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

Recent nt ne new p progress o

• n

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

Strong constraint -- no two electrons on the same site

The he m minim inimal m l mode

• del

l

• - e
• - exte

xtende nded t-J d t-J H Hamiltonia iltonian n

• - hole-hole repulsive Jastrow factor!

tij = n.n.(t), 2nd n.n.(t’), 3rd n.n.(t’’) hopping J = n.n. AF spin-spin interaction

! tij ! ci,"

† !

c j," + h.c.

( )

i, j,"

#

+ J " Si • " S j

<i, j>

#

! = ˆ P

J

1" nRi#nRi$

( )

Ri

%

!0

Use a variational Monte Carlo approach to satisfy the constraint

ˆ P

J ) 1 ( P

d ↓ ↑

− ∏ =

i i i

n n

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

Vojta, Adv. in Phys. ‘09

Magnetic period vs Doping à Half-doped Stripe ! x = εS = 1/aS

Neutr utron sc

• n scatte

ttering ring – pr

– probing spin or

• bing spin orde

dering ring

Yamada, Fujita, Tranquada, …

Yamada da’s plot s plot à à Bond-­‑centered ¡ ¡ ¡ ¡ Site-­‑centered ¡ ¡ ¡ ¡

Example ¡of ¡x=1/8 ¡doping, ¡ ¡ magnetic ¡modulation ¡period ¡is ¡1/x ¡ Charge ¡modulation ¡is ¡1/2x ¡ ¡

For vertical stripes

Soft X-r Soft X-ray sc y scatte ttering ring – pr

– probing c

• bing cha

harge or

• rde

dering ring

Abbamonte, et al., Nature Physics ’05

La La1.8

.875Ba

Ba0.1

.125CuO

uO4

4

Charge correlation in (H,0,L) plane Stripe orbital pattern Modulation period: 4a0

aS = 2aC

Evide Evidenc nce f for bond-c

• r bond-cente

ntered e d ele lectr tronic

• nic c

cluste luster r gla lass sta ss state te with unidir with unidirectiona tional 4 l 4a0 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

• rmation m

tion mecha hanism nism of

• f

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

• ng lattic

ttice c coupling

• upling

à à a c com

• mpe

peting inte ting interaction? tion? ne nesting v sting vector? tor? – – Mosk Moskie iewic wicz, ’9 z, ’99 2.

• 2. Hub

ubba bard or t-J d or t-J m mode

• del f

l favor

• rs pha

s phase se se sepa paration a tion and long-r nd long-rang nge C Coulom

• ulomb

b inte interaction fr 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
• mpe

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

• del

l – – HF --- Za F --- Zaana nan, P n, Poilb

• ilbla

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
• bayashi 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)

vk /uk = Ek !"k #k , #k = #(coskx ! cosky) "k = !2(coskx + cosky)! 4 $ tv coskx cosky ! 2 $$ tv(cos2kx + cos2ky)! µv , Ek = "k

2 + #k 2

RVB = Pd (uk + vkCk,!

+ k

"

C#k,$

+

) % & ' ( ) * 0 = Pd BCS

) 1 ( P

d ↓ ↑

− ∏ =

i i i

n n

The Gutzwiller operator Pd enforces no doubly occupied

sites for hole-doped systems

.

P

Ne RVB = Pd

vk uk Ck,!

+ C"k,# + k

$

% & ' ( ) *

Ne/2

0 = Pd ai, jCi,!

+ C j,# + i, j

$

% & ' ( ) *

Ne/2

) 1 ( P

d ↓ ↑

− ∏ =

i i i

n n

• 2. Make Bogoliubov transformation,
• 1. Solve the BdG equations and obtain the eigenvectors,

How to include the inhomogeneity into the wave function?

• 3. Construct the trial wave function,

Matrix size: 2N x 2N reduced to N x N

Himeda et al., PRL ‘02

• 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

ci! " fi ci# " di

†

Mean-field Hamiltonian AF-RVB stripe [In-Phase-Δ and Anti-Phase-m] Constr

• nstruc

uct stripe t stripe | |Ψ0> f > for hole

• r hole-dope
• doped sta

d state tes s Charge: Spin: Pair:

Bond-centered

aC, aP, as

Charge period Pair period Spin period

Hij! = ! tij" i+# , j

!=n,nn,nnn

"

+ !i +" mi(!1)

Ri

x+Ri y ! µ

( )! ij

H MF = ci!

†

ci"

( )

Hij! #ij # ji $H ji" % & ' ' ( ) * * cj! cj"

†

% & ' ' ( ) * *

!i = !v cos ! QC ! Ri ! ! R0

( )

" # $ %

mi = mv sin ! QS ! Ri ! ! R0

( )

" # $ %

!i,i+x = !v

M cos

! QP ! Ri " ! R0

( )

# $ % & " !v

C

!i,i+y = "!v

M cos

! QP ! Ri " ! R0

( )+ Qy

# $ % & + !v

C

! Q! = Qx,Qy

( ) = 0, 2!

a! ! " # $ % & , ! = C,P,S ! R0 = 0, 1 2 ! " # $ % & , ! Ri = Ri

x,Ri y

( )

3 new parameters,

¡ ¡ ¡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

• del

Ht!J = !t ci"

† c j" + h.c.

( )

<i, j>,"

#

+ J Si i S j

<i, j>

#

Si i S j = gs(i)gs( j) Si i S j ci!

† c j!

= gt! (i)gt! ( j) ci!

† c j!

( ) ( )

( )( )( )

1 ( ) 1 1 1 2 ( ) 2

i i t i i i i i i i s i i i

n n g i n n n n n n n g i n n n

σ σ ↑ ↓ ↑ ↓ ↑ ↓

− = − − − − = −

Me Mean-f n-fie ield de ld decoupling…

• upling…

( ) ( )

* * , , ,

3 ( ) ( ) . . ( ) ( ) 8

t J t t ij s s ij ij ij ij i j i j i j

H t g i g j c c J g i g j m m

σ σ σ σ

χ χ χ

− < > < >

⎛ ⎞ = − + − + Δ Δ − ⎜ ⎟ ⎝ ⎠

∑ ∑

† ij i j ij ij

c c

σ σ σ σ σ

χ χ χ = = ∑

ij i j i j z i i

c c c c m S

↓ ↑ ↑ ↓

Δ = − =

1

i i

n x = −

1 2

i i i

x n m

σ

σ − = +

Wha What a t are the the pe periods of riods of the these se c colle

• llectiv

tive excita itations? tions?

Fluctuating charge, spin, and pair modes:

i i

x x x δ → +

i i

m m m δ → +

ij ij

δ Δ → Δ+ Δ

In the case of and (for h-doped cases), The excitation will have c, s,p coupled modes due to

m =

Δ ≠

Collective mode pattern: è Period: spin = 2 × charge ; pair = charge è Most favorable stripe pattern?

2 2 /2

~

i i q q

x m x m δ δ δ δ

−

→ ~

i ij q q

x x δ δ δ δ

−

Δ Δ → Δ Δ

aC = aP = aS 2

Charge period Pair period Spin period

tw two dif

• different stripe

nt stripe pa patte tterns rns

C.P. Chou, N. Fukushima, and T.K. Lee, PRB 78, 134530 (2008)

è

Pair field: 1. 2 × period 2. ΔC = 0

hole density pairing amplitude spin density

Himeda, et al., PRL ’02

AF-RVB Stripe

(Anti-Phase-m, In-Phase-Δ)

“AntiPhase” Stripe

(Anti-Phase-m, Anti-Phase-Δ)

Optim Optimiz ized e d ene nergy f gy for 8

• r 8a0 A

AR-R

• RVB

VB stripe stripe

C.P. Chou, N. Fukushima, and T.K. Lee, PRB 78, 134530 (2008)

Random Pattern

J/t=0.3

iPEPS a iPEPS also --- sa lso --- same type type Stripe Stripe

Corboz, et al., PRB 84, 041108 (2011)

Red number: hole density Black number: magnetization Bond size: pairing strength (J/t=0.4)

iPEPS = infinite Projected Entanglement Pair State

The stripe they found in the pure t-J model: 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. For us, energies are too close to tell!

x=1/6 x=1/8

For e

• r exte

xtende nded t-J d t-J m mode

• dels

ls, stripe , stripe sta state tes s ha have v very low e ry low ene nergie gies, b , but m ut may be y be not not enough to r nough to repla place the the unif uniform

• rm g

ground

• und

sta state tes! s!

Wha What e t extr xtra m mic icrosc

• scopic
• pic ing

ingredie dient is r nt is requir quired to d to

• bta
• btain stripe

in stripes or ha s or half-dope lf-doped stripe d stripes? s?

Electron-Phonon Interaction

Zhou, et al., a chapter in "Treatise of High Temperature Superconductivity", ed. by J.R. Schrieffer ’07

Strong coupling of charge order to the lattice!

Isotope Isotope Ef Effect t

1 1

effective mass:

Mishchenko, et al.,PRL ‘08

Mass-renormalization due to electron –phonon interaction

New <nh> Bare t*/t

Self-Consistent VMC

Optimize Ht*-J

Half-­‑doped ¡Stripe ¡has ¡been ¡stabilized! ¡

εS ¡= ¡1/aS ¡= ¡δ ¡= ¡εC/2 ¡= ¡1/2aC ¡ C.P. ¡Chou ¡and ¡T.K. ¡Lee, ¡PRB ¡81, ¡060503(R) ¡(2010) ¡ aC ¡× ¡δ ¡= ¡0.5 ¡ aS ¡× ¡δ ¡= ¡1.0 ¡ ¡ (t’,t’’,J)/t ¡= ¡(-­‑0.2,0.1,0.3) ¡ ΔE ¡= ¡E(Stripe)-­‑E(RVB) ¡ Λ ¡= ¡0.25 ¡ Mass ¡renormalization ¡by ¡ electron-­‑phonon ¡coupling ¡

Long r Long rang nge C Coulom

• ulomb – f

b – favor

• rs 8

s 8a stripe stripe!

δ=1/8 VC=t C.P. Chou and T.K. Lee, PRB 81, 060503(R) (2010)

ΔE = E(Stripe)-E(RVB)

! tij ! ci,!

† !

c j,! + h.c.

( )

i, j,!

"

+ J ! Si • ! S j

<i, j>

"

+VC ˆ ni

h ˆ

nj

h

! Ri ! ! Rj

i# j

"

Ele Electr tron-dope

• n-doped

d Cupr uprate tes

Armitage, Fournier, and Greene, RMP ‘10

(t’,t’’)/t=(-0.2,0.1)

Hole

• le-dope
• doped

Ele Electr tron-dope

• n-doped

Particle-hole Asymmetric!

(t’,t’’)/t=(0.1,-0.05)

Same t-J model except t’ and t’’ have sign changed

Inhom Inhomog

• gene

neous c

• us cha

harge distrib distribution ? ution ?

• 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 ’05 …

• 2. 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, ’06 …

• 3. 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 p to now, no c , no conse

• nsensus y

nsus yet! t!

Wha What k t kind of ind of pe period f riod for stripe

• r stripe sta

state tes? s?

Fluctuating charge, spin, and pair modes:

i i

x x x δ → +

i i

m m m δ → +

ij ij

δ Δ → Δ+ Δ

In the case of and (for e-doped cases), the significant contributions to the total energy are

m ≠ Δ ≠

~

i ij

x δ δ Δ Δ

Period: spin = charge = pair

~

i i

m x m δ δ

“In-Pha “In-Phase se-m

• m” stripe

” stripe in e in ele lectr tron-dope

• n-doped

d case ses? s?

Pha Phase se D Dia iagram (| (|t’/ ’/t| t|=0 =0.1 .1) )

Λ = 0.25 (t’’,J)/t=(-t’/2,0.3)

Electron doping Hole doping

No half-doped stripe in electron-doped cases ! “pha “phase se se sepa paration” !? tion” !?

24 × 24

Gla Glassy Stripe ssy Stripe Cha harge P Patte ttern rn

(t’’,J)/t=(-t’/2,0.3) 24 × 24

Electron doping Hole doping

Λ = 0.25

Gla Glassy Stripe ssy Stripe Cha harge P Patte ttern rn

Stronger Particle-hole asymmetry !

Half-dope lf-doped stripe d stripe f for h-dope

• r h-doped b

d but not f ut not for e

• r e-dope
• doped !

d ! Λ = 0.25

“Pha “Phase se Se Sepa paration” (t tion” (t’/ ’/t=0 t=0.1 .1) )

Electron-rich phase v.s. Electron-poor phase (AF+SC state: |<M>| = 0.269)

In-Phase-m Stripe: |<M>| = 0.265

AF-RVB Stripe In-Phase-m Stripe Λ = 0.25 (t’’,J)/t=(-t’/2,0.3) 24 × 24

Sum Summarie ries s

• 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 electron-

phonon interaction is proposed :

ü In the hole-doped regime, A weak electron-phonon interaction is enough to stabilize the half- doped 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!

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

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

what caused this strong phase fluctuation? competing interaction?

Giant Nernst effect and quantum diamagnetism

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: ¡

P

Ne RVB = Pd

vk uk Ck,!

+ C"k,# + k

$

% & ' ( ) *

Ne/2

0 = Pd ai, jCi,!

+ C j,# + i, j

$

% & ' ( ) *

Ne/2

For Ne electrons RVB = Pd (uk + vkCk,!

+ k

"

C#k,$

+

) % & ' ( ) * 0 = Pd BCS

!dRVB = ˆ P

d

uk + vkck"

† c#k$ †

( )

k

%

Yokoyama and Shiba, JPSJ 1988

ci! " fi ci# " di

†

!" # f

(df )

h ! d

(df )

! " 0 (df ) ! " df

(df )

In df space,

vac (df ) ,d + vac (df ) , f + vac (df ) , f +d + vac (df )

+ + df + 0 0 + df ↑ ↓ ⇔ ↓ ↑ ⇔ ! + " # h + h df + 0 # d + d ! + h " h + ! df + d " d + df

! + h " h + ! 0 + d " d + 0

!dRVB

(df ) = ˆ

P

d

ukdk

† + vk fk †

( )

k

"

(df )

In the “df” representation

!dRVB

(df ) = ˆ

P

G

ukdk

† + vk fk †

( )

k

"

(df )

it has a problem:

Examine this wave function !dRVB

g (df ) = ˆ

P

d

gukdk

† + vk fk †

( )

k

"

(df ) (df )

We must introduce a fugacity factor : g

F µg

( ) = E Ne

( )! µgNe

dF = 0 ! µg = "E "Ne

ˆ P

J =

1! 1! ! r

ij " # v ! $

( )

% !

Rj , ! Ri+ ! $

& ' ( ) * + ˆ n !

Ri h ˆ

n !

Rj h

,

• .

/ 1

i< j

2

! rij = ! Ri ! ! Rj with PBC

! ! = 1st, 2nd, 3rd neighbors

!dRVB

g

= ˆ P

J ˆ

P

d !dBCS g

† † R i i R R i

c c ϕ

↑ + ↓

Δ = ∑ ∑

1 2 1 2 1 2

† † † , ,

ˆ ˆ

R R i i R j j R R R i j

c c c c ϕ ϕ∗

↑ + ↓ ↓ + ↑ ≠

Δ Δ = ∑

∑

1 2 1 2 1 2 1 2

† † † † † † † † ,

ˆ ˆ

R R i i R j R j i i R j j R R R i j

c c c c f d f d ϕ ϕ

+ + ↑ + ↓ ↓ + ↑ ≠ ≠

Δ Δ = − →

∑ ∑

( )

( )

1

j i j i

ik R R R R R k

k e N ϕ ϕ

− = −

=

∑

!" = 1 2!0

2

#dRVB

g

ˆ !† ˆ ! $ ˆ !† ˆ !† #dRVB

g

( )

! k

( ) =< c+

k"c+ #k$ >

BCS value is 1 PRB 85, 054510 (2012).

Sum Summarie ries

• An algorithm for the grand-canonical wave function is re-

introduced.

• 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. --- 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.

Thank you for your attention!

!"!Ne !1