Polarizable atoms interacting with a charged wire do so through an inverse-square potential, V = −g/r2. This system is known to realize scale invariance in a nontrivial way and to be subject to ambiguities associated with the choice of boundary condition at the origin, often termed the problem of ‘fall to the center’. Point-particle effective field theory (PPEFT) provides a systematic framework for determining the boundary condition in terms of the properties of the source residing at the origin. We apply this formalism to the charged-wire/polarizable-atom problem, finding a result that is not a self-adjoint extension because of absorption of atoms by the wire. We explore the RG flow of the complex coupling constant for the dominant low-energy effective interactions, finding flows whose character is qualitatively different when g is above or below a critical value, gc. Unlike the self-adjoint case, (complex) fixed points exist when g > gc, which we show correspond to perfect absorber (or perfect emitter) boundary conditions. We describe experimental consequences for wire-atom interactions and the possibility of observing the anomalous breaking of scale invariance.
Effective Field Theories Renormalization Group Nonperturbative Effects
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
L. Landau and E. Lifshitz, Mechanics, Elsevier Science (1982).Google Scholar
S.P. Alliluev, The problem of collapse to the center in quantum mechanics, JETP34 (1972) 8.ADSGoogle Scholar
R. Jackiw, Delta function potentials in two-dimensional and three-dimensional quantum mechanics, in Diverse topics in theoretical and mathematical physics, World Scientific (1991), pp. 25-42.Google Scholar
K.S. Gupta and S.G. Rajeev, Renormalization in quantum mechanics, Phys. Rev.D 48 (1993) 5940 [hep-th/9305052] [INSPIRE].
S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski, J. McGuire and U. van Kolck, Singular potentials and limit cycles, Phys. Rev.A 64 (2001) 042103 [quant-ph/0010073] [INSPIRE].
S.A. Coon and B.R. Holstein, Anomalies in Quantum Mechanics: the 1/r2Potential, Am. J. Phys.70 (2002) 513 [quant-ph/0202091] [INSPIRE].
M. Bawin and S.A. Coon, The Singular inverse square potential, limit cycles and selfadjoint extensions, Phys. Rev.A 67 (2003) 042712 [quant-ph/0302199] [INSPIRE].
E.J. Mueller and T.-L. Ho, Renormalization Group Limit Cycles in Quantum Mechanical Problems, [cond-mat/0403283].
E. Braaten and D. Phillips, The Renormalization group limit cycle for the 1/r2potential, Phys. Rev.A 70 (2004) 052111 [hep-th/0403168] [INSPIRE].
F. Werner, Trapped cold atoms with resonant interactions: unitary gas and three-body problem, Theses, Université Pierre et Marie Curie — Paris VI, Paris France (2008).Google Scholar
D. Bouaziz and M. Bawin, Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling, Phys. Rev.A 89 (2014) 022113 [arXiv:1402.5325] [INSPIRE].
L. Platter, Few-Body Systems and the Pionless Effective Field Theory, in proceedings of the 6th International Workshop on Chiral Dynamics (CD09), Bern, Switzerland, 6-10 July 2009, p. 104 [PoS(CD09)104] [arXiv:0910.0031] [INSPIRE].