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Relating pseudogaps and pairing fluctuations in underdoped cuprates
PHYSICA® ELSEVIER
Physica C 341-348 (2000) 139 140 www.elsevier.nl/Iocate/physc
Relating Pseudogaps and Pairing Fluctuations in Underdoped Cuprates Jan R. Engelbrecht and A. Nazarenko Department of Physics, Boston College, Chestnut Hill, MA 02167 We show that pseudogap behavior in the Attractive Hubbard Model is related to 2D-XY critical pairing fluctuations and argue that in the underdoped cuprates which have dramatically enhanced anisotropy, this precursor pairing mechanism also leads to the observed pseudogap behavior.
The short coherence lengths and twodimensional nature of superconductivity in the cuprates, suggests that it is important to understand strong pairing correlations. We have been investigating models of highly correlated superconductors for some time[I-3] and have developed a good understanding of how the BCS/Eliashberg scenario is modified. We discuss how pre-cursor pairing in the normal state leads to a suppression of density of states, referred to as pseudogap behavior. This pseudogap phenomena is now observed in many experiments on underdoped cuprates and our strong-pairing scenario is a leading explanation of this behavior. We study a simple model of a correlated superconductor - the 2D, attractive Hubbard Model:
H = -t ~
(c~cy~ +he) -IUl~_,nivni,.
(1)
i
For moderate U, away from half-filling, this describes degenerate fermions with pre-cursor (swave) pairing effects in the normal state leading to properties very similar to the underdoped cuprates. (This correspondence is improved when we consider &wave attraction [2]). We use selfconsistent diagrammatics, numerically solving for the dressed propagator and the four-point pairing vertex through the coupled equations
E(k)=U2EG(q-k)x(q);
H(q) x(q)= 1 _ [Ulii(q) (2)
where G(k: ikn) = [ikn - e(k) + # - E(k, ikn)] -1 and II(q) = y~ G ( q - k ) G ( k ) the particle-particle bubble defined in terms of this dressed propagator. Here the sums (with proper factors) represent a trace over intermediate momenta, frequen0921-4534/00/$ - see front matter © 2000 Elsevier Science B.M Pl! S0921-4534(00)00416-0
cies and, where appropriate, spins. These equations define the self-consistent T-matrix approximarion (STA) which is related to the fluctuation exchange approximation [4] (FEA) of Bickers and Scalapino and later Serene's group. The superconducting instability is signaled by a diverging uniform static pair susceptibility
X-l(0) ~ 1 -IV[ H(q=0, i ~ , = 0 ) = 0.
(3)
In Mean-Field or l-loop approximations where the dressed Green's functions are replaced by non-interacting Go's, (3) has a solution at some TcMF > 0 in any dimension and for finite systems. Exact calculations for a finite system, however, cannot give a diverging X(0). Moreover, the infinite 2D system has a transition to algebraic order rather than long-range order (LRO). An important improvement[5,3] of the STA is that (3) has no solution for finite systems. Upon cooling, X becomes sharply peaked about q = 0 but never diverges since the strong pair fluctuations feed back into the self-energy through (2). The fine balance captured by the self-consistent coupled equations prevents the incorrect prediction of a phase transition in a finite system. An even more remarkable result[3] of the STA is that, near Tc, the peak in X increases with increasing system size and finite-size scaling shows that X indeed diverges at some Tc in the thermodynamic limit. Moreover, X does not diverge as a power law but according to
x(q = 0) ~ e x p ( A / v @ - Tc).
(4)
The STA captures 2D-XY critical behavior of the Kosterlitz-Thouless (KT) transition expected All rights reserved.
140
JR. Engelbrecht,A. Nazarenko/Physica C 341-348 (2000) 139-140
for this model! The correlation length is extracted from the equal-time pair correlation function through x(r, r=0) .~ e x p ( - r / ~ ( T ) ) / r C. For large systems, ~(T) scales with the exponential K T temperature-dependence of (4) rather than a power-law expected for transitions to LRO, and further satisfies the scaling relation X "" ~2-n.
o2o
£ //~,~
/A, //
o ls
~-
[
Z" 0.10
o.o 1°° i
102
0 . 0 0
o @ csc~ t ,0-, o% o I L ~, X
% 10~2
I0-310-I
0 16x16 ~ 32x32 o 64x64 s 128xl 28 100
0.10 T
~ %
% iO'
'
102
Figure 1. Pair susceptibility data, for different system sizes, U - - - 4 t and ¼-filling collapses onto a scaling function through finite-size scaling. Fig. 1 shows KT scaling in X for sufficiently large system sizes and the remarkable success of the finite-size scaling hypothesis in collapsing data from nearly an order of magnitude in linear dimension onto a single scaling function. A KT fit through our data estimates the transition temperature T K T '~ 0.049t in very good agreement with the value estimated from Monte Carlo[6]. We find critical scaling occurs over a wide temperature range reminiscent[7] of quantum 2D XY Monte Carlo results. The degree to which ~(T) and X display XY scaling, and the success of finite-size scaling is remarkable and quite surprising. It is very unusual for an approximate Green's function technique based on fermions to correctly describe anomalous dimensions near criticality. (One traditionally may use an effective (bosonic) order-parameter theory and an ~-expansion to approximate critical exponents.) Using the Pad~ method we analytically continue our Matsubara Green's functions to retarded frequencies to obtain the density of states,
~
T=03
T=0.30 ~ T=o2 j T=o17
t
-'~--.5=o.14
1
~ i ~
/ - -
-3
'
..... '
0
!
3
Figure 2. Density of states for U = - 4 at ¼-filling. N ( ~ ) = - 2 I m }-~k G ( k , w + iO+). Fig. 2 shows how, in the normal state, the density of states near the Fermi energy is strongly suppressed upon cooling below T* ,~ 0.35. The wide range of temperatures above Tc over which this pseudogap behavior develops coincides with the 2D XY critical scaling regime of the pair susceptibility. It is natural to conclude that 2D XY critical pairing fluctuations lead to this loss of spectral weight in our model. Given the strong pairing in the cuprates together with the dramatic reduction in Pc/Pab upon underdoping, we believe that the observed normal state pseudogap behavior is related to strong, critical 2D-XY pairing fluctuations over a large temperature range.
REFERENCES 1. CAR S~ de Melo et al, Phys. Rev. Left. 71, 3202 (1993) and JR Engelbrecht et al, in Strongly Correlated Electronic Materials, ed. K Bedell et al (Addison Wesley, 1994). 2. JR Engelbrecht et al, Phys. Rev. B57, 16223 (1998). 3. JR Engelbrecht et al, cond-mat/9806223. 4. NE Bickers et al, Phys. Rev. Lett. 62, 961 (1989) and JW Serene et al, Phys. Rev. B44, 3391 (1991). 5. JJ Deisz et al, Phys. Rev. Lett. 80,373 (1998). 6. A Moreo et al, Phys. Rev. Lett. 66, 946 (1991). 7. HQ Ding et al, Phys. Rev. B42, 6827 (1990).
Physica C 341-348 (2000) 139 140 www.elsevier.nl/Iocate/physc
Relating Pseudogaps and Pairing Fluctuations in Underdoped Cuprates Jan R. Engelbrecht and A. Nazarenko Department of Physics, Boston College, Chestnut Hill, MA 02167 We show that pseudogap behavior in the Attractive Hubbard Model is related to 2D-XY critical pairing fluctuations and argue that in the underdoped cuprates which have dramatically enhanced anisotropy, this precursor pairing mechanism also leads to the observed pseudogap behavior.
The short coherence lengths and twodimensional nature of superconductivity in the cuprates, suggests that it is important to understand strong pairing correlations. We have been investigating models of highly correlated superconductors for some time[I-3] and have developed a good understanding of how the BCS/Eliashberg scenario is modified. We discuss how pre-cursor pairing in the normal state leads to a suppression of density of states, referred to as pseudogap behavior. This pseudogap phenomena is now observed in many experiments on underdoped cuprates and our strong-pairing scenario is a leading explanation of this behavior. We study a simple model of a correlated superconductor - the 2D, attractive Hubbard Model:
H = -t ~
(c~cy~ +he) -IUl~_,nivni,.
(1)
i
For moderate U, away from half-filling, this describes degenerate fermions with pre-cursor (swave) pairing effects in the normal state leading to properties very similar to the underdoped cuprates. (This correspondence is improved when we consider &wave attraction [2]). We use selfconsistent diagrammatics, numerically solving for the dressed propagator and the four-point pairing vertex through the coupled equations
E(k)=U2EG(q-k)x(q);
H(q) x(q)= 1 _ [Ulii(q) (2)
where G(k: ikn) = [ikn - e(k) + # - E(k, ikn)] -1 and II(q) = y~ G ( q - k ) G ( k ) the particle-particle bubble defined in terms of this dressed propagator. Here the sums (with proper factors) represent a trace over intermediate momenta, frequen0921-4534/00/$ - see front matter © 2000 Elsevier Science B.M Pl! S0921-4534(00)00416-0
cies and, where appropriate, spins. These equations define the self-consistent T-matrix approximarion (STA) which is related to the fluctuation exchange approximation [4] (FEA) of Bickers and Scalapino and later Serene's group. The superconducting instability is signaled by a diverging uniform static pair susceptibility
X-l(0) ~ 1 -IV[ H(q=0, i ~ , = 0 ) = 0.
(3)
In Mean-Field or l-loop approximations where the dressed Green's functions are replaced by non-interacting Go's, (3) has a solution at some TcMF > 0 in any dimension and for finite systems. Exact calculations for a finite system, however, cannot give a diverging X(0). Moreover, the infinite 2D system has a transition to algebraic order rather than long-range order (LRO). An important improvement[5,3] of the STA is that (3) has no solution for finite systems. Upon cooling, X becomes sharply peaked about q = 0 but never diverges since the strong pair fluctuations feed back into the self-energy through (2). The fine balance captured by the self-consistent coupled equations prevents the incorrect prediction of a phase transition in a finite system. An even more remarkable result[3] of the STA is that, near Tc, the peak in X increases with increasing system size and finite-size scaling shows that X indeed diverges at some Tc in the thermodynamic limit. Moreover, X does not diverge as a power law but according to
x(q = 0) ~ e x p ( A / v @ - Tc).
(4)
The STA captures 2D-XY critical behavior of the Kosterlitz-Thouless (KT) transition expected All rights reserved.
140
JR. Engelbrecht,A. Nazarenko/Physica C 341-348 (2000) 139-140
for this model! The correlation length is extracted from the equal-time pair correlation function through x(r, r=0) .~ e x p ( - r / ~ ( T ) ) / r C. For large systems, ~(T) scales with the exponential K T temperature-dependence of (4) rather than a power-law expected for transitions to LRO, and further satisfies the scaling relation X "" ~2-n.
o2o
£ //~,~
/A, //
o ls
~-
[
Z" 0.10
o.o 1°° i
102
0 . 0 0
o @ csc~ t ,0-, o% o I L ~, X
% 10~2
I0-310-I
0 16x16 ~ 32x32 o 64x64 s 128xl 28 100
0.10 T
~ %
% iO'
'
102
Figure 1. Pair susceptibility data, for different system sizes, U - - - 4 t and ¼-filling collapses onto a scaling function through finite-size scaling. Fig. 1 shows KT scaling in X for sufficiently large system sizes and the remarkable success of the finite-size scaling hypothesis in collapsing data from nearly an order of magnitude in linear dimension onto a single scaling function. A KT fit through our data estimates the transition temperature T K T '~ 0.049t in very good agreement with the value estimated from Monte Carlo[6]. We find critical scaling occurs over a wide temperature range reminiscent[7] of quantum 2D XY Monte Carlo results. The degree to which ~(T) and X display XY scaling, and the success of finite-size scaling is remarkable and quite surprising. It is very unusual for an approximate Green's function technique based on fermions to correctly describe anomalous dimensions near criticality. (One traditionally may use an effective (bosonic) order-parameter theory and an ~-expansion to approximate critical exponents.) Using the Pad~ method we analytically continue our Matsubara Green's functions to retarded frequencies to obtain the density of states,
~
T=03
T=0.30 ~ T=o2 j T=o17
t
-'~--.5=o.14
1
~ i ~
/ - -
-3
'
..... '
0
!
3
Figure 2. Density of states for U = - 4 at ¼-filling. N ( ~ ) = - 2 I m }-~k G ( k , w + iO+). Fig. 2 shows how, in the normal state, the density of states near the Fermi energy is strongly suppressed upon cooling below T* ,~ 0.35. The wide range of temperatures above Tc over which this pseudogap behavior develops coincides with the 2D XY critical scaling regime of the pair susceptibility. It is natural to conclude that 2D XY critical pairing fluctuations lead to this loss of spectral weight in our model. Given the strong pairing in the cuprates together with the dramatic reduction in Pc/Pab upon underdoping, we believe that the observed normal state pseudogap behavior is related to strong, critical 2D-XY pairing fluctuations over a large temperature range.
REFERENCES 1. CAR S~ de Melo et al, Phys. Rev. Left. 71, 3202 (1993) and JR Engelbrecht et al, in Strongly Correlated Electronic Materials, ed. K Bedell et al (Addison Wesley, 1994). 2. JR Engelbrecht et al, Phys. Rev. B57, 16223 (1998). 3. JR Engelbrecht et al, cond-mat/9806223. 4. NE Bickers et al, Phys. Rev. Lett. 62, 961 (1989) and JW Serene et al, Phys. Rev. B44, 3391 (1991). 5. JJ Deisz et al, Phys. Rev. Lett. 80,373 (1998). 6. A Moreo et al, Phys. Rev. Lett. 66, 946 (1991). 7. HQ Ding et al, Phys. Rev. B42, 6827 (1990).