Fermi-Liquid description of spin-charge separation & application to cuprates
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Fermi-Liquid description of spin-charge separation & application to cuprates
T.K. Ng (HKUST)
Also: Ching Kit Chan & Wai Tak Tse (HKUST)
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Aim:
To understand the relation between SBMFT (gauge theory) approach to High-Tc cuprates and traditional Fermi-liquid theory applied to superconductors.
General phenomenology of superconductors with spin-charge separation
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Content:
1) U(1) gauge theory & Fermi-liquid superconductor
a)superconducting state b)pseudo-gap state
2)Fermi-liquid phenomenology of superconductors with spin-charge separation
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SBMFT for t-J model
ijijijijji
jijijjii
iiiiii
ji jijijjii
SS
ccbbbccb
ccbb
SSJchbccbtH
8
3.
)1(
...,, ,
Slave-bosonMFT
ijijiijjiijcccccc ,,
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Q1: What is the corresponding low energy (dynamical) theory?
Expect: Fermi liquid (superconductor) when <b>0
Derive low energy effective Hamiltonian in SBMFT and compare with Fermi liquid theory: what are the quasi-particles?
...8
3.
ijijijijji
jijijjii
SS
ccbbbccb
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Time-dependent slave-boson MFT
Idea: We generalized SBMFT to time-dependent regime, studying Heisenberg equation of motion of operators like
k
qkqkqkqkk
qkqkk
b
ccccq
ccq
2/2/2/2/
2/2/
)(
,)(
(TK Ng: PRB2004)
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Time-dependent slave-boson MFT
ccbb
cccctccccJcc
bccbtccccJcc
ccHcc
ab
babababa
babababa
baba
''
''''
''''
)(
)(
],[
Decoupling according to SBMFT
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Time-dependent slave-boson MFT
Similar equation of motion can also be obtained for boson-like function
The equations can then be linearized to obtain a set of coupled linear Transport equations for
kkkbqq
),(),(
kkbq
),(
and constraint field )(q
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Landau Transport equation
The boson function
can be eliminated to obtain coupled linear transport equations for fermion functions
kb
)(),( qqkk
q
k
k
kkq
q
q
k
k
b
q
q
b
q
q
t
)(
)(
.......
......
........
)(
)(
'
'
'
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Landau Transport equation
The constraint field is eliminated by the requirement )(q
0)()( qq bbbq
Notice: The equation is in general a second order differential equation in time after eliminating the boson and constraint field, i.e. non-fermi liquid form.
i.e. no doubly occupancy in Gaussian fluctuations
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Landau Transport equation
The constraint field is eliminated by the requirement )(q
0)()( qq bbbq
Surprising result: After a gauge transformation the resulting equations becomes first order in time-derivative and are of the same form as transport equations derived for Fermi-liquid superconductors (Leggett) with Landau interaction functions given explicitly.
i.e. no doubly occupancy in Gaussian fluctuations
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Landau Transport equation
Gauge transformation that does the trick
)||(, ii i
ii
i
ii ebbecc
Interpretation: the transformed fermion operators represents quasi-particles in Landau Fermi liquid theory!
)/(
...sinsin)1()(~)( '2
'
aJxtxtz
kkzz
tqqVqf
kk
Landau interaction: (F0s) (F1s)
(x= hole concentration)
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Recall: Fermi-Liquid superconductor (Leggett)
Assume: 1) H = HLandau + H BCS
2) TBCS << TLandau
',''
','
)()()()()(*
|.|~
)()(~
kkkkkk
kkkLandau
kkkkBCS
qqqfqqm
kqH
qqgH
Notice: fkk’(q) is non-singular in q0 in Landau FermiLiquid theory.
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Recall: Fermi-Liquid superconductor (Leggett)
Assume: 1) H = HLandau + H BCS
2) TBCS << TLandau
Important result: superfluid density given by
f(T) ~ quasi-particle contribution, f(0)=0, f(TBCS)=1
1+F1s ~ current renormalization ~ quasi-particle charge
)(1
)()1(1)1(
* 1
11
)0(
TfF
TfFF
m
m
s
ssss
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Fermi-Liquid superconductor (Leggett)
)()1(*
~)(1
)0(
BCS
ssBCSs
TOF
m
mT
xzF s ~1 1
superfluid density << gap magnitude (determined by s(0)
More generally,
(x = hole concentration)
In particular
);,()1(*
~);,(01
TqKFm
mTqK
sBCS
(K=current-current response function)
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U(1) slave-boson description of pseudo-gap state
Superconductivity is destroyed by transition from <b>0 to <b>=0 state in slave-boson theory (either U(1) or SU(2))
Question:
Is there a corresponding transition in Fermi liquid language?
T
x
Phase diagram in SBMFT
<b>0 0
<b>=0 0
<b>=0 =0
<b>0 =0
Tb
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U(1) slave-boson description of pseudo-gap state
The equation of motion approach to SBMFT can be generalized to the <b>=0 phase (Chan & Ng (PRB2006))
Frequency and wave-vector dependent Landau interaction.
All Landau parameters remain non-singular in the limit q,0 except F1s.
(b = boson current-current response function)
<b>0 1+F1s(0,0)0
iqbqqF dbs 22
1 ~),(),(1
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U(1) slave-boson description of pseudo-gap state
Recall: Fermi-liquid superconductor
s 0 either when
(i) f(T) 1 (T Tc) (BCS mean-field transition)
(ii) 1+F1s 0 (quasi-particle charge 0 , or spin-charge separation)
Claim: SBMFT corresponds to (ii)(i.e. pseudo-gap state = superconductor with spin-charge separation)
)(1
)()1(1)1(
*1
11
)0(
TfF
TfFF
m
m
s
ssss
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Phenomenology of superconductors with spin-charge separation
22
2
2
1 )()()0(),(1
TzqTqF
ds
What can happen when 1+F1 (q0,0)=0?
Expect at small q and :
1) d>0 (stability requirement)
2) 1+F1sz (T=0 value) when >>
Kramers-Kronig relation 221 )(
)(),(Im
T
TzqF
s
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Phenomenology of superconductor with spin-charge separation
),()),(1(~),(01
qKqFqK
(transverse) current-current response function at T<<BCS (no quasi-particle contribution)
Ko(q,)=current current response for BCS superconductor (without Landau interaction)
1)=0, q small2
0)0,0()()0,( qKTqK
d
Diamagnetic metal!
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Phenomenology of superconductor with spin-charge separation
iT
zK
i
K
)(
)0,0(~
)(
),0()( 0
(transverse) current-current response function at T<<BCS (no quasi-particle contribution)
2)q=0, small (<<BCS)
Or
)()0,0(),0(
0 Ti
izKK
Drude conductivity with density of carrier = (T=0) superfluid densityand lifetime 1/. Notice there is no quasi-particle contributionconsistent with a spin-charge separation picture
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Phenomenology of superconductor with spin-charge separation
)0,0()](Re[1
00
zKd
Notice:
More generally,
if we include only contribution from F1(0,), i.e. the lost of spectral weight in superfluid density is converted to normal conductivitythrough frequency dependence of F1.
~ T=0 superfluid density
)0,0(~)0,0())0,0(),0((~)0,0(),0(
)],0(Im[1)](Re[
1
0011
00
zKKFFKK
Kdd
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Effective GL action
Effective action of the spin-charge separated superconductor state ~ Ginzburg-Landau equation for Fermi Liquid superconductor with only F0s and F1s -1 (Ng & Tse:
Cond-mat/0606479)
))1(
),0(,))1(1(
)1(,(
)(242
)()(
*2
1
0
0
0
1
1
2
*
22
s
s
ss
i
s
F
TF
Fe
Am
T
mF
s << Separation in scale of amplitude & phase fluctuation!
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Effective G-L Action
T<<BCS, (neglect quasi-particles contribution)
,)()(242
)()(
*2
1
1
,))(1(242
)()(
*2
1
222
*
22
2
1
2
1*
22
Am
T
mL
qF
AFm
T
mL
ds
s
amplitude fluctuation small but phase rigidity lost!Strongly phase-disordered superconductor
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Pseudo-gap & KT phases
Recall:
sKT
sss
mkT
F
*
2
)0(
1
4~
;)1(~
Assume 1+F1s~x at T=0 1+F1s 0 at T=Tb
)0(2
1
41~
)(0~
)()(~1
s
bKT
b
bb
amkT
T
TT
TTTTaF
~ fraction of Tb
(Tc~TKT)
(Tb)
x
T
T*
KT phase(weak phase disorder)
SC
Spin-chargeseparation? (strong phase-disorder)
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Application to pseudo-gap state
3 different regimes
1)Superconductor (1+F1s0, T<TKT)
2)Paraconductivity regime (1+F1s0, TKT<T<Tb)
- strong phase fluctuations, KT physics, pseudo-gap
3) Spin-charge separation regime (1+F1s=0)
- Diamagnetic metal, Drude conductivity, pseudo-gap
(Tc~TKT)
(Tb)
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Beyond Fermi liquid phenomenology
Notice more complicated situations can occur with spin-charge separation:
For example: statistics transmutation
1) spinons bosons holons fermions (Slave-fermion mean-field theory, Spiral antiferromagnet, etc.)
2) spinons bosons holons bosons + phase string
non-BCS superconductor, CDW state, etc…. (ZY Weng)
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Electron & quasi-particles
Problem of simple spin-charge separation picture: Appearance of Fermi arc in photo-emission expt. in normal state
Question: What is the nature of these peaks observed in photo-emission expt.?
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Electron & quasi-particles
Recall that the quasi-particles are described by “renormalized” spinon operators which are not electron operators in SBMFT
Quasi-particle fermi surface ~ nodal point of d-wave superconductor and this picture does not change when going to the pseudo-gap state where only change is in the Landau parameter F1s.
Problem: how does fermi arc occurs in photoemission expt.?
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Electron & quasi-particles
A possibility: weak effective spinon-holon attraction which does not destroy the spin-separation transition!
Ng:PRB2005: formation of Fermi arc/pocket in electron Greens function spectral function in normal state (<b>=0) when spin-charge binding is included.Dirac nodal point is recovered in the superconducting state
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Electron & quasi-particles
A possibility: weak effective spinon-holon attraction which does not destroy the spin-separation transition!
Notice: peak in electron spectral function quasi-particle peak in spin-charge separated state in this picture
It reflects “resonances” at higher energy then quasi-particle energy (where spin-charge separation takes place)
Notice: Landau transports equation due with quasi-particles, not electrons.
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Summary
Based on SBMFT, We develop a “Fermi-liquid” description of spin-charge separation
Pseudo-gap state = d-wave superconductor with spin-charge separation in this picture ~ a superconductor with vanishing phase stiffness
Notice: other possibilities exist with spin-charge separation