1. Fermi-Liquid description of spin-charge separation & application to cuprates T.K. Ng (HKUST) Also: Ching Kit Chan & Wai Tak Tse (HKUST)

2. Aim:  To understand the relation between SBMFT (gauge theory) approach to High-T c cuprates and traditional Fermi-liquid theory applied to superconductors.   General phenomenology of superconductors with spin-charge separation

3. 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

4. SBMFT for t-J model Slave-boson MFT

5. Q1: What is the corresponding low energy (dynamical) theory? Expect: Fermi liquid (superconductor) when 0  Derive low energy effective Hamiltonian in SBMFT and compare with Fermi liquid theory: what are the quasi-particles?

6. Time-dependent slave-boson MFT  Idea: We generalized SBMFT to time- dependent regime, studying Heisenberg equation of motion of operators like (TK Ng: PRB2004)

7. Time-dependent slave-boson MFT Decoupling according to SBMFT

8. 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

9. Landau Transport equation The boson function can be eliminated to obtain coupled linear transport equations for fermion functions

10. Landau Transport equation The constraint fieldis eliminated by the requirement 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

11. Landau Transport equation The constraint fieldis eliminated by the requirement 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

12. Landau Transport equation Gauge transformation that does the trick Interpretation: the transformed fermion operators represents quasi-particles in Landau Fermi liquid theory! Landau interaction: (F 0s ) (F 1s ) (x= hole concentration)

13. Recall: Fermi-Liquid superconductor (Leggett) Assume: 1) H = H Landau + H BCS 2) T BCS << T Landau Notice: f kk’ (q) is non-singular in q  0 in Landau Fermi Liquid theory.

14. Recall: Fermi-Liquid superconductor (Leggett) Assume: 1) H = H Landau + H BCS 2) T BCS << T Landau Important result: superfluid density given by f(T) ~ quasi-particle contribution, f(0)=0, f(T BCS )=1 1+F 1s ~ current renormalization ~ quasi-particle charge

15. Fermi-Liquid superconductor (Leggett)  superfluid density << gap magnitude (determined by  s (0 ) More generally, (x = hole concentration) In particular (K=current-current response function)

16. U(1) slave-boson description of pseudo-gap state Superconductivity is destroyed by transition from  0 to =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  0   0 =0   0 =0  =0  0  =0 TbTb

17. U(1) slave-boson description of pseudo-gap state The equation of motion approach to SBMFT can be generalized to the =0 phase (Chan & Ng (PRB2006))  Frequency and wave-vector dependent Landau interaction. All Landau parameters remain non-singular in the limit q,  0 except F 1s. (  b = boson current-current response function)   0  1+F 1s (0,0)  0

18. U(1) slave-boson description of pseudo-gap state Recall: Fermi-liquid superconductor  s  0 either when (i)f(T)  1 (T  T c ) (BCS mean-field transition) (ii) 1+F 1s  0 (quasi-particle charge  0, or spin-charge separation) Claim: SBMFT corresponds to (ii) (i.e. pseudo-gap state = superconductor with spin-charge separation)

19. Phenomenology of superconductors with spin-charge separation What can happen when 1+F 1 (q  0,  0)=0? Expect at small q and  : 1)  d >0 (stability requirement) 2) 1+F 1s  z (T=0 value) when  >>  Kramers-Kronig relation 

20. Phenomenology of superconductor with spin-charge separation (transverse) current-current response function at T<<  BCS (no quasi-particle contribution) K o (q,  )=current current response for BCS superconductor (without Landau interaction) 1)  =0, q small  Diamagnetic metal!

21. Phenomenology of superconductor with spin-charge separation (transverse) current-current response function at T<<  BCS (no quasi-particle contribution) 2)q=0,  small (<<  BCS ) Or Drude conductivity with density of carrier = (T=0) superfluid density and lifetime 1/ . Notice there is no quasi-particle contribution consistent with a spin-charge separation picture

22. Phenomenology of superconductor with spin-charge separation Notice: More generally, if we include only contribution from F 1 (0,  ), i.e. the lost of spectral weight in superfluid density is converted to normal conductivity through frequency dependence of F 1. ~ T=0 superfluid density

23. Effective GL action Effective action of the spin-charge separated superconductor state ~ Ginzburg-Landau equation for Fermi Liquid superconductor with only F 0s and F 1s  -1 (Ng & Tse: Cond- mat/0606479)  s <<   Separation in scale of amplitude & phase fluctuation!

24. Effective G-L Action T<<  BCS, (neglect quasi-particles contribution)  amplitude fluctuation small but phase rigidity lost!  Strongly phase-disordered superconductor

25. Pseudo-gap & KT phases Recall: Assume 1+F 1s ~x at T=0 1+F 1s  0 at T=T b ~ fraction of T b (T c ~T KT ) (T b ) x T T* KT phase (weak phase disorder) SC Spin- charge separati on? (strong phase- disorder)

26. Application to pseudo-gap state 3 different regimes 1)Superconductor (1+F 1s  0, T<T KT ) 2)Paraconductivity regime (1+F 1s  0, T KT <T<T b ) - strong phase fluctuations, KT physics, pseudo-gap 3) Spin-charge separation regime (1+F 1s =0) - Diamagnetic metal, Drude conductivity, pseudo-gap (T c ~T KT ) (T b )

27. 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)

28. 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.?

29. 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 F 1s. Problem: how does fermi arc occurs in photoemission expt.?

30. 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 ( =0) when spin-charge binding is included. Dirac nodal point is recovered in the superconducting state

31. 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.

32. 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