Fermi-Liquid description of spin-charge separation

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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
Slave-boson MFT
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Q1 What is the corresponding low energy
(dynamical) theory?

• Expect Fermi liquid (superconductor) when ltbgt?0
• ? Derive low energy effective Hamiltonian in
  SBMFT and compare with Fermi liquid theory what
  are the quasi-particles?

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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

(TK Ng PRB2004)
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Time-dependent slave-boson MFT
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
and constraint field
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Landau Transport equation
The boson function
can be eliminated to obtain coupled linear
transport equations for fermion functions
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Landau Transport equation
The constraint field
is eliminated by the requirement
i.e. no doubly occupancy in Gaussian fluctuations
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.
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Landau Transport equation
The constraint field
is eliminated by the requirement
i.e. no doubly occupancy in Gaussian fluctuations
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.
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Landau Transport equation
Gauge transformation that does the trick
Interpretation the transformed fermion operators
represents quasi-particles in Landau Fermi liquid
theory!
Landau interaction
(F1s)
(F0s)
(x hole concentration)
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Recall Fermi-Liquid superconductor (Leggett)
Assume 1) H HLandau H BCS
2) TBCS ltlt TLandau
Notice fkk(q) is non-singular in q?0 in Landau
Fermi Liquid theory.
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Recall Fermi-Liquid superconductor (Leggett)
Assume 1) H HLandau H BCS
2) TBCS ltlt TLandau Important result
superfluid density given by
f(T) quasi-particle contribution, f(0)0,
f(TBCS)1 1F1s current renormalization
quasi-particle charge
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Fermi-Liquid superconductor (Leggett)
In particular
(x hole concentration)

• superfluid density ltlt gap magnitude (determined
  by ?s(0)
• More generally,

(Kcurrent-current response function)
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U(1) slave-boson description of pseudo-gap state
Superconductivity is destroyed by transition from
ltbgt?0 to ltbgt0 state in slave-boson theory
(either U(1) or SU(2)) Question Is there a
corresponding transition in Fermi liquid
language?
Phase diagram in SBMFT
T
Tb
ltbgt0 ? 0
ltbgt0 ? ?0
ltbgt?0 ? 0
ltbgt?0 ? ?0
x
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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 ltbgt0 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) ?ltbgt?0 ? 1F1s(0,0)?0
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U(1) slave-boson description of pseudo-gap state
Recall Fermi-liquid superconductor

• ?s ?0 either when
• f(T) ?1 (T ?Tc) (BCS mean-field transition)
• (ii) 1F1s ?0 (quasi-particle charge ?0 , or
  spin-charge separation)
• Claim SBMFT corresponds to (ii)
• (i.e. pseudo-gap state superconductor with
  spin-charge separation)

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Phenomenology of superconductors with spin-charge
separation
What can happen when 1F1 (q?0,??0)0? Expect at
small q and ?

• 1) ?dgt0 (stability requirement)
• 2) 1F1s?z (T0 value) when ?gtgt?
• Kramers-Kronig relation ?

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Phenomenology of superconductor with spin-charge
separation
(transverse) current-current response function at
Tltlt?BCS (no quasi-particle contribution)

• Ko(q,?)current current response for BCS
  superconductor (without Landau interaction)
• 1)?0, q small

? Diamagnetic metal!
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Phenomenology of superconductor with spin-charge
separation

• (transverse) current-current response function at
  
• Tltlt?BCS (no quasi-particle contribution)

• 2)q0, ? small (ltlt?BCS)

Or
Drude conductivity with density of carrier
(T0) superfluid density and lifetime 1/?. Notice
there is no quasi-particle contribution consistent
with a spin-charge separation picture
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Phenomenology of superconductor with spin-charge
separation
Notice
T0 superfluid density
More generally,
if we include only contribution from F1(0,?),
i.e. the lost of spectral weight in superfluid
density is converted to normal conductivity throug
h frequency dependence of F1.
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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)
?s ltlt ? ?Separation in scale of amplitude phase
fluctuation!
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Effective G-L Action
Tltlt?BCS, (neglect quasi-particles contribution)

• amplitude fluctuation small but phase rigidity
  lost!
• Strongly phase-disordered superconductor

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Pseudo-gap KT phases
T
Recall
Spin-charge separation? (strong phase-disorder)
T
(Tb)
Assume 1F1sx at T0 1F1s? 0 at
TTb
KT phase (weak phase disorder)
(TcTKT)
fraction of Tb
SC
x
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Application to pseudo-gap state
3 different regimes
(Tb)
1)Superconductor (1F1s?0, TltTKT) 2)Paraconducti
vity regime (1F1s?0, TKTltTltTb) - strong
phase fluctuations, KT physics, pseudo-gap 3)
Spin-charge separation regime (1F1s0) -
Diamagnetic metal, Drude conductivity, pseudo-gap
(TcTKT)
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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!
NgPRB2005 formation of Fermi arc/pocket in
electron Greens function spectral function in
normal state (ltbgt0) 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