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Serguei Brazovskii and Natasha Kirova Natal 2012

Serguei Brazovskii and Natasha Kirova Natal 2012 Physics of synthetic conductors as low dimensional correlated electronic systems. Lecture 8. Physics of solitons in quasi one-dimensional conductors: from old theories to new events.

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Serguei Brazovskii and Natasha Kirova Natal 2012

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  1. SergueiBrazovskii and Natasha Kirova Natal 2012 Physics of synthetic conductors as low dimensional correlated electronic systems. Lecture 8 Physics of solitons in quasi one-dimensional conductors: from old theories to new events.

  2. 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 orderings in organic conductors, New accesses to Charge Density Waves,New inhomogeneous-nonstationarystrongly correlated systems Theory inspirations: V.Fateev, S.Novikov, E.Belokolos. Theory collaborations: I.Dzyaloshinskii, N.Kirova, I.Krichever, S.Matveenko Joint work with experimental groups of Grenoble (Monceau) , Paris (C.Brun), Moscow (F.Nad, Yu.Latyshev, et al).

  3. Helicoidal polyacethelene PA - multi-scale material: cells -> spirals -> threads -> crystalline fibers -> polymer chains -> p-electrons -> Peierls-dimerization -> solitons -> confinement Realisation of the Gross-Neveumodel and the KdVeqs.

  4. STM scan of a CDW with asolitonBrunWang, Monceau & S.B. Realisation of the chiral Gross-Neveu model and the NLSE At the (red) front line the defected chainis displaced by half of the period. Along the defected chain the whole period ±2is missed or gained Soliton profile along the defected chain - doped by one electron. The CDW adopts itself by half a period and its amplitude vanishesto accommodate the electron or the hole.

  5. Direct observation of solitons and their arrays in tunnelingLatyshev, Monceau, SB, et al 2004-2006 creation of soliton at Es=2D/p All features scale with (T) ! peak 2Dfor inter-gap creation of e-h pairs absolutethreshold at low Vt≈0.2 oscillating fine structure Realization of the instanton – the process of a direct transformation of one electron into one soliton.

  6. Eg=2 - unbound pair of kinks Exciton=breather=two-kinks bound state Peierlsspin gap Drude peaks 1D Mott-Hubburd state. 1 electron per site . Charge degrees of freedom can be gapful: H~ (/4) [v (x )2 + (t )2 /v] - Ucos (2 -2) • Degeneracy πoriginates solitons with chargese Optical evidences for soltons = holons – charged spinless particles.

  7. Ferroelectric charge ordering: mixed symmetry breaking via dimerizations of sites and bonds with lifting the inversion. . Ferroelectric anomaly in the electric perittivity (T) in an organic crystal.F.Nad, P.Monceau & SB - 2001. Charge ordering phase transition is driven by the Wigner crystallization of electrons (the Mott state instability). It is seen via the ferroelectricity= spontaneous electric polarisation – the sign of breaking inversion symmetry. It gives rise to as many as 4 types of solitons – all observed.

  8. Solitons in general : propagating isolated profiles – from the tsunami wave (Russel 1834) to magnetic domain walls and to hypothetic 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: superconductivity, 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 Symmetry breaking : degenerate equivalent ground states. simplest Soliton= kink between them

  9. Landau theory of phase transitions with symmetry breakings and Ginsburg-Landau approach to their deformed states a>0 , h=0 Static states – extrema (minima) of Our case: ζand ∂ζ are not small, electrons bring anomalous charges, energies and spectra are not accessed by the order parameter alone.

  10. E 2kF F π/2a -kF 2kF -2kF 2kF kF -π/2a E Charge Density Waves - three models in one: Half-filled band - 1 e per unit cell, ±2kF= ± π/a – the same point – only one degree of freedom in Δ; Incommensurate 2kF : complex field Δ= Δ1+i Δ2 3. Small 2kF: quadratic dispersion near the band bottom – Shroedinger rather than Dirac H.

  11. Origins of multi-periodicities and of solitons at the periodicities’ branching. From half filling Zeeman splitting From bottom

  12. Microscopics of local and instantaneous electronic states in CDWs. BCS-like Peierls-Fröhlich model for theCDW. 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 for co-observation of electrons and solitons: Small phonon frequency: experimentally ωph <0.1Δ

  13. 2ᴂ0– no gap at this wave number Only one (finite number in general) forbidden gap – Contrarily to common expectations of the band theory that gaps opens at any repetition of the Bragg vector 2ᴂ0

  14. Statements of Novikov theory for spectra in 1D. For a Shroedinger, Dirac, Toda, etc. eqs. for eigenvalues, there exist (multi) periodic potentials such that the number N of allowed and forbidden (gaps) bands is finite. All the rest follows from this statement ! The game takes place at the Rieman surface in the complex plane of the energy E: W2= (E-E1) (E-E2)…(E-E2N) where En are the bands rims. One periodicity is associated with each gap. Periodicities are not locked, their relative displacements, while adjusting the shapes, take no energy – hence the zero modes. When a gap collapses E2m → E2m+1 then the corresponding periodicity is degenerated to a single soliton. The soliton shape gives the reflectionless potential for the Hamiltonian. The potentials satisfy NLPDEs from the hierarchy of:KdV for Shroedinger, mKdV for the real field Dirac, NLSE for the complex field Dirac. ∂t u=δI/δu where I(u, ∂xu) is a linear combination of first N from the infinite series of integrals of motion In.Eigenvalues E also form another series of integrals of motion. Functional eqs. for an extremum of any linear combination of E(k) and In are reduced to algebraic connections. Parameters are the integrals over loops from the torus of the Rieman surface. – Key to our goals.

  15. Two infinite sets of integrals of motion: In{D} and eigenvalues of H Each term is an integral of motion of the mKdV equation tD + x3D+6 D 2xD = 0 hence the total energy is invariant. Expecting physically only the translational degeneracy, we insist the traveling solution D(x,t) = D(x-ct) We know the answer before obtaining it. This simple-minded anzats applies only to a single periodicity

  16. One way to build the family of solutions – diverging kinks (x D)2 +c D 2 -D 4 -2A D -B=0 One trapped electron – stable intermediate position Two trapped electrons – diverging pair of kinks

  17. Very important model with confinement of solitons is “illigitimately” exactly solvable: The symmetry lifting term ~Δ in the energy density does not belong to integrals of motion of the mKdV eq. Energy difference per unit lengthis a constant confinement force F.

  18. Periodic solutions: stripes as lattices of amplitude solitons CDW or SC under supercritical Zeeman splitting. plotted: Solitonic lattice of the order parameter, Density of unpaired spins = mid-gap states is distributed near the gap zeros. If melted, each element becomes a particle - Amplitude Soliton = Spinon 1 0 -1

  19. Soliton upon a periodic structure: two limiting regimes. Soliton embedded into a rare array of waves Surfers’ hunt: Overlapping soliton modulates sequence of waves. Whatever regime or soliton’s position, the array shift across the soliton is always half of the period .The soliton is topologically nontrivial but is not topologically stable. It is stable energetically accommodating the unpaired spin ½ at the split off level. The charge is variable: evolves from e to zero.

  20. Evolution from real to complex fields: From KdV to NLSE. A modulated dense array of waves with slowly varying parameters. Withem method of averaging over fast oscillations yields the overlapping function – complex Ψ(x,t) describing the local amplitude and the local displacement (the phase shift) within the wave packet. Related to an “incommensurate” Charge Density Wave – soliton results from adding an unpaired spin.

  21. Another family: chordus solitons in the complex plane of Δ 2 V1 V2 2q E - The n- filled spit off state E0 is intragap but not the midgap ! Charge conjugation symmetry is broken. Noninteger, variable charge q. If no constraints on θ, the equillibrium solution for just one electron occupying the split off sate is θ=π hence q=0: particle with spin ½ and no charge – the spinon.

  22. Joint effect of build-in ∆iand spontaneous ∆econtributions to gap ∆. - from the build-in site dimerization – inequivalence of sites A and B. - from spontaneous dimerization of bonds ∆i=∆b - thePeierls effect. SOILITONS WITH NONINTEGER VARIABLE CHARGES – new life in 2000’s: What does constrain the chiral angle and makes the charge noninteger? 1. Strict constraint from two interfering dimerizations. 2. Soft universal constraint from the additional integral of motion – the energy term linear in gradient iA(Δ*∂xΔ-Δ∂xΔ*) - Like vector potential A×current in a superconductor or stress shifting the wave number in a CDW.

  23. Noninteger variable charge: sources and problems. n=2 – spin degeneracy of filled band states n0 - filling of the split-off state. Compensation of r0(x) by local dilatations ~1/L of L delocalized states. Picture of a classical motion of the solitonΔ(x-vt): Fraction 1-2q/p of the charge e moves and gives the current,Fraction 2q/p of the charge e is homogeneously distributed over the whole length L, it gives no current. Problem: after quantization of the soliton, the whole complex of Bose field Δ and the fermions becomes a wave function distributed over L – will the local and the delocalized charges recombine?

  24. SPIN EXCITATIONS CARRY CHARGE CURRENTS: ONE- DIMENSIONAL HUBBARD MODEL. S. Brazovskii, S. Matveenko, P. Nozieres Current carrying states for 1D interacting electrons via exact results for the Hubbard model and the bosonization. The key tool is a discovered possibility to calculate the electric current from the BA solution. Away from half filling both spin and charge excitations carry charge currents, proportional to their momenta. That contradicts to the common spin-charge separation concept.The paradox resolution: Weak coupling, bosonization: account for Fermi velocity dispersion and reexamining the current operator. Strong repulsion, BA: drag of holes by the spinon as reflected in their momenta displacements via scattering phase shifts.

  25. Summary for currents. Spinons: u << 1 : j = 2p cos(πρ/2) u >> 1 : j = 2p sin(πρ)/πρ U=U/t ρ – band filling

  26. Continuous symmetries. The Instanton. • Solitons are stable energetically but not topologically. • Special significance: allowance for a direct transformation of one • electron into one soliton. (Only 22 allowed for dimerization kinks) D q 2π SOLITON E 2 V1 V2 2q E - Sequence of chordus solitons develops from bare θ=0 through the amplitude soliton at 2θ=πto the full phase slip2θ=2π. Intra-gap split-off state E evolves from 0 to -0 providing spectral flow across gap together with the electrons’ conversion. • Vn() - selftrapping branches of the total energy for chordus solitonswith the intragap state fillings n =1and n=2.

  27. Phase mode action : (u- phase velocity) Boundary condition : thediscontinuity isenforced bythechordussoliton forming around xs=0 Integrate out φ(x,t) from exp(-Ssnd[φ,θ]) - arrive for θ(t) at the typical action for the problem of quantum dissipation Caldeira-Leggett action! AS edge - the power law

  28. Probabilities to create combined topological defects: AS creates the p- discontinuity δφ=along its world line: (0<t<T, x=0) compensating for the sign change of the amplitude : x + -/+ +/- t + To be topologically allowed = to have a finite action S, the line must terminate with half integer space-time vortices located at (0,0) , (0,T) : their circulation provides the jump A→-A combined with φ→φ+π which leaves invariant the order parameter Aexp(iφ). The phase action as a function of time T: Contrary to usual 2p- vortices, the connecting line is the physical singularity which string tension gives Total action S=Score+Sphase , min{S}~ln(T) hence the power law for I(Ω)~exp(-S)

  29. Major puzzle and inspiration: amplitude solitons has been observed within the long range ordered phase at T<TcObstacle : confinement. Changing the minima on one chain would lead to a loss of interchain ordering energy ~ total length. Need to activate other modes to cure the defect ! Unifying observation : combination of a discrete and continuous symmetries Complex Order ParameterO= A exp[i] ; A - amplitude ,  - phase Ground State with an odd number of particles: In 1D - Amplitude SolitonO(x=-)  - O(x=) performed via A -A at arbitrary =cnst Favorable in energy in comparison with an electron, but: Prohibited to be created dynamically even in 1D Prohibited to exist even stationary at D>1 RESOLUTION – Combined Symmetry : A -A combined with →φ+π – semi-vortex of phase rotation compensates for the amplitude sign change

  30. confined spin SPIN-GAP cases - superconductivity or incommensurate CDW. Bosonisation language : 1D~()2 -Vcos(2)} +()2 V - from the backward exchange scattering of electrons In 1D : Spinon as a soliton + hence s=1/2 + Gapless charge sound in . CDW order parameter~ †- +  †- ~ exp [i] cos Its amplitude cos changes the sign along the allowed  soliton Same for the superconductivity with another (gauge) phase  At higher D : allowed mixed configuration + , s=1/2 , e=1 spin solitoncharged wings Spinon as a soliton + semi-integer -vortex of 

  31. Half filled band with repulsion. SDW rout to the doped Mott-Hubbard insulator. 1D~()2 -Ucos(2)} +()2 U - Umklapp amplitude (Dzyaloshinskii & Larkin ; Luther & Emery).  - chiral phase of charge displacements  - chiral phase of spin rotations. Degeneracy of the ground state: +π = translation by one site Excitations in 1D : holon as a soliton in , spin sound in  Higher D : A hole in the AFM environment. Staggered magnetization AFM=SDW order parameter: OSDW ~ cos() exp{i(Qx+)} , amplitude A= cos()changes the sign To survive in D>1 : The soliton in  : cos - cosenforces a  rotation in  to preserve OSDW

  32. + - + +/- -/+ - + + Kink-roton complexes as nucleuses of melted lattices: FFLO phase for superconductors or strips for doped AFMs. Defect is embedded into the regular stripe structure (black lines).+/- are the alternating signs of the order parameter amplitude. Termination points of a finite segment L (red color) of the zero line must be encircled by semi-vortices of the p rotation (blue circles)to resolve the signs conflict.The minimal segment would correspond to the spin carrying kink. Vortices cost ~EphaselogL is less than the gain ~-DL for the string formation at long L. Can we shrink to the atomic scale? For smallest L~"unit cell", it is still valid in quasi 1D : Ephase~Tc< For isotropic SCs, Ephase~Ef– strong coupling is necessary.

  33. Spatial Line Nodes and Fractional Vortex Pairs in the FFLO Vortex State of Superconductors D. F. Agterberg, Z. Zheng, and S. Mukherjee 2008 Vortex molecules in coherently coupled two-component Bose-Einstein condensates K. Kasamatsu, M.Tsubota, and M. Ueda 2004 Last step: reformulate these results inversely – unpaired spin creates the vortex pair at NO rotation/MF.

  34. TOPOLOGICAL COUPLINGOF DISLOCATIONS ANDVORTICES IN INCOMMENSURATE Spin DENSITY WAVES N. Kirova, S. Brazovskii, 2000 An attempt to rehabilitate the Density Waves against more fascinating symmetries: He3 (Volovik et al), skyrmions in QHE (Yu.Bychkov et al, B. Doucot et al for bi-layers) ISDW order parameter:OSDW~m cos(Qx+) m – staggered magnetization vector Three types of self mapping for the OSDW : 1. normal dislocation, 2 translation: +2, mm 2. normal m - vortex, 2 rotation: mR2m,  3. combined object : +, mRm = -m Coulomb energy favors splitting the phase dislocation at a smaller cost of creating spin semi-vortices. Effect of rotational anisotropy: String tension binds semi-vortices

  35. SUMMARY • Existence of solitons is proved experimentally in single- or bi-electronic processes of 1D regimes in quasi 1D materials. • They feature self-traping of electrons into mid-gap states and • separation of spin and charge into spinons and holons, • sometimes with their reconfinement at essentially different scales. • Topologically unstable configurations are of particular importance • allowing for direct transformation of electrons into solitons. • Continuously broken symmetries allow for solitons to enter • D>1 world of long range ordered states: SC, ICDW, SDW. • They take forms of amplitude kinks topologically bound to • semi-vortices of gapless modes – half integer rotons • These combined particles substitute for electrons certainly in quasi-1D systems – valid for both charge- and spin- gaped cases • The description is extrapolatable to strongly correlated isotropic cases. Here it meets the picture of fragmented stripe phases

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