Solitons in quasi 1D electronic systems and beyondSolitons in quasi-1D electronic systems and beyond.

ECRYS – 2017, Carges, France

Serguei BrazovskiiSerguei Brazovskii

CNRS‐LPTMS, University Paris‐Sud,, y ,University Paris‐Saclay, Orsay, France

Microscopic solitons in quasi-1D electronic systems with symmetry breakings:

Born in theories of late 70’s, Guessed in experiments of early 80’s,

Persuading observations in 2000’s, Visualization in 2010’s

Why the resent explosion of interest ? New optically and electronically active polymers,

New ferroelectric Mott state in organic conductors.New STM and tunneling observations in Charge Density Waves.

New high magnetic field access to spin polarized FFLO-like states of superconductor, cold atoms, CDW, and the spin-Peierls state.

Further motivations from the new area:Transformations of symmetry broken electronic states by

impacts of optical pumping and electrostatic field-effect doping.

The tidal arrayThe tidal array

Solitons in general : propagating isolated profiles –from the tsunami wave to magnetic domain walls and

3

from the tsunami wave to magnetic domain walls and to stripes and patterns in correlated electronic systems

Solitons in perspective of quasi-1D electronic systemsSelf-localized excitations exploring degeneracies of ground states

ith t l b k twith a spontaneously broken symmetry: superconductor, CDW, SDW, charge-order Mott insulator, electronic ferroelectric.S lit i d/ h t l i f tiSolitons carry spin and/or charge – separately, even in fractions. They bring associated spectral features (electronic mid-gap states, vibrational modes)They can take over normal electrons as charge or spin carriers.They give definitions of holons and spinons in 1D correlated systems.Instantons : related transient processes for creations of solitons or ptheir pairs, for conversion of electrons or excitons into solitons

S mmetr breakingSymmetry breaking :degenerate equivalent ground states.i l t litsimplest soliton=

kink between them

New chapter of recent years:

Visualization of a soliton in a 1D BCS system:

Visualization of microscopic solitons.

Visualization of a soliton in a 1D BCS system:Cooper pairing in attractive cold Fermi-atoms.

M.W. Zwierlein, MIT “We create long-lived solitons in a strongly interacting e e , e c eate o g ed so to s a st o g y te act gsuperfluid of fermionic atoms and directly observe their motion.”

Observation of a soliton in a truly 1D physics, Han Woong Yeom Lab., IBS, Postech, PohangThe array of Indium chains at Si 111 surfaceThe array of Indium chains at Si 111 surface.Ground state: 2-fold commensurate CDW : dimerization.

EE

EF

-kF kF kF-kF

Nb chains with a spontaneous Peierls dimerization, STM.IBS, Postech, Korea. wait for T.-H. Kim talk

Strong defect immobilizes the soliton at the lower chain

Excellent theory fittingwith ξ= 3 4nm=9a /with ξ= 3.4nm=9a

)/cos()/tanh( axx

/Fv

Amplified central view

Hopping of solitons’ was noticed – mobile intrinsic defect !

STM view of the CDW2 in NbSe3STM is tuned to see only the CDW

The chain bearingthe soliton

S. B., C. Brun, Z.-Z. Wang, and P. Monceau, Phys Rev Let (2012)

Another presentation of same scanAt the (red) front line the defected chainAt the (red) front line, the defected chainis displaced by half of the period.Along this chain, the whole period 2is missed (or may be gained).A particle with the charge 2e ? No,1e - might be created by one electron.g yGo to the raw data:

Soliton profile along the defected chain. The CDW adopts itself by half a period shift and its amplitude vanishesshift and its amplitude vanishes, presumably to accommodate electron or hole to the mid-gap state.

Profile along the defected chainit t i hb

The CDW is perforated here and there by nodes of amplitude

vs its nearest neighbor

Defected chain vs theory

))/arctan(/2sin()/tanh(

lxxx

))((Phase stretching by over long tails allows the amplitude kinkto adapt to a long range 3D order. General concept:

l i ll b d l f h ki k d h h lf itopologically bound complex of the kink core and the half-integer vortex.

STM in NbSe3, Brun, Wang, Monceau & SB, PRL 2012

Singlet-ground-state gapful systems: SuperConductors SCs and Charge Density Waves CDWs.Standard BCS-Bogolubov view:Spectra : E(k)= ±(∆2+(vfk)2)1/2 , ∆exp[-1/]States = linear combinations of :electrons and holes at ±p for SC ap upap+vpa-p

+

electrons at –p and p+2p for CDW a u a +v aelectrons at –p and p+2pf for CDW ap upap+vpap-2pF

Is it always true? Proved “yes” for typical SCsIs 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 modern experimentsCertainly incomplete for CDWs as proved by many modern experiments.Certainly inconsistent for 1D and even quasi 1D systems as proved theoretically.

Guilty and Most Wanted : solitons and their arrays.

CaC6 NbSe3

Figures: pair-breaking gaps from conventional tunneling experiments.

Resolving the intragap richness by special, intrinsic, tunneling experiments in NbSe3 (Latyshev, Monceau, SB, et al

The gap interval (-Δ, Δ) may become partly filled by:

Absence of the true long range order 2/1 EAbsence of the true long range order(the pseudogap),Anomalous lower energy excitations

2/1 E

Anomalous lower energy excitations - solitons

Pseudogaps in optics 13

Optical absorption,Blue Bronze, L. Degiorgi, et al

Solitons in the Peierls system = CDW of bonds dimerization, SSH picture:polyacetylene Nb chains etc

W

polyacetylene, Nb chains, etc.

Δ0

Δ0-Δ0

Δ0

0

Charged solitons q=e, s=0-Δ0

Spin soliton, q=0, s=1/2Δ0

0Spin soliton, q 0, s 1/2

tanh [x/ ] W = (2/) E =0

-Δ0

14

0 tanh [x/0], Wtot = (2/)0 Eel =0

SB for CDW; SSH, Takayama-LinLiu-Maki for polyacetylene

The way to create solutions in the bulk – diverging kinks

0

Eb

Intragap levels of thedeveloping bound state

0-Eb

One trapped electron –stable intermediate position

Two trapped electrons –diverging pair of kinks

N t t i i f (CH)

Killing argument for kinks: global lifting of symmetry.

Nature present -- cis-isomer of (CH)x : build-in slight inequivalence of bondshence lifting of ground state degeneracy,

f f

Cis-(CH)x :Nonsymmetric dependenceof GS energy on dimerisation

hence confinement of solitons

Confinement – the linear growth of the

A

Ideal chain

attraction energy while the particle diverge.

B

Energy difference per unit lengthis a constant confinement force Fis a constant confinement force F.

2 kinks cannot diverge far away anymore.They form a loose bound state – a particle with the charge 2e -the bipolaronthe bipolaron

N i di tNew ingredient –confinement energy

Confinement of kinks pair into 2e charged (bipolaron) or neutral (exciton) complexor neutral (exciton) complex.Symmetry determined picture of optical differences for trans- and cis- isomers S. B. and N. K.

17

Photoconductivity trans-(CH)x vs photoluminescence cis-(CH)xalso new optical features due to hybridization of mid-gap states

Only luminescence of cis-(CH)x vs only photoconductivityIn trans- (CH)x

Broad line covered by multiple phonons repetitions - vibrations of the lattice dressing

Same data, justcleaned from phonons

repetitions vibrations of the lattice dressing of the selft-traped state

trans-PA, T=7K, pump at 2.54eVRemnant luminescence is 50 times less then that of cis PA50 times less then that of cis-PA

18

Arrays of amplitude solitons:FFLO state in spin-polarized quasi-1D superconductorsBuzdin et al, Machida et al;, ;CDW superstructure in HMF (in low-gap organic conductors).Soliton lattice in spin-Peierls systems in magnetic field(NMR – C. Berthier et al, Grenoble; neutrons - J.P. Pouget et al;(NMR C. Berthier et al, Grenoble; neutrons J.P. Pouget et al;Fukuyama et al in theory)Solitonic lattice as a support for superconductivity nearSDW insulator-metal transition (Jerome Pasquier S B ):SDW insulator metal transition (Jerome, Pasquier, S.B.): 1

CDW or SC under slightly supercritical Zeeman splitting. plotted:

0

p g pSolitonic lattice of the order parameter,Unpaired spins = mid-gap states density distributed near the gap zeros.

-1

y g pIf melted, each element becomes a particle - Amplitude Soliton = Spinon

Equivalent picture in Peierls systems: charged spinless solitons introduced by dopping or field effect SB, Gordunin, Kirova for CDW; Buzdin & Tugushev, Machida for SC

Combination of charge doping and spin injection: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.SB, Dzyaloshinskii, Kirova

SOILITONS WITH NONINTEGER VARIABLE CHARGES – new life in 2000’s:Combination of  two interfering dimerizations: electronic ferroelectricity.

22bs

Joint effect of two contributions to the gap ∆. ∆s and ∆b from dimerizations of sites and of bonds. One is the build-in another – spontaneousOne is the build in, another spontaneous.

221

Kiiv

Tr xF

21

2021

221

)(,

cnst

Kivi

TrxF

Nontrivial chiral angle 0<2θ<π of the soliton trajectory corresponds tothe noninteger electric charge q= eθ/πthe noninteger electric charge q eθ/π.Interpretation: this is the FE domain wall, q=polarization jump

A realization with proving for the ferroelectricity and the solitons:charge-transfer complexes with the neutral-ionic transitions.Recall N. Kirova talk.Mainly Y. Tokura school,Mainly Y. Tokura school,

with particular contributions from Koshihara, Okamoto, …Theories from Nagaosa, Yonemitsu, Ishihara

Most fascinating in applications if achieved:a substituted polyacetylene, (AB)x polymer.SB and N. Kirova: The necessary polymers do exist: (since 1999 Kyoto Osaka Utah Nanjing Hong Kong)(since 1999, Kyoto-Osaka-Utah, Nanjing, Hong Kong).By today – complete optical characterization,

indirect proof for spontaneous bonds dimerizationindirect proof for spontaneous bonds dimerization via spectral signatures of solitons.

Still a missing link : no idea was to check for the Ferroelectricity:To be tried ? and discovered !

23

Electronic ferroelectricityprovoked by a phase transition to the Mott insulator stateMonceau, Nad, SB, et al, 2000

Access to switching on/off of the Mott state and to the Zoo of solitons.

(TMTCF)2X, 1980 – Bechgaard, Jerome

SC- superconductivityAF- AFM = SDWSP- Spin-PeierlsLL- Luttinger (Tomonaga!) liquidMI M tt i l tMI- Mott insulator

Red line TCO – 2000’s revolution:Ferroelectricity (Monceau, Nad, SB, et al)= Charge Ordering (Brown et al)

spontaneous good metals on top of

Facility to see Solitons:

spontaneousMott insulators

good metals on top of hidden Mott insulators Access to switching on/off of the

Mott state and to the Zoo of solitons.

Views and interpretations:FerroElectric Mott-Hubbard state,mixed site/bond 4KF CDW, non-

Facility to see Solitons:purely 1D regime -TCO150K is 10 times b 3D l t imixed site/bond 4KF CDW, non

symmetrically pinned Wigner crystal,charge ordering = disproportionation

above 3D electronic transitions.

What does conduct in these “narrow gap semiconductors” ?

Organic conductors: low scales of gaps allow for their determination from thermal activation and comparison with optical features.

PF6 Conductance normalized to RT

What does conduct in these narrow gap semiconductors ?NOT the electrons, even in the polaronic version !

PF6AsF6SbF6

Conductance , normalized to RT- Ahrenius plot LogG(1/T). Gaps for thermal activation ofGaps c for thermal activation ofconductance within 500-2000K.exp(-s /T) , s

Contrarily to normal semiconductors –NO GAP, or a very small gap s, in spin susceptibility χ(T).Clearlest example for conduction by charged spinless solitons - holons.c a ged sp ess so o s o o s

Collective description for the 1D Mott state on the dimerized lattice.Spin degrees of freedom are gapless, split-off and not important for a while. Charge degrees of freedom: phase =(x,t)

Two fold commensurability: pinning potential potential and its degeneracy

2KF CDW/SDW ~cos (+ πx/2a) 4KF CDW~ cos (2+ πx/a)

Two fold commensurability: pinning potential potential and its degeneracy.Wave function distriburtion at EF (for the site dimerisation):+ o - o + o - o

- +

=0,π: =1 at good sites and =0 at bad sites =π/2: =0 at good sites and = =1 at bad sites

- +

H~ (/4) [v (x )2 + (t )2 /v ] - Ucos(2) + Wcos (4) without interactions =1 and v=vFThe energy HU= -Ucos (2) is doubly degenerate between = 0 and = . It allows for phase solitons = holons with charge e : CO vacancy defects.

φ=0φ =π

COMBINED MOTT - HUBBARD STATEH~ [v (x )2 + (t )2 /v ] + HU

Chiral phase =(x t) for electrons near +/-KChiral phase =(x,t) for electrons near +/-KFU: amplitude of the Umklapp scattering

Dzyaloshinskii & Larkin, Luther & Emeryy y

Other views: commensurate Wigner crystal, 4KF CDW (Pouget et al)

2 types of dimerization 2 types of dimerization 2 interfering sources for two-fold commensurability 2 contributions to the Umklapp interaction:Site dimerization : HU

s=-Us cos 2 (spontaneous)U s ( p )Bond dimerization : HU

b=-Ub sin 2 (build-in)At presence of both site and bond types

HU= -Uscos 2 -Ubsin 2 = -Ucos (2-2) Us0 0 phase = “mean displacement of all electrons”shifts from =0 to = , hence the gigantic FE polarizationshifts from 0 to , hence the gigantic FE polarization.

HU= -Ucos (2-2) Double degeneracy : = and = +. Allows for phase ± solitons – spinless holons with charge ± e.

φ=0φ =π

Purely on-chain solitons, exist as conducting quasiparticles both above and below the TFE.

Spontaneous can change sign between different FE domains.Then electronic system must also adjust its ground state from to -.

φ quasiparticles both above and below the TFE.

y j gHence the domain boundary Us-Us requires for thephase soliton of the increment =-2which will concentrate the non integer charge q= 2e/ per chainwhich will concentrate the non integer charge q=-2e/ per chain.

φ =- φ=

alpha- solitons are walls betweendomains of opposite FE polarizationsφ domains of opposite FE polarizations

They are on-chain conducting particles only above TFE.Below TFE they aggregate into macroscopic walls. FE y gg g pThey do not conduct any more, but determine the FE depolarization dynamics (Nad, Monceau, SB)

Another evidence for dynamics of soltons = holons: the activated conductivity with gapless spin susceptibility.Optical gap in the quasi-Mott state of organic conductors

E =2 - unbound

Optical gap in the quasi Mott state of organic conductorsas a creation of pairs of solitons in the quantum sin-Gordon theory.

Exciton = two-kinks bound state

Eg 2 unboundpair of kinks

Peierlsspin gap

Drudepeaks

Optical conductivity (absorption): M. Dressel, L. Degiorgi, et al

Effects of subsequent symmetry lifting.Topologically confined solitons. Spin-Charge reconfinement.

Reincarnation of the electron.Reincarnation of the electron. Another present from the Nature: second symmetry lowing transition

of the tetramerization or of the spin-Peierls

SCN

ReO4Ahrenius plots for the cobductivity.Inclination gives the activation energy f h i

Spin-charge reconfinement below TAO of tetramerisation.

E h d f t l i ll l d lit i b th

for charge carriers

Enhanced gap comes from topologically coupled π- solitons in both sectors of the charge and the spin. The last is weakly localized. What does it mean ?

]2/(exp[~ iSpin degrees of freedom enter the game:

θ - spin chiral phase, θ′/ π = smooth spin density

Schematic illustrations for  effects of the tetramerizationI i l f b d b t d it Inequivalence of bonds  = , ‐‐ between good sites endorses ordering of spin singlets.l h b l b dAlso it prohibits translations by one distance

which were explored by the =π soliton.But its combination with the defected unpaired spin(θ=π soliton which shifts the sequence of singlets)is still allowed as the selfmapping –connction of equivalent ground states

Further symmetry lifting of lattice tetramerization or of spin-Peierls ordermixes charge and spin: additional energy Vcos ()cos θ - on top of ~Ucos (2)Vcos ()cos θ on top of Ucos (2) and θ -- chiral phases counting the charge and the spin ′/ π and θ′/ π = smooth charge and spin densitiescos θ sign instructs the CDW to make spin singlets over sorter bonds

Major effects of the small V - term :Opens spin gap 2σ :

spI

cos θ sign instructs the CDW to make spin singlets over sorter bonds

triplet pair of θ= solitons at =cnst• Prohibits = solitons –

now bound in pairs by spin strings ch

In

now bound in pairs by spin strings• Allows for combined spin-charge

topologically bound solitons:{ = θ=} – leaves the V term invariant

harg{ = , θ=} – leaves the V term invariant

Quantum numbers of the compound particle --charge e spin ½ but differently localized:

e

spIcharge e , spin ½ but differently localized:

charge e , = sharply within ħvF/

spin ½ , θ = loosely within ħvF/σ

In

FROM SOLITONS TO STRIPES at HIGHER DIMENSIONSFINITE TEMPERATURE, ENSEMBLES OF SOLITONS, PHASE TRANSITIONS OF SOLITONS’ CONFINEMENT

AND OF THEIR AGGREGATION INTO STRIPES.

Can the solitons cross the border to the higher D world ?Ca t e so to s c oss t e bo de to t e g e o d

Are they allowed to bring their anomalies like spin charge separation or mid gap states?spin-charge separation or mid-gap states?

Password : confinement.

As topological objects connecting degenerate vacuums,solitons acquire an infinite energy unless they q gy yreduce or compensate their topological charges.

FROM SOLITONS TO STRIPES at HIGHER DIMENSIONSFINITE TEMPERATURE, ENSEMBLES OF SOLITONS, PHASE TRANSITIONS OF SOLITONS’ CONFINEMENT

AND OF THEIR AGGREGATION INTO STRIPES.

S l i d d li f i i l d lSolution and modeling for a statistical modelof ensembles of solitons

T. Bohr, S.Teber, P. Karpov, and SB 1983-2016, , p ,

2D, 3D ordered states of quasi-1D systems with a double Z2 degeneracy of the ground state:

t ith th l tti di i tisystems with the lattice dimerization(CH)x , atomic chains, spin-Peierls state, charge ordering

In-phase In-phaseOut of phase

Interaction of chains brings confinement of solitons into pairs.In-between the kinks, the interchain correlation is broken, h th fi t W F| |hence the confinement energy Wcnf = F|x| ;

T > T1: Solitons exist as individual entities; concentration <<1T > T1: Solitons exist as individual entities; concentration <<1. Interchain coupling: attraction of kinks towards binding them into pairs at the same chain or to walls at neighboring chains .

T1 >T > T2 Confined pairs of solitonsT1 >T > T2. Confined pairs of solitons.

T2 >T . Aggregation of solitons into domain walls immensed into the liquid of confined pairs of solitons.

T<T2 : macroscopic domain walls are formed from transversely

Monte-Carlo modeling: Peter Karpov and SB, 2016

T=2.1J

T T2 : macroscopic domain walls are formed from transversely aggregated kinks by a second phase transition at T2.

T=2.0J

Distractive effects of Coulomb interactions

V = 0 02J VC = 0.04JVC = 0.02J VC 0.04J

VC = 0.1J VC = 0.5J

Continuous symmetries. Complex order parameter.Superconductors, AFM, SDWs, incommensurate CDWs.

The higher continuous degeneracyThe higher continuous degeneracyallows for existence and creation of isolated solitons,for direct conversion of a single electron into a single soliton.

In 1D, a single electron becomes unstable with respect to its , a s g e e ect o beco es u stab e t espect to tsselftrapping - transformation into a soliton.

The soliton is stable energetically but not any more topologicallyThe soliton is stable energetically but not any more topologically

Modeling for the amplitude -soliton at the unhindered phase.

Localisation of the self-trapped electron

(A,ϕ) trajectory of the CDWorder parameter Acos(Qx+ϕ)

This is the state of a 1D CDW or 1D superconductor with one added unpaired electronwith one added unpaired electron.

Solitons are stable energetically but not topologically.Special significance: allowance for a direct transformation of one electron  into one soliton.(Only 22 allowed for dimerization kinks)

The path from the single electron to the amplitude soliton.

2

E2π

V1

E2πSOLITON

E

V2

Sequence of chordus solitons develops from bare chiral angle θ=0th h th lit d lit t 2θ t th f ll h li 2θ 2

-

through the amplitude soliton at 2θ=π to the full phase slip 2θ=2π. Intra-gap split-off state E evolves from 0 to -0 providing spectral flow

across the gap together with the electrons’ conversion.Vn() - selftrapping branches of the total energy for chordus solitonswith the intragap state fillings n =1 and n=2.

C. Brun egt al, Soliton in NbSe3,defected chain vs its neighbors

))/arctan(/2sin()/tanh( lxxx ))/arctan(/2sin()/tanh( lxxx

A li d li D 1 Compensating phase shiftAmplitude soliton, D=1 Compensating phase shift,augmented in D>1

Unifying observation :b f d dcombination of a discrete and continuous symmetries.

Complex Order Parameter O= A exp[i] ; A ‐ amplitude ,  ‐ phase

Ground State with an odd number of particles:The 1D ‐ Amplitude Soliton O(x=‐)  ‐ O(x=)performed via  A ‐A at arbitrary =cnstp y

Favorable in energy in comparison with an electron, butProhibited to be created dynamically even in 1DProhibited to be created dynamically even in 1DProhibited to exist even stationary at D>1RESOLUTION :Combined Symmetry U1/Z2 of the order parameter Aexp(iφ)A ‐A combined with →φ+π – semi-vortex of phase rotation

compensates for the amplitude sign change

SB+N. Kirova, 2000+

confinedspin

Spinon as a soliton + semi-integer -vortex of

spin

Spinon as a soliton semi integer vortex of

ispin carryingcore

halfflux lines

Quasi 1d view : spinon as a - Josephson junction in theQuasi 1d view : spinon as a Josephson junction in the superconducting wire (applications: Yakovenko et al).2D view : pair of - vortices shares the common core

bearing unpaired spinbearing unpaired spin.3D view : half-flux vortex stabilized by the confined spin.Updown view: nucleus of melted FFLO phase in spin-polarized SC

Kink‐roton complexes asnucleuses of melted lattices:

+

FFLO phase for superconductors or strips for doped AFMs.

+ - +/- -/+ - +

+D f t i b dd d i t th l t i t t (bl k li )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 b i l d b i ti f th t ti (bl i l )be encircled by semi-vortices of the 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 ~-Lfor 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.

At presence of unpaired spins, the vortex created by rotation (magnetic field) splits into two semi vortices(magnetic field) splits into two semi-vortices.

Spatial Line Nodes and Fractional Vortex Pairs in the FFLO Vortex State of SuperconductorsD. F. Agterberg, Z. Zheng, and S. Mukherjee 2008

Vortex molecules in coherently coupled two-Vortex molecules in coherently coupled twocomponent Bose-Einstein condensatesK. Kasamatsu, M.Tsubota, and M. Ueda 2004

Last step: reformulate these results inversely –unpaired spin creates the vortex pair at NO rotation/MF.

Propagating hole as an amplitude soliton.Its motion permutes AFM sublattices ↑,↓ creating a string of the reversed order parameter:creating a string of the reversed order parameter:staggered magnetization. It blocks the direct propagation.

Bulaevskii,

Addi th i ti it t th t i d h l th t ti

Khomskii, Nagaev.Brinkman and Rice.

Adding the semi-vorticity to the string end heals the permutationthus allowing for propagation of the combined particle.

sro

tatio

ns

spin flipping

b h l ti

spin

by hole motion

Alternative view:Nucleus of the stripe phase or the minimal element of its melt.

Half filled band with repulsion. SDW rout to the doped  Mott‐Hubbard insulator.

1D~()2 ‐Ucos(2)+()2

U ‐ Umklapp amplitude  atio

ns

spin flipping(Dzyaloshinskii & Larkin ; Luther & Emery). ‐ chiral phase of charge displacements ‐ chiral phase of spin rotations. sp

in ro

t spin flipping

by hole motion

p pDegeneracy of the ground state:

+π =  translation by one site

Staggered magnetization   AFM=SDW order parameter:OSDW ~ cos() exp{i(Qx+)} , amplitude A= cos() changes the signTo survive in D>1 :  The  soliton in  : cos ‐ cos

enforces  a  rotation in  to preserve OSDW

TOPOLOGICAL COUPLING OF DISLOCATIONS AND VORTICESIN INCOMMENSURATE Spin DENSITY WAVES

N. Kirova, S. Brazovskii, 2000

An attempt to rehabilitate the Density Waves against more fascinating symmetries:He3 , skyrmions in QHE

ISDW d t O (Q + )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, mm2. normal m - vortex, 2 rotation:

RmR2m, 3. combined object :+, m R m = -m

Coulomb energy favors splitting the phase dislocation at a smaller cost of creatingspin semi-vortices.

Effect of rotational anisotropy:String tension binds semi-vortices

OUTLOOK

•Existence of electronic solitons is proved experimentally for CDWs in several classes of quasi-1D conductorsCDWs in several classes of quasi-1D conductors.

•Solitons feature self-trapping of electrons into mid-gap states and separation of spin and charge into eitherstates and separation of spin and charge into either spinons or holons, sometimes with their reconfinement.C i l b k i ll f li•Continuously broken symmetries allow for solitons to enterD>1 world of long range ordered states: SC, ICDW, SDW.

•Solitons take forms of amplitude kinks, topologically bound tosemi-vortices of gapless modes – half integer rotons.

•These combined particles substitute for electrons: certainly in quasi-1D systems – for both charge- and spin- gaped cases.Th d i ti i t l t bl t t l l t d i t i•The description is extrapolatable to strongly correlated isotropic

cases: from dopped AFM insulators to FFLO in superconductors. Here it meets the picture of fragmented stripe phases.

Theory inspirations: E. Belokolos, A. Buzdin, V. Fateev, H. Fukuyama, K. Maki, , , , y , ,Matchida, S. Novikov, R. Schrieffer, ….Theory collaborations:I Dzyaloshinskii N Kirova I Krichever S Matveenko V Yakovenko

Sources : Joint work with experimental groups ofG bl (P M ) O (C B & Z Z W C P i & D J )

I. Dzyaloshinskii, N. Kirova, I. Krichever, S. Matveenko, V. Yakovenko

Grenoble (P. Monceau), Orsay (C. Brun & Z.Z. Wang; C. Pasquier & D. Jerome), Moscow (F. Nad, Yu. Latyshev, A. Sinchenko), Ljubljana (D.Mihailovic), Seoul (J.W. Park), Tokyo (K. Miyano, H. Okamoto).