Quantum hierarchical Fokker-Planck equations with U(1) gauge fields: Application to the Aharonov-Bohm ring

TL;DR

The paper develops gauge- and rotationally invariant non-Markovian quantum dynamics for a 3D subsystem coupled to a time-dependent $U(1)$ gauge field via the $U(1)$-HEOM, and its Wigner-space counterpart $U(1)$-QHFPE. By applying this formalism to an Aharonov-Bohm ring, it demonstrates equilibrium distributions, linear response spectra, and non-Markovian persistent currents, highlighting qualitative differences between rotationally invariant baths (RISB) and Caldeira–Leggett (CL) baths. The results show that persistent currents can persist in dissipative environments at low temperature when the bath is non-Markovian, while Markovian approximations fail to capture this effect, and that RISB better describe rotational spectral features than CL in nanoscale rings. The framework provides a rigorous tool for studying gauge-field effects in topological and nanoscale quantum systems and suggests avenues for future quantum simulations and symmetry-consistent bath modeling.

Abstract

We investigate a three-dimensional subsystem under a time-dependent U(1) gauge field coupled to rotationally invariant environments. To capture the dynamic behavior of the subsystem under thermal excitations and dissipations, it is imperative to treat the bath in a non-Markovian and nonperturbative manner. This is because quantum noise is constrained by the uncertainty principle, which dictates the relationship between the noise correlation time and the amplitude of the energy fluctuation. To this end, we derive the hierarchical equations of motion (HEOM) incorporating the gauge field, enabling a rigorous investigation of the dynamics of the reduced subsystem. Transforming the HEOM into the Wigner representation yields quantum hierarchical Fokker-Planck equations [U(1)-QHFPE] with U(1) gauge fields. These equations incorporate vector fields into the damping operators while preserving both gauge invariance and rotational symmetry. To demonstrate the practical use of the formalism, the effects of a heat bath in the Aharonov-Bohm (AB) ring. Our investigation includes simulations of the equilibrium distribution, linear absorption spectra, and AB currents under thermal conditions. Within a rotationally invariant system-bath (RISB) model, we predict the emergence of a persistent current even in dissipative environments, provided the bath is non-Markovian and the temperature is sufficiently low. We also assessed the validity of the Caldeira-Leggett model in this context.

Figures (5)

• Figure 1: Schematic of an AB ring system. We consider the single flow case, in which a single charge rotates in one direction along the ring. The magnetic field is confined only inside the ring and is set to zero outside, as shown in Table \ref{['tab:MagVec']}.
• Figure 2: The equilibrium distribution in the momentum space for the single charge flow obtained from the RISB model [Eq. \ref{['eqn:FP_RISB']}] (blue bars) and the CL model [Eq. \ref{['eqn:FP_CL']}] (red curves). Here we set $\eta = 0.01$, $\beta = 0.2$ for magnetic flux (a) $\bar{\Phi} = 0$ and (b) $\bar{\Phi}= 1$ for $\bar{\Phi}\equiv \Phi/\Phi_0$. In the case of the RISB model, unlike the CL model, which exhibits a continuous distribution, it shows a discretized distribution that exists only at integer values. However, in both cases, the distribution follows a Gaussian profile centered at $\bar{\Phi}$.
• Figure 3: Rotational spectrum without magnetic flux ($i.e. \bar{\Phi}=0$) are shown for the RISB (blue curves) and CL (red curves) cases under fixed $\beta = 0.2$, with (a) $\eta = 0.01$, (b) $0.1$, and (c) $1.0$ for fixed $\beta = 0.2$.
• Figure 4: Rotational spectrum under the influence of the vector potential. Each spectrum is obtained with $\eta = 0.01$ and $\beta = 0.2$ at various magnetic fluxes: (a) $\bar{\Phi} = 0.1$, (b) $0.2$, (c) $0.3$, (d) $0.4$, and (e) $0.5$. The case for $\bar{\Phi} = 0$ is presented in Fig. \ref{['fig:ProjCurr']} (a).
• Figure 5: Persistent current computed from the U(1)-QHFPE is plotted as a function of the magnetic flux $\bar{\Phi}$ for very weak ($\eta = 1.0 \times 10^{-3}$, blue curves) and the strong ($\eta = 1.0$, red curves) S-B coupling cases at the low temperature $\beta = 2.5$. The orange line represents the high-temperature case ($\beta = 0.2$) with the weak S-B coupling ($\eta = 1.0 \times 10^{-3}$): The persistent current vanishes at this high-temperature regime, regardless of the coupling strength, aligning with the behavior of the orange line. Similarly, Markovian results—relying on high-temperature approximations—also show no persistent current.(not shown.)