[Paper Review] Scale Invariance Breaking and Discrete Phase Invariance in Few-Body Problems

Satoshi Ohya|arXiv (Cornell University)|Jan 14, 2026

Quantum Mechanics and Non-Hermitian Physics0 citations

TL;DR

The paper analyzes a novel breaking pattern of continuous scale invariance to discrete phase invariance in inverse-square systems and presents three few-body examples where circularly distributed S-matrix poles emerge.

ABSTRACT

Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.

Motivation & Objective

• Investigate how continuous scale invariance can break to discrete phase invariance in scale-invariant quantum mechanics.
• Characterize the S-matrix restructuring, including log-periodicity along the imaginary axis and circular pole distributions.
• Provide exact solutions for the one-body inverse-square problem in the intermediate window and extend to select few-body scenarios with codimension-two magnetic fluxes.

Proposed method

• Analyze the one-body Hamiltonian H = -d^2/dr^2 + λ/r^2 on the half-line and identify upper and lower critical λ values.
• Derive the boundary condition ensuring probability conservation, yielding a one-parameter self-adjoint extension parameter g.
• Solve the eigenvalue problem exactly using the f(z) and (z) basis built from Hankel functions to obtain bound-state and scattering solutions.
• Derive the S-matrix S(E) and show its discrete phase invariance in the intermediate window λ ∈ (-1/4, 3/4).
• Demonstrate that S(E) exhibits circularly distributed simple poles on Riemann sheets in the complex energy plane.
• Extend the analysis to few-body problems that map to inverse-square-like settings with a codimension-two magnetic flux, including the Aharonov-Bohm context.

Experimental results

Research questions

• RQ1 Does continuous scale invariance break to discrete phase invariance in the intermediate λ window for the inverse-square potential on the half-line?
• RQ2 What is the exact form and analytic structure of the S-matrix in this intermediate window, and how do poles distribute in the complex energy plane?
• RQ3 How do the inverse-square potential findings generalize to few-body problems with codimension-two magnetic flux?
• RQ4 What boundary conditions and self-adjoint extensions are required to maintain probability conservation in this setting?

Key findings

• In the intermediate window -1/4 < λ < 3/4 - continuous scale invariance is broken to discrete phase invariance.
• The S-matrix is exactly solvable and shows log-periodicity along the imaginary axis with circularly distributed simple poles on Riemann sheets.
• There exists a one-parameter family of boundary conditions (parameter g) that preserves probability conservation and introduces an energy scale E0.
• The bound-state spectrum can exhibit a single bound state for g > 0, with energy E0 = -|E0|, and, in general, complex pole structures appear on higher Riemann sheets.
• In few-body problems with codimension-two magnetic flux (Aharonov-Bohm type), the same discrete phase invariance mechanism arises, leading to similar resonant features in scattering.