fermi-liquid description of spin-charge separation & application to cuprates t.k. ng (hkust)...
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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)
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
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
SBMFT for t-J model
ijijijijji
jijijjii
iiiiii
ji jijijjii
SS
ccbbbccb
ccbb
SSJchbccbtH
8
3.
)1(
...,, ,
Slave-bosonMFT
ijijiijjiijcccccc ,,
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
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)
Time-dependent slave-boson MFT
ccbb
cccctccccJcc
bccbtccccJcc
ccHcc
ab
babababa
babababa
baba
''
''''
''''
)(
)(
],[
Decoupling according to SBMFT
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
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
)(
)(
.......
......
........
)(
)(
'
'
'
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
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
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)
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.
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
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)
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
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
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
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
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!
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
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
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!
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
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)
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)
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)
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.?
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.?
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
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.
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