Theta dependence, sign problems and topological interference

TL;DR

The $\theta$-angle in gauge theories creates a sign problem in Euclidean formulations and a geometric phase in Minkowski space, leading to rich $\theta$-dependence of the vacuum structure that the authors analyze through circle-compactified, semiclassical deformed Yang–Mills. In SU(2) and related quantum-mechanical models, $\theta$ induces interference among monopole-like defects, causing a two-branched vacuum structure with a CP-symmetric phase at $\theta=\pi$ and a small-mass-gap regime generated by magnetic bions. The work systematically develops a dual long-distance description on $\mathbb{R}^3\times S^1$, derives explicit mass gaps, string tensions, and deconfinement temperatures as functions of $\theta$, and reveals deep connections to Aharonov-Bohm effects and Berry phases, linking 4d topological physics to lower-dimensional topological terms. Through perturbative/nonperturbative resurgence arguments and the continuity to deformed theories, the results provide a controlled framework to understand $\theta$-dependence and CP realization, with implications for lattice studies and cross-dimensional topological phenomena.

Abstract

In a Euclidean path integral formulation of gauge theory and quantum mechanics, the theta-term induces a sign problem, and relatedly, a complex phase for the fugacity of topological defects; whereas in Minkowskian formulation, it induces a topological (geometric) phase multiplying ordinary path-amplitudes. In an SU(2) Yang-Mills theory which admits a semi-classical limit, we show that the complex fugacity generates interference between Euclidean path histories, i.e., monopole-instanton events, and radically alters the vacuum structure. At theta=0, a mass gap is due to the monopole-instanton plasma, and the theory has a unique vacuum. At theta=pi, the monopole induced mass gap vanishes, despite the fact that monopole density is independent of theta, due to destructive topological interference. The theory has two options: to remain gapless or to be gapped with a two-fold degenerate vacua. We show the latter is realized by the magnetic bion mechanism, and the two-vacua are realization of spontaneous CP-breaking. The effect of the theta-term in the circle-compactified gauge theory is a generalization of Aharonov-Bohm effect, and the geometric (Berry) phase. As theta varies from 0 to pi, the gauge theory interpolates between even- and odd-integer spin quantum anti-ferromagnets on two spatial dimensional bi-partite lattices, which have ground state degeneracies one and two, respectively, as it is in gauge theory at theta=0 and theta=pi.

Figures (7)

• Figure 1: The $\theta$ angle (in)dependence of observables in large-$N$ limit of gauge theory. For extensive observables, such as vacuum energy density, the $\theta$ dependence is present at $N=\infty$. The Hilbert space and the mass gap exhibits $\theta$ independence at $N = \infty$. The figure is for $N=5$. At $N=\infty$, $m(\theta)$ becomes a straight horizontal line.
• Figure 2: Field configuration as a function of Euclidean time and the equivalent dilute gas of instantons and topological molecules. In the textbook treatment, usually, only instantons are accounted for. Topological molecules such as $[{\cal I} {\cal I}], [ \overline {\cal I} \overline {\cal I}], [{\cal I} \overline {\cal I}]$ despite being rarer, are nonetheless present. There are some effects for which instantons do not contribute, and the leading semi-classical contribution arise from molecular instantons. The topological molecules are also crucial in order to make sense of the continuum theory in connection with large-orders in perturbation theory.
• Figure 3: The plot of the integrand over the quasi-zero mode (separation between two instanton events) for $g \ll 1$. The saddle point of the integral is located at $r_{\rm b{\cal I}}= \log\left( \frac{32}{g}\right)$. Since the separation between these two (correlated) instanton events $r_{\rm b{\cal I}}$ is much larger than the instanton size, each instanton is individually sensible. Since $r_{\rm b{\cal I}}$ is exponentially smaller than the typical inter-instanton separation, these pairs cannot be viewed as two uncorrelated single instanton events. Due to this reason, we interpret the resulting structure as a topological molecule, with size $r_{\rm b{\cal I}}$.
• Figure 4: The $\theta$ angle in the $T_N(\theta)$-model is the equivalent of Aharonov-Bohm flux $\Phi$ in units of the flux quantum $\Phi_0$, with identification $\frac{\theta}{2\pi} \equiv \frac{\Phi}{\Phi_0}$.
• Figure 5: The dilute gas of monopole-instantons and bions. In Euclidean space where monopole-instantons are viewed as particles, the correlated instanton events should be viewed as molecules. Despite the fact that the density of monopole-instantons is independent of $\theta$, at $\theta=\pi$, the effect of the monopole-instanton events dies off due to destructive topological interference, and the properties of dYM theory are determined by a dilute bion plasma.