Global inconsistency, 't Hooft anomaly, and level crossing in quantum mechanics

TL;DR

The paper investigates global inconsistency as a milder counterpart to 't Hooft anomalies by studying solvable quantum-mechanical systems with topological angles. It develops both operator and path-integral formalisms to detect anomalies and global inconsistency, linking central extensions of symmetry to energy-level behavior and phase structure. Through one- and two-particle models on S^1, it shows how level crossings and symmetry realizations at high-symmetry points encode the underlying topological constraints, including mixed anomalies and bulk inflow or 2D bulk terms. The findings illuminate how nonperturbative symmetry constraints manifest in the spectrum and dynamics of simple quantum systems, offering templates for applying these ideas to more complex QFTs and bridging anomaly inflow, global inconsistency, and spectral features.

Abstract

An 't Hooft anomaly is the obstruction for gauging symmetries, and it constrains possible low-energy behaviors of quantum field theories by excluding trivial infrared theories. Global inconsistency condition is recently proposed as a milder condition but is expected to play an almost same role by comparing high symmetry points in the theory space. In order to clarify the consequence coming from this new condition, we discuss several quantum mechanical models with topological angles and explicitly compute their energy spectra. It turns out that the global inconsistency can be saturated not only by the ground-state degeneracy at either of high symmetry points but also by the level crossing (phase transition) separating those high symmetry points.

Figures (4)

• Figure 1: The schematic figure illustrating the global inconsistency in the space of coupling constants $\vec{g}$. In the original theory $\mathcal{T}$, symmetry $G$ exists at generic couplings $\vec{g}$ and it is enhanced by $H$ at $\vec{g}_1$ and $\vec{g}_2$. To gauge the symmetry $G$, $\mathcal{T}$ is coupled to the topological $G$-gauge theory with the discrete parameter $\vec{k}$. In $(\mathcal{T}/G)_{\vec{k}_1}$, the symmetry is absent except at $\vec{g}=\vec{g}_1$. In $(\mathcal{T}/G)_{\vec{k}_2}$, the symmetry is absent except at $\vec{g}=\vec{g}_2$.
• Figure 2: Energy levels as functions of $\theta$ with $\lambda=0.5$ in \ref{['eq:cos_int']} for $\mathbb{Z}_4$ and $\mathbb{Z}_3$ symmetric cases, respectively. Each color corresponds to different $\mathbb{Z}_n$ charge. (a) Every state forms a pair at $\theta=\pi, 3\pi, 5\pi$, which is a consequence of the t' Hooft anomaly. (b) Not every state forms a pair at $\theta=\pi, 3\pi, 5\pi$. But, a singlet state at $\theta=0$ are not continuously connected to a singlet state at $\theta=\pi$, which is a consequence of the global inconsistency.
• Figure 3: Energy spectra as functions of $\theta$ with $m_1=1$, $m_2=1/2$ and $\lambda=1$. Color of lines indicates the $U(1)$ charge of states. (a) All the levels are degenerate at $(\theta_1,\theta_2)=(\pi,0)$ due to the 't Hooft anomaly. (b) A singlet state at $(\theta_1,\theta_2)=(0,0)$ must be connected to a degenerate state at $(\theta_1,\theta_2)=(\pi,\pi)$ and vice versa due to the global inconsistency. (c) Singlet states at $(\theta_1,\theta_2)=(0,0)$ are connected to singlet states at $(\theta_1,\theta_2)=(\pi,-\pi)$.
• Figure 4: Phase diagram on $(\theta_1,\theta_2)$-plane with $\lambda=0.2$. Each line represents a level crossing (phase transition). The phase structure is $2\pi$ periodic along $\theta_1$ and $\theta_2$ axises.