The Solitons

Goldberg points out, however, that despite any apparent similarity between PFA supported solitons and the soliton solutions of differential equations such as the... 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

In Ref. [4], the soliton lattice configuration and energy within the SSH model were found numerically. Analytical expressions for these quantities can be obtained in the weak-coupling limit, when the gap 2A() is much smaller than the width of the re-electron band 4/0. At this point it is useful to define the lattice correlation length ... 

The single-electron spectrum for the antisoliton solution, A,U)=-Av(x ), is exactly the same as for the soliton, except that now the wave function of the midgap state is given by... 

Up until now we have only discussed the situation where the Coulomb interaction between the electrons is neglected. As we mentioned in the Introduction, the soliton concept does survive electron-electron interactions, although the energetics differ from the above. We will return to this in Section 3.4. 

The two intragap stales g (x) arc the symmetric and antisymmetric superpositions of the midgap stales localized near the soliton and the anlisoliton ... 

The energy splitting 2 e between the intragap states decreases exponentially with the soliton-antisoliton separation, so that for f. [Pg.50]

The first term in this equation describes the suppression of the probability of the fluctuation with the correlator Eq. (3.22) (the weight />[//(a)] of the disorder configuration is exp (— J da/2 (x))), while the second term stems from the condition that the energy c+[t/(x)] of the lowest positive-energy single-electron state for the disorder realization t/(x) equals c. The factor /< is a Lagrange multiplier. It can be shown that the disorder fluctuation //(a) that minimizes A [//(a)] has the form of the soliton-anlisolilon pair configuration described by [48] ... 

Thus, the soliton-anlisolilon separation R is fixed by the condition t + (/ )=a. 

From Eq. (3.28), one finds that the oscillator strength is proportional to the square of the soliton-anlisolilon separation R. Using this, Eq. (3.33) reduces to the following form for the average absorption coefficient at low photon eneigy ... 

The linear and nonlinear optical properties of the conjugated polymeric crystals are reviewed. It is shown that the dimensionality of the rr-electron distribution and electron-phonon interaction drastically influence the order of magnitude and time response of these properties. The one-dimensional conjugated crystals show the strongest nonlinearities their response time is determined by the diffusion time of the intrinsic conjugation defects whose dynamics are described within the soliton picture. 

Here we outline a dynamical description (42) of the polymerisation of the polydiacetylenes. The approach relies much on the one used (43,44) in the theory of non radiative transitions in crystals and the soliton description of the defects in the lD-or-ganic semiconductors. 

In the solitonic sector, the CFL chiral Lagrangian [27, 28] gives us the scaling behavior of the coefficient of the Skyrme term and thus shows that the mass of the soliton is of the order of... 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

In this way, we can relate duality to quark-hadron continuity. We considered duality, which is already present at zero chemical potential, between the soliton and the vector mesons a fundamental property of the spectrum of QCD which should persists as we increase the quark chemical potential. Should be noted that differently than in [42] we have not subtracted the energy cost to excite a soliton from the Fermi sea. Since we are already considering the Lagrangian written for the excitations near the Fermi surface we would expect not to consider such a corrections. In any event this is of the order //, [42] and hence negligible with respect to Msoiiton. 

We have shown that the vector mesons in the CFL phase have masses of the order of the color superconductive gap, A. On the other hand the solitons have masses proportional to F%/A and hence should play no role for the physics of the CFL phase at large chemical potential. We have noted that the product of the soliton mass and the vector meson mass is independent of the gap. This behavior reflects a form of electromagnetic duality in the sense of Montonen and Olive [29], We have predicted that the nucleon mass times the vector meson mass scales as the square of the pion decay constant at any nonzero chemical potential. In the presence of two or more scales provided by the underlying theory the spectrum of massive states shows very different behaviors which cannot be obtained by assuming a naive dimensional analysis. 

The solitonic charges are mobile along the polymer chain by rearrangement of double and single bonds ... 

At high doping levels, when the coulombic attractions to the counterions are screened, the bipolarons or the solitons may become highly... 

A sequence may form and eventually meet a B sequence, as shown, but in doing so, a free radical, called a soliton, is produced. The soliton is a relatively stable electron with an unpaired spin and is located in a nonbonding state in the energy gap, midway between the conduction and valence bands. It is the presence of these neutral solitons which gives frany-polyacetylene the characteristics of an intrinsic semiconductor with conductivities of 10 to 10 (f2 cm) ... 

Many phenomena such as dislocations, electronic structures of polyacetylenes and other solids, Josephson junctions, spin dynamics and charge density waves in low-dimensional solids, fast ion conduction and phase transitions are being explained by invoking the concept of solitons. Solitons are exact analytical solutions of non-linear wave equations corresponding to bell-shaped or step-like changes in the variable (Ogurtani, 1983). They can move through a material with constant amplitude and velocity or remain stationary when two of them collide they are unmodified. The soliton concept has been employed in solid state chemistry to explain diverse phenomena. 

Two lower states of the frans-(CH) are energetically degenerated as follows from symmetry conditions. Theory shows that electron excitation invariably includes the lattice distortion leading to polaron or soliton formations. If polarons have analogs in the three dimensional (3D) semiconductors, the solitons are nonlinear excited states inherent only to ID systems. This excitation may travel as a solitary wave without dissipation of the energy. So the 1-D lattice defines the electronic properties of the polyacetylene and polyconjugated polymers. 

These and other results on absorption and luminescence were the reason why the soliton model was proposed for the photoconduction process in trans-polyacetylene [103-110],... 

The main scheme is shown in Fig. 17. The photogenerated electron hole pairs transfer to the soliton-antisoliton pairs in 10 13s. Two kinks appeared in the polymer structure, which separates the degenerated regions. Due to the degeneration, two charged solitons may move without energy dissipation in the electric field and cause the photoconductivity. The size of the soliton was defined as 15 monomer links with the mass equal to the mass of the free electron. In the scheme in Fig. 17, the localized electron levels in the forbidden gap correspond to the free ( + ) and twice occupied ( — ) solitons. The theory shows the suppression of the interband transitions in the presence of the soliton. For cis-(CH)n the degeneration is absent, the soliton cannot be formed and photoconductivity practically does not exist. 

A lot of research was carried out for proving existence of the soliton in trans-(CH)n by various methods - induced absorption and luminescence, electron spin resonance, transit photogeneration and so on. The references may be found in the monograph [14]. 

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