Dynamics of the Fermion-Rotor System

TL;DR

This work analyzes a simple yet subtle impurity model—the fermion-rotor system in $d=1+1$—where purely right-moving fermions interact with a localized rotor. The authors show the rotor acts as a twist operator, enabling consistent scattering that respects symmetries and yielding dressed fermions that propagate as free single-particle states; they extend the construction to multiple rotors and unequal charges, framing it as a UV completion of boundary states in chiral theories and connecting to BCFT concepts including the rotation matrix ${\cal R}_{ij}$ and the $3450$ model. Through explicit correlation-function calculations, they derive how rotor insertions modify correlators, demonstrate cluster decomposition, and reveal the mild nonlocality of the twist operators. A mod $2$ anomaly is identified, descending from the 4d Witten anomaly, with Majorana zero modes whose presence depends on the boundary-state class (vector vs axial) and on the parity of $N$; the circle compactification analysis makes this connection precise. Altogether, the paper provides a coherent framework linking monopole-fermion scattering, twist operators, BCFT boundary states, and anomaly structure in a lower-dimensional, chirally coupled system, with concrete illustrations exemplified by the $3450$ model.

Abstract

We explore the dynamics of the fermion-rotor system, a simple impurity model in d=1+1 dimensions that consists of a collection of purely right-moving fermions interacting with a quantum mechanical rotor localised at the origin. This was first introduced by Polchinski as a toy model for monopole-fermion scattering and is surprisingly subtle, with ingoing and outgoing fermions carrying different quantum numbers. We show that the rotor acts as a twist operator in the low-energy theory, changing the quantum numbers of excitations that have previously passed through the origin to ensure scattering consistent with all symmetries. We further show how generalisations of this model with multiple rotors and unequal charges can be viewed as a UV-completion of boundary states for chiral theories, including the well-studied 3450 model. We compute correlation functions between ingoing and outgoing fermions and show that fermions dressed with the rotor degree of freedom act as local operators and create single-particle states, generalising an earlier result obtained in a theory with a single rotor and equal charges. Finally, we point out a mod 2 anomaly in these models that descends from the Witten anomaly in 4d

Figures (2)

• Figure 1: The function $f(x)$ is a regularised step function, while $f'(x)$ is a regularised delta-function, with the length scale $r_0$ acting as the regulator.
• Figure 2: The space-time diagram of fermion-rotor insertion.