EPJ manuscript No.
(will be inserted by the editor)
Renormalized quasiparticles in antiferromagnetic states of the
Hubbard model
J. Bauer and A.C. Hewson
Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
November 4, 2018
Abstract. We analyze the properties of the quasiparticle excitations of metallic antiferromagnetic states in
a strongly correlated electron system. The study is based on dynamical mean field theory (DMFT) for the
infinite dimensional Hubbard model with antiferromagnetic symmetry breaking. Self-consistent solutions
of the DMFT equations are calculated using the numerical renormalization group (NRG). The low energy
behavior in these results is then analyzed in terms of renormalized quasiparticles. The parameters for
these quasiparticles are calculated directly from the NRG derived self-energy, and also from the low energy
fixed point of the effective impurity model. From these the quasiparticle weight and the effective mass are
deduced. We show that the main low energy features of the k-resolved spectral density can be understood
in terms of the quasiparticle picture. We also find that Luttinger’s theorem is satisfied for the total electron
number in the doped antiferromagnetic state.
PACS. 7 1.10.Fd,71.27.+a
1 Introduction
The nature of the metallic antiferromagnetically ordered
state in strongly correlated systems has been subject of
study for over two decades, but still remains to be fully un-
derstood. Interest in this topic has been stimulated by the
fact that the high temperature superconductivity of the
cuprates emerges from the doping of an antiferromagnetic
insulating compound, such as La2CuO4 [1,2]. The sim-
plest models to describe the electrons in the CuO2 planes
of the cuprates are the two dimensional t-J-model and
Hubbard model. Much of the initial effort went into the
study of a single hole state of these models in an antifer-
romagnetic background. For the motion of this hole, there
is a competition between the gain in kinetic energy from
the hopping and its disruptive effect on the antiferromag-
netic order, and consequent loss of potential energy. As a
result a hole excitation becomes a quasiparticle or mag-
netic polaron, heavily dressed by antiferromagnetic spin
fluctuations (see review article by Dagotto [3] and refer-
ences therein). Much of this work, however, relied on exact
diagonalization or quantum Monte Carlo methods, which
are limited to small clusters and very few hole excitations,
and cannot be readily extended to study the many-hole,
finite doping situation.
More recent studies capable of describing finite dop-
ing have concentrated on the relation between the antifer-
romagnetic fluctuations and superconducting order (for
a review see [4] and the references therein). One of the
main motivations is to understand whether the exchange
of these types of fluctuations can provide a purely elec-
tronic mechanism for inducing superconductivity. Here, in
this paper, we focus on the metallic antiferromagnetism,
the doped state with long range antiferromagnetic order.
Our interest is to examine how well the low energy ex-
citations in this ordered state can be described in terms
renormalized quasiparticles. To tackle this problem we use
the infinite dimensional Hubbard model.
The simplification in the infinite dimensional limit is
that the electron self-energy becomes local in character,
with no wavevector dependence [5,6]. The self-energy then
depends only on the frequency, as is the case for impurity
models, allowing the lattice problem to be cast in the form
of a self-consistent impurity model. There are several rea-
sonably accurate techniques for solving this effective im-
purity problem, a very accurate one for the zero and low
temperature regime being the numerical renormalization
group approach (NRG) [7].
Recently we studied the effect of a magnetic field on
the quasiparticle excitations in the strong correlation regime
of the infinite dimensional Hubbard model using the NRG
method [8]. We also extended a form of renormalized per-
turbation theory (RPT), originally developed for impu-
rity models [9], to this model, and used it to calculate
the local dynamic spin susceptibilities, obtaining results
in good overall agreement with those from the NRG. In
this paper we extend this combination of renormalization
techniques, NRG and RPT within dynamical mean field
theory, to look at the low energy excitations of the infi-
nite dimensional Hubbard model in a staggered field, and
in antiferromagnetic broken symmetry states. Extensive
calculations of the antiferromagnetic states in the Hub-
http://arxiv.org/abs/0704.0243v2
2 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
bard model using the DMFT-NRG approach have already
been reported in the paper of Zitzler, Pruschke and Bulla
[10]. We confirm their results for the phase diagram and
extend the calculations and analysis to the description of
the low energy renormalized excitations, and how these
can be described within the framework of a renormalized
perturbation theory.
2 Antiferromagnetic Broken Symmetry in
In considering the response of the Hubbard model [11] to
a staggered magnetic field and antiferromagnetic order,
we take the case of a bipartite lattice, which consists of
two sublattices A and B such that the nearest neighbors
of a site in the A sublattice are on the B sublattice and
vice versa. The Hamiltonian for the Hubbard model can
be written in the form,
i,j,σ
(tijc
A,i,σcB,j,σ + h.c.) + U
nα,i,↑nα,i,↓
A,i,σcA,i,σ + µ−σc
B,i,σcB,i,σ), (1)
where the hopping matrix element is taken as tij = −t
between nearest sites i and j only, and zero otherwise,
and α = A,B. A staggered field His
His =
H for i ∈ A sublattice
−H for i ∈ B sublattice (2)
has been included so that µσ = µ+σh, where h = gµBH/2
with the Bohr magneton µB. The non-interacting part
of the Hamiltonian H0,µ can be diagonalized in terms of
Bloch states and then expressed in the form,
H0,µ =
k,σMk,σCk,σ. (3)
where C
k,σ = (c
A,k,σ, c
B,k,σ), and the matrix Mk,σ is
given by
Mk,σ =
−µσ εk
εk −µ−σ
. (4)
The k sums run over a reduced Brillouin zone, and the
energy of the Bloch state is εk =
j tije
i(Ri−Rj)·k. The
free Green’s function matrix G0k,σ(ω) is given by (ω −
Mk,σ)
−1. The poles of the free Green’s function give the
elementary single particle excitations, which are given by
E0k,±(U = 0) = −µ0(h)±
h2 + ε2
, (5)
where µ0(h) is the chemical potential of the noninteracting
system in a staggered field. This illustrates that the elec-
tronic excitations are split into two subbands for a finite
staggered field.
Notice that we have adopted a special choice of ba-
sis {cA,k,σ, cB,k,σ} here [12,10]. Another common basis to
study antiferromagnetic and spin density wave symmetry
(SDW) breaking is {ck,σ, ck+q0,σ}, where q0 is the recip-
rocal lattice vector for commensurate SDW ordering. The
bases can be related by a linear transformation,
k+q0,σ
cA,k,σ
cB,k,σ
. (6)
For the latter basis the matrix Mk,σ would be diagonal
in the kinetic energy term and the symmetry breaking
is offdiagonal. For our study in the DMFT framework the
A−B-sublattice basis is, however, more convenient and we
will use it throughout the rest of this paper. It is possible,
of course, to relate the obtained quantities with the help
of (6) to the {ck,σ, ck+q0,σ} basis.
We can generalize the equations to the interacting prob-
lem by introducing a self-energy Σα,k,σ(ω), so that the
matrix Green’s function can be written in the form
Gk,σ(ω)=
ζA,k,σ(ω)ζB,k,σ(ω)− ε2k
ζB,k,σ(ω) −εk
−εk ζA,k,σ(ω)
where ζα,k,σ(ω) = ω + µσ − Σα,k,σ(ω). As we are dealing
with the infinite dimensional limit of the model, we take
the self-energy to be local so we can drop the k index. This
is the reason why the self-energy has a single site index
α = A,B and no offdiagonal terms appear in equation (7).
The symmetry of the bipartite lattice gives ΣB,σ(ω) =
ΣA,−σ(ω) ≡ Σ−σ(ω) and hence
ζB,−σ(ω) = ζA,σ(ω) ≡ ζσ(ω),
where we have simplified the notation. To determine these
quantitiesΣσ(ω) it is sufficient to focus on the A sublattice
only.
Summing the first component in the Green’s function
in equation (7) over k we obtain the Green’s function for
a site on the A sublattice, Glocσ (ω),
Glocσ (ω) =
ζ−σ(ω)
ζσ(ω)ζ−σ(ω)
ρ0(ε)
ζσ(ω)ζ−σ(ω)− ε
, (8)
where ρ0(ε) is the density of states of the non-interacting
system in the absence of the staggered field.
In the DMFT this local Green’s function, and the self-
energy Σσ(ω), are identified with the corresponding quan-
tities for an effective impurity model [12]. This implies
that the Green’s function G0,σ(ω) for the effective impu-
rity in the absence of an interaction at the impurity site
is given by
G−10,σ(ω) = Glocσ (ω)−1 +Σσ(ω). (9)
We can take the form of this impurity model to correspond
to an Anderson model [13] in a magnetic field,
HAM =
εd,σd
σdσ + Und,↑nd,↓ (10)
(Vk,σd
σck,σ + V
k,σdσ) +
εk,σc
k,σck,σ,
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 3
where εd,σ = εd−σh is the energy of the localized level at
an impurity site in a magnetic fieldH , U the interaction at
this local site, and Vk,σ the hybridization matrix element
to a band of conduction electrons of spin σ with energy
εk,σ. As we are focusing on an A site as the impurity we
take H = Hs.
The one-electron Green’s function for the impurity site
of this model is given by
Gimpσ (ω) =
ω − εdσ −Kσ(ω)−Σσ(ω)
, (11)
where
Kσ(ω) =
|Vk,σ|2
ω − εk,σ
. (12)
If this impurity Green’s function is equated to the local
lattice Green’s function Glocσ (ω), we identify εdσ = −µσ
and from equation (9), Kσ(ω) is given by
Kσ(ω) = ω + µσ − G−10,σ(ω). (13)
The function Kσ(ω) plays the role of the effective medium
and has to be calculated self-consistently.
The self-consistent calculations for Kσ(ω) can usually
be performed iteratively. Starting from a conjectured form
for Kσ(ω), the NRG method is used to calculate the self-
energy of the effective Anderson model, from which the
impurity Green’s function Gimpσ (ω) in (11) and the local
Green’s function for the lattice Glocσ (ω) in (8) can be de-
duced. If these two Green’s functions do not agree, then
equation (9) is used to derive a new starting value for
Kσ(ω) and the process continued until self-consistency is
achieved.
To find antiferromagnetic solutions, we calculated self-
consistent solutions for a decreasing sequence of staggered
magnetic fields to see if broken symmetry solutions of this
type exist as the staggered field is reduced to zero. For
the non-interacting density of states ρ0(ε) we take the
Gaussian form ρ0(ε) = e
−(ε/t∗)2/
πt∗, corresponding to
an infinite dimensional hypercubic lattice. It is useful to
define an effective bandwidth W = 2D for this density of
states via D, the point at which ρ0(D) = ρ0(0)/e
2, giving
2t∗ corresponding to the choice in reference [14].
In all the results we present here we take the value W =
4. In the NRG calculations we have used the improved
method [15,16] of evaluating the response functions with
the complete Anders-Schiller basis [17], and also determine
the self-energy from a higher order Green’s function [18].
In figure 1 we show the self-consistently calculated lo-
cal spectral density for the spin-up (upper panel) and spin-
down electrons (lower panel) at an A site with U = 3 and
5% hole doping (from the state at half-filling) for various
values of an applied staggered field. The staggered mag-
netic field induces a sublattice magnetization,
(nA,↑ − nA,↓), (14)
so that these spectra are quite different. For this set of
parameters, this difference persists as the staggered field
−4 −2 0 2 4
h =0.05
h =0.1
−4 −2 0 2 4
h =0.05
h =0.1
Fig. 1. (Color online) The spectral densities for the spin-up
electrons (upper panel) and spin-down electrons (lower panel)
at the A site for various values of the applied staggered field
for U = 3 and x = 0.95
is reduced to zero so that we have a spontaneous sub-
lattice magnetization corresponding to spontaneous anti-
ferromagnetic order. For the case away from half filling,
δ 6= 0, we have to keep adjusting the chemical potential
when iterating for a self-consistent solution. It shows a
slightly oscillatory behavior when iterating for a specific
filling x, and we follow the procedure described in refer-
ence [10]. This feature is related to the fact that the calcu-
lations are for a metastable ground state and instabilities
to more complicated ground states for antiferromagnetic
ordering than the homogeneous, commensurate Néel state,
which forms the basis for these DMFT calculations, can
occur [19,20,21,22,23,24,25,26,10]. As far as phase sep-
aration in the ground state is concerned, the results of
our calculations are generally in line with the conclusions
in [10] as they are carried out within the same frame-
work. The focus of this work is, however, the analysis of
generic quasiparticle properties in a doped antiferromag-
netic state. We consider the approach as a valid, approxi-
mate starting point for this endeavor, but modifications to
the results presented here can occur for calculations based
4 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
on a more complicated ground states not accessible within
the DMFT framework. For a more extensive discussion of
the applicability of the DMFT in this situation we refer
to the earlier work [10].
From results of this type of calculation, we have built
up a global antiferromagnetic/paramagnetic phase dia-
gram as a function of the doping δ and the on-site inter-
action U . This phase diagram is shown in figure 2, where
the value of the corresponding sublattice magnetization is
shown in a false color plot. We have added a line separat-
ing the spontaneously ordered and paramagnetic regimes.
0 0.05 0.1 0.15 0.2
0.1 0.2 0.3 0.4
Fig. 2. (Color online) Phase diagram showing the doping and
the U dependence of the sublattice magnetization mA as de-
duced from the DMFT-NRG calculations.
At half filling (δ = 0 axis) the spontaneous magnetiza-
tion increases with U . We can see that the antiferromag-
netic order from the half filled case persists when holes
are added. The value of the critical doping δc at which
the antiferromagnetism disappears depends on the on-site
interaction U . We expect that for small U the critical dop-
ing δc will increase with U since a tendency to order only
appears when an on-site interaction is present. From the
mapping to the t−J model we also expect that for large U
the antiferromagnetic coupling J decreases and therefore
the order is destroyed more easily. The values of U are,
however, not large enough to display this trend.
If we compare these results with the phase diagram
given by Zitzler et al. [10] we see that they are in very good
agreement. In their case the antiferromagnetic region was
mapped out to values of U ≃ 4.5. As the iterations tend
to oscillate, as discussed before, there is a problem of ob-
taining a self-consistent antiferromagnetic solution in the
large U regime. We have managed to extend the diagram
to somewhat larger values of U by stabilising the calcula-
tions by averaging the effective medium over a number of
iterations.
3 Local Quasiparticle Parameters
To examine the nature of the low energy excitations, we
will assume that the self-energy Σσ(ω) is non-singular at
ω = 0 so that, at least asymptotically, it can be expanded
in powers of ω. This assumption is not expected to be valid
close to the quantum critical point when the magnetic
order sets in, but to be a reasonable assumption otherwise.
We also assume that the imaginary part of the self-energy
vanishes which is confirmed by the numerical results of the
DMFT-NRG calculations. We will retain terms to order
ω only for the moment. The higher order corrections will
be considered later. We then find for ζσ(ω),
ζσ(ω) = ω(1−Σ′σ(0)) + µσ −Σσ(0) (15)
= z−1σ (ω + µ̃0,σ), (16)
where
µ̃0,σ = zσ(µ−Σσ(0)), and z−1σ = 1−Σ′σ(0). (17)
The interacting Green’s function (7) has poles at the roots
of the quadratic equation,
ζσ(ω)ζ−σ(ω)− ε2k = 0. (18)
The solutions of this equation are
E0k,± = −µ̃±
+∆µ̃2, (19)
where ε̃k =
z↑z↓εk, ∆µ̃ = (µ̃0,↑ − µ̃0,↓)/2, and µ̃ =
(µ̃0,↑ + µ̃0,↓)/2. This has the same form as for the non-
interacting system in a staggered field (5), so we can in-
terpret these excitations as quasiparticles coupled to an
effective staggered magnetic field h̃s = ∆µ̃/gµB, with µ̃
playing the role of a quasiparticle chemical potential. This
equation gives the dispersion relation for these single par-
ticle excitations, which can be regarded as constituting a
renormalized band, or bands as there are two branches.
The term magnetic polaron is sometimes used to describe
these single particle excitations in states of magnetic or-
der, because of the analogy with the motion of a particle
in a lattice to which it is strongly coupled, where the ex-
citation is termed a polaron.
The corresponding density of states of these free local
quasiparticles on the sublattice is
ρ̃0,σ(ω)=
ω + µ̃− σ∆µ̃
ω + µ̃+ σ∆µ̃
(ω + µ̃)2 −∆µ̃2
for |ω + µ̃| > |∆µ̃|, and is zero otherwise. In the case of
a half-filled band µ̃ = 0 and there is a gap at the Fermi
level εF = 0.
To determine this local quasiparticle density of states
in the presence of the symmetry breaking staggered mag-
netic field we need to calculate zσ and µ̃0,σ for each spin
type. Using the NRG we can do this in two ways. As the
DMFT-NRG calculations give us the self-energyΣσ(ω) di-
rectly, we only need its value, and that of its first derivative
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 5
at ω = 0, to deduce both zσ and µ̃0,σ using equation (17).
However, because the model is solved using an effective
impurity model, we can also deduce these quantities indi-
rectly from the many-body energy levels of the impurity
on approaching the low energy fixed point [27]. This sec-
ond method gives us not only a check on the results of the
direct method, but also allows to deduce some informa-
tion about the quasiparticle interactions, as we shall show
in the next section.
3.1 Calculation of Renormalized Parameters
To describe how the renormalized parameters are deduced
from the energy levels of the NRG calculation, we need to
outline how the NRG calculations are carried out. Fol-
lowing the procedure introduced by Wilson [28], the con-
duction band is logarithmically discretized and the model
then converted into the form of a one dimensional tight
binding chain, coupled via an effective hybridization Vσ to
the impurity at one end. In this representation Kσ(ω) =
|Vσ|2g(N)0,σ (ω), where g
0,σ (ω) is the one-electron Green’s
function for the first site of the isolated conduction elec-
tron chain of length N . The impurity Green’s function for
this discretized model then takes the form,
Gimpσ (ω) =
ω − εdσ − |Vσ|2g(N)0,σ (ω)−Σσ(ω)
. (21)
We can find the quasiparticle excitations of this model
by expanding the self-energy Σσ(ω) in the denominator of
this equation to first order in ω, and write the result in
the form,
Gimpσ (ω) =
ω − ε̃dσ − |Ṽσ |2g(N)0,σ (ω) +O(ω2)
, (22)
where
ε̃dσ = zσ[εdσ +Σσ(0)], |Ṽσ|2 = zσ|Vσ|2. (23)
We can then define a free quasiparticle propagator, G̃0,σ(ω),
0,σ (ω) =
ω − ε̃dσ − |Ṽσ |2g(N)0,σ (ω)
, (24)
and interpret zσ as the local quasiparticle weight.
In the NRG calculation the many-body excitations are
calculated iteratively, starting at the impurity site, and
increasing the chain length N by one site with each itera-
tion. When the matrices become too large to handle, only
the lowest 500-1500 states are kept at each iteration. The
many-body energy levels for the Nth iteration and the
set of quantum numbers M , EM (N), depend on the chain
length N and the discretization parameter Λ > 1. When
N becomes large these energy levels go to zero as Λ−N/2.
We now conjecture that the lowest single particle Eσp (N)
and single hole excitations Eσh (N) determined from the
NRG many-body excitations correspond to quasiparticle
excitations. If this is the case then they should correspond
to the poles of the quasiparticle Green’s function given in
equation (24), with values of Ṽσ and ε̃dσ, which are inde-
pendent of N as N → ∞. We can test this hypothesis by
substituting the values, ω = Eσp (N) and ω = E
h (N), into
the equation,
ω − ε̃dσ − |Ṽσ|2g(N)0,σ (ω) = 0, (25)
and deduce values of Ṽσ and ε̃dσ, which will in general
depend upon N . From these we can deduce zσ = |Ṽσ/Vσ|2
and µ̃0,σ = −ε̃dσ, which will also depend upon N , but if
the lowest single particle excitations of the system do cor-
respond to free quasiparticles, the values of zσ and µ̃0,σ
will become independent of N for large N . It should be
noted that we need both the particle and hole excitations
for each spin to determine the four renormalized parame-
ters. The parameters corresponding to spin up involve the
particle excitations with spin up and the hole excitations
with spin down.
That parameters can be found, which are independent
ofN for largeN , can be seen in figure 3, where we take the
results of a Kσ(ω) and µσ from the antiferromagnetic self-
consistent solution for the Hubbard model with U = 3 and
10% doping, using a value for the discretization parameter
Λ = 1.8.
0 10 20 30 40 50
Fig. 3. (Color online) The N-dependence of the renormalized
parameters zσ and µ̃0,σ for U = 3 and x = 0.9.
It can be seen that after about 25 iterations all the values
deduced for zσ and µ̃0,σ become independent of N . In the
next section, where we compare these results with the cor-
responding values deduced directly from the self-energy,
we get further confirmation that the values deduced re-
ally do describe the quasiparticle excitations of the lattice
model.
When two or more quasiparticles are excited from the
interacting ground state, there will be an interaction be-
tween them. For the Anderson impurity model this inter-
action will be local and can be expressed as Ũ , a renor-
malized value of the original interaction of the ‘bare’ par-
6 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
ticles. The value of Ũ can be deduced by looking at low-
est lying two-particle excitations derived from NRG cal-
culation. These could either be two-particle excitations,
E↑,↓pp (N), two-hole excitations, E
hh (N) or a particle-hole
excitation E
ph (N). By looking at the difference between
a two-particle excitation and two single particle excita-
tions, E↑,↓pp (N) − E↑p(N) − E↓p(N), as a function of N
we can deduce an effective interaction Ũ↑,↓pp (N) between
these two quasiparticles, as has been described fully ear-
lier for the standard Anderson model [27]. In a similar way
we can deduce an effective interaction between two holes,
hh (N), or a particle and hole, −Ũ
ph (N). To be able to
define a single quasiparticle interaction Ũ , not only must
Ũ↑,↓pp (N), Ũ
hh (N) and Ũ
ph (N), give values which are in-
dependent of N for large N , these values must be equal
so Ũ↑,↓pp = Ũ
hh = Ũ
ph = Ũ .
0 10 20 30 40 50
Fig. 4. (Color online) The N-dependence of the renormal-
ized particle-particle, particle-hole and hole-hole interactions
for U = 6 and x = 0.9, showing that they converge to a unique
value Ũ .
In figure 4 we give the values of Ũ↑,↓pp (N), Ũ
hh (N) and
ph (N) as deduced from DMFT-NRG calculation for the
Hubbard model in an antiferromagnetic state with U = 6,
10% doping and Λ = 1.8. It can be seen that the three
sets of results settle down to a common value Ũ .
We can go further and identify Ũ with the local quasi-
particle 4-vertex interaction for the effective impurity model,
Ũ = z↑z↓Γ↑,↓,↓,↑(0, 0, 0, 0), (26)
where Γ↑,↓,↓,↑(ω1, ω2, ω3, ω4) is the total 4-vertex at the
impurity site, which is equal to the same quantity for a
site in the lattice model. With this interpretation it is
possible to identify these parameters with those used in a
renormalized perturbation expansion. The parameters, V ,
εd,σ and U , together with g
0,σ(ω), specify the effective im-
purity model. The renormalized parameters, Ṽ , ε̃d,σ and
Ũ , together with gN0,σ(ω), can be used as an alternative
way of specifying this model. The renormalized perturba-
tion theory (RPT) is set up by expanding the self-energy
to order ω, as earlier, but retaining all the higher order
correction terms in a remainder term,
Σσ(ω) = Σσ(0) + ωΣ
σ(0) +Σ
σ (ω), (27)
where Σremσ (ω) is the remainder term. On substituting
this into the equation for the impurity Green’s function in
equation (11), we can deduce a general expression for the
quasiparticle Green’s function in the form,
G̃impσ (ω) =
ω − ε̃dσ − K̃σ(ω)− Σ̃σ(ω)
, (28)
where K̃σ(ω) = zσKσ(ω) and Σ̃σ(ω) = zσΣ
σ (ω) plays
the role of a renormalized self-energy. A diagrammatic
perturbation theory can then be carried out for Σ̃σ(ω)
in terms of the free quasiparticle propagators, with ad-
ditional diagrams arising from counter terms, which are
required to prevent over-counting (renormalization condi-
tions) [29,9,30]. This form of perturbation theory is valid
for all energy scales but is particularly effective for calcu-
lating the low energy terms arising from the quasiparticle
interactions. For the symmetric Anderson impurity model
it has been shown that this perturbation theory taken to
second order in Ũ , gives the exact spin and charge suscep-
tibilities at T = 0, and the exact T 2 contribution to the
conductivity [9].
Because, within DMFT, the self-energy for the lat-
tice is the same as that for the effective impurity, we can
equally well use the effective impurity model to calculate
it. This means that we can use the renormalized pertur-
bation theory for the effective impurity model to estimate
the correction terms to the free quasiparticle picture aris-
ing from the quasiparticle interactions.
3.2 Local Quasiparticle Weight
We now consider the values of the local quasiparticle weight
factor zσ, commonly known also as the wavefunction renor-
malization factor. This is an important factor in deter-
mining the parameters needed to describe the low energy
behavior of the system. When there is no k-dependence
of the self-energy, as is the case for infinite dimensional
models and DMFT, the effective mass of the quasiparti-
cles in the paramagnetic state is proportional to 1/zσ. We
show later that in the antiferromagnetic state the expres-
sion is more complicated and depends both on zσ and the
renormalized chemical potential µ̃0,σ. We have determined
zσ from the NRG results by the two methods described
and give the values of zσ deduced for both spin types as a
function of doping in figure 5. The results are for the case
U = 3, where there is antiferromagnetic order and the ex-
ternal staggered field has been set to zero. It can be seen
that there is a reasonable agreement between the values
obtained by the two different methods of calculation. Visi-
ble differences can be attributed to the inaccuracies when
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 7
0 0.05 0.1 0.15 0.2
 from Σ
 from Σ
 from FP
 from FP
Fig. 5. (Color online) The local quasiparticle weight zσ as de-
duced directly from the self-energy and also from the impurity
fixed point (FP) for U = 3 and various dopings.
numerically computing the derivative of the self-energy,
whose calculation involves a broadening procedure. When
the system is doped but still ordered we have z↑ 6= z↓, and
the renormalization effects are stronger for the minority
(down) spin particles on the sublattice. This is similar to
the results we found for a doped Hubbard model in a para-
magnetic state in the presence of a strong uniform mag-
netic field [8]. For a certain range of dopings the values of
z↑ and z↓ do not vary much. The tendency is that z↓ first
decreases and later increases, whereas z↑ decreases over
the whole range until both of them merge at the doping
point where the antiferromagnetic order disappears.
The results for the corresponding case with U = 6, a
value which is larger than the bandwidth, are shown in
figure 6.
0 0.05 0.1 0.15 0.2
 from Σ
 from Σ
 from FP
 from FP
Fig. 6. (Color online) The local quasiparticle weight zσ as
deduced directly from the self-energy and from the impurity
fixed point (FP) for U = 6 and various dopings.
On the whole the behavior is quite similar to that for
the case U = 3, only that the renormalization effects
are more pronounced. For a range of dopings the local
quasiparticle weights do not change much and have the
same tendency as described above. The implications for
the spectral quasiparticle weight and the effective mass
enhancement will be discussed in detail later.
3.3 Renormalized chemical potential
In figure 7 we give the results for the renormalized chemi-
cal potential, µ̃0,σ [defined in equation (17) and (23)], for
the two spin types in the spontaneously ordered antifer-
romagnetic states for U = 3 and U = 6 for a range of
dopings.
0 0.05 0.1 0.15
 from Σ
 from Σ
 from FP
 from FP
0 0.05 0.1 0.15
 from Σ
 from Σ
 from FP
 from FP
Fig. 7. (Color online) The renormalized chemical potential
µ̃0,σ as deduced directly from the self-energy and from the
impurity fixed point (FP) for various dopings for U = 3 (upper
panel) and U = 6 (lower panel).
The values calculated by the two different methods can be
seen to be in good agreement here, as well. We have added
the values for the half filled case. These were calculated
from the self-energy in the gap at ω = 0. The general
8 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
behavior of the values for µ̃0,σ for the case with U = 6 is
very similar to the case with smaller U
The renormalized chemical potential µ̃0,σ is an impor-
tant parameter in specifying the form of the local sublat-
tice quasiparticle spectral density ρ̃0σ(ω). From equation
(20) it can be seen that, as ω → −µ̃0,σ, ρ̃0,σ(ω) behaves
asymptotically as
ρ̃0,σ(ω) ∼
ω + µ̃0,σ
, (29)
so the quasiparticle density of states has a square root
singularity at ω = −µ̃0,σ. On the other hand, however, as
ω → −µ̃0,−σ, ρ̃0,σ(ω) behaves as
ρ̃0,σ(ω) ∼
ω + µ̃0,−σ, (30)
so the quasiparticle density of states goes to zero at ω =
−µ̃0,−σ. Between the two points, ω = −µ̃0,σ and ω =
−µ̃0,−σ, the quasiparticle density of states has a gap of
magnitude 2∆µ̃. As can be seen in figure 7 this free quasi-
particle gap decreases with increasing doping and closes
in the paramagnetic state. If we take into account the
values at half filling we see a strong reduction of 2∆µ̃,
when doping the system. We also see that µ̃0,↑ drops to
small negative values for finite hole doping, which corre-
sponds to the fact that the Fermi level then lies within the
lower band. These features will be seen clearly in the fig-
ures presented in the next section, where we compare the
quasiparticle densities of states with the full local spectral
densities calculated from the DMFT-NRG.
3.4 The Quasiparticle Interaction
The quasiparticles can be further characterized by an ef-
fective interaction Ũ as described before. In figure 8 we
plot the doping dependence of the renormalized interac-
tion over a range of dopings and U = 3 and U = 6.
We can see that in both cases the values decrease with
increasing doping. Hence, the effective quasiparticle inter-
action is stronger for a smaller hole density. For a certain
range of dopings, however, Ũ does not vary much. We can
also see that the ratio Ũ/U for the effective interaction as-
sume smaller values the larger the bare U becomes. Also
the absolute value of Ũ , i.e. without the scaling with U as
in figure 8, is smaller for larger bare U for the full range
of dopings. This effect of smaller quasiparticle interactions
for the stronger coupling case can be seen as sharper quasi-
particle peaks for larger U as will be discussed in the next
section.
4 Spectra and Quasiparticle Bands
4.1 Local Spectra
In this section we examine how well the local sublattice
quasiparticle density of states ρ̃0,σ(ω), evaluated from equa-
tion (20) with the renormalised parameters, describes the
0.05 0.1 0.15
 U / U for U =3
 U / U for U =6
Fig. 8. (Color online) The renormalised quasiparticle interac-
tion Ũ/U as deduced from the impurity fixed point for various
dopings and U = 3, 6.
low energy features seen in the local spectral density ρσ(ω)
calculated from the DMFT-NRG. At half filling there is
a gap at the Fermi level, so there are no single particle
excitations in the immediate neighbourhood of the Fermi
level, and this is not a very interesting case to consider.
We look in detail at the case of 10% doping where the
Fermi level lies at the top of the lower band, and consider
the two cases U = 3 and U = 6. In the upper panel of
figure 9 we compare the spectral density ρ↑(ω) with the
corresponding quantity z↑ρ̃0,↑(ω), from the quasiparticle
density of states.
We see that the behavior near the Fermi level (ω = 0),
and the singular feature seen in the lower branch of ρ↑(ω),
are well reproduced by the quasiparticle density of states.
Above the Fermi level there is a peak in the quasiparti-
cle density of states similar to that in the full spectrum
but somewhat more pronounced. Above the Fermi level
and below the upper peak there is a pseudo-gap region. In
the free quasiparticle spectrum it is a definite gap. In the
spectrum calculated from the direct NRG evaluation it ap-
pears as a pseudo-gap, with rather small spectral weight
just above the Fermi level. From the direct DMFT-NRG
calculations, due to the broadening features introduced to
obtain a continuous spectrum, it is not always possible
to say definitively whether there is a true gap above the
Fermi level or not. To resolve this question we can ap-
peal to the renormalised perturbation theory to look at
the corrections to the quasiparticle density of states aris-
ing from the quasiparticle interactions. A calculation of
the imaginary part of the renormalised self-energy Σ̃σ(ω)
to order Ũ2 should be sufficient to settle this issue. The
imaginary part of the second order diagram for the renor-
malised self-energy in the limit T → 0 for ω > 0 is given
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 9
−2 −1 0 1 2 3 4
−2 −1 0 1 2 3 4
Fig. 9. (Color online) The free local quasiparticle spectrum
(dashed line) in comparison with DMFT-NRG spectrum for
x = 0.9 and U = 3 for the spin-up electrons (upper panel) and
spin-down electrons (lower panel).
ImΣ̃(2)σ (ω) = πŨ
dε2 ρ̃0,σ(ε1)ρ̃0,−σ(ω − ε1 + ε2)
× ρ̃0,−σ(ε2)θ(ω − ε1 + ε2), (31)
where ρ̃0,σ(ε) is the free quasiparticle density of states.
The integration area is a triangle in the (ε1, ε2)-plane as
shown in figure 10.
To analyze the behavior of ImΣ̃
σ (ω) in the regime |µ̃0,↑| <
ω < |µ̃0,↓| we have to study where the integrand is non-
zero taking into account that ρ̃0,σ(ε) = 0 for |µ̃0,↑| <
ε < |µ̃0,↓|. The only non-zero contribution comes from
the small shaded region in figure 10, which leads to the
estimate,
ImΣ̃(2)σ (ω) ≃ πŨ2ρ̃0,σ(0)ρ̃0,−σ(−ω)ρ̃0,−σ(0)µ̃20,↑. (32)
When µ̃0,↑ is small, which occurs when the lower edge of
the gap in the quasiparticle density of states is very near
the Fermi level, this contribution to the imaginary part
PSfrag replacements
−ω + |µ̃0,↑|
|µ̃0,↑| |µ̃0,↓|
Fig. 10. (Color online) Integration region in the (ε1, ε2)-plane
for the imaginary part of the self-energy. The original triangle
region (0, ω,−ω) for integration in equation (31) is reduced in
the gap region, |µ̃0,↑| < ω < |µ̃0,↓|, to the small shaded region
shown in the figure.
of the renormalized self-energy will be finite but small. It
decreases with ω due the behavior of ρ̃0,−σ(−ω). Based on
this argument we conclude that there is a small, but finite
imaginary part of the self-energy in the free quasiparticle
gap 2∆µ̃, when it lies above the Fermi level, giving rise
to a finite spectral weight there. However, this spectral
weight is very small close to the lower edge of the free
quasiparticle density of states, when this edge lies only
just above the Fermi level.
In the lower panel of figure 9 we compare the ρ↓(ω)
with z↓ρ̃0,↓(ω). We see that in this case also the quasi-
particle density of states reproduces well the spectrum in
the region of the Fermi level and the peak structure in
the lower band, which is non-singular in this case. The
position of the peak above the Fermi level is also well re-
produced, but the peak in the free quasiparticle density
of states is singular, whereas that in the DMFT-NRG re-
sults is not. We would expect to lose this singularity in the
free quasiparticle density of states once the quasiparticle
scattering is taken into account and the renormalized self-
energy is included. It is possible also that the peak above
the Fermi level in the DMFT-NRG spectrum should be
sharper, as there is some tendency for the broadening in-
troduced in this approach to flatten peaked features in
regions away from the Fermi level. The spectral weight
in the pseudo-gap is even smaller than in the case for the
spin-up electrons, particularly in the region of the gap that
lies closest to the Fermi level. This is qualitatively in line
with the conclusions based on the renormalized pertur-
bation theory estimate of the effects of the quasiparticle
scattering.
We see very similar features in the spectra for the case
U = 6 and also 10% doping shown in figure 11.
Here, the peaks near the Fermi level are a bit sharper. The
observations made on the comparison of the quasiparticle
and DMFT-NRG spectra apply equally well to this case.
In addition to the low energy features charge peaks cor-
responding to the Hubbard bands appear. The lower one
can be identified in the full spectra, whereas the upper
10 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
−2 −1 0 1 2 3
−2 −1 0 1 2 3
Fig. 11. (Color online) The free quasiparticle spectrum
(dashed line) in comparison with DMFT-NRG spectrum for
x = 0.9 and U = 6 for the spin-up electrons (upper panel) and
spin-down electrons (lower panel).
Hubbard peak is not seen on the energy scale shown. The
quasiparticle density of states does not contain informa-
tion about these features at higher energy.
4.2 k-resolved Spectra
We can learn more about the low energy single parti-
cle excitations by looking at the spectral density of the
Green’s function Gk,σ(ω) in equation (7) for a given wave-
vector k. With the self-energies Σσ(ω) calculated within
the DMFT-NRG approach all elements of this matrix can
be evaluated. The local spectra and self-energies are spin-
dependent in the doped broken symmetry state, however,
the free quasiparticle bands E0
k,± [equation (19)] do not
depend on the spin. Here, we focus on the diagonal part
of Gk,σ(ω) corresponding to the A sublattice,
Gk,σ(ω) =
ζ−σ(ω)
ζσ(ω)ζ−σ(ω)− ε2k
. (33)
The weights of the quasiparticle excitations in this case
depend on the spin corresponding to the sublattice prop-
erties. We note that one can also analyze the quasiparticle
bands differently, for instance, from the k-resolved spectra
and the diagonal form of Gk,σ(ω). The form of the quasi-
particle bands remains unchanged then, but the weights
differ and do not depend on the spin σ in that case.
We first of all look at the Fermi surface which is the
locus of the k-points at the Fermi level (ω = 0) where
the Green’s function has poles. The conduction electron
energy εkF at these point is given by
ε2kF = (µ↑ −Σ↑(0))(µ↓ −Σ↓(0)). (34)
By Luttinger’s theorem, the volume of the Fermi sur-
face for the interacting system must equal that for the
non-interacting system with the same density. As the self-
energy depends only on ω, the two Fermi surfaces must
also have the same shape, and therefore must be identical.
The Fermi surface of the non-interacting system is given
by εkF = µ0, where µ0 is the chemical potential of the
non-interacting system in the absence of any applied field
for the given density. For this to be identical with that
given in equation (34),
(µ↑ −Σ↑(0))(µ↓ −Σ↓(0)) = µ20. (35)
We can check that this relation indeed holds from our
results for Σσ(ω) and µσ, independent of the value of U ,
or in the case of an applied staggered field, independent of
the field value. This relation implies that the total number
of electrons per site n can be calculated from an integral
over the non-interacting density of states,
n = 2
ρ0(ω)dω, (36)
where in the hole doped case µ0 = −
µ̄↑µ̄↓ and µ̄σ =
µσ −Σσ(0).
To relate this result to the quasiparticle picture, we
expand the self-energy in equation (33) to first order in
ω, but retain the remainder term, ΣRσ (ω) as in equation
(27). The Green’s function can be rewritten in the form,
G̃k,σ(ω) =
ζ̃−σ(ω)
ζ̃σ(ω)ζ̃−σ(ω)− ε̃2k
, (37)
where ζ̃σ(ω) = ω + µ̃0,σ − Σ̃σ(ω). We define a quasipar-
ticle Green’s function G̃k,σ(ω) via zσG̃k,σ(ω) = Gk,σ(ω).
The renormalized self-energy vanishes, Σ̃σ(ω) = 0, for the
free quasiparticle Green’s function G̃
k,σ(ω), which can be
separated into two independent branches of free quasipar-
ticles,
k,σ(ω) =
uσ+(εk)
ω − E0
uσ−(εk)
ω − E0
, (38)
where E0
k,± was defined in equation (19) and the weights
are given by
uσ±(εk) =
1∓ σ ∆µ̃√
∆µ̃2 + ε̃2
. (39)
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 11
This is similar in form to mean field theory, which would
correspond to putting zσ = 1, and ∆µ̃ = Ummf , where
mmf is the mean field sublattice magnetization. The spin
dependent contribution in (39) which arises from the sec-
ond term is most marked in the region near the Fermi
level. It should be noted that the quasiparticle excitations
k,± and weights u
±(εk) here are defined by expanding
the self-energy at ω = 0. This is so that they correspond
to the free quasiparticles in the renormalized perturbation
theory which have an infinite lifetime.
The spectral density ρ̃
(ω) for this free quasiparticle
Green’s function is a set of delta-functions,
k,σ(ω) = u
+(εk)δ(ω−E0k,+)+uσ−(εk)δ(ω−E0k,−). (40)
On the Fermi surface E0k,− = 0, which is consistent with
the result for the Fermi surface given in equation (34).
Summing over k gives the local quasiparticle density of
states in equation (20). We define the quasiparticle num-
ber ñ as the integral of the sum of the spin up and spin
down quasiparticle density of states up to the Fermi level,
dω(ω + µ̃)
(ω + µ̃)2 −∆µ̃2
(ω + µ̃)2 −∆µ̃2
If we change the variable of integration to ω′, where
z↑z↓ =
(ω + µ̃)2 −∆µ̃2,
the integration can be shown to be identical with that
in equation (36), using the fact that µ0 = −
µ̄↑µ̄↓. We
then have an alternative statement of Luttinger’s theorem
in the form ñ = n. This can also be found by summing
both spin components in (40), integrating over ω and then
converting the k-summation to an integral over the free
electron density of states ρ0(ω). We can check in our nu-
merical results that the relation in this form holds. The
occupation number n can be calculated both from a direct
evaluation of the number operator in the ground state, and
also by integrating the sum of the spectral densities ρσ(ω)
of the full local Green’s function to the Fermi level. The
value of ñ is similarly determined from the integral over
the total quasiparticle density of states, ρ̃σ(ω). All three
results were found to be in good agreement, to within one
or two percent deviation at the most.
Before discussing the k-resolved spectra in detail we
would like to ask what the spectral weight wqp of a quasi-
particle excitation at the Fermi level in the lower band is,
such that the Green’s function reads there
Gqp(ω) =
ω − E0
. (42)
To calculate wqp, we can not focus on the spin depen-
dent local sublattice quantities, but have to sum over both
sublattices or equivalently the two spin components. The
reason for this is that the antiferromagnetically ordered
state does not possess any net magnetization and has on
average as many spin up polarized as spin down electrons.
The division in the A and B sublattices is convenient for
the DMFT calculations but somewhat artificial. In our
case with hole doping the Fermi level lies within the lower
band, which for the free quasiparticles is denoted by E0
The corresponding weight on the Fermi surface defined by
(34) is then given by
wqp =
−(εkF) =
z↑ + z↓
(z↑ − z↓)∆µ̃
2|µ̃|
, (43)
where the average of the renormalized chemical potential
µ̃ and the difference ∆µ̃ were defined below equation (19).
From the definition of ∆µ̃ we can see that the second term
in (43) is spin rotation invariant. The spectral quasipar-
ticle weight wqp on the Fermi surface depends not only
on the renormalization factors zσ, but also on the renor-
malized chemical potentials µ̃0,σ. The same result for the
weight (43) can be obtained from the diagonal form of
Gk,σ(ω) and the spectral weight of the lower band. The
weight wqp corresponds to the spectral weight Z at the
Fermi level as for example given in references [3,31,32].
The first term of the result for wqp is like the arithmetic
average of zσ. From figures 5 and 6 we can see that z↑ > z↓
and from figure 7 that µ̃0,↓ < µ̃0,↑ < 0. Therefore the sec-
ond term in (43) gives a positive contribution to the spec-
tral weight. At the end of the section in figure 18 we show
values of wqp in comparison with the arithmetic average
of zσ.
In order to understand better the properties of the
quasiparticle bands, we now compare the quasiparticle
spectrum with the k-resolved spectral density ρk,σ(ω) de-
rived from the DMFT-NRG results. In figure 12 we make
a comparison for the case of 12.5% doping with U = 3
for the Green’s function Gk,σ(ω) given in equation (33),
ρk,σ(ω) = −ImGk,σ(ω+)/π, where ω+ = ω+ iη, with η →
0, with that derived for the free quasiparticles, zσ ρ̃
k,σ(ω)
from equation (40). The delta-functions of the free quasi-
particle results are indicated by arrows with the height of
the arrow indicating the value of the corresponding spec-
tral weight. The plots as a function of ω are shown for a
sequence values of εk and, where the peaks in ρk,σ(ω)
get very narrow and high in the vicinity of the Fermi
level, they have been truncated. It can be seen that the
free quasiparticle results give a reasonable picture of the
form of ρk,σ(ω), particularly in the immediate region of
the Fermi level. There is considerable variation along the
curves in the way the overall spectral weight is distributed
between the excitations below and above the pseudo-gap
as a function of εk. This is most marked in the region near
the Fermi level for the spin-up electrons (upper panel)
where most of the spectral weight is in the lower band and
it is much reduced in the upper band, whereas the opposite
is the case for the spin-down electrons. This is reflected
in the expression of the quasiparticle weights uσ±(εk) in
equation (39). For instance, u
−(εk) corresponding to the
lower band E0
k,− becomes maximal near the Fermi en-
ergy, whereas u
+(εk) goes to zero there. The finite width
of the quasiparticle peaks in ρk,σ(ω) can be described by
a RPT, when we take into account the renormalized self-
12 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
Fig. 12. (Color online) The spectral density ρk,σ(ω) for the
spin-up electrons (upper panel) and spin-down (lower panel)
plotted as a function of ω and a sequence of values of εk for
U = 3 and 12.5% doping. Also shown with arrows are the
positions of the free quasiparticle excitations, with the height
of the arrow indicating the corresponding weight.
energy Σ̃σ(ω) in equation (37). If we, for instance, use
the the second order approximation in Ũ , which was illus-
trated in the last section (31), we get a similar behavior
for small ω as seen for ρk,σ(ω) in figure 12.
From the positions of the peaks in the ρk,σ(ω) spectra
we can deduce two branches of an effective dispersion Ek,±
for single particle excitations and compare it with the ones
for the free quasiparticles E0
k,±. We give the results for
U = 3 in figure 13.
It can be seen that E0
k,− tracks the peak in the lower
band closely over a wide range of εk, −1.5 < εk < 1.5
(note the bandwidth W = 4). This is not the case in the
upper band, where E0
k,+ tracks the peak closely only in
the lowest section that lies closest to the Fermi level. As
one can see from the dotted line the Fermi level lies in
the lower band and intersects the lower band twice. This
corresponds to the two values with opposite sign ε±
can be see from equation (34).
The corresponding results for the k-resolved spectra
for U = 6 and also 12.5% doping are shown in figure 14.
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 13. (Color online) A plot of the peak positions Ek,± in
the spectral density ρk,σ(ω) (full line) as a function of εk for
U = 3 and 12.5% doping compared with the free quasiparticle
dispersion E0k (dashed line).
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
Fig. 14. (Color online) The spectral density ρk,σ(ω) for the
spin-up electrons (upper panel) and spin-down (lower panel)
plotted as a function of ω and a sequence of values of εk for
U = 6 and 12.5% doping. Also shown with arrows are the
positions of the free quasiparticle excitations, with the height
of the arrow indicating the corresponding weight.
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 13
In order to compare well with the case U = 3 we have
chosen an identical range for ω and εk, although the large
spectral peaks near the energy are very close together in
this presentation. It can be seen that the overall features
are very similar to those seen for U = 3. For the spin
up spectrum (upper panel) the peaks for the lower band
have most of the weight near the Fermi energy, whereas
the upper band is suppressed there, and vice versa for
the opposite spin direction. The lower bands are tracked
well by the free quasiparticles, and we can see that the
bands for the larger value of U are significantly flatter.
This is also clearly visible in the following figure 15, where
we again compare the quasiparticle band with the peak
position of the full spectra. On the range shown the lower
band Ek,− completely coincides with the free quasiparticle
band E0
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 15. (Color online) A plot of the peak positions Ek,± in the
spectral density ρk,σ(ω) (full line) as a function of εk for U = 6
and 12.5% doping compared with the free quasiparticle disper-
sion E0k (dashed line). On the range shown the lower band Ek,−
completely coincides with the free quasiparticle band E0k,−.
From the k-resolved spectra in figures 12 and 14 we can
also extract the width of the quasiparticle peak ∆qp in
the spectral density ρk,σ(ω) (majority spin σ =↑). Its in-
verse 1/∆qp gives a measure of the quasiparticle lifetime.
The results for ∆qp for the lower band Ek,− for the two
cases U = 3, 6 and 12.5% doping are shown in figure 16 as
function of εk.
This plot brings out more clearly the feature that can
be seen already in figures 12 and 14 (upper panel) that
the width increases sharply when we move away from the
Fermi level and the values for the width ∆qp for U = 6
are significantly smaller than those for U = 3. This is in
line with the fact that the local quasiparticle interaction
Ũ is smaller for the larger value of the bare interaction U
as commented on earlier. The free quasiparticle picture is
therefore even more appropriate in the case with stronger
interaction. To numerical accuracy the width vanishes at
and is finite for the interval ε−
< εk < ε
which lies
within the lower band but above the Fermi level.
−1 −0.5 0 0.5 1
Fig. 16. (Color online) A plot of the width of the peaks ∆qp in
the spectral density of the majority spin ρk,↑(ω) as a function
of εk for U = 3 (dashed line) and U = 6 (full line) and 12.5%
doping.
Another quasiparticle property, the effective mass en-
hancement m∗/m, can be extracted by calculating the
derivative of E0k,− in (19) with respect to εk, which yields
when evaluated at the Fermi energy (34),
µ̃0,↑µ̃0,↓
. (44)
The effective mass enhancement therefore does not only
depend on zσ, but also on the renormalized chemical po-
tentials µ̃0,σ. The general trend for m
∗/m as function of
U can be seen in figure 17 for the case of 7.5% doping.
0.2 0.4 0.6 0.8 1
Fig. 17. The ratio m∗/m according to (44) plotted over a
range of t2/U for 7.5% doping.
The effective mass increases sharply for large U as the
hole motion is energetically more costly in the ordered
background. The fact that the lower band for U = 6 seen
in figure 15 is flatter than in the case U = 3 in figure 13
14 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
can be attributed to the larger effective mass. We find a
similar behavior of m∗/m as function of U for different
filling factors from the ones shown in figure 16. The trend
is that the effective mass enhancement is less pronounced
for larger doping, which is intuitively understandable by
the quasiparticle motion in an ordered background.
In the DMFT framework for the paramagnetic state
as well as the case with homogeneous magnetic field, the
quasiparticle spectral weight wqp and the inverse of the ef-
fective mass enhancement m/m∗ can be described simply
by the renormalization factor zσ. In figure 18 we show a
comparison of the spectral quasiparticle weight wqp (43),
the arithmetic, (z↑ + z↓)/2, and geometric,
z↑z↓, aver-
age of the renormalization factors, and the inverse of the
effective mass, m/m∗, (44) for U = 3 for various dopings.
0 0.05 0.1 0.15
m/ m*
Fig. 18. (Color online) Comparison of the spectral quasiparti-
cle weight wqp from equation (43), the arithmetic, (z↑+ z↓)/2,
and geometric,
z↑z↓, average of the renormalization factors,
and the inverse of the effective mass, m/m∗ from equation (44),
for U = 3 and a range of dopings.
As seen in this case with antiferromagnetic symmetry break-
ing these quantities take a different form (43) and (44)
and have distinct values. For different values of U the
behaviour is qualitatively similar. As a first approxima-
tion the quasiparticle spectral weight wqp corresponds to
the arithmetic average of the renormalization factors zσ,
whilst m/m∗ relates to the geometric average. In general,
one can, however, not omit the dependence on the renor-
malized chemical potential as it gives a significant contri-
bution as can be seen in figure 18. This can be understood
for example for the limit of zero doping. The system then
becomes an antiferromagnetically ordered insulator with
spectral gap. The weights zσ tend to finite values, but the
effective mass must diverge. This is found in equation (44)
since µ̃0,↑ → 0 for δ → 0, and the trend can be seen in
figure 18.
5 Conclusions
We have studied the field induced and spontaneous anti-
ferromagnetic ordering in the hole doped Hubbard model
with DMFT-NRG calculations at T = 0. A phase diagram
separating antiferromagnetic and paramagnetic solutions
for different values of doping and interactions U ranging
from zero to about 1.5 times the bandwidth W has been
established and is in agreement with earlier results by
Zitzler et al. [10]. Our main objective has been to ana-
lyze the properties of the quasiparticle excitations in the
metallic antiferromagnetic state. We presented two differ-
ent ways of calculating the parameters zσ and µ̃0,σ, which
define the renormalized quasiparticles, and the two sets
of results have been shown to be in agreement. We have
also been able to deduce the effective on-site quasiparticle
interaction Ũ from the NRG low lying excitations. The
low energy properties of the local spectral function can
be understood in terms of the free quasiparticle picture.
We have used the second order perturbation expansion in
powers of Ũ to estimate the spectral weight in the pseudo-
gap region above the Fermi level.
We have been able to compare the position of the
peaks found in the k-dependent spectral functions with
the dispersion relation for the free quasiparticles. The free
quasiparticle dispersion gives a very good fit to the posi-
tion of these peaks in the lower band which intersects the
Fermi level. The quasiparticle lifetime, as deduced from
the widths of the peaks in the spectrum, increases for
stronger interactions. This is consistent with the fact that
the on-site quasiparticle interaction Ũ , which gives the
quasiparticles a finite lifetime, decreases with increase of
U in the same range. We have also shown how the spec-
tral quasiparticle weight at the Fermi level wqp and the
effective mass can be deduced from the parameters zσ and
µ̃0,σ. The effective mass is found to increase with the in-
teraction, and it diverges in the limit of zero doping whilst
wqp remains finite.
We have found that Luttinger’s theorem for the total
electron density in the antiferromagnetically ordered state
holds within the numerical accuracy for the range of dop-
ings and interactions studied. This is a further indication
that many aspects of Fermi liquid description may hold in
situations with symmetry breaking.
It is not easy to make a direct comparison of our re-
sults with earlier work [3] analyzing the quasiparticle exci-
tations in an metallic antiferromagnet as these have been
mainly based on the t − J-model for one or two holes
in a finite cluster. However, at a semiquantitative level,
the overall trend in our results seems to be similar to the
results surveyed by Dagotto, where the effective quasipar-
ticle bandwidth Weff is found to decrease with decreasing
J . This is line with our results if we identify Weff ∼ m/m∗
and J ∼ t2/U (see figure 17). Our values for the spectral
quasiparticle weight wqp are qualitatively similar to those
presented as the wavefunction renormalization Z in the
review article by Dagotto (see fig 27 [3]), and also the
ones reported more recently [32].
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 15
Acknowledgment
We wish to thank N. Dupuis, D.M. Edwards, W. Koller, D.
Meyer and A. Oguri for helpful discussions and W. Koller
and D. Meyer for their contributions to the development
of the NRG programs. We also acknowledge stimulating
discussions with G. Sangiovanni. One of us (J.B.) thanks
the Gottlieb Daimler and Karl Benz Foundation, the Ger-
man Academic exchange service (DAAD) and the EPSRC
for financial support.
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	Introduction
	Antiferromagnetic Broken Symmetry in DMFT
	Local Quasiparticle Parameters
	Spectra and Quasiparticle Bands
	Conclusions

  We analyze the properties of the quasiparticle excitations of metallic
antiferromagnetic states in a strongly correlated electron system. The study is
based on dynamical mean field theory (DMFT) for the infinite dimensional
Hubbard model with antiferromagnetic symmetry breaking. Self-consistent
solutions of the DMFT equations are calculated using the numerical
renormalization group (NRG). The low energy behavior in these results is then
analyzed in terms of renormalized quasiparticles. The parameters for these
quasiparticles are calculated directly from the NRG derived self-energy, and
also from the low energy fixed point of the effective impurity. They are found
to be in good agreement. We show that the main low energy features of the $\bf
k$-resolved spectral density can be understood in terms of the quasiparticle
picture. We also find that Luttinger's theorem is satisfied for the total
electron number in the doped antiferromagnetic state.

Introduction
The nature of the metallic antiferromagnetically ordered
state in strongly correlated systems has been subject of
study for over two decades, but still remains to be fully un-
derstood. Interest in this topic has been stimulated by the
fact that the high temperature superconductivity of the
cuprates emerges from the doping of an antiferromagnetic
insulating compound, such as La2CuO4 [1,2]. The sim-
plest models to describe the electrons in the CuO2 planes
of the cuprates are the two dimensional t-J-model and
Hubbard model. Much of the initial effort went into the
study of a single hole state of these models in an antifer-
romagnetic background. For the motion of this hole, there
is a competition between the gain in kinetic energy from
the hopping and its disruptive effect on the antiferromag-
netic order, and consequent loss of potential energy. As a
result a hole excitation becomes a quasiparticle or mag-
netic polaron, heavily dressed by antiferromagnetic spin
fluctuations (see review article by Dagotto [3] and refer-
ences therein). Much of this work, however, relied on exact
diagonalization or quantum Monte Carlo methods, which
are limited to small clusters and very few hole excitations,
and cannot be readily extended to study the many-hole,
finite doping situation.
More recent studies capable of describing finite dop-
ing have concentrated on the relation between the antifer-
romagnetic fluctuations and superconducting order (for
a review see [4] and the references therein). One of the
main motivations is to understand whether the exchange
of these types of fluctuations can provide a purely elec-
tronic mechanism for inducing superconductivity. Here, in
this paper, we focus on the metallic antiferromagnetism,
the doped state with long range antiferromagnetic order.
Our interest is to examine how well the low energy ex-
citations in this ordered state can be described in terms
renormalized quasiparticles. To tackle this problem we use
the infinite dimensional Hubbard model.
The simplification in the infinite dimensional limit is
that the electron self-energy becomes local in character,
with no wavevector dependence [5,6]. The self-energy then
depends only on the frequency, as is the case for impurity
models, allowing the lattice problem to be cast in the form
of a self-consistent impurity model. There are several rea-
sonably accurate techniques for solving this effective im-
purity problem, a very accurate one for the zero and low
temperature regime being the numerical renormalization
group approach (NRG) [7].
Recently we studied the effect of a magnetic field on
the quasiparticle excitations in the strong correlation regime
of the infinite dimensional Hubbard model using the NRG
method [8]. We also extended a form of renormalized per-
turbation theory (RPT), originally developed for impu-
rity models [9], to this model, and used it to calculate
the local dynamic spin susceptibilities, obtaining results
in good overall agreement with those from the NRG. In
this paper we extend this combination of renormalization
techniques, NRG and RPT within dynamical mean field
theory, to look at the low energy excitations of the infi-
nite dimensional Hubbard model in a staggered field, and
in antiferromagnetic broken symmetry states. Extensive
calculations of the antiferromagnetic states in the Hub-
http://arxiv.org/abs/0704.0243v2
2 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
bard model using the DMFT-NRG approach have already
been reported in the paper of Zitzler, Pruschke and Bulla
[10]. We confirm their results for the phase diagram and
extend the calculations and analysis to the description of
the low energy renormalized excitations, and how these
can be described within the framework of a renormalized
perturbation theory.
2 Antiferromagnetic Broken Symmetry in
In considering the response of the Hubbard model [11] to
a staggered magnetic field and antiferromagnetic order,
we take the case of a bipartite lattice, which consists of
two sublattices A and B such that the nearest neighbors
of a site in the A sublattice are on the B sublattice and
vice versa. The Hamiltonian for the Hubbard model can
be written in the form,
i,j,σ
(tijc
A,i,σcB,j,σ + h.c.) + U
nα,i,↑nα,i,↓
A,i,σcA,i,σ + µ−σc
B,i,σcB,i,σ), (1)
where the hopping matrix element is taken as tij = −t
between nearest sites i and j only, and zero otherwise,
and α = A,B. A staggered field His
His =
H for i ∈ A sublattice
−H for i ∈ B sublattice (2)
has been included so that µσ = µ+σh, where h = gµBH/2
with the Bohr magneton µB. The non-interacting part
of the Hamiltonian H0,µ can be diagonalized in terms of
Bloch states and then expressed in the form,
H0,µ =
k,σMk,σCk,σ. (3)
where C
k,σ = (c
A,k,σ, c
B,k,σ), and the matrix Mk,σ is
given by
Mk,σ =
−µσ εk
εk −µ−σ
. (4)
The k sums run over a reduced Brillouin zone, and the
energy of the Bloch state is εk =
j tije
i(Ri−Rj)·k. The
free Green’s function matrix G0k,σ(ω) is given by (ω −
Mk,σ)
−1. The poles of the free Green’s function give the
elementary single particle excitations, which are given by
E0k,±(U = 0) = −µ0(h)±
h2 + ε2
, (5)
where µ0(h) is the chemical potential of the noninteracting
system in a staggered field. This illustrates that the elec-
tronic excitations are split into two subbands for a finite
staggered field.
Notice that we have adopted a special choice of ba-
sis {cA,k,σ, cB,k,σ} here [12,10]. Another common basis to
study antiferromagnetic and spin density wave symmetry
(SDW) breaking is {ck,σ, ck+q0,σ}, where q0 is the recip-
rocal lattice vector for commensurate SDW ordering. The
bases can be related by a linear transformation,
k+q0,σ
cA,k,σ
cB,k,σ
. (6)
For the latter basis the matrix Mk,σ would be diagonal
in the kinetic energy term and the symmetry breaking
is offdiagonal. For our study in the DMFT framework the
A−B-sublattice basis is, however, more convenient and we
will use it throughout the rest of this paper. It is possible,
of course, to relate the obtained quantities with the help
of (6) to the {ck,σ, ck+q0,σ} basis.
We can generalize the equations to the interacting prob-
lem by introducing a self-energy Σα,k,σ(ω), so that the
matrix Green’s function can be written in the form
Gk,σ(ω)=
ζA,k,σ(ω)ζB,k,σ(ω)− ε2k
ζB,k,σ(ω) −εk
−εk ζA,k,σ(ω)
where ζα,k,σ(ω) = ω + µσ − Σα,k,σ(ω). As we are dealing
with the infinite dimensional limit of the model, we take
the self-energy to be local so we can drop the k index. This
is the reason why the self-energy has a single site index
α = A,B and no offdiagonal terms appear in equation (7).
The symmetry of the bipartite lattice gives ΣB,σ(ω) =
ΣA,−σ(ω) ≡ Σ−σ(ω) and hence
ζB,−σ(ω) = ζA,σ(ω) ≡ ζσ(ω),
where we have simplified the notation. To determine these
quantitiesΣσ(ω) it is sufficient to focus on the A sublattice
only.
Summing the first component in the Green’s function
in equation (7) over k we obtain the Green’s function for
a site on the A sublattice, Glocσ (ω),
Glocσ (ω) =
ζ−σ(ω)
ζσ(ω)ζ−σ(ω)
ρ0(ε)
ζσ(ω)ζ−σ(ω)− ε
, (8)
where ρ0(ε) is the density of states of the non-interacting
system in the absence of the staggered field.
In the DMFT this local Green’s function, and the self-
energy Σσ(ω), are identified with the corresponding quan-
tities for an effective impurity model [12]. This implies
that the Green’s function G0,σ(ω) for the effective impu-
rity in the absence of an interaction at the impurity site
is given by
G−10,σ(ω) = Glocσ (ω)−1 +Σσ(ω). (9)
We can take the form of this impurity model to correspond
to an Anderson model [13] in a magnetic field,
HAM =
εd,σd
σdσ + Und,↑nd,↓ (10)
(Vk,σd
σck,σ + V
k,σdσ) +
εk,σc
k,σck,σ,
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 3
where εd,σ = εd−σh is the energy of the localized level at
an impurity site in a magnetic fieldH , U the interaction at
this local site, and Vk,σ the hybridization matrix element
to a band of conduction electrons of spin σ with energy
εk,σ. As we are focusing on an A site as the impurity we
take H = Hs.
The one-electron Green’s function for the impurity site
of this model is given by
Gimpσ (ω) =
ω − εdσ −Kσ(ω)−Σσ(ω)
, (11)
where
Kσ(ω) =
|Vk,σ|2
ω − εk,σ
. (12)
If this impurity Green’s function is equated to the local
lattice Green’s function Glocσ (ω), we identify εdσ = −µσ
and from equation (9), Kσ(ω) is given by
Kσ(ω) = ω + µσ − G−10,σ(ω). (13)
The function Kσ(ω) plays the role of the effective medium
and has to be calculated self-consistently.
The self-consistent calculations for Kσ(ω) can usually
be performed iteratively. Starting from a conjectured form
for Kσ(ω), the NRG method is used to calculate the self-
energy of the effective Anderson model, from which the
impurity Green’s function Gimpσ (ω) in (11) and the local
Green’s function for the lattice Glocσ (ω) in (8) can be de-
duced. If these two Green’s functions do not agree, then
equation (9) is used to derive a new starting value for
Kσ(ω) and the process continued until self-consistency is
achieved.
To find antiferromagnetic solutions, we calculated self-
consistent solutions for a decreasing sequence of staggered
magnetic fields to see if broken symmetry solutions of this
type exist as the staggered field is reduced to zero. For
the non-interacting density of states ρ0(ε) we take the
Gaussian form ρ0(ε) = e
−(ε/t∗)2/
πt∗, corresponding to
an infinite dimensional hypercubic lattice. It is useful to
define an effective bandwidth W = 2D for this density of
states via D, the point at which ρ0(D) = ρ0(0)/e
2, giving
2t∗ corresponding to the choice in reference [14].
In all the results we present here we take the value W =
4. In the NRG calculations we have used the improved
method [15,16] of evaluating the response functions with
the complete Anders-Schiller basis [17], and also determine
the self-energy from a higher order Green’s function [18].
In figure 1 we show the self-consistently calculated lo-
cal spectral density for the spin-up (upper panel) and spin-
down electrons (lower panel) at an A site with U = 3 and
5% hole doping (from the state at half-filling) for various
values of an applied staggered field. The staggered mag-
netic field induces a sublattice magnetization,
(nA,↑ − nA,↓), (14)
so that these spectra are quite different. For this set of
parameters, this difference persists as the staggered field
−4 −2 0 2 4
h =0.05
h =0.1
−4 −2 0 2 4
h =0.05
h =0.1
Fig. 1. (Color online) The spectral densities for the spin-up
electrons (upper panel) and spin-down electrons (lower panel)
at the A site for various values of the applied staggered field
for U = 3 and x = 0.95
is reduced to zero so that we have a spontaneous sub-
lattice magnetization corresponding to spontaneous anti-
ferromagnetic order. For the case away from half filling,
δ 6= 0, we have to keep adjusting the chemical potential
when iterating for a self-consistent solution. It shows a
slightly oscillatory behavior when iterating for a specific
filling x, and we follow the procedure described in refer-
ence [10]. This feature is related to the fact that the calcu-
lations are for a metastable ground state and instabilities
to more complicated ground states for antiferromagnetic
ordering than the homogeneous, commensurate Néel state,
which forms the basis for these DMFT calculations, can
occur [19,20,21,22,23,24,25,26,10]. As far as phase sep-
aration in the ground state is concerned, the results of
our calculations are generally in line with the conclusions
in [10] as they are carried out within the same frame-
work. The focus of this work is, however, the analysis of
generic quasiparticle properties in a doped antiferromag-
netic state. We consider the approach as a valid, approxi-
mate starting point for this endeavor, but modifications to
the results presented here can occur for calculations based
4 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
on a more complicated ground states not accessible within
the DMFT framework. For a more extensive discussion of
the applicability of the DMFT in this situation we refer
to the earlier work [10].
From results of this type of calculation, we have built
up a global antiferromagnetic/paramagnetic phase dia-
gram as a function of the doping δ and the on-site inter-
action U . This phase diagram is shown in figure 2, where
the value of the corresponding sublattice magnetization is
shown in a false color plot. We have added a line separat-
ing the spontaneously ordered and paramagnetic regimes.
0 0.05 0.1 0.15 0.2
0.1 0.2 0.3 0.4
Fig. 2. (Color online) Phase diagram showing the doping and
the U dependence of the sublattice magnetization mA as de-
duced from the DMFT-NRG calculations.
At half filling (δ = 0 axis) the spontaneous magnetiza-
tion increases with U . We can see that the antiferromag-
netic order from the half filled case persists when holes
are added. The value of the critical doping δc at which
the antiferromagnetism disappears depends on the on-site
interaction U . We expect that for small U the critical dop-
ing δc will increase with U since a tendency to order only
appears when an on-site interaction is present. From the
mapping to the t−J model we also expect that for large U
the antiferromagnetic coupling J decreases and therefore
the order is destroyed more easily. The values of U are,
however, not large enough to display this trend.
If we compare these results with the phase diagram
given by Zitzler et al. [10] we see that they are in very good
agreement. In their case the antiferromagnetic region was
mapped out to values of U ≃ 4.5. As the iterations tend
to oscillate, as discussed before, there is a problem of ob-
taining a self-consistent antiferromagnetic solution in the
large U regime. We have managed to extend the diagram
to somewhat larger values of U by stabilising the calcula-
tions by averaging the effective medium over a number of
iterations.
3 Local Quasiparticle Parameters
To examine the nature of the low energy excitations, we
will assume that the self-energy Σσ(ω) is non-singular at
ω = 0 so that, at least asymptotically, it can be expanded
in powers of ω. This assumption is not expected to be valid
close to the quantum critical point when the magnetic
order sets in, but to be a reasonable assumption otherwise.
We also assume that the imaginary part of the self-energy
vanishes which is confirmed by the numerical results of the
DMFT-NRG calculations. We will retain terms to order
ω only for the moment. The higher order corrections will
be considered later. We then find for ζσ(ω),
ζσ(ω) = ω(1−Σ′σ(0)) + µσ −Σσ(0) (15)
= z−1σ (ω + µ̃0,σ), (16)
where
µ̃0,σ = zσ(µ−Σσ(0)), and z−1σ = 1−Σ′σ(0). (17)
The interacting Green’s function (7) has poles at the roots
of the quadratic equation,
ζσ(ω)ζ−σ(ω)− ε2k = 0. (18)
The solutions of this equation are
E0k,± = −µ̃±
+∆µ̃2, (19)
where ε̃k =
z↑z↓εk, ∆µ̃ = (µ̃0,↑ − µ̃0,↓)/2, and µ̃ =
(µ̃0,↑ + µ̃0,↓)/2. This has the same form as for the non-
interacting system in a staggered field (5), so we can in-
terpret these excitations as quasiparticles coupled to an
effective staggered magnetic field h̃s = ∆µ̃/gµB, with µ̃
playing the role of a quasiparticle chemical potential. This
equation gives the dispersion relation for these single par-
ticle excitations, which can be regarded as constituting a
renormalized band, or bands as there are two branches.
The term magnetic polaron is sometimes used to describe
these single particle excitations in states of magnetic or-
der, because of the analogy with the motion of a particle
in a lattice to which it is strongly coupled, where the ex-
citation is termed a polaron.
The corresponding density of states of these free local
quasiparticles on the sublattice is
ρ̃0,σ(ω)=
ω + µ̃− σ∆µ̃
ω + µ̃+ σ∆µ̃
(ω + µ̃)2 −∆µ̃2
for |ω + µ̃| > |∆µ̃|, and is zero otherwise. In the case of
a half-filled band µ̃ = 0 and there is a gap at the Fermi
level εF = 0.
To determine this local quasiparticle density of states
in the presence of the symmetry breaking staggered mag-
netic field we need to calculate zσ and µ̃0,σ for each spin
type. Using the NRG we can do this in two ways. As the
DMFT-NRG calculations give us the self-energyΣσ(ω) di-
rectly, we only need its value, and that of its first derivative
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 5
at ω = 0, to deduce both zσ and µ̃0,σ using equation (17).
However, because the model is solved using an effective
impurity model, we can also deduce these quantities indi-
rectly from the many-body energy levels of the impurity
on approaching the low energy fixed point [27]. This sec-
ond method gives us not only a check on the results of the
direct method, but also allows to deduce some informa-
tion about the quasiparticle interactions, as we shall show
in the next section.
3.1 Calculation of Renormalized Parameters
To describe how the renormalized parameters are deduced
from the energy levels of the NRG calculation, we need to
outline how the NRG calculations are carried out. Fol-
lowing the procedure introduced by Wilson [28], the con-
duction band is logarithmically discretized and the model
then converted into the form of a one dimensional tight
binding chain, coupled via an effective hybridization Vσ to
the impurity at one end. In this representation Kσ(ω) =
|Vσ|2g(N)0,σ (ω), where g
0,σ (ω) is the one-electron Green’s
function for the first site of the isolated conduction elec-
tron chain of length N . The impurity Green’s function for
this discretized model then takes the form,
Gimpσ (ω) =
ω − εdσ − |Vσ|2g(N)0,σ (ω)−Σσ(ω)
. (21)
We can find the quasiparticle excitations of this model
by expanding the self-energy Σσ(ω) in the denominator of
this equation to first order in ω, and write the result in
the form,
Gimpσ (ω) =
ω − ε̃dσ − |Ṽσ |2g(N)0,σ (ω) +O(ω2)
, (22)
where
ε̃dσ = zσ[εdσ +Σσ(0)], |Ṽσ|2 = zσ|Vσ|2. (23)
We can then define a free quasiparticle propagator, G̃0,σ(ω),
0,σ (ω) =
ω − ε̃dσ − |Ṽσ |2g(N)0,σ (ω)
, (24)
and interpret zσ as the local quasiparticle weight.
In the NRG calculation the many-body excitations are
calculated iteratively, starting at the impurity site, and
increasing the chain length N by one site with each itera-
tion. When the matrices become too large to handle, only
the lowest 500-1500 states are kept at each iteration. The
many-body energy levels for the Nth iteration and the
set of quantum numbers M , EM (N), depend on the chain
length N and the discretization parameter Λ > 1. When
N becomes large these energy levels go to zero as Λ−N/2.
We now conjecture that the lowest single particle Eσp (N)
and single hole excitations Eσh (N) determined from the
NRG many-body excitations correspond to quasiparticle
excitations. If this is the case then they should correspond
to the poles of the quasiparticle Green’s function given in
equation (24), with values of Ṽσ and ε̃dσ, which are inde-
pendent of N as N → ∞. We can test this hypothesis by
substituting the values, ω = Eσp (N) and ω = E
h (N), into
the equation,
ω − ε̃dσ − |Ṽσ|2g(N)0,σ (ω) = 0, (25)
and deduce values of Ṽσ and ε̃dσ, which will in general
depend upon N . From these we can deduce zσ = |Ṽσ/Vσ|2
and µ̃0,σ = −ε̃dσ, which will also depend upon N , but if
the lowest single particle excitations of the system do cor-
respond to free quasiparticles, the values of zσ and µ̃0,σ
will become independent of N for large N . It should be
noted that we need both the particle and hole excitations
for each spin to determine the four renormalized parame-
ters. The parameters corresponding to spin up involve the
particle excitations with spin up and the hole excitations
with spin down.
That parameters can be found, which are independent
ofN for largeN , can be seen in figure 3, where we take the
results of a Kσ(ω) and µσ from the antiferromagnetic self-
consistent solution for the Hubbard model with U = 3 and
10% doping, using a value for the discretization parameter
Λ = 1.8.
0 10 20 30 40 50
Fig. 3. (Color online) The N-dependence of the renormalized
parameters zσ and µ̃0,σ for U = 3 and x = 0.9.
It can be seen that after about 25 iterations all the values
deduced for zσ and µ̃0,σ become independent of N . In the
next section, where we compare these results with the cor-
responding values deduced directly from the self-energy,
we get further confirmation that the values deduced re-
ally do describe the quasiparticle excitations of the lattice
model.
When two or more quasiparticles are excited from the
interacting ground state, there will be an interaction be-
tween them. For the Anderson impurity model this inter-
action will be local and can be expressed as Ũ , a renor-
malized value of the original interaction of the ‘bare’ par-
6 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
ticles. The value of Ũ can be deduced by looking at low-
est lying two-particle excitations derived from NRG cal-
culation. These could either be two-particle excitations,
E↑,↓pp (N), two-hole excitations, E
hh (N) or a particle-hole
excitation E
ph (N). By looking at the difference between
a two-particle excitation and two single particle excita-
tions, E↑,↓pp (N) − E↑p(N) − E↓p(N), as a function of N
we can deduce an effective interaction Ũ↑,↓pp (N) between
these two quasiparticles, as has been described fully ear-
lier for the standard Anderson model [27]. In a similar way
we can deduce an effective interaction between two holes,
hh (N), or a particle and hole, −Ũ
ph (N). To be able to
define a single quasiparticle interaction Ũ , not only must
Ũ↑,↓pp (N), Ũ
hh (N) and Ũ
ph (N), give values which are in-
dependent of N for large N , these values must be equal
so Ũ↑,↓pp = Ũ
hh = Ũ
ph = Ũ .
0 10 20 30 40 50
Fig. 4. (Color online) The N-dependence of the renormal-
ized particle-particle, particle-hole and hole-hole interactions
for U = 6 and x = 0.9, showing that they converge to a unique
value Ũ .
In figure 4 we give the values of Ũ↑,↓pp (N), Ũ
hh (N) and
ph (N) as deduced from DMFT-NRG calculation for the
Hubbard model in an antiferromagnetic state with U = 6,
10% doping and Λ = 1.8. It can be seen that the three
sets of results settle down to a common value Ũ .
We can go further and identify Ũ with the local quasi-
particle 4-vertex interaction for the effective impurity model,
Ũ = z↑z↓Γ↑,↓,↓,↑(0, 0, 0, 0), (26)
where Γ↑,↓,↓,↑(ω1, ω2, ω3, ω4) is the total 4-vertex at the
impurity site, which is equal to the same quantity for a
site in the lattice model. With this interpretation it is
possible to identify these parameters with those used in a
renormalized perturbation expansion. The parameters, V ,
εd,σ and U , together with g
0,σ(ω), specify the effective im-
purity model. The renormalized parameters, Ṽ , ε̃d,σ and
Ũ , together with gN0,σ(ω), can be used as an alternative
way of specifying this model. The renormalized perturba-
tion theory (RPT) is set up by expanding the self-energy
to order ω, as earlier, but retaining all the higher order
correction terms in a remainder term,
Σσ(ω) = Σσ(0) + ωΣ
σ(0) +Σ
σ (ω), (27)
where Σremσ (ω) is the remainder term. On substituting
this into the equation for the impurity Green’s function in
equation (11), we can deduce a general expression for the
quasiparticle Green’s function in the form,
G̃impσ (ω) =
ω − ε̃dσ − K̃σ(ω)− Σ̃σ(ω)
, (28)
where K̃σ(ω) = zσKσ(ω) and Σ̃σ(ω) = zσΣ
σ (ω) plays
the role of a renormalized self-energy. A diagrammatic
perturbation theory can then be carried out for Σ̃σ(ω)
in terms of the free quasiparticle propagators, with ad-
ditional diagrams arising from counter terms, which are
required to prevent over-counting (renormalization condi-
tions) [29,9,30]. This form of perturbation theory is valid
for all energy scales but is particularly effective for calcu-
lating the low energy terms arising from the quasiparticle
interactions. For the symmetric Anderson impurity model
it has been shown that this perturbation theory taken to
second order in Ũ , gives the exact spin and charge suscep-
tibilities at T = 0, and the exact T 2 contribution to the
conductivity [9].
Because, within DMFT, the self-energy for the lat-
tice is the same as that for the effective impurity, we can
equally well use the effective impurity model to calculate
it. This means that we can use the renormalized pertur-
bation theory for the effective impurity model to estimate
the correction terms to the free quasiparticle picture aris-
ing from the quasiparticle interactions.
3.2 Local Quasiparticle Weight
We now consider the values of the local quasiparticle weight
factor zσ, commonly known also as the wavefunction renor-
malization factor. This is an important factor in deter-
mining the parameters needed to describe the low energy
behavior of the system. When there is no k-dependence
of the self-energy, as is the case for infinite dimensional
models and DMFT, the effective mass of the quasiparti-
cles in the paramagnetic state is proportional to 1/zσ. We
show later that in the antiferromagnetic state the expres-
sion is more complicated and depends both on zσ and the
renormalized chemical potential µ̃0,σ. We have determined
zσ from the NRG results by the two methods described
and give the values of zσ deduced for both spin types as a
function of doping in figure 5. The results are for the case
U = 3, where there is antiferromagnetic order and the ex-
ternal staggered field has been set to zero. It can be seen
that there is a reasonable agreement between the values
obtained by the two different methods of calculation. Visi-
ble differences can be attributed to the inaccuracies when
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 7
0 0.05 0.1 0.15 0.2
 from Σ
 from Σ
 from FP
 from FP
Fig. 5. (Color online) The local quasiparticle weight zσ as de-
duced directly from the self-energy and also from the impurity
fixed point (FP) for U = 3 and various dopings.
numerically computing the derivative of the self-energy,
whose calculation involves a broadening procedure. When
the system is doped but still ordered we have z↑ 6= z↓, and
the renormalization effects are stronger for the minority
(down) spin particles on the sublattice. This is similar to
the results we found for a doped Hubbard model in a para-
magnetic state in the presence of a strong uniform mag-
netic field [8]. For a certain range of dopings the values of
z↑ and z↓ do not vary much. The tendency is that z↓ first
decreases and later increases, whereas z↑ decreases over
the whole range until both of them merge at the doping
point where the antiferromagnetic order disappears.
The results for the corresponding case with U = 6, a
value which is larger than the bandwidth, are shown in
figure 6.
0 0.05 0.1 0.15 0.2
 from Σ
 from Σ
 from FP
 from FP
Fig. 6. (Color online) The local quasiparticle weight zσ as
deduced directly from the self-energy and from the impurity
fixed point (FP) for U = 6 and various dopings.
On the whole the behavior is quite similar to that for
the case U = 3, only that the renormalization effects
are more pronounced. For a range of dopings the local
quasiparticle weights do not change much and have the
same tendency as described above. The implications for
the spectral quasiparticle weight and the effective mass
enhancement will be discussed in detail later.
3.3 Renormalized chemical potential
In figure 7 we give the results for the renormalized chemi-
cal potential, µ̃0,σ [defined in equation (17) and (23)], for
the two spin types in the spontaneously ordered antifer-
romagnetic states for U = 3 and U = 6 for a range of
dopings.
0 0.05 0.1 0.15
 from Σ
 from Σ
 from FP
 from FP
0 0.05 0.1 0.15
 from Σ
 from Σ
 from FP
 from FP
Fig. 7. (Color online) The renormalized chemical potential
µ̃0,σ as deduced directly from the self-energy and from the
impurity fixed point (FP) for various dopings for U = 3 (upper
panel) and U = 6 (lower panel).
The values calculated by the two different methods can be
seen to be in good agreement here, as well. We have added
the values for the half filled case. These were calculated
from the self-energy in the gap at ω = 0. The general
8 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
behavior of the values for µ̃0,σ for the case with U = 6 is
very similar to the case with smaller U
The renormalized chemical potential µ̃0,σ is an impor-
tant parameter in specifying the form of the local sublat-
tice quasiparticle spectral density ρ̃0σ(ω). From equation
(20) it can be seen that, as ω → −µ̃0,σ, ρ̃0,σ(ω) behaves
asymptotically as
ρ̃0,σ(ω) ∼
ω + µ̃0,σ
, (29)
so the quasiparticle density of states has a square root
singularity at ω = −µ̃0,σ. On the other hand, however, as
ω → −µ̃0,−σ, ρ̃0,σ(ω) behaves as
ρ̃0,σ(ω) ∼
ω + µ̃0,−σ, (30)
so the quasiparticle density of states goes to zero at ω =
−µ̃0,−σ. Between the two points, ω = −µ̃0,σ and ω =
−µ̃0,−σ, the quasiparticle density of states has a gap of
magnitude 2∆µ̃. As can be seen in figure 7 this free quasi-
particle gap decreases with increasing doping and closes
in the paramagnetic state. If we take into account the
values at half filling we see a strong reduction of 2∆µ̃,
when doping the system. We also see that µ̃0,↑ drops to
small negative values for finite hole doping, which corre-
sponds to the fact that the Fermi level then lies within the
lower band. These features will be seen clearly in the fig-
ures presented in the next section, where we compare the
quasiparticle densities of states with the full local spectral
densities calculated from the DMFT-NRG.
3.4 The Quasiparticle Interaction
The quasiparticles can be further characterized by an ef-
fective interaction Ũ as described before. In figure 8 we
plot the doping dependence of the renormalized interac-
tion over a range of dopings and U = 3 and U = 6.
We can see that in both cases the values decrease with
increasing doping. Hence, the effective quasiparticle inter-
action is stronger for a smaller hole density. For a certain
range of dopings, however, Ũ does not vary much. We can
also see that the ratio Ũ/U for the effective interaction as-
sume smaller values the larger the bare U becomes. Also
the absolute value of Ũ , i.e. without the scaling with U as
in figure 8, is smaller for larger bare U for the full range
of dopings. This effect of smaller quasiparticle interactions
for the stronger coupling case can be seen as sharper quasi-
particle peaks for larger U as will be discussed in the next
section.
4 Spectra and Quasiparticle Bands
4.1 Local Spectra
In this section we examine how well the local sublattice
quasiparticle density of states ρ̃0,σ(ω), evaluated from equa-
tion (20) with the renormalised parameters, describes the
0.05 0.1 0.15
 U / U for U =3
 U / U for U =6
Fig. 8. (Color online) The renormalised quasiparticle interac-
tion Ũ/U as deduced from the impurity fixed point for various
dopings and U = 3, 6.
low energy features seen in the local spectral density ρσ(ω)
calculated from the DMFT-NRG. At half filling there is
a gap at the Fermi level, so there are no single particle
excitations in the immediate neighbourhood of the Fermi
level, and this is not a very interesting case to consider.
We look in detail at the case of 10% doping where the
Fermi level lies at the top of the lower band, and consider
the two cases U = 3 and U = 6. In the upper panel of
figure 9 we compare the spectral density ρ↑(ω) with the
corresponding quantity z↑ρ̃0,↑(ω), from the quasiparticle
density of states.
We see that the behavior near the Fermi level (ω = 0),
and the singular feature seen in the lower branch of ρ↑(ω),
are well reproduced by the quasiparticle density of states.
Above the Fermi level there is a peak in the quasiparti-
cle density of states similar to that in the full spectrum
but somewhat more pronounced. Above the Fermi level
and below the upper peak there is a pseudo-gap region. In
the free quasiparticle spectrum it is a definite gap. In the
spectrum calculated from the direct NRG evaluation it ap-
pears as a pseudo-gap, with rather small spectral weight
just above the Fermi level. From the direct DMFT-NRG
calculations, due to the broadening features introduced to
obtain a continuous spectrum, it is not always possible
to say definitively whether there is a true gap above the
Fermi level or not. To resolve this question we can ap-
peal to the renormalised perturbation theory to look at
the corrections to the quasiparticle density of states aris-
ing from the quasiparticle interactions. A calculation of
the imaginary part of the renormalised self-energy Σ̃σ(ω)
to order Ũ2 should be sufficient to settle this issue. The
imaginary part of the second order diagram for the renor-
malised self-energy in the limit T → 0 for ω > 0 is given
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 9
−2 −1 0 1 2 3 4
−2 −1 0 1 2 3 4
Fig. 9. (Color online) The free local quasiparticle spectrum
(dashed line) in comparison with DMFT-NRG spectrum for
x = 0.9 and U = 3 for the spin-up electrons (upper panel) and
spin-down electrons (lower panel).
ImΣ̃(2)σ (ω) = πŨ
dε2 ρ̃0,σ(ε1)ρ̃0,−σ(ω − ε1 + ε2)
× ρ̃0,−σ(ε2)θ(ω − ε1 + ε2), (31)
where ρ̃0,σ(ε) is the free quasiparticle density of states.
The integration area is a triangle in the (ε1, ε2)-plane as
shown in figure 10.
To analyze the behavior of ImΣ̃
σ (ω) in the regime |µ̃0,↑| <
ω < |µ̃0,↓| we have to study where the integrand is non-
zero taking into account that ρ̃0,σ(ε) = 0 for |µ̃0,↑| <
ε < |µ̃0,↓|. The only non-zero contribution comes from
the small shaded region in figure 10, which leads to the
estimate,
ImΣ̃(2)σ (ω) ≃ πŨ2ρ̃0,σ(0)ρ̃0,−σ(−ω)ρ̃0,−σ(0)µ̃20,↑. (32)
When µ̃0,↑ is small, which occurs when the lower edge of
the gap in the quasiparticle density of states is very near
the Fermi level, this contribution to the imaginary part
PSfrag replacements
−ω + |µ̃0,↑|
|µ̃0,↑| |µ̃0,↓|
Fig. 10. (Color online) Integration region in the (ε1, ε2)-plane
for the imaginary part of the self-energy. The original triangle
region (0, ω,−ω) for integration in equation (31) is reduced in
the gap region, |µ̃0,↑| < ω < |µ̃0,↓|, to the small shaded region
shown in the figure.
of the renormalized self-energy will be finite but small. It
decreases with ω due the behavior of ρ̃0,−σ(−ω). Based on
this argument we conclude that there is a small, but finite
imaginary part of the self-energy in the free quasiparticle
gap 2∆µ̃, when it lies above the Fermi level, giving rise
to a finite spectral weight there. However, this spectral
weight is very small close to the lower edge of the free
quasiparticle density of states, when this edge lies only
just above the Fermi level.
In the lower panel of figure 9 we compare the ρ↓(ω)
with z↓ρ̃0,↓(ω). We see that in this case also the quasi-
particle density of states reproduces well the spectrum in
the region of the Fermi level and the peak structure in
the lower band, which is non-singular in this case. The
position of the peak above the Fermi level is also well re-
produced, but the peak in the free quasiparticle density
of states is singular, whereas that in the DMFT-NRG re-
sults is not. We would expect to lose this singularity in the
free quasiparticle density of states once the quasiparticle
scattering is taken into account and the renormalized self-
energy is included. It is possible also that the peak above
the Fermi level in the DMFT-NRG spectrum should be
sharper, as there is some tendency for the broadening in-
troduced in this approach to flatten peaked features in
regions away from the Fermi level. The spectral weight
in the pseudo-gap is even smaller than in the case for the
spin-up electrons, particularly in the region of the gap that
lies closest to the Fermi level. This is qualitatively in line
with the conclusions based on the renormalized pertur-
bation theory estimate of the effects of the quasiparticle
scattering.
We see very similar features in the spectra for the case
U = 6 and also 10% doping shown in figure 11.
Here, the peaks near the Fermi level are a bit sharper. The
observations made on the comparison of the quasiparticle
and DMFT-NRG spectra apply equally well to this case.
In addition to the low energy features charge peaks cor-
responding to the Hubbard bands appear. The lower one
can be identified in the full spectra, whereas the upper
10 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
−2 −1 0 1 2 3
−2 −1 0 1 2 3
Fig. 11. (Color online) The free quasiparticle spectrum
(dashed line) in comparison with DMFT-NRG spectrum for
x = 0.9 and U = 6 for the spin-up electrons (upper panel) and
spin-down electrons (lower panel).
Hubbard peak is not seen on the energy scale shown. The
quasiparticle density of states does not contain informa-
tion about these features at higher energy.
4.2 k-resolved Spectra
We can learn more about the low energy single parti-
cle excitations by looking at the spectral density of the
Green’s function Gk,σ(ω) in equation (7) for a given wave-
vector k. With the self-energies Σσ(ω) calculated within
the DMFT-NRG approach all elements of this matrix can
be evaluated. The local spectra and self-energies are spin-
dependent in the doped broken symmetry state, however,
the free quasiparticle bands E0
k,± [equation (19)] do not
depend on the spin. Here, we focus on the diagonal part
of Gk,σ(ω) corresponding to the A sublattice,
Gk,σ(ω) =
ζ−σ(ω)
ζσ(ω)ζ−σ(ω)− ε2k
. (33)
The weights of the quasiparticle excitations in this case
depend on the spin corresponding to the sublattice prop-
erties. We note that one can also analyze the quasiparticle
bands differently, for instance, from the k-resolved spectra
and the diagonal form of Gk,σ(ω). The form of the quasi-
particle bands remains unchanged then, but the weights
differ and do not depend on the spin σ in that case.
We first of all look at the Fermi surface which is the
locus of the k-points at the Fermi level (ω = 0) where
the Green’s function has poles. The conduction electron
energy εkF at these point is given by
ε2kF = (µ↑ −Σ↑(0))(µ↓ −Σ↓(0)). (34)
By Luttinger’s theorem, the volume of the Fermi sur-
face for the interacting system must equal that for the
non-interacting system with the same density. As the self-
energy depends only on ω, the two Fermi surfaces must
also have the same shape, and therefore must be identical.
The Fermi surface of the non-interacting system is given
by εkF = µ0, where µ0 is the chemical potential of the
non-interacting system in the absence of any applied field
for the given density. For this to be identical with that
given in equation (34),
(µ↑ −Σ↑(0))(µ↓ −Σ↓(0)) = µ20. (35)
We can check that this relation indeed holds from our
results for Σσ(ω) and µσ, independent of the value of U ,
or in the case of an applied staggered field, independent of
the field value. This relation implies that the total number
of electrons per site n can be calculated from an integral
over the non-interacting density of states,
n = 2
ρ0(ω)dω, (36)
where in the hole doped case µ0 = −
µ̄↑µ̄↓ and µ̄σ =
µσ −Σσ(0).
To relate this result to the quasiparticle picture, we
expand the self-energy in equation (33) to first order in
ω, but retain the remainder term, ΣRσ (ω) as in equation
(27). The Green’s function can be rewritten in the form,
G̃k,σ(ω) =
ζ̃−σ(ω)
ζ̃σ(ω)ζ̃−σ(ω)− ε̃2k
, (37)
where ζ̃σ(ω) = ω + µ̃0,σ − Σ̃σ(ω). We define a quasipar-
ticle Green’s function G̃k,σ(ω) via zσG̃k,σ(ω) = Gk,σ(ω).
The renormalized self-energy vanishes, Σ̃σ(ω) = 0, for the
free quasiparticle Green’s function G̃
k,σ(ω), which can be
separated into two independent branches of free quasipar-
ticles,
k,σ(ω) =
uσ+(εk)
ω − E0
uσ−(εk)
ω − E0
, (38)
where E0
k,± was defined in equation (19) and the weights
are given by
uσ±(εk) =
1∓ σ ∆µ̃√
∆µ̃2 + ε̃2
. (39)
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 11
This is similar in form to mean field theory, which would
correspond to putting zσ = 1, and ∆µ̃ = Ummf , where
mmf is the mean field sublattice magnetization. The spin
dependent contribution in (39) which arises from the sec-
ond term is most marked in the region near the Fermi
level. It should be noted that the quasiparticle excitations
k,± and weights u
±(εk) here are defined by expanding
the self-energy at ω = 0. This is so that they correspond
to the free quasiparticles in the renormalized perturbation
theory which have an infinite lifetime.
The spectral density ρ̃
(ω) for this free quasiparticle
Green’s function is a set of delta-functions,
k,σ(ω) = u
+(εk)δ(ω−E0k,+)+uσ−(εk)δ(ω−E0k,−). (40)
On the Fermi surface E0k,− = 0, which is consistent with
the result for the Fermi surface given in equation (34).
Summing over k gives the local quasiparticle density of
states in equation (20). We define the quasiparticle num-
ber ñ as the integral of the sum of the spin up and spin
down quasiparticle density of states up to the Fermi level,
dω(ω + µ̃)
(ω + µ̃)2 −∆µ̃2
(ω + µ̃)2 −∆µ̃2
If we change the variable of integration to ω′, where
z↑z↓ =
(ω + µ̃)2 −∆µ̃2,
the integration can be shown to be identical with that
in equation (36), using the fact that µ0 = −
µ̄↑µ̄↓. We
then have an alternative statement of Luttinger’s theorem
in the form ñ = n. This can also be found by summing
both spin components in (40), integrating over ω and then
converting the k-summation to an integral over the free
electron density of states ρ0(ω). We can check in our nu-
merical results that the relation in this form holds. The
occupation number n can be calculated both from a direct
evaluation of the number operator in the ground state, and
also by integrating the sum of the spectral densities ρσ(ω)
of the full local Green’s function to the Fermi level. The
value of ñ is similarly determined from the integral over
the total quasiparticle density of states, ρ̃σ(ω). All three
results were found to be in good agreement, to within one
or two percent deviation at the most.
Before discussing the k-resolved spectra in detail we
would like to ask what the spectral weight wqp of a quasi-
particle excitation at the Fermi level in the lower band is,
such that the Green’s function reads there
Gqp(ω) =
ω − E0
. (42)
To calculate wqp, we can not focus on the spin depen-
dent local sublattice quantities, but have to sum over both
sublattices or equivalently the two spin components. The
reason for this is that the antiferromagnetically ordered
state does not possess any net magnetization and has on
average as many spin up polarized as spin down electrons.
The division in the A and B sublattices is convenient for
the DMFT calculations but somewhat artificial. In our
case with hole doping the Fermi level lies within the lower
band, which for the free quasiparticles is denoted by E0
The corresponding weight on the Fermi surface defined by
(34) is then given by
wqp =
−(εkF) =
z↑ + z↓
(z↑ − z↓)∆µ̃
2|µ̃|
, (43)
where the average of the renormalized chemical potential
µ̃ and the difference ∆µ̃ were defined below equation (19).
From the definition of ∆µ̃ we can see that the second term
in (43) is spin rotation invariant. The spectral quasipar-
ticle weight wqp on the Fermi surface depends not only
on the renormalization factors zσ, but also on the renor-
malized chemical potentials µ̃0,σ. The same result for the
weight (43) can be obtained from the diagonal form of
Gk,σ(ω) and the spectral weight of the lower band. The
weight wqp corresponds to the spectral weight Z at the
Fermi level as for example given in references [3,31,32].
The first term of the result for wqp is like the arithmetic
average of zσ. From figures 5 and 6 we can see that z↑ > z↓
and from figure 7 that µ̃0,↓ < µ̃0,↑ < 0. Therefore the sec-
ond term in (43) gives a positive contribution to the spec-
tral weight. At the end of the section in figure 18 we show
values of wqp in comparison with the arithmetic average
of zσ.
In order to understand better the properties of the
quasiparticle bands, we now compare the quasiparticle
spectrum with the k-resolved spectral density ρk,σ(ω) de-
rived from the DMFT-NRG results. In figure 12 we make
a comparison for the case of 12.5% doping with U = 3
for the Green’s function Gk,σ(ω) given in equation (33),
ρk,σ(ω) = −ImGk,σ(ω+)/π, where ω+ = ω+ iη, with η →
0, with that derived for the free quasiparticles, zσ ρ̃
k,σ(ω)
from equation (40). The delta-functions of the free quasi-
particle results are indicated by arrows with the height of
the arrow indicating the value of the corresponding spec-
tral weight. The plots as a function of ω are shown for a
sequence values of εk and, where the peaks in ρk,σ(ω)
get very narrow and high in the vicinity of the Fermi
level, they have been truncated. It can be seen that the
free quasiparticle results give a reasonable picture of the
form of ρk,σ(ω), particularly in the immediate region of
the Fermi level. There is considerable variation along the
curves in the way the overall spectral weight is distributed
between the excitations below and above the pseudo-gap
as a function of εk. This is most marked in the region near
the Fermi level for the spin-up electrons (upper panel)
where most of the spectral weight is in the lower band and
it is much reduced in the upper band, whereas the opposite
is the case for the spin-down electrons. This is reflected
in the expression of the quasiparticle weights uσ±(εk) in
equation (39). For instance, u
−(εk) corresponding to the
lower band E0
k,− becomes maximal near the Fermi en-
ergy, whereas u
+(εk) goes to zero there. The finite width
of the quasiparticle peaks in ρk,σ(ω) can be described by
a RPT, when we take into account the renormalized self-
12 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
Fig. 12. (Color online) The spectral density ρk,σ(ω) for the
spin-up electrons (upper panel) and spin-down (lower panel)
plotted as a function of ω and a sequence of values of εk for
U = 3 and 12.5% doping. Also shown with arrows are the
positions of the free quasiparticle excitations, with the height
of the arrow indicating the corresponding weight.
energy Σ̃σ(ω) in equation (37). If we, for instance, use
the the second order approximation in Ũ , which was illus-
trated in the last section (31), we get a similar behavior
for small ω as seen for ρk,σ(ω) in figure 12.
From the positions of the peaks in the ρk,σ(ω) spectra
we can deduce two branches of an effective dispersion Ek,±
for single particle excitations and compare it with the ones
for the free quasiparticles E0
k,±. We give the results for
U = 3 in figure 13.
It can be seen that E0
k,− tracks the peak in the lower
band closely over a wide range of εk, −1.5 < εk < 1.5
(note the bandwidth W = 4). This is not the case in the
upper band, where E0
k,+ tracks the peak closely only in
the lowest section that lies closest to the Fermi level. As
one can see from the dotted line the Fermi level lies in
the lower band and intersects the lower band twice. This
corresponds to the two values with opposite sign ε±
can be see from equation (34).
The corresponding results for the k-resolved spectra
for U = 6 and also 12.5% doping are shown in figure 14.
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 13. (Color online) A plot of the peak positions Ek,± in
the spectral density ρk,σ(ω) (full line) as a function of εk for
U = 3 and 12.5% doping compared with the free quasiparticle
dispersion E0k (dashed line).
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
−1 −0.5 0 0.5 1
−1.125
−0.75
−0.375
0.375
1.125
Fig. 14. (Color online) The spectral density ρk,σ(ω) for the
spin-up electrons (upper panel) and spin-down (lower panel)
plotted as a function of ω and a sequence of values of εk for
U = 6 and 12.5% doping. Also shown with arrows are the
positions of the free quasiparticle excitations, with the height
of the arrow indicating the corresponding weight.
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 13
In order to compare well with the case U = 3 we have
chosen an identical range for ω and εk, although the large
spectral peaks near the energy are very close together in
this presentation. It can be seen that the overall features
are very similar to those seen for U = 3. For the spin
up spectrum (upper panel) the peaks for the lower band
have most of the weight near the Fermi energy, whereas
the upper band is suppressed there, and vice versa for
the opposite spin direction. The lower bands are tracked
well by the free quasiparticles, and we can see that the
bands for the larger value of U are significantly flatter.
This is also clearly visible in the following figure 15, where
we again compare the quasiparticle band with the peak
position of the full spectra. On the range shown the lower
band Ek,− completely coincides with the free quasiparticle
band E0
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 15. (Color online) A plot of the peak positions Ek,± in the
spectral density ρk,σ(ω) (full line) as a function of εk for U = 6
and 12.5% doping compared with the free quasiparticle disper-
sion E0k (dashed line). On the range shown the lower band Ek,−
completely coincides with the free quasiparticle band E0k,−.
From the k-resolved spectra in figures 12 and 14 we can
also extract the width of the quasiparticle peak ∆qp in
the spectral density ρk,σ(ω) (majority spin σ =↑). Its in-
verse 1/∆qp gives a measure of the quasiparticle lifetime.
The results for ∆qp for the lower band Ek,− for the two
cases U = 3, 6 and 12.5% doping are shown in figure 16 as
function of εk.
This plot brings out more clearly the feature that can
be seen already in figures 12 and 14 (upper panel) that
the width increases sharply when we move away from the
Fermi level and the values for the width ∆qp for U = 6
are significantly smaller than those for U = 3. This is in
line with the fact that the local quasiparticle interaction
Ũ is smaller for the larger value of the bare interaction U
as commented on earlier. The free quasiparticle picture is
therefore even more appropriate in the case with stronger
interaction. To numerical accuracy the width vanishes at
and is finite for the interval ε−
< εk < ε
which lies
within the lower band but above the Fermi level.
−1 −0.5 0 0.5 1
Fig. 16. (Color online) A plot of the width of the peaks ∆qp in
the spectral density of the majority spin ρk,↑(ω) as a function
of εk for U = 3 (dashed line) and U = 6 (full line) and 12.5%
doping.
Another quasiparticle property, the effective mass en-
hancement m∗/m, can be extracted by calculating the
derivative of E0k,− in (19) with respect to εk, which yields
when evaluated at the Fermi energy (34),
µ̃0,↑µ̃0,↓
. (44)
The effective mass enhancement therefore does not only
depend on zσ, but also on the renormalized chemical po-
tentials µ̃0,σ. The general trend for m
∗/m as function of
U can be seen in figure 17 for the case of 7.5% doping.
0.2 0.4 0.6 0.8 1
Fig. 17. The ratio m∗/m according to (44) plotted over a
range of t2/U for 7.5% doping.
The effective mass increases sharply for large U as the
hole motion is energetically more costly in the ordered
background. The fact that the lower band for U = 6 seen
in figure 15 is flatter than in the case U = 3 in figure 13
14 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model
can be attributed to the larger effective mass. We find a
similar behavior of m∗/m as function of U for different
filling factors from the ones shown in figure 16. The trend
is that the effective mass enhancement is less pronounced
for larger doping, which is intuitively understandable by
the quasiparticle motion in an ordered background.
In the DMFT framework for the paramagnetic state
as well as the case with homogeneous magnetic field, the
quasiparticle spectral weight wqp and the inverse of the ef-
fective mass enhancement m/m∗ can be described simply
by the renormalization factor zσ. In figure 18 we show a
comparison of the spectral quasiparticle weight wqp (43),
the arithmetic, (z↑ + z↓)/2, and geometric,
z↑z↓, aver-
age of the renormalization factors, and the inverse of the
effective mass, m/m∗, (44) for U = 3 for various dopings.
0 0.05 0.1 0.15
m/ m*
Fig. 18. (Color online) Comparison of the spectral quasiparti-
cle weight wqp from equation (43), the arithmetic, (z↑+ z↓)/2,
and geometric,
z↑z↓, average of the renormalization factors,
and the inverse of the effective mass, m/m∗ from equation (44),
for U = 3 and a range of dopings.
As seen in this case with antiferromagnetic symmetry break-
ing these quantities take a different form (43) and (44)
and have distinct values. For different values of U the
behaviour is qualitatively similar. As a first approxima-
tion the quasiparticle spectral weight wqp corresponds to
the arithmetic average of the renormalization factors zσ,
whilst m/m∗ relates to the geometric average. In general,
one can, however, not omit the dependence on the renor-
malized chemical potential as it gives a significant contri-
bution as can be seen in figure 18. This can be understood
for example for the limit of zero doping. The system then
becomes an antiferromagnetically ordered insulator with
spectral gap. The weights zσ tend to finite values, but the
effective mass must diverge. This is found in equation (44)
since µ̃0,↑ → 0 for δ → 0, and the trend can be seen in
figure 18.
5 Conclusions
We have studied the field induced and spontaneous anti-
ferromagnetic ordering in the hole doped Hubbard model
with DMFT-NRG calculations at T = 0. A phase diagram
separating antiferromagnetic and paramagnetic solutions
for different values of doping and interactions U ranging
from zero to about 1.5 times the bandwidth W has been
established and is in agreement with earlier results by
Zitzler et al. [10]. Our main objective has been to ana-
lyze the properties of the quasiparticle excitations in the
metallic antiferromagnetic state. We presented two differ-
ent ways of calculating the parameters zσ and µ̃0,σ, which
define the renormalized quasiparticles, and the two sets
of results have been shown to be in agreement. We have
also been able to deduce the effective on-site quasiparticle
interaction Ũ from the NRG low lying excitations. The
low energy properties of the local spectral function can
be understood in terms of the free quasiparticle picture.
We have used the second order perturbation expansion in
powers of Ũ to estimate the spectral weight in the pseudo-
gap region above the Fermi level.
We have been able to compare the position of the
peaks found in the k-dependent spectral functions with
the dispersion relation for the free quasiparticles. The free
quasiparticle dispersion gives a very good fit to the posi-
tion of these peaks in the lower band which intersects the
Fermi level. The quasiparticle lifetime, as deduced from
the widths of the peaks in the spectrum, increases for
stronger interactions. This is consistent with the fact that
the on-site quasiparticle interaction Ũ , which gives the
quasiparticles a finite lifetime, decreases with increase of
U in the same range. We have also shown how the spec-
tral quasiparticle weight at the Fermi level wqp and the
effective mass can be deduced from the parameters zσ and
µ̃0,σ. The effective mass is found to increase with the in-
teraction, and it diverges in the limit of zero doping whilst
wqp remains finite.
We have found that Luttinger’s theorem for the total
electron density in the antiferromagnetically ordered state
holds within the numerical accuracy for the range of dop-
ings and interactions studied. This is a further indication
that many aspects of Fermi liquid description may hold in
situations with symmetry breaking.
It is not easy to make a direct comparison of our re-
sults with earlier work [3] analyzing the quasiparticle exci-
tations in an metallic antiferromagnet as these have been
mainly based on the t − J-model for one or two holes
in a finite cluster. However, at a semiquantitative level,
the overall trend in our results seems to be similar to the
results surveyed by Dagotto, where the effective quasipar-
ticle bandwidth Weff is found to decrease with decreasing
J . This is line with our results if we identify Weff ∼ m/m∗
and J ∼ t2/U (see figure 17). Our values for the spectral
quasiparticle weight wqp are qualitatively similar to those
presented as the wavefunction renormalization Z in the
review article by Dagotto (see fig 27 [3]), and also the
ones reported more recently [32].
J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 15
Acknowledgment
We wish to thank N. Dupuis, D.M. Edwards, W. Koller, D.
Meyer and A. Oguri for helpful discussions and W. Koller
and D. Meyer for their contributions to the development
of the NRG programs. We also acknowledge stimulating
discussions with G. Sangiovanni. One of us (J.B.) thanks
the Gottlieb Daimler and Karl Benz Foundation, the Ger-
man Academic exchange service (DAAD) and the EPSRC
for financial support.
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	Introduction
	Antiferromagnetic Broken Symmetry in DMFT
	Local Quasiparticle Parameters
	Spectra and Quasiparticle Bands
	Conclusions
