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Fermi liquid theory in two dimensions
PflYSICA
Physica B 186-188 (1993) 971-974 North-Holland
Fermi liquid theory in two dimensions Sudhakar Yarlagadda and Susumu Kurihara NTT Basic Research Labs, Musashino-Shi, Tokyo, Japan We examine some of the issues concerning the validity of Fermi liquid theory in two dimensions. Contrary to previous claims, we find that the quasiparticle residue is nonzero and that the quasiparticle interaction function in the forwardscattering case is nondivergent for all values of the repulsive potential.
In the recent past, the question of the validity of Landau's Fermi liquid theory in two dimensions has generated some controversy. On one hand, Anderson [1] proposed that the two-dimensional Hubbard model exhibits a non-Fermi-liquid-like behavior because the quasiparticle residue assumes value zero and the interaction function of quasiparticles of identical momenta becomes singular. Following this work, Stamp [2] argues that singular behavior in the quasiparticle interaction function in the forward scattering case and value zero of the quasiparticle residue are consistent with each other. On the other hand, the results reported in refs. [3,4,5] all point in the direction that the Fermi liquid theory does not break down in two dimensions. In the light of these studies, in the present paper we develop a general framework to examine some of the controversial issues of Fermi liquid theory in two dimensions in a system with repulsive interactions. The rest of the paper is organized as follows. Starting with a pairing Hamiitonian for electrons of opposite spin, we obtain the ground state energy in terms of the pair susceptibility. Next, on examining the divergence conditions of the pair susceptibility, we find that, in addition to the pair formation condition in three dimensions, there are new criteria for condensation in two dimensions. To study the validity of Fermi liquid theory in view of this novel pairing phenomenon, we obtain consistently the quasiparticle energy and the quasiparticle interaction function, within the framework of random phase approximation (RPA), by taking the functional derivative of the ground state energy with respect to the electronic occupation number.
Then we show explicitly that the interaction function for two quasiparticles of the same momentum is convergent. Also, using the same arguments as for the interaction function, we show that the real part of the frequency derivative of the self-energy at the Fermi surface has no singular behavior. We begin our analysis by considering the following pairing Hamiltonian involving interactions among electrons of opposite spin: +
n = ~ eka~ak~ +
q-A E
+
Vk,pak+q/2ra_k+q/z,ta p+q/2~ap+q/2t
(1)
k,p.q +
where ek and a ~ (ak~) are the energy and creation (destruction) operator of an electron with momentum k and spin ~r. Furthermore, A is a coupling constant and Vk,, is a separable interaction potential of the form V k , p : OlUkVp with a = 1 ( - 1 ) for repulsive (attractive) interactions. Next, we define the two-electron creation operator as +
+
C~ = ~ vkak+q/2ra k+q/zs
(2)
k
where q and 2k are the center-of-mass momentum and relative momentum of the two-electron system. Then, by using the integration-over-the-coupling-constant algorithm [6], the shift of the ground state energy E c of the system with respect to its noninteracting value E o° at 0 K can be expressed as follows: 1
EG - E ° = a
JdA~ I(01C~ln>l=
(3)
q,n
Correspondence to: Sudhakar. Yarlagadda, Material Science Lab, N'F'F Basic Research Labs, 9-11, Midori-Cho 3-Chome, Musashino-Shi, Tokyo 180, Japan.
0
where In) are the eigenstates with 10) being the ground state. First, we note that the right hand side of
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
972
S. Yarlagadda, S. Kurihara / Fermi liquid theory in two dimensions
the above equation can be expressed in terms of the imaginary part of the lowest order pair susceptibility {) Xp,~ which is defined in terms of the electronic OCCU¢r pation number n k as follows:
form of the lowest-order pair susceptibility X°p,,(q, w - 2/z) can be replaced by its time-ordered equivalent Xp,~(q, -0 o) - 2p,) as defined below: )~Oair(q,
¢O - -
Z/z)
1 -- n'~k+q/2 -- n~--k+q/2 e k + q / 2 -]- 6 k _ q / 2 -1- O) - -
k
2 # + b7
"
(4)
=--
~
f
d O V,.,G~ (k + q/2, g2 - w)
(7)
x G~ ( - k + q/Z, - g 2 )
In the above equation,/z is the chemical potential and "0 is an infinitesimal positive constant. Next, we consider ladder diagrams for the vertex corrections due to electron-electron interactions in the lowest-order pair susceptibility and obtain the following RPA expression for the ground state energy: E~
--
E ° = Im[ q~ f -dw ln[1 + 'IT
o
Xpair(q,
o)
o
- 2/z)]]
where Go is the noninteracting electron Green's function. Then we take functional derivatives with respect to the occupation number n~ of the ground state energy expressed in terms of the time-ordered pair susceptibility Xpa,r -° and obtain the following expressions for the quasiparticle energy E~ and the Landau interaction function f ~ ' :
.
(5)
E~=~mm + I m
-+ ~0 0 ~r 1 Xpair(q, O)--2/Z)
0
We will now examine the term 1 + Xpair occurring in eq. (5) for zeros so as to determine the conditions for pair formation. Here and in what follows, we assume that the interaction potential Vk,k has a cutoff energy larger than the Fermi energy. We first note t h a t XOair diverges at the usual three-dimensional values of q = 0 and w = O. In addition to this, in two dimensions, X°,~ diverges for all q ~- 2pv and the divergence condition is given (also see refs. [3,4,7,8] for similar results) as follows:
+
q, -,o - %
+ ~)]
(8)
and [ 2VIp
n
*l/2,1p-kl/2~
f;~' = t~e~ - - ~ - - - -
....
'
1 + Xpair(P + k, -2/x)
) + G~,~'
(9)
where
?
q2
w + ~mm - 2 / z = 0.
× c,, ~ ( - p
g.~ (~)
(6) o
However, for a repulsive potential, for q > 2pv, the value of X0pair~ O0, whereas for q < 2pv the value of o Xp~ir~ - ~ . In contrast to this, for an attractive potential the sign of the divergence gets flipped in these two momentum regions. The sign of the divergence changes at q = 2pv because of the restriction on electron occupation imposed by the Pauli exclusion principle. Thus, in the weak-coupling regime, eq. (6) can be taken to be the pairing condition for particles with repulsive (attractive) interactions when the inequality q < 2 p v ( q > 2 p v ) is satisfied. However, for intermediate- and strong-coupling cases, the condition for pair formation depends on the strength of interaction and also on the form of the interaction potential. In light of the new conditions for pair condensation in two dimensions, we will now examine the validity of the Fermi liquid theory at 0 K. To this end, we will first cast the ground state energy in a more convenient form [9]. Now in eq. (5), since the integration frequency ¢o ranges over nonnegative values, the retarded
0 'rr
[1 + Xpair(q, w - 2/x)]
X G o " ( - p + q, - t o x Go~'(-k+
-
e~, + tz)
q, - w - e k + p,)] .
(10)
The expression for the self-energy in eq. (8) is similar to that obtained due to paramagnons [10]. In the next equation, i.e. eq. (9), the first term on the right hand side is real and represents the effective interaction between two opposite-spin quasiparticles leading to the particles retaining their momenta. Also, in this equation the second term G ~ ' corresponds to a more complicated interaction whereby like-spin particles exchange their momenta and opposite-spin particles preserve their momenta. For repulsive interactions, we will first examine the Landau interaction function, in the forward-scattering
S. Yarlagadda, S. Kurihara / Fermi liquid theory in two dimensions case, for any possible singularities. The first term in eq. (9) is always convergent because Xpair~0appeanng' in the denominator is always positive and finite. The function g ~ ' ( t o ) defined in eq. (10) does not have any poles in the first quadrant of the complex to plane. By integrating around the edge of the first quadrant in the to plane, we transform G ~ ~ into the following convenient form:
973
We will now study the nature of the quasiparticle residue Z. By taking the derivative of the quasiparticle energy given by eq. (8) and using the same arguments that led to the derivation of G~,7~ given by eq. (11), we obtain 1
1 - ~ =Re
OE(p, S2) , ~ o 0g2
w/2
=fd
G.-7 ~ = i J dto g~;~(ioJ)
o
o
v
~r2 [1 + Xp~,r(2pr cos ~b, -2p.)]
0+
+lm[ f di'° v' w- [1 +
iO +
iO +
I-p+q/2l,l-p+q/21
1
× [1 + X p a i r ( q , O i t o -- 2/,/~)] 2 [ E p + q -- ~ + i t o ] 2
]
1
~
]
x [ e _ . + q - t* + ko] 2 "
"
- 2/x)l (13)
(11) In the above expression, the integrand does not have any singularities for nonzero values of the frequency. Furthermore, at large values of to the integral is convergent. So, the only region where a possible singularity might result is at to = 0 +. For to = 0 ÷, the denominator of the integrand does not have any zeros in the first quadrant of the complex momentum q plane. Next, we assume a constant value for the interaction potential of the form V,, k = V O ( k c - k) with k c >>PF the cutoff momentum k c > P v . Then again, upon performing a contour integration along the circumference of the first quadrant of the q plane and carrying out the integration with respect to the angle between the momenta p and q, we obtain g~7~(i0 + ) 2mZV2fdq[ "~"
1
[1+ X op a i r ( l q ,
2
Then, based on the same analysis that established the convergence of f p7 ~, we conclude that the real part of the frequency derivative of the quasiparticle energy at the Fermi surface, as given by the above equation, converges too. The fact that Z is nonzero has also been shown by the authors of refs. [3,4] by considering the imaginary part of the self-energy. Furthermore, Fabrizio et al. [5] find that in the zero-density limit Z-+I. In conclusion, we say that we have developed a general framework to analyse the Fermi liquid properties in the context of pair formation. We find that, within RPA, the quasiparticle residue is nonzero and the Landau interaction function in the forward-scattering case is nonsingular for any value of the interaction strength of the repulsive potential. Our RPA results are exact for k c >>Pv. Thus based on our analysis there is definitely reason to doubt the claims of ref. [1] that the Fermi liquid theory breaks down in two dimensions. Further analysis of the behavior of the quasiparticle self-energy and the Landau interaction function will be presented elsewhere [11].
--2//,)] 2
2k c
+ [1 + 0
0
,
Xpair(q,
_2/z)]2
]
(12)
where Xpair(lq,--2/.L) is always positive and diverges logarithmically for values of q2/8m + pZr/2m smaller than the cutoff energy k2/2m imposed by the repulsive potential. From eq. (12), it is clear that at to = 0 ÷, the integrand converges for all values of the momentum iq. Hence, the quasiparticle interaction function in the forward-scattering case is convergent for all values of the repulsive interaction strength.
References
[1] P.W. Anderson, Phys. Rev. Lett. 64 (1990) 1839; 65 (1990) 2306; 66 (1991) 3226. [2] P.C.E. Stamp, Phys. Rev. Lett. 68 (1992) 2180. [3] J.R. Engelbrecht and M. Randeria, Phys. Rev. B 45 (1992) 12419; Phys. Rev. Lett. 65 (1990) 1032. [4] H. Fukuyama, O. Narikiyo and Y. Hasegawa, J. Phys. Soc. Jpn. 60 (1991) 372; 60 (1991) 2013. [5] M. Fabrizio, A. Parola and E. Tosatti, Phys. Rev. B 44 (1991) 1033.
974
S. Yarlagadda, S. Kurihara / Fermi liquid theory in two dimensions
[6] D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966) p. 296. [7] S. Schmitt-Rink, C.M. Varma and A.E. Ruekenstein, Phys. Rev. Lett. 63 (1989) 445. [8] J. Serene, Phys. Rev. B 40 (1989) 10 873. [9] T.M. Rice, Ann. Phys. 31 (1965) 100; C.S. Ting, T.K. Lee and J.J. Quinn, Phys. Rev. Lett. 34 (1975) 870; S.
Yarlagadda and G.F. Giuliani, Phys. Rev. B 40 (1989) 5432; Surf. Sci. 229 (1990) 410; to be published. [10] S. Doniach and E.H. Sondheimer, Green's Functions for Solid State Physicists (Benjamin, New York, 1974) p. 172. [11] S. Yarlagadda and S. Kurihara, to be published.
Physica B 186-188 (1993) 971-974 North-Holland
Fermi liquid theory in two dimensions Sudhakar Yarlagadda and Susumu Kurihara NTT Basic Research Labs, Musashino-Shi, Tokyo, Japan We examine some of the issues concerning the validity of Fermi liquid theory in two dimensions. Contrary to previous claims, we find that the quasiparticle residue is nonzero and that the quasiparticle interaction function in the forwardscattering case is nondivergent for all values of the repulsive potential.
In the recent past, the question of the validity of Landau's Fermi liquid theory in two dimensions has generated some controversy. On one hand, Anderson [1] proposed that the two-dimensional Hubbard model exhibits a non-Fermi-liquid-like behavior because the quasiparticle residue assumes value zero and the interaction function of quasiparticles of identical momenta becomes singular. Following this work, Stamp [2] argues that singular behavior in the quasiparticle interaction function in the forward scattering case and value zero of the quasiparticle residue are consistent with each other. On the other hand, the results reported in refs. [3,4,5] all point in the direction that the Fermi liquid theory does not break down in two dimensions. In the light of these studies, in the present paper we develop a general framework to examine some of the controversial issues of Fermi liquid theory in two dimensions in a system with repulsive interactions. The rest of the paper is organized as follows. Starting with a pairing Hamiitonian for electrons of opposite spin, we obtain the ground state energy in terms of the pair susceptibility. Next, on examining the divergence conditions of the pair susceptibility, we find that, in addition to the pair formation condition in three dimensions, there are new criteria for condensation in two dimensions. To study the validity of Fermi liquid theory in view of this novel pairing phenomenon, we obtain consistently the quasiparticle energy and the quasiparticle interaction function, within the framework of random phase approximation (RPA), by taking the functional derivative of the ground state energy with respect to the electronic occupation number.
Then we show explicitly that the interaction function for two quasiparticles of the same momentum is convergent. Also, using the same arguments as for the interaction function, we show that the real part of the frequency derivative of the self-energy at the Fermi surface has no singular behavior. We begin our analysis by considering the following pairing Hamiltonian involving interactions among electrons of opposite spin: +
n = ~ eka~ak~ +
q-A E
+
Vk,pak+q/2ra_k+q/z,ta p+q/2~ap+q/2t
(1)
k,p.q +
where ek and a ~ (ak~) are the energy and creation (destruction) operator of an electron with momentum k and spin ~r. Furthermore, A is a coupling constant and Vk,, is a separable interaction potential of the form V k , p : OlUkVp with a = 1 ( - 1 ) for repulsive (attractive) interactions. Next, we define the two-electron creation operator as +
+
C~ = ~ vkak+q/2ra k+q/zs
(2)
k
where q and 2k are the center-of-mass momentum and relative momentum of the two-electron system. Then, by using the integration-over-the-coupling-constant algorithm [6], the shift of the ground state energy E c of the system with respect to its noninteracting value E o° at 0 K can be expressed as follows: 1
EG - E ° = a
JdA~ I(01C~ln>l=
(3)
q,n
Correspondence to: Sudhakar. Yarlagadda, Material Science Lab, N'F'F Basic Research Labs, 9-11, Midori-Cho 3-Chome, Musashino-Shi, Tokyo 180, Japan.
0
where In) are the eigenstates with 10) being the ground state. First, we note that the right hand side of
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
972
S. Yarlagadda, S. Kurihara / Fermi liquid theory in two dimensions
the above equation can be expressed in terms of the imaginary part of the lowest order pair susceptibility {) Xp,~ which is defined in terms of the electronic OCCU¢r pation number n k as follows:
form of the lowest-order pair susceptibility X°p,,(q, w - 2/z) can be replaced by its time-ordered equivalent Xp,~(q, -0 o) - 2p,) as defined below: )~Oair(q,
¢O - -
Z/z)
1 -- n'~k+q/2 -- n~--k+q/2 e k + q / 2 -]- 6 k _ q / 2 -1- O) - -
k
2 # + b7
"
(4)
=--
~
f
d O V,.,G~ (k + q/2, g2 - w)
(7)
x G~ ( - k + q/Z, - g 2 )
In the above equation,/z is the chemical potential and "0 is an infinitesimal positive constant. Next, we consider ladder diagrams for the vertex corrections due to electron-electron interactions in the lowest-order pair susceptibility and obtain the following RPA expression for the ground state energy: E~
--
E ° = Im[ q~ f -dw ln[1 + 'IT
o
Xpair(q,
o)
o
- 2/z)]]
where Go is the noninteracting electron Green's function. Then we take functional derivatives with respect to the occupation number n~ of the ground state energy expressed in terms of the time-ordered pair susceptibility Xpa,r -° and obtain the following expressions for the quasiparticle energy E~ and the Landau interaction function f ~ ' :
.
(5)
E~=~mm + I m
-+ ~0 0 ~r 1 Xpair(q, O)--2/Z)
0
We will now examine the term 1 + Xpair occurring in eq. (5) for zeros so as to determine the conditions for pair formation. Here and in what follows, we assume that the interaction potential Vk,k has a cutoff energy larger than the Fermi energy. We first note t h a t XOair diverges at the usual three-dimensional values of q = 0 and w = O. In addition to this, in two dimensions, X°,~ diverges for all q ~- 2pv and the divergence condition is given (also see refs. [3,4,7,8] for similar results) as follows:
+
q, -,o - %
+ ~)]
(8)
and [ 2VIp
n
*l/2,1p-kl/2~
f;~' = t~e~ - - ~ - - - -
....
'
1 + Xpair(P + k, -2/x)
) + G~,~'
(9)
where
?
q2
w + ~mm - 2 / z = 0.
× c,, ~ ( - p
g.~ (~)
(6) o
However, for a repulsive potential, for q > 2pv, the value of X0pair~ O0, whereas for q < 2pv the value of o Xp~ir~ - ~ . In contrast to this, for an attractive potential the sign of the divergence gets flipped in these two momentum regions. The sign of the divergence changes at q = 2pv because of the restriction on electron occupation imposed by the Pauli exclusion principle. Thus, in the weak-coupling regime, eq. (6) can be taken to be the pairing condition for particles with repulsive (attractive) interactions when the inequality q < 2 p v ( q > 2 p v ) is satisfied. However, for intermediate- and strong-coupling cases, the condition for pair formation depends on the strength of interaction and also on the form of the interaction potential. In light of the new conditions for pair condensation in two dimensions, we will now examine the validity of the Fermi liquid theory at 0 K. To this end, we will first cast the ground state energy in a more convenient form [9]. Now in eq. (5), since the integration frequency ¢o ranges over nonnegative values, the retarded
0 'rr
[1 + Xpair(q, w - 2/x)]
X G o " ( - p + q, - t o x Go~'(-k+
-
e~, + tz)
q, - w - e k + p,)] .
(10)
The expression for the self-energy in eq. (8) is similar to that obtained due to paramagnons [10]. In the next equation, i.e. eq. (9), the first term on the right hand side is real and represents the effective interaction between two opposite-spin quasiparticles leading to the particles retaining their momenta. Also, in this equation the second term G ~ ' corresponds to a more complicated interaction whereby like-spin particles exchange their momenta and opposite-spin particles preserve their momenta. For repulsive interactions, we will first examine the Landau interaction function, in the forward-scattering
S. Yarlagadda, S. Kurihara / Fermi liquid theory in two dimensions case, for any possible singularities. The first term in eq. (9) is always convergent because Xpair~0appeanng' in the denominator is always positive and finite. The function g ~ ' ( t o ) defined in eq. (10) does not have any poles in the first quadrant of the complex to plane. By integrating around the edge of the first quadrant in the to plane, we transform G ~ ~ into the following convenient form:
973
We will now study the nature of the quasiparticle residue Z. By taking the derivative of the quasiparticle energy given by eq. (8) and using the same arguments that led to the derivation of G~,7~ given by eq. (11), we obtain 1
1 - ~ =Re
OE(p, S2) , ~ o 0g2
w/2
=fd
G.-7 ~ = i J dto g~;~(ioJ)
o
o
v
~r2 [1 + Xp~,r(2pr cos ~b, -2p.)]
0+
+lm[ f di'° v' w- [1 +
iO +
iO +
I-p+q/2l,l-p+q/21
1
× [1 + X p a i r ( q , O i t o -- 2/,/~)] 2 [ E p + q -- ~ + i t o ] 2
]
1
~
]
x [ e _ . + q - t* + ko] 2 "
"
- 2/x)l (13)
(11) In the above expression, the integrand does not have any singularities for nonzero values of the frequency. Furthermore, at large values of to the integral is convergent. So, the only region where a possible singularity might result is at to = 0 +. For to = 0 ÷, the denominator of the integrand does not have any zeros in the first quadrant of the complex momentum q plane. Next, we assume a constant value for the interaction potential of the form V,, k = V O ( k c - k) with k c >>PF the cutoff momentum k c > P v . Then again, upon performing a contour integration along the circumference of the first quadrant of the q plane and carrying out the integration with respect to the angle between the momenta p and q, we obtain g~7~(i0 + ) 2mZV2fdq[ "~"
1
[1+ X op a i r ( l q ,
2
Then, based on the same analysis that established the convergence of f p7 ~, we conclude that the real part of the frequency derivative of the quasiparticle energy at the Fermi surface, as given by the above equation, converges too. The fact that Z is nonzero has also been shown by the authors of refs. [3,4] by considering the imaginary part of the self-energy. Furthermore, Fabrizio et al. [5] find that in the zero-density limit Z-+I. In conclusion, we say that we have developed a general framework to analyse the Fermi liquid properties in the context of pair formation. We find that, within RPA, the quasiparticle residue is nonzero and the Landau interaction function in the forward-scattering case is nonsingular for any value of the interaction strength of the repulsive potential. Our RPA results are exact for k c >>Pv. Thus based on our analysis there is definitely reason to doubt the claims of ref. [1] that the Fermi liquid theory breaks down in two dimensions. Further analysis of the behavior of the quasiparticle self-energy and the Landau interaction function will be presented elsewhere [11].
--2//,)] 2
2k c
+ [1 + 0
0
,
Xpair(q,
_2/z)]2
]
(12)
where Xpair(lq,--2/.L) is always positive and diverges logarithmically for values of q2/8m + pZr/2m smaller than the cutoff energy k2/2m imposed by the repulsive potential. From eq. (12), it is clear that at to = 0 ÷, the integrand converges for all values of the momentum iq. Hence, the quasiparticle interaction function in the forward-scattering case is convergent for all values of the repulsive interaction strength.
References
[1] P.W. Anderson, Phys. Rev. Lett. 64 (1990) 1839; 65 (1990) 2306; 66 (1991) 3226. [2] P.C.E. Stamp, Phys. Rev. Lett. 68 (1992) 2180. [3] J.R. Engelbrecht and M. Randeria, Phys. Rev. B 45 (1992) 12419; Phys. Rev. Lett. 65 (1990) 1032. [4] H. Fukuyama, O. Narikiyo and Y. Hasegawa, J. Phys. Soc. Jpn. 60 (1991) 372; 60 (1991) 2013. [5] M. Fabrizio, A. Parola and E. Tosatti, Phys. Rev. B 44 (1991) 1033.
974
S. Yarlagadda, S. Kurihara / Fermi liquid theory in two dimensions
[6] D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966) p. 296. [7] S. Schmitt-Rink, C.M. Varma and A.E. Ruekenstein, Phys. Rev. Lett. 63 (1989) 445. [8] J. Serene, Phys. Rev. B 40 (1989) 10 873. [9] T.M. Rice, Ann. Phys. 31 (1965) 100; C.S. Ting, T.K. Lee and J.J. Quinn, Phys. Rev. Lett. 34 (1975) 870; S.
Yarlagadda and G.F. Giuliani, Phys. Rev. B 40 (1989) 5432; Surf. Sci. 229 (1990) 410; to be published. [10] S. Doniach and E.H. Sondheimer, Green's Functions for Solid State Physicists (Benjamin, New York, 1974) p. 172. [11] S. Yarlagadda and S. Kurihara, to be published.