Scale Invariance Breaking and Discrete Phase Invariance in Few-Body Problems

TL;DR

This work identifies a novel symmetry-breaking pattern in quantum few-body systems: continuous scale invariance can break to discrete phase invariance, a complexified analogue of log-periodicity. Focusing on the inverse-square potential in the intermediate window $\lambda_{*}<\lambda<\lambda_{**}$, the authors derive an exact S-matrix that is log-periodic along the imaginary axis and features circularly distributed simple poles on the S-matrix Riemann surface. They then show that three few-body problems — the one-body Aharonov-Bohm problem, a two-body problem in two dimensions, and a one-dimensional three-body problem — realize the same discrete phase invariance in two angular-momentum channels, tying topology to symmetry breaking. The results illuminate how discrete phase invariance can shape resonance-like scattering and offer a framework for exploring pole structures on higher Riemann sheets in systems with fluxes in configuration space. Together, these findings broaden the understanding of scale-symmetry breaking patterns in quantum mechanics and suggest new avenues for observing novel resonant phenomena in flux-bearing few-body setups.

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.

Figures (2)

• Figure 1: "Phases" of the inverse-square-potential problem. There exist two critical values of the coupling constant $\lambda$: the upper critical value $\lambda_{\ast\ast}=3/4$ and the lower critical value $\lambda_{\ast}=-1/4$. These two values are determined by the boundary behavior of energy eigenfunction. Near the boundary, solutions to the eigenvalue equation $H\psi_{E}(r)=E\psi_{E}(r)$ are well approximated by the zero-energy solutions given by $r^{1/2\pm\sqrt{\lambda+1/4}}$. If these solutions are square integrable near the boundary, the general solution behaves as a linear combination $Ar^{1/2+\sqrt{\lambda+1/4}}+Br^{1/2-\sqrt{\lambda+1/4}}$ near the boundary. But $r^{1/2+\sqrt{\lambda+1/4}}$ and $r^{1/2-\sqrt{\lambda+1/4}}$ have different scaling dimensions, such a linear combination must introduce a scale, which breaks continuous scale invariance. On the other hand, if one of these solutions is non-square integrable even in a small interval $0<r<\varepsilon$, there is no chance to introduce a scale so that continuous scale invariance remains intact. For $\lambda>\lambda_{\ast\ast}=3/4$, the solution $r^{1/2-\sqrt{\lambda+1/4}}$ is non-square integrable near the boundary and hence continuous scale invariance is never broken. This is the continuous-scale-invariant phase above the upper critical value. For $\lambda<\lambda_{\ast\ast}$, on the other hand, both the solutions are square integrable near the boundary and hence continuous scale invariance is broken. But for $\lambda=\lambda_{\ast}<-1/4$, the linear combination $Ar^{1/2+i\sqrt{-1/4-\lambda}}+Br^{1/2-i\sqrt{-1/4-\lambda}}$ becomes invariant (up to an overall factor) under the discrete scale transformation $r\mapsto\mathop{\mathrm{e}}\nolimits^{t}r$ with $t=n\pi/\sqrt{-1/4-\lambda}$, where $n$ is an integer. This is the discrete-scale-invariant phase below the lower critical value $\lambda_{\ast}=-1/4$. Furthermore, for $\lambda_{\ast}<\lambda<\lambda_{\ast\ast}$, the linear combination $Ar^{1/2+\sqrt{\lambda+1/4}}+Br^{1/2-\sqrt{\lambda+1/4}}$ becomes invariant (up to an overall factor) under the discrete phase transformation $r\mapsto\mathop{\mathrm{e}}\nolimits^{t}r$ with $t=in\pi/\sqrt{\lambda+1/4}$. This is the discrete-phase-invariant phase in the intermediate window.
• Figure 2: Typical simple-pole distribution on the Riemann surface of the S-matrix \ref{['eq:32']} for $g>0$. Blue dots represent the simple poles, which are distributed along the blue circle of radius $\kappa_{0}$. Red dots represent the branch point at $k=0$. Branch cuts are chosen along the negative imaginary axis and represented by red lines. The whole Riemann surface is given by gluing adjacent sheets along the branch cut. In this figure, $\nu$ is chosen as the irrational number $\nu=1/\sqrt{2}$ so that the simple poles are distributed at $k=i\kappa_{0}\mathop{\mathrm{e}}\nolimits^{in\pi/\nu}=\kappa_{0}\mathop{\mathrm{e}}\nolimits^{i(1/2+\sqrt{2}n)\pi}$.