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That, in principle, solitons may act as carriers of matter, and eventually electricity, seems to be a clear consequence of the fact that soliton-bearing equations possess solutions in the form of “waves of translation”. It is also an experimental fact1. Soliton-mediated electric transport was predicted (SSH theory) and experimentally verified with polymers like trans-polyacetylene. In the SSH theory the backbone lattice is taken harmonic hence linear in the inter-atomic interactions. The nonlinearity and the solitons come from the degeneracy of the ground state and mismatch of the corresponding two spatial configurations. The latter brings a topological (kink-like) soliton which acts as the electron carrier. The SSH theory predicts subsonic motions2.
Davydov3 introduced the concept of electro-soliton to predict electron transport along bio-(macro)molecules but his predictions did not clearly match experiments and were shown not to survive above 10K4. Davydov’s theory also builts upon a backbone harmonic lattice and solitons (with subsonic motions) came from an appropriate handling of the nonlinearity of the electron-lattice phonons interaction forming polarons (Landau, Pekar, Fröhlich, Holstein, Feynman and others).
Recently another soliton-mediated form of electric conduction has been predicted building on the coupling of the electron (quantum) dynamics to the nonlinear elasticity of the lattice (taken with anharmonic e.g. Morse interactions). Accordingly, the new prediction builds upon the addition of the backbone lattice solitons to the polaron-like effect, hence to the electron-soliton (phonon) interaction, thus using yet another source of nonlinearity. The newly predicted motions have, generally, supersonic velocity5.
As a means of uncovering the basic elements of the theory it was first developed for a driven-dissipative anharmonic lattice (with Toda and Morse interactions), within a purely classical framework, as a bifurcation process to a supersonic and non-Ohmic current from the base state of Ohmic (linear) conduction. The non-Ohmic soliton-mediated current showed striking and intriguing features, not all unexpected. On the one hand, in the presence of an external electric field, for high values of the field’s strength the current is proportional to the field (Ohm’s linear law). Then as the field’s strength is lowered the current takes off and attains a constant, field-independent value down to vanishing field strength (non-Ohmic law) (Fig. 1). On the other hand, for a given field strength value, at high enough temperatures the non-Ohmic electric current does not vary appreciably, as it occurs with an Ohmic current. Then as the temperature is lowered, and the solitons dominate the dynamics in the lattice the current increases dramatically with the temperature decrease (Fig. 2) 6, 7, 8.
The above specified features have been established on solid ground (albeit only for 1D systems) by considering classically the lattice dynamics (with Morse interactions) while treating the electron-lattice vibrations interaction using the standard quantum mechanical “tight-binding” approximation. Interaction between the lattice and the electrons arises (as in the polaron) due to the dependence of the electron transfer-matrix elements on the relative, time-dependent distance between neighboring lattice units. It has also being shown that electron trapping by solitons and the formation of electron-soliton dynamic bound states, called “solectrons”, is a universal phenomenon with strong similarities (and expected differences) between the classical and quantum approaches9. Further there is the enhancement of transport in the new soliton-mediated process that can be identified with a genuine high peak in the decay of velocity correlations following the (classical or quantum Kubo-Greenwod-Green) linear response theory10. The solectron quasi-particle appears as a natural extension to nonlinear lattices on the one hand of the polaron in harmonic (linear) lattices and on the other hand of the electro-soliton also built upon a harmonic lattice5.
A new form of electron pairing (satisfying Pauli’s principle) has also been shown possible mediated by solitons. This was done by including in the electron Hamiltonian the Coulomb repulsion using Hubbard’s (local) approximation. There is soliton-mediated pairing both in momentum space and in real space. The two electrons move together with the soliton velocity, and the soliton acts a long-range “correlator” for the two electrons even when, far from each other, they are at quite large separation distances11. It has been shown that traveling pairs of lattice solitons serve as carriers for the paired electrons realizing coherent transport of the two correlated electrons. It has also been found a dynamical narrowing of the states, that is, starting from an initial double-peak profile of the electron probability distribution, a single-peak profile is eventually adopted going along with enhancement of localization of the paired electrons12. There is a range of parameter values where supersonic transport of paired electrons is achieved, thus generalizing the bipolaron concept introduced by Alexandrov and Mott13.
To establish the range where the solitons in an anharmonic lattice (with, e.g., Morse interactions) appear and hence where the above described features are to be found in a material the specific heat at constant volume/length was calculated (Fig. 3). The soliton range is above the Dulong-Petit range, otherwise called multi-phonon range, on the way to melting. Similar results have been found by analyzing the dynamic structure factor as a function of temperature. As earlier noted, for the time being, the theory refers to 1D lattices. As it seems difficult to study solitons in 2D, work is now in progress following an idea first used by Pauli (and subsequently by many other authors) to account for hopping-like transport processes using a (probability density) master equation. As thermal heating excites phonons and solitons, using Pauli’s approach we have been able to show that lattice solitons and (with added electrons) solectrons are stable from 0K up to real high temperatures in the above mentioned range that for bio-(macro)molecules include the physiological or room temperature range (ca. 300K). The evolution of solitons has been monitored by observing the evolution of the core electron density profiles surrounding the centers of the atoms originally placed at lattice sites. This is a useful alternative to the observation of just point trajectories along the purely mechanical lattice. The addition of electrons also permits to follow the conducting electron densities visualizing the solectrons14, 15.
The results so far obtained offer a promising line of research. On the one hande, for a strongly anisotropic (quasi 1D) system like a “molecular” wire we foresee interest in the supersonic form of electron transfer. Experimental verification of the features above described is needed. On the other hand, being aware of the impossibility of genuine electric superconduction in 1D, a form o long range ordering, it is necessary to explore in depth the dynamics in 2D. Questions like could bisolectrons (as Bosons) have an Einstein condensation? Could percolation in 2D be a mechanism underlying superconduction? To answer to these questions is not easy but seems worth trying.


1. NEKORKIN, V. I. and VELARDE, M. G. Synergetic Phenomena in Active Lattices. Patterns, Waves, Solitons, Chaos, Springer, Berlin (2002) and references therein.
2. HEEGER, A. J., KIVELSON, S., SCHRIEFFER, J. R. and SU, W.P., Solitons in Conducting Polymers, Rev. Mod. Phys. 60, 781-850 (1988).
3. DAVYDOV,A. S., Solitons in Molecular Systems, 2nd ed., Reidel, Dordrecht (1991) and references therein.
4. CHRISTIANSEN, A. L. and SCOTT, A. C. (Eds.), Davydov’s Solitons Revisited. Self-trapping of Vibrational Energy in Protein, Plenum Press, New York (1990) and references therein.
5. VELARDE, M. G., From polaron to solectron: The addition of nonlinear elasticity to quantum mechanics and its possible effect upon electric transport, J. Appl. Comput. Maths. (2009, to appear).
6. VELARDE, M. G., EBELING, W. and CHETVERIKOV, A. P., On the possibility of electric conduction mediated by dissipative solitons, Int. J. Bifurcation Chaos 15, 245-251 (2005).
7. VELARDE, M. G., EBELING, W., HENNIG, D. and NEISSNER, C., On soliton-mediated fast electric conduction in a nonlinear lattice with Morse interactions, Int. J. Bifurcation Chaos 16, 1035-1039, (2006).
8. MAKAROV, V.A., VELARDE, M.G., CHETVERIKOV, A.P. and EBELING, W., Anharmonicity and its significance to non-Ohmic electric conduction, Phys. Rev. E 73, 066626-1-12 (2006).
9. VELARDE, M.G., EBELING, W., CHETVERIKOV, A.P. and HENNIG, D., Electron trapping by solitons. Classical versus quantum mechanical approach, Int. J. Bifurcation Chaos 18, 521-526 (2008).
10. CHETVERIKOV, A. P., EBELING, W., RÖPKE, G. and VELARDE, M. G., Anharmonic excitations, time correlations and electric conductivity, Contrib. Plasma Phys. 47, 465-478 (2007).
11. VELARDE, M. G. and NEISSNER, C., Soliton-mediated electron pairing, Int. J. Bifurcation Chaos, 18, 885-890 (2008).
12. HENNIG, D., VELARDE, M.G., EBELING, W. and CHETVERIKOV, A.P., Compounds of paired-electrons lattice solitons moving with supersonic velocity, Phys. Rev. E 78, 066606-1-9 (2008).
13. ALEXANDROV, A. S. and MOTT, N. F., Polarons and Bipolarons, World Scientific, London (1996) and references therein.
14. VELARDE, M.G., EBELING, W. and CHETVERIKOV, A.P., Thermal solitons and solectrons in 1D anharmonic lattices up to physiological temperatures, Int. J. Bifurcation Chaos 18, 3815-3823 (2008).
15. CHETVERIKOV, A.P., EBELING, W. and M.G. VELARDE, Thermal Solitons and Solectrons in nonlinear conducting chains, Int. J. Quantum Chem. (2009, to appear)


Fig. 1. Anharmonic Morse lattice. Current-field characteristics: In the absence of solitons, the lines JD and Ji correspond to the Drude-Ohm currents for electrons and lattice ions, respectively. For high values of the electric field strength the current, Je, follows Ohm’s law but as the field strength lowers the current becomes solectronic, and field-independent down to vanishing field strength. The quantity E0 corresponds to the field imparting a reference value, v0, to the electrons in the absence of solitons.

Fig. 2. Anharmonic Morse lattice. Current-temperature characteristics: At high enough temperatures (still in the solid phase) the current, Je, is not significantly affected by temperature as with the Drude-Ohm law which is not affected at all. Then as the temperature falls down to the soliton dominated region the current dramatically increases. The zero value of T is a reference value that can be taken as the value corresponding to, e.g., Cv = 0.75 (see Fig. 3). Both curves are drawn for the ratio E/E0 = 0.5 in Fig. 1.

Fig. 3. Anhamonic Morse lattice. Upper curve: specific heat at constant volumen/length vs temperature; lower curve: energy ratio. Solitons dominate in the range 0.1 < T < 1. This range for bio-(macro)molecules embraces the physiological or room temperature range (ca. 300K). The lowest quantum (Debye) range (Cv is proportional to TD, D=1 here) is not shown. It is for temperatures below the flat part which is the Dulong-Petit plateau. Cv = 1 and 0.5, correspond to the (standard) solid and fluid phases, respectively. T is made dimensionless with a unit/scale combining the frequency of the harmonic approximation, and the binding strength and stiffness of the Morse potential.
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