Holstein mechanism in single-site model with unitary evolution

TL;DR

The paper investigates intrinsic nonadiabatic polaron dynamics in a time-driven, single-site Holstein model under unitary evolution. By applying a Lang-Firsov transformation and tracking the driven boson displacement $\beta(t)$, it reveals a dynamical phase transition from non-Markovian (power-law) to Markovian (exponential) relaxation reflected in the polaron shift, boson energy, and reduced density-matrix dynamics. A degeneracy-driven ground-state manifold triggers a crossover to open-system behavior with a dissipative gap, where internal environment scrambling and Haar-random-like dynamics emerge at late times. These results connect polaron physics, phase-space evolution, and information-theoretic features of entanglement, and suggest future work on Lindbladian spectra and squeezing diagnostics.

Abstract

We investigate the Holstein mechanism in a single-electron (one-site) system, where unitary evolution intrinsically involves both fermion and boson operators under nonadiabatic conditions. The resulting unitary dynamics and boson-frequency dependence reveal a quantum phase transition, evidenced by distinct short-time (power-law decay) and long-time (exponential decay) behaviors, which are manifested in the polaronic shift, bosonic energy, and dynamics of reduced density matrix. This observation is consistent with a non-Markovian to Markovian transition.

Figures (6)

• Figure 1: Expectation of boson operator on the single electron basis state. at long-time.
• Figure 2: Trace of the classical part of position operator $x_{2}(t)$. The orange and green lines fit the early and latter stages using power-law and exponential decay, respectively.
• Figure 3: Expectations of $\Delta x^2$ as a function of time ((a)-(b)) and frequency ((c)). Above the critical frequency $\omega_{c}=1$, the real part star to decline and nonzero imaginary part emerges. (d)The same with (a)-(b) but replacing the $S$ by $S'$.
• Figure 4: First diagonal element of the reduced density matrix $\rho_{e}$ and its time derivative. The vertical dashed line indicate the time where the degenerated ground states of $H$ emege.
• Figure 5: (a) The four diagonal elements (populations) of $\rho_{b}$ for unitary evolution and pure total density, obtained by setting an infinitesimally small threshold value for judging the degenerate ground state. (b)-(c) Populations of $\rho_{b}$ and $\rho_{e}$ when a finite threshold value is used. The non-unitary evolution and mixed total density appear at a critical time $t_{c}\approx 800$, as indicated by the vertical cyan dashed line. (d) Purities of $\rho_{tot}$ (blue), $\rho_{e}$ (red), and $\rho_{b}$ (green) for finite threshold value (left axis) and infinitesimally threshold value (right axis). The green dashed line indicates when the population starts to deviate from its logarithmic increase, which signifies the emergence of a Markovian process. The left red arrow indicates the lowered purity due to entanglement between subsystems, while the right red arrow indicates the lowered purity due to entanglement between the subsystem and the bath. Obviously, the effect of the latter dominates the former. That is why non-Markovian type fluctuations outweigh the Markovian dynamics in the late stage.