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Two-particle self-consistent theory for spin and charge fluctuations in the Hubbard model
Physica C 235-240 (1994) 2235-2236 North-Holland
PHYSICA
T w o - p a r t i c l e self-consistent theory for spin and charge fluctuations in the H u b b a r d m o d e l . Y. M. Vilk, Liang Chen, and A.-M. S. Tremblay a. aD6partment de physique and CRPS Universit~ de Sherbrooke, Sherbrooke, Quebec, Canada J1K 2R1 A theory which is self-consistent at the two-particle level is presented for both spin and charge fluctuations in the H~!bbard model. It is in quantitative agreement with Monte Carlo data at least up to intermediate coupling (U ,,, St). It includes both short-wavelength quantum renormalization effects, and long-wavelength thermal fluctuations which can destroy long-range order in two dimensions. This last effect leads to a small energy scale, as often observed in high temperature superconductors. The theory is conse,'ving, satisfies the Pauli principle and includes three-particle correlations necessary to account for the incipient MoLt transition.
1. I N T R O D U C T I O N
We have recently introduced[I] a simple twoparticle self-consistent approach to the twodimensional Hubbard model that gives, without adjustable parameter, quantitative agreement with Monte Carlo data for spin and charge structure factors and susceptibilities at all fillings up to quite strong coupling. The approach takes into account not only the short-range quantum effects, but also the long-range thermal fluctuations that destroy antiferromagnetic long-range order in two-dimensions at any finite temperature (Mermin-Wagner theorem). This is the key physical ingredient which !cads to a small energy scale, and associated large correlation length, in the magnetic fluctuations. 2. T H E O R Y
Using an equation of motion approach with a decoupling which becomes exact in the limit where particle and hole are created on the same site[l], we have found that the spin and charge structure factors have RPA functional forms with a value for the renormalized spin U~p interaction that is different from the renormalized charge U¢h *This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds pour la formation de chercheurs et l'aide h la recherche from the Government of Que'bec (FCAR), the Canadian Institute of Advanced Research (CIAR), and the Killam Foundation
interactions. The fluctuation-dissipation theorem (FDT) is used to establish self-consistency. The resulting equations are
g o, (,:
=
u,p = aT~ (o)u
((ni.,~,} (ni.,,o,) )
(1)
~TT (0)= o
(z)
9~ (t,r) = -'n f '(2rr) - ~ ~ [&,~ (,~ - 11~'¢('-") T \o(iw.~ ,i) & v (q-') = -~ F_,~., ~-(t,,,i=,)xoC~,..m.¢)
t,~x
the last equativa being the FDT for spin. One also necds the analogous relation between gch and Sch as well as the charge F D T (with U.,p -.* - U c n ) The above approach is conserving, as can be seen from the RPA form of the equations, and the Pauli principle is satisfied (gTT (0) = 0). 3. C O M P A R I S O N S WITH CARLO SIMULATIONS
MONTE
Tile absence of a magnetic phase transition at any finite temperature in 2D follows immediately from tile above approach. In Llle regime in which the temperature is larger than the small energy scale associated with the proximity to tile phase transition, the correlation length grows exponeutially ( c~ e c°'~'tlr reflecting the logarithmic divergence associated ~vith long-raI~ge thermal fluctuations in two dimensions. When the correlation length becomes comparable with systeln size, finite-size effects are expected to become
0921-4534/94/$07.00 © 1994 - ElsevierScience B.V. All rights reserved. SSDI 0921-4534(94)01683-6
YM. Vilk et at./Physica C 235-240 (1994) 2235-2236
2236
•
I
U=4 n=l
' n = 0.20
n =0.4S
0.4
I~
( 1J=8
2
Monte C a r l o [~
I0
g
m~O.O
8x8 lOxlO o,
12x12
!
!
,
•
.-oa6
'~ A
. :0.60 U = 4 ~ n 0.94 /
•
(a)[ A
%=4.7 ,
0.0
I
0.5
n ,,
A
n=
LO
_
~ U=8.
)l~
n-0.19(d)-
4x4 6x6
A
n = 0.20 0.45
.'~
1.0
T
Figure 1. Telnperature dependence of S.,p(rr, ,'r) at half-fillillg n = 1. The solid line is our theory; symbols are Monte Carlo data from Ref.[2]
important in simulations. The largest size effects are thus expected in S,p (Tr, ,'r) at half-filling (n = 1). This is shown in Fig.1. The Monte Carlo data[2] follow our theoretical curve (solid line) until they saturate to a size-dependent value. We checked that finite-size effects for S., v (q") away from the antifcrromagtmtic wave vector ~"= (~r, ,'r) are much slnaller, so that, even for 8 × 8 systems, theory and simulations agree very well for all other values of if' (not shown). Figures 2b and 2c show that, even for relatively strong coupling (U -~ 8), the theory agrees very well with both spin S,p (q-') and charge Sea (q-') structure factors. Ilowever the theory should eventually break down for U ~ c~. Fig. 2a shows that our theory reproduces the i m p o r t a n t qualitative fact that the charge structure factor S~h (~'; n) depends on filling in a nonmonotonous manner. The decrease of Sch (q-') towards half-iiiling is a signature of the incipient Mott transition. The effect can l)e seen because our approach takes ialto accotillt, hotll three-particle correlations and the Pauli prillciple. Writing the Pauli principle as a sum-rule Ei[Sch(q-')+ Ssp(q-')] = 2E¢S0(q-), the paran,e-
°'°(o,o) ( O)q
(o,o)
--,.o
¢/1,~
A
n = 0.33 n=0.80
t "(;'2'1 °(o,o)
(o,o)
Figure 2. Wave vector (q-') dependence of the spin and charge structure factors for different sets of parameters. Solid lines are our theory; symbols are our Monte Carlo data. Monte Carlo data for n = 1 and U = 8 are for 6 x 6 cluster and T = 0.5; all other data are for 8 x 8 cluster and T = 0.2. Error bars are shown only when significant.
t,er 69T1 , which partially takes into account threeparticle correlations, must increase close to halffilling in order to reduce S¢h (q-') and compensate for tim increase in the contribution of the sFia structure factor. Bulut et al. [3] have fitted a number of experiments in IITSC by fine-tuning the value o f / . r ~ , close to a magnetic instability (6U .-, 0). In our approach, a wide range of bare values of U naturally renormalizes to such a situation.
REFERENCES 1. Y.M. \;ilk, Liang Chen, and A.-M.S. Tremblay, Pllys. Roy. B 49, 13 267 (1994) o. . . . S R. ~:l.:,~ i--,, c, _,__. , and R. L. augar,'-" ,, ~,,tc, tJ.d . ,.3Ca/~lJl|iO E. Y. I,olt, Jr., J. E. Gubernatis, and R. T. Scal(qtar. Pl:ys. Rev. B 4 0 , 5 0 6 (1989). 3. N. Bulut, D. llone, D. J. Scalapino, and N. E. Bickers, Phys. Rev. Lett., 64, 2723 (1990).
PHYSICA
T w o - p a r t i c l e self-consistent theory for spin and charge fluctuations in the H u b b a r d m o d e l . Y. M. Vilk, Liang Chen, and A.-M. S. Tremblay a. aD6partment de physique and CRPS Universit~ de Sherbrooke, Sherbrooke, Quebec, Canada J1K 2R1 A theory which is self-consistent at the two-particle level is presented for both spin and charge fluctuations in the H~!bbard model. It is in quantitative agreement with Monte Carlo data at least up to intermediate coupling (U ,,, St). It includes both short-wavelength quantum renormalization effects, and long-wavelength thermal fluctuations which can destroy long-range order in two dimensions. This last effect leads to a small energy scale, as often observed in high temperature superconductors. The theory is conse,'ving, satisfies the Pauli principle and includes three-particle correlations necessary to account for the incipient MoLt transition.
1. I N T R O D U C T I O N
We have recently introduced[I] a simple twoparticle self-consistent approach to the twodimensional Hubbard model that gives, without adjustable parameter, quantitative agreement with Monte Carlo data for spin and charge structure factors and susceptibilities at all fillings up to quite strong coupling. The approach takes into account not only the short-range quantum effects, but also the long-range thermal fluctuations that destroy antiferromagnetic long-range order in two-dimensions at any finite temperature (Mermin-Wagner theorem). This is the key physical ingredient which !cads to a small energy scale, and associated large correlation length, in the magnetic fluctuations. 2. T H E O R Y
Using an equation of motion approach with a decoupling which becomes exact in the limit where particle and hole are created on the same site[l], we have found that the spin and charge structure factors have RPA functional forms with a value for the renormalized spin U~p interaction that is different from the renormalized charge U¢h *This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds pour la formation de chercheurs et l'aide h la recherche from the Government of Que'bec (FCAR), the Canadian Institute of Advanced Research (CIAR), and the Killam Foundation
interactions. The fluctuation-dissipation theorem (FDT) is used to establish self-consistency. The resulting equations are
g o, (,:
=
u,p = aT~ (o)u
((ni.,~,} (ni.,,o,) )
(1)
~TT (0)= o
(z)
9~ (t,r) = -'n f '(2rr) - ~ ~ [&,~ (,~ - 11~'¢('-") T \o(iw.~ ,i) & v (q-') = -~ F_,~., ~-(t,,,i=,)xoC~,..m.¢)
t,~x
the last equativa being the FDT for spin. One also necds the analogous relation between gch and Sch as well as the charge F D T (with U.,p -.* - U c n ) The above approach is conserving, as can be seen from the RPA form of the equations, and the Pauli principle is satisfied (gTT (0) = 0). 3. C O M P A R I S O N S WITH CARLO SIMULATIONS
MONTE
Tile absence of a magnetic phase transition at any finite temperature in 2D follows immediately from tile above approach. In Llle regime in which the temperature is larger than the small energy scale associated with the proximity to tile phase transition, the correlation length grows exponeutially ( c~ e c°'~'tlr reflecting the logarithmic divergence associated ~vith long-raI~ge thermal fluctuations in two dimensions. When the correlation length becomes comparable with systeln size, finite-size effects are expected to become
0921-4534/94/$07.00 © 1994 - ElsevierScience B.V. All rights reserved. SSDI 0921-4534(94)01683-6
YM. Vilk et at./Physica C 235-240 (1994) 2235-2236
2236
•
I
U=4 n=l
' n = 0.20
n =0.4S
0.4
I~
( 1J=8
2
Monte C a r l o [~
I0
g
m~O.O
8x8 lOxlO o,
12x12
!
!
,
•
.-oa6
'~ A
. :0.60 U = 4 ~ n 0.94 /
•
(a)[ A
%=4.7 ,
0.0
I
0.5
n ,,
A
n=
LO
_
~ U=8.
)l~
n-0.19(d)-
4x4 6x6
A
n = 0.20 0.45
.'~
1.0
T
Figure 1. Telnperature dependence of S.,p(rr, ,'r) at half-fillillg n = 1. The solid line is our theory; symbols are Monte Carlo data from Ref.[2]
important in simulations. The largest size effects are thus expected in S,p (Tr, ,'r) at half-filling (n = 1). This is shown in Fig.1. The Monte Carlo data[2] follow our theoretical curve (solid line) until they saturate to a size-dependent value. We checked that finite-size effects for S., v (q") away from the antifcrromagtmtic wave vector ~"= (~r, ,'r) are much slnaller, so that, even for 8 × 8 systems, theory and simulations agree very well for all other values of if' (not shown). Figures 2b and 2c show that, even for relatively strong coupling (U -~ 8), the theory agrees very well with both spin S,p (q-') and charge Sea (q-') structure factors. Ilowever the theory should eventually break down for U ~ c~. Fig. 2a shows that our theory reproduces the i m p o r t a n t qualitative fact that the charge structure factor S~h (~'; n) depends on filling in a nonmonotonous manner. The decrease of Sch (q-') towards half-iiiling is a signature of the incipient Mott transition. The effect can l)e seen because our approach takes ialto accotillt, hotll three-particle correlations and the Pauli prillciple. Writing the Pauli principle as a sum-rule Ei[Sch(q-')+ Ssp(q-')] = 2E¢S0(q-), the paran,e-
°'°(o,o) ( O)q
(o,o)
--,.o
¢/1,~
A
n = 0.33 n=0.80
t "(;'2'1 °(o,o)
(o,o)
Figure 2. Wave vector (q-') dependence of the spin and charge structure factors for different sets of parameters. Solid lines are our theory; symbols are our Monte Carlo data. Monte Carlo data for n = 1 and U = 8 are for 6 x 6 cluster and T = 0.5; all other data are for 8 x 8 cluster and T = 0.2. Error bars are shown only when significant.
t,er 69T1 , which partially takes into account threeparticle correlations, must increase close to halffilling in order to reduce S¢h (q-') and compensate for tim increase in the contribution of the sFia structure factor. Bulut et al. [3] have fitted a number of experiments in IITSC by fine-tuning the value o f / . r ~ , close to a magnetic instability (6U .-, 0). In our approach, a wide range of bare values of U naturally renormalizes to such a situation.
REFERENCES 1. Y.M. \;ilk, Liang Chen, and A.-M.S. Tremblay, Pllys. Roy. B 49, 13 267 (1994) o. . . . S R. ~:l.:,~ i--,, c, _,__. , and R. L. augar,'-" ,, ~,,tc, tJ.d . ,.3Ca/~lJl|iO E. Y. I,olt, Jr., J. E. Gubernatis, and R. T. Scal(qtar. Pl:ys. Rev. B 4 0 , 5 0 6 (1989). 3. N. Bulut, D. llone, D. J. Scalapino, and N. E. Bickers, Phys. Rev. Lett., 64, 2723 (1990).