Diapositive 1

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 p for SC or of
electrons at p and p2pf 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. Certain
ly 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 directions on-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 2p 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
Eg2? - 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
f)
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 f)
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 ?0.
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 tunneling in NbSe3 Latyshev, Monceau, Orlov,
S.B., et al 2004-2006
creation of the amplitude soliton at Eas2D/p
peak 2D for inter-gap creation of e-h pairs
Absolute threshold at low Vt0.2? bi-particle
channel
All features scale with ?(T)
2Easlt2D -- true pair-breaking threshold VtltltEas -
- spinless charge injection threshold
9
Major puzzle and inspiration amplitude solitons
has been observed within the long range ordered
phase at TltTc Obstacle  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
Parameter O A expi? A - amplitude , ? -
phase Ground State with an odd number of
particles In 1D - Amplitude Soliton O(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 Dgt1 RESOLUTION Combined Symmetry
A ?-A combined with ??fp semi-vortex of
phase rotation compensates for the amplitude
sign change
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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 s1/2 gapless charge sound in ?.

• Singlet Superconductivity order parameter
• OSC ??? ?-?? ??? ?-? cos? exp i?
• - Its amplitude cos? changes the sign along the
  allowed ? soliton
• ? ? ??? s1/2 ?? ??
• ?spin soliton ? ?wings of supercurents?

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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 of
half-flux vortex line, its center confines the
spin. Best view nucleus of melted FFLO phase in
spin-polarized SC.
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Solitonic lattices in CDWs or stripes in doped
AFM or FFLO in SC
FFLO in superconductors SC with imbalanced spin
population FFFuldeFerrell 1964,
LOLarkinOvchinnikov 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
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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
2000s 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 p rotation (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 -DL for the string formation at long
enough L. In quasi 1D it is still valid for
smallest L EphaseTclt? For isotropic SCs -
EphaseEF strong coupling DEF is necessary.
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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.
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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 N2 no such a thing as
4p, etc. vortex.
But splitting of N1 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 N1/2 circulation, hence
splitting into ½ vortices with a joint
spin-carrying core
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TOPOLOGICAL COUPLING OF DISLOCATIONS AND
VORTICES 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
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• 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
• Dgt1 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.

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Inverse rout from stripes to solitons
1D ? quasi 1D ? 2D,3D route to dopping of AFM
insulator. Aggregation of holes into stripes.
Equivalence for spin-gap cases Fulde-Ferell-Larki
n-Ovchinnikov FFLO phase in superconductors Solito
nic lattices in CDW above magnetic breakdown
LANL, FSU HMFNL Solitonic lattices in
spin-Peierls GeCuO in HMF - Grenoble
20
Thermally activated interchain collective
tunneling current I(U) in the subgap region
Ult?. U is shown in units of ?. The plots
correspond to temperatures T/ ? 1/4,1/6,1/8
21
Major puzzle and insight The amplitude
solitons has been observed at the long range
ordered phase TltTc Obstacle  confinement.As
topological objects connecting degenerate
vacuums,solitons acquire an infinite energy
unless they reduce or compensate their
topological charges.
Symmetry breaking degenerate equivalent ground
states. simplest Soliton kink between them
Energy as a function of configuration. Two-fold
degeneracy or cross-section of the
axial-symmetry shape
Simple changing of the minima on one chain would
lead to loss of interchain ordering energy
total length. Need to activate other modes to
cure the defect !
22

• 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).
• ? - phase of charge displacements
• ? - chiral phase of spin rotations.
• Degeneracy of the ground state
• ? ??p 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 ? AFMSDW order
  parameter
• OSDW cos(?) exp?i(Qx?) , amplitude A cos(?)
  changes the sign
• To survive in Dgt1  The ? soliton in ? cos ?
  ? - cos ? enforces the ? rotation in ? to
  preserve OSDW

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Nagaev et al , Brinkman and Rice
In our language Propagating hole is the
amplitude soliton. Its motion permutes AFM
sublattices ?,? - creating a string of reversed
order parameter - staggered magnetization. It
blocks propagation of holes unless spins are
allowed to rotate.
Adding the semi-vorticity to the string end heals
the permutation, thus allowing for propagation
of the combined particle.
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This easy plane XY picture - exact equivalence
of solitons in ICDW as observed by tunneling at
TltTc. For a spin-isotropic XYZ AFM, a better
choice is the monopol T. Morinari, YITP
Alternative viewNucleus of the stripe phase or
the minimal element of its melt.
25
Competitor to the above scenario of split ½
vortices with a joint spin-carrying core A
single vortex with a half-filled mid-gap core
extrapolation of Caroli-De Gennes-Matricon
staircase to single zero-energy level.
Pro compatible with quasi-1D limit, similar
local energy scales Contra LogL growing energy
requires for rotation or MF or for high
temperature above the BKT transition.
Still keep in mind this possibility that
rotation/MF can erase the FFLO stripes by placing
unpaired spins to the vortex core.