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Marginal Fermi-liquid model and Ward-Takahashi relations
PhysicsLettersA 161 (1992) 553—555 North-Holland
PHYSICS LETTERS A
Marginal Fermi-liquid model and Ward—Takahashi relations Tadashi Toyoda a,b and Yasushi Takahashi b
a
TheoreticalPhysics Institute, University ofAlberta, Edmonton, Canada T6G 2J1 Nagoya Shoka Daigaku, Sagamine, Nisshin, Aichi-gun, Aichi 470-01, Japan’
Received 30 September 1991; accepted for publication 18 November 1991 Communicated by A.A. Maradudin
It is shown that the vanishing of the quasi-particle weight at the Fermi surface is equivalent to the divergence of the vertex function using a finite-temperature Ward—Takahashi relation.
Recently Varma et al. [1] introduced a model electron polarizability and examined the normal state properties of the Cu—O high temperature superconductors [2], which seem to be quite different from the known Fermi-liquid properties. Using the model polarizability Varma et al. and Pelzer [3] calculated the proper self-energy of the electron Green function and found that the quasi-particle weight Z(k) vanishes at the Fermi surface. This means that the electron number distribution in the momentum space has no discontinuous gap at the Fermi momentum, i.e., the Fermi surface does not seem to exist any more. Hence, Varma et al. called their many-electron system the “marginal Fermi liquids”. We shall show in this note that the divergence of the vertex function is equivalent to the vanishing of the quasi-particle weight by the help of the finitetemperature Ward—Takahashi relation between the electron self-energy and the vertex function. The basic assumption in the calculation [1,3] of the electron self-energy is that the polarizability can be treated as a boson propagator and the coupling between the polarizability-boson (P-boson) and the electron is sufficiently weak to allow a perturbative calculation. In other words, they assumed that the electron system can be described in terms of an effective Hamiltonian that consists of the free electron term, the electron—boson coupling and a free boson term. Calculating the lowest order correction due to Permanent address.
the exchange of the P-boson to the electron Green function, they obtained the proper self-energy. Although the details of the effective Hamiltonian are not given, it is possible to assume some fundamental symmetries of the Hamiltonian rather safely. Then one can derive the corresponding Ward— Takahashi relations.In this paper we shall derive such Ward—Takahashi relations for an effective Hamiltonian that describes the model of Varma et al. The motivation of this work is the analogy between the renormalization in quantum electrodynamics [4] and the renormalization of the electron wavefunction in the marginal Fermi liquids found by Varma et al. In quantum electrodynamics the wavefunction renormalization factor is directly related to the renormalization factor of the electromagnetic vertex part through the Ward—Takahashi [5,61 relation associated with the local conservation of the electron charge. It has been shown that in the non-relativistic SchrOdinger electron field one can derive similar Ward—Takahashi relations for Matsubara temperature Green functions [7]. Therefore, if the effective Hamiltonian for the marginal Fermi-liquid model satisfies the local conservation of the electron density, we may expect that in analogy with quantum electrodynamics the renormalization of the electron wave function can be directly related to that of the electron—boson coupling constant. We first define the effective Hamiltonian and some
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
553
Volume 161, number 6
PHYSICS LETFERS A
basic requirements on it. We assume the effective Hamiltonian (1) electron term, H1~~ is the coupling between the electrons and the P-bosons and HB is the Hamiltonian for the free P-bosons. The free electron term can be written as 3x ~(t(x) [~(V)—~uJw(x), (2) Hei = d where ~v(x) is the electron Schrodinger field and ~t is the chemical potential. For simplicity we omit the spin variables. The perturbative calculation of the electron self-energy employed by Varma et al. [1] and by Pelzer [3] implies HHeI+Hint+HB, Hei is the free
where
$
20 January 1992
and /1= 1 /kBT with kB the Boltzmann constant. Let us now define the electron—boson vertex function in terms of G and I’: A(p;q)=-_1+ ~JG(k)G(k+q) k
xT’(p+q, k; k+q,p) (7) Note that this electron—boson vertex function is a proper vertex function, because it contains the not renormalized boson propagator through r. For instance a chain type (or RPA type) contribution to the boson propagator renormalization is included. The Ward—Takahashi relation (4) with q=0 gives .
[1—1(p)] Hint=gjd3xwt(x)w(x)co(x),
(3)
where g is the coupling constant and ~(x) is the quantized boson field for the P-boson. It is obvious that it commutes with the electron number density operator ~vt(x)W(x). This effective Hamiltonian leads to the following finite-temperature Ward—Takahashi relation associated with the local conservation ofthe electron number density [7], [G~’(k+q)—G~’(k)] xG(k)G(k+q)T~k,p+q;p,k+q),
[
(p)
—
~z]
.
(4)
(5)
The definitions of G, I and 1’ are the same as those given in ref. [7]. Throughout this paper we use the notation P~(.P,1Wm), q~(q,iwm), k~(k,iw,),
$
(2~)s$dkffi~~ ,
~$
(6)
554
G(k)G(kt)r(k,pt;p, kt),
A(fl, ~Wm 0,
(8)
icon)
(9)
icon
This relation is valid for finite w~.To make the analytic continuation with respect to the Matsubara frequencies, we assume Wm>O and w,,>0 [9]. Taking the co,,—’ 0 limit we obtain AQ,, co; 0, 0)= 1—
(10)
8w
where 1R is the proper self-energy for the finite ternperature real-time retarded one-particle Green function. The proper self-energyobtained by Varma et al. is [1] 2N2(0)[wln(x/cv,)—~itix] (11) 1(j,,w)—’g where x=max(IwI, T), co 1 is an ultraviolet cutoff and N( 0) is the unrenormalized one-particle density ofstates. Thisexpressiongives [1] ,
1ôReII 8w ____
where Wm, co,, and w1 are the Matsubara frequencies
k
where we have defined pt = (p, lWm + ico,,) and kt= (k, iW 1+iw~).The definition ofA given by (7) together with (8) yields the finite temperature Ward— Takahashi relation
=
where G(p) is the one-particle Matsubara temperature Green function, 1”( k, p+ q; p, k+ q) is the fourfermion vertex part [7,8], 1(k) is the proper selfenergy of the electron Green function and G0(p) is the free electron Green function, G0(P) —‘ =
— —
~nI
WI
I
I(~rt,)_~~eQ) —~tI
(12)
Volume 161, number 6
PHYSICS LETFERS A
near the Fermi surface e(p) ji. The Ward— Takahashi relation (10) shows that the result (12) also leads to —
I (01 I I. Ie(p)—~uI
(13)
Thus, the vertex function diverges when the electron renormalization factor Z(k), i.e. the quasi-particle weight, vanishes at the Fermi surfaces. This means a strongly enhanced effective coupling between the electrons and the P-bosons. It is natural to expect that such a divergent vertex function leads to formation of bound states in the vicinity of the Fermi surface. On the other hand, if we consider consistency of the canonical quantum field theory ofthe many-electron system, the vanishing of the renormalization factor Z(k) causes difficulties in keeping the equal time canonical anti-commutation relation ofthe Heisenberg field operators. Although we have not included an electron—electron interaction term Heiei in the Hamiltonian (1), as long as Hd~commutes with ~ (x) ~(x) the fi nite-temperature Ward—Takahashi relation (4) is valid [7]. In concluding this Letter we remark that when one obtains a divergent vertex function for a many-fer-
20 January 1992
mion system, there is a “marginal Fermi-liquid” effect, i.e. the vanishing of the quasi-particle weight at the Fermi momentum, provided the model Hamiltonian gives the finite temperature Ward—Takahashi relation (4). We would like to thank Dr. Chao Zhang for many valuable comments and stimulating discussions. T.T. would like to thank Dr. Frank Pelzer for helpful comments on his work. This work is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. References [1] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989)1996. [2] D.M. Ginsberg, ed., The physical properties of high temperature superconductors (World Scientific, Singapore, 1987). [3] F. Pelzer, Phys. Rev. 844 (1991) 293. [4] D. Lurié, Particles and fields (Interscience, New York, 1968). [5] [6] [7] [8]
J.C. Ward, Phys. Rev. 78 (1950) 1824. Y. Takahashi, Nuovo Cimento 6 (1957) 370. T. Toyoda, Ann. Phys. (NY) 173 (1987) 226. A.A. Abrikosov, L.P. Gor’kov and I.Ye. Dzyloshinskii, Quantum field theoretical methods in statistical physics (Pergamon, Oxford, 1965).
[9] T. Toyoda, Phys. Rev. A 39 (1989) 2659.
555
PHYSICS LETTERS A
Marginal Fermi-liquid model and Ward—Takahashi relations Tadashi Toyoda a,b and Yasushi Takahashi b
a
TheoreticalPhysics Institute, University ofAlberta, Edmonton, Canada T6G 2J1 Nagoya Shoka Daigaku, Sagamine, Nisshin, Aichi-gun, Aichi 470-01, Japan’
Received 30 September 1991; accepted for publication 18 November 1991 Communicated by A.A. Maradudin
It is shown that the vanishing of the quasi-particle weight at the Fermi surface is equivalent to the divergence of the vertex function using a finite-temperature Ward—Takahashi relation.
Recently Varma et al. [1] introduced a model electron polarizability and examined the normal state properties of the Cu—O high temperature superconductors [2], which seem to be quite different from the known Fermi-liquid properties. Using the model polarizability Varma et al. and Pelzer [3] calculated the proper self-energy of the electron Green function and found that the quasi-particle weight Z(k) vanishes at the Fermi surface. This means that the electron number distribution in the momentum space has no discontinuous gap at the Fermi momentum, i.e., the Fermi surface does not seem to exist any more. Hence, Varma et al. called their many-electron system the “marginal Fermi liquids”. We shall show in this note that the divergence of the vertex function is equivalent to the vanishing of the quasi-particle weight by the help of the finitetemperature Ward—Takahashi relation between the electron self-energy and the vertex function. The basic assumption in the calculation [1,3] of the electron self-energy is that the polarizability can be treated as a boson propagator and the coupling between the polarizability-boson (P-boson) and the electron is sufficiently weak to allow a perturbative calculation. In other words, they assumed that the electron system can be described in terms of an effective Hamiltonian that consists of the free electron term, the electron—boson coupling and a free boson term. Calculating the lowest order correction due to Permanent address.
the exchange of the P-boson to the electron Green function, they obtained the proper self-energy. Although the details of the effective Hamiltonian are not given, it is possible to assume some fundamental symmetries of the Hamiltonian rather safely. Then one can derive the corresponding Ward— Takahashi relations.In this paper we shall derive such Ward—Takahashi relations for an effective Hamiltonian that describes the model of Varma et al. The motivation of this work is the analogy between the renormalization in quantum electrodynamics [4] and the renormalization of the electron wavefunction in the marginal Fermi liquids found by Varma et al. In quantum electrodynamics the wavefunction renormalization factor is directly related to the renormalization factor of the electromagnetic vertex part through the Ward—Takahashi [5,61 relation associated with the local conservation of the electron charge. It has been shown that in the non-relativistic SchrOdinger electron field one can derive similar Ward—Takahashi relations for Matsubara temperature Green functions [7]. Therefore, if the effective Hamiltonian for the marginal Fermi-liquid model satisfies the local conservation of the electron density, we may expect that in analogy with quantum electrodynamics the renormalization of the electron wave function can be directly related to that of the electron—boson coupling constant. We first define the effective Hamiltonian and some
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
553
Volume 161, number 6
PHYSICS LETFERS A
basic requirements on it. We assume the effective Hamiltonian (1) electron term, H1~~ is the coupling between the electrons and the P-bosons and HB is the Hamiltonian for the free P-bosons. The free electron term can be written as 3x ~(t(x) [~(V)—~uJw(x), (2) Hei = d where ~v(x) is the electron Schrodinger field and ~t is the chemical potential. For simplicity we omit the spin variables. The perturbative calculation of the electron self-energy employed by Varma et al. [1] and by Pelzer [3] implies HHeI+Hint+HB, Hei is the free
where
$
20 January 1992
and /1= 1 /kBT with kB the Boltzmann constant. Let us now define the electron—boson vertex function in terms of G and I’: A(p;q)=-_1+ ~JG(k)G(k+q) k
xT’(p+q, k; k+q,p) (7) Note that this electron—boson vertex function is a proper vertex function, because it contains the not renormalized boson propagator through r. For instance a chain type (or RPA type) contribution to the boson propagator renormalization is included. The Ward—Takahashi relation (4) with q=0 gives .
[1—1(p)] Hint=gjd3xwt(x)w(x)co(x),
(3)
where g is the coupling constant and ~(x) is the quantized boson field for the P-boson. It is obvious that it commutes with the electron number density operator ~vt(x)W(x). This effective Hamiltonian leads to the following finite-temperature Ward—Takahashi relation associated with the local conservation ofthe electron number density [7], [G~’(k+q)—G~’(k)] xG(k)G(k+q)T~k,p+q;p,k+q),
[
(p)
—
~z]
.
(4)
(5)
The definitions of G, I and 1’ are the same as those given in ref. [7]. Throughout this paper we use the notation P~(.P,1Wm), q~(q,iwm), k~(k,iw,),
$
(2~)s$dkffi~~ ,
~$
(6)
554
G(k)G(kt)r(k,pt;p, kt),
A(fl, ~Wm 0,
(8)
icon)
(9)
icon
This relation is valid for finite w~.To make the analytic continuation with respect to the Matsubara frequencies, we assume Wm>O and w,,>0 [9]. Taking the co,,—’ 0 limit we obtain AQ,, co; 0, 0)= 1—
(10)
8w
where 1R is the proper self-energy for the finite ternperature real-time retarded one-particle Green function. The proper self-energyobtained by Varma et al. is [1] 2N2(0)[wln(x/cv,)—~itix] (11) 1(j,,w)—’g where x=max(IwI, T), co 1 is an ultraviolet cutoff and N( 0) is the unrenormalized one-particle density ofstates. Thisexpressiongives [1] ,
1ôReII 8w ____
where Wm, co,, and w1 are the Matsubara frequencies
k
where we have defined pt = (p, lWm + ico,,) and kt= (k, iW 1+iw~).The definition ofA given by (7) together with (8) yields the finite temperature Ward— Takahashi relation
=
where G(p) is the one-particle Matsubara temperature Green function, 1”( k, p+ q; p, k+ q) is the fourfermion vertex part [7,8], 1(k) is the proper selfenergy of the electron Green function and G0(p) is the free electron Green function, G0(P) —‘ =
— —
~nI
WI
I
I(~rt,)_~~eQ) —~tI
(12)
Volume 161, number 6
PHYSICS LETFERS A
near the Fermi surface e(p) ji. The Ward— Takahashi relation (10) shows that the result (12) also leads to —
I (01 I I. Ie(p)—~uI
(13)
Thus, the vertex function diverges when the electron renormalization factor Z(k), i.e. the quasi-particle weight, vanishes at the Fermi surfaces. This means a strongly enhanced effective coupling between the electrons and the P-bosons. It is natural to expect that such a divergent vertex function leads to formation of bound states in the vicinity of the Fermi surface. On the other hand, if we consider consistency of the canonical quantum field theory ofthe many-electron system, the vanishing of the renormalization factor Z(k) causes difficulties in keeping the equal time canonical anti-commutation relation ofthe Heisenberg field operators. Although we have not included an electron—electron interaction term Heiei in the Hamiltonian (1), as long as Hd~commutes with ~ (x) ~(x) the fi nite-temperature Ward—Takahashi relation (4) is valid [7]. In concluding this Letter we remark that when one obtains a divergent vertex function for a many-fer-
20 January 1992
mion system, there is a “marginal Fermi-liquid” effect, i.e. the vanishing of the quasi-particle weight at the Fermi momentum, provided the model Hamiltonian gives the finite temperature Ward—Takahashi relation (4). We would like to thank Dr. Chao Zhang for many valuable comments and stimulating discussions. T.T. would like to thank Dr. Frank Pelzer for helpful comments on his work. This work is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. References [1] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989)1996. [2] D.M. Ginsberg, ed., The physical properties of high temperature superconductors (World Scientific, Singapore, 1987). [3] F. Pelzer, Phys. Rev. 844 (1991) 293. [4] D. Lurié, Particles and fields (Interscience, New York, 1968). [5] [6] [7] [8]
J.C. Ward, Phys. Rev. 78 (1950) 1824. Y. Takahashi, Nuovo Cimento 6 (1957) 370. T. Toyoda, Ann. Phys. (NY) 173 (1987) 226. A.A. Abrikosov, L.P. Gor’kov and I.Ye. Dzyloshinskii, Quantum field theoretical methods in statistical physics (Pergamon, Oxford, 1965).
[9] T. Toyoda, Phys. Rev. A 39 (1989) 2659.
555