# Periodic instantons

Periodic instantons are finite energy solutions of Euclidean-time field equations which communicate (in the sense of quantum tunneling) between two turning points in the barrier of a potential and are therefore also known as bounces. Vacuum instantons, normally simply called instantons, are the corresponding zero energy configurations in the limit of infinite Euclidean time. For completeness we add that ``sphalerons´´ are the field configurations at the very top of a potential barrier. Vacuum instantons carry a winding (or topological) number, the other configurations do not. Periodic instantons werde discovered with the explicit solution of Euclidean-time field equations for double-well potentials and the cosine potential with non-vanishing energy[1] and are explicitly expressible in terms of Jacobian elliptic functions (the generalization of trigonometrical functions). Periodic instantons describe the oscillations between two endpoints of a potential barrier between two potential wells. The frequency ${\displaystyle \Omega }$ of these oscillations or the tunneling between the two wells is related to the bifurcation or level splitting ${\displaystyle \Delta E}$ of the energies of states or wave functions related to the wells on either side of the barrier, i.e. ${\displaystyle \Omega =\Delta E/\hbar }$. One can also interpret this energy change as the energy contribution to the well energy on either side originating from the integral describing the overlap of the wave functions on either side in the domain of the barrier.

Evaluation of ${\displaystyle \Delta E}$ by the path integral method requires summation over an infinite number of widely separated pairs of periodic instantons -- this calculation is therefore said to be that in the ``dilute gas approximation´´.

Periodic instantons have meanwhile been found to occur in numerous theories and at various levels of complication. In particular they arise in investigations of the following topics.

(1) Quantum mechanics and path integral treatment of periodic and anharmonic potentials.[1][2][3][4]

(2) Macroscopic spin systems (like ferromagnetic particles) with phase transitions at certain temperatures.[5][6][7] The study of such systems was started by D.A. Garanin and E.M. Chudnovsky[8][9] in the context of condensed matter physics, where half of the periodic instanton is called a ``thermon´´.[10]

(3) Two-dimensional abelian Higgs model and four-dimensional electro-weak theories.[11][12]

(4) Theories of Bose-Einstein condensation and related topics in which tunneling takes place between weakly-linked macroscopic condensates confined to double-well potential traps.[13][14]

## References

1. Liang, Jiu-Qing; Müller-Kirsten, H.J.W.; Tchrakian, D.H. (1992). "Solitons, bounces and sphalerons on a circle". Physics Letters B. 282 (1–2): 105–110. doi:10.1016/0370-2693(92)90486-N. ISSN 0370-2693.
2. Liang, Jiu-Qing; Müller-Kirsten, H. J. W. (1992). "Periodic instantons and quantum-mechanical tunneling at high energy". Physical Review D. 46 (10): 4685–4690. doi:10.1103/PhysRevD.46.4685. ISSN 0556-2821.
3. J.-Q. Liang and H.J.W. Müller-Kirsten: Periodic instantons and quantum mechanical tunneling at high energy, Proc. 4th Int. Symposium on Foundations of Quantum Mechanics, Tokyo 1992, Jpn. J. Appl. Phys., Series 9 (1993) 245-250.
4. Liang, J.-Q.; Müller-Kirsten, H. J. W. (1994). "Nonvacuum bounces and quantum tunneling at finite energy". Physical Review D. 50 (10): 6519–6530. doi:10.1103/PhysRevD.50.6519. ISSN 0556-2821.
5. Liang, J.-Q; Müller-Kirsten, H.J.W; Zhou, Jian-Ge; Pu, F.-C (1997). "Quantum tunneling at excited states and macroscopic quantum coherence in ferromagnetic particles". Physics Letters A. 228 (1–2): 97–102. doi:10.1016/S0375-9601(97)00071-6. ISSN 0375-9601.
6. Liang, J.-Q.; Müller-Kirsten, H. J. W.; Park, D. K.; Zimmerschied, F. (1998). "Periodic Instantons and Quantum-Classical Transitions in Spin Systems". Physical Review Letters. 81 (1): 216–219. arXiv:cond-mat/9805209. doi:10.1103/PhysRevLett.81.216. ISSN 0031-9007.
7. Zhang, Yunbo; Nie, Yihang; Kou, Supeng; Liang, Jiuqing; Müller-Kirsten, H.J.W.; Pu, Fu-Cho (1999). "Periodic instanton and phase transition in quantum tunneling of spin systems". Physics Letters A. 253 (5–6): 345–353. arXiv:cond-mat/9901325. doi:10.1016/S0375-9601(99)00044-4. ISSN 0375-9601.
8. Chudnovsky, E. M.; Garanin, D. A. (1997). "First- and Second-Order Transitions between Quantum and Classical Regimes for the Escape Rate of a Spin System". Physical Review Letters. 79 (22): 4469–4472. arXiv:cond-mat/9805060. doi:10.1103/PhysRevLett.79.4469. ISSN 0031-9007.
9. Garanin, D. A.; Chudnovsky, E. M. (1997). "Thermally activated resonant magnetization tunneling in molecular magnets:Mn12Acand others". Physical Review B. 56 (17): 11102–11118. arXiv:cond-mat/9805057. doi:10.1103/PhysRevB.56.11102. ISSN 0163-1829.
10. Chudnovsky, Eugene M. (1992). "Phase transitions in the problem of the decay of a metastable state". Physical Review A. 46 (12): 8011–8014. doi:10.1103/PhysRevA.46.8011. ISSN 1050-2947.
11. Khlebnikov, S.Yu.; Rubakov, V.A.; Tinyakov, P.G. (1991). "Periodic instantons and scattering amplitudes". Nuclear Physics B. 367 (2): 334–358. doi:10.1016/0550-3213(91)90020-X. ISSN 0550-3213.
12. Cherkis, Sergey A.; O’Hara, Clare; Zaitsev, Dmitri (2016). "A compact expression for periodic instantons". Journal of Geometry and Physics. 110: 382–392. arXiv:1509.00056. doi:10.1016/j.geomphys.2016.09.008. ISSN 0393-0440.
13. Zhang, Y.-B.; Müller-Kirsten, H.J.W. (2001). "Instanton approach to Josephson tunneling between trapped condensates". The European Physical Journal D. 17 (3): 351–363. arXiv:cond-mat/0110054. doi:10.1007/s100530170010. ISSN 1434-6060.
14. Zhang, Yunbo; Müller-Kirsten, H. J. W. (2001). "Periodic instanton method and macroscopic quantum tunneling between two weakly linked Bose-Einstein condensates". Physical Review A. 64 (2). arXiv:cond-mat/0012491. doi:10.1103/PhysRevA.64.023608. ISSN 1050-2947.