Quantum phase transition of sub-Ohmic spin-boson models: An approach by the multiple Davydov D2 Ansatz

TL;DR

This work advances the numerical study of sub-Ohmic spin-boson models by applying a time-independent, multi-Davydov D$_2$ variational Ansatz to three coupling scenarios: diagonal-only, dual baths with diagonal and off-diagonal couplings, and a single bath with both couplings. The approach yields critical couplings in agreement with established methods for the diagonal case, reveals first-order quantum phase transitions in the two-bath setup due to coupling competition, and introduces a rotational transformation that maps the mixed-coupling single-bath problem to a diagonal form, reducing complexity and enabling intuitive understanding. The results underscore the versatility and efficiency of the multi-D$_2$ Ansatz in capturing entanglement and bath-induced effects (via observables like $ig\

Abstract

The ground state properties and quantum phase transitions of sub-Ohmic spin-boson models are investigated using the multiple Davydov D2 Ansatz in conjunction with the variational principle. Three variants of the model are studied: (i) a single bath with diagonal coupling, (ii) two independent baths with diagonal and off-diagonal couplings, and (iii) a single bath with simultaneous diagonal and off-diagonal couplings. For the purely diagonal model, the multiple Davydov D2 Ansatz yields critical coupling strengths that are consistent with other methodologies, validating its accuracy and efficiency. In the two-bath model, the competition between diagonal and off-diagonal couplings drives a first-order transition for both symmetric and asymmetric spectral exponents, with von-Neumann entropy showing a continuous peak only under exact symmetry. Finally, for a single bath with simultaneous diagonal and off-diagonal couplings, we demonstrate that a rotational transformation maps the system to an equivalent purely diagonal model, enabling simpler and intuitive physical interpretation and reduced computational complexity.

Figures (6)

• Figure 1: Illustration of the ground state energy of the SBM excluding any external bias. A doubly degenerate localised phase is separated from a non-degenerate delocalised phase by a quantum phase transition. The system energy, represented by the y-axis, uses the energy of the system when $\langle \sigma_z \rangle = 0$ as a benchmark.
• Figure 2: Top plot: $\langle\sigma_z\rangle$ (blue squares) and $\langle\sigma_x\rangle$ (red circles) as a function of $\beta$. Bottom plot: von-Neumann entropy $S_{v-N}$ (blue squares) and ground state energy $E_g$ (red circles) as a function of $\beta$. This is obtained with $s=\bar{s}=0.25$, $\alpha=0.02$, $\varepsilon=\Delta=0$. The dotted line denotes the transition point $\beta_c=0.02$.
• Figure 3: The derivative of the ground-state energy $E_g$ is displayed with respect to the off-diagonal coupling strength $\beta$. This is obtained with $s=\bar{s}=0.25$, $\alpha=0.02$, $\varepsilon=\Delta=0$.
• Figure 4: Top plot: $\langle\sigma_z\rangle$ (blue squares) and $\langle\sigma_x\rangle$ (red circles) as a function of $\beta$. Bottom plot: $S_{v-N}$ (blue squares) and ground state energy $E_g$ (red circles) as a function of $\beta$. This is obtained with $s=0.1$, $\bar{s}=0.4$, $\alpha=0.02$, $\varepsilon=\Delta=0$. The dotted line denotes the transition point $\beta_c=0.087$.
• Figure 5: (a) $\langle \tilde{\sigma}_z \rangle$ and (b) $\langle \tilde{\sigma}_x \rangle$ as a function of $\tilde{\alpha}$ or $4\alpha$. The red legend represents the expectation values obtained from the pre-rotated Hamiltonian in Eq. (19) and rotated into the post-rotation frame. The blue legend represents the expectation values obtained from the post-rotated Hamiltonian in Eq. 20.