Transient Dynamical Phase Diagram of the Spin-Boson Model at Finite Temperature

Abstract

We present numerically exact inchworm quantum Monte Carlo results for the real-time dynamics of the spin polarization in the sub-Ohmic spin-boson model at finite temperature. We focus in particular on the localization and coherence behavior of the model, extending our previous study at low temperature [Phys. Rev. Lett. 134, 056502 (2025)]. As the temperature increases, the system becomes less localized and less coherent. The loss of coherence, which is controlled by two independent mechanisms -- a smooth damping-driven crossover and a sharp frequency-driven transition -- exhibits a nontrivial temperature dependence. While both types of coherence loss occur at lower coupling in the high temperature regime, the frequency exhibits a sharper drop at high temperatures and this drop is observed for all values of the sub-Ohmic exponent, in contrast to the zero-temperature case. We discuss the full temperature-dependent dynamical phase diagram of the system and the interplay between coherence and localization across a wide range of physical parameters.

Figures (14)

• Figure 1: Time evolution of $\langle\sigma_z(t)\rangle$ as a function of coupling $\alpha$ deep in the sub-Ohmic regime (top panels) and close to the Ohmic regime (bottom panels) at low temperature (left panels) and at high temperature (right panels). Dashed black lines denote the analytical prediction for peaks in the dynamics, based on kehrein1996.
• Figure 2: Time-evolution of $\langle\sigma_z(t)\rangle$ as a function of temperature $T$ deep in the sub-Ohmic regime at $s=0.2$ (top panels), at $s=0.5$ (center panels), and close to the Ohmic regime at $s=0.8$ (bottom panels) at different fixed values of coupling $\alpha$ (from left to right: below, near, and above $\alpha^*$). Dashed black lines denote the analytical prediction for peaks in the dynamics, based on kehrein1996.
• Figure 3: The offset fit coefficient $c$, which characterizes the localization transition. Top panels: Offset as function of temperature for different values of $\alpha$ for $s=0.2$ (left), $s=0.5$ (center), and $s=0.8$ (right). Bottom panels: Offset as function of $\alpha$ for different temperatures for $s=0.2$ (left), $s=0.5$ (center), and $s=0.8$ (right).
• Figure 4: Left: the renormalized oscillation frequency, $\Omega/\Delta$. Right: the ratio between $\Omega$ and the corresponding damping coefficient $\gamma_1$. Temperatures (top to bottom): $\beta\Delta=20$, $\beta\Delta=4$, and $\beta\Delta=1$.
• Figure 5: Oscillation frequency $\Omega$ as function of coupling $\alpha$ for three values of $s$ at different temperatures: $\beta\Delta = 20$ (left), $\beta\Delta=4$ (center) and $\beta\Delta = 1$ (right). The symbols represent numerical data (solid lines are guides to the eye) and the dashed lines of the corresponding color are the analytical predictions from kehrein1996. The approximate theory and numerics agree at weak coupling, but the regime in which they agree becomes narrower at higher temperature. The numerically observed sharp drop to zero frequency at the frequency-drivel decoherence transition is not predicted by the theory.