Non-perturbative conserving approximations and Luttinger’s sum rule
Jutta Ortloff, Matthias Balzer, Michael Potthoff
Institut für Theoretische Physik und Astrophysik,
Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Weak-coupling conserving approximations can be constructed by truncations of the Luttinger-
Ward functional and are well known as thermodynamically consistent approaches which respect
macroscopic conservation laws as well as certain sum rules at zero temperature. These properties
can also be shown for variational approximations that are generated within the framework of the self-
energy-functional theory without a truncation of the diagram series. Luttinger’s sum rule represents
an exception. We analyze the conditions under which the sum rule holds within a non-perturbative
conserving approximation. Numerical examples are given for a simple but non-trivial dynamical two-
site approximation. The validity of the sum rule for finite Hubbard clusters and the consequences
for cluster extensions of the dynamical mean-field theory are discussed.
PACS numbers: 71.10.-w, 71.10.Fd
I. INTRODUCTION
Continuous symmetries of a Hamiltonian imply the
existence of conserved quantities: The conservation of
total energy, momentum, angular momentum, spin and
particle number is enforced by a not explicitly time-
dependent Hamiltonian which is spatially homogeneous
and isotropic and invariant under global SU(2) and U(1)
gauge transformations. For the treatment of a macro-
scopically large quantum system of interacting fermions,
approximations are inevitable in general. Approxima-
tions, however, may artificially break symmetries and
thus lead to unphysical violations of conservations laws.
Baym and Kadanoff1,2 have analyzed under which cir-
cumstances an approximation for time-dependent corre-
lation functions, and for one- and two-particle Green’s
functions in particular, respect the mentioned macro-
scopic conservation laws. They were able to give cor-
responding rules for a proper construction of approxima-
tions, namely criteria for selecting suitable classes of dia-
grams, within diagrammatic weak-coupling perturbation
theory. Weak-coupling approximations following these
rules and thus respecting conservation laws are called
“conserving”. Frequently cited examples for conserving
approximations are the Hartree-Fock or the fluctuation-
exchange approximation.1,3,4
Baym2 has condensed the method of constructing con-
serving approximations into a compact form: A con-
serving approximation for the one-particle Green’s func-
tion G is obtained by using Dyson’s equation G =
1/(G−10 − Σ) with (the free, U = 0, Green’s function
G0 and) a self-energy Σ = ΣU [G] given by a univer-
sal functional. Apart from G, the universal functional
ΣU must depend on the interaction parameters U only.
Furthermore, the functional must satisfy a vanishing-curl
condition or, alternatively, must be derivable from some
(universal) functional ΦU [G] as TΣU [G] = δΦU [G]/δG
(the temperature T is introduced for convenience). In
short, “Φ-derivable” approximations are conserving.
Φ-derivable approximations have been shown2 to ex-
hibit several further advantageous properties in addition.
One of these concerns the question of thermodynamical
consistency. There are different ways to determine the
grand potential of the system from the Green’s function
which do not necessarily yield the same result when us-
ing approximate quantities. On the one hand, Ω may be
calculated by integration of expectation values, accessi-
ble by G, with respect to certain model parameters. For
example, Ω may be calculated by integration of the av-
erage particle number, as obtained from the trace of G,
with respect to the chemical potential µ. On the other
hand, Ω can be obtained as Ω = Φ + Tr lnG − Tr(ΣG)
without integration. A Φ-derivable approximation con-
sistently gives the same result for Ω in both ways.
At zero temperature T = 0 there is another non-trivial
theorem which is satisfied by any Φ-derivable approxima-
tion, namely Luttinger’s sum rule.5,6 This states that the
volume in reciprocal space that is enclosed by the Fermi
surface is equal to the average particle number. The orig-
inal proof of the sum rule by Luttinger and Ward5 is
based on the existence of Φ in the exact theory and is
straightforwardly transferred to the case of a Φ-derivable
approximation. This also implies that other Fermi-liquid
properties, such as the linear trend of the specific heat at
low T and Fermi-liquid expressions for the T = 0 charge
and the spin susceptibility are respected by a Φ-derivable
approximation.
There is a perturbation expansion5,7 which gives the
Luttinger-Ward functional ΦU [G] in terms of closed
skeleton diagrams (see Fig. 1). As a manageable Φ-
= + + +Φ
FIG. 1: Diagrammatic representation of the Luttinger-Ward
functional ΦU [G]. Double lines stand for the interacting one-
particle Green’s function G, dashed lines represent the ver-
tices U .
http://arxiv.org/abs/0704.0249v2
derivable approximation must specify a (universal) func-
tional ΦU [G] that can be evaluated in practice, one usu-
ally considers truncations of the expansion and sums up
a certain subclass of skeleton diagrams only. This, how-
ever, means that the construction of conserving approx-
imations is restricted to the weak-coupling limit.
One purpose of the present paper is to show that it is
possible to construct Φ-derivable approximations for lat-
tice models of correlated fermions with local interactions
which are non-perturbative, i.e. do not employ trunca-
tions of the skeleton-diagram expansion. The idea is to
employ the self-energy-functional theory (SFT).8,9,10 The
SFT constructs the Luttinger-Ward functional ΦU [G], or
its Legendre transform FU [Σ], in an indirect way, namely
by making contact with an exactly solvable reference sys-
tem. Thereby, the exact functional dependence of FU [Σ]
becomes available on a certain subspace of self-energies
which is spanned by the self-energies generated by the
reference system.
The obvious question is whether those non-
perturbative Φ-derivable approximations have the
same properties as the weak-coupling Φ-derivable ap-
proximations suggested by Baym and Kadanoff. This
requires the discussion of the following points:
(i) Macroscopic conservations laws. For fermionic lat-
tice models, conservation of energy, particle number and
spin have to be considered. Besides the static thermody-
namics, the SFT concept concentrates on the one-particle
excitations. For the approximate one-particle Green’s
function, however, it is actually simple to prove that the
above conservation laws are respected. A short discus-
sion is given in Appendix A.
(ii) Thermodynamical consistency. This issue has al-
ready been addressed in Ref. 11. It has been shown that
the µ derivative of the (approximate) SFT grand poten-
tial (including a minus sign) equals the average particle
number 〈N〉 as obtained by the trace of the (approxi-
mate) Green’s function. The same holds for any one-
particle quantity coupling linearly via a parameter to the
Hamiltonian, e.g. for the average total spin 〈S〉 coupling
via a field of strength B.
(iii) Luttinger sum rule. This is the main point to
be discussed in the present paper. There are different
open questions: First, it is straightforward to prove that
weak-coupling Φ-derivable approximations respect the
sum rule as one can directly take over the proof for the
exact theory. For approximations constructed within the
SFT, a different proof has to be given. Second, it turns
out that a non-perturbative Φ-derivable approximation
respects the sum rule if and only if the sum rule holds
for the reference system that is used within the SFT.
As the original and thereby the related reference system
may be studied in the strong-coupling regime, this raises
the question which reference system does respect the sum
rule, i.e. which approximation is consistent with the sum
rule. Third, it will be particularly interesting to study
reference systems which generate dynamical impurity ap-
proximations (DIA)8,9 and variational cluster approxi-
mations (VCA),10,12 as these consist of a finite number of
degrees of freedom. Does the Luttinger sum rule hold for
finite systems? Do the DIA and the VCA respect the sum
rule? What is the simplest approximation consistent with
the sum rule? Note that finite reference systems consist-
ing of a few sites only have been shown9,13,14,15,16,17,18,19
to generate approximations which qualitatively capture
the main physics correctly. Finally, it is important to un-
derstand these issues in order to understand whether and
how a violation of the sum rule is possible within cluster
extensions20,21,22,23 of the dynamical mean-field theory
(DMFT).24,25,26,27,28 Note that the SFT comprises the
DMFT and certain29 cluster extensions and that possi-
ble violations of the sum rule in the two-dimensional lat-
tice models have been reported,30,31,32 including a study
using the dynamical cluster approximation (DCA).33
The paper is organized as follows: A brief general dis-
cussion of the Luttinger sum rule is given in the next
section, and a form of the sum rule specific to systems
with a finite number of spatial degrees of freedom is
derived. Sec. III clarifies the status of the sum rule
with respect to non-perturbative approximations gener-
ated within the SFT framework. The results are elu-
cidated by several numerical examples obtained for the
most simple but non-trivial non-perturbative conserving
approximation in Sec. IV. Violations of the sum rule in fi-
nite systems and their consequences are discussed in Sec.
V. Finally, Sec. VI summarizes our main conclusions.
II. LUTTINGER SUM RULE
A system of interacting electrons on a lattice is gen-
erally described by a Hamiltonian H(t,U) = H0(t) +
H1(U) consisting of a one-particle part H0(t) and an
interaction H1(U) with one-particle and interaction pa-
rameters t and U , respectively. As a prototype, let
us consider the single-band Hubbard model34,35,36 on a
translationally invariant D dimensional lattice consist-
ing of L sites with periodic boundary conditions. The
Hamiltonian is given by:
iσcjσ +
niσni−σ . (1)
Here, i = 1, ..., L refers to the sites, σ =↑, ↓ is the spin
projection, ciσ (c
iσ) annihilates (creates) an electron in
the one-electron state |iσ〉, and niσ = c
ciσ. Fourier
transformation diagonalizes the hopping matrix t and
yields the dispersion ε(k). There are L allowed k points
in the first Brillouin zone.
Let G = Gt,U denote the one-electron Green’s func-
tion of the model H(t,U). In case of the Hubbard model,
its elements are given by Gij(ω) = 〈〈ciσ ; c
jσ〉〉ω . In the
absence of spontaneous symmetry breaking, the Green’s
function is spin-independent and diagonal in reciprocal
space. It can be written as Gk(ω) = 1/(ω + µ − ε(k) −
Σk(ω)) where µ is the chemical potential and Σk(ω) the
self-energy. We also introduce the notation Σt,U for
the self-energy, and Gt,0 = 1/(ω + µ − t) for the free
(non-interacting) Green’s function which exhibits the de-
pendence on the model parameters but suppresses the
frequency dependence. Dyson’s equation then reads as
Gt,U = 1/(G
t,0 −Σt,U ).
The Luttinger sum rule5,6 states that
〈N〉 = 2
Θ(Gk(0)) (2)
where N =
niσ is the particle-number operator,
〈N〉 its (T = 0) expectation value, and Θ the Heavy-
side step function. The factor 2 accounts for the two
spin directions. Since Gk(0)
−1 = µ − ε(k) − Σk(0), the
sum gives the number of k points enclosed by the in-
teracting Fermi surface which, for L → ∞, is defined via
µ−ε(k)−Σk(0) = 0. In the thermodynamic limit the sum
rule therefore equates the average particle number with
the Fermi-surface volume (apart from a factor (2π)D/L).
Note that, as Θ(Gk(0)) = Θ(1/Gk(0)), the sum rule Eq.
(2) also includes the so-called Luttinger volume37 which
(for L → ∞) is enclosed by the zeros of Gk(0).
The standard proof of the sum rule can be found in
Ref. 5. It is based on diagrammatic perturbation theory
to all orders which is used to construct the Luttinger-
Ward functional ΦU [G] as the sum of renormalized closed
skeleton diagrams (see Fig. 1). We emphasize that the
original proof straightforwardly extends also to finite sys-
tems. For L < ∞ the sum in Eq. (2) is discrete. Actually,
the proof is performed for finite L first, and the thermo-
dynamic limit (if desired) can be taken in the end. The
limit T → 0, on the other hand, is essential and is re-
sponsible for possible violations of the sum rule (see Sec.
Below we need an alternative but equivalent formu-
lation of the sum rule. We start from the following
(Lehmann) representation for the Green’s function:
Gk(ω) =
αm(k)
ω + µ− ωm(k)
. (3)
Here, ωm(k)−µ are the (real) poles and αm(k) the (real
and positive) weights. For real frequencies ω, it is then
easy to verify the identity:
Θ(Gk(ω)) =
Θ(ω+µ−ωm(k))−
Θ(ω+µ−ζn(k))
where ζn(k) − µ is the n-th (real) zero of the Green’s
function, i.e. Gk(ζn(k)− µ) = 0.
For temperature T = 0 we have 〈N〉 =
dω(−1/π)ImGk(ω + i0
+) and thus 〈N〉 =
αm(k)Θ(µ−ωm(k)). Hence, the Luttinger sum
rule reads:
αm(k)Θ(µ− ωm(k))
Θ(µ− ωm(k))−
Θ(µ− ζn(k))
This form of the sum rule is convenient for the discussion
of finite systems with L < ∞.
III. SELF-ENERGY-FUNCTIONAL THEORY
AND LUTTINGER SUM RULE
Within the self-energy-functional theory (SFT),8,9,10
the grand potential Ω is considered as a functional of the
self-energy:
Ωt,U [Σ] = Tr ln
t,0 −Σ
+ FU [Σ] . (6)
Here, the trace Tr of a quantity A is defined as TrA ≡
eiωn0
Ak(iωn) where iωn = i(2n+ 1)πT are
the fermionic Matsubara frequencies, and the functional
FU [Σ] is the Legendre transform of the Luttinger-Ward
functional ΦU [G]. The self-energy functional (6) is sta-
tionary at the physical self-energy, δΩt,U [Σt,U ]/δΣ = 0,
and, if evaluated at the physical self-energy, yields the
physical value for the grand potential: Ωt,U [Σt,U ] =
Ωt,U ≡ −T ln tr exp(−β(H(t,U)−µN)) where β = 1/T .
Comparing with the self-energy functional
Ωt′,U [Σ] = Tr ln
t′,0 −Σ
+ FU [Σ] (7)
of a reference system with the same interaction but a
modified one-particle part, i.e. with the Hamiltonian
H(t′,U), the not explicitly known but only U -dependent
functional FU [Σ] can be eliminated:
Ωt,U [Σ] = Ωt′,U [Σ] + Tr ln
t,0 −Σ
− Tr ln
t′,0 −Σ
An approximation is constructed by searching for a sta-
tionary point of the self-energy functional on the sub-
space of trial self-energies spanned by varying the one-
particle parameters t′:
∂Ωt,U [Σt′,U ]
= 0 . (9)
Inserting a trial self-energy into Eq. (8) yields
Ωt,U [Σt′,U ] = Ωt′,U +Tr ln
t,0 −Σt′,U
− Tr lnGt′,U .
The decisive point is that the r.h.s. can be evaluated ex-
actly for a reference system which is exactly solvable.
Apart from the free Green’s function Gt,0, it involves
quantities of the reference system only.
This strategy to generate approximations has several
advantages: (i) Contrary to the usual conserving approxi-
mations, the exact functional form of Ωt,U [Σ] is retained.
Any approximation is therefore non-perturbative by con-
struction. On the level of one-particle excitations, macro-
scopic conservation laws are respected as shown in Ap-
pendix A. (ii) With Ωt,U [Σt′,U ] evaluated at the station-
ary point t′ = t′s, an approximate but explicit expression
for a thermodynamical potential is provided. As all phys-
ical quantities derive from this potential, the approxima-
tion is thermodynamically consistent in itself (see Ref.
11 for details). (iii) As different reference systems gen-
erate different approximations, the SFT provides a uni-
fying framework that systematizes a class of “dynamic”
approximations (see Refs. 29,38 for a discussion).
In the following we discuss the question whether or
not a dynamic approximation respects the Luttinger sum
rule. For this purpose consider first the Tr ln(· · · ) terms
in Eq. (10). These can be evaluated using the ana-
lytical and causal properties of the Green’s functions
as described in Ref. 9 (see Eq. (4) therein). Using
−T ln(1 + exp(−ω/T )) → ωΘ(−ω) for T → 0 yields:
Tr ln
t,0 −Σt′,U
(ωm(k)− µ)Θ(µ− ωm(k))
(ζn(k)− µ)Θ(µ− ζn(k)) . (11)
Analogously, we have
Tr lnGt′,U = 2
(ω′m(k)− µ)Θ(µ− ω
m(k))
(ζn(k)− µ)Θ(µ− ζn(k)) .
Note that the reference system is always assumed to be
in the same macroscopic state as the original system,
i.e. it is considered at the same temperature and, more
importantly here, at the same chemical potential µ. Fur-
thermore, it has been used that, by construction of the
approximation, the self-energy and hence its poles at
ζn(k) − µ are the same for both, the original and the
reference system. This implies that the second terms on
the r.h.s. of Eq. (11) and (12), respectively, cancel each
other in Eq. (10). Finally, a (large but) finite system
(L < ∞) and a finite reference system are considered.
Hence, the set of poles of the Green’s function and of the
self-energy as well as sums over k are discrete and finite.
Taking the µ derivative on both sides of Eq. (10) then
yields:
∂Ωt,U [Σt′,U ]
∂Ωt′,U
Θ(µ− ωm(k))
Θ(µ− ω′m(k)) . (13)
Here we have assumed the ground state of the refer-
ence system to be non-degenerate with respect to the
particle number. From the (zero-temperature) Lehmann
representation39 it is then obvious that, within a sub-
space of fixed particle number, the µ-dependence of the
Green’s function is the same as its ω-dependence, i.e.
G(ω) = G̃(ω+µ) with a µ-independent function G̃. Via
the Dyson equation of the reference system, this prop-
erty can also be inferred for the self-energy and, via the
Dyson equation of the original system, for the (approx-
imate) Green’s function of the original system. Conse-
quently, the poles of (G−1
t,0 −Σt′,U )
−1 and of Gt′,U are
linearly dependent on µ, i.e. ωm(k) and ω
m(k) in Eqs.
(11) and (12) are independent of µ.
We once more exploit the fact that the self-energy of
the original system is identified with the self-energy of the
reference system. Using Eq. (4) one immediately arrives
〈N〉 = 〈N〉′ + 2
Θ(Gk(0))− 2
(0)) . (14)
This is the final result: The Luttinger sum rule for
the original system, Eq. (2), is satisfied if and only if
it is satisfied for the reference system, i.e. if 〈N〉′ =
(0)).
A few remarks are in order. For the reference sys-
tem, the status of the Luttinger sum rule is that of a
general theorem (as long as the general proof is valid);
〈N〉′ and G′
(0) represent exact quantities. The above
derivation shows that the theorem is “propagated” to the
original system irrespective of the approximation that is
constructed within the SFT. This propagation also works
in the opposite direction. Namely, a possible violation
of the exact sum rule for the reference system would im-
ply a violation of the sum rule, expressed in terms of
approximate quantities, for the original system.
Eq. (14) holds for any choice of t′. Note, however, that
stationarity with respect to the variational parameters t′
is essential for the thermodynamical consistency of the
approximation. In particular, consistency means that the
average particle number 〈N〉 = −∂Ωt,U [Σt′,U ]/∂µ on the
l.h.s. can be obtained as the trace of the Green’s function.
Stationarity is thus necessary to get the sum rule in the
form (5).
There are no problems to take the thermodynamic
limit (if desired) on both sides of Eq. (14) (after divi-
sion of both sides by the number of sites L). The k sums
turn into integrals over the unit cell of the reciprocal lat-
tice. For a D-dimensional lattice the D − 1-dimensional
manifolds of k points with Gk(0) = ∞ or Gk(0) = 0 form
Fermi or Luttinger surfaces, respectively.
For the above derivation, translational symmetry has
been assumed for both, the original as well as the refer-
ence system. Nothing, however, prevents us from repeat-
ing the derivation in case of systems with reduced (or
completely absent) translational symmetries. One sim-
ply has to re-interprete the wave vector k as an index
which, combined with m, refers to the elements of the
diagonalized Green’s function matrix G. The exact sum
rule, Eq. (5), generalizes accordingly. The result (14) re-
mains valid (with the correct interpretation of k) for an
original system with reduced translational symmetries.
It is also valid for the case of a translationally symmet-
ric original Hamiltonian where, due to the choice of a
reference system with reduced translational symmetries,
the symmetries of the (approximate) Green’s function of
the original system are (artificially) reduced. A typical
example is the variational cluster approximation (VCA)
where the reference system consists of isolated clusters of
finite size.
IV. TWO-SITE DYNAMICAL-IMPURITY
APPROXIMATION
While the Hartree-Fock approximation may be con-
sidered as the most simple weak-coupling Φ-derivable
approximation, the most simple non-perturbative Φ-
derivable approximation is given by the dynamical-
impurity approximation (DIA). This shall be demon-
strated in the following for the single-band Hubbard
model (1) as the original system to be investigated. The
DIA is generated by a reference system consisting of a
decoupled set of single-impurity Anderson models with
a finite number of sites ns and is known
8 to recover
the dynamical mean-field theory in the limit ns → ∞.
As long as the Luttinger sum rule holds for the single-
impurity reference system, the DIA must yield a one-
particle Green’s function and a self-energy respecting the
sum rule.
The Hamiltonian of the reference system is H(t′,U) =∑L
i=1 H
i with
H ′i =
iσciσ +
niσni−σ
aikσ +
ciσ + h.c.) .
For a homogeneous phase, the variational parameters
′ = ({ε
0 , ε
}) can be assumed to be indepen-
dent of the site index i: ε0 ≡ ε
0 , εk ≡ ε
, Vk ≡ V
For the sake of simplicity, we consider the two-site DIA
(ns = 2), i.e. a single bath site per correlated site only.
In this case there are three independent variational pa-
rameters only: the on-site energies of the correlated and
of the bath site, ε0 and εc ≡ εk=2, respectively, as well as
the hybridization strength V ≡ Vk=2. As the reference
0 0.2 0.4 0.6 0.8 1
filling  n
U=W=4
FIG. 2: Filling dependence of the variational parameters
at their respective optimized values and of the chemical po-
tential. Calculations for the Hubbard model with a semi-
elliptical free density of states of band width W = 4 and
interaction strength U = W = 4 using the two-site DIA.
system consists of replicated identical impurity models
which are spatially decoupled, the trial self-energy is lo-
cal and site-independent, Σij(ω) = δijΣ(ω).
Calculations have been performed for the Hub-
bard model with a one-particle dispersion ε(k) =
e−ik(Ri−Rj)tij such that the density of one-
particle energies D(ε) is semi-elliptic. For |ε| ≤ W/2,
D(ε) =
δ(ε− ε(k)) =
(W/2)2 − ε2 . (16)
The free band width is set to W = 4. This serves as the
energy scale.
The computation of the SFT grand potential is per-
formed as described in Ref. 9. Stationary points of the
resulting function Ω(ε0, εc, V ) ≡ Ωt,U [Σε0,εc,V ] are ob-
tained via iterated linearizations of its gradient. There
is a unique non-trivial stationary point (with V 6= 0).
Fig. 2 shows the variational parameters at this point as
functions of the filling n. For the entire range of fillings,
the ground state of the reference system lies in the in-
variant subspace with Ntot =
iσciσ + a
iσaiσ) = 2.
The parameters as well as the chemical potential are
smooth functions of n. We have checked that the ther-
modynamical consistency condition n = −L−1∂Ω/∂µ =∫ 0
ρ(ω)dω is satisfied within numerical accuracy. Here
ρ(ω) = D(ω + µ− Σ(ω)) (17)
is the interacting local density of states (DOS).
At half-filling the values of the optimized on-site en-
ergies are consistent with particle-hole symmetry. With
ε0 − µ = −U/2 and εc − µ = 0 the reference system is
in the Kondo regime with a well-formed local moment at
the correlated site. The finite hybridization strength V
leads, for U = W , to a finite DOS ρ(ω = 0) > 0 and thus
0 0.2 0.4 0.6 0.8 1
filling  n
2S-DMFT
two-site DIA
FIG. 3: Quasi-particle weight z as a function of the filling
within the two-site DIA (full lines) and the two-site DMFT40
(dashed lines). Calculations for U = W and U = 2W .
to a metallic Fermi liquid as it is expected for the Hub-
bard model within a (dynamical) mean-field description.
Due to the simple structure of the self-energy generated
by the two-site reference system, however, quasi-particle
damping effects are missing.
Decreasing the filling from n = 1 to n = 0 drives the
reference system more and more out of the Kondo regime.
While εc stays close to the chemical potential, the on-site
energy of the correlated site ε0 crosses µ close to quar-
ter filling and lies above µ eventually. Note that ε0 = 0
within the DMFT, i.e. for ns → ∞, while for finite ns
there is a clear deviation from ε0 = 0 which is necessary
to ensure thermodynamical consistency. For fillings very
close to n = 0, the grand potential Ωt,U [Σε0,εc,V ] be-
comes almost independent of Σ. This implies that it be-
comes increasingly difficult to locate the stationary point
with the numerical algorithm used. The slight upturn of
ε0 below n = 0.01 (see Fig. 2) might be a numerical ar-
tifact.
It is instructive to compare the parameters with those
of the two-site DMFT (2S-DMFT).40 The 2S-DMFT is
a simplified version of the DMFT where a mapping onto
the two-site single impurity Anderson model is achieved
by means of a simplified self-consistency equation. As-
suming ε0 = 0 as in the full DMFT, there are two pa-
rameters left (εc and V ) which are fixed by considering
the first non-trivial order in the low- and in the high-
frequency expansion of the self-energy and the Green’s
function in the DMFT self-consistency equation. Al-
though being well motivated, this approximation is es-
sentially ad hoc. One therefore has to expect that the 2S-
DMFT is thermodynamically inconsistent and exhibits a
violation of Luttinger’s sum rule. A comparison of the
DIA for ns = 2 with the 2S-DMFT is thus ideally suited
to demonstrate the advantages gained by constructing
approximations within the variational framework of the
First of all, there are differences in fact. At half-filling
the 2S-DMFT predicts the hybridization to be somewhat
larger than the two-site DIA while the value for εc is
again fixed by particle-hole symmetry. Deviations grow
with decreasing filling. Contrary to the two-site DIA, V
monotonously increases and is larger in the entire filling
range, ε0 = 0 by construction, and εc even diverges for
n → 0 within the 2S-DMFT (see Ref. 40). On the other
hand, the system is essentially uncorrelated in the limit
n → 0. Strong differences in the parameters, which en-
ter the self-energy only, therefore do not necessarily im-
ply strongly different physical quantities. This is demon-
strated by Fig. 3 which shows the quasi-particle weight
calculated via
dΣ(ω = 0)
as a function of the filling. While there are obvious differ-
ences when comparing the results from the two-site DIA
with those of the 2S-DMFT, the qualitative trend of z is
very similar in both approximations. Both approxima-
tions also compare well with the full DMFT: There is a
quadratic behavior of z(n) for n → 1 in the Fermi-liquid
phase (U = W ) and a linear trend when approaching the
Mott phase (U = 2W ). The critical interaction strength
for the Mott transition is found to be Uc ≈ 1.46W for
the two-site DIA and Uc = 1.5W within the 2S-DMFT.
For details on the Mott transition see Refs. 9,40.
In case of a local and site-independent self-energy, the
Luttinger sum rule can be written in the form41
µ = µ0 +Σ(ω = 0) , (19)
where µ0 is the chemical potential of the free (U = 0)
system at the same particle density. Eq. (19) implies that
not only the enclosed volume but also the shape of the
Fermi surface remains unchanged when switching on the
interaction. Using Eq. (17) this immediately implies41
ρ(0) = D(µ0) = ρ0(0) , (20)
i.e., in case of a correlated metal, the value of the inter-
acting local density of states at ω = 0 is independent of
U and thus fixed to the value of the density of states of
the non-interacting system at the same filling.
The interacting and the non-interacting DOS are plot-
ted in Fig. 4 for different fillings and for U = W and
U = 2W . The impurity self-energy of the two-site ref-
erence system is an analytical function of ω except for
two first-order poles on the real axis. Via Eq. (17) this
two-pole structure implies that the DOS consists of three
peaks the form of which is essentially given by the non-
interacting DOS. At half-filling the three peaks are eas-
ily identified as the lower and the upper Hubbard band
and the quasi-particle resonance as it is characteristic
for a (dynamical) mean-field description.27 For U = W
the resonance still has a significant weight. The weight
decreases upon approaching the critical interaction, and
the resonance has disappeared in the Mott insulator for
-4 -2 0 2 4 6
-6 -4 -2 0 2 4 6 8 10
n=1.0
n=0.75
n=0.5
n=0.25
n=0.0
FIG. 4: Interacting local density of states ρ(ω) (solid lines)
for different fillings as indicated. Calculations using the two-
site DIA for U = W (left) and U = 2W (right). For n = 0.25,
n = 0.5 and n = 0.75 the non-interacting DOS ρ0(ω) is shown
for comparison (dashed lines). Note that ρ(0) = ρ0(0). The
dotted line for U = 2W in the top panel is the DOS for
n = 0.99.
U = 2W . Hole doping of the Mott insulator is accom-
plished by the reappearance of the resonance at ω = 0
which preempts the creation of holes in the lower Hub-
bard band.42 As can be seen in the spectrum for n = 0.99
in the top panel (dotted line), the quasi-particle res-
onance appears within the Mott-Hubbard gap. With
decreasing filling, the upper Hubbard band gradually
shifts to higher excitation energies and loses weight. This
weight is transfered to the low-energy part of the spec-
trum. For lower fillings where the Kondo regime has been
left, one would actually expect that the quasi-particle res-
onance disappears by merging with the lower Hubbard
band. This, however, cannot be described with the sim-
ple two-pole structure of the self-energy. One therefore
should interprete the gap around ω = −1 at n = 0.25
as an artifact of the approximation. Furthermore, the
widths of the Hubbard bands is considerably underes-
timated as damping effects are missing completely. The
filling-dependent spectral-weight transfer across the Hub-
bard gap as well as the energy positions of the main
peaks, however, are in overall agreement with general
expectations.34,43
It is worth emphasizing that this simple two-site
0 0.2 0.4 0.6 0.8 1
-0.02
filling  n
Hubbard-I
2S-DMFT
two-site DIA
FIG. 5: Numerical results for the difference between the
volume enclosed by the Fermi surface VFS and the filling n as
a function of n for U = W = 4. The Luttinger sum rule (VFS−
n = 0) is exactly respected by the two-site DIA. Results for
the 2S-DMFT and the Hubbard-I approximation are shown
for comparison. Dashed line: difference between the filling n
and the average occupation of the correlated (impurity) site
in the reference system at stationarity for the two-site DIA.
dynamical-impurity approximation exactly fulfills the
Luttinger sum rule. In Fig. 4 this can be seen by compar-
ing with the DOS of the non-interacting system (dashed
lines). The non-interacting DOS cuts the interacting one
at ω = 0 which shows that Eq. (20) is satisfied. Note
that this is trivial for n = 1 as this is already enforced
by particle-hole symmetry. Off half-filling, however, the
pinning of the DOS to its non-interacting value at ω = 0
is a consequence of Φ-derivability and thereby a highly
non-trivial feature.
In contrast, the 2S-DMFT does show a violation of
Luttinger’s sum rule which, however, must be attributed
to the ad hoc nature of the approximation. Fig. 5 shows
0 0.2 0.4 0.6 0.8 1
filling  n
2S-DMFT
two-site DIA
FIG. 6: Filling dependence of the compressibility κ for U =
W as obtained within the 2S-DMFT via κ = ∂n/∂µ (solid
line) and via a general Fermi-liquid relation (Eq. (22), dashed
line). Using the two-site DIA identical results are obtained
for both cases.
the difference between the volume enclosed by the Fermi
surface
VFS =
Θ(µ−ε(k)−Σ(0)) = 2
dεD(ε+µ−Σ(0))
and the filling n as a function of the filling. As can be
seen, there is an artificial violation of the sum rule for the
2S-DMFT which is of the order of a few per cent while
for the Φ-derivable two-site DIA the sum rule is fully re-
spected. Note that, unlike the DMFT and also unlike
the simplified 2S-DMFT, the two-site DIA predicts a fill-
ing which slightly differs from the average occupation of
the correlated impurity site in the reference system (see
dashed line in Fig. 5). For a finite number of bath sites ns
this appears to be necessary to fulfill the Luttinger sum
rule. The figure also shows the result obtained within
the Hubbard-I approximation.34 Here a very strong (ar-
tificial) violation of up to 100 % (for n close to half-filling)
is obtained. This should be considered as a strong draw-
back which is typical for uncontrolled mean-field approx-
imations.
There are more relations which, analogously to the
Luttinger sum rule, can be derived by means of per-
turbation theory to all orders6 in the exact theory and
which are respected by weak-coupling conserving approx-
imations. For example, the compressibility, defined as
κ = ∂n/∂µ, can be shown to be related to the interact-
ing DOS and the self-energy at the Fermi edge via
κ = 2ρ(0)
∂Σ(0)
. (22)
Fig. 6 shows that for the 2S-DMFT it makes a difference
whether κ is calculated as the µ-derivative of the filling
or via Eq. (22). Again, this must be attributed to the
fact that the 2S-DMFT is not a Φ-derivable approxima-
tion. Contrary, the two-site DIA does respect the general
Fermi-liquid property (22) and thus yields the same re-
sult in both cases (see Fig. 6).
V. VIOLATION OF LUTTINGER’S SUM RULE
IN FINITE SYSTEMS
The preceding section has demonstrated that the two-
site DIA satisfies the Luttinger sum rule. According to
Eq. (14), we can conclude that the Luttinger sum rule
must hold for the corresponding reference system, i.e. for
the two-site single-impurity Anderson model. Of course,
this can be verified more directly by evaluating Eq. (5).
In case of a finite system or a system with reduced trans-
lational symmetries, the Green’s function is a matrix with
elements Gαβ(ω) where α refers to the one-particle basis
states, and the Luttinger sum rule reads:
α(k)m Θ(µ−ω
m ) =
Θ(µ−ω(k)m )−
Θ(µ− ζ(k)n ) .
Here the index k labels the elements of the diagonalized
Green’s function, i.e. Eq. (5) is generalized by replacing
(k, σ) → k. In case of an impurity model, Eq. (23) ac-
tually represents the Friedel sum rule.44,45 For the two-
site single-impurity Anderson model, the different one-
particle excitation energies ω
m − µ, the zeros of the
Green’s function ζ
n − µ and the weights α
m are eas-
ily determined by full diagonalization. We find that Eq.
(23) is satisfied in the entire parameter space (except for
V = 0, see below).
Note that a violation of the sum rule occurs when, as
a function of a model parameter x, a zero of the Green’s
function crosses ω = 0 for x = xc. At xc the number of
negative zeros counted by the second term on the r.h.s.
changes by one while the first term as well as the l.h.s.
remain constant since (unlike a pole) a zero of the Green’s
function is generically not connected with a change of the
ground state (level crossing). This implies that the sum
rule would be violated for x < xc or for x > xc.
The case V = 0 is exceptional. Within the two-site
DIA this corresponds to the Mott insulator (see Fig. 4,
topmost panel for U = 2W ). For V = 0 the reference
system consists of two decoupled sites, and the Green’s
function becomes diagonal in the site index. There is
no zero of the local Green’s function corresponding to
the uncorrelated site. We can thus concentrate on the
correlated site where the local Green’s function exhibits
a zero at η−µ = ε0+U/2. In the sector with one electron
at the correlated site (ε0 < µ < ε0 + U), the second
term on the r.h.s. changes by two at µ = µc = ε0 + U/2
because of the two-fold degenerate ground state. In this
case Luttinger’s sum rule in the form (23) is violated for
µ < µc and for µ > µc. This “violation”, however, is a
trivial one which immediately disappears if the ground-
state degeneracy is lifted by applying a weak field term,
for example.
Fig. 7 shows a phase diagram of the single-impurity
Anderson model with ns = 4 sites as obtained by
full diagonalization. The diagram covers the entire
range of the total particle number N =
〈c†σcσ〉 +∑
k=2〈a
akσ〉 from N = 0 to N = 2ns = 8. A non-
degenerate ground state is enforced by applying a small
but finite magnetic field. No violation of the Luttinger
sum rule is found. We have repeated the same calculation
also for ns = 10 using the Lanczos technique.
46 Again,
the sum rule is found to be always satisfied (We have
performed calculations for different U and bath parame-
ters). This might have been expected as the (ns → ∞)
Anderson model can generally be classified as a (local)
Fermi liquid.47
The situation is less clear in the case of correlated lat-
tice models such as the Hubbard or the t-J model. For
two dimensions there are several numerical studies using
high-temperature expansion,30 quantum Monte-Carlo,31
extended DMFT,32,48 and dynamical cluster approxima-
tion (DCA)33 which indicate a violation in the strongly
correlated metallic phase close to half-filling. For studies
of large clusters or studies directly working in the thermo-
-1 0 1 2 3
SIAM n  =4s
FIG. 7: Phase diagram µ vs. ε of the single-impurity An-
derson model with ns = 4 sites. Total particle numbers are
indicated by Roman figures. Results have been obtained by
full diagonalization for the following model parameters. One-
particle energies: ε0 = 0 (correlated site), εk = ε + (k − 3)
with k = 2, 3, 4 (uncorrelated bath sites). Hubbard interac-
tion: U = 2ε. Hybridization strength: Vk = 0.1 for k = 2, 3, 4.
To lift Kramers degeneracy in case of an odd particle number,
a weak (ferromagnetic) field of strength b = 0.001 is coupled
to the local spins. The dashed line marks the particle-hole
symmetric case. Luttinger’s sum rule is found to be satisfied
in the entire parameter space.
dynamic limit, a definite conclusion on the validity of the
sum rule is difficult to obtain as finite-temperature or ar-
tificial broadening effects etc. must be controlled numer-
ically. Contrary, full diagonalization of Hubbard clusters
consisting of a few sites only can provide exact results.
While their direct relevance for the thermodynamic limit
is less clear, it is important to note that reference sys-
tems with a finite number of sites or a finite number of
correlated sites provide the basis for a number of cluster
approaches within the SFT framework. Via Eq. (14) their
properties are transferred to the approximate treatment
of lattice models in the thermodynamic limit.
The validity of Eq. (23) has been checked for Hubbard
clusters of different size and in different geometries. The
µ vs. U phase diagram for an L = 4-site open Hubbard
chain with nearest-neighbor hopping in Fig. 8 shows a
representative example. Again, a small but finite field
term is added to avoid a ground-state degeneracy. As the
chemical potential, for fixed U , is moved off the particle-
hole symmetric point µ = U/2 and exceeds certain criti-
cal values (red lines), the particle number N [as obtained
from the l.h.s. of Eq. (23)] changes from N = L down to
(up to) N = 0 (N = 2L). A critical µ value indicates a
change of the ground state (level crossing) that is accom-
panied by a change of the ground-state particle number.
In the one-particle Green’s function this is characterized
by a pole ω
m −µ crossing ω = 0. The blue lines indicate
-5 0 5 10 15
Hubbard L=4
FIG. 8: Phase diagram µ vs. U of the Hubbard model with
L = 4 sites (open chain) as obtained by full diagonalization.
Nearest-neighbor hopping t = −1. A weak (ferromagnetic)
field of strength b = 0.01 is applied to lift Kramers degener-
acy. The dashed line marks the particle-hole symmetric case.
Particle numbers (l.h.s. of Eq. (23)) are indicated by Roman
figures. R.h.s. of Eq. (23): Arabic figures. Luttinger’s sum
rule is found to be violated for sufficiently strong U . Uc1, Uc,2:
critical interactions.
those chemical potentials at which a zero of the Green’s
function ζ
n − µ crosses ω = 0. Whenever this happens
the r.h.s. of Eq. (23) changes while the l.h.s. is constant.
Fig. 8 shows that this occurs several times in the N = L
sector. At the particle-hole symmetric point µ = U/2
the Luttinger sum rule is obeyed while it is violated in a
wide region of the parameter space corresponding to half-
filling N = L. However, a critical interaction strength
Uc turns out to be necessary. The value for Uc strongly
varies for different cluster sizes and geometries but has
always been found to be positive and finite. Note that
for L = 4 the sum rule is fulfilled for any particle num-
ber N 6= L. Qualitatively similar results can be found for
the L = 2-site Hubbard cluster where calculations can be
done even analytically. Again, a violation of the sum rule
is found in the half-filled sector beyond a certain critical
This has already been noticed by Rosch49 and was used
in combination with a strong-coupling expansion to ar-
gue that a violation of the sum rule generically occurs
for a Mott insulator. Stanescu et al.50 have shown quite
generally that the sum rule is fulfilled when particle-hole
symmetry is present (the Luttinger surface is the same
as the Fermi surface of the non-interacting system) but
violated in the Mott insulator away from particle-hole
symmetry. It is interesting to note that these arguments
cannot be used to construct a violation of the sum rule
within DMFT or for a single-impurity Anderson model:
For an (almost) particle-hole symmetric case and model
parameters describing a Mott insulator (within DMFT),
-4 -2 0 2 4 6 8
Luttinger
sum rule
particle
number
U=16t
FIG. 9: Ground-state particle number (red Roman figures,
l.h.s. of Eq. (23)) and prediction by the Luttinger sum rule
(blue Arabic figures, r.h.s. of Eq. (23)) as functions of the
chemical potential for a L = 9-site Hubbard cluster with pe-
riodic boundary conditions. Arabic numbers are only given
when different from Roman ones. Calculations using the
Lanczos method and a finite but small magnetic field and
finite but small on-site potentials to lift ground-state degen-
eracies.
an odd number of sites ns (with ns → ∞) must be consid-
ered and thus a magnetic field is needed to lift Kramers
degeneracy. Even an infinitesimal field, however, leads
(at zero temperature) to a finite and even large polar-
ization corresponding a well-formed but unscreened local
moment. This polarization is incomplete for any finite
U as the DMFT predicts a small but finite double occu-
pancy for a Mott insulator. Still there is a proximity to
the fully polarized band insulator which finally results in
a weakly correlated state and thus in a situation which
is unlikely to show a violation of the sum rule.
We have also considered Hubbard clusters with L = 9
and L = 10 sites by using the Lanczos technique.46
Calculations have been performed for different Lanczos
depths lmax to ensure that the results are independent
of lmax. Fig. 9 displays an example for L = 9 and a
highly symmetric cluster geometry with periodic bound-
ary conditions and a well-defined reciprocal space. To lift
ground-state degeneracies resulting from spatial symme-
tries as well as the Kramers degeneracy, small but fi-
nite on-site potentials and a small magnetic-field term
are included in the cluster Hamiltonian. Fig. 10 shows
an example for L = 10 sites without any spatial sym-
metries. Kramers degeneracy for odd N is removed by
applying a small magnetic field. With the figures we com-
pare the expressions on the left-hand and the right-hand
side of Eq. (23). Obviously, the sum rule is respected in
-4 -2 0 2 4 6 8
Luttinger
sum rule
particle
number
U=16t
FIG. 10: The same as Fig. 9 but for 10 sites.
most cases. Violations are seen for half-filling N = L,
i.e. in the “Mott-insulating phase”, which is consistent
with Ref. 49. However, the sum rule is also violated
in the “metallic phase” close to half-filling, namely for
N = L − 1 (Fig. 9, L = 9) and N = L − 1, L − 2 (Fig.
10, L = 10). This nicely corresponds to the generally
observed trend30,31,32,33,48 for violations in the slightly
doped metallic regime. We have also verified that the
sum rule is restored by lowering U .
Fig. 9 and 10 demonstrate that the sum rule is violated
in the whole µ range corresponding to N = L − 1. This
is an important point as it shows that it is irrelevant
whether the T = 0 limit is approached by holding 〈N〉
fixed and adjusting µ = µ(T ) or by fixing µ and let 〈N〉 =
〈N〉(T ) be T -dependent. A violation of the sum rule is
found in both cases.
Kokalj and Prelovs̆ek51 have demonstrated that viola-
tions of the sum rule can also be found for the t-J model
on a finite number of sites. Our result provides an ex-
plicit example showing that not only for t-J51 but also
for Hubbard clusters a violation can be found when the
chemical potential is set to µ = limT→0 µ(T ) with µ(T )
obtained for given 〈N〉 = const. Anyway, the original
proof5 does not depend on this choice for µ but appears
to work for any µ.
The results raise the question which assumptions used
in the original proof of the theorem are violated or where
the proof breaks down. Note that the recently proposed
alternative topological proof52 assumes a Fermi-liquid
state from the very beginning and thus cannot be applied
to a finite system. Using weak symmetry-breaking fields,
a more or less trivial breakdown due to ground-state de-
generacy has been excluded. An analysis of the ground
state of the L = 2 and L = 4 Hubbard clusters which
are accessible with exact (analytical or numerical) meth-
ods has shown that, for model parameters where the sum
rule is violated, the interacting ground state can never-
theless be adiabatically connected to the non-interacting
one. This excludes level crossing as a potential cause for
the breakdown. While we cannot make a definite state-
ment, it appears at least plausible that the violation of
the sum rule results from a non-commutativity of two
limiting processes, the infinite skeleton-diagram expan-
sion and the limit T → 0.
Using a functional-integral formalism, the Luttinger-
Ward functional at finite T can also be constructed in a
non-perturbative way, i.e. avoiding an infinite summation
of diagrams, as has been shown recently.53 Formally, the
Luttinger sum rule can be obtained by exploiting a gauge
invariance of the Luttinger-Ward functional [see Ref. 53]:
∂(iωn)
ΦU [G(iωn)] = 0 . (24)
If at all, this invariance can only be shown for T = 0
where iωn becomes a continuous variable. Unfortunately,
the non-perturbative construction of ΦU requires a T > 0
formalism. Hence, the validity of the sum rule depends
on question whether the limit T → 0 commutes with the
frequency differentiation. Necessary and sufficient condi-
tions for this assumption are not easily worked out. An
understanding of the main reason for the possible break-
down of the sum rule in finite systems, very similar to
the case of Mott insulators, is therefore not yet available
(see also the discussion in Ref. 49).
VI. CONCLUSIONS
Φ-derivable approximations are conserving, thermody-
namically consistent and, for T = 0, formally respect cer-
tain non-trivial theorems such as the Luttinger sum rule.
As the construction of the Luttinger-Ward functional Φ
is by no means trivial and may conflict with the limit
T → 0 or different other limiting processes, however, the
validity of the sum rule may be questioned. Violations of
the sum rule can be found in fact for the case of strongly
correlated electron systems. For Mott insulators and fi-
nite systems in particular, a breakdown is documented
easily.
This implies that a general approximation for the
spectrum of one-particle excitations (of the one-particle
Green’s function) may violate the sum rule for two pos-
sible reasons, namely because (i) the sum rule is violated
in the exact theory, or (ii) the approximation generates
an artificial violation.
Within the usual weak-coupling conserving approxima-
tions, such as the fluctuation-exchange approximation,
the sum rule always holds as the formal steps in the gen-
eral proof of the sum rule can be carried over to the
approximation – but with the important simplification
of a limited class of diagrams. This also implies that
weak-coupling conserving approximations, when applied
beyond the weak-coupling regime, might erroneously pre-
dict the sum rule to hold.
The present paper has focussed on non-perturbative
conserving approximations. Non-perturbative approxi-
mations, constructed within the framework of the self-
energy-functional theory and referring to a certain ref-
erence system, are Φ-derivable and consequently respect
certain macroscopic conservation laws and are thermo-
dynamically consistent. Whether or not the sum rule
holds within the approximate approach, however, cannot
be answered generally. We found that Luttinger’s sum
rule holds within an (SFT) approximation if and only if
it holds exactly in the corresponding reference system.
The reference system that leads to the most simple
but non-trivial example for a non-perturbative conserv-
ing approximation consists of a single correlated and
a single bath site. For this two-site system, we have
found the sum rule to be valid in the entire parameter
space. Consequently, the resulting two-site dynamical-
impurity approximation (DIA) – opposed to more ad hoc
approaches like the two-site DMFT – fully respects the
sum rule as could be demonstrated in different ways. In
view of the simplicity of the approximation this is a re-
markable result. Since the sum rule dictates the low-
frequency behavior of the one-particle Green’s function,
important mean-field concepts, such as the emergence of
a quasi-particle resonance at the Fermi edge, are quali-
tatively captured correctly, even away from the particle-
hole symmetric case. This qualifies the two-site DIA for a
quick but rough estimate of mean-field physics, including
phases with spontaneously broken symmetries.
Full diagonalization and the Lanczos method have
been employed to show that also the single-impurity An-
derson model with a finite number of ns > 2 sites respects
the sum rule. Consequently, this property is transferred
to an ns-site DIA. For ns → ∞ the full dynamical mean-
field theory is recovered which is thereby recognized as
the prototypical non-perturbative conserving approxima-
tion. Clearly, in the case of the DMFT, Φ-derivability is
well known27 and obvious, for example, when construct-
ing the DMFT with the help of the skeleton-diagram ex-
pansion.
Using as a trial self-energy the self-energy of a cluster
with L > 1 correlated sites, generates an approximation
where short-range spatial correlations are included up to
the cluster extension. These variational cluster approxi-
mations provide a first step beyond the mean-field con-
cept. Again, whether or not the sum rule is respected
within the VCA depends on the reference system itself.
For the L = 2 Hubbard cluster, analytical calculations
straightforwardly show that violations of the sum rule
occur at half-filling, beyond a certain critical interaction
strength. In the thermodynamic limit, this would corre-
spond to the Mott-insulating regime. Applying the Lanc-
zos method to larger clusters, has shown, however, that
a breakdown of the Luttinger sum rule is also possible
for fillings off half-filling. For sufficiently strong U , the
sum rule is violated in the whole N = L − 1-particle
sector. This would correspond to a (strongly correlated)
metallic state in the thermodynamic limit. Whether or
not a VCA calculation is consistent with the sum rule,
then depends on the set of cluster hopping parameters t′
which make the self-energy functional stationary. First
VCA calculations54 for the D = 2 Hubbard model at low
doping and using clusters with up to L = 10 sites do
predict a violation in fact.
It is by no means clear a priori what happens in a clus-
ter approach using additional bath degrees of freedom as
variational parameters, as e.g. in the cellular DMFT.21,22
The usual periodization of the self-consistent C-DMFT
self-energy, however, should be avoided when testing the
sum rule as this introduces an additional (though physi-
cally motivated) approximation. Instead, Eq. (5) must be
used with k re-interpreted as an index referring to the el-
ements of the self-consistent diagonalized lattice Green’s
function.
Employing the dynamical cluster approximation
(DCA)20 represents an alternative which directly oper-
ates in reciprocal space. From a real-space perspective,
the DCA is equivalent with the cellular DMFT but ap-
plied to a modified model H = H(t,U) → H(t,U) with
modified hopping parameters which are invariant under
superlattice translations as well as under translations on
the cluster.29,55 In the limit L → ∞ the replacement
t → t becomes irrelevant. Analogous to the C-DMFT,
the sum rule then holds within the DCA if and only if
it holds for the individual cluster at self-consistently de-
termined cluster parameters. Note, however, that this
requires that (besides the DCA self-energy) the modified
hopping t instead of the physical hopping has to be con-
sidered in the computation of the volume enclosed by
the Fermi (Luttinger) surface of the lattice model. This
is exactly what is usually done in DCA calculations.
Within this context and in view of the violations found
for finite Hubbard clusters, it is possible to understand
why a non-perturbative cluster approximation, like the
VCA,54 or a cluster extension of the DMFT, like the
DCA,33 can produce results that are inconsistent with
Luttinger’s theorem.
Acknowledgments
We thank Robert Eder and Achim Rosch for valu-
able discussions. The work is supported by the Deutsche
Forschungsgemeinschaft within the Forschergruppe FOR
APPENDIX A: MACROSCOPIC CONSERVATION
OF ENERGY, PARTICLE NUMBER AND SPIN
The one-particle Green’s function as obtained within
an approximation generated by the choice of a reference
system respects the macroscopic conservation laws which
result from symmetries of the system with respect to con-
tinuous transformation groups:
Energy conservation is apparently respected as by con-
struction the approximate SFT Green’s function depends
on a single frequency only, i.e. is invariant under time
translations.
Conservation of the total particle number and spin is
respected if the approximate G transforms in the same
way as the exact Green’s function under global U(1) and
SU(2) gauge transformations. Consider a general trans-
formation of the form
c†α → c
with unitary S such that the interaction part H1(U)
of the Hamiltonian is invariant (α refers to the states
of the one-particle basis). In a diagrammatic approach,
the invariance of H1(U) implies that the corresponding
conservation law is respected “locally” at each vertex.
Hence, for a conserving approximation in the sense of
Baym and Kadanoff, the transformation behavior of the
free Green’s function is then propagated by the diagram
rules to the full Green’s function. Consequently, the lat-
ter must transform under S in the same way as the exact
G, i.e.
Gαβ → Gαβ =
. (A2)
Consider now the case of the SFT. One has to show
that the approximate Green’s function G for the trans-
formed system with Hamiltonian H is given by G =
† if G is the approximate Green’s function of the
model H . Applying the transformation (A1) to H , one
finds H = H0(t) +H1(U) → H = H0(t) +H1(U) with
t = StS†. Again, S is assumed to leave the interaction
part invariant.
The Green’s function G of the transformed model is
(approximately) constructed via
from the free Green’s function of the transformed model
and the SFT self-energy which is the self-energy of the
reference system H ′ = H0(t
s) +H1(U) at the stationary
point t
For the transformed problem H , the stationary point
s is determined from the SFT Euler equation:
)ω;αβ
= 0 .
As an ansatz to solve the Euler equation we take
= St′1S
† (A5)
with t′1 to be determined. The transformation law (A2)
for the exact Green’s function of the reference system
= GSt′
S†,U = SGt′1,US
†. This also holds for
the free Green’s function. Using the Dyson equation of
the reference system we can deduce Σ
= SΣt′
Furthermore, for the free Green’s function of the trans-
formed original model we have G
t,0 = SGt,0S
†. Using
these results, we see that Eq. (A4) is equivalent to
t,0 −Σt′1,U
∂(Σt′
,U )ω;αβ
= 0 .
But this is just the Euler equation for the original
model which is solved by t′1 = t
s. Remembering the
ansatz made, we now have for the stationary point
s = St
†. Inserting this into Eq. (A3) gives G =
S(G−1
t,0 − Σt′s,U )
† = SGS† which is the desired re-
sult.
Note that one has to ensure that the stationary point
for the transformed problem t
s = St
† lies within the
space of one-particle parameters characteristic for the ref-
erence system. For models with local interaction part
and for local (and also global) gauge transformations,
however, this is always easily satisfied.
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http://arxiv.org/abs/cond-mat/0407172
http://arxiv.org/abs/cond-mat/0602656

  Weak-coupling conserving approximations can be constructed by truncations of
the Luttinger-Ward functional and are well known as thermodynamically
consistent approaches which respect macroscopic conservation laws as well as
certain sum rules at zero temperature. These properties can also be shown for
variational approximations that are generated within the framework of the
self-energy-functional theory without a truncation of the diagram series.
Luttinger's sum rule represents an exception. We analyze the conditions under
which the sum rule holds within a non-perturbative conserving approximation.
Numerical examples are given for a simple but non-trivial dynamical two-site
approximation. The validity of the sum rule for finite Hubbard clusters and the
consequences for cluster extensions of the dynamical mean-field theory are
discussed.

Non-perturbative conserving approximations and Luttinger’s sum rule
Jutta Ortloff, Matthias Balzer, Michael Potthoff
Institut für Theoretische Physik und Astrophysik,
Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Weak-coupling conserving approximations can be constructed by truncations of the Luttinger-
Ward functional and are well known as thermodynamically consistent approaches which respect
macroscopic conservation laws as well as certain sum rules at zero temperature. These properties
can also be shown for variational approximations that are generated within the framework of the self-
energy-functional theory without a truncation of the diagram series. Luttinger’s sum rule represents
an exception. We analyze the conditions under which the sum rule holds within a non-perturbative
conserving approximation. Numerical examples are given for a simple but non-trivial dynamical two-
site approximation. The validity of the sum rule for finite Hubbard clusters and the consequences
for cluster extensions of the dynamical mean-field theory are discussed.
PACS numbers: 71.10.-w, 71.10.Fd
I. INTRODUCTION
Continuous symmetries of a Hamiltonian imply the
existence of conserved quantities: The conservation of
total energy, momentum, angular momentum, spin and
particle number is enforced by a not explicitly time-
dependent Hamiltonian which is spatially homogeneous
and isotropic and invariant under global SU(2) and U(1)
gauge transformations. For the treatment of a macro-
scopically large quantum system of interacting fermions,
approximations are inevitable in general. Approxima-
tions, however, may artificially break symmetries and
thus lead to unphysical violations of conservations laws.
Baym and Kadanoff1,2 have analyzed under which cir-
cumstances an approximation for time-dependent corre-
lation functions, and for one- and two-particle Green’s
functions in particular, respect the mentioned macro-
scopic conservation laws. They were able to give cor-
responding rules for a proper construction of approxima-
tions, namely criteria for selecting suitable classes of dia-
grams, within diagrammatic weak-coupling perturbation
theory. Weak-coupling approximations following these
rules and thus respecting conservation laws are called
“conserving”. Frequently cited examples for conserving
approximations are the Hartree-Fock or the fluctuation-
exchange approximation.1,3,4
Baym2 has condensed the method of constructing con-
serving approximations into a compact form: A con-
serving approximation for the one-particle Green’s func-
tion G is obtained by using Dyson’s equation G =
1/(G−10 − Σ) with (the free, U = 0, Green’s function
G0 and) a self-energy Σ = ΣU [G] given by a univer-
sal functional. Apart from G, the universal functional
ΣU must depend on the interaction parameters U only.
Furthermore, the functional must satisfy a vanishing-curl
condition or, alternatively, must be derivable from some
(universal) functional ΦU [G] as TΣU [G] = δΦU [G]/δG
(the temperature T is introduced for convenience). In
short, “Φ-derivable” approximations are conserving.
Φ-derivable approximations have been shown2 to ex-
hibit several further advantageous properties in addition.
One of these concerns the question of thermodynamical
consistency. There are different ways to determine the
grand potential of the system from the Green’s function
which do not necessarily yield the same result when us-
ing approximate quantities. On the one hand, Ω may be
calculated by integration of expectation values, accessi-
ble by G, with respect to certain model parameters. For
example, Ω may be calculated by integration of the av-
erage particle number, as obtained from the trace of G,
with respect to the chemical potential µ. On the other
hand, Ω can be obtained as Ω = Φ + Tr lnG − Tr(ΣG)
without integration. A Φ-derivable approximation con-
sistently gives the same result for Ω in both ways.
At zero temperature T = 0 there is another non-trivial
theorem which is satisfied by any Φ-derivable approxima-
tion, namely Luttinger’s sum rule.5,6 This states that the
volume in reciprocal space that is enclosed by the Fermi
surface is equal to the average particle number. The orig-
inal proof of the sum rule by Luttinger and Ward5 is
based on the existence of Φ in the exact theory and is
straightforwardly transferred to the case of a Φ-derivable
approximation. This also implies that other Fermi-liquid
properties, such as the linear trend of the specific heat at
low T and Fermi-liquid expressions for the T = 0 charge
and the spin susceptibility are respected by a Φ-derivable
approximation.
There is a perturbation expansion5,7 which gives the
Luttinger-Ward functional ΦU [G] in terms of closed
skeleton diagrams (see Fig. 1). As a manageable Φ-
= + + +Φ
FIG. 1: Diagrammatic representation of the Luttinger-Ward
functional ΦU [G]. Double lines stand for the interacting one-
particle Green’s function G, dashed lines represent the ver-
tices U .
http://arxiv.org/abs/0704.0249v2
derivable approximation must specify a (universal) func-
tional ΦU [G] that can be evaluated in practice, one usu-
ally considers truncations of the expansion and sums up
a certain subclass of skeleton diagrams only. This, how-
ever, means that the construction of conserving approx-
imations is restricted to the weak-coupling limit.
One purpose of the present paper is to show that it is
possible to construct Φ-derivable approximations for lat-
tice models of correlated fermions with local interactions
which are non-perturbative, i.e. do not employ trunca-
tions of the skeleton-diagram expansion. The idea is to
employ the self-energy-functional theory (SFT).8,9,10 The
SFT constructs the Luttinger-Ward functional ΦU [G], or
its Legendre transform FU [Σ], in an indirect way, namely
by making contact with an exactly solvable reference sys-
tem. Thereby, the exact functional dependence of FU [Σ]
becomes available on a certain subspace of self-energies
which is spanned by the self-energies generated by the
reference system.
The obvious question is whether those non-
perturbative Φ-derivable approximations have the
same properties as the weak-coupling Φ-derivable ap-
proximations suggested by Baym and Kadanoff. This
requires the discussion of the following points:
(i) Macroscopic conservations laws. For fermionic lat-
tice models, conservation of energy, particle number and
spin have to be considered. Besides the static thermody-
namics, the SFT concept concentrates on the one-particle
excitations. For the approximate one-particle Green’s
function, however, it is actually simple to prove that the
above conservation laws are respected. A short discus-
sion is given in Appendix A.
(ii) Thermodynamical consistency. This issue has al-
ready been addressed in Ref. 11. It has been shown that
the µ derivative of the (approximate) SFT grand poten-
tial (including a minus sign) equals the average particle
number 〈N〉 as obtained by the trace of the (approxi-
mate) Green’s function. The same holds for any one-
particle quantity coupling linearly via a parameter to the
Hamiltonian, e.g. for the average total spin 〈S〉 coupling
via a field of strength B.
(iii) Luttinger sum rule. This is the main point to
be discussed in the present paper. There are different
open questions: First, it is straightforward to prove that
weak-coupling Φ-derivable approximations respect the
sum rule as one can directly take over the proof for the
exact theory. For approximations constructed within the
SFT, a different proof has to be given. Second, it turns
out that a non-perturbative Φ-derivable approximation
respects the sum rule if and only if the sum rule holds
for the reference system that is used within the SFT.
As the original and thereby the related reference system
may be studied in the strong-coupling regime, this raises
the question which reference system does respect the sum
rule, i.e. which approximation is consistent with the sum
rule. Third, it will be particularly interesting to study
reference systems which generate dynamical impurity ap-
proximations (DIA)8,9 and variational cluster approxi-
mations (VCA),10,12 as these consist of a finite number of
degrees of freedom. Does the Luttinger sum rule hold for
finite systems? Do the DIA and the VCA respect the sum
rule? What is the simplest approximation consistent with
the sum rule? Note that finite reference systems consist-
ing of a few sites only have been shown9,13,14,15,16,17,18,19
to generate approximations which qualitatively capture
the main physics correctly. Finally, it is important to un-
derstand these issues in order to understand whether and
how a violation of the sum rule is possible within cluster
extensions20,21,22,23 of the dynamical mean-field theory
(DMFT).24,25,26,27,28 Note that the SFT comprises the
DMFT and certain29 cluster extensions and that possi-
ble violations of the sum rule in the two-dimensional lat-
tice models have been reported,30,31,32 including a study
using the dynamical cluster approximation (DCA).33
The paper is organized as follows: A brief general dis-
cussion of the Luttinger sum rule is given in the next
section, and a form of the sum rule specific to systems
with a finite number of spatial degrees of freedom is
derived. Sec. III clarifies the status of the sum rule
with respect to non-perturbative approximations gener-
ated within the SFT framework. The results are elu-
cidated by several numerical examples obtained for the
most simple but non-trivial non-perturbative conserving
approximation in Sec. IV. Violations of the sum rule in fi-
nite systems and their consequences are discussed in Sec.
V. Finally, Sec. VI summarizes our main conclusions.
II. LUTTINGER SUM RULE
A system of interacting electrons on a lattice is gen-
erally described by a Hamiltonian H(t,U) = H0(t) +
H1(U) consisting of a one-particle part H0(t) and an
interaction H1(U) with one-particle and interaction pa-
rameters t and U , respectively. As a prototype, let
us consider the single-band Hubbard model34,35,36 on a
translationally invariant D dimensional lattice consist-
ing of L sites with periodic boundary conditions. The
Hamiltonian is given by:
iσcjσ +
niσni−σ . (1)
Here, i = 1, ..., L refers to the sites, σ =↑, ↓ is the spin
projection, ciσ (c
iσ) annihilates (creates) an electron in
the one-electron state |iσ〉, and niσ = c
ciσ. Fourier
transformation diagonalizes the hopping matrix t and
yields the dispersion ε(k). There are L allowed k points
in the first Brillouin zone.
Let G = Gt,U denote the one-electron Green’s func-
tion of the model H(t,U). In case of the Hubbard model,
its elements are given by Gij(ω) = 〈〈ciσ ; c
jσ〉〉ω . In the
absence of spontaneous symmetry breaking, the Green’s
function is spin-independent and diagonal in reciprocal
space. It can be written as Gk(ω) = 1/(ω + µ − ε(k) −
Σk(ω)) where µ is the chemical potential and Σk(ω) the
self-energy. We also introduce the notation Σt,U for
the self-energy, and Gt,0 = 1/(ω + µ − t) for the free
(non-interacting) Green’s function which exhibits the de-
pendence on the model parameters but suppresses the
frequency dependence. Dyson’s equation then reads as
Gt,U = 1/(G
t,0 −Σt,U ).
The Luttinger sum rule5,6 states that
〈N〉 = 2
Θ(Gk(0)) (2)
where N =
niσ is the particle-number operator,
〈N〉 its (T = 0) expectation value, and Θ the Heavy-
side step function. The factor 2 accounts for the two
spin directions. Since Gk(0)
−1 = µ − ε(k) − Σk(0), the
sum gives the number of k points enclosed by the in-
teracting Fermi surface which, for L → ∞, is defined via
µ−ε(k)−Σk(0) = 0. In the thermodynamic limit the sum
rule therefore equates the average particle number with
the Fermi-surface volume (apart from a factor (2π)D/L).
Note that, as Θ(Gk(0)) = Θ(1/Gk(0)), the sum rule Eq.
(2) also includes the so-called Luttinger volume37 which
(for L → ∞) is enclosed by the zeros of Gk(0).
The standard proof of the sum rule can be found in
Ref. 5. It is based on diagrammatic perturbation theory
to all orders which is used to construct the Luttinger-
Ward functional ΦU [G] as the sum of renormalized closed
skeleton diagrams (see Fig. 1). We emphasize that the
original proof straightforwardly extends also to finite sys-
tems. For L < ∞ the sum in Eq. (2) is discrete. Actually,
the proof is performed for finite L first, and the thermo-
dynamic limit (if desired) can be taken in the end. The
limit T → 0, on the other hand, is essential and is re-
sponsible for possible violations of the sum rule (see Sec.
Below we need an alternative but equivalent formu-
lation of the sum rule. We start from the following
(Lehmann) representation for the Green’s function:
Gk(ω) =
αm(k)
ω + µ− ωm(k)
. (3)
Here, ωm(k)−µ are the (real) poles and αm(k) the (real
and positive) weights. For real frequencies ω, it is then
easy to verify the identity:
Θ(Gk(ω)) =
Θ(ω+µ−ωm(k))−
Θ(ω+µ−ζn(k))
where ζn(k) − µ is the n-th (real) zero of the Green’s
function, i.e. Gk(ζn(k)− µ) = 0.
For temperature T = 0 we have 〈N〉 =
dω(−1/π)ImGk(ω + i0
+) and thus 〈N〉 =
αm(k)Θ(µ−ωm(k)). Hence, the Luttinger sum
rule reads:
αm(k)Θ(µ− ωm(k))
Θ(µ− ωm(k))−
Θ(µ− ζn(k))
This form of the sum rule is convenient for the discussion
of finite systems with L < ∞.
III. SELF-ENERGY-FUNCTIONAL THEORY
AND LUTTINGER SUM RULE
Within the self-energy-functional theory (SFT),8,9,10
the grand potential Ω is considered as a functional of the
self-energy:
Ωt,U [Σ] = Tr ln
t,0 −Σ
+ FU [Σ] . (6)
Here, the trace Tr of a quantity A is defined as TrA ≡
eiωn0
Ak(iωn) where iωn = i(2n+ 1)πT are
the fermionic Matsubara frequencies, and the functional
FU [Σ] is the Legendre transform of the Luttinger-Ward
functional ΦU [G]. The self-energy functional (6) is sta-
tionary at the physical self-energy, δΩt,U [Σt,U ]/δΣ = 0,
and, if evaluated at the physical self-energy, yields the
physical value for the grand potential: Ωt,U [Σt,U ] =
Ωt,U ≡ −T ln tr exp(−β(H(t,U)−µN)) where β = 1/T .
Comparing with the self-energy functional
Ωt′,U [Σ] = Tr ln
t′,0 −Σ
+ FU [Σ] (7)
of a reference system with the same interaction but a
modified one-particle part, i.e. with the Hamiltonian
H(t′,U), the not explicitly known but only U -dependent
functional FU [Σ] can be eliminated:
Ωt,U [Σ] = Ωt′,U [Σ] + Tr ln
t,0 −Σ
− Tr ln
t′,0 −Σ
An approximation is constructed by searching for a sta-
tionary point of the self-energy functional on the sub-
space of trial self-energies spanned by varying the one-
particle parameters t′:
∂Ωt,U [Σt′,U ]
= 0 . (9)
Inserting a trial self-energy into Eq. (8) yields
Ωt,U [Σt′,U ] = Ωt′,U +Tr ln
t,0 −Σt′,U
− Tr lnGt′,U .
The decisive point is that the r.h.s. can be evaluated ex-
actly for a reference system which is exactly solvable.
Apart from the free Green’s function Gt,0, it involves
quantities of the reference system only.
This strategy to generate approximations has several
advantages: (i) Contrary to the usual conserving approxi-
mations, the exact functional form of Ωt,U [Σ] is retained.
Any approximation is therefore non-perturbative by con-
struction. On the level of one-particle excitations, macro-
scopic conservation laws are respected as shown in Ap-
pendix A. (ii) With Ωt,U [Σt′,U ] evaluated at the station-
ary point t′ = t′s, an approximate but explicit expression
for a thermodynamical potential is provided. As all phys-
ical quantities derive from this potential, the approxima-
tion is thermodynamically consistent in itself (see Ref.
11 for details). (iii) As different reference systems gen-
erate different approximations, the SFT provides a uni-
fying framework that systematizes a class of “dynamic”
approximations (see Refs. 29,38 for a discussion).
In the following we discuss the question whether or
not a dynamic approximation respects the Luttinger sum
rule. For this purpose consider first the Tr ln(· · · ) terms
in Eq. (10). These can be evaluated using the ana-
lytical and causal properties of the Green’s functions
as described in Ref. 9 (see Eq. (4) therein). Using
−T ln(1 + exp(−ω/T )) → ωΘ(−ω) for T → 0 yields:
Tr ln
t,0 −Σt′,U
(ωm(k)− µ)Θ(µ− ωm(k))
(ζn(k)− µ)Θ(µ− ζn(k)) . (11)
Analogously, we have
Tr lnGt′,U = 2
(ω′m(k)− µ)Θ(µ− ω
m(k))
(ζn(k)− µ)Θ(µ− ζn(k)) .
Note that the reference system is always assumed to be
in the same macroscopic state as the original system,
i.e. it is considered at the same temperature and, more
importantly here, at the same chemical potential µ. Fur-
thermore, it has been used that, by construction of the
approximation, the self-energy and hence its poles at
ζn(k) − µ are the same for both, the original and the
reference system. This implies that the second terms on
the r.h.s. of Eq. (11) and (12), respectively, cancel each
other in Eq. (10). Finally, a (large but) finite system
(L < ∞) and a finite reference system are considered.
Hence, the set of poles of the Green’s function and of the
self-energy as well as sums over k are discrete and finite.
Taking the µ derivative on both sides of Eq. (10) then
yields:
∂Ωt,U [Σt′,U ]
∂Ωt′,U
Θ(µ− ωm(k))
Θ(µ− ω′m(k)) . (13)
Here we have assumed the ground state of the refer-
ence system to be non-degenerate with respect to the
particle number. From the (zero-temperature) Lehmann
representation39 it is then obvious that, within a sub-
space of fixed particle number, the µ-dependence of the
Green’s function is the same as its ω-dependence, i.e.
G(ω) = G̃(ω+µ) with a µ-independent function G̃. Via
the Dyson equation of the reference system, this prop-
erty can also be inferred for the self-energy and, via the
Dyson equation of the original system, for the (approx-
imate) Green’s function of the original system. Conse-
quently, the poles of (G−1
t,0 −Σt′,U )
−1 and of Gt′,U are
linearly dependent on µ, i.e. ωm(k) and ω
m(k) in Eqs.
(11) and (12) are independent of µ.
We once more exploit the fact that the self-energy of
the original system is identified with the self-energy of the
reference system. Using Eq. (4) one immediately arrives
〈N〉 = 〈N〉′ + 2
Θ(Gk(0))− 2
(0)) . (14)
This is the final result: The Luttinger sum rule for
the original system, Eq. (2), is satisfied if and only if
it is satisfied for the reference system, i.e. if 〈N〉′ =
(0)).
A few remarks are in order. For the reference sys-
tem, the status of the Luttinger sum rule is that of a
general theorem (as long as the general proof is valid);
〈N〉′ and G′
(0) represent exact quantities. The above
derivation shows that the theorem is “propagated” to the
original system irrespective of the approximation that is
constructed within the SFT. This propagation also works
in the opposite direction. Namely, a possible violation
of the exact sum rule for the reference system would im-
ply a violation of the sum rule, expressed in terms of
approximate quantities, for the original system.
Eq. (14) holds for any choice of t′. Note, however, that
stationarity with respect to the variational parameters t′
is essential for the thermodynamical consistency of the
approximation. In particular, consistency means that the
average particle number 〈N〉 = −∂Ωt,U [Σt′,U ]/∂µ on the
l.h.s. can be obtained as the trace of the Green’s function.
Stationarity is thus necessary to get the sum rule in the
form (5).
There are no problems to take the thermodynamic
limit (if desired) on both sides of Eq. (14) (after divi-
sion of both sides by the number of sites L). The k sums
turn into integrals over the unit cell of the reciprocal lat-
tice. For a D-dimensional lattice the D − 1-dimensional
manifolds of k points with Gk(0) = ∞ or Gk(0) = 0 form
Fermi or Luttinger surfaces, respectively.
For the above derivation, translational symmetry has
been assumed for both, the original as well as the refer-
ence system. Nothing, however, prevents us from repeat-
ing the derivation in case of systems with reduced (or
completely absent) translational symmetries. One sim-
ply has to re-interprete the wave vector k as an index
which, combined with m, refers to the elements of the
diagonalized Green’s function matrix G. The exact sum
rule, Eq. (5), generalizes accordingly. The result (14) re-
mains valid (with the correct interpretation of k) for an
original system with reduced translational symmetries.
It is also valid for the case of a translationally symmet-
ric original Hamiltonian where, due to the choice of a
reference system with reduced translational symmetries,
the symmetries of the (approximate) Green’s function of
the original system are (artificially) reduced. A typical
example is the variational cluster approximation (VCA)
where the reference system consists of isolated clusters of
finite size.
IV. TWO-SITE DYNAMICAL-IMPURITY
APPROXIMATION
While the Hartree-Fock approximation may be con-
sidered as the most simple weak-coupling Φ-derivable
approximation, the most simple non-perturbative Φ-
derivable approximation is given by the dynamical-
impurity approximation (DIA). This shall be demon-
strated in the following for the single-band Hubbard
model (1) as the original system to be investigated. The
DIA is generated by a reference system consisting of a
decoupled set of single-impurity Anderson models with
a finite number of sites ns and is known
8 to recover
the dynamical mean-field theory in the limit ns → ∞.
As long as the Luttinger sum rule holds for the single-
impurity reference system, the DIA must yield a one-
particle Green’s function and a self-energy respecting the
sum rule.
The Hamiltonian of the reference system is H(t′,U) =∑L
i=1 H
i with
H ′i =
iσciσ +
niσni−σ
aikσ +
ciσ + h.c.) .
For a homogeneous phase, the variational parameters
′ = ({ε
0 , ε
}) can be assumed to be indepen-
dent of the site index i: ε0 ≡ ε
0 , εk ≡ ε
, Vk ≡ V
For the sake of simplicity, we consider the two-site DIA
(ns = 2), i.e. a single bath site per correlated site only.
In this case there are three independent variational pa-
rameters only: the on-site energies of the correlated and
of the bath site, ε0 and εc ≡ εk=2, respectively, as well as
the hybridization strength V ≡ Vk=2. As the reference
0 0.2 0.4 0.6 0.8 1
filling  n
U=W=4
FIG. 2: Filling dependence of the variational parameters
at their respective optimized values and of the chemical po-
tential. Calculations for the Hubbard model with a semi-
elliptical free density of states of band width W = 4 and
interaction strength U = W = 4 using the two-site DIA.
system consists of replicated identical impurity models
which are spatially decoupled, the trial self-energy is lo-
cal and site-independent, Σij(ω) = δijΣ(ω).
Calculations have been performed for the Hub-
bard model with a one-particle dispersion ε(k) =
e−ik(Ri−Rj)tij such that the density of one-
particle energies D(ε) is semi-elliptic. For |ε| ≤ W/2,
D(ε) =
δ(ε− ε(k)) =
(W/2)2 − ε2 . (16)
The free band width is set to W = 4. This serves as the
energy scale.
The computation of the SFT grand potential is per-
formed as described in Ref. 9. Stationary points of the
resulting function Ω(ε0, εc, V ) ≡ Ωt,U [Σε0,εc,V ] are ob-
tained via iterated linearizations of its gradient. There
is a unique non-trivial stationary point (with V 6= 0).
Fig. 2 shows the variational parameters at this point as
functions of the filling n. For the entire range of fillings,
the ground state of the reference system lies in the in-
variant subspace with Ntot =
iσciσ + a
iσaiσ) = 2.
The parameters as well as the chemical potential are
smooth functions of n. We have checked that the ther-
modynamical consistency condition n = −L−1∂Ω/∂µ =∫ 0
ρ(ω)dω is satisfied within numerical accuracy. Here
ρ(ω) = D(ω + µ− Σ(ω)) (17)
is the interacting local density of states (DOS).
At half-filling the values of the optimized on-site en-
ergies are consistent with particle-hole symmetry. With
ε0 − µ = −U/2 and εc − µ = 0 the reference system is
in the Kondo regime with a well-formed local moment at
the correlated site. The finite hybridization strength V
leads, for U = W , to a finite DOS ρ(ω = 0) > 0 and thus
0 0.2 0.4 0.6 0.8 1
filling  n
2S-DMFT
two-site DIA
FIG. 3: Quasi-particle weight z as a function of the filling
within the two-site DIA (full lines) and the two-site DMFT40
(dashed lines). Calculations for U = W and U = 2W .
to a metallic Fermi liquid as it is expected for the Hub-
bard model within a (dynamical) mean-field description.
Due to the simple structure of the self-energy generated
by the two-site reference system, however, quasi-particle
damping effects are missing.
Decreasing the filling from n = 1 to n = 0 drives the
reference system more and more out of the Kondo regime.
While εc stays close to the chemical potential, the on-site
energy of the correlated site ε0 crosses µ close to quar-
ter filling and lies above µ eventually. Note that ε0 = 0
within the DMFT, i.e. for ns → ∞, while for finite ns
there is a clear deviation from ε0 = 0 which is necessary
to ensure thermodynamical consistency. For fillings very
close to n = 0, the grand potential Ωt,U [Σε0,εc,V ] be-
comes almost independent of Σ. This implies that it be-
comes increasingly difficult to locate the stationary point
with the numerical algorithm used. The slight upturn of
ε0 below n = 0.01 (see Fig. 2) might be a numerical ar-
tifact.
It is instructive to compare the parameters with those
of the two-site DMFT (2S-DMFT).40 The 2S-DMFT is
a simplified version of the DMFT where a mapping onto
the two-site single impurity Anderson model is achieved
by means of a simplified self-consistency equation. As-
suming ε0 = 0 as in the full DMFT, there are two pa-
rameters left (εc and V ) which are fixed by considering
the first non-trivial order in the low- and in the high-
frequency expansion of the self-energy and the Green’s
function in the DMFT self-consistency equation. Al-
though being well motivated, this approximation is es-
sentially ad hoc. One therefore has to expect that the 2S-
DMFT is thermodynamically inconsistent and exhibits a
violation of Luttinger’s sum rule. A comparison of the
DIA for ns = 2 with the 2S-DMFT is thus ideally suited
to demonstrate the advantages gained by constructing
approximations within the variational framework of the
First of all, there are differences in fact. At half-filling
the 2S-DMFT predicts the hybridization to be somewhat
larger than the two-site DIA while the value for εc is
again fixed by particle-hole symmetry. Deviations grow
with decreasing filling. Contrary to the two-site DIA, V
monotonously increases and is larger in the entire filling
range, ε0 = 0 by construction, and εc even diverges for
n → 0 within the 2S-DMFT (see Ref. 40). On the other
hand, the system is essentially uncorrelated in the limit
n → 0. Strong differences in the parameters, which en-
ter the self-energy only, therefore do not necessarily im-
ply strongly different physical quantities. This is demon-
strated by Fig. 3 which shows the quasi-particle weight
calculated via
dΣ(ω = 0)
as a function of the filling. While there are obvious differ-
ences when comparing the results from the two-site DIA
with those of the 2S-DMFT, the qualitative trend of z is
very similar in both approximations. Both approxima-
tions also compare well with the full DMFT: There is a
quadratic behavior of z(n) for n → 1 in the Fermi-liquid
phase (U = W ) and a linear trend when approaching the
Mott phase (U = 2W ). The critical interaction strength
for the Mott transition is found to be Uc ≈ 1.46W for
the two-site DIA and Uc = 1.5W within the 2S-DMFT.
For details on the Mott transition see Refs. 9,40.
In case of a local and site-independent self-energy, the
Luttinger sum rule can be written in the form41
µ = µ0 +Σ(ω = 0) , (19)
where µ0 is the chemical potential of the free (U = 0)
system at the same particle density. Eq. (19) implies that
not only the enclosed volume but also the shape of the
Fermi surface remains unchanged when switching on the
interaction. Using Eq. (17) this immediately implies41
ρ(0) = D(µ0) = ρ0(0) , (20)
i.e., in case of a correlated metal, the value of the inter-
acting local density of states at ω = 0 is independent of
U and thus fixed to the value of the density of states of
the non-interacting system at the same filling.
The interacting and the non-interacting DOS are plot-
ted in Fig. 4 for different fillings and for U = W and
U = 2W . The impurity self-energy of the two-site ref-
erence system is an analytical function of ω except for
two first-order poles on the real axis. Via Eq. (17) this
two-pole structure implies that the DOS consists of three
peaks the form of which is essentially given by the non-
interacting DOS. At half-filling the three peaks are eas-
ily identified as the lower and the upper Hubbard band
and the quasi-particle resonance as it is characteristic
for a (dynamical) mean-field description.27 For U = W
the resonance still has a significant weight. The weight
decreases upon approaching the critical interaction, and
the resonance has disappeared in the Mott insulator for
-4 -2 0 2 4 6
-6 -4 -2 0 2 4 6 8 10
n=1.0
n=0.75
n=0.5
n=0.25
n=0.0
FIG. 4: Interacting local density of states ρ(ω) (solid lines)
for different fillings as indicated. Calculations using the two-
site DIA for U = W (left) and U = 2W (right). For n = 0.25,
n = 0.5 and n = 0.75 the non-interacting DOS ρ0(ω) is shown
for comparison (dashed lines). Note that ρ(0) = ρ0(0). The
dotted line for U = 2W in the top panel is the DOS for
n = 0.99.
U = 2W . Hole doping of the Mott insulator is accom-
plished by the reappearance of the resonance at ω = 0
which preempts the creation of holes in the lower Hub-
bard band.42 As can be seen in the spectrum for n = 0.99
in the top panel (dotted line), the quasi-particle res-
onance appears within the Mott-Hubbard gap. With
decreasing filling, the upper Hubbard band gradually
shifts to higher excitation energies and loses weight. This
weight is transfered to the low-energy part of the spec-
trum. For lower fillings where the Kondo regime has been
left, one would actually expect that the quasi-particle res-
onance disappears by merging with the lower Hubbard
band. This, however, cannot be described with the sim-
ple two-pole structure of the self-energy. One therefore
should interprete the gap around ω = −1 at n = 0.25
as an artifact of the approximation. Furthermore, the
widths of the Hubbard bands is considerably underes-
timated as damping effects are missing completely. The
filling-dependent spectral-weight transfer across the Hub-
bard gap as well as the energy positions of the main
peaks, however, are in overall agreement with general
expectations.34,43
It is worth emphasizing that this simple two-site
0 0.2 0.4 0.6 0.8 1
-0.02
filling  n
Hubbard-I
2S-DMFT
two-site DIA
FIG. 5: Numerical results for the difference between the
volume enclosed by the Fermi surface VFS and the filling n as
a function of n for U = W = 4. The Luttinger sum rule (VFS−
n = 0) is exactly respected by the two-site DIA. Results for
the 2S-DMFT and the Hubbard-I approximation are shown
for comparison. Dashed line: difference between the filling n
and the average occupation of the correlated (impurity) site
in the reference system at stationarity for the two-site DIA.
dynamical-impurity approximation exactly fulfills the
Luttinger sum rule. In Fig. 4 this can be seen by compar-
ing with the DOS of the non-interacting system (dashed
lines). The non-interacting DOS cuts the interacting one
at ω = 0 which shows that Eq. (20) is satisfied. Note
that this is trivial for n = 1 as this is already enforced
by particle-hole symmetry. Off half-filling, however, the
pinning of the DOS to its non-interacting value at ω = 0
is a consequence of Φ-derivability and thereby a highly
non-trivial feature.
In contrast, the 2S-DMFT does show a violation of
Luttinger’s sum rule which, however, must be attributed
to the ad hoc nature of the approximation. Fig. 5 shows
0 0.2 0.4 0.6 0.8 1
filling  n
2S-DMFT
two-site DIA
FIG. 6: Filling dependence of the compressibility κ for U =
W as obtained within the 2S-DMFT via κ = ∂n/∂µ (solid
line) and via a general Fermi-liquid relation (Eq. (22), dashed
line). Using the two-site DIA identical results are obtained
for both cases.
the difference between the volume enclosed by the Fermi
surface
VFS =
Θ(µ−ε(k)−Σ(0)) = 2
dεD(ε+µ−Σ(0))
and the filling n as a function of the filling. As can be
seen, there is an artificial violation of the sum rule for the
2S-DMFT which is of the order of a few per cent while
for the Φ-derivable two-site DIA the sum rule is fully re-
spected. Note that, unlike the DMFT and also unlike
the simplified 2S-DMFT, the two-site DIA predicts a fill-
ing which slightly differs from the average occupation of
the correlated impurity site in the reference system (see
dashed line in Fig. 5). For a finite number of bath sites ns
this appears to be necessary to fulfill the Luttinger sum
rule. The figure also shows the result obtained within
the Hubbard-I approximation.34 Here a very strong (ar-
tificial) violation of up to 100 % (for n close to half-filling)
is obtained. This should be considered as a strong draw-
back which is typical for uncontrolled mean-field approx-
imations.
There are more relations which, analogously to the
Luttinger sum rule, can be derived by means of per-
turbation theory to all orders6 in the exact theory and
which are respected by weak-coupling conserving approx-
imations. For example, the compressibility, defined as
κ = ∂n/∂µ, can be shown to be related to the interact-
ing DOS and the self-energy at the Fermi edge via
κ = 2ρ(0)
∂Σ(0)
. (22)
Fig. 6 shows that for the 2S-DMFT it makes a difference
whether κ is calculated as the µ-derivative of the filling
or via Eq. (22). Again, this must be attributed to the
fact that the 2S-DMFT is not a Φ-derivable approxima-
tion. Contrary, the two-site DIA does respect the general
Fermi-liquid property (22) and thus yields the same re-
sult in both cases (see Fig. 6).
V. VIOLATION OF LUTTINGER’S SUM RULE
IN FINITE SYSTEMS
The preceding section has demonstrated that the two-
site DIA satisfies the Luttinger sum rule. According to
Eq. (14), we can conclude that the Luttinger sum rule
must hold for the corresponding reference system, i.e. for
the two-site single-impurity Anderson model. Of course,
this can be verified more directly by evaluating Eq. (5).
In case of a finite system or a system with reduced trans-
lational symmetries, the Green’s function is a matrix with
elements Gαβ(ω) where α refers to the one-particle basis
states, and the Luttinger sum rule reads:
α(k)m Θ(µ−ω
m ) =
Θ(µ−ω(k)m )−
Θ(µ− ζ(k)n ) .
Here the index k labels the elements of the diagonalized
Green’s function, i.e. Eq. (5) is generalized by replacing
(k, σ) → k. In case of an impurity model, Eq. (23) ac-
tually represents the Friedel sum rule.44,45 For the two-
site single-impurity Anderson model, the different one-
particle excitation energies ω
m − µ, the zeros of the
Green’s function ζ
n − µ and the weights α
m are eas-
ily determined by full diagonalization. We find that Eq.
(23) is satisfied in the entire parameter space (except for
V = 0, see below).
Note that a violation of the sum rule occurs when, as
a function of a model parameter x, a zero of the Green’s
function crosses ω = 0 for x = xc. At xc the number of
negative zeros counted by the second term on the r.h.s.
changes by one while the first term as well as the l.h.s.
remain constant since (unlike a pole) a zero of the Green’s
function is generically not connected with a change of the
ground state (level crossing). This implies that the sum
rule would be violated for x < xc or for x > xc.
The case V = 0 is exceptional. Within the two-site
DIA this corresponds to the Mott insulator (see Fig. 4,
topmost panel for U = 2W ). For V = 0 the reference
system consists of two decoupled sites, and the Green’s
function becomes diagonal in the site index. There is
no zero of the local Green’s function corresponding to
the uncorrelated site. We can thus concentrate on the
correlated site where the local Green’s function exhibits
a zero at η−µ = ε0+U/2. In the sector with one electron
at the correlated site (ε0 < µ < ε0 + U), the second
term on the r.h.s. changes by two at µ = µc = ε0 + U/2
because of the two-fold degenerate ground state. In this
case Luttinger’s sum rule in the form (23) is violated for
µ < µc and for µ > µc. This “violation”, however, is a
trivial one which immediately disappears if the ground-
state degeneracy is lifted by applying a weak field term,
for example.
Fig. 7 shows a phase diagram of the single-impurity
Anderson model with ns = 4 sites as obtained by
full diagonalization. The diagram covers the entire
range of the total particle number N =
〈c†σcσ〉 +∑
k=2〈a
akσ〉 from N = 0 to N = 2ns = 8. A non-
degenerate ground state is enforced by applying a small
but finite magnetic field. No violation of the Luttinger
sum rule is found. We have repeated the same calculation
also for ns = 10 using the Lanczos technique.
46 Again,
the sum rule is found to be always satisfied (We have
performed calculations for different U and bath parame-
ters). This might have been expected as the (ns → ∞)
Anderson model can generally be classified as a (local)
Fermi liquid.47
The situation is less clear in the case of correlated lat-
tice models such as the Hubbard or the t-J model. For
two dimensions there are several numerical studies using
high-temperature expansion,30 quantum Monte-Carlo,31
extended DMFT,32,48 and dynamical cluster approxima-
tion (DCA)33 which indicate a violation in the strongly
correlated metallic phase close to half-filling. For studies
of large clusters or studies directly working in the thermo-
-1 0 1 2 3
SIAM n  =4s
FIG. 7: Phase diagram µ vs. ε of the single-impurity An-
derson model with ns = 4 sites. Total particle numbers are
indicated by Roman figures. Results have been obtained by
full diagonalization for the following model parameters. One-
particle energies: ε0 = 0 (correlated site), εk = ε + (k − 3)
with k = 2, 3, 4 (uncorrelated bath sites). Hubbard interac-
tion: U = 2ε. Hybridization strength: Vk = 0.1 for k = 2, 3, 4.
To lift Kramers degeneracy in case of an odd particle number,
a weak (ferromagnetic) field of strength b = 0.001 is coupled
to the local spins. The dashed line marks the particle-hole
symmetric case. Luttinger’s sum rule is found to be satisfied
in the entire parameter space.
dynamic limit, a definite conclusion on the validity of the
sum rule is difficult to obtain as finite-temperature or ar-
tificial broadening effects etc. must be controlled numer-
ically. Contrary, full diagonalization of Hubbard clusters
consisting of a few sites only can provide exact results.
While their direct relevance for the thermodynamic limit
is less clear, it is important to note that reference sys-
tems with a finite number of sites or a finite number of
correlated sites provide the basis for a number of cluster
approaches within the SFT framework. Via Eq. (14) their
properties are transferred to the approximate treatment
of lattice models in the thermodynamic limit.
The validity of Eq. (23) has been checked for Hubbard
clusters of different size and in different geometries. The
µ vs. U phase diagram for an L = 4-site open Hubbard
chain with nearest-neighbor hopping in Fig. 8 shows a
representative example. Again, a small but finite field
term is added to avoid a ground-state degeneracy. As the
chemical potential, for fixed U , is moved off the particle-
hole symmetric point µ = U/2 and exceeds certain criti-
cal values (red lines), the particle number N [as obtained
from the l.h.s. of Eq. (23)] changes from N = L down to
(up to) N = 0 (N = 2L). A critical µ value indicates a
change of the ground state (level crossing) that is accom-
panied by a change of the ground-state particle number.
In the one-particle Green’s function this is characterized
by a pole ω
m −µ crossing ω = 0. The blue lines indicate
-5 0 5 10 15
Hubbard L=4
FIG. 8: Phase diagram µ vs. U of the Hubbard model with
L = 4 sites (open chain) as obtained by full diagonalization.
Nearest-neighbor hopping t = −1. A weak (ferromagnetic)
field of strength b = 0.01 is applied to lift Kramers degener-
acy. The dashed line marks the particle-hole symmetric case.
Particle numbers (l.h.s. of Eq. (23)) are indicated by Roman
figures. R.h.s. of Eq. (23): Arabic figures. Luttinger’s sum
rule is found to be violated for sufficiently strong U . Uc1, Uc,2:
critical interactions.
those chemical potentials at which a zero of the Green’s
function ζ
n − µ crosses ω = 0. Whenever this happens
the r.h.s. of Eq. (23) changes while the l.h.s. is constant.
Fig. 8 shows that this occurs several times in the N = L
sector. At the particle-hole symmetric point µ = U/2
the Luttinger sum rule is obeyed while it is violated in a
wide region of the parameter space corresponding to half-
filling N = L. However, a critical interaction strength
Uc turns out to be necessary. The value for Uc strongly
varies for different cluster sizes and geometries but has
always been found to be positive and finite. Note that
for L = 4 the sum rule is fulfilled for any particle num-
ber N 6= L. Qualitatively similar results can be found for
the L = 2-site Hubbard cluster where calculations can be
done even analytically. Again, a violation of the sum rule
is found in the half-filled sector beyond a certain critical
This has already been noticed by Rosch49 and was used
in combination with a strong-coupling expansion to ar-
gue that a violation of the sum rule generically occurs
for a Mott insulator. Stanescu et al.50 have shown quite
generally that the sum rule is fulfilled when particle-hole
symmetry is present (the Luttinger surface is the same
as the Fermi surface of the non-interacting system) but
violated in the Mott insulator away from particle-hole
symmetry. It is interesting to note that these arguments
cannot be used to construct a violation of the sum rule
within DMFT or for a single-impurity Anderson model:
For an (almost) particle-hole symmetric case and model
parameters describing a Mott insulator (within DMFT),
-4 -2 0 2 4 6 8
Luttinger
sum rule
particle
number
U=16t
FIG. 9: Ground-state particle number (red Roman figures,
l.h.s. of Eq. (23)) and prediction by the Luttinger sum rule
(blue Arabic figures, r.h.s. of Eq. (23)) as functions of the
chemical potential for a L = 9-site Hubbard cluster with pe-
riodic boundary conditions. Arabic numbers are only given
when different from Roman ones. Calculations using the
Lanczos method and a finite but small magnetic field and
finite but small on-site potentials to lift ground-state degen-
eracies.
an odd number of sites ns (with ns → ∞) must be consid-
ered and thus a magnetic field is needed to lift Kramers
degeneracy. Even an infinitesimal field, however, leads
(at zero temperature) to a finite and even large polar-
ization corresponding a well-formed but unscreened local
moment. This polarization is incomplete for any finite
U as the DMFT predicts a small but finite double occu-
pancy for a Mott insulator. Still there is a proximity to
the fully polarized band insulator which finally results in
a weakly correlated state and thus in a situation which
is unlikely to show a violation of the sum rule.
We have also considered Hubbard clusters with L = 9
and L = 10 sites by using the Lanczos technique.46
Calculations have been performed for different Lanczos
depths lmax to ensure that the results are independent
of lmax. Fig. 9 displays an example for L = 9 and a
highly symmetric cluster geometry with periodic bound-
ary conditions and a well-defined reciprocal space. To lift
ground-state degeneracies resulting from spatial symme-
tries as well as the Kramers degeneracy, small but fi-
nite on-site potentials and a small magnetic-field term
are included in the cluster Hamiltonian. Fig. 10 shows
an example for L = 10 sites without any spatial sym-
metries. Kramers degeneracy for odd N is removed by
applying a small magnetic field. With the figures we com-
pare the expressions on the left-hand and the right-hand
side of Eq. (23). Obviously, the sum rule is respected in
-4 -2 0 2 4 6 8
Luttinger
sum rule
particle
number
U=16t
FIG. 10: The same as Fig. 9 but for 10 sites.
most cases. Violations are seen for half-filling N = L,
i.e. in the “Mott-insulating phase”, which is consistent
with Ref. 49. However, the sum rule is also violated
in the “metallic phase” close to half-filling, namely for
N = L − 1 (Fig. 9, L = 9) and N = L − 1, L − 2 (Fig.
10, L = 10). This nicely corresponds to the generally
observed trend30,31,32,33,48 for violations in the slightly
doped metallic regime. We have also verified that the
sum rule is restored by lowering U .
Fig. 9 and 10 demonstrate that the sum rule is violated
in the whole µ range corresponding to N = L − 1. This
is an important point as it shows that it is irrelevant
whether the T = 0 limit is approached by holding 〈N〉
fixed and adjusting µ = µ(T ) or by fixing µ and let 〈N〉 =
〈N〉(T ) be T -dependent. A violation of the sum rule is
found in both cases.
Kokalj and Prelovs̆ek51 have demonstrated that viola-
tions of the sum rule can also be found for the t-J model
on a finite number of sites. Our result provides an ex-
plicit example showing that not only for t-J51 but also
for Hubbard clusters a violation can be found when the
chemical potential is set to µ = limT→0 µ(T ) with µ(T )
obtained for given 〈N〉 = const. Anyway, the original
proof5 does not depend on this choice for µ but appears
to work for any µ.
The results raise the question which assumptions used
in the original proof of the theorem are violated or where
the proof breaks down. Note that the recently proposed
alternative topological proof52 assumes a Fermi-liquid
state from the very beginning and thus cannot be applied
to a finite system. Using weak symmetry-breaking fields,
a more or less trivial breakdown due to ground-state de-
generacy has been excluded. An analysis of the ground
state of the L = 2 and L = 4 Hubbard clusters which
are accessible with exact (analytical or numerical) meth-
ods has shown that, for model parameters where the sum
rule is violated, the interacting ground state can never-
theless be adiabatically connected to the non-interacting
one. This excludes level crossing as a potential cause for
the breakdown. While we cannot make a definite state-
ment, it appears at least plausible that the violation of
the sum rule results from a non-commutativity of two
limiting processes, the infinite skeleton-diagram expan-
sion and the limit T → 0.
Using a functional-integral formalism, the Luttinger-
Ward functional at finite T can also be constructed in a
non-perturbative way, i.e. avoiding an infinite summation
of diagrams, as has been shown recently.53 Formally, the
Luttinger sum rule can be obtained by exploiting a gauge
invariance of the Luttinger-Ward functional [see Ref. 53]:
∂(iωn)
ΦU [G(iωn)] = 0 . (24)
If at all, this invariance can only be shown for T = 0
where iωn becomes a continuous variable. Unfortunately,
the non-perturbative construction of ΦU requires a T > 0
formalism. Hence, the validity of the sum rule depends
on question whether the limit T → 0 commutes with the
frequency differentiation. Necessary and sufficient condi-
tions for this assumption are not easily worked out. An
understanding of the main reason for the possible break-
down of the sum rule in finite systems, very similar to
the case of Mott insulators, is therefore not yet available
(see also the discussion in Ref. 49).
VI. CONCLUSIONS
Φ-derivable approximations are conserving, thermody-
namically consistent and, for T = 0, formally respect cer-
tain non-trivial theorems such as the Luttinger sum rule.
As the construction of the Luttinger-Ward functional Φ
is by no means trivial and may conflict with the limit
T → 0 or different other limiting processes, however, the
validity of the sum rule may be questioned. Violations of
the sum rule can be found in fact for the case of strongly
correlated electron systems. For Mott insulators and fi-
nite systems in particular, a breakdown is documented
easily.
This implies that a general approximation for the
spectrum of one-particle excitations (of the one-particle
Green’s function) may violate the sum rule for two pos-
sible reasons, namely because (i) the sum rule is violated
in the exact theory, or (ii) the approximation generates
an artificial violation.
Within the usual weak-coupling conserving approxima-
tions, such as the fluctuation-exchange approximation,
the sum rule always holds as the formal steps in the gen-
eral proof of the sum rule can be carried over to the
approximation – but with the important simplification
of a limited class of diagrams. This also implies that
weak-coupling conserving approximations, when applied
beyond the weak-coupling regime, might erroneously pre-
dict the sum rule to hold.
The present paper has focussed on non-perturbative
conserving approximations. Non-perturbative approxi-
mations, constructed within the framework of the self-
energy-functional theory and referring to a certain ref-
erence system, are Φ-derivable and consequently respect
certain macroscopic conservation laws and are thermo-
dynamically consistent. Whether or not the sum rule
holds within the approximate approach, however, cannot
be answered generally. We found that Luttinger’s sum
rule holds within an (SFT) approximation if and only if
it holds exactly in the corresponding reference system.
The reference system that leads to the most simple
but non-trivial example for a non-perturbative conserv-
ing approximation consists of a single correlated and
a single bath site. For this two-site system, we have
found the sum rule to be valid in the entire parameter
space. Consequently, the resulting two-site dynamical-
impurity approximation (DIA) – opposed to more ad hoc
approaches like the two-site DMFT – fully respects the
sum rule as could be demonstrated in different ways. In
view of the simplicity of the approximation this is a re-
markable result. Since the sum rule dictates the low-
frequency behavior of the one-particle Green’s function,
important mean-field concepts, such as the emergence of
a quasi-particle resonance at the Fermi edge, are quali-
tatively captured correctly, even away from the particle-
hole symmetric case. This qualifies the two-site DIA for a
quick but rough estimate of mean-field physics, including
phases with spontaneously broken symmetries.
Full diagonalization and the Lanczos method have
been employed to show that also the single-impurity An-
derson model with a finite number of ns > 2 sites respects
the sum rule. Consequently, this property is transferred
to an ns-site DIA. For ns → ∞ the full dynamical mean-
field theory is recovered which is thereby recognized as
the prototypical non-perturbative conserving approxima-
tion. Clearly, in the case of the DMFT, Φ-derivability is
well known27 and obvious, for example, when construct-
ing the DMFT with the help of the skeleton-diagram ex-
pansion.
Using as a trial self-energy the self-energy of a cluster
with L > 1 correlated sites, generates an approximation
where short-range spatial correlations are included up to
the cluster extension. These variational cluster approxi-
mations provide a first step beyond the mean-field con-
cept. Again, whether or not the sum rule is respected
within the VCA depends on the reference system itself.
For the L = 2 Hubbard cluster, analytical calculations
straightforwardly show that violations of the sum rule
occur at half-filling, beyond a certain critical interaction
strength. In the thermodynamic limit, this would corre-
spond to the Mott-insulating regime. Applying the Lanc-
zos method to larger clusters, has shown, however, that
a breakdown of the Luttinger sum rule is also possible
for fillings off half-filling. For sufficiently strong U , the
sum rule is violated in the whole N = L − 1-particle
sector. This would correspond to a (strongly correlated)
metallic state in the thermodynamic limit. Whether or
not a VCA calculation is consistent with the sum rule,
then depends on the set of cluster hopping parameters t′
which make the self-energy functional stationary. First
VCA calculations54 for the D = 2 Hubbard model at low
doping and using clusters with up to L = 10 sites do
predict a violation in fact.
It is by no means clear a priori what happens in a clus-
ter approach using additional bath degrees of freedom as
variational parameters, as e.g. in the cellular DMFT.21,22
The usual periodization of the self-consistent C-DMFT
self-energy, however, should be avoided when testing the
sum rule as this introduces an additional (though physi-
cally motivated) approximation. Instead, Eq. (5) must be
used with k re-interpreted as an index referring to the el-
ements of the self-consistent diagonalized lattice Green’s
function.
Employing the dynamical cluster approximation
(DCA)20 represents an alternative which directly oper-
ates in reciprocal space. From a real-space perspective,
the DCA is equivalent with the cellular DMFT but ap-
plied to a modified model H = H(t,U) → H(t,U) with
modified hopping parameters which are invariant under
superlattice translations as well as under translations on
the cluster.29,55 In the limit L → ∞ the replacement
t → t becomes irrelevant. Analogous to the C-DMFT,
the sum rule then holds within the DCA if and only if
it holds for the individual cluster at self-consistently de-
termined cluster parameters. Note, however, that this
requires that (besides the DCA self-energy) the modified
hopping t instead of the physical hopping has to be con-
sidered in the computation of the volume enclosed by
the Fermi (Luttinger) surface of the lattice model. This
is exactly what is usually done in DCA calculations.
Within this context and in view of the violations found
for finite Hubbard clusters, it is possible to understand
why a non-perturbative cluster approximation, like the
VCA,54 or a cluster extension of the DMFT, like the
DCA,33 can produce results that are inconsistent with
Luttinger’s theorem.
Acknowledgments
We thank Robert Eder and Achim Rosch for valu-
able discussions. The work is supported by the Deutsche
Forschungsgemeinschaft within the Forschergruppe FOR
APPENDIX A: MACROSCOPIC CONSERVATION
OF ENERGY, PARTICLE NUMBER AND SPIN
The one-particle Green’s function as obtained within
an approximation generated by the choice of a reference
system respects the macroscopic conservation laws which
result from symmetries of the system with respect to con-
tinuous transformation groups:
Energy conservation is apparently respected as by con-
struction the approximate SFT Green’s function depends
on a single frequency only, i.e. is invariant under time
translations.
Conservation of the total particle number and spin is
respected if the approximate G transforms in the same
way as the exact Green’s function under global U(1) and
SU(2) gauge transformations. Consider a general trans-
formation of the form
c†α → c
with unitary S such that the interaction part H1(U)
of the Hamiltonian is invariant (α refers to the states
of the one-particle basis). In a diagrammatic approach,
the invariance of H1(U) implies that the corresponding
conservation law is respected “locally” at each vertex.
Hence, for a conserving approximation in the sense of
Baym and Kadanoff, the transformation behavior of the
free Green’s function is then propagated by the diagram
rules to the full Green’s function. Consequently, the lat-
ter must transform under S in the same way as the exact
G, i.e.
Gαβ → Gαβ =
. (A2)
Consider now the case of the SFT. One has to show
that the approximate Green’s function G for the trans-
formed system with Hamiltonian H is given by G =
† if G is the approximate Green’s function of the
model H . Applying the transformation (A1) to H , one
finds H = H0(t) +H1(U) → H = H0(t) +H1(U) with
t = StS†. Again, S is assumed to leave the interaction
part invariant.
The Green’s function G of the transformed model is
(approximately) constructed via
from the free Green’s function of the transformed model
and the SFT self-energy which is the self-energy of the
reference system H ′ = H0(t
s) +H1(U) at the stationary
point t
For the transformed problem H , the stationary point
s is determined from the SFT Euler equation:
)ω;αβ
= 0 .
As an ansatz to solve the Euler equation we take
= St′1S
† (A5)
with t′1 to be determined. The transformation law (A2)
for the exact Green’s function of the reference system
= GSt′
S†,U = SGt′1,US
†. This also holds for
the free Green’s function. Using the Dyson equation of
the reference system we can deduce Σ
= SΣt′
Furthermore, for the free Green’s function of the trans-
formed original model we have G
t,0 = SGt,0S
†. Using
these results, we see that Eq. (A4) is equivalent to
t,0 −Σt′1,U
∂(Σt′
,U )ω;αβ
= 0 .
But this is just the Euler equation for the original
model which is solved by t′1 = t
s. Remembering the
ansatz made, we now have for the stationary point
s = St
†. Inserting this into Eq. (A3) gives G =
S(G−1
t,0 − Σt′s,U )
† = SGS† which is the desired re-
sult.
Note that one has to ensure that the stationary point
for the transformed problem t
s = St
† lies within the
space of one-particle parameters characteristic for the ref-
erence system. For models with local interaction part
and for local (and also global) gauge transformations,
however, this is always easily satisfied.
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http://arxiv.org/abs/cond-mat/0407172
http://arxiv.org/abs/cond-mat/0602656
