Spin-orbit coupled spin-boson model : A variational analysis

TL;DR

This paper studies a one-dimensional spin-orbit coupled particle dissipatively coupled to a sub-ohmic bath using a variational polaron approach. By transforming to a polaron frame and variationally optimizing bath displacements, it reveals a dissipation-driven restructuring of the energy–momentum dispersion from double minima at finite momentum to a single minimum at zero momentum, along with a bath-induced continuous magnetization transition in momentum-conserving settings and a prominent entanglement signature between spin and environment. In a harmonic trap, the ground state becomes a cat-like superposition of opposite-momentum components below a critical bath coupling, with a sharp first-order transition lifting quasi-degeneracy and reducing entanglement above the transition. The work highlights how environmental coupling can control transport and intra-particle entanglement in SO-coupled systems, with potential relevance for graphene-like materials and engineered cold-atom or trapped-ion platforms.

Abstract

The spin-boson (SB) model is a standard prototype for quantum dissipation, which we generalize in this work, to explore the dissipative effects on a one-dimensional spin-orbit (SO) coupled particle in the presence of a sub-ohmic bath. We analyze this model by extending the well-known variational polaron approach, revealing a localization transition accompanied by an intriguing change in the spectrum, for which the doubly degenerate minima evolves to a single minimum at zero momentum as the system-bath coupling increases. For translational invariant system with conserved momentum, a continuous magnetization transition occurs, whereas the ground state changes discontinuously. We further investigate the transition of the ground state in the presence of harmonic confinement, which effectively models a quantum dot-like nanostructure under the influence of the environment. In both the scenarios, the entanglement entropy of the spin-sector can serve as a marker for these transitions. Interestingly, for the trapped system, a cat-like superposition state corresponds to maximum entanglement entropy below the transition, highlighting the relevance of the present model for studying the effect of decoherence on intra-particle entanglement in the context of quantum information processing.

Figures (4)

• Figure 1: Dissipation induced transition: Changes in the dispersion of the lower energy branch with increasing coupling strength $\alpha$ for sub-ohmic dissipation with $s=0.4$. From the top to the bottom curve: $\alpha = 0.001,\, 0.005,\, 0.010,\, 0.015,\, 0.020,\, {\rm and}\, 0.025$, respectively. The transition occurs at $\alpha_{c} \sim 0.0187$, and colorscale represents the Magnetization $M$. Colorscale map of the (b) magnetization $M$ and (c) Entanglement entropy $S_{en}$, in the $k$-$\alpha$ plane. (d) Variation of Entanglement entropy $S_{en}$ (left axis) and magnetization $M$ (right axis) with coupling $\alpha$ for fixed values of $k$, denoted by the dashed and dotted lines in (b,c). In all figures, the cut-off frequency is chosen as $\omega_{c} = 10$. In these and the rest of the figures, the energies and momenta are in the units of $q^2$ and $q$, respectively.
• Figure 2: Ground state properties: (a) Variation of momentum $k_{min}$ (top) and Magnetization $M$ (bottom) corresponding to the minimum energy state with increasing $\alpha$ for different values of $s$. Variation of the entanglement entropy $S_{en}$ (left axis) and Magnetization $M$ (right axis) corresponding to the ground state energy with increasing $\alpha$ for $s=0.4$. The jump at $\alpha_{c}$ indicates a first-order transition. In all figures, the cut-off frequency is chosen as $\omega_{c} = 10$.
• Figure 3: Ground state properties in the presence of a harmonic trap: (a) Variation of the momentum $k_{min}$ (blue) and Magnetization $M$ (red) corresponding to the minimum energy state with increasing $\alpha$. (b) Variation of the entanglement entropy $S_{en}$ corresponding to the ground state with increasing $\alpha$. Variation of (c) effective magnetization $\epsilon = B_{+}-B_{-}$ and (d) splitting energy $\tilde{\Delta} = 2|\Delta|e^{-k^2l^2}/\mathcal{N}$, with increasing $\alpha$. The inset in (d) indicates a zoom region near $\alpha=\alpha_{c}$. The jump at $\alpha_{c}\sim 0.0162$ indicates a first-order transition. Parameters chosen: $s=0.4$, $\omega_{0}=0.1$, and $\omega_{c} = 10$.
• Figure 4: Wigner distributions of the ground state: Colorscale plot of $\mathcal{W}(\tilde{x},\tilde{k})$ at different values of coupling constant $\alpha$, (a,b) before, and (c,d) after the point of transition, $\alpha_{c} \sim 0.0162$. Parameters chosen: $s=0.4$, $\omega_{0}=0.1$, and $\omega_{c} = 10$.