Librations with large periods in tunneling: Efficient calculation and applications to trigonal dimers

A. Yu. Anikin   ; S. Yu. Dobrokhotov   ; I. A. Nosikov

Teoretičeskaâ i matematičeskaâ fizika, Tome 213 (2022) no. 1, pp. 163-190

Résumé

In studying tunnel asymptotics for lower energy levels of the Schrödinger operator (such as the energy splitting in a symmetric double well or the width of a spectral band in a periodic problem), there naturally arise librations, i.e., periodic solutions of a classical system with inverted potential, which reach the boundary of the domain of possible motions twice during the period. In the limit, they give double-asymptotic solutions with two symmetric unstable equilibria (instantons). The tunnel asymptotics can be written in two ways: either in terms of the action on the instanton and the linearized dynamics in its neighborhood or in terms of a certain libration, called a tunnel libration. The second way is more constructive, since when used in numerical calculations, it reduces to two operations: finding a libration with a given energy and calculating the Floquet coefficients for a given libration. To apply this approach in practice, we propose to find librations with a given energy by using a numerical variational method that generalizes the ideas of the nudged elastic band method. As an application, we find the asymptotics for the widths of the lower spectral bands and gaps, expressed in terms of tunnel libration in a four-dimensional system describing the dimer in a trigonal-symmetric field, which was proposed by M. I. Katsnelson.

Détail

Keywords: tunneling, Schrödinger operator spectrum, variational principle, quantum dimer.

Bibliographie

[1] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. 3, Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1974 | MR

[2] M. V. Fedoryuk, “Asimptotika diskretnogo spektra operatora $w''(x)-\lambda^2p(x)w(x)$”, Matem. sb., 68(110):1 (1965), 81–110 | MR | Zbl

[3] C. Herring, “Critique of the Heitler–London method of calculating spin couplings at large distances”, Rev. Modern Phys., 34:4 (1962), 631–645 | DOI | MR

[4] E. Harrell, “Double wells”, Commun. Math. Phys., 75:3 (1980), 239–261 | DOI | MR | Zbl

[5] S. Coleman, “The uses of instantons”, The Whys of Subnuclear Physics (Erice, Sicily, 23 July – 10 August, 1977), The Subnuclear Series, 15, ed. A. Zichichi, Springer, New York, 1979, 805–941 ; E. Gildener, A. Patrascioiu, “Pseudoparticle contributions to the energy spectrum of a one-dimensional system”, Phys. Rev. D, 16:2 (1977), 423–430 ; A. M. Polyakov, “Quark confinement and topology of gauge theories”, Nucl. Phys. B, 120:3 (1977), 429–458 ; G. Jona-Lasinio, F. Martinelli, E. Scoppola, “New approach to the semiclassical limit of quantum mechanics. I. Multiple tunnelings in one dimension”, Commun. Math. Phys., 80:2 (1981), 223–254 ; J. M. Combes, P. Duclos, R. Seiler, “Krein's formula and one dimensional multiple well”, J. Funct. Anal., 52:2 (1983), 257–301 ; Т. Ф. Панкратова, “Квазимоды и расщепление собственных значений”, Докл. АН СССР, 276:4 (1984), 795–798 | DOI | DOI | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | MR

[6] A. G. Alenitsyn, “Rasscheplenie spektra, porozhdennoe potentsialnym barerom v zadachakh s simmetrichnym potentsialom”, Differents. uravneniya, 18:11 (1982), 1971–1975 | MR | Zbl

[7] B. Helffer, J. Sjöstrand, “Multiple wells in the semi-classical limit I”, Commun. Part. Differ. Equ., 9:4 (1984), 337–408 | DOI | Zbl

[8] B. Helffer, J. Sjöstrand, “Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation”, Ann. Inst. H. Poincaré Phys. Théor., 42:2 (1985), 127–212 | MR | Zbl

[9] V. P. Maslov, “Globalnaya eksponentsialnaya asimptotika reshenii tunnelnykh uravnenii i zadachi o bolshikh ukloneniyakh”, Tr. MIAN SSSR, 163 (1984), 150–180 | MR | Zbl

[10] B. Simon, “Semiclassical analysis of low lying eigenvalues. II. Tunneling”, Ann. Math., 120:1 (1984), 89–118 | DOI | MR | Zbl

[11] B. Simon, “Semiclassical analysis of low lying eigenvalues. III. Width of the ground state band in strongly coupled solids”, Ann. Phys., 158:2 (1984), 415–420 | DOI | MR | Zbl

[12] A. Outassourt, “Comportement semi-classique pour l'opérateur de Schrödinger à potentiel périodique”, J. Func. Anal., 72:1 (1987), 65–93 | DOI | MR

[13] A. Martinez, “Estimations de l'effet tunnel pour le double puits I”, J. Math. Pures Appl. (9), 66:2 (1987), 195–215 | MR | Zbl

[14] S. Yu. Dobrokhotov, V. N. Kolokoltsov, V. P. Maslov, “Rasscheplenie nizhnikh energeticheskikh urovnei uravneniya Shredingera i asimptotika fundamentalnogo resheniya uravneniya $hu_t=h^2\Delta u/2-V(x)u$”, TMF, 87:3 (1991), 323–375 | DOI | MR | Zbl

[15] J. Brüning, S. Yu. Dobrokhotov, E. S. Semenov, “Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem”, Regul. Chaotic Dyn., 11:2 (2006), 167–180 | DOI | MR | Zbl

[16] A. Yu. Anikin, “Libratsii i rasscheplenie nizhnikh urovnei operatora Shredingera s potentsialom tipa dvoinoi yamy v mnogomernom sluchae”, TMF, 175:2 (2013), 193–205 | DOI | DOI | MR | Zbl

[17] A. Yu. Anikin, S. Yu. Dobrokhotov, M. I. Katsnelson, “Nizhnyaya chast spektra dvumernogo operatora Shredingera s periodicheskim po odnoi peremennoi potentsialom i prilozheniya k kvantovym dimeram”, TMF, 188:2 (2016), 288–317 | DOI | DOI | MR

[18] V. V. Kozlov, S. V. Bolotin, “Libratsiya v sistemakh so mnogimi stepenyami svobody”, PMM, 42:2 (1978), 245–250 | DOI | MR | Zbl

[19] S. Yu. Dobrokhotov, V. N. Kolokoltsov, “Ob amplitude rasschepleniya nizhnikh energeticheskikh urovnei operatora Shredingera s dvumya simmetrichnymi yamami”, TMF, 94:3 (1993), 426–434 | DOI | MR | Zbl

[20] S. Yu. Dobrokhotov, V. N. Kolokol'tsov, “The double-well splitting of the low energy levels for the Schrödinger operator of discrete $\phi^4$-models on tori”, J. Math. Phys., 36:3 (1995), 1038–1053 | DOI | MR

[21] A. Yu. Anikin, “Asymptotic behaviour of the Maupertuis action on a libration and a tunneling in a double well”, Rus. J. Math. Phys., 20:1 (2013), 1–10 | DOI | MR

[22] A. Anikin, M. Rouleux, “Multidimensional tunneling between potential wells at non degenerate minima”, Days on Diffraction 2014 (St. Petersburg, May 26–30, 2014), eds. O. V. Motygin, A. P. Kiselev, L. I. Goray, A. Ya. Kazakov, A. S. Kirpichnikova, IEEE, New York, 2014, 17–22 | DOI

[23] S. Yu. Dobrokhotov, A. Yu. Anikin, “Tunneling, librations and normal forms in a quantum double well with a magnetic field”, Nonlinear Physical Systems. Spectral Analysis, Stability and Bifurcations, Mechanical Engineering and Solid Mechanics Series, eds. O. N. Kirillov, D. E. Pelinovsky, John Wiley and Sons, Hoboken, NJ; ITSE, London, 2014, 85–110 | DOI | MR

[24] A. Yu. Anikin, M. A. Vavilova, “Kvaziklassicheskaya asimptotika nizhnikh spektralnykh zon operatora Shredingera s trigonalno-simmetrichnym periodicheskim potentsialom”, TMF, 202:2 (2020), 264–277 | DOI

[25] H. Jónsson, G. Mills, K. W. Jacobsen, “Nudged elastic band method for finding energy paths of transitions”, Classical and Quantum Dynamics in Condensed Phase Simulations (Lerici, 7–18 July, 1997), World Sci., Singapore, 1998, 385–404 | DOI

[26] G. Henkelman, B. P. Uberuaga, H. Jónsson, “A climbing image nudged elastic band method for finding saddle points and minimum energy paths”, J. Chem. Phys., 113:22 (2000), 9901–9904 | DOI

[27] D. M. Einarsdóttir, A. Arnaldsson, F. Óskarsson, H. Jónsson, “Path optimization with application to tunneling”, Applied Parallel and Scientific Computing. PARA 2010, Lecture Notes in Computer Science, 7134, eds. K. Jónasson, Springer, Berlin, Heidelberg, 2012, 45–55 ; V. Ásgeirsson, A. Arnaldsson, H. Jónsson\, “Efficient evaluation of atom tunneling combined with electronic structure calculations”, J. Chem. Phys., 148:10 (2018), 102334, 10 pp. | DOI | DOI

[28] M. I. Katsnelson, M. van Schilfgaarde, V. P. Antropov, B. N. Harmon, “Ab initio instanton molecular dynamics for the description of tunneling phenomena”, Phys. Rev. A, 54:6 (1996), 4802–4809 | DOI

[29] M. I. Katsnelson, Graphene. Carbon in Two Dimensions, Cambridge Univ. Press, Cambridge, 2012

[30] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, Editorial URSS, M., 2002 | DOI | MR

[31] G. Zeifert, V. Trelfall, Variatsionnoe ischislenie v tselom, RKhD, M., 2000

[32] E. A. Koddington, N. Levinson, Teoriya obyknovennykh differentsialnykh uravnenii, URSS, M., 2007

[33] C. Fusco, A. Fasolino, T. Janssen, “Nonlinear dynamics of dimers on periodic substrates”, Eur. Phys. J. B, 31:1 (2003), 95–102 ; E. Pijper, A. Fasolino, “Mechanisms for correlated surface diffusion of weakly bonded dimers”, Phys. Rev. B, 72:16 (2005), 165328, 5 pp. | DOI | DOI

[34] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982 | MR | MR | Zbl

[35] H. C. Andersen, “Molecular dynamics simulations at constant pressure and/or temperature”, J. Chem. Phys., 72:4 (1980), 2384–2393 | DOI