Solitons in quasi one dimensional conductors and beyond

1. Solitons in quasi one dimensional conductors and beyond Solitons/instantons in electronic properties:Born in theories of late 70’s, Found in experiments of early 80’s. Why in 2000’s ? New conducting polymers,New events in organic conductors, New accesses to Charge Density Waves,New inhomogeneous-nonstationarystrongly correlated systems

2. Singlet ground state gapful systems: SuperConductorsSCs and Charge Density Waves CDWs. Standard BCS-Bogolubov view: Spectra : E(k)= ±(∆2+(vfp)2)1/2 States = linear combinations of :electrons and holes at ±p for SC electrons at –p and p+2pf for CDW CaC6 Figures: pair-breaking gaps from tunneling experiments. Is it always true? Proved “yes” for typical SCs.Questionable for strong coupling : High-Tc SC,real space pairs,cold atoms, bi-polarons. Certainly incomplete for CDWsas proved by many modern experiments. Certainly inconsistent for 1D and even quasi 1D systems as proved theoretically. NbSe3 Guilty and Most Wanted : solitons and their arrays.

3. Solitons in general : propagating isolated profiles (Russel 1834) – from the tsunami waveto magnetic domain walls and (?) to hypothetical holons and spinons in strongly correlated systems Solitons in our perspective of electronic systems : Nonlinear self-localized excitations on top of a ground state with a spontaneously broken symmetry: Peierls state, Csuperconductivity, CDW,SDW,AFM Solitons carry a charge or a spin – separately, even in fractions. They bring associated spectral features (e.g. mid-gap states) Instantons: related transient processes for creations of solitons or their pairs, for conversion of electrons or excitons into solitons

4. Symmetry breaking : degenerate equivalent ground states. simplest soliton= kink between them Figures : energy as a function of configuration. two-fold degeneracy or cross-section of the axial-symmetry Mexican hat shape

5. Tsunami soliton appears as a particle for 1D fermions with spontaneous mass generation: Gross-Neveu model in field theory, Charge Density Waves in SSPh. These Hamiltonians are associated to the KdV nonlinear equation. Miura transformation applied to Tsunami profile gives the shape of a soliton created by pair breaking in multi-electronic system with a spontaneous mass generation. Tsunami is a solitary wave solution of the KdV equation, as well as harmless quasiperiodic sea waves. They also are mapped upon superstructures formed by electrons in solids. Nonlinear Schrödinger eq. describes pulses carrying information in fiber-optics. It also gives microscopics of spin-carrying solitons of variable charge in electronic states with broken continuous symmetries: chiral for charge density waves of guage for superconductors (the FFLO phases). Sine-Gordon eq. describes the locking and friction of undulating elastic surfaces. It also gives the p-solitons as anomalous elementary particle carrying either charge (holons) or spin (spinons).

6. somevocabulary: • Electronic Crystals (Wigner X) translational symmetry is broken. • Charge Density Waves CDW – crystal of singlet electron pairs; translational=chiralsymmetry is broken • AntiferromagnetsAFM translational and time reversal symmetries are broken. Spin Density Waves SDW chiral symmetry and time reversal symmetries are broken. • Mott Insulator: hypothetic phase with a (pseudo)gap generated without usual symmetry breakings. Symmetry breaking : degenerate equivalent ground states. simplest Soliton= kink between them Figure : energy as a function of configuration. two-fold degeneracy or cross-section of the axial-symmetry shape

7. Microscopics of local and instantaneous electronic states in CDWs. BCS-like Peierls-Fröhlich model for incommensurate CDWs - ICDW. Exact static solutions – solitons- of multi-electronic models. Adiabatic generalization to dynamic processes – instantons. Incommensurate CDW : Acos(Qx+φ) Q=2Kf Order parameter : Δ(x,t)~ Aexp(iφ) Electronic states Ψ= Ψ+exp(iKfx+iφ/2) + Ψ-exp(-iKfx-iφ/2) Justification of the mean-field BCS, and co-observation of electrons and solitons: Small phonon frequency: experimentally ωph <0.1Δ

8. CDW : Acos(Qx+φ) Q=2Kf Order parameter : Δ(x,t)~ Aexp(iφ) Electronic states Ψ= Ψ+exp(iKfx) + Ψ-exp(-iKfx) or after chiral transform Ψ= Ψ+exp(iKfx+iφ/2) + Ψ-exp(-iKfx-iφ/2) Dimerization case - Z2 : phase φ of  is fixed Ground state double degeneracy allows for topological solitons = kinks =trajectories connecting equivalent vacuums (+/-1). Major properties of kinks: • Energy < : selftrapping of electrons into kinks (22) • They bear mid-gap states = zero fermionic mode 3. They carry either charge or spin

9. Fatal effect upon kinks: lifting of degeneracy, hence confinement. Trivial but spectacular case: global lifting. Nature present:cis-isomer of (CH)xBuild-in inequivalence of bonds = nonzero starting “mass”. Confinement of kinks pairs into 2e charged (bipolaron) or neutral (exciton). Firework of experimental proofs Phases of ’s are fixed, no degeneracy S. B. and N. Kirova: Interpretation of experiments and « accidental » exact solution for the Gross-Neveu model with confinement.

10. Joint effect of build-in ∆biand spontaneous ∆spcontributions to gap ∆. - from the build-in site dimerization – inequivalence of sites A and B. - from spontaneous dimerization of bonds ∆in=∆b - generic Peierls effect. SOILITONS WITH NONINTEGER VARIABLE CHARGES – new life in 2000’s: Orthogonal mixing of static and dynamic mass generations. Realizations: Modified polyacetylene (CRCR)x, organic ferroelectrics. Theories for solitons with variable charges: S.B. & N.K. , E.Mele and M.Rice Nontrivial chiral angle 0<2θ<π of the soliton trajectory corresponds to the noninteger electric charge q= eθ/π.

12. Origin of solitons: the isolated wave in a channel or in the Ocean – tsunami. Mathematically, and with Natal’s views, most relevent case: Waves over the shallow water described by the Korteweg–de Vries equation (KdV) t u + x3 u +6 ux u = 0 φ(x,t) = f(x-ct) -c xu + x3 u +3xu2= 0 -c u + 2xu +3u2= A=0 (xu)2 -c u2+ u3=B=0 (xu)2+c u2- u3 =min

13. modified KdV , Miura transformation φ=u2± xu t u + x3 u § 6 u2x u = 0 φ(x,t) = f(x-ct) -c xu + x3 u §2 xu3= 0 -c u + 2xu §2 u3= A -c u + 2xu §2 u3= A (xu)2 -c u2§ u4 -2Au-B=0 - 1/2(xu)2 +(u2-1)2=0 1/2(xu)2 +(u2-1)2=min Two infinite sets of integrals of motion:

21. Topological soliton

22. Polaron

27. Non degenerate ground state