Quantum criticality in sub-Ohmic systems with three competing terms: beyond conventional spin-boson physics

Abstract

Quantum phase transitions (QPTs) in the spin-boson model with/without the rotating-wave approximation (RWA) are systematically investigated through variational calculations using a sub-Ohmic bath with high spectral density. Four cases involving different system-environment interactions are examined, where transition points and critical exponents are accurately determined across varying tunneling strengths. Contrary to prior work, a rich phase diagram is revealed in the tunneling-coupling plane even at the low spectral exponent $s<1/2$, with a novel U(1)-symmetric phase being identified. As coupling increases, a multi-stage QPT sequence arises for the tunneling $0<Δ< Δ^*=0.074(1)$, whereas a single transition occurs beyond this range. Furthermore, an odd-parity phase is found to emerge even under the positive tunneling, exhibiting distinct characteristics relative to the prototype model.

Figures (13)

• Figure 1: In the diagonal coupling case, the symmetry parameter $\langle\hat{\Pi}\rangle$ in (a) and the spin magnetization $|\langle\sigma_z\rangle|$ in (b) are plotted against the coupling strength $\alpha$ for different values of the tunneling amplitude $\Delta$. The insets show the total number of the excitations $\langle \hat{N}_{\rm ex} \rangle$ and the transition points $\alpha_c$, and dashed line represents a power-law fit.
• Figure 2: Quantum entanglement ($S_{\rm v-N}$) and quantum fluctuation ($QF = \Delta X_b\Delta P_b-1/4$) are presented as a function of the coupling strength $\alpha$ for different tunneling $\Delta$. In the insets, the slope of the curves $S_{\rm v-N}$ as well as the frequency-dependent $QF(\omega_k)$ for different coupling $\alpha$ are shown. Besides, dashed lines represent linear and power-law fits in (a) and its inset, respectively. In (b), $QF_{\rm max}$ denotes the peak value of the curve $QF(\omega_k)$.
• Figure 3: (a) The logarithmic value of the ratio between two average coherent-state weights $\log_{10}(\overline{A}/\overline{B})$ is plotted with respect to the coupling $\alpha$ for different tunneling $\Delta$. (b) Average displacement coefficients $\overline f_k$ (pluses) and $\overline g_k$ (triangles) are displayed for the delocalized phase, the transition point, and the localized phase, taking $\alpha=0.010,~0.0347,$ and $0.075$ as examples. Dashed lines represent the fits with the optimal displacement formula.
• Figure 4: (a) In the off-diagonal coupling case under the tunneling $\Delta=0.05$, the variance of the ground-state energy $\delta E_g^{(2)}$ is displayed with respect to the coupling $\alpha$. The critical point $\alpha_c=0.0210(3)$ is marked by the dashed line. (b) Absolute values of the position $|\langle \hat{x}_{k=0}\rangle|$ and moment $|\langle \hat{p}_{k=0}\rangle|$ are plotted on a linear-log scale for the bath mode with the lowest frequency $\omega_{\rm min}$. In the inset, frequency-dependent behaviors are given for $\alpha=0.0208$ and $0.0213$. Besides, the ground-state energy $E_g$, spin coherence $\langle \sigma_x \rangle$, and spin magnetization $\langle \sigma_{y,z} \rangle$ in (c), and symmetry parameter $\langle\hat{\Pi}\rangle$ , von Neumann entropy $S_{\rm v-N}$, and maximum of the quantum fluctuation $QF_{\rm max}$ in (d) are shown, in comparison with those of the diagonal coupling case and rotated model.
• Figure 5: In the RW coupling case, numerical results obtained from NVM1, NVM2, and VMPS are presented for the spin magnetization $|\langle \sigma_z \rangle|$ in (a) and the spin coherence $\langle \sigma_x \rangle$ in (b), under the strong tunneling $\Delta=0.1$. The difference of the ground-state energies $\Delta E_{\text{g}}$ and the parity $\langle\hat{\Pi}\rangle$ are shown in the insets. (c) The von Neumann entropy $S_{\rm v-N}$ as well as the maximum of the quantum fluctuation $QF_{\rm max}$ is plotted along with the total number of excitations $\langle \hat{N}_{\rm ex} \rangle$ in the inset. (d) The average displacement coefficients $\overline f_k$ and $\overline g_k$ are shown for $\alpha = 0.01, 0.02, 0.03$, and $0.10$ (from top to bottom).