Recent new progress on variational approach for strongly correlated t-j model

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1 Recent new progress on variational approach for strongly correlated t-j 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

2 The minimal model -- extended t-j Hamiltonian ( )!# t ij!c i,"!c j," + h.c. + J i, j," # <i, j> " S i " S 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 ( )! = ˆP % J 1" n n Ri! # R i $ 0 R i P d = (1 n n i i i ) ˆP J -- hole-hole repulsive Jastrow factor!

3 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

4 Neutron scattering probing spin ordering Magnetic period vs Doping à Half-doped Stripe! Yamada s plot à Bond- centered Site- centered For vertical stripes x = ε S = 1/a S Yamada, Fujita, Tranquada, Example of x=1/8 doping, magnetic modulation period is 1/x Charge modulation is 1/2x Vojta, Adv. in Phys. 09

5 Soft X-ray scattering probing charge ordering La Ba CuO 4 a S = 2a C Charge correlation in (H,0,L) plane Stripe orbital pattern Modulation period: 4a 0 Abbamonte, et al., Nature Physics 05

6 Evidence for bond-centered electronic cluster glass state with unidirectional 4a0 domains! V-shape LDOS! Kohsaka, et al., Science 07

7 Three main views about the formation mechanism of stripes: 1.Usual CDW or SDW needs a nesting wave vector and strong lattice coupling à a competing interaction? nesting vector? Moskiewicz, 99 2.Hubbard or t-j model favors phase separation and long-range Coulomb interaction frustrates the phase separation and leads to stripes à stability? why half-doped? Emery, Kivelson, Lin, Fradkin, et. al., and C. Di Castro, et al. 3.Competition between kinetic and exchange energies in Hubbard or t-j model HF --- Zaanan, Poilblanc and Rice. DMRG --- White and Scalapino (t or 2nd n.n. hopping suppresses stripes) VMC --- Kobayashi and Yamada; Miyazaki et al.; Himeda, Kato and Ogata (t stabilizes stripes) To treat these competing states quantitatively and reliably ---- variational approach!

8 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,! C # k,$ )* & k ) 0 = P d BCS The Gutzwiller operator P d enforces no doubly occupied sites for hole-doped systems (1 n n ) P d % v P Ne RVB = P k d $ C i & ' u k,! k k N = (1 n e /2 n + ( i i + C " k,# ) * P d = i i i ) % 0 = P d ' $ a i, j C i,! & i, j + + C j,# ( * ) N e /2 0 v k / u k = E k! " k # k, # k = #(cos k x! cos k y ) " k =!2(cos k x + cos k y )! 4 t v $ cos k x cos k y! 2 t v $$ (cos2k x + cos2k y )! µ v, E k = " k 2 + # k 2.

9 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..

10 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 c i! " f i c i# " d i

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

12 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?

13 Gutzwiller s approximation in t-j model H t!j =!t # ( c i" c j" + h.c. ) + J 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 i! c j! = g t! (i)g t! ( j) c i! c j! 0 Mean-field decoupling 3 * * H = t g () i g ( j) ( χ + cc..) J g () i g ( j) 8 ( χ χ +Δ Δ ) mm t J tσ tσ ijσ s s ij ij ij ij i j < i, j>, σ < i, j> ( ) g () i = 1 n tσ gs () i = n iσ ni 2n n i i i n i ( 1 n ) ( 1 n )( 1 n )( ni 2n n ) i i i i i χ χ ijσ iσ jσ ij n i = σ c c = χ = 1 x i ijσ 0 Δ = m n ij i j i j i iσ = c c S z i 0 c c 1 xi = + σ m 2 i 0

14 What are the periods of these collective excitations? Fluctuating charge, spin, and pair modes: x x x i + δ i mi m δ mi + Δij Δ+ δδij m = 0 Δ 0 In the case of and (for h-doped cases), The excitation will have c, s,p coupled modes due to 2 2 ~ δxiδmi δxqδm q/2 ~ ΔδxiδΔij ΔδxqδΔ q Collective mode pattern: è Period: spin = 2 charge ; pair = charge è Most favorable stripe pattern? Charge period Pair period a C = a P = a S 2 Spin perio

15 two different stripe patterns AF-RVB Stripe (Anti-Phase-m, In-Phase-Δ) AntiPhase Stripe (Anti-Phase-m, Anti-Phase-Δ) hole density spin density è Pair field: 1. 2 period 2. Δ C = 0 pairing amplitude Himeda, et al., PRL 02 C.P. Chou, N. Fukushima, and T.K. Lee, PRB 78, (2008)

16 Optimized energy for 8a 0 AR-RVB stripe J/t=0.3 Random Pattern C.P. Chou, N. Fukushima, and T.K. Lee, PRB 78, (2008)

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

18 For extended t-j models, stripe states have very low energies, but may be not enough to replace the uniform ground states! What extra microscopic ingredient is required to obtain stripes or half-doped stripes?

19 Electron-Phonon Interaction Strong coupling of charge order to the lattice! Isotope Effect Zhou, et al., a chapter in "Treatise of High Temperature Superconductivity", ed. by J.R. Schrieffer 07

20 Mass-renormalization due to electron phonon interaction Mishchenko, et al.,prl 08 effective mass: Self-Consistent VMC Bare t*/t New <n h > Optimize H t*-j

21 C.P. Chou and T.K. Lee, PRB 81, (R) (2010) Half- doped Stripe has been stabilized! ΔE = E(Stripe)- E(RVB) Mass renormalization by electron- phonon coupling a C δ = 0.5 a S δ = 1.0 Λ = 0.25 (t,t,j)/t = (- 0.2,0.1,0.3) ε S = 1/a S = δ = ε C /2 = 1/2a C

22 Long range Coulomb favors 8a stripe! ( )!" t ij!c i,!!c j,! + h.c. + J i, j,!! S i S! ˆn h h " j +V i ˆn " j C! R i! R! i# j j <i, j> δ=1/8 V C =t E = E(Stripe)-E(RVB) C.P. Chou and T.K. Lee, PRB 81, (R) (2010)

23 Electron-doped Cuprates Particle-hole Asymmetric! Same t-j model except t and t have sign changed Hole-doped (t,t )/t=(-0.2,0.1) Electron-doped Armitage, Fournier, and Greene, RMP 10 (t,t )/t=(0.1,-0.05)

24 Inhomogeneous charge distribution? 1. Many indirect experiments on the spatial inhomogeneity Bakharev, et al., PRL 04 ; Kang, et al., Nat. Mater. 07 ; Dai, et al., PRB 05 ; Zamborszky, et al., 04 ; Sun, et al., PRL 04 ; Kang, et al., PRB A number of theoretical predictions (extended Hubbard model) on the filled or in-phase stripes Aichhorn, et al., EPL 05 ; Sadori, et al., PRL 00 ; Valenzuela, There are also experimental evidences (chemical potential shift) against phase separation Niestemski, et al. Nature 07; Harima, et al., PRB 03 ; Yagi, et al., PRB 06 Up to now, no consensus yet!

25 What kind of period for stripe states? Fluctuating charge, spin, and pair modes: x x x i + δ i mi m δ mi + Δij Δ+ δδij m 0 Δ 0 In the case of and (for e-doped cases), the significant contributions to the total energy are ~ mδxδm i i ~ Δδx δδ i ij Period: spin = charge = pair In-Phase-m stripe in electron-doped cases?

26 Phase Diagram ( t /t =0.1) No half-doped stripe in electron-doped cases! phase separation!? Glassy Stripe Charge Pattern (t,j)/t=(-t /2,0.3) Electron doping Hole doping = 0.25

27 Stronger Particle-hole asymmetry! Half-doped stripe for h-doped but not for e-doped! Glassy Stripe Charge Pattern Electron doping Hole doping = (t,j)/t=(-t /2,0.3) Λ = 0.25

28 Phase Separation (t /t=0.1) Electron-rich phase v.s. Electron-poor phase (AF+SC state: <M> = 0.269) AF-RVB Stripe In-Phase-m Stripe In-Phase-m Stripe: <M> = (t,j)/t=(-t /2,0.3) Λ = 0.25

29 Summaries 1. Collective excitations with spin, charge, and pair coupled naturally leads to appropriate lowest energy stripe patterns (AF-RVB) in the t-j-type model based on variational approach. 2. Long-range Coulomb interaction does not produce half-dopes stripes but it favors 8a at 1/8 3. A model to renormalize the mass due to electronphonon interaction is proposed : ü In the hole-doped regime, A weak electron-phonon interaction is enough to stabilize the halfdoped vertical stripes for doping 1/8. ü In the electron-doped regime, No stable half-doped stripe is found, phase separation seems more favorable. Thank you for your attention!

30 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

31 Giant Nernst effect and quantum diamagnetism what caused this strong phase fluctuation? competing interaction? Emery and Kivelson, Nature, 1995 Strong phase fluctuation for cuprates, specially in the underdoped regime due to small number of holes, is that enough? Wang et al, PRB,2006

32 Un6l now, almost all of the varia6onal calcula6ons for the t- J model, except Yokoyama and Shiba (1988), have been using the canonical ensemble, a wave func6on with a fixed number of par6cles: % + + ( For N e electrons RVB = P d '"(u k + v k C k,! C # k,$ )* & k ) 0 = P d BCS % v P Ne RVB = P k d $ C & ' u k,! k k + + C " k,# ( ) * N e /2 % 0 = P d ' $ a i, j C i,! & i, j + + C j,# ( * ) N e /2 0

33 ( )! drvb = ˆP % d u k + v k c k" c # k$ 0 k c i! " f i c i# " d i In df space, vac (df ),d + vac (df ), f + vac (df ), f + d + vac (df ) h! d (df )! " 0 (df )! " df (df )!" # f (df )

34 In the df representation ( )! drvb (df ) = ˆP " d u k d k + v k f k 0 (df ) k! + h " h +! 0 + d " d df df! + h " h +! df + d " d + df! + " # h + h df + 0 # d + d

35 Examine this wave function it has a problem: ( )! drvb (df ) = ˆP " G u k d k + v k f k 0 (df ) k We must introduce a fugacity factor : g g! drvb ( ) = ˆP " (df ) d gu k d k + v k f k 0 (df ) k

36 F ( µ ) = E( N ) g e! µ g N e df = 0! µ g = "E "N e! r ij =! R i!! R j with PBC, ˆP J = 1! 1! r! " & ij # v! 2 - '( $ i< j. ( ) %! R j, R! i +! $ ) *+ ˆn h R! h ˆn! i R j!! = 1st, 2nd, 3rd neighbors / 0 1

37 ! g drvb = ˆP g ˆPd J! dbcs

38 !" = 1 ( g ˆ! ˆ! $ ˆ! ˆ! g # ) drvb 2! 0 2 # drvb Δ = ΔΔ ˆ ˆ = ˆ ˆ 0 ϕr c c i i+ R R i R R i i R j j R R, R 0 i, j ϕ ϕ c c c c R= Rj R= i N k ΔΔ = 1 ik Rj i ϕ k e ( ) ϕ ϕ c c c c f d + f + d R1 R2 i i+ R i i R 1 j j+ R2 1 j R2 j R, R 0 i j ϕ! k ( R ) ( ) =< c + k" c+ # k$ >

39 PRB 85, (2012). BCS value is 1

40 Thank you for your attention! Summaries An algorithm for the grand-canonical wave function is reintroduced. Projected BCS wave function is modified by a fugacity factor. Mott physics has produced a very large phase fluctuation, (Tesanovic) especially at UD for two main reasons 1. very low charge carrier density 2. the particle number distribution function is no longer a Gaussian.!"!N e! instability of SC -- AFM, CPCDW (Cooper pair CDW), stripe,.. Chemical potential ( and probably tunneling conductance) as a function of hole density agrees with experiments on cuprates fairly well.