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IIP‐UFRN, Natal, 2013, lecture 2.S lf i f l d li i h i l d lSelftraping of electons and solitons in the Peierls model.
Landau theory of phase transitions with symmetry breakingsand Ginsburg‐Landau approach to their deformed states
2242
21
41
21 lbahH
242a>0 , h=0
Static states – extrema(minima) of
dVH
0222 lbah
0 lbah
Our case: ζ and ∂ζ are not small vacuum electrons bring anomalous chargesOur case: ζ and ∂ζ are not small, vacuum electrons bring anomalous charges, energies and spectra are not accessed by the order parameter alone.
Dream – a metal Reality ‐ insulator
CH
CH
CH
C CH H
Peierls effect ‐ one dimentional chain of equidistant atoms is unstable with respect to the dimerisation: Spontaneus symmetry braking results in an “insulating state” the gap is open
EE
Spontaneus symmetry braking results in an insulating state , the gap is open.
FDexp(‐
3
‐kF kF kF‐kF
EE
2kF2kF
2kF‐2kF
F
F
2kF
))(),(( xx π/2a ‐kF‐π/2a kF
xikxik FF ee
kv
H F*
)(
xkixki FF exexx 22 )(*)()( ee
kv
HF xik )(x
Charge Density Waves ‐ three models in one:1. Half‐filled band ‐ 1 e per unit cell, ±2kF= ± π/a – the same point –
only one degree of freedom in Δ; 2 Incommensurate 2k : complex field Δ= Δ +iΔ2. Incommensurate 2kF : complex field Δ Δ1+iΔ23. Small 2kF : quadratic dispersion near the band bottom –
Shroedinger rather than Dirac H.
Microscopics of local and instantaneous electronic states in CDWs .BCS‐like Peierls‐Fröhlich model for the CDW.Exact static solutions solitons of multi electronic modelsExact static solutions – solitons‐ of multi‐electronic models. Adiabatic generalization to dynamic processes – instantons.
Incommensurate CDW : Acos(Qx+j) Q=2kfIncommensurate CDW : Acos(Qx j) Q 2kfOrder parameter : Δ(x,t)~ Aexp(ij)Electronic states y = y +exp(ikfx+ij /2) + y-exp(-ikfx-ij /2)
2
22
* 1t
x dxKiTr
22 t
phx
dxi
Justification of the mean‐field BCS, and for co‐observation of electrons and solitons: Small phonon frequency: experimentally ωph <0.1Δ
Homogeneous ground state of the Peierls‐Frohlich insulator.
22 kvE Fk
dkkvkvLEW FFE
el 22)(22
2
L
We subtract the energy of the parent metal where kvE Fk
2gLWlat
F
Fel Log
vLW
2
2
FLnCKW 2
2 WMin2
0
For the ground state Δ=Δ0 should be found selfconsistently
/1g
2
F
Lnv
CL 2 FvL
Min
/10
eF Fvg
lSpontaneous deformation forms the potential well U for an electron
Self‐trapping of an added electron in 1D ‐ polaron
Energy functional:U0
p p
22
2*
*
2* KUU
mPdxW
22 m
~1/2m*l2 ~ ‐U0 ~KU02l/2
K‐ elasticity contantΨ – electron wave functionm*‐ electron effective mass
Min over U0 : U0~1/Kl
E 1/2 *l2 U +KU 2l/2E=1/2m l2 ‐U0+KU02l/2
11
W
1/ll lKllm
W2
12
12*
Minimum W0 at l0~K/m* , W0< E0 <0
1/ll=∞ ‐ always unstable
7
At D>1 long range Coulomb interactions (ionic crystals) are necessary,to get self‐trapping without the barfrier, hence the name “polaron”
Add a single electron or hole near the gap rims, ±Δ0, allow for deformations of the gap: amplitude Δ=Δ0+δ and phase j dj /dx= j’ if applicable. Check for stability: )( C ec o stab ty
HLLL ˆ,, 00 )(; x
2
2
2
4
udxvL F
2
20
22
dx
vL
F
2
0)ˆ( EEH 0|| 2
)(ˆ)(* xHxdxH 22
ˆ0
2Fv
mpH
0,0)( EEH 0||
1||,0|| 22 dx )()(
20
20000 )/cosh(
1||//
x
Topological soliton !
0)()()()( *1 xxxx Charge of trapped electron is exactly compensated
Many‐body problem, a finite concentration of added particles.Low band filling: Shroedinger electrons on the deformable lattice
min, elU WWUW 2
221 Ug
dxW U 0)()ˆ(,ˆˆ 2 xEHUpH E2 g
)())((*}{ 2 xUxdxUEW )())((*}{ xUxdxUEW xEE
el
2 )()(*)(,0)()( 2 xxxxEU Ex
02)(4 UUE
0)()( xU
02)(4 EEE UUE
0)()(2
EE x
g
)()( BxUAx
Ansatz: any term has the same structure
0)()( xU
0,1)()(
2 EE
EEE
BA
BxUAx 0)()(2
EE x
g
2g
1)()( EEE LBxUdxAxdx
02)(4 UUE
022)(4 UBUUUUEUA EE
02)(4 EEE UUE
06,024 UUUBEA EE
‐ The stationary KdV eq.
ii
Band near half‐filling: Dirac electrons
ii
a
xiaxix
2exp
2exp)(
E
el EWaiai exexx )(*)()(
latel WWWW
kvF
))(),(( xx
kv
HF
F
xik
)sin()()cos()(2)( xkxivxkxux FEFEE
EE gxvxuxW )(),(),( 2
2
E
EEEEEEEE xvxuxuxvxuxvxvxui )()()()()()()()( **'*'*
)(21
21 **
2 xFdxduvvu
ig EEE
EEEE
)()()( * xvxvxF EEE
0)(2 EE vxQEv 2)(xQ 0)( EE vxQEv )(xQ
A simple‐minded way to get a luck of exact solution:
EEE iEvuu EEE iEuvv EEE EEE
0)(2 EE uxPEu 2)(xP
0)(2 EE vxQEv 2)(xQ
And the self-consistency condition
)(21
21 **
2 xFdxduvvu
ig EEE
EEEE
)()()( * xvxvxF EEE
Ansatz: any term has the same structure
)()()(2 EEE BxAxF
dxdx
,0,12 EE B
gA
g
0t6 2 Works if:
0cnst6 2
02' 242 CBA 02' 242 CBA B=0 for a kink
Two infinite sets of integrals of motion:
xi
ofseigenvalue...,))(2(, 242
dxdx x xiofseigenvalue
WWWW EW latel WWWW
E
el EW
Each term is an integral of motion of the mKdV equation
hence the total energy is invariant Expecting physically only the translational
06 32 xxt
0''6''' 2 A
hence the total energy is invariant. Expecting physically only the translational degeneracy, we insist the traveling solution (x,t) = (x-ct)
0''6''' 2 AWe know the answer before obtaining it.
This simple minded anzats applies only to a single periodicityThis simple‐minded anzats applies only to a single periodicityA deeper view: may be hidden, even multiple, degeneracy
)/tanh( 00 x0
0 FvTopological soliton
1)(xu 0E
The single intragap state is at the midgap – “zero fermionic mode”
210)/cosh(2)(
00 xxub
ikx1xik )/tanh(
00 E 2,1,00 v
ikxk e
Lv
2ikx
kk e
Exik
Lu )/tanh(
2000
I t t f tImportant features: kink‐like (topologucal soliton) shape of deformations. The mid‐gap state at E0=0 Its wave function does not contain the component (v0=0) which enters self‐
consistently Eq.; hence any filling number νis allowed (s=0,1/2,0) Delocalized states (uk,vk) density is diluted which sums up to the compensating
h h l l h b ( ) / hcharge ‐1: the total soliton charge becomes (‐1,0,1) – spin/charge separation These states show scattering phases which alone contribute to the soliton energy
Peculiarity: weakly decaying phase shifts δk of u‐states affect the momentum quantizationaffect the momentum quantization
,1)(tank
xk
nknLkn 22 10
k
122}{12 202
dkkdnoshiftskW
12414
12}{12
2020
00
dkdkk
dkkdk
noshiftskW kn
nc
tl tti2142
121
00
00
dk
dkdk
kk kk
compensatelattice1
00
200
dkk
One way to build the family of solutions – diverging kinks
0)( (x )2 +c 2 ‐ 4 ‐2A ‐B=0 0)(
2tanh
2tanh1)( 0
xxx coth0x
One trapped electron –stable intermediate position
Two trapped electrons –diverging pair of kinks
Polaron
0)(
0/2cosh211
0 1)(x
x
ixxp ,12
)()( 2/100
4/5
2/1220
0
20
2
22/)(1)(
xx
00 7.0
bE
2 00 0
Eb
Intragap levelsof bound state
07.02
bE
0b
‐Eb
created by the polaron
Polaron’s energy :
00 9.022
pW00 1.0 pWE
Almost compensated gain and cost of energies
1
0.75
0.5
0.25
50-5
0
-0.25
-0.5
Exact shapes:
-0.75
-1
Exact shapes:equilibrium polaron (upper thick line); pair of solitons with E=0.01) (lower thick line). Thi li t h f ti l fl t ti (l t 5)Thin lines ‐ exact shapes of optimal fluctuations (lecture 5): necessary to create these states by tunneling. Their shapes are much less pronounced in comparison with the final states which facilitates the tunneling.
N
Fatal effect upon kinks: lifting of degeneracy, hence confinement.
Nature present:cis-isomer of (CH)xBuild-in inequivalence of bonds = nonzero starting “mass”.
Confinement – the linear growth of the attraction energy while the particle divergeConfinement the linear growth of the attraction energy while the particle diverge.
Confinement of kinks pairs into 2e charged (bipolaron) or neutral (exciton).
W
Energy difference per unit length is a constant confinement force F.0-0
)()( tNon degenerate ground state
)()( xx ie conste
2 )(xW i 2
)(g
W ilat
22
F
e
F
eF
F vvLog
vLW
21
2 2
FFF
2*
ex K
iTr
Phases of ’s are fixed,
dexi no degeneracy
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.
2tanh
2tanh1)( 0
xxx
cosh0 bE
0
cos
bE0
e
tanhcossincos4
40E
2 kinks cannot diverge far away anymore. They form a loose bound state – a particle with the charge 2e ‐ the bipolaron
New ingredient –fconfinement energy
Towards the lattice dynamics: kinetic energy and effective mass of a soliton
L i
22
* 11x dxi
Tr
Lagrangian:
222 t
phx
x dxgi
Tr
vtx
vtxtx tanh),( 0
22
2),(
kin gtx
dxW
2
21 vMW skin
phg 2
*44 20
30M
0* *220
2220 mv
MphFph
S
20*
Fvm
Complex order parameter Incommensurate CDWsComplex order parameter , Incommensurate CDWs.Noninteger variable charges.
2
22
21
21)()()(),();(),(
gxxxvxuxxW EE
E
EEEE xuxvxvxui )()()()( **
)()()()()( ** xvxvxuxux
)()()()()( **i
)()()()()(1 xvxvxuxux EEEE
)()()()()(2 xvxuxuxvxi EEEE
0)(2 **1
vvuuW 0)(2 21
EEE
EE vvuu
g
1 ** W 0)(12 **22
2
EEE
EE uvvu
igW
Now – two independent self‐consistency eds.
Anzats: family of chordus solitons in the complex plane of Δ
// eq 2q
V
V1
‐
E
V2
cos001 E)tanh()( 002 xkkx
sin00 kVn() ‐ selftrapping branches of total energy for chordus solitons with intragap state fillings n =1 and n=2.
The ‐ 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.
Sequence of chordus solitons develops from the bare θ=0 through the amplitude soliton AS at 2θ=π to the full phase slip 2θ=2π. Intra‐gap split‐off state E evolves from 0 to ‐0 providing the spectral flow across the gap together with the electrons’ conversion.
)tanh( 001 xkik0
0 FvTopological soliton
The single intragap state is at the midgap “zero fermionic mode”
2/)( 0k
xub
The single intragap state is at the midgap – zero fermionic mode
cos001 E sin00 k
)cosh()(
0xkb
ikxk ev 1ikxxikk )/tanh( 0000 k L2ikx
kk e
ELu )(
20000
Important features:1. kink‐like (topological soliton) shape of deformations.2 Intragap may not be mid‐gap state at E0≠02. Intragap, may not be mid gap ,state at E0≠03. Its wave function does contain the component which enters self‐consisteny Eq.;
hence the filling affects the equilibrium shape.4. Delocalized states (uk,vk) density is diluted which sums up to the compensating ( k, k) y p p g
charge 0>q>‐15. These states show scattering phases which alone contribute to the soliton energy
Noninteger variable charge: sources and problems.
)( 0k kxx 00 )()( sin00 k)(cosh2
)(0
20
0 xkx kk
k ELx
0
)(
)()()(
sin00 k
=2 – spin degeneracy of filled band states
)()()( 0000 xxxk
k
p g y0 ‐ filling of the split‐off state.Compensation of 0(x) by local dilatations ~1/L of L delocalized statesL delocalized states.Picture of a classical motion of the soliton Δ(x‐vt): Fraction 1‐2 of the charge emoves and gives the current,
f h h h l d b dFraction 2 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 q pthe fermions becomes a wave function distributed over L– will the local and the delocalized charges recombine?
The total energy via its density
kx 20
20
2 )(
)()()( 00 xExxw kkc
)()( 00 xExwb xkgk
ggxwlat
02
020
2 cosh)(
)()()( 00Lkk
kc
Cos)()( 0000
xwwwxw cblat
The amplitude soliton, θ=π/2 – the energy is zero! – wrong result.The same tric of phase shifts as for the real‐field model
12
recovers the missed term:
The energyW (2/ π)is here but it is totally delocalized;
LLxw 12Sin)( 0
The energy Ws (2/ π)is here, but it is totally delocalized;it does not move with the soliton.Where does it all come from?
Trick around the problem
ddW
ddWdx
ddW 2
2
1
1
sin00
ddW
Integrate with the boundary condition W(0) = ν ∆ corresponding to
ddd 21
Integrate with the boundary condition W(0) = ν0∆0, corresponding to electrons at the momentum k = 0 for an undeformed superstructure,
sincos)( 00W
00
0 sin)(WWS
0
0
SOILITONS WITH NONINTEGER VARIABLE CHARGES – new life in 2000’s:What can constrain the chiral angle and makes the charge noninteger?1 Strict constraint from two interfering dimerizations1. Strict constraint from two interfering dimerizations.
21
22
Joint effect of build‐in ∆i and spontaneous ∆e contributions to gap ∆.Joint effect of build in ∆i and spontaneous ∆e contributions to gap ∆.‐ from the build‐in site dimerization – inequivalence of sites A and B.‐ from spontaneous dimerization of bonds ∆i=∆b ‐ the Peierls effect.
22
21
21
K
iiii
Trx
x
21
2021 )(, cnst
Combined Peiels effect in diatomic linear chain =π/2
Joint effect of extrinsic ∆e and intrinsic ∆i contributions to dimerization gap ∆.
∆e comes from the build‐in site dimerization – non‐equivalence of sites A and B.
∆i comes from spontaneous dimerization of bonds, the Peierls effect.
R`
R
R`
R
∆i comes from spontaneous dimerization of bonds, the Peierls effect.
E
R R
E
22 F
Fie
kF‐kF kF‐kF
33
Threshold effect : ∆i WILL NOT be spontaneously generated
if ∆e already exceeds the wanted optimal Peierls gap.
2. Soft universal constraint from the additional integral of motion –the energy term linear in gradient iA(Δ*∂xΔ‐Δ∂xΔ*)
k l d‐ Like vector potential A×current in a superconductor or stress shifting the wave number in a CDW.
E
dxd
dxdiB
vdxW
F
**
2
FB
1
2sinsin2cos2)( 2
00
BW
Does not affect the equilibrium points for Homogeneous CDW
2
eqS 00 E 0 B
2
**22
2
21 )()(
)(),(),( igc
gx
dxxxxW
E
F xxiv *12
*2
*12
*21
*1 )()(
2
1
2
1
*2
1
)(
)(ˆ
Eix
xx
iH
222 )(
xix
02 **
* ic 02*
ic02 *
222*1
gic
gE
i **
02221 ggE
1
2
dxLiI **
2 8dx
LI
21
1
Two infinite sets of integrals of motion:
xi *
...,)(,|| **2
dxidxxi
WWWW EW
and eigenvalues of
latel WWWW
E
el EW
Each term is an integral of motion of the NLSE equation
02 2 xxti
g q
hence the total energy is invariant. Expecting physically only the translational degeneracy, we insist the traveling solution (x t) = (x-ct)
0'||2'' 2 ic
Its solution is our former chordus soliton.
(x,t) (x ct)
We could have known the answer before obtaining it.