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Spin fluctuation effects in the normal state of Hubbard systems
ELSEVIER
Physica B 230-232 (1997) 929-931
Spin fluctuation effects in the normal state of Hubbard systems J. C o s t a - Q u i n t a n a * , F. L 6 p e z - A g u i l a r , A. P 6 r e z - N a v a r r o , L. P u i g - P u i g Departament de Fisica, Grup d'Electromagnetisme, Universitat Autdnoma de Barcelona, Edifici Cn, Bellaterra, E-08193 Barcelona, Spain
Abstract
We obtain and discuss the self-energy and the renormalized electronic structure of the normal state in strongly correlated electron systems within the pseudogap regime. The considered effective potentials are found from spin fluctuations within a Hubbard single band model. We find that the self-energy tends to violate the Luttinger theorem as the bandwidths are narrowed and also as the Hubbard U increases. This is an indication that a new fundamental state should be used to describe the system under these conditions. We have used an antiferromagnetic fundamental state and found that in this case the new self-energy does satisfy the Luttinger theorem. Keywords: High temperature superconductivity; Superconductivity; Spin fluctuations; Luttinger theorem; Hubbard model
High-Tc superconductors are complex systems where electron-electron interactions play a fundamental role. There are several anomalous behaviors in the normal state, and the presence of an antiferromagnetic phase near the superconducting one seems to be an important characteristic. In this paper we study the self-energy effects on the normal state which come up from a spin fluctuation effective interaction. In previous works [1] we have analyzed how some band conditions and vertex effects may be sufficient to yield a superconducting state, with coupling provided by a spin fluctuation effective potential acting in a non-magnetic normal state. A conclusion drawn from these previous works [2] is that, as the bandwidths are narrowed in order to approach the values for which superconductivity is found, the normal-state self-energy unstabilizes so it no longer verifies the Luttinger theorem [3]. This is manifested by the
* Corresponding author.
appearence of several cuts of the imaginary part of the self-energy with the w-axis. This unstable behavior of the normal state appears before the superconducting solutions are found. This may be an indication that the above non-magnetic normal state is no longer valid although the superconducting state is still not reached. From the physical point of view it is clear that this band parameter region may correspond to the antiferromagnetic state. Thus, in the present work we have explored whether a new fundamental state which describes the antiferromagnetic state can result in a self-energy satisfying the Luttinger theorem. The starting Hamiltonian in the non-magnetic normal state is a Hubbard-like one. Treating the problem perturbatively we define, as in previous works [ 1, 2], an effective interaction for singlet and triplet states considering spin fluctuations. In Ref. [2] we took into account both normal and superconducting states with vertex effects and both exchange and direct self-energies at T :~ 0. Let us now focus on a simpler
0921-4526/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved Pll S092 1-4526(96)00688-6
J. Costa-Quintana et al./ Physica B 230-232 (1997) 929-931
930
case which also violates the Luttinger theorem in some cases: we consider just the exchange self-energy on the normal state, at T = 0, with the spin fluctuation interaction between parallel spins, without vertex effects. We assume that the starting non-interacting density of states (DOS) corresponds to the so-called pseudogap regime, so that there is a pseudogap around the Fermi level. We describe it by two Lorentzian peaks, one at each side of EF, with a separation from their maxima to EF given by 2 and ~ and widths by A and ~, respectively. With this DOS and assuming that the k dependence is small, we find the following expression for the parallel spin Berk-Schrieffer-like effective interaction (see Ref. [1]):
0.50 0.25
>~ 0.00 -0.25 -0,50 0.5 ff I . . . . . . . . . . .
~'w 0.0
~s/
¢¢J
-0.5
....
I ....
~ ,' I,,,
, [,,,,
0.50 0.25
~, o.oo : W-o.25
f',/',,
,
-0.50
v~A((D) = U
20~------'~ 0) ~- 0(,02 -""""'~
(1)
'
where O1, 0 2 and f are dependent on the band parameters as shown in Ref. [1]. The resulting exchange self-energy at T = 0 is then given by:
S(~o) =
F
dx JV'(x)
o¢
f 1 + 202 o9 + O 2
{
f 1 201 O) + 01 -- X
+ O(x)V yA( o x)). -
--
X
(2) A self-consistent process is performed since S depends on the DOS JV', which also depends on S. In Figs. l ( a ) - ( c ) the resulting real and imaginary parts of the self-energy are shown. When narrowing the bandwidth of the non-interacting system, we find the breakdown of the Luttinger theorem under similar conditions as in Ref. [2]. For wide bandwidths, the imaginary part has the usual behavior with a welldefined Fermi level, whereas below a given bandwidth, it has several cuts with the og-axis. The same transition from a well-behaved self-energy to the one which violates the Luttinger theorem occurs when increasing U. This means that in this case the interacting system does not have a single well-defined Fermi level. This is an indication that the fundamental state used in this case is not appropriate for these bandwidths and U.
-10
-5
0 ~(~v)
5
10
Fig. 1. Real part (solid line) and imaginary part (dashed line) of the self-energy in the non-magnetic normal state for different bandwidths: (a) ~ = 0.1eV and 2 = 0.3eV; (b) ~ = 0.2eV and 2 = 0 . 4 e V , (c) ~ = 0.3eV and 2 = 0.5eV. In all cases U = 1.5eV.
Let us now see how to introduce in our formalism a new (antiferromagnetic) fundamental state. The procedure is similar to that given, for instance, in Ref. [4]. We introduce the spin-density operator into the Hubbard model Hamiltonian, taking the z direction for the spin, and we make a Bogolyubov transformation for the operators appearing in the Hamiltonian, as follows: ak,~r = blkCk, a Jr- Vkdk,~r,
ak+Q,o = sgn(a)(--vkck, o + ukdk,~),
(3)
with 'j2
I E-~kJ}
(4)
and Ek = (e~ + A2)1/2, with A being the antiferromagnetic gap. Q is a vector which connects opposite faces of the Fermi surface. The new fundamental state IF )
J. Costa-Quintana et al./ Physica B 230-232 (1997) 929 931
(a)I @
is then defined by ck,~lF) :
cl~,~lF) =
0.
0.05
(5)
.
!
i
~'~ 0.00
In mean field approximation, this transformation diagonalizes the Hamiltonian. Next, we go beyond a mean field approximation and use a perturbative method over the new Hamiltonian which, as in Ref. [4], is assumed to have a Hubbard-like structure with a modified U. With this new fundamental state, and following similar procedures as in our previous works and Ref. [4], we find a new effective interaction which is formally the same as Eq. (1) but with a different definition of y, f21, and f22, namely, y'2 =/32 = (7 -- iV) 2,
(6)
931
.... - .......
"
-
. ............
-o.o5 I
. . . .
I
. . . .
(b)
.... (c)
0.2 &~' 0.0
,
I
......
I
. . . .
I
;',
I ....
I ......
I ....
1
: ,
,
~
...............
,,,,- . . . . . . . . . . . . .
rn -0.2
,
0112 = fi2 _ Un(1 - n)fl
....
I ....
I .....
I
....
_1[.
x [l-~x/ff-A2+Az]rl ~
= 2 / 2 - ~ ,t,2
,
~(eV)
(7)
(8)
where r = ~ - i~ and l = 2 - iA. Thus, the self-energy expression (2) is also valid in this case provided the new (the primed) values of f21, f22, and y are used. The resulting real and imaginary parts of this selfenergy are given in Fig. 2. We note that the good behavior of the self-energy in the new antiferromagnetic state is preserved, even for bandwidths as small as those of Fig. 2(c) (which corresponds to the clearest case of Luttinger theorem violation shown in Fig. 1). Therefore, the new chosen fundamental state yields again a self-energy which satisfies the Luttinger theorem with values of the band parameters closer to those for which superconductivity is found.
Fig. 2. The same as Fig. 1 but with the new antiferromagnetic normal state.
This work has been financed by the DGICYT (PB93-1249) and by the CIRIT (1995SGR 00039). A.P.N. acknowledges his grant given by the C.U.R. de la Generalitat de Catalunya.
References [1] L. Puig-Puig and F. Ldpez-Aguilar, Phys. Rev. B 52 (1995) 17 143; Europhys. Lett. 33 (1966) 135. [2] L. Puig-Puig and F. Ldpez-Aguilar, J. Condens. Matter, submitted. [3] J.M. Luttinger, Phys. Rev. 119 (1960) 1153. [4] A.V. Chubukov and D.M. Frenkel, Phys. Rev. B 46 (1992) 11 884; J.R. Schrieffer, X.G. Wen and S.C. Zhang, Phys. Rev. B 39 (1989) 11 663.
Physica B 230-232 (1997) 929-931
Spin fluctuation effects in the normal state of Hubbard systems J. C o s t a - Q u i n t a n a * , F. L 6 p e z - A g u i l a r , A. P 6 r e z - N a v a r r o , L. P u i g - P u i g Departament de Fisica, Grup d'Electromagnetisme, Universitat Autdnoma de Barcelona, Edifici Cn, Bellaterra, E-08193 Barcelona, Spain
Abstract
We obtain and discuss the self-energy and the renormalized electronic structure of the normal state in strongly correlated electron systems within the pseudogap regime. The considered effective potentials are found from spin fluctuations within a Hubbard single band model. We find that the self-energy tends to violate the Luttinger theorem as the bandwidths are narrowed and also as the Hubbard U increases. This is an indication that a new fundamental state should be used to describe the system under these conditions. We have used an antiferromagnetic fundamental state and found that in this case the new self-energy does satisfy the Luttinger theorem. Keywords: High temperature superconductivity; Superconductivity; Spin fluctuations; Luttinger theorem; Hubbard model
High-Tc superconductors are complex systems where electron-electron interactions play a fundamental role. There are several anomalous behaviors in the normal state, and the presence of an antiferromagnetic phase near the superconducting one seems to be an important characteristic. In this paper we study the self-energy effects on the normal state which come up from a spin fluctuation effective interaction. In previous works [1] we have analyzed how some band conditions and vertex effects may be sufficient to yield a superconducting state, with coupling provided by a spin fluctuation effective potential acting in a non-magnetic normal state. A conclusion drawn from these previous works [2] is that, as the bandwidths are narrowed in order to approach the values for which superconductivity is found, the normal-state self-energy unstabilizes so it no longer verifies the Luttinger theorem [3]. This is manifested by the
* Corresponding author.
appearence of several cuts of the imaginary part of the self-energy with the w-axis. This unstable behavior of the normal state appears before the superconducting solutions are found. This may be an indication that the above non-magnetic normal state is no longer valid although the superconducting state is still not reached. From the physical point of view it is clear that this band parameter region may correspond to the antiferromagnetic state. Thus, in the present work we have explored whether a new fundamental state which describes the antiferromagnetic state can result in a self-energy satisfying the Luttinger theorem. The starting Hamiltonian in the non-magnetic normal state is a Hubbard-like one. Treating the problem perturbatively we define, as in previous works [ 1, 2], an effective interaction for singlet and triplet states considering spin fluctuations. In Ref. [2] we took into account both normal and superconducting states with vertex effects and both exchange and direct self-energies at T :~ 0. Let us now focus on a simpler
0921-4526/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved Pll S092 1-4526(96)00688-6
J. Costa-Quintana et al./ Physica B 230-232 (1997) 929-931
930
case which also violates the Luttinger theorem in some cases: we consider just the exchange self-energy on the normal state, at T = 0, with the spin fluctuation interaction between parallel spins, without vertex effects. We assume that the starting non-interacting density of states (DOS) corresponds to the so-called pseudogap regime, so that there is a pseudogap around the Fermi level. We describe it by two Lorentzian peaks, one at each side of EF, with a separation from their maxima to EF given by 2 and ~ and widths by A and ~, respectively. With this DOS and assuming that the k dependence is small, we find the following expression for the parallel spin Berk-Schrieffer-like effective interaction (see Ref. [1]):
0.50 0.25
>~ 0.00 -0.25 -0,50 0.5 ff I . . . . . . . . . . .
~'w 0.0
~s/
¢¢J
-0.5
....
I ....
~ ,' I,,,
, [,,,,
0.50 0.25
~, o.oo : W-o.25
f',/',,
,
-0.50
v~A((D) = U
20~------'~ 0) ~- 0(,02 -""""'~
(1)
'
where O1, 0 2 and f are dependent on the band parameters as shown in Ref. [1]. The resulting exchange self-energy at T = 0 is then given by:
S(~o) =
F
dx JV'(x)
o¢
f 1 + 202 o9 + O 2
{
f 1 201 O) + 01 -- X
+ O(x)V yA( o x)). -
--
X
(2) A self-consistent process is performed since S depends on the DOS JV', which also depends on S. In Figs. l ( a ) - ( c ) the resulting real and imaginary parts of the self-energy are shown. When narrowing the bandwidth of the non-interacting system, we find the breakdown of the Luttinger theorem under similar conditions as in Ref. [2]. For wide bandwidths, the imaginary part has the usual behavior with a welldefined Fermi level, whereas below a given bandwidth, it has several cuts with the og-axis. The same transition from a well-behaved self-energy to the one which violates the Luttinger theorem occurs when increasing U. This means that in this case the interacting system does not have a single well-defined Fermi level. This is an indication that the fundamental state used in this case is not appropriate for these bandwidths and U.
-10
-5
0 ~(~v)
5
10
Fig. 1. Real part (solid line) and imaginary part (dashed line) of the self-energy in the non-magnetic normal state for different bandwidths: (a) ~ = 0.1eV and 2 = 0.3eV; (b) ~ = 0.2eV and 2 = 0 . 4 e V , (c) ~ = 0.3eV and 2 = 0.5eV. In all cases U = 1.5eV.
Let us now see how to introduce in our formalism a new (antiferromagnetic) fundamental state. The procedure is similar to that given, for instance, in Ref. [4]. We introduce the spin-density operator into the Hubbard model Hamiltonian, taking the z direction for the spin, and we make a Bogolyubov transformation for the operators appearing in the Hamiltonian, as follows: ak,~r = blkCk, a Jr- Vkdk,~r,
ak+Q,o = sgn(a)(--vkck, o + ukdk,~),
(3)
with 'j2
I E-~kJ}
(4)
and Ek = (e~ + A2)1/2, with A being the antiferromagnetic gap. Q is a vector which connects opposite faces of the Fermi surface. The new fundamental state IF )
J. Costa-Quintana et al./ Physica B 230-232 (1997) 929 931
(a)I @
is then defined by ck,~lF) :
cl~,~lF) =
0.
0.05
(5)
.
!
i
~'~ 0.00
In mean field approximation, this transformation diagonalizes the Hamiltonian. Next, we go beyond a mean field approximation and use a perturbative method over the new Hamiltonian which, as in Ref. [4], is assumed to have a Hubbard-like structure with a modified U. With this new fundamental state, and following similar procedures as in our previous works and Ref. [4], we find a new effective interaction which is formally the same as Eq. (1) but with a different definition of y, f21, and f22, namely, y'2 =/32 = (7 -- iV) 2,
(6)
931
.... - .......
"
-
. ............
-o.o5 I
. . . .
I
. . . .
(b)
.... (c)
0.2 &~' 0.0
,
I
......
I
. . . .
I
;',
I ....
I ......
I ....
1
: ,
,
~
...............
,,,,- . . . . . . . . . . . . .
rn -0.2
,
0112 = fi2 _ Un(1 - n)fl
....
I ....
I .....
I
....
_1[.
x [l-~x/ff-A2+Az]rl ~
= 2 / 2 - ~ ,t,2
,
~(eV)
(7)
(8)
where r = ~ - i~ and l = 2 - iA. Thus, the self-energy expression (2) is also valid in this case provided the new (the primed) values of f21, f22, and y are used. The resulting real and imaginary parts of this selfenergy are given in Fig. 2. We note that the good behavior of the self-energy in the new antiferromagnetic state is preserved, even for bandwidths as small as those of Fig. 2(c) (which corresponds to the clearest case of Luttinger theorem violation shown in Fig. 1). Therefore, the new chosen fundamental state yields again a self-energy which satisfies the Luttinger theorem with values of the band parameters closer to those for which superconductivity is found.
Fig. 2. The same as Fig. 1 but with the new antiferromagnetic normal state.
This work has been financed by the DGICYT (PB93-1249) and by the CIRIT (1995SGR 00039). A.P.N. acknowledges his grant given by the C.U.R. de la Generalitat de Catalunya.
References [1] L. Puig-Puig and F. Ldpez-Aguilar, Phys. Rev. B 52 (1995) 17 143; Europhys. Lett. 33 (1966) 135. [2] L. Puig-Puig and F. Ldpez-Aguilar, J. Condens. Matter, submitted. [3] J.M. Luttinger, Phys. Rev. 119 (1960) 1153. [4] A.V. Chubukov and D.M. Frenkel, Phys. Rev. B 46 (1992) 11 884; J.R. Schrieffer, X.G. Wen and S.C. Zhang, Phys. Rev. B 39 (1989) 11 663.