Theory without small parameter:
How should we proceed?
• Identify important physical principles and laws
to constrain non-perturbative approximation
schemes
– From weak coupling (kinetic)
– From strong coupling (potential)
• Benchmark against “exact” (numerical) results.
• Check that weak and strong coupling approaches
agree at intermediate coupling.
• Compare with experiment
Two-Particle Self-Consistent Approach (U < W)
- How it works
• General philosophy
– Drop diagrams
– Impose constraints and sum rules (No adjustable
parameter)
• Conservation laws
• Pauli principle ( <ns2> = <ns> )
• Local moment and local density sum-rules
• Get for free:
• Mermin-Wagner theorem
• Kanamori-Brückner screening
• Consistency between one- and two-particle SG = U<ns n-s>
Vilk, AMT J. Phys. I France, 7, 1309 (1997); Allen et al.in Theoretical methods for
strongly correlated electrons also cond-mat/0110130
(Mahan, third edition)
Benchmarks
Proof that it works (comparisons with QMC)
Notes:
-F.L.
parameters
n = 0.20
n = 0.45
U=8
n = 1.0
0.4
U=8
2
n = 0.26 (a)
n = 0.60 U = 4
n = 0.94
n Ssp(q)
n Sch(q)
-Self also
Fermi-liquid
0.0
(c)
n = 0.20
n = 0.45
(b)
0
2
n = 0.19
n = 0.33
(d)
U>0
b=5
8X8
(
n = 0.80
U=4
0.4
(0.0) (
0.0
(0,0) q 
(0,0)
0
QMC + cal.: Vilk et al. P.R. B 49, 13267 (1994)
q
Tmf
TX
16
U=4
n=1
n=1
Monte Carlo
12
4x4
sp
S ()
6x6
8x8
8
10x10
12x12
 ~ expC(T ) / T 
4

0
0.0
0.2
0.4
0.6
0.8
T
Calc.: Vilk et al. P.R. B 49, 13267 (1994)
QMC: S. R. White, et al. Phys. Rev. 40, 506 (1989).
O ( N  )
A.-M. Daré, Y.M. Vilk and A.-M.S.T Phys. Rev. B 53, 14236 (1996)
1.0
Double occupancy :
0.25
U=2
0.20
0.15
0.10
U=4
0.05
0.00
0
1
This work, Nc=36
This work, Nc=64
Moukouri Jarrell, Nc=36
Moukouri Jarrell, Nc=64
2
3
S. Moukouri and M. Jarrell, PRL 87, 167010 (2001)
Kyung et al. PRL 90, 099702 (2003),
4
T
Proofs...
U=+4
b
(,0)
TPSC
Monte Carlo
Many-Body
(0,0)
Flex
(0,)
( 4 , 2)
( 4, 4)
( 2 , 2)
(0,  )
(0,0)
-5.00
0.00
5.00
-5.00
0.00
5.00
/t
Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).
-5.00
0.00
5.00
Benchmark: d-wave superconductivity
dx -y -wave susceptibility for 6x6 lattice
2
b=4
L=6
2
0.6
2
U=4
b=4
b=3
b=2
b=1
0.5
d
3
b
2
1
cd
4
U = 0,
0.7
x -y
U=0
U=4
U=6
U=8
U = 10
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
1.00.10
0.8
0.6
0.4
0.2
0.0
cd
2
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
1.0
Doping
QMC: symbols.
Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)
dx -y -wave susceptibility for 6x6 lattice
2
b
2
1
L=6
2
2
U=4
b=4
b=3
b=2
b=1
0.5
d
3
b=4
0.6
cd
4
U = 0,
0.7
x -y
U=0
U=4
U=6
U=8
U = 10
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
1.00.10
0.8
0.6
0.4
0.2
0.0
cd
2
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
1.0
Doping
QMC: symbols.
Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)
0.7
U = 0,
dx -y -wave susceptibility for 6x6 lattice
2
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
1.00.10
0.8
0.6
0.4
0.2
0.0
b=4
L=6
2
x -y
2
0.6
U=0
U=4
U=6
U=8
U = 10
cd
2
0.5
d
3
b
2
1
b=4
b=3
b=2
b=1
0.4
cd
4
U=4
0.3
0.2
0.1
0.0
0.0
QMC: symbols.
Solid lines analytical.
0.2
0.4
0.6
0.8
Doping
Kyung, Landry, A.-M.S.T. PRB (2003)
1.0
0.08
U=4
U=6
0.07
0.06
Tc
0.05
0.04
DCA
Maier et al. cond-mat/0504529
0.03
0.02
0.01
0.00
0.00
L = 256
0.05
0.10
0.15
0.20
Doping
Kyung, Landry, A.-M.S.T. PRB (2003)
0.25
Pseudogap near optimal doping in edoped cuprates
TPSC

H  ijt i,j c 
c

c
U  i n in i
i j
jc i 
t’’
fixed
t’
U
t
t’=-0.175t, t’’=0.05t
t=350 meV, T=200 K
Weak coupling U<8t
n=1+x – electron filling
15% doping: EDCs along the Fermi surface
TPSC
Exp
Umin< U< Umax
Umax also from CPT
Armitag
e et al.
PRL
87,
147003;
88,
257001
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004
15% doped case: EDCs in two directions
TPSC
Exp
Exp
Hankevych, Kyung,
A.-M.S.T., PRL (2004).
Fermi surface plots
Hubbard repulsion U has to…
U=5.75
U=6.25
be not too large
15%
U=6.25
10%
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004
U=5.75
increase for
smaller doping
B.Kyung et al.,PRB 68, 174502 (2003)
Recent confirmation with KR slave-bosons
Yuan, Yuan, Ting, cond-mat/0503056
Hot spots from AFM quasi-static scattering
Pseudogap temperature and QCP
Prediction
Matsui et al. PRL (2005)
Verified theo.T* at x=0.13
with ARPES
ΔPG≈10kBT* comparable with optical measurements
Hankevych, Kyung, A.-M.S.T., PRL 2004 : Expt: Y. Onose et al., PRL (2001).
Observation
Matsui et al. PRL 94, 047005 (2005)
Reduced, x=0.13
AFM 110 K, SC 20 K
Weak and strong-coupling mechanism
for pseudogap, see poster by
Hankevych, Kyung, Daré, Sénéchal,
Tremblay
Liang Chen
Yury Vilk
Steve Allen
François Lemay
Samuel Moukouri
David Poulin
Hugo Touchette J.-S. Landry
M. Boissonnault
Alexis Gagné-Lebrun
A-M.T. Alexandre Blais
Vasyl Hankevych
K. LeHur
C. Bourbonnais
R. Côté
Sébastien Roy
Sarma Kancharla
Bumsoo Kyung
D. Sénéchal
Maxim Mar’enko
C’est fini…
enfin