulations, and we comment on the consequences for impurity moments in critical magnets.

Figure 1

(a) Phase diagram as a function of the bath exponent $s$ and the spin-boson coupling $\alpha$. The line of second-order quantum phase transitions (red) between the CR and L phases terminates at the point $(s^{*},{\alpha }_{\text{d}}^{*})$, such that the CR phase disappears for $s<s^{*}$; for details see text. (b) Local moment $m_{\mathrm{loc}}^2={\lim }_{\beta \to \infty }⟨S^x(\beta /2)S^x(0)⟩$ for different $\alpha$. We only show data that are converged in temperature. For each $\alpha$, the corresponding horizontal line in the lower right corner marks the onset of the CR phase where $m_{\mathrm{loc}}^2=0$ for $s>s({\alpha }_{\mathrm{QC}})$. The dashed line corresponds to $m_{\mathrm{loc}}^2(\alpha \to \infty )=S^2/3=1/12$.

Figure 2

Schematic RG flow for (a) $0<s<s^{*}$ and (b) $s^{*}<s<1$.

Figure 3

(a) Correlation length ${\xi }_x/\beta$ as a function of $\alpha$ for different temperatures and $s=0.75$. The insets show close-ups of the crossings near the CR and QC fixed points. (b) Data collapse for ${\xi }_x/\beta$ near the QC fixed point. The inset shows a detailed view of the critical region. Finite-size scaling yields $1/\nu =0.192(2)$ [33].

Figure 4

(a) Inverse correlation-length exponent $1/\nu$ associated with the QC fixed point as a function of $s$. $\nu$ diverges for both $s\to 1$ and $s\to s^{*}$. The dashed lines show the predictions (8) and (9) based on fixed-point duality; for the latter, we fit $\sqrt{A_1{B}_0}=0.72(2)$. (b) Magnetization exponent ${\beta }^{\prime }$ calculated from $1/\nu$ via the hyperscaling relation ${\beta }^{\prime }/\nu =(1-s)/2$.

Figure 5

Fixed-point duality. (a) Location of the two intermediate-coupling fixed points CR and QC, as determined from crossing points of $T^s{\chi }_x$ [33], as a function of the bath exponent $s$. The black dashed line indicates the prediction (4) of the perturbative RG for ${\alpha }_{\mathrm{CR}}$. (b) Close to $s^{*}$, the fixed-point collision is well approximated by $s=s^{*}+(B_0/A_1){\ln }^2(\alpha /{\alpha }_{\text{d}}^{*})$ from which we extract $s^{*}=0.6540(2)$, ${\alpha }_{\text{d}}^{*}=0.317(1)$ where the fixed points disappear, and $A_1/B_0=17.7(2)$. (c) Product of the two fixed-point couplings as a function of $s$: This is approximately constant despite the couplings varying over several orders of magnitude.

Figure 6

(a) Weak-coupling and (b) its dual strong-coupling beta function, as in Eqs. (3) and (6). (c) Reflection symmetry of $\tilde{\beta }(\ln \alpha )$ around ${\alpha }_{\mathrm{d}}^{*}$. The zeros of $\beta (\alpha )$ disappear for $s<s^{*}$.