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Soliton-mediated electron transfer and electric transport arising from coupling electron quantum mechanics to nonlinear elasticity in anharmonic crystal lattices M.G. Velarde, W. Ebeling and A. P. Chetverikov Abstract After recalling features of solitons in the Toda (more precisely an adapted Morse-Toda) lattice a succint discussion is provided about the stability of such soli- tons when the lattice is heated up to physiological temperatures for values of param- eters typical of bio-macro-molecules. Then the discussion is focused on the soliton trapping of an added excess (originally free) electron thus creating the solectron electric carrier. Results are presented for 1d- and 2d-anharmonic crystal lattices. 1 Background: Solitons The study of anharmonic lattices owes much to the seminal work done by Fermi, Pasta and Ulam [13]. They tried numerically albeit with no success to explain equipartition of energy (of paramount importance in statistical mechanics) by using the first few non-Hookean corrections to linear elasticity as a mechanism to allow energy sharing and exchange between harmonic modes otherwise non-interacting. The difficulty was clarified by Zabusky and Kruskal [36, 37] who studied solitary waves, their overtaking collisions in such anharmonic lattices and their continuum counterpart. In view of their remarkable particle-like behavior, these waves reap- pearing unaltered following collisions, the hallmark of their dynamics, they denoted them by solitons (solit/solitary wave; on/like in electron, proton, etc.). In fact, before M. G. Velarde Instituto Pluridisciplinar, Paseo Juan XXIII, 1, Madrid-28040, Spain, e-mail: mgve- [email protected] W. Ebeling Institut f¨ ur Physik, Humboldt-Universit¨ at Berlin, Newtonstrasse, 15, Berlin-12489, Germany, e- mail: [email protected] A. P. Chetverikov Dept. of Physics, Saratov State University, Astrakhanskaya, 83, Saratov-410012, Russia, e-mail: [email protected] 1

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Soliton-mediated electron transfer and electrictransport arising from coupling electronquantum mechanics to nonlinear elasticity inanharmonic crystal lattices

M. G. Velarde, W. Ebeling and A. P. Chetverikov

Abstract After recalling features of solitons in the Toda (more precisely an adaptedMorse-Toda) lattice a succint discussion is provided aboutthe stability of such soli-tons when the lattice is heated up to physiological temperatures for values of param-eters typical of bio-macro-molecules. Then the discussionis focused on the solitontrapping of an addedexcess (originally free) electron thus creating the solectronelectric carrier. Results are presented for 1d- and 2d-anharmonic crystal lattices.

1 Background: Solitons

The study of anharmonic lattices owes much to the seminal work done by Fermi,Pasta and Ulam [13]. They tried numerically albeit with no success to explainequipartition of energy (of paramount importance in statistical mechanics) by usingthe first few non-Hookean corrections to linear elasticity as a mechanism to allowenergy sharing and exchange between harmonic modes otherwise non-interacting.The difficulty was clarified by Zabusky and Kruskal [36, 37] who studied solitarywaves, their overtaking collisions in such anharmonic lattices and their continuumcounterpart. In view of their remarkable particle-like behavior, these waves reap-pearing unaltered following collisions, the hallmark of their dynamics, they denotedthem by solitons (solit/solitary wave; on/like in electron, proton, etc.). In fact, before

M. G. VelardeInstituto Pluridisciplinar, Paseo Juan XXIII, 1, Madrid-28040, Spain, e-mail: [email protected]

W. EbelingInstitut fur Physik, Humboldt-Universitat Berlin, Newtonstrasse, 15, Berlin-12489, Germany, e-mail: [email protected]

A. P. ChetverikovDept. of Physics, Saratov State University, Astrakhanskaya, 83,Saratov-410012, Russia, e-mail:[email protected]

1

2 M. G. Velarde, W. Ebeling and A. P. Chetverikov

the discovery of the soliton, Visscher and collaborators numerical computations [25]had revealed “soliton-like” mediated behavior and enhanced heat transport. Solitarywaves and solitons, found also in other realms of science, appear as potential “uni-versal” carriers of almost anything [11, 22] (like surf waves/non-topological solitonsin the ocean or bores/topological solitons in rivers). Of particular interest to us hereis the model-lattice invented by Toda [28] for which analytical, exact solutions areknown.

Let us recall how solitons appear in the anharmonic Toda lattice. Consider anone-dimensional (1d) lattice of units (equal masses,m andm = 1 for simplicity) in-teracting with their nearest-neighbors via a potentialU(x). Then, classically, for thedisplacement of thenth-lattice unit/particle from its equilibrium position, Newton’sequations are

xn =U ′ (xn+1− xn)−U ′ (xn − xn−1) , (1)

wherexn denotes displacement (depending on circumstances it is of interest to fo-cus on local lattice deformations or on gradient of displacements) of the unit at site“n”. A dash indicates a derivative with respect to the argument. No on-site dynam-ics or structure is considered. There are cases of, e.g., biological interest where anintra-unit dynamics is added. If rather than actual unit-displacements, relative dis-placements,ξn = xn+1− xn, are considered, then Eqs. (1) become

ξn =U ′ (ξn+1)−2U ′ (ξn)+U ′ (ξn−1) . (2)

The paradigmatic interaction potential introduced by Todais

U (ξn) =−ab

[

e−b(ξn−σ)+b(ξn −σ)−1]

, (3)

whereσ is the mean equilibrium interparticle distance;a > 0 andb > 0 are param-eters;b accounts for the non-Hookean stiffness of the “springs” in the lattice; thelast term (−1) is added for computational convenience and need not to be included.Note that withab finite for b → 0, the function (3) becomes the harmonic potential(linear Hookean “springs” for a standard crystal lattice) and ω2

0 ≡ ab/m defines theangular frequency of vibrations in the harmonic limit. In the extreme opposite caseb → ∞, the potential (3) approaches the hard-rod/sphere limit (fluid-like system).Note also that under an external force or for finite temperatures the lattice constantequilibrium distance may not correspond to the minimum of the potential. Fig. 1shows the Toda potential adequately compared with Morse andLennard-Jones po-tentials of current use in Physics and Chemistry. In what follows consideration willbe given only to strong interparticle compressions such that ξ ≤ σ/2 (for simplicityσ = 1). Materials are usually stronger when compressed and weaker when stretched.In view of this, the fact that the attractive part of Toda’s potential (3) is unphysicalis of no concern to the study here.

For any finite value ofb, in the infinite lattice, the equations of motion (2) possessa one-parameter family of soliton solutions

ξn = σ −1(1/b)ℓn[

1+sinh2κsech2 (κn∓sinhκ)ωt]

. (4)

Title Suppressed Due to Excessive Length 3

Inverting the logarithm it is just the sech2 for e−b(ξn−σ). This exponential is relatedto the force (3) and characterizes the strength of the solitonic pulse. The parameterκ controls the wave velocity and by the same token the wave amplitude (highersolitons travel faster),

vsoliton(κ) =±ω0 (sinhκ)/κ , (5)

which in dimensionless units shows its supersonic character as the linear sound ve-locity is here given byvsound = ω0 (positive and negative signs merely give directionof wave propagation). When a periodic hence finite lattice is considered the exactsolution of the equations of motion is a periodic “cnoidal” wave formed with Ja-cobian elliptic functions and complete elliptic integralsof the first and second kind[28]. It can be shown that in thecontinuum limit the solution of the discrete latticecan be approximated by thecnoidal solution of the Boussinesq-Korteweg de Vriesequation [1, 20, 23] and in another limit by the solitary wavesolution in the formof sech2. When the lattice has fixed constant length, as expansion is not permitted,experiences internal stress (pressure). If, however, the lattice length is free but noexternal force to it is applied (like compression or stretching at a free end), it canbe shown that the lattice expands as it vibrates. The solitary wave is acompressionpulse, and cnoidal waves cause expansion with, however, high compression at eachperiodic wave “peak” (or maximum). The exact wave dispersion relation of the Todalattice is known explicitely.

Incidentally, the Toda lattice cannot sustain a thermal gradient although it permitsa temperature difference, hence it is “transparent” to heat(solitons with exponentialinteraction like (3) run freely in the Toda lattice). This problem does not arise withLennard-Jones interactions. In view of this, use is to be done of animperfect Todalattice and, recalling that interest here focused only on rather-strong lattice com-

-1

-0.5

0

0.5

1

1.5

2

2.5

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3

Toda 3

L-J - - - -

Morse ——–

U

r

Fig. 1 Rescaled representation with a common minimum (usingr rather them

x) of the Toda potential(

U =UT = ab

[

e−bσ(r−1)+bσ (r−1)−1])

, Morse po-

tential

(

U =UM = a2b

[

(

e−bσ(r−1)−1)2

−1

])

and Lennard-Jones potential(

U =UL−J =U0

[

1r12 − 1

r6 −1])

.

4 M. G. Velarde, W. Ebeling and A. P. Chetverikov

pressions, this can be achieved by substituting (3) with anadapted (non-integrable,hence imperfect) Toda-Morse lattice whose solutions and corresponding featuresshould not differ significantly from the exact Toda solutions given above [3, 10, 26].Thus rather than (3) we shall use:

UM(ξ ) = D[

e−B(ξ−σ)−1]2

. (6)

The specific heat at constant length/volume of the Toda lattice was obtained longago [28]. The high-temperature limitCL = 0,5 corresponds well with the fluid-like, hard-sphere phase. Then aroundT = 1, CL ≈ 0,75 it is thesoliton range (Tunit: 2D; see below for further scalings). Well belowT = 1 = Ttransition, phonons(Fourier modes) control the thermodynamics (and dynamics)of the system. Sim-ilar phenomena can be observed in the dynamical structure factor (DSF) (typicalfor inelastic thermal neutron scattering experiments, 4A ∼ 5meV ∼ 60K even up to0.3A ∼ 0.1eV ∼ 103K). The latter is the double Fourier transform of the density-density correlation. WhenT is well belowT = Ttransition a single phonon peak ap-pears that provides the linear sound velocity. As the transition temperature is ap-proached from below the phonon spectrum gets multipeaked with many phononsor highly deformed phonons showing up (multiphonon range),until a much higherpeak clearly emerges above a messy background. It corresponds to the soliton withsupersonic velocity (5). Both the specific heat and DSF pointto the significant roleplayed by strong lattice compressions leading to solitons [3].

0.5

0.6

0.7

0.8

0.9

1

0.0001 0.001 0.01 0.1 1 10 T

Cv

Fig. 2 Specific heat at constant length/volume inkB units. Ttransition = 1, for which hereCL ≡Cv = 0.75. Cv = 1 is the Dulong-Petit (Einstein) value (solid phase; harmonic interaction) andCv = 0,5 corresponds to the fluid-like phase (hard-rod interaction).Missing in the figure is the lowtemperature values arising from genuinely quantum mechanics (T d Debye law withd denotingspace dimension).

Title Suppressed Due to Excessive Length 5

2 Electron capture and electron transfer

2.1 The solectron concept

If we now consider that lattice units are atoms with electrons and we add anex-cess electron we can consider two possibilities, one is electrontransfer (ET) froma donor (D) to an acceptor (A) as schematized in Fig. 3 [33]. The other possibilityis electric transport or current in the pressence of an external electric field. In bothcases we have to follow the time evolution of the electron coupled to that of thelattice units, one affects the other. In the simplest form wecan use the tight bindingapproximation (TBA) hence placing the electron at a latticesite and allowing elec-tron hopping to nearby sites. In the TBA the time evolution ofthe electron followsthe Schrodinger equation (for the lattice “space”) augmented with the coupling withthe anharmonic lattice, that reduces to

ihcn = Encn − (Vn,n−1cn−1+Vn+1,ncn+1) , (7)

where the coupling of electron (normalized) probability density (amplitude,|cn|2)to lattice variables implicit in theVn,m appears. The choice

Vn,n−1 =V0e−α(ξn−ξn−1), (8)

is of current use dealing with, e.g., biomolecules. The parameterα is an inversecharacteristic “length” scale.

To have a universal description suffices to make quantities dimensionless by in-troducing suitable scales/units:τ = V0/hωM, α = α/B, andV = V0/2D thus us-

ing the depth of the Morse potential as unit/scale;ωM =(

DB2/M)1/2

, M denoteslattice units mass (typical parameter values for some biomolecules [14, 15] are:B = 4.45A−1, α = 1.75B, D = V0 = 0.1eV , ωM = 3.1012s−1, V0/h = 0.6 ·1014s−1,τ = 10). Then we can rewrite (7) as

icn =−τ[

e−α(ξn−ξn−1)cn−1+ e−α(ξn+1−ξn)cn+1

]

. (9)

Fig. 3 ET along a biomolecule modeled by a lattice. The excess electron(wave functionΨ is emit-ted from siteD (donor) by appropriate energy supply and travels along the bridge or “backbone”lattice made ofanharmonic elements down to the siteA (acceptor).

6 M. G. Velarde, W. Ebeling and A. P. Chetverikov

The parameterτ shows explicitely the time scale of electron motions while the timet corresponds to the slower time scale of the lattice vibrations. The latter obey theequations (2) augmented with the coupling to electronhopping motions or bettersaid, electron probability coefficients,

ξn =[

1− e(ξn+1−ξn)]

e−(ξn+1−ξn)−[

1− e−(ξn−ξn−1)]

e−(ξn−ξn−1)−

−αV[

(

c∗n+1cn + cn+1c∗n)

e−α(ξn+1−ξn)+(

c∗ncn−1+ cnc∗n−1

)

e−α(ξn−ξn−1)]

. (10)

Clearly, the interplay between electron and lattice vibrations has now a genuinelynew element, the soliton-mediated effect. This permits to consider the compoundelectron-soliton “quasiparticle”, due to its “universal”carrier character, as a newphysical entity which is the “solectron” one way of providing electron “surfing” ona subsonic/supersonic sound (longitudinal lattice soliton) wave [2, 29].

2.2 Soliton electron trapping

Consider an electron placed at site “n” in a lattice. Then let alone the electron,its evolution is dictated by equation (9) withα = 0. Figure 4 shows how from aninitially peaked probability density as time proceeds the probability spreads downto a uniform distribution over all lattice sites and hence ends up by beingcompletelydelocalized.

Other evolution possibilities have been explored [16, 17].Taking now Eq. (2) andlaunching as an initial condition a soliton at a certain lattice site and then switching-on the electron-lattice interaction hence switching-on Eq. (10), for an initial condi-tion of the electron completelydelocalized, and then operating Eq. (9) in full, thestriking result found is illustrated in Fig. 5. Subsequently, after trapping the electron,

0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100

-0.005 0

0.005 0.01

0.015 0.02

0.025 0.03

0.035

lattice

n

t

lattice

0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100

0 0.02 0.04 0.06 0.08

0.1 0.12 0.14

electron

n

t

electron

Fig. 4 Soliton and electron taken separately (no interaction). Left figure: lattice soliton time evolu-tion starting at siten = 200. Right figure: electron probability density time evolution. From initial“localization” at siten = 200 the electron ends upcompletely delocalized, i.e., the proability den-sity is spread “uniformly” everywhere along the lattice.

Title Suppressed Due to Excessive Length 7

the compound or bound state soliton-electron, i.e., thesolectron proceeds movingunaltered along the lattice.

When two solitons which are allowed to collide in their evolution along the latticeare launched and the electron starts being trapped and carried by one of the solitons,e.g. by the one moving left-to-right, then as the collision proceeds and “finishes”, theelectron may leave the first soliton and reappear trapped andcarried by the secondsoliton. Accordingly, the electron may change both partnerand direction of motionafter the collision. Another striking result also observednumerically is the electronprobability density splitting thus illustrating how quantum mechanically the electron(in probability sense) can move simultaneously! in both directions [31].

If an external electric field is acting there is current, it suffices to add to Eq. (9)the term (−nEcn) and then to compute

j = i ∑(

c∗n+1cn − c∗ncn−1)

, (11)

which then depends on the external field strength. This is apparently so but notalways in reality. Indeed, for high enough field strength it can be seen that the lat-ter forces the electron to follow Ohm’s law. But as the field strength becomes lowenough it is rather the soliton which commands the electric current which becomesfield-independent thus remaining constant as the field strength tends to zero. Thisstriking result is not unexpected if we recall what the “soliton” wave does to a surfer.

3 Heated crystal lattices

So far no mention was done of temperature other than when referring to the spe-cific heat (Fig. 2). Strictly speaking the results describedabove hold at zero-K.

0 50 100 150 200 250 300 350 400 0 50

100 150

200 250

300 350

400

-0.01-0.005

0 0.005

0.01 0.015

0.02 0.025

0.03 0.035

lattice

n

t

lattice

0 50 100 150 200 250 300 350 400 0 50

100 150

200 250

300 350

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

electron

n

t

electron

Fig. 5 Interaction of asoliton with a completely delocalized electron. Right figure: the electronafter following evolution dictated by Schrodinger equation ends upcompletely delocalized withprobability density spread like “dust” over the entire lattice. Then at such time instant the soliton islaunched taking as initial condition for the electron the final state of the right picture in Fig. 4. Leftfigure: the soliton after gathering the electron dusty probability density eventually reconstructs theelectron probability density in a kind of “vacuum cleaner” process.

8 M. G. Velarde, W. Ebeling and A. P. Chetverikov

Let us now consider that the system is heated-up from zero-K to the soliton range(T ≈ 0.1− 1) defined in Fig. 2. The heating can be done by a suitable thermalbath satisfying Einstein’s relation between noise strength and equivalent tempera-ture in K. As we shall continue restricting consideration to1d lattices let us recallthe Hamiltonian,Ha, usingxn as lattice coordinates. Then we have

Ha =m2

N

∑n=1

v2n +

12

N

∑n,i=1

U (xn,xi) , (12)

wherevn denote velocities, andU corresponds to the Morse potential (6) (Fig. 1).Then in the presence of random forces (hence nonzero temperature) and also exter-nal forces the evolution of lattice particles is given by theLangevin equations

ddt

vn +1m

∂Ha

∂xn=−γ0vn +

2Dv ξn(t). (13)

The stochastic forces√

2Dv ξn(t) define a time delta correlated Gaussian whitenoise. The parameterγ0 describes the standard friction frequency acting from thebath. The Einstein relation isDv = kBT γ0/m, whereT denotes absolute temperatureandkB is Boltzmann’s constant.

In order to visualize the solitons we can focus attention to the “atomic” den-sity. We assume that each lattice unit is surrounded by a Gaussian electron densityproviding a screened ion core of widths = 0.35σ . Then the total atomic electrondensity, defining a lattice unit, is given by

ρ(x) = ∑n

1√2πs

exp

[

− (x− xn(t))2

2s2

]

. (14)

Hence we assume that the atom is like a spherical object with continuous electrondensity concentrated around each lattice site. In regions where the atoms overlap,the electron density is enhanced. This permits easy visualization of soliton-like ex-citations based on the colors in a density plot. This is of course a rough approxima-tion. Figure 6 shows the result of simulations for the temperaturesT = 0.005 andT = 1. The diagonal stripes correspond to regions of enhanced density which arerunning along the lattice. This is a sign of solitonic excitations. Checking the slopewe see excitations which over 10 time units move withsupersonic velocity. We havesolitonic excitations living about 10-50 time units corresponding to 1−3ps. Besidesthey survive even atT = 1 which is well above the physiological temperature (about300K which is aboveT ≈ 0.1 with D ≃ 0.1eV ). Due to lack of space we shall notdiscuss here the solectron survival as we heat the lattice. The formation of solectronoccurs as indicated above and does survive as a compound up tosuch temperatures.For details we refer the reader to the analyses presented in Refs. [4, 5, 9, 12, 32].

Title Suppressed Due to Excessive Length 9

4 Two-dimensional soliton-like excitations

Let us now extend the study to the case of a two-dimensional (2d) lattice still withMorse interactions. The lattice Hamiltonian (12) becomes now

Ha =m2 ∑

nv2

n +12 ∑

i, jV (ri, r j). (15)

The subscripts locate atoms at lattice sitesn with coordinates (i, j) and the sum-mations run from 1 toN. As before the characteristic distance determining the repul-sion between the particles in the lattice isσ . We limit ourselves to nearest-neighborsonly using the relative distancer = |rn− rk|. The Morse potential (Fig. 1) if for con-venience, expressed as

UM = D{exp[2B(r−σ)]−2exp[−B(r−σ)]} . (16)

In order to avoid unphysical cumulative interaction effects, a suitable cut-off rulesout a stronger interaction than due arising from the influence of particles outsidethe first neighborhood of each particle. In fact, rather thana cut-off we consider theinteraction with a smooth decay to zero as distance increases. Hence rather than (16)for the 2d lattice we take

40

0.6

x/σ

0.7

4 8 12 16

0.8

0.9

1

ρ‘

0.4

0.6

0.8

1

0

4

8

12

16

100 140

180 220

260 t

x/σ

ρ‘

1

2

3

0 40 80 120 160 x/σ

ρ‘

0

2

4

6

t

0

40

80

120

160

100 140

180 220

260

x/σρ‘

Fig. 6 Visualisation of soliton-like running excitations. Densityρ ′ = ρ√

2πs refers to electrons inlattice atoms (color coding in arbitrary units).N = 200,Bσ = 1,s = 0.35σ . For two temperatures(given in units of 2D) we have: Upper set of figures: (i)T = 0.005: Only harmonic lattice vibrationsshow up with no evidence of soliton-like excitations; and bottom set of figures: (ii)T = 1: Besidesmany excitations also a few strong solitons appear running with velocity around 1.3vsound . In bothcases a snapshot of the distribution for a certain time instant and the actual time evolution of thedistribution are displayed.

10 M. G. Velarde, W. Ebeling and A. P. Chetverikov

UM(r) = 2D{exp[−2b(r−σ)]−2exp[−b(r−σ)]} ··{1+exp[(r−d)/2ν ]}−1 . (17)

As a rule the cut-off “interaction radius” is supposed to be equal to 1.5σ , togetherwith parameter valuesd = 1.35σ andν = 0.025. Beyond the cut-off radius the po-tential is set to zero. To study, at varying temperature, thenonlinear excitations ofthe lattice and the possible electron transport in a latticeit is sufficient to know thelattice (point) particles coordinates at each time and the potential interaction of lat-tice deformations with electrons. The former are obtained by solving the equationsof motion of each particle (15) under the influence of all possible forces. The latterinclude forces between particles which are supposed to be ofthe Morse kind andthe friction and random forces accounting for a Langevin model bath in the heatedlattice. We use complex coordinatesZ = x+ iy, wherex andy are Cartesian coordi-nates for eachr. Then the Langevin equations (13) for the lattice units,n, becomenow

d2Zn

dt2 = ∑k

Fnk(Znk)znk +

[

−γZn

dt+√

2Dv (ξnx + iξny)

]

, (18)

whereγ, Dv andξnx,y have earlier defined roles.Znk = Zn − Zk, thenznk = (Zn −Zk)/|Zn −Zk| is a unit vector defining the direction of the interaction forceFnk. Tohave dimensionless variables we consider the spatial coordinates normalized to thelengthσ . The energy is scaled with 2D. The interaction forceFnk is given by

Fnk = Fnk(|Znk|) =−dV (r)dr

|r=|Znk|. (19)

In view of the above only those lattice units with coordinates Zk, satisfying the con-dition |Zn −Zk| < 1.5, are taken into account in the sum in Eq. (18). In computersimulations the interaction of particles is considered to take place inside a rectangu-lar cellLx ·Ly with periodic boundary conditions andLx,y , depending on the symme-try of an initial distribution of units and their numberN. For illustration we considera distribution corresponding to the minimum of potential energy for an equilibriumstate of atriangular lattice 10σ · 10

3/2σ for N = 100 or 20σ · 20√

3/2σ forN = 400.

As in the preceding Section, we will assume that the atomic electrons may berepresented by a Gaussian distribution centered on each lattice site:

ρ(Z, t) = ∑|Z−Zi(t)|<1.5

exp

[

−|Z−Zi(t)|22λ 2

]

. (20)

In Fig. 7 we show the evolution of one localized soliton-likeexcitation in a tri-angular Morse lattice. The initial form of the excitation isa small piece of a planesoliton-like wave with a front oriented along one of symmetry axes of a triangularlattice and a velocity directed along an other axis (x-axis here). The density dis-tribution (left column), and the cumulated during the time mentioned at each rowamplitude-filtered density-distribution (right column) are presented for three time

Title Suppressed Due to Excessive Length 11

Fig. 7 Propagation of soliton-like excitation in a triangular lattice. Left column: density of theatom cores. Right column: a cumulated representation at final time. In order to study the evolutionof perturbations we changed the initial positions of the atomsat t = 0 in a small region. Parametervalues:N = 400,bσ = 4, λ = 0.3, T = 0.01.

instants. We observe the transformation of the initial piece of a plane wave to asoliton-like horseshoe-shaped supersonic excitation. Asthe excitation travels a dis-tance of about 16 units in the time internalt = 8, its supersonic velocity is 16/8= 2in units of the sound velocity in 1D lattice. Note thatvsound = 1 for 1d-lattices. Ina 2d-triangular lattice the sound velocity is

√2 ≃ 1.4 times higher than the sound

velocity in a 1d-lattice.In a subsequent set of simulations we studied two solitons excited initially, the

left one propagating to the right, the right one to the left. The parameter values arethe same as in the one-soliton case above. We have observed first a transformation ofthe initial pieces of a plane wave to a 2d horseshoe-shaped soliton-like wave frontsand then both moving head-on against each other. Looking at Fig. 8 we observethat the two localized and supersonic excitations pass through each other withoutchanging their form. This is a signature of solitons. They are not solitons in therigorous mathematical sense because we do not prove that they are exact stationarywaves (indeed our 2d excitations are “long lasting” transitory waves) but clearlytheir behavior is like that of surface solitons observed in fluids [7, 8, 24].

We have also studied the role of heating and hence observing excitations at finitetemperatures. As we have no space to discuss this problem here we refer the readerto Ref. [6].

12 M. G. Velarde, W. Ebeling and A. P. Chetverikov

Fig. 8 Triangular lattice: Head-on collision of two oppositely moving solitons in the intervalt =0−8, both with the same parameters values:N = 400,bσ = 4, λ = 0.3, T = 0.01.

Title Suppressed Due to Excessive Length 13

5 Conclusion

We have succintly discussed features of solitons in 1d and 2danharmonic crystallattices and, in particular, some consequences of heating the system. We have alsodiscussed features of coupling lattice solitons to addedexcess electrons leading tothe formation of bound states or compounds electron-soliton, denoted solectrons.Recently, we have also studied, albeit only in the 1d case, the formation of electronpairs (with opposite spins satisfying Pauli’s exclusion principle and experiencingCoulomb repulsion using Hubbard’s local approximation) [18, 30, 34, 35]. Finally,three recent experiments have provided collateral verification of the major predic-tions of the theory here presented. One [27] provides evidence ofballistic transportin synthetic DNA and the other two [19, 21] (see also [9, 22]) provide evidenceof electron“surfing” on sound waves in piezoelectricGaAs. Further details aboutcomparison between theory and experiments would be provided elsewhere.

Acknowledgements The authors are grateful to A.S. Alexandrov, E. Brandas, L. Brizhik, L.Cruzeiro, F. de Moura, J. Feder, D. Hennig, R. Lima, R. Miranda,R. McNeil, G. Ropke and G.Vinogradov for enlightening discussions. This research has beensponsored by the Spanish Minis-terio de Ciencia e Innovacion, under Grants EXPLORA-FIS2009-06585-E and MAT2011-26221.

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