# Characterization of Quantum and Classical Critical Points for an Integrable Two-Qubit Spin–Boson Model

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Dynamics

#### 3.1. Conditions

#### 3.2. Decoherence-Free Subspace

#### 3.3. Ohmic Regime

#### Mixing Subspaces

## 4. Quantum Phase Transitions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Model and Its Symmetries

## Appendix B. Observables’ Mapping

## Appendix C. Ground State of the Single-Impurity Spin–Boson Model

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**Figure 1.**Time behavior of both the single-spin and the net magnetizations for $s=1$, ${\alpha}_{a}=1/2$, and three levels of isotropy: (

**a**) minimum, (

**b**) intermediate, (

**c**) maximum. The two spins start from $\left(\right|++\rangle +|+-\rangle )/\sqrt{2}$, while the bath from the thermal state.

**Figure 2.**Dependence of $\Lambda ={\lambda}_{0}^{a}-{\lambda}_{0}^{b}$ on $\alpha ={\alpha}_{a}={\alpha}_{b}/k$ for: (

**a**) ${\gamma}_{x}=2{\gamma}_{y}={10}^{-2}{\omega}_{c}$ and ${\Omega}_{1}=2{\Omega}_{2}$$={10}^{-7}{\omega}_{c}$; (

**b**) ${\gamma}_{x}=2{\gamma}_{y}={10}^{-3}{\omega}_{c}$ and ${\Omega}_{1}=2{\Omega}_{2}={10}^{-9}{\omega}_{c}$; (

**c**) ${\gamma}_{x}=2{\gamma}_{y}={10}^{-3}{\omega}_{c}$ and ${\Omega}_{1}=2{\Omega}_{2}$$={10}^{-2}{\omega}_{c}$; with $k=0$ (solid red line), $k=0.25$ (dashed green line), $k=0.5$ (dotted blue line), $k=0.75$ (dot-dashed cyan line). The horizontal black line represents $\Lambda =0$.

**Figure 3.**Dependence of $\Lambda ={\lambda}_{0}^{a}-{\lambda}_{0}^{b}$ on $\alpha ={\alpha}_{a}={\alpha}_{b}/k$ and k for ${\gamma}_{x}=2{\gamma}_{y}={10}^{-2}{\omega}_{c}$ and ${\Omega}_{1}=2{\Omega}_{2}$$={10}^{-7}{\omega}_{c}$. The white strip identifies the critical points where the level crossing occurs. As explicitly shown in the figure, the red (blue) phase corresponds to the GS ${|GS\rangle}_{a}$ (${|GS\rangle}_{b}$).

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**MDPI and ACS Style**

Grimaudo, R.; Messina, A.; Nakazato, H.; Sergi, A.; Valenti, D.
Characterization of Quantum and Classical Critical Points for an Integrable Two-Qubit Spin–Boson Model. *Symmetry* **2023**, *15*, 2174.
https://doi.org/10.3390/sym15122174

**AMA Style**

Grimaudo R, Messina A, Nakazato H, Sergi A, Valenti D.
Characterization of Quantum and Classical Critical Points for an Integrable Two-Qubit Spin–Boson Model. *Symmetry*. 2023; 15(12):2174.
https://doi.org/10.3390/sym15122174

**Chicago/Turabian Style**

Grimaudo, Roberto, Antonino Messina, Hiromichi Nakazato, Alessandro Sergi, and Davide Valenti.
2023. "Characterization of Quantum and Classical Critical Points for an Integrable Two-Qubit Spin–Boson Model" *Symmetry* 15, no. 12: 2174.
https://doi.org/10.3390/sym15122174