Recent nt ne new p progress o on n variationa nal a l - PowerPoint PPT Presentation

Recent nt ne new p progress o on n variationa nal a l approach f h for s strong ngly ly correla lated t t-J -J mo model Ting-Kuo Lee Institute ¡of ¡Physics, ¡Academia ¡Sinica, ¡Taipei, ¡Taiwan ¡ Chun-­‑Pin ¡Chou ¡ Brookhaven ¡National ¡Lab, ¡Long ¡Island Stat. Phys. , Taipei, July 29, 2013

The he m minim inimal m l mode odel l -- e -- exte xtende nded t-J d t-J H Hamiltonia iltonian n ( ) " " † ! # # ! c j , " + h . c . + J S i • t ij ! c i , " S j i , j , " < i , j > t ij = n.n.(t), 2nd n.n.(t’), 3rd n.n.(t’’) hopping J = n.n. AF spin-spin interaction Strong constraint -- no two electrons on the same site Use a variational Monte Carlo approach to satisfy the constraint ( ) % ! = ˆ 1 " n R i # n R i $ ! 0 P P ( 1 n n ) = ∏ − J d i i ↑ ↓ i R i ˆ P -- hole-hole repulsive Jastrow factor! J

Outline Outline Use variational approach Part I: To study strongly correlated superconducting state with spatial inhomogeneity : a theory for stripes in high temperature superconductors Part II: To study phase fluctuation of the strongly correlated superconducting state

Neutr utron sc on scatte ttering ring – pr – probing spin or obing spin orde dering ring Magnetic period vs Doping à Half-doped Stripe ! Bond-­‑centered ¡ ¡ ¡ ¡ Yamada da’s plot s plot à à Site-­‑centered ¡ ¡ ¡ ¡ Example ¡of ¡x=1/8 ¡doping, ¡ ¡ magnetic ¡modulation ¡period ¡is ¡1/x ¡ For vertical stripes Charge ¡modulation ¡is ¡1/2x ¡ ¡ Yamada, Fujita, x = ε S = 1/a S Tranquada, … Vojta, Adv. in Phys. ‘09

Soft X-r Soft X-ray sc y scatte ttering ring – pr – probing c obing cha harge or orde dering ring La La 1.8 .875 Ba Ba 0.1 .125 CuO uO 4 a S = 2a C 4 Charge correlation in (H,0,L) plane Stripe orbital pattern Modulation period: 4a 0 Abbamonte, et al ., Nature Physics ’05

Evide Evidenc nce f for bond-c or bond-cente ntered e d ele lectr tronic onic c cluste luster r gla lass sta ss state te with unidir with unidirectiona tional 4 l 4a 0 dom domains! ins! V-sha -shape pe LD LDOS ! OS ! Kohsaka, et al ., Science ’07

Thr hree m main vie in views a ws about the bout the f form ormation m tion mecha hanism nism of of stripe stripes: s: 1. 1. Usua sual C l CDW or SD W or SDW ne W needs a ds a ne nesting w sting wave v vector a tor and str nd strong la ong lattic ttice c coupling oupling à a c com ompe peting inte ting interaction? tion? ne nesting v sting vector? tor? à – – Mosk Moskie iewic wicz, ’9 z, ’99 2. 2. Hub ubba bard or t-J d or t-J m mode odel f l favor ors pha s phase se se sepa paration a tion and long-r nd long-rang nge C Coulom oulomb b interaction fr inte tion frustr ustrate tes the s the pha phase se se sepa paration a tion and le nd leads to stripe ds to stripes s à à sta stability? bility? why ha why half-dope lf-doped? d? – Em – Emery ry, K , Kiv ivelson, Lin, F lson, Lin, Fradk dkin, in, et. a t. al ., a ., and C nd C. D . Di C i Castr stro, , et a t al . . 3. 3. Com ompe petition be tition betw tween k n kine inetic tic a and e nd excha hang nge e ene nergie gies in H s in Hub ubba bard or t-J d or t-J m mode odel l – – HF --- Za F --- Zaana nan, P n, Poilb oilbla lanc nc a and R nd Ric ice…. …. – D – DMR MRG --- White G --- White a and Sc nd Scala lapino (t pino (t’ or 2 ’ or 2nd n.n. hopping suppr nd n.n. hopping suppresse sses stripe s stripes) s) – VMC – VMC --- K --- Koba obayashi a shi and Y nd Yamada da; Miy ; Miyaza zaki i et a t al .; H .; Him imeda da, K , Kato a to and Og nd Ogata ta (t (t’ ’ sta stabiliz bilizes stripe s stripes)… s)… To treat these competing states quantitatively and reliably ---- variational approach!

For a translational invariant state, it is straightforward to construct a variational wave function for a projected d-wave state or resonanting valence bond state (by P. W. Anderson) % ( " RVB = P d ( u k + v k C k , ! + + * 0 = P d BCS C # k , $ ) ' & ) k The Gutzwiller operator P d enforces no doubly occupied sites for hole-doped systems P ( 1 n n ) = ∏ − d i i ↑ ↓ i P ( 1 n n ) = ∏ − N e /2 N e /2 % ( % ( v k $ d $ + C " k , # i i + C j , # N e RVB = P d + ↑ 0 = P d ↓ + P C k , ! a i , j C i , ! 0 i ' * ' * & ) & ) u k k i , j v k / u k = E k ! " k , # k = # (cos k x ! cos k y ) # k " k = ! 2(cos k x + cos k y ) ! 4 $ t v cos k x cos k y ! 2 $$ t v (cos2 k x + cos2 k y ) ! µ v , 2 + # k E k = " k 2 .

How to include the inhomogeneity into the wave function? 1. Solve the BdG equations and obtain the eigenvectors, 2. Make Bogoliubov transformation, 3. Construct the trial wave function, Himeda et al ., PRL ‘02 Matrix size: 2N x 2N reduced to N x N 4. To optimize energy, change to a new set of parameters, back to steps 1,2,3..

Himeda et al ., PRL ‘02 There is a problem with this construction as d-wave pairing has a node, hence there is a possibility to hit a singular value. A new way to express the wave function… Yokoyama and Shiba, JPSJ 1988 c i ! " f i c i # " d i †

Constr onstruc uct stripe t stripe | | Ψ 0 > f > for hole or hole-dope -doped sta d state tes s ( ) ! ij Mean-field Hamiltonian y ! µ " x + R i H ij ! = ! t ij " i + # , j + ! i + " m i ( ! 1) R i ! = n , nn , nnn % ( % ( ( ) 3 new parameters, # ij c j ! H ij ! Pair period ' * ' * H MF = † c i ! c i " Charge period ' * ' * # ji $ H ji " † c j " a C , a P , a s & ) & ) Spin period AF-RVB stripe [In-Phase- Δ and Anti-Phase-m] ! R i ! ! ! ( ) Charge: Bond-centered " $ ! i = ! v cos Q C R 0 # % ! ! ! ( ) " $ m i = m v sin R i ! Q S R 0 Spin: # % ! $ ! ( ) = 0, 2 ! ! ! ! ( ) Q ! = Q x , Q y & , ! = C , P , S M cos # # % ! i , i + x = ! v R i " & " ! v C Q P R 0 " a ! % $ Pair: ! ! ! ( ) + Q y ! ! ! $ M cos ( ) R 0 = 0, 1 # % ! i , i + y = "! v R i " & + ! v C Q P R 0 R i = R i x , R i y & , # $ " % 2

¡ ¡ ¡The ¡stripe ¡states ¡will ¡involve ¡charge ¡density ¡ and ¡spin ¡density ¡ ¡modula6on, ¡it ¡will ¡also ¡involve ¡ ¡pair ¡field. ¡What ¡is ¡the ¡rela6on ¡between ¡these ¡ three ¡quan66es? ¡Their ¡rela6ve ¡periods? ¡

Gutzwille Gutzwiller’ r’s a s appr pproxim ximation in t-J tion in t-J m mode odel ( ) † c j " + h . c . # # H t ! J = ! t + J c i " S i i S j < i , j > , " < i , j > S i i S j = g s ( i ) g s ( j ) S i i S j 0 † c j ! † c j ! = g t ! ( i ) g t ! ( j ) c i ! c i ! 0 Me Mean-f n-fie ield de ld decoupling… oupling… 3 ⎛ ⎞ ( ) ( * * ) H t g ( ) i g ( ) j c c . . J g i g ( ) ( ) 8 j m m ∑ ∑ = − χ + − χ χ + Δ Δ − ⎜ ⎟ t J t t ij s s ij ij ij ij i j − σ σ σ ⎝ ⎠ i j , , i j , < > σ < > c c c c † c c Δ = − n 1 n χ = ( ) − ij i j i j ij i j i i ↓ ↑ ↑ ↓ σ σ σ g ( ) i 1 n ( ) 0 0 = − t i σ σ ( )( )( ) 1 n 1 n n 2 n n − − − = ∑ z m S i χ χ i i i i ↑ ↓ ↑ ↓ = ij ij σ i i 0 n σ g i ( ) i = 1 x − s n 2 n n − n 1 x n i m = − = + σ i i i ↑ ↓ i i i i σ 2

Wha What a t are the the pe periods of riods of the these se c colle ollectiv tive excita itations? tions? Fluctuating charge, spin, and pair modes: x x x m m m Δ → Δ + Δ δ → + δ → + δ ij ij i i i i m = 0 In the case of and (for h-doped cases), 0 Δ ≠ The excitation will have c, s,p coupled modes due to 2 2 ~ x m x m ~ x x δ δ → δ δ Δ δ δ Δ → Δ δ δ Δ i i q q /2 − i ij q q − Pair period Collective mode pattern: Charge period a C = a P = a S è Period: spin = 2 × charge ; pair = charge Spin period 2 è Most favorable stripe pattern?

tw two dif o different stripe nt stripe pa patte tterns rns AF-RVB Stripe “AntiPhase” Stripe (Anti-Phase-m, In-Phase- Δ ) (Anti-Phase-m, Anti-Phase- Δ ) hole density spin density è Pair field: 1. 2 × period 2. Δ C = 0 Himeda, et al ., PRL ’02 pairing amplitude C.P. Chou, N. Fukushima, and T.K. Lee, PRB 78, 134530 (2008)

Optim Optimiz ized e d ene nergy f gy for 8 or 8a 0 A AR-R -RVB VB stripe stripe J/t=0.3 Random Pattern C.P. Chou, N. Fukushima, and T.K. Lee, PRB 78, 134530 (2008)

iPEPS a iPEPS also --- sa lso --- same type type Stripe Stripe iPEPS = infinite Projected Entanglement Pair State The stripe they found in the pure t-J model: x=1/8 same pattern as the AF-RVB stripe è Period: spin = 2 × charge ; pair = charge The amplitude of modulation is a bit larger than our results. These are their ground states. x=1/6 For us, energies are too close to Red number: hole density tell! Black number: magnetization Bond size: pairing strength (J/t=0.4) Corboz, et al ., PRB 84, 041108 (2011)