Critical Charge and Current Fluctuations across a Voltage-Driven Phase Transition

Abstract

We investigate bias-driven non-equilibrium quantum phase transitions in a paradigmatic quantum-transport setup: an interacting quantum dot coupled to non-interacting metallic leads. Using the Random Phase Approximation, which is exact in the limit of a large number of dot levels, we map out the zero-temperature non-equilibrium phase diagram as a function of interaction strength and applied bias. We focus our analysis on the behavior of the charge susceptibility and the current noise in the vicinity of the transition. Remarkably, despite the intrinsically non-equilibrium nature of the steady state, critical charge fluctuations admit an effective-temperature description, $T_{\text{eff}}(T,V)$, that collapses the steady-state behavior onto its equilibrium form. In sharp contrast, current fluctuations exhibit genuinely non-equilibrium features: the fluctuation-dissipation ratio becomes negative in the ordered phase, corresponding to a negative effective temperature for the current degrees of freedom. These results establish current noise as a sensitive probe of critical fluctuations at non-equilibrium quantum phase transitions and open new directions for exploring voltage-driven critical phenomena in quantum transport systems.

Figures (6)

• Figure 1: (a) Schematic representation of the quantum transport setup. (b) Order parameter $\phi$ as a function of the inverse interaction strength $\lambda^{-1}$ for both zero $(T=0)$ and finite $(T>0)$ temperature. (c) Conductance $G=dI/dV$ as a function of voltage $V$ for several different temperatures. (d) Non-equilibrium phase diagram as a function of interaction strength $\lambda^{-1}$, temperature $T$, and bias voltage $V$, showing the continuous transition between the ordered and disordered phases.
• Figure 2: Charge Susceptibility across the non-equilibrium phase transition. (a-c) Real part (a), imaginary part (b) of the retarded susceptibility $\chi_{\rm RPA}^R$, and the Keldysh component (c) $\chi_{\rm RPA}^K$ as a function of frequency $\omega$. Curves are shown for $T=0$ at $V<V_c$, $V=V_c$, and $V>V_c$ (orange scale), and for $T>0$ at $V=V_c (T)$. (d) Zero-frequency retarded $\chi^{\rm RPA}_R(\omega=0)$ and Keldysh $\chi^{\rm RPA}_K(\omega=0)$ components as a function of voltage, showing the critical divergence at $V_c$. (e) Fluctuation-dissipation ratio (FDR) as a function of frequency $\omega$, using the same parameters and color-coding as (a-c). (f) Effective temperature $T_{\rm eff}/T$, derived from the zero-frequency FDR, plotted as a function of $V/T$ for several lead temperatures $T$.
• Figure 3: (a) Order parameter $\phi$ as a function of the effective temperature $T^{\chi}_{\rm eff}$. (b) The $V–T$ phase diagram. The color-coded points indicate the $(V,T)$ values plotted in (a). The inset shows the effective temperature along the critical line $V_c(T)$. (c) Ratio of effective temperatures derived from current noise $T^{\mathcal{S}_0}_{\rm eff}$ and charge susceptibility $T^{\chi}_{\rm eff}$ in the non-interacting limit, shown as a function of voltage $V$ and temperature $T$.
• Figure 4: RPA Current Noise Correlations. Panels (a-c) show the $T=0$ case for several voltages ($V=0$, $V<V_c$, $V=V_c=0.7\Lambda$, $V>V_c$): (a) Imaginary part of the Keldysh component $\Im \mathcal{S}^K_{\rm RPA}$ as a function of frequency $\omega$. (b) Imaginary part of the advanced response function $\Im \mathcal{S}^A_{\rm RPA}$ as a function of frequency $\omega$. (c) The fluctuation-dissipation ratio $\Im \mathcal{S}^R_{\rm RPA}/\Im \mathcal{S}^K_{\rm RPA}$ as a function of frequency $\omega$. Panels (d-f) show the $V=0$(equilibrium) case for several temperatures:(d) Imaginary part of the Keldysh component $\Im \mathcal{S}^K_{\rm RPA}$ as a function of frequency $\omega$. (e) Imaginary part of the advanced response function $\Im \mathcal{S}^A_{\rm RPA}$ as a function of frequency $\omega$. (f) The fluctuation-dissipation ratio $\Im \mathcal{S}^R_{\rm RPA}/\Im \mathcal{S}^K_{\rm RPA}$ as a function of frequency $\omega$.
• Figure 5: Zero-frequency effective temperature derived from current noise $T^{\mathcal{S}_{\rm RPA}}_{\rm eff}$ as a function of voltage $V$ and temperature $T$. The gray crosses indicate a region where results are omitted due to numerical precision issues.