Does a bound bipolaron ever break rotational symmetry?
For $U \ge 0$ and $\psi \in H^1(\mathbb{R}^6)$ with $\|\psi\|_2 = 1$, written $\psi(x, y)$ with $x, y \in \mathbb{R}^3$, let $$\rho_\psi(x) = \int_{\mathbb{R}^3} \bigl(|\psi(x,y)|^2 + |\psi(y,x)|^2\bigr)\,dy$$ and let $$\mathcal{P}_U(\psi) = \iint \Bigl(|\nabla_x \psi|^2 + |\nabla_y \psi|^2 + \frac{U\,|\psi|^2}{|x - y|}\Bigr)\,dx\,dy - \iint \frac{\rho_\psi(x)\,\rho_\psi(y)}{|x - y|}\,dx\,dy$$ be the Pekar–Tomasevich energy of the bipolaron. Let $E_2(U) = \inf \mathcal{P}_U$ over such $\psi$, and let $$E_1 = \inf_{\phi \in H^1(\mathbb{R}^3),\ \|\phi\|_2 = 1} \Bigl\{ \int |\nabla \phi|^2 - \iint \frac{|\phi(x)|^2\,|\phi(y)|^2}{|x - y|}\,dx\,dy \Bigr\}$$ be the energy of a single polaron. A minimizer of $\mathcal{P}_U$ exists whenever $E_2(U) < 2E_1$ (Lewin), and $E_2(U) = 2E_1$ for all sufficiently large $U$ (Frank, Lieb, Seiringer and Thomas). Frank, Lieb and Seiringer proved that for all sufficiently small $U$ the minimizer is unique up to translations and a constant phase, and after a translation satisfies $\psi(Rx, Ry) = \psi(x, y)$ for every $R \in O(3)$.
Problem. Is it true that for every $U \ge 0$ with $E_2(U) < 2E_1$, every minimizer $\psi$ of $\mathcal{P}_U$ satisfies, after a common translation of its two coordinates, $$\psi(Rx, Ry) = \psi(x, y) \quad \text{for every } R \in O(3)?$$ Equivalently, does rotational symmetry breaking occur anywhere in the binding regime?
Source: Frank, Lieb and Seiringer (2013), after Theorem 1, for repulsion near the critical value at which the bipolaron ceases to be bound. Stated by AIM — Open Applied Problems for the whole binding regime.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, polarons, symmetry breaking
References
@article{frank2013,
author = {Frank, Rupert L. and Lieb, Elliott H. and Seiringer, Robert},
title = {Symmetry of bipolaron bound states for small {C}oulomb repulsion},
journal = {Comm. Math. Phys.},
volume = {319},
number = {2},
year = {2013},
pages = {557--573},
doi = {10.1007/s00220-012-1604-y},
eprint = {1201.3954},
archivePrefix = {arXiv},
}
@article{frank2011,
author = {Frank, Rupert L. and Lieb, Elliott H. and Seiringer, Robert and Thomas, Lawrence E.},
title = {Stability and absence of binding for multi-polaron systems},
journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.},
volume = {113},
year = {2011},
pages = {39--67},
doi = {10.1007/s10240-011-0031-5},
}
@article{lewin2011,
author = {Lewin, Mathieu},
title = {Geometric methods for nonlinear many-body quantum systems},
journal = {J. Funct. Anal.},
volume = {260},
number = {12},
year = {2011},
pages = {3535--3595},
doi = {10.1016/j.jfa.2010.11.017},
}
@misc{aim-bipolaron-rotation-symmetry,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does a bound bipolaron ever break rotational symmetry?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/202-bipolaron-rotation-symmetry.md},
note = {Entry 202 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00411, "Does a bound bipolaron ever break rotational symmetry?", clopen.solutions.