Tunable quantum criticality and pseudocriticality across the fixed-point annihilation in the anisotropic spin-boson model

TL;DR

The paper investigates how fixed-point annihilation in the anisotropic spin-boson model shapes quantum criticality, using exact wormhole QMC to track two sequential fixed-point collisions as the bath exponent $s$ is varied. It uncovers a rich RG structure with CR1/CR2 fixed points, QC1/QC2 critical points, and L1/L2 localized phases, leading to symmetry-enhanced first-order transitions, continuous Landau-nonparadigm transitions, and pronounced pseudocritical scaling near the collision. The authors map phase diagrams across anisotropy, quantify critical exponents via finite-size scaling, and demonstrate slow RG flows that produce drifting exponents, providing a controlled platform to study deconfined-like critical phenomena in a $(0+1)$-D impurity setting. The results have broad implications for understanding tunable quantum criticality in dissipative quantum systems and may inform the study of related Bose-Fermi Kondo and impurity models in open quantum matter.

Abstract

Spin-boson models are simple examples of quantum dissipative systems, but also serve as effective models in quantum magnetism and exhibit nontrivial criticality. Recently, they have been established as a platform to study the nontrivial renormalization-group (RG) scenario of fixed-point annihilation, in which two intermediate-coupling RG fixed points collide and generate an extremely slow RG flow near the collision. For the Bose Kondo model, a single $S=1/2$ spin where each spin component couples to an independent bosonic bath with power-law spectrum $\propto ω^s$ via dissipation strengths $α_i$, $i\in\{x,y,z\}$, such phenomena occur sequentially for the U(1)-symmetric model at $α_z=0$ and the SU(2)-symmetric case at $α_z = α_{xy}$, as the bath exponent $s<1$ is tuned. Here we use an exact wormhole quantum Monte Carlo method for retarded interactions to explore how this nontrivial fixed-point structure affects the phase diagram and phase transitions of the anisotropic model. In particular, we show how fixed-point annihilation within a symmetry-enhanced critical manifold leads to (i) a continuous order-to-order transition beyond the Landau paradigm, (ii) a symmetry-enhanced first-order transition, and (iii) pseudocriticality, which can be tuned into each other via the bath exponent $s$. We extract critical exponents at the continuous transition, but also find scaling behavior at the symmetry-enhanced first-order transition, for which the inverse correlation-length exponent is given by the bath exponent $s$. Moreover, we provide direct numerical evidence for pseudocritical scaling on both sides of the fixed-point collision, which manifests in an extremely slow drift of the correlation-length exponent. We also study the crossover away from the SU(2)-symmetric case and determine the phase boundary of an extended U(1)-symmetric critical phase.

Figures (23)

• Figure 1: Fixed-point structure at intermediate spin-boson couplings $\alpha_{xy}$ and for bath exponents $0<s<1$. Nontrivial fixed points only occur for (a) $\alpha_z=0$ and (b) $\alpha_z=\alpha_{xy}$. As a function of $s$, the two fixed points CR1/2 and QC1/2 collide and annihilate each other at $s^\ast_{1/2}$; the precise values at which the collisions occur are determined in Fig. \ref{['fig:FPduality']}. The black dashed lines indicate the predictions \ref{['eq:cr1']} and \ref{['eq:cr2']} of the perturbative RG for $\alpha_\mathrm{CR1}$ and $\alpha_\mathrm{CR2}$. Results for the SU(2)-symmetric case presented in (b) are taken from Ref. PhysRevLett.130.186701.
• Figure 2: (a), (b) Estimation of the coordinates $(s^\ast_{1/2},\alpha^\ast_{1/2})$, at which the intermediate-coupling fixed points collide, by fitting their evolution to $s(\alpha_{xy}) = s^\ast + (b/a) \ln^2(\alpha_{xy} / \alpha^\ast)$. Final estimates for $s^\ast_{1/2}$ and $\alpha^\ast_{1/2}$ are stated in (a) and (b), whereas $b_1 / a_1 = 0.0532(2)$ and $b_2 / a_2 = 0.0562(5)$. (c),(d) Numerical test of the fixed-point duality. If the duality was exact, the product $\alpha_\mathrm{CR1/2}\times\alpha_\mathrm{QC1/2}$ would be constant. Results for the SU(2)-symmetric case presented in (b) and (d) are taken from Ref. PhysRevLett.130.186701.
• Figure 3: Schematic illustration of the RG flow for the anisotropic spin-boson model as a function of the dissipation strengths $\alpha_{xy}$ and $\alpha_z$ within the different regimes tuned by the bath exponent $s$. Stable (unstable) fixed points (within their symmetry sector) are marked by crosses (circles) and the direction of the RG flow is indicated by the arrowheads. (a) For $s>1$, the free-spin fixed point $\mathrm{F}$ is stable for all $\alpha_i$, as indicated by the green shaded area. (b) For $s^\ast_1 < s < 1$, $\mathrm{F}$ becomes unstable. There appear two pairs of intermediate-coupling fixed points, one pair on the diagonal line ($\alpha_z = \alpha_{xy}$) and the other at $\alpha_z = 0$, each consisting of a critical phase CR1/2 and a quantum critical point QC1/2. Note that $\mathrm{CR2}$ and $\mathrm{QC2}$ are unstable towards perturbations that break the SU(2) spin symmetry and lead to the stable fixed points of the localized phases $\mathrm{L}$ and $\mathrm{L1}$ as well as to the stable critical phase $\mathrm{CR1}$. The yellow (gray) shaded area indicates the extent of the critical phase $\mathrm{CR1}$ ($\mathrm{CR2}$) and the red line the separatrix between the $\mathrm{CR1}$ and $\mathrm{L1}$ phases. (c) For $s^\ast_2 < s < s^\ast_1$, the fixed points $\mathrm{CR1}$ and $\mathrm{QC1}$ have disappeared via fixed-point annihilation. Only $\mathrm{CR2}$ and $\mathrm{QC2}$ remain, which are still unstable towards symmetry-breaking perturbations leading to $\mathrm{L}$ and $\mathrm{L1}$. (d) For $0<s < s^\ast_2$, also the second pair of fixed points has disappeared, so that the localized phases $\mathrm{L}$, $\mathrm{L1}$, and $\mathrm{L2}$ are the only stable phases within their symmetry sectors.
• Figure 4: Finite-temperature analysis of the rescaled susceptibility $T^s \chi_{xy}$ as a function of the dissipation strength $\alpha_{xy}$ for fixed anisotropies (a) $\alpha_z=0$ and (b) $\alpha_z = \alpha_{xy}/2$ at $s=0.8$. (a) At $\alpha_z=0$, $T^s \chi_{xy}$ exhibits two clean crossings, as highlighted in the two insets, which correspond to the critical fixed point $\mathrm{CR1}$ and the quantum critical fixed point $\mathrm{QC1}$ and are marked by the vertical dashed lines. (b) At $\alpha_z = \alpha_{xy}/2$, we observe a clear crossing at strong couplings, which corresponds to the critical coupling $\alpha^\mathrm{c}_{xy}$ between the critical and the localized phase. However, the crossings at weak couplings do not converge to a fixed value, but significantly drift with decreasing temperature (see inset); this is consistent with the absence of a weak-coupling fixed point at finite anisotropies.
• Figure 5: Phase diagrams as a function of the bath exponent $s$ and the spin-boson coupling $\alpha_{xy}$ for different spin anisotropies (a) $\alpha_z=0$, (b) $\alpha_z=\alpha_{xy}/2$, (c) $\alpha_z = \alpha_{xy}$, and (d) $\alpha_z > \alpha_{xy}$. The red crosses in (a) and (c) indicate the positions of the fixed-point collisions estimated in Fig. \ref{['fig:FPduality']}, whereas the red ellipse in (b) marks the region where the critical coupling is supposed to disappear.