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arXiv:2603.25508 · Journal of High Energy Physics

What happens to wavepackets of fermions when scattered by the Maldacena-Ludwig wall?

Yuji Tachikawa, Keita Tsuji, Masataka Watanabe

Published Oct 12, 2026• 0 citations
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Abstract

A bstract We study wavepackets of exotic excitations after two-dimensional fermions are scattered by the boundary condition constructed by Maldacena and Ludwig, which turns elementary excitations into exotic fractionally-charged objects. They are of interest in the s-wave approximation of the fermion-monopole scattering in four-dimensional QED and of the multi-channel Kondo effect. Our main result is an explicit expression for the outgoing state of a pair of such particles. We show analytically and numerically that the expectation value 〈 N 〉 of the total number of fermions and anti-fermions diverges when the wavepacket is localized to a point. The energy and charge densities remain localized, and the charge integrated locally around each wavepacket is finite and fractional, as expected from the known action of the boundary condition.

DOI
10.1007/jhep10(2026)140

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A bstract We study wavepackets of exotic excitations after two-dimensional fermions are scattered by the boundary condition constructed by Maldacena and Ludwig, which turns elementary excitations into exotic fractionally-charged objects. They are of interest in the s-wave approximation of the fermion-monopole scattering in four-dimensional QED and of the multi-channel Kondo effect. Our main result is an explicit expression for the outgoing state of a pair of such particles. We show analytically and numerically that the expectation value 〈 N 〉 of the total number of fermions and anti-fermions diverges when the wavepacket is localized to a point. The energy and charge densities remain localized, and the charge integrated locally around each wavepacket is finite and fractional, as expected from the known action of the boundary condition.
Fermion number fractionization in quantum field theory on a finite interval is studied for a (1 + 1) dimensional fermion-soliton system with explicit charge conjugation symmetry breaking. The effects of boundary conditions on the fractional fermion number and the connection with the corresponding open space problem are investigated. It is argued that the open space fractional charges can be correctly reproduced from the finite interval results only through a careful definition of what is meant by the soliton charge. This definition of the charge distinguishes between the fermionic and boundary induced charges in the system, and isolates the soliton from possibly other charged topological objects in the system. It therefore gives a true measure of the localized fractional fermion number induced on the soliton of interest. It is then rigorously proven that the corresponding charge fluctuations vanish, and hence that the induced fractional charge on the soliton is a quantum observable.
In the presence of extended defects, familiar incoming particles can scatter into exotic outgoing states created by twist operators. We show that one possible mechanism driving these "categorical scattering" processes is the presence of localized 't Hooft anomalies on the defect's worldvolume. Defect anomalies trap non-trivial charges at junctions between the symmetry lines and the interface, opening new transmission channels that would naively appear to violate selection rules. After outlining the general mechanism, we investigate several concrete examples with defects, interfaces, and boundaries. For models of massless chiral fermions already studied in the literature, we show that the emergence of twist operators can be understood as a consequence of defect anomalies. We then introduce new massive integrable theories in which a similar phenomenon occurs, and we explicitly solve the associated scattering problem, obtaining new integrable solutions. Finally, we construct lattice spin chains with defects where similar physics is expected to arise.
A bstract We explore the dynamics of the fermion-rotor system, a simple impurity model in d = 1 +1 dimensions consisting 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, generalizing 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.
Boundary condition and reflection anomaly in 2+12+12+1 dimensions
It is known that the 2+1 2 + 1 d single Majorana fermion theory has an anomaly of the reflection, which is canceled out when 16 copies of the theory are combined. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition that varies at a single point, and find that there is a problem with one Majorana fermion. The problem is the absence of a corresponding outgoing wave to a specific incoming wave into the boundary, which leads to the non-conservation of the energy. For 16 Majorana fermions, it is possible to connect every incoming wave to an outgoing wave without breaking the reflection symmetry. In addition, we discuss the connection with the fermion-monopole scattering in 3+1 3 + 1 dimensions.
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We study an exotic Kondo effect in a system consisting of two one-dimensional XX Heisenberg ferromagnetic spin-1/2 chains (denoted by α=u,d for up and down chains) coupled to a quantum dot consisting of two localized spin-1/2 magnets. Using the Jordan–Wigner transformation on the Heisenberg Hamiltonian of the two chains, this system can be expressed in terms of non-interacting spinless fermionic quasiparticles. As a result, the Hamiltonian of the whole system is expressed as an Anderson model for spin-1/2 fermions interacting with a spin-1/2 impurity. Thus, we study the scattering of fermionic quasiparticles (propagating along spin chains) by a pair of localized magnetic impurities. At low temperature, the localized spin-1/2 magnets are shielded by the chain “spins” via the Kondo effect. We calculate the Kondo temperature TK and derive the temperature dependence of the entropy, the specific heat and the “magnetic susceptibility” of the dot for T≫TK. Our results can be generalized to the case of antiferromagnetic XX chains.