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