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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-T c 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 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 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

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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

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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: (F 0s ) (F 1s ) (x= hole concentration)

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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.

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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

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

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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

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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 =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

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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 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)

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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

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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!

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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

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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

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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 F 0s and F 1s -1 (Ng & Tse: Cond- mat/0606479) s << Separation in scale of amplitude & phase fluctuation!

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

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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)

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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 )

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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 F 1s. 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 ( =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

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