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Charge fluctuations in the 2D Hubbard model
PHYSICA ELSEVIER
Physica C 341-348 (2000) 249-250
www.elsevier.nl/locate/physc
Charge fluctuations in the 2D H u b b a r d model R.Citro, S. Di Matteo and M.Marinaro Dipartimento di Scienze Fisiche "E.R. Caianiello", Universit& di Salerno and Unit& INFM,Italy The charge susceptibility of the 2D Hubbard model is calculated by means of a Bethe-Salpeter equation within a cumulant expansion method. It is shown that in the considered approach an instability in the charge susceptibility appears and a new collective mode is generated. The energy spectrum of this excitation is evaluated numerically and a gap is found at k -- (0, 0) which is a function of the Coulomb repulsion U and the doping. For any fixed U a critical value of the doping xc, at which the spectrum becomes gapless, can be determined.
One of the important issues in describing the normal state of copper-oxide compounds, is the study of the spectrum of charge and spin fluctuations. Strong dynamical spin fluctuations are observed for a wide region of concentration of charge carriers and this can lead to the superconducting pairing of the latter. On the other hand, near optimal doping the charge degrees of freedom play a major role whereas spin degrees of freedom follow the charge dynamics. Thus, it is widely believed that charge degrees of freedom dominate the onset of the charge instability at high-doping whereas in the low-doping regime spin fluctuations contribute to the formation of AF background. Within this scenario, the critical fluctuations can mediate a singular interaction between quasiparticles which is responsible for non-Fermiliquid behavior in the metallic phase. The theoretical approaches for calculating dynamical spin and charge susceptibilities (or dynamical structure form-factors) can be divided into two main groups. In the first group, the Fermi liquid with a weak Coulomb interaction (U < t) is studied. The MMP phenomenological model of an AF Fermi liquid[l], the nested Fermiliquid model[2], the quantum critical point approach[3] and a number of others all belong to this group. In the second group, the strong correlation limit, U >> t, which is usually studied on the basis of the one-band t - J-model, is considered[4]. We propose a strong-coupling approach based on a generalized cumulant expansion for the Hubbard model to calculate the charge susceptibil-
ity )/c(k, w) that mediates an effective interaction among quasi-particles. In the developed approach the Coulomb repulsion of electrons of the Hubbard model is taken into account in a zero order approximation while the hopping matrix elements are considered as a perturbation. New elements of the theory that, characterize the approach are the local many-particle irreducible Green functions, or Kubo cumulants, whose introduction is essential to describe the spin and charge fluctuations of a strong correlated system. Since the zero order Hamiltonian contains the Coulomb interaction, the ordinary Wick theorem proposed in the weak coupling field theory for disentangling chronological averages, is not valid. Instead a generalized Wick theorem has been proposed that permits to evaluate the one-particle Matsubara Green functions by means of a nonstandard diagram technique[5]. The summation of only chain diagrams leads to the ordinary Dyson equation with hopping matrix element as self-energy. This approximation leads to the Hubbard I type of Green's function, denoted by G(a)(k, iw,~), in which the energy spectrum consists of two Hubbard subbands. In order to evaluate the charge susceptibility we first introduce the F vertex function in the particle-hole (p-h) channel which is obtained by summing a class of irreducible diagrams (ladder diagrams) which describe pair hole-particle fluctuation. The Bethe-Salpeter equation for the F vertex is:
re(k) = rcC°)(k°)+rc(°)(k°)n(k)r~(k),
0921-4534/00/$ - see front matter {~ 2000 Elsevier Science B.V. All rights reserved. Pll S0921-4534(00)00464-0
(1)
250
R. Citro et al./Physica C 341-348 (2000) 249-250
where F c(°) is the irreducible vertex function (i.e. a two-particle cumulant)[6] listed in Table.1 and H is the polarization: 2 H(k) = - 2 t t k - -~
4~ 35
2 (1) (k + q)G (U(q). tk+qtqG
3D
(2)
25
Here we have introduced the shorthand notation k ° = iwn and k = (k, iw~), where Wn are the fermion Matsubara frequencies; tk is the Fourier transform of the nearest-neighbor hopping term and t is the hopping amplitude. The difference between the classical RPA and our ladder generalized RPA consists in the character of the bare 'vertex, which in our case is determined by pairing fluctuations and by a renormalized hopping. We find that, after performing the analytical continuation in Eq.(1), the vertex exhibits a complex pole whose real part is determined by the following equation: [F~(°)(w)] -1 - ReH(k,w) = 0
2~ 8
1,5
----." X:Xc
\ \,,
0,5 01)
~,o)
(~,0)
(~)
(0.0)
F
U~
M
r
Figure 1. Charge spectrum at varying x, for U/t = 3. and T / t = 0.01.
(3) Irreducible vertex in the p-h channel]
while the imaginary part, describing the damping effects, is small and goes to zero in the limit of small frequencies. The charge susceptibility is given by xc(k) = xHF(k) + (1 + I~I(k))2rC(k), where X HE is the usual Hartree Fock term and l:i is still a polarization insertion as (2) without hopping terms. In Fig.1 we show the spectrum of the charge excitations along the path F ~ M~ ~ M ~ F in the first Brillouin zone for U / t = 3, T / t = 0.01 at varying doping x. As shown, the spectrum exhibits a gap Ac at small frequencies around the point (0, 0), whose value depends on U and x. Thus, at a fixed value of U it is possible to determine a critical value of the doping xc (that for U/t = 3 is xc - 0.1) at which the gap vanishes. In conclusion, in the considered approach, the Hubbard model exhibits a new collective excitation which appear as an instability in the charge fluctuations. The influence of this collective mode on the single-particle particle spectrum is under study. We believe that the existence of such a collective mode can be responsible for a non-Fermi-liquid behavior in the metallic phase of HTSC.
rc(°)(i~n) = < n~ >5 [ ~ : +1 ~
~--=v]
Table. 1 REFERENCES
1. A.J. Millis, H. Monien, D. Pines, Phys. Rev. B 42, 167 (1990) 2. A. Virosztek, J. Ruvalds, Phys. Rev. B 42, 4064 (1990) 3. S. Caprara et al., Physica C 317, 230 (1999) 4. T.Tanamoto et al., J. Phys. Soc. Jpn. 60, 3072 (1991) 5. V.A. Moskalenko, L.Z. Kon, Cond. Matt. Phys. 1, 23 (1998) 6. F. Mancini, M. Marinaro, Y. Nakano, Physica S 159, 330 (1989)
Physica C 341-348 (2000) 249-250
www.elsevier.nl/locate/physc
Charge fluctuations in the 2D H u b b a r d model R.Citro, S. Di Matteo and M.Marinaro Dipartimento di Scienze Fisiche "E.R. Caianiello", Universit& di Salerno and Unit& INFM,Italy The charge susceptibility of the 2D Hubbard model is calculated by means of a Bethe-Salpeter equation within a cumulant expansion method. It is shown that in the considered approach an instability in the charge susceptibility appears and a new collective mode is generated. The energy spectrum of this excitation is evaluated numerically and a gap is found at k -- (0, 0) which is a function of the Coulomb repulsion U and the doping. For any fixed U a critical value of the doping xc, at which the spectrum becomes gapless, can be determined.
One of the important issues in describing the normal state of copper-oxide compounds, is the study of the spectrum of charge and spin fluctuations. Strong dynamical spin fluctuations are observed for a wide region of concentration of charge carriers and this can lead to the superconducting pairing of the latter. On the other hand, near optimal doping the charge degrees of freedom play a major role whereas spin degrees of freedom follow the charge dynamics. Thus, it is widely believed that charge degrees of freedom dominate the onset of the charge instability at high-doping whereas in the low-doping regime spin fluctuations contribute to the formation of AF background. Within this scenario, the critical fluctuations can mediate a singular interaction between quasiparticles which is responsible for non-Fermiliquid behavior in the metallic phase. The theoretical approaches for calculating dynamical spin and charge susceptibilities (or dynamical structure form-factors) can be divided into two main groups. In the first group, the Fermi liquid with a weak Coulomb interaction (U < t) is studied. The MMP phenomenological model of an AF Fermi liquid[l], the nested Fermiliquid model[2], the quantum critical point approach[3] and a number of others all belong to this group. In the second group, the strong correlation limit, U >> t, which is usually studied on the basis of the one-band t - J-model, is considered[4]. We propose a strong-coupling approach based on a generalized cumulant expansion for the Hubbard model to calculate the charge susceptibil-
ity )/c(k, w) that mediates an effective interaction among quasi-particles. In the developed approach the Coulomb repulsion of electrons of the Hubbard model is taken into account in a zero order approximation while the hopping matrix elements are considered as a perturbation. New elements of the theory that, characterize the approach are the local many-particle irreducible Green functions, or Kubo cumulants, whose introduction is essential to describe the spin and charge fluctuations of a strong correlated system. Since the zero order Hamiltonian contains the Coulomb interaction, the ordinary Wick theorem proposed in the weak coupling field theory for disentangling chronological averages, is not valid. Instead a generalized Wick theorem has been proposed that permits to evaluate the one-particle Matsubara Green functions by means of a nonstandard diagram technique[5]. The summation of only chain diagrams leads to the ordinary Dyson equation with hopping matrix element as self-energy. This approximation leads to the Hubbard I type of Green's function, denoted by G(a)(k, iw,~), in which the energy spectrum consists of two Hubbard subbands. In order to evaluate the charge susceptibility we first introduce the F vertex function in the particle-hole (p-h) channel which is obtained by summing a class of irreducible diagrams (ladder diagrams) which describe pair hole-particle fluctuation. The Bethe-Salpeter equation for the F vertex is:
re(k) = rcC°)(k°)+rc(°)(k°)n(k)r~(k),
0921-4534/00/$ - see front matter {~ 2000 Elsevier Science B.V. All rights reserved. Pll S0921-4534(00)00464-0
(1)
250
R. Citro et al./Physica C 341-348 (2000) 249-250
where F c(°) is the irreducible vertex function (i.e. a two-particle cumulant)[6] listed in Table.1 and H is the polarization: 2 H(k) = - 2 t t k - -~
4~ 35
2 (1) (k + q)G (U(q). tk+qtqG
3D
(2)
25
Here we have introduced the shorthand notation k ° = iwn and k = (k, iw~), where Wn are the fermion Matsubara frequencies; tk is the Fourier transform of the nearest-neighbor hopping term and t is the hopping amplitude. The difference between the classical RPA and our ladder generalized RPA consists in the character of the bare 'vertex, which in our case is determined by pairing fluctuations and by a renormalized hopping. We find that, after performing the analytical continuation in Eq.(1), the vertex exhibits a complex pole whose real part is determined by the following equation: [F~(°)(w)] -1 - ReH(k,w) = 0
2~ 8
1,5
----." X:Xc
\ \,,
0,5 01)
~,o)
(~,0)
(~)
(0.0)
F
U~
M
r
Figure 1. Charge spectrum at varying x, for U/t = 3. and T / t = 0.01.
(3) Irreducible vertex in the p-h channel]
while the imaginary part, describing the damping effects, is small and goes to zero in the limit of small frequencies. The charge susceptibility is given by xc(k) = xHF(k) + (1 + I~I(k))2rC(k), where X HE is the usual Hartree Fock term and l:i is still a polarization insertion as (2) without hopping terms. In Fig.1 we show the spectrum of the charge excitations along the path F ~ M~ ~ M ~ F in the first Brillouin zone for U / t = 3, T / t = 0.01 at varying doping x. As shown, the spectrum exhibits a gap Ac at small frequencies around the point (0, 0), whose value depends on U and x. Thus, at a fixed value of U it is possible to determine a critical value of the doping xc (that for U/t = 3 is xc - 0.1) at which the gap vanishes. In conclusion, in the considered approach, the Hubbard model exhibits a new collective excitation which appear as an instability in the charge fluctuations. The influence of this collective mode on the single-particle particle spectrum is under study. We believe that the existence of such a collective mode can be responsible for a non-Fermi-liquid behavior in the metallic phase of HTSC.
rc(°)(i~n) = < n~ >5 [ ~ : +1 ~
~--=v]
Table. 1 REFERENCES
1. A.J. Millis, H. Monien, D. Pines, Phys. Rev. B 42, 167 (1990) 2. A. Virosztek, J. Ruvalds, Phys. Rev. B 42, 4064 (1990) 3. S. Caprara et al., Physica C 317, 230 (1999) 4. T.Tanamoto et al., J. Phys. Soc. Jpn. 60, 3072 (1991) 5. V.A. Moskalenko, L.Z. Kon, Cond. Matt. Phys. 1, 23 (1998) 6. F. Mancini, M. Marinaro, Y. Nakano, Physica S 159, 330 (1989)