Quantum Brownian Motion: proving that the Schmid transition belongs to the Berezinskii-Kosterlitz-Thouless universality class

Abstract

We investigate the equilibrium properties of a quantum Brownian particle moving in a periodic potential, specifically addressing the nature of the dissipation-driven Schmid transition in the Ohmic regime. By employing World-Line Monte Carlo in the path-integral formalism and introducing a specific binary order parameter, we demonstrate that the transition belongs to the Berezinskii-Kosterlitz-Thouless universality class. This classification is substantiated through finite-size scaling analysis that reveals the characteristic logarithmic decay of the correlation functions associated with the order parameter at the critical point. Quantum phase transition turns out to be extremely fragile: it disappears in both over- and sub-Ohmic dissipation regimes. Crucially, we find that the presence of the periodic potential does not alter the localization properties in the sub-Ohmic and super-Ohmic regimes, where the system exhibits the same qualitative behavior as the free quantum Brownian particle. These findings highlight that the emergence of critical behavior is strictly governed by the low-frequency form of the environmental spectral function, which determines the long-range temporal decay of the dissipative kernel.

Figures (6)

• Figure 1: Scaling function $G(\alpha, \beta)$ as a function of $\beta$ (in units of $1/E_C$) for $E_J/E_C=0.4$ (a), $E_J/E_C=0.5$ (b), $E_J/E_C=0.6$ (c), $E_J/E_C=0.7$ (d), for different values of the coupling $\alpha$ around the expected critical point for the Schmid transition. Estimated critical couplings are $\alpha_c \approx 1.147$, $\alpha_c \approx 1.104$, $\alpha_c \approx 1.07$, $\alpha_c \approx 1.037$ respectively. The legends indicate the value of the jump $\Psi_c$ used to generate the curves.
• Figure 2: Correlation function $\langle S(\tau)S(0)\rangle$ (blue dots) and corresponding logarithmic fit (red line) as functions of the imaginary time (in units of $\hbar/E_C$) for $E_J/E_C=0.4$ (a), $E_J/E_C=0.5$ (b), $E_J/E_C=0.6$ (c), $E_J/E_C=0.7$ (d), at $\beta E_C=50000$ for critical couplings $\alpha_c$ estimated via $G(\alpha, \beta)$ scaling argument. It should be noted that deviations from logarithmic decay only appear at short time and at $\tau \approx \beta\hbar/2$ due to finite size effects, i.e., finite $\beta$.
• Figure 3: $\sigma^2$ at $E_J/E_C = 0.5$ as a function of $\beta$ (in units of $1/E_C$) for the spectral function \ref{['eqn:spectral_form2']} at $s=1.1$ (a) and $s=0.9$ (b), for different values of the coupling $\alpha$. The lower cutoff is set to ${\tilde{\omega}}=0.5$ (circles) and ${\tilde{\omega}}=0.2$ (triangles).
• Figure 4: Critical coupling at finite size (blue dots) and corresponding logarithmic fit $\alpha_c + \frac{D}{2\ln\beta + E}$ (red lines) as functions of $\beta$ (in units of $1/E_C$) for $E_J/E_C = 0.4$ (a), $E_J/E_C = 0.5$ (b), $E_J/E_C = 0.6$ (c), $E_J/E_C = 0.7$ (d). The fitted $\alpha_c$ values are in good agreement with the Schmid critical coupling reported in the literature and calculated in the main text.
• Figure 5: (a) Order parameter $m^2$ as a function of $\alpha-\alpha_c$ for different values of $E_J$ (in units of $E_C$), computed at $\beta E_C = 50000$. The values of $\alpha_c$ are determined as discussed in the main text. Note the suppression of $m^2$ near the transition as $E_J/E_C$ decreases. (b) Order parameter $m^2$ as a function of $\beta$ (in units of $1/E_C$), evaluated exactly at the critical coupling $\alpha_c$. The jump in the order parameter corresponds to the asymptotic limit $\beta \to \infty$. These data confirm the reduction of the discontinuity for smaller $E_J/E_C$ values.