Vertex symmetry and the asymptotic frequency dependence of the ...

PHYSICAL REVIEW B
VOLUME 55, NUMBER 4
15 JANUARY 1997-II
Vertex symmetry and the asymptotic frequency dependence of the self-energy
J. J. Deisz
Department of Physics, Georgetown University, Washington, D.C. 20057
D. W. Hess
Complex Systems Theory Branch, Naval Research Laboratory, Washington, D.C. 20375-5345
J. W. Serene
Department of Physics, Georgetown University, Washington, D.C. 20057
Received 5 August 1996
We show that the exact 1/ n dependence of the self-energy for the Hubbard model for large n is obtained
entirely from the second-order skeleton diagram evaluated with the exact propagator. Explicit calculation
demonstrates that contributions from higher-order skeleton diagrams cancel exactly as a consequence of the
crossing symmetry of the renormalized four-point vertex function. From an analysis of the skeleton diagram
expansion and direct numerical calculation, we show that an imperfect cancellation of contributions from
higher-order skeleton diagrams appears at third and fourth order in U for the shielded potential approximation
and the fluctuation exchange approximation, respectively. By itself, this indication of crossing symmetry
violation of the vertex function says little about the accuracy of the resulting self-energy and propagator,
particularly at low frequency. S0163-18299602647-1
In the propagator-renormalized perturbation theory of
Luttinger and Ward, 1 the self-energy is expressed as a sum of
skeleton diagrams composed of the renormalized singleparticle
propagator and the bare vertex function. A reformulation
of this theory allows the self-energy to be written more
compactly as the sum of only two terms: the Hartree-Fock
diagram, and a diagram involving one bare and one fully
renormalized vertex function. 2,3 As in the theory of Luttinger
and Ward, only the renormalized single-particle propagator
appears in this expression for the self-energy. Although these
formulations are formally equivalent, natural approximation
schemes that begin from one formulation can yield very different
self-energies from those obtained using natural approximations
schemes that begin from the other.
An especially appealing aspect of approximation schemes
based on the formalism of Luttinger and Ward is that the
resulting approximate Green’s functions and correlation
functions obey conservation laws derived from symmetries
of the Hamiltonian. 4 On the other hand, a renormalized vertex
function calculated from commonly used approximations
of this kind generally does not obey crossing symmetry, 5,6 a
consequence of the Pauli exclusion principle. These approximations
include the shielded potential 7 and fluctuation exchange
approximations. 8 While violation of conservation
laws leads to unphysical results for the q→0,→0 limit of
correlation functions for conserved quantities, 5,9 less is
known about the signatures and consequences of violation of
crossing symmetry.
In this paper, we demonstrate explicitly that for the
single-band Hubbard model the properties of the renormalized
vertex function under interchange of labels is directly
reflected in the amplitude of the asymptotic 1/ n dependence
of the self-energy. Our proof involves three steps: 1 We
show that, to order 1/ n , the high-frequency behavior of the
exact Hubbard model self-energy is obtained entirely from
the Hartree and second-order skeleton diagrams evaluated
with the exact propagator. 2 We identify all skeleton diagrams
of order 3 and higher that have a 1/ n dependence at
high frequency, and express their contribution in terms of the
renormalized vertex function. 3 We show that the total
1/ n contribution from higher-order diagrams sums to zero
provided the renormalized vertex function obeys crossing
symmetry. The renormalized vertex functions for the fluctuation
exchange approximation FEA and shielded potential
approximation do not obey crossing symmetry; thus contributions
from skeleton diagrams of third and higher orders to
the 1/ n behavior of the self-energy do not vanish. Numerical
calculation shows that these contributions are as large as
20% of that from the second-order diagram for the rather
weak-coupling interaction strength UW/4, where W is the
bandwidth. We argue that this quantitative measure of vertex
symmetry violation says little about the accuracy of quasiparticle
properties obtained in these approximation schemes.
It is straightforward to generalize these results to other lattice
models with a local two-body interaction.
In the theory of Luttinger and Ward, the single-particle
Green’s function G is related to the self-energy by Dyson’s
equation,
G 1 k, n G 0 1 k, n k, n ,
and is in turn expressed in terms of G by
k, n 1 G
2 Gk, n ,
2
where G is a scalar functional represented by closed
Feynman diagrams involving the exact propagator, and is
invariant under all symmetries of the Hamiltonian. To avoid
double counting, G must contain only skeleton diagrams.
In a fully symmetrized perturbation theory there is
1
0163-1829/97/554/20896/$10.00 55 2089 © 1997 The American Physical Society

2090 J. J. DEISZ, D. W. HESS, AND J. W. SERENE
55
k, n 0 k k
k
i n i n 2 •••. 5
Here we have included a n -independent term 0 (k), in the
expansion; such a term cannot occur in the expansion for
G. 10 We relate the high- n behavior of G and by substituting
Eqs. 3 and 5 into Eq. 1, and identifying terms of the
same order in 1/ n . Using G 1 o (k, n )i n k with
k k , where k and are the noninteracting singleparticle
energy and renormalized chemical potential, respectively,
we obtain the following relations:
Gk1,
Gk k 0 k,
Gk k 0 k 2 k.
6a
6b
6c
From the expression for G in the Heisenberg representation,
Gk,T e K c k e K c k † ,
it is straightforward to show that
7
FIG. 1. a Hartree and b second-order self-energy diagrams
for the Hubbard model. The entire frequency-independent contribution
to the Hubbard model self-energy is obtained by evaluating the
diagram a with the exact propagator, while the entire 1/ n dependence
of the Hubbard model self-energy is obtained by evaluating
the diagram b with the exact propagator.
exactly one first-order skeleton diagram the Hartree diagram
and one second-order skeleton diagram for the selfenergy;
both are shown in Fig. 1.
The behavior of G and at high frequency n depends on
their smoothness in imaginary time . The functional dependence
of G and at high frequency can be connected to
discontinuities in G and and their derivatives at 0 by
repeated integration by parts of the Fourier integral,
where
Gk, n
0
Gk,e
i n d
Gk
••• 1p1 G p k
i n
i n p1
1p1
i n p1
0
e
i n p1 Gk,
p1 d, 3
G p k d p Gk,
d p d p Gk,
d
→0 p , 4 →0
and we have used G(k, )G(k,0 ). The final term in
Eq. 3 approaches zero faster than 1/ n p1 , because
G(k,) is infinitely differentiable on the open interval
(0,). We can represent the high- n behavior of by an
expansion analogous to Eq. 3,
G p kK,c k p ,c k † ,
where KHN, and K,c k p is the usual commutator iterated
p times K,K, ...K,c k ... and defined to be
c k for p0. In passing we note that the p-fold ‘‘nested’’
commutator is also proportional to frequency moments of the
single-particle spectral function A(k,), 11
p
p Ak,d1 p1 G p k.
For the Hubbard model it is straightforward to evaluate Eq.
8 for p0, 1, and 2; 11,12 the results are
Gk1,
Gk k Un/2,
Gk k 2 U k U 2 n/2.
8
9
10a
10b
10c
Here n is the exact density of the interacting system. Upon
substitution of Eqs. 10b and 10c into Eqs. 6b and 6c,
the exact n -independent and 1/ n contributions to are
obtained; in r space these are given by
o rU n 2 r,0 ,
rU 21 n 2 n 2 r,0.
11
12
Direct evaluation of the asymptotic n dependence of the
two lowest-order skeleton graphs for leads to forms similar
to Eqs. 11 and 12. The first-order skeleton diagram for
is the Hartree diagram shown in Fig. 1a, and yields a
frequency-independent contribution to the self-energy given
by

55 VERTEX SYMMETRY AND THE ASYMPTOTIC . . .
2091
H rU ñ 2 r,0 ,
13
where ñ is the density calculated from the trace of
G(k, n ). If ñ is equal to the exact density either because
G is the true Green’s function or as a consequence of special
circumstances such as the particle-hole symmetry of the halffilled
Hubbard model, then the Hartree contribution to
gives the exact n -independent component. This is not a surprising
result. The Hartree diagram is the only skeleton diagram
that is independent of the external frequency, so it must
fully account for o (r).
In the (r,) representation, the second-order skeleton diagram,
shown in Fig. 1b, is
2 r,U 2 G 2 r,Gr,.
14
The single-particle propagator is discontinuous only for
r0, with G(0 )1ñ/2 and G(0 )ñ/2. This yields
2 rU 21 ñ 2 ñ 2 r,0.
15
Evaluation of Eq. 15 with the exact propagator ensures that
ñ is the true density, and comparison with Eq. 12 shows
that
exact 2 G exact .
16
Thus, when evaluated with the exact G, the second-order
skeleton diagram, shown in Fig. 1b, gives the entire 1/ n
contribution to .
This result is perhaps surprising given that there is an
infinite set of third- and higher-order skeleton diagrams that
individually have asymptotic 1/ n dependences. All members
of this class of diagrams have a single unbroken Green’s
function line joining the two external vertices along with
either particle-hole fluctuation or particle-particle fluctuation
diagrams as shown in Fig. 2. The 0 discontinuity results
from the unbroken Green’s function line for r0; the
particle-hole and particle-particle fluctuation diagrams are
continuous, and their values at (r0,0) scale the amplitude
of the discontinuity for each such self-energy diagram.
When there is no unbroken Green’s-function line connecting
the two bare external vertices in a given skeleton diagram,
integration over intermediate coordinates smooths any discontinuity
from individual Green’s functions leaving a function
that is continuous at 0. Such diagrams appear at
fourth and higher orders, and contribute to order 1/ n 2 and
higher; they will not be considered further here.
Equation 16 implies that the contributions to from
third- and higher-order terms cancel exactly. To see how this
cancellation arises, consider the class of diagrams in Fig. 2
with renormalized particle-hole and particle-particle vertices,
ph and pp . 13 Using Fig. 2 and the labeling conventions
established there, we obtain
n3 U 2 dx 1 dx 2 dx 3 dx 4 Gx 1 Gx 2 Gx 3 Gx 4
↓↓;↓↓ ph ↑↓;↑↓ ph ↑↓;↑↓ pp x 1 x 2 ;x 3 x 4 ,
17
FIG. 2. The set of all a particle-hole and b particle-particle
fluctuation diagrams for the self-energy that individually have a
1/ n frequency dependence at high frequency. The net contribution
to the 1/ n dependence is zero when ph and pp satisfy vertex
labeling symmetries see text.
where x(r,) and dx r 0 d. The relative minus sign
between particle-hole and particle-particle contributions reflects
the opposite sense in of the unbroken G. A twist of
the corners labeled ‘‘2’’ and ‘‘4’’ in pp yields ph , i.e.,
↑↓;↑↓ pp (x 1 x 2 ;x 3 x 4 ) ↑↓;↑↓ ph (x 1 x 2 ;x 3 x 4 ), and the expression
for n3 becomes
n3 U 2 dx 1 dx 2 dx 3 dx 4 Gx 1 Gx 2 Gx 3
Gx 4 ↓↓;↓↓ ph x 1 x 2 ;x 3 x 4 . 18
Using the symmetry of the vertex function under the interchange
of outgoing labels ‘‘crossing symmetry’’,
↓↓;↓↓ ph (x 1 x 2 ;x 3 x 4 ) ↓↓;↓↓ ph (x 1 x 2 ;x 4 x 3 ) leads us to the result
n3 n3 ,or n3 0.
It is important to observe that the result n3 0 reflects
the form of the self-energy at high frequency, and is
entirely a consequence of the symmetry of the vertex function
under the interchange of labels. If this relation holds for
a vertex function obtained in a particular approximation, no
direct statement regarding the accuracy of G particularly for
low frequencies is possible from this information alone.
The essential symmetries of the fully renormalized vertex
used in this proof are not always satisfied by vertex functions
obtained in approximate calculations. One such approximation
scheme of current interest is the fluctuation exchange
approximation FEA, 8 which has received considerable attention
as a means for exploring the relationship between

2092 J. J. DEISZ, D. W. HESS, AND J. W. SERENE
55
FIG. 3. Ratio of the discontinuity at 0 in the self-energy to
the discontinuity obtained from the second-order diagram,
/ 2 , evaluated with the renormalized propagator at temperature
T0.1t for the fluctuation exchange approximation and the
particle-hole fluctuation approximation. For the exact solution to
the Hubbard model / 2 is identically unity. The deviation
from unity shown is a consequence of the violation of crossing
symmetry of the renormalized two-particle vertex function resulting
from the omission of skeleton diagrams from the vertex function.
For the particle-hole fluctuation approximation fluctuation exchange
approximation the lowest-order omission in the skeleton
diagram expansion for the self-energy is at order U 3 (U 4 ), resulting
in a leading departure of / 2 from unity that is linear quadratic
in U.
anomalous quasiparticle excitations, superconductivity, and
antiferromagnetic correlations in layered copper-oxide compounds.
The absence of these symmetries in the FEA is demonstrated
in Fig. 3, where / 2 is plotted as a function
of U for half-filling in two dimensions. 14 For modest coupling,
e.g., U/t2, a deviation from unity of order 20%
occurs. The deviation is quadratic in U which is easily understood
by substituting into Eq. 17 the second-order parts
of the FEA particle-hole and particle-particle vertices:
↑↓;↑↓ U 2 (x 1 x 4 )(x 2 x 3 )G(x 1 x 2 )G(x 2 x 1 )
↓↓;↓↓ ph ph
and ↑↓;↑↓ pp U 2 (x 1 x 2 )(x 3 x 4 )G 2 (x 1 x 3 ). This yields
a fourth-order contribution to the discontinuity that is equal
to
4 U 4 dx 1 dx 2 2Gx 1 Gx 2 Gx 2 x 1 Gx 1 x 2
Gx 2 Gx 1 G 2 x 1 G 2 x 2 x 1 G 2 x 2 .
19
Further calculation shows that 4 would be cancelled exactly
by including all the second-order vertex diagrams omitted
in the FEA. Examination of the explicit expressions for
the second-order parts of the FEA vertex function shows that
without including these diagrams the vertex function possesses
neither of the two symmetries used to show
n3 0.
The shielded potential or particle-hole approximation reduces
the set of FEA diagrams by omitting the particleparticle
ladder diagrams. As a consequence, contributions to
appear at third order,
FIG. 4. Dependence of / 2 on a filling, and b temperature
in the fluctuation exchange approximation for the Hubbard
model in two dimensions. The filling dependence is strong as is
expected since the many-body correlations represented by become
weak in the dilute limit for a local repulsive interaction. However,
the temperature variation is weak even though fluctuation
propagators are strongly renormalized in this temperature range.
3 U 3 dx G 2 xG 2 x.
20
Thus, as shown Fig. 3, the departure from unity of the normalized
is linear in U. 15
Clearly, / 2 provides a direct indication of the violation
of vertex symmetry and the order at which the violation
first occurs; but does it characterize the accuracy of quasiparticle
properties obtained in these approximation
schemes? To consider this question, we display the filling
dependence of / 2 in Fig. 4a. Since multiparticle correlations
become weaker in the dilute limit for a repulsive
local interaction, it is not surprising that the deviations from
unity are smallest at lower fillings. At the same time, it has
been shown numerically that FEA results for G are poorer
away from half-filling. 8 This provides at least one case where
the size of the departure of / 2 from unity does not
correlate with the accuracy of the renormalized G.
It is also worth evaluating the dependence of / 2 on
other parameters which strongly affect the FEA’s behavior
and, possibly, accuracy. Figure 4b displays the temperature
dependence of / 2 at half-filling in two dimensions.
For U2t and near the lower temperature shown, the spin
fluctuation propagator becomes nearly singular around
q(,) and m 0, producing a non-Fermi-liquid
self-energy. 16 This dramatic temperature-driven change in
the self-energy at low energies does not result in a strong
temperature dependence of / 2 .
Likewise, as shown in Fig. 5, there is little dependence on
lattice dimension, d, for a fixed ratio of U to td 1/2 . This
scaling of U removes the trivial dependence that arises from
the increase of the bandwidth with dimensionality. Thus we

55 VERTEX SYMMETRY AND THE ASYMPTOTIC . . .
2093
FIG. 5. Dependence of / 2 on lattice dimension d for
n1 and T0.1. The Hubbard interaction U has been scaled by
the factor td 1/2 which is proportional to the large-d expression for
the bandwidth. The / 2 shown are nearly independent of d.
see no sign of an enhancement of crossing symmetry violation
as the lattice dimension is lowered, although it is expected
that the FEA will become less accurate.
It has been argued 17 that the appearance of a satellite
structure in the single-particle spectral function for the
Anderson lattice model ALM is a consequence of the highfrequency
behavior of the self-energy. For the simplest form
of the Anderson lattice model, the only two-body interaction
occurs between f electrons, and the self-energy is given entirely
by skeleton diagrams involving only f -electron propagators.
As in the case of the Hubbard model, the contributions
of skeleton diagrams of order 3 and higher to the
discontinuity of the self-energy at 0 vanishes, and the
high-frequency behavior of f is given entirely by the
second-order diagram evaluated with the fully renormalized
f -electron propagator. The leading high-frequency dependence
of the self-energy is given by Eq. 15, with ñ as the
f -electron density. For the symmetric ALM, the average
f -electron occupancy is constrained to be unity by particlehole
symmetry, and we find that at high frequency the exact
self-energy goes as →U 2 /4. This exact result turns out to
be the same as that obtained for the atomic model in Sec. III
of Ref. 17, which places all f -electron spectral weight in
satellite peaks at U/2, a result that is strictly valid only in
the U→ limit. Although it is most accurate for small U,
second-order perturbation theory also duplicates the exact
high-energy result as a consequence of particle-hole
symmetry. 17 Thus, while f -electron satellites at U/2 are
consistent with the high-energy behavior of the exact selfenergy,
the existence of qualitatively different approximations
with the same high-frequency behavior as the exact
self-energy shows that this signature of crossing symmetry
places little if any constraint on the accuracy of the selfenergy
at low energy.
In summary, we have demonstrated a property of the exact
self-energy of the Hubbard model; the leading asymptotic
frequency dependence of the self-energy is provided entirely
by the Hartree and second-order skeleton diagrams. The
frequency-independent contribution to the self-energy is obtained
from the Hartree diagram as expected. The leading
1/ n behavior is obtained entirely from the second-order
skeleton diagram, and reflects a perfect cancellation of the
contributions to order 1/ n of an infinite class of higher-order
diagrams, which we have shown depend only on the labeling
symmetry of the renormalized vertex function. Numerical
calculation shows substantial deviations in the fluctuation exchange
and shielded potential approximations, where the
vertex functions do not obey these symmetries. However,
these numerical results suggest that this indicator of
crossing-symmetry violation in and of itself does not provide
information on the accuracy of the resulting self-energy and
propagator, particularly at low energy.
ACKNOWLEDGMENTS
We thank M. Steiner for bringing Ref. 12 to our attention.
J.D. and J.S. were supported by National Science Foundation
Grant No. ASC-9504067; D.W.H. was supported by the Office
of Naval Research. Numerical work presented was supported
in part by a grant of computer time from the Department
of Defense High-Performance Computing Shared
Resource Center, Naval Research Laboratory Connection
Machine Facility CM-5. D.H. would like to acknowledge the
kind hospitality of the Universität Bayreuth during the preparation
of this manuscript.
1 J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 1960.
2 See, for example, A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,
Methods of Quantum Field Theory in Statistical
Physics Dover, New York, 1975, pp. 87 and 88.
3 C. de Dominicis and P. C. Martin, J. Math. Phys. 5, 141964.
4 G. Baym, Phys. Rev. 127, 1391 1962.
5 N. E. Bickers and S. R. White, Phys. Rev. B 43, 8044 1991.
6 See, for example, D. J. Amit, J. W. Kane and H. Wagner, Phys.
Rev. 175, 326 1968.
7 G. Baym and L. Kadanoff, Quantum Statistical Mechanics Benjamin,
New York, 1962.
8 N. E. Bickers, D. J. Scalapino, and S. R. White, Phys. Rev. Lett.
62, 961 1989.
9 G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 1961.
10 J. M. Luttinger, Phys. Rev. 121, 942 1961.
11 S. R. White, Phys. Rev. B 44, 4670 1991.
12 W. Nolting, Z. Phys. 255, 251972.
13 This argument may also be cast in terms of local, equal-time
susceptibilities. The contribution of the diagrams of Fig. 2 to the
0 discontinuity can then be written as
n3 U 2 1 4 0,0 1 4 0,0 pp 0,0n,
where
0,0
0,0 c † c c † c ,
,↑,↓
ix,y,z,,↑,↓
pp 0,0c ↑ † c ↓ † c ↓ c ↑ ;
i i c † c c † c ,

2094 J. J. DEISZ, D. W. HESS, AND J. W. SERENE
55
and all fermion operators have identical space and time indices.
These local, equal-time susceptibilities can be expressed in
terms of two parameters, c ↑ † c ↑ c ↓ † c ↓ and n, and n3 is easily
seen to be zero.
14 We use the algorithm reported by J. J. Deisz, D. W. Hess, and J.
W. Serene, in Recent Progress in Many-Body Theories, edited
by E. Schachinger, H. Mitter, and H. Sormann Plenum, New
York, 1995, Vol. 4 to calculate the FEA self-energy and, in
particular, its discontinuity at 0.
15 The same conclusions follow, but with an opposite sign for
3 , for the T approximation, where only particle-particle ladder
diagrams are included in .
16 J. J. Deisz, D. W. Hess, and J. W. Serene, Phys. Rev. Lett. 76,
1312 1996.
17 M. Steiner, R. C. Albers, D. J. Scalapino, and L. J. Sham, Phys.
Rev. B 43, 1637 1991.