Solitons from CDWs to FFLO in superconductors Serguei Brazovskii

1. Solitons from CDWs to FFLO in superconductors Serguei Brazovskii CNRS, Orsay, France and Landau Institute, Moscow, Russia Motivation: What Charge Density Waves can tell to : Doped Mott Insulators and Spin-Polarized Superconductors SC. Content: Expanding observations of solitons in quasi 1D CDW conductors. Theory: from solitons in 1D models to vortex-like elementary excitations in ordered CDWs and superconductors. Sources : Joint work with experimental groups of Grenoble (Monceau) and Moscow (IREE - F.Nad; Yu.Latyshev, et al); other experiments : STM (Brun and Wang, Takaishi), optics (Degiorgi, Dressel);solitonic lattices in HMF – several ECRYS 08 talks; earlier theories (Kirova and S.B., Buzdin, Machida, Artemenko; … et al);new modeling (A.Rojo – see the poster); New dimension – strongly non-equilibrium CDWs (Mihailovic talk).

2. Singlet ground state gapful systems: SuperConductors SCs and CDWs. Figures: deparing gaps from tunneling experiments. Standard BCS-Bogolubov view: Spectra : E(k)= ±(∆2+(vfk)2)1/2 States : linear combinations of :electrons and holes at ±pfor SC or of electrons at –p and p+2pf for CDWs CaS6 Is it always true? Proved “yes” for typical SCs.Questionable for strong coupling : High-Tc, real space pairs,cold atoms, bi-polarons. Certainly incomplete for CDWs as proved by many experiments. Certainly inconsistent for 1D and even quasi 1D systems as proved theoretically. NbSe3 – two CDWs Guilty and Most Wanted : solitons and their arrays.

3. Incommensurate CDW = ICDW at first sight – a semiconductor with free electrons or holes near the gap edges ±0.Gap performs the functions: 1. 0 - in kinetics and thermodynamics (conductivity, spin susceptibility, heat capacitance, NMR).2. 0 – in dynamic (photoemission, external tunneling). 3. 20 - in optics or in internal tunneling. 4. 0 – threshold for electronic pockets from doping or injection (FET). Nothing of this standard picture takes place in ICDWs: 1. Activation energies from transport in directionson-chain and inter-chain differ by several times (TaS3 or Blue Bronze : 200K and 800K); 2. Activations for spins and from relaxation are in between - 600K; 3. Optical absorption peaks at 20, but is deeply spread below; 4. Thresholds for charge transfer are as low as the on-chain activation, i.e. as the interchain decoupling scale Tc; 5. Charge injection is accommodated into the extended ground state via phase slip processes, rather than in formation of Fermi pockets. Static phase slip - a 2π soliton has been directly observed by STM.

4. Solitons’ workshop in organic conductors like (TMTCF)2X Discovery of charge ordering and related ferroelectricity in 2000-01 Nad, Monceau and S.B.; S. Brown et al - Access to switching on/off of the Mott state and to the Zoo of solitons. Exciton = 2 kinks bound state Eg=2 - unbound pair of kinks Peierls spin gap Drude peaks Interpretation of optics on conducting TMTCF in terms of firm expectations for Charge Ordering (Mott insulator) state. (Dressel and Degiorgi groups).

5. Incommensurate Charge Density Wave – ICDW ~cos(Qx + φ) Charge Ordering in organic Mott state was a crystal of electrons. Conventional CDW is a crystal of electron pairs. Its lowest energy current carrier may be a charge 2e defect of adding/missing one period at a defected chain. It is the ±2 soliton of the ICDW order O= Acos(2KFx + φ) Visualization of the 2 soliton = 2e prefabricated electrons’ pair C. Brun and Z.Z. Wang STM scan of NbSe3 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 –a pair of electrons/holes is accommodated to the ground state.

6. What comes if a singlet pair is broken into spin ½ components ? NOT an expectedly liberated electron-hole pair at ±, but two spin carrying “amplitude solitons” – zeros of the order parameter distributed over ξ0. This creature substitutes for unpaired electron (S.B. 1978-80) : Amplitude soliton with energy 2∆/3 , total charge 0, spin ½ This is the CDW realization of the SPINON Oscillating electronic density, Overlap soliton A(x), Midgap state =spin distribution Analogies and aggregated forms: FFLO unit for spin-polarized superconductors Unit of CDW superstructure in HMF (experiments on organics) Kink in the polyacetylene. Soliton lattice unit in spin-Peierls systems in HMF (seen by NMR)

7. Can we see the soliton bearing one unpaired electron? Expect to have a half-period amplitude kink – the elementary stripe fragment. Success for a dimerized system : Local Valence Structures in Quasi-1D Halogen-Bridged Complex Ni0.05Pd0.95Br by STM Shinya Takaishi, et al, 2004. «the first time the spin soliton has been visualized in real space» white arrow: 1D chains direction blue arrow: chain with spin soliton

8. Indirect observation of solitons and their arrays by tunnelingin NbSe3 Latyshev, Monceau, Orlov, S.B., et al 2004-2006 creation of the amplitude soliton at Eas=2D/p peak 2Dfor inter-gap creation of e-h pairs Absolute threshold at low Vt≈0.2 :bi-particle channel All features scale with (T) 2Eas<2D -- true pair-breaking threshold Vt<<Eas -- spinless charge injection threshold

9. 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

10. Singlet Superconductivity order parameter OSC~ -+ - ~ cos exp [i] - Its amplitude cos changes the sign along the allowed  soliton  s=1/2+ spin soliton wings of supercurents 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 .

11. spin carrying core half flux lines Resulting Spin - Roton complex 1D view : spinon as a - Josephson junction in the superconducting wire (V.Yakovenko et al). 2D view: pair of - vortices shares the common core bearing unpaired spin stabilizing the state. 3D view: ring ofhalf-flux vortex line, its center confines the spin. Best view: nucleus of melted FFLO phase in spin-polarized SC.

12. Solitonic lattices in CDWs or stripes in doped AFM or FFLO in SC FFLO in superconductors SC with imbalanced spin population : FF=Fulde&Ferrell 1964, LO=Larkin&Ovchinnikov 1964 1. Homogeneous phase: Fill excess spins to states above the gap 2. Modulated phase: wave number Q≠0 FF: ~exp(iQx) & LO: ~ cos(Qx) erases a mismatching at some (all in quasi-1D)parts of the FS. Valid for both suggestions FF and LO

13. 3. Build a structure of local walls so strong as to create intra-gap states which are able to accommodate access spins.Able to evolve into the LO (not to FF – gap passes trrough zeros),Proved by theory in quasi-1D. Similar to CDWs Zeeman breakdown in HMF: Experimemts 2000’s on ICDWs (Brooks, Kartsovnik, Singleton); on spin-Peierls (Berthier,Horvatic et al). 1 CDW or SC under slightly supercritical Zeeman splitting. plotted: Solitonic lattice of the order parameter, Unpaired spins = mid-gap states density distributed near the gap zeros. If melted, each element becomes a particle - Amplitude Soliton = Spinon 0 -1

14. + - + +/- -/+ - + + 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 protation (blue circles)to resolve the signs conflict.The minimal segment corresponds to the spin carrying kink. Vortices cost ~EphaseLog(L) is equilibrated by the gain ~-D*L for the string formation at long enough L. In quasi 1D it is still valid for smallest L : Ephase~Tc< For isotropic SCs - Ephase~EF– strong coupling D~EFis necessary.

15. In absence of microscopic theory for strong coupling vortices(with a single intra-gap state) - use numeric modeling of,still phenomenological, models. And it works ! At presence of unpaired spins, vortex created by rotation (magnetic field) splits into two semi-vortices. K. Kasamatsu et al 2004 Last step: reformulate these results inversely – unpaired spin creates the vortex pair even at NO orbital Magnetic Field.

16. Upper chains φ: 0→ Defected chain with spin-carrying amplitude soliton – the spinon Lower chains φ: →2 Energetics behind the vortex splitting: 2pN vortex energy ~N2 hence ½ of it is gained by splitting in 2 vortices with vorticity 2pN/2. Would always work for N=2 – no such a thing as 4p, etc. vortex. But splitting of N=1 vortex into two ½ ones is prohibited, hence expect a single vortex with a half-filled mid-gap core – generalization of Caroli-De Gennes-Matricon staircaseto the single zero-energy level. But the node in order parameter amplitude allows for “prohibited” N=1/2 circulation, hence splitting into ½ vortices with a joint spin-carrying core”:

17. TOPOLOGICAL COUPLINGOF DISLOCATIONS ANDVORTICES IN INCOMMENSURATE Spin DENSITY WAVES N. Kirova and S. Brazovskii, 2000 ISDW order parameter:OSDW~m cos(Qx+) m – staggered magnetization vector Three types of self mapping for 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 dislocationat a smaller cost of creating spin semi-vortices. Effect of rotational anisotropy: String tension binds semi-vortices

18. SUMMARY • Existence of solitons is proved experimentally in single- or bi-electronic processes of CDWs in several 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. • Solitons 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.