The $θ$-vacuum from functional renormalisation

TL;DR

This paper develops a functional renormalisation group framework to study topological properties of a $U(1)$-symmetric quantum mechanical system with a $\theta$-term. By embedding the symmetry into $\mathbb{C}$ and decomposing into topological sectors, the authors construct sector-specific effective potentials $V_n(r^2,\theta)$ and assemble the full potential via a sector-minimization principle, validating the approach against exact Schrödinger-equation results. They show that topology information is encoded in UV boundary conditions and devise a regulator strategy that avoids topology freezing, enabling accurate computation of vacuum energy, energy levels, cusps in the effective potential, and topological susceptibility within the local potential approximation. The results robustly reproduce rotor limits at large coupling and demonstrate non-analytic cusp structures, providing a solid stepping stone toward applying fRG to theta-vacua in gauge theories such as the Schwinger model or QCD. Overall, the work establishes a controlled non-perturbative method for incorporating topology in quantum mechanical systems and informs future non-perturbative treatments of topological sectors in higher-dimensional theories.

Abstract

We study topological properties of a quantum mechanical system with $U(1)$-symmetry within the functional renormalisation group (fRG) approach. These properties include the vacuum energy structure and the topological susceptibility. Our approach works with a complexification of the flow equation, and specifically we embed the original symmetry into the complex plane, $U(1)\rightarrow \mathbb{C}$. We compute the effective potential of a given topological sector by restricting ourselves to field configurations with a given generalised non-trivial Chern-Simons numbers. The full potential is directly constructed from these sector potentials. Our results compare well with the benchmark results obtained from solving the corresponding Schrödinger equation.

Figures (15)

• Figure 1: Effective potential $\Delta V(r^2,\theta)=V(r^2,\theta)-V(r^2,0)$, \ref{['eq:DeltaV']} of $U(1)$-symmetric quantum mechanics. It is obtained from the functional renormalisation group approach with Litim-type regulator, see \ref{['sec:NumericalResults']}.
• Figure 2: $\theta$-parameter dependence of the energy levels for each $n$. The left-hand side panel exhibits the energy levels \ref{['eq:EgInfty']} which corresponds to $g\to \infty$. The solid lines indicate the ground-state energy in the system. In the right-hand side panel, we depict the energy levels obtained by solving the Schrödinger equation \ref{['eq:RadialSchroedinger']} for $g=30$.
• Figure 3: The additional potential energy $V_\textrm{top}$ with respect to the $\theta$-parameter at fixed $r^2 = 1$, which corresponds to the "vacuum energy" at finite "temperature" $1/\beta$. The dashed lines denote the potential energy with different energy levels $n$ with respect to $\theta$. The orange and the red lines denote the potentials with $\beta = 10$ and $\beta = 10^2$, respectively.
• Figure 4: The additional potential $V_\textrm{top}(r^2,\theta)$ realised from the summation of all winding configurations. The red line denotes the potential evaluated at $r^2 = 1$, which corresponds to the ground-state energy (the left-hand side panel of \ref{['fig:thetaVacuumExact']}), and the blue line denotes the potential in terms of the field variable with fixed $\theta = 2 \pi$.
• Figure 5: The $r$-dependence of the 'topological' potential energy $V_\textrm{top}(r^2,\theta$ fixed $\theta = 2 \pi$ and the limit $\beta \rightarrow \infty$ (solid blue line), which corresponds to the blue line in \ref{['fig:SummedVtopAtUV']}. The dashed lines denote the potential energy \ref{['eq:VtopAtUV']} with different energy levels $n$, evaluated at $\theta = 2 \pi$.