Efficient treatment of the high-frequency tail of the self-energy function and its relevance for multiorbital models
ABSTRACT In this paper, we present an efficient and stable method to determine the one-particle Green's function in the hybridization-expansion continuous-time (CT-HYB) quantum Monte Carlo method within the framework of the dynamical mean-field theory. The high-frequency tail of the impurity self-energy is replaced with a noise-free function determined by a dual-expansion around the atomic limit. This method does not depend on the explicit form of the interaction term. More advantageous, it does not introduce any additional numerical cost to the CT-HYB simulation. We discuss the symmetries of the two-particle vertex, which can be used to optimize the simulation of the four-point correlation functions in the CT-HYB. Here, we adopt it to accelerate the dual-expansion calculation, which turns out to be especially suitable for the study of material systems with complicated band structures. As an application, a two-orbital Anderson impurity model with a general on-site interaction form is studied. The phase diagram is extracted as a function of the Coulomb interactions for two different Hund's coupling strengths. In the presence of the hybridization between different orbitals, for smaller interaction strengths, this model shows a transition from metal to band-insulator. Increasing the interaction strengths, this transition is replaced by a crossover from Mott-insulator to band-insulator behavior.
An Efficient treatment of the high-frequency tail of the self-energy function and its
relevance to multi-orbital models
Gang Li1, ∗and Werner Hanke1
1Institut f¨ ur Theoretische Physik und Astrophysik,
Universit¨ at W¨ urzburg, 97074 W¨ urzburg, Germany
In this paper, we present an efficient and stable method to determine the one-particle Green’s
functions in the hybridization-expansion continuous-time (CT-HYB) Quantum Monte Carlo method
within the framework of the dynamical mean-field theory (DMFT). The high-frequency tail of the
impurity self-energy is replaced by a noise-free function determined by a dual-expansion around the
atomic limit. This method does not depend on the explicit form of the interaction term. More
advantageous, it does not introduce any numerical cost to the CT-HYB simulation. We discuss
the symmetries of the two-particle vertex, which can be used to optimize the simulation of the
4-point correlation functions in the CT-HYB. Here, we adapt it to accelerate the dual-expansion
calculation, which turns to be especially suitable for the study of material systems with complicated
band structures. As an application, a 2-orbital Anderson impurity model with a general interaction
form is studied, the phase diagram is given as a function of the Coulomb interactions for two different
Hund’s coupling strengths. In the presence of the hybridization between different orbitals, for smaller
interaction strengths, this model shows a transition from metal to band-insulator. Increasing the
interaction strengths, this transition is replaced by a crossover from Mott insulator to band-insulator.
PACS numbers: 71.10.Fd, 71.27.+a, 71.30.+h
The study of electronic structure of transition metal
and heavy-fermion materials is one of the most active
field in condensed matter physics.
lated d- and f-electrons cannot be fully described by ef-
fective one-particle methods, such as the local-density
approximation (LDA) to the density-functional theory
(DFT). Here, the DMFT can be a powerful tool, espe-
cially when the momentum dependence of the self-energy
is negligible, regardless of the electron-electron interac-
The central problem of the DMFT is to solve an effec-
tive impurity model. In real materials, such a model usu-
ally contains both inter- and intra-orbital interactions,
as well as the hybridization between different orbitals.
They account for the competitions between the magnetic,
charge, and orbital fluctuations. Thus, an efficient impu-
rity solver, which can handle all the interactions and hy-
bridizations is of obvious importance. Among the avail-
able impurity solvers1–5, the numerically exact Quan-
tum Monte Carlo (QMC) methods were widely used.
The recent development of the continuous-time Quan-
tum Monte Carlo (CT-QMC) methods6–9further sup-
ports the DMFT for the study of realistic materials in
the sense that lower temperature region can be reached
and more orbitals can be investigated.
For realistic material calculations based on the CT-
QMC solvers, correctly resolving the high-frequency be-
havior of the impurity self-energy Σimp(iωn) is of cru-
cial importance. On the one hand, due to the iterative
nature of the DMFT equations, Σimp(iωm) determines
the Weiss-function at each iteration and, in the end, the
converged solution of the DMFT. On the other hand,
Σimp(iωn) strongly influences the determination of the
The highly corre-
total particle number, which has a direct connection to
In this paper, we show how to determine the impurity
self-energy for a rather general multi-orbital model in
an efficient and stable manner within the CT-HYB. The
direct simulation in the Matsubara-frequency space and
careful treatment of the self-energy high-frequency tail
make this method especially suitable for studying the
material systems with complex band structures.
This paper is organized as follows: Sec. II explains
how the ’dual transformation’ can be employed to effec-
tively simulate the one-particle Green’s function in the
CT-HYB. Additionally, it is shown how the simulation
of the two-particle Green’s function χ can be straight-
forwardly carried out as the Wick’s theorem still holds.
The symmetry of χ is discussed in detailed in this sec-
tion. In Sec. III, we make use of the CT-HYB to study a
2-orbital Hubbard model with a general interaction term.
For readers who are especially interested in our CT-HYB
implementation and the self-energy correction scheme,
Sec. II is the primary option. If mainly the phase dia-
gram of the 2-orbital model is of interests, one may skip
Sec. II and go to Sec. III which is self-contained. Con-
clusions and outlook can be found in Sec. IV.
To explain our implementation of the CT-HYB in a
concrete framework, here we take a 2-orbital model as
an example, i.e.
Hloc = H∆+ HV+ Hint− µ
arXiv:1109.4056v1 [cond-mat.str-el] 19 Sep 2011
tween orbitals. For the interaction part, a general form
σ(n1σ− n2σ) represents the crystal splitting,
and HV = V?
1σc2σ+ h.c.) is the hybridization be-
Coulomb interactions, as well as the spin flip and pair-
ing hopping processes. As an impurity solver for the
DMFT, CT-HYB employs essentially the same idea as
all the other CT-QMC impurity solvers, i.e. it expands
the impurity effective action around a certain limit and
evaluates the expansion terms via stochastic sampling.
Here, we will only present the expressions relevant to
this work. For a more detailed review of the CT-QMC
methods, we suggest ref. 10.
In the CT-HYB, the expansion of the ”impurity +
bath” action Stot= Sloc+Sbath+Shybaround the atomic
limit is carried out by integrating out the bath degrees of
freedom. Sloc,Sbathare the actions for the local and the
bath Hamiltonian, respectively. Shyb is the hybridiza-
tion between them11, which is expanded order by order.
The contraction of the bath operator bσ,b†
Wick’s theorem as the bath is non-interacting. This re-
sults in a determinant DetCkwith the hybridization func-
tion ∆(τ,τ?) as matrix elements. The full hybridization
matrix usually can be decoupled into block diagonal form
with respect to certain conserved quantum numbers, for
example, the total particle number n, the spin σz and
cluster momenta K. The final expression of the parti-
tion function can then be written as,
Z = ZbZloc
Here, ka is the expansion order (also the dimension of
the determinant matrix) for the ”a” flavor, where flavor
represents spin, orbital, or cluster momenta. Tr(Ck) =
a group of ’kinks’9, i.e. cluster operators, in the interval
of [0,β). From now on, we always work with the diago-
nal form of the hybridization function. The evaluation of
Tr(Cka) can be carried out in two ways. One can either
express the cσ,c†
of Hloc, or employ the Krylov implementation. The for-
mer one benefits from the diagonal form of the time evo-
lution operators e−Hlocτ. The Krylov implementation, on
the other hand, works in the particle-number basis, for
which e−Hlocτbecomes a sparse matrix. It uses the effi-
cient Krylov-space method, which allows to simulate up
to typically 7 orbital problems at acceptable numerical
costs. In this work, the first implementation is used, in
which we diagonalize Hlocwith respect to the conserved
quantum numbers12. The trace of the fermion operators
a(τk)? is the cluster trace of
σoperators as matrices in the eigenbasis
is evaluated by first searching for non-zero overlap be-
tween different eigenstates with respect to the group of
the cluster operators. The nonzero trace is, then, calcu-
lated along the trajectory found.
A. One-particle Green’s function
The impurity Green’s function is obtained by remov-
ing one row and column from the determinantal matrix.
Ga(iωn) simply relates to the inverse of M−1= ∆ by9,12
Ga(iωn) = −1
Alternatively, one can simulate the impurity Green’s
function from the cluster trace at each Monte Carlo
For each specific configuration Ck sampled in the CT-
HYB, this expression has the following form
The explicit form of the determinant is given in Eq. (3),
l+1are the left and right list of cluster operators
cα(τ) with the constraint τl+1< τ < τl. The partition
function corresponding to the configuration Ck is given
By combining the above two equations, we have
with the notation: Tm,n
iωn+ En− Ep
The ratio ZCk/Z is the probability of configuration Ck
being sampled in the Monte Carlo simulation.
When kais small, we measure Gimpdirectly from the
cluster trace12, i.e. Eq. (8). Although this scheme is
a not very fast, it is more stable than Eq. (4). When
ka is large and Eq. (4) is used in the simulation, the
high-frequency parts of Gimpconverge much slower and
contains more statistical errors than the low-frequency
parts.As a result, the corresponding self-energy can
be fluctuating at large ωn. Such fluctuations originates
from the large data noise of Gimp(τ) at τ ≈ β/2. As al-
ready pointed out in the introduction, the correct high-
frequency behavior of Σimp(iωn) is crucial for the CT-
HYB. Thus, special attention has to be payed to get rid
of the noises in the self-energy data.
To the best of our knowledge, there are three schemes
have been proposed to this problem. 1). Noise filtering:
one can either smooth the noises at τ ≈ β/2 by averaging
Gimp(τ) over a small range of τ, see ref.8,9, or apply the
orthogonal polynomial filtering routine recently proposed
by Boehnke et al.13to achieve a smooth Gimp(τ) for all
τ ∈ [0,β). The order of the orthogonal polynomial has
to be chosen carefully in order to achieve the best filter-
ing. 2). Replacing the high-frequency tail of Σimp(iωn)
with some well-behaving function: This function can be
either the self-energy, calculated from a weak-coupling
perturbation expansion, or the moment-expansion of the
Green’s function14,15.Such a replacement provides a
smoothly behaved high-frequency tail of the self-energy
function. However the corresponding expression usually
becomes complicated in the multi-orbital case and relies
on the explicit form of the interaction term. 3). Mea-
suring Gimp(τ) from an improved estimator which is ob-
tained through its equation of motion16. This method
becomes advantageous for the density-density type inter-
action, for which the ’segment picture’8can be used. For
general type interactions, numerical cost has to be payed
to measure additional correlators.
Here, we propose a simple and stable scheme which
does not rely on any direct noise filtering of Gimp(τ),
and does not introduce any cost to the run-time sim-
ulation. This method does not depend on the explicit
form of the interaction term and remains efficient in the
multi-orbital calculations. The basic idea is to determine
an approximate self-energy function by performing the
perturbation expansion around the atomic limit, using
the ’dual-transformation’. As we will see later on, such
method generates systematic improvements to the atomic
self-energy. The first-order expansion term already gives
considerable corrections and reproduces the correct high-
frequency behavior of Σimp(iωn).
The expansion around the atomic limit has been
studied17.In the strong-coupling region, this method
yields results comparable to the numerical exact QMC
results. Here we use an elegant and different way, i.e.
the ”dual transformation”18. This transformation has
been used in the construction of the dual fermion (DF)
method, which gives an action well behaving in both the
weak- and strong-coupling limit. Thus, our perturbation
expansion actually also works in the weak-coupling re-
gion. The impurity model has the following action
S[c∗,c] = Simp[c∗,c] +
In the ’dual transformation’, new variables f∗,f are in-
troduced to rewrite the hybridization term in the follow-
The complex number α can be arbitrary in the above ex-
pression. In ref. 18, it is taken as the impurity Green’s
function. This makes the correlator of the dual variables,
i.e. Gd= −?faf∗
function, which decreases as 1/iωnfor large ωn. For sim-
plicity, we take α as one. Although in this case, the dual
variables can not be interpreted as fermions, the impurity
Green’s function remains the same.
Integrating out the c-variable, the full action becomes
a functional which only depends on variables f∗,f, i.e.
of dual variables turns out to be the reducible four-point
correlations of the atomic system, i.e. V(4)
Since the dual transformation is mathematically ex-
act, the two different actions which depend on only c-
variables, i.e. Eq. (10), and f-variables, i.e. Eq. (12),
are equivalent. Thus, we can obtain an exact relation be-
tween the correlators for Gaand Gd
the two actions with respect to ∆a. This yields:
a? behaves like the one-particle Green’s
Z = ZfZb
0is given as [Gat
a]−1. The effective interaction
23, with Gatthe atomic Green’s
a − ∆−1
variables. The expression of χimp
literature, e.g. ref.19–21. If the interaction of the dual
variables in Eq. (12) is neglected, the atomic self-energy
will be recovered. This can be seen by inserting Gd
Eq. (13), we have
ais obtained from the Dyson equation, Gd
ais the self-energy function of the dual
12;34can be found in the
Ga(iωn) = Gat
a/(1 − ∆aGat
Then, from the Dyson equation, we immediately see that
= iωn+ µ − ∆a− G−1
= iωn+ µ − 1/Gat
Thus, one can imagine the interaction term in Eq. (12)
will generate systematic corrections to the atomic self-
By including the interaction and further restricting the
calculation of Σd
ato the first order, we have
In this equation, only the element Vd,(4)
quired. Additionally, this calculation can be further ac-
celerated by employing the look-up routine and the sym-
metry of χat
12;34, which is shown in Sec. IIB. By doing
so, the perturbation expansion remains very efficienct in
As a benchmark, we first apply the dual expansion
scheme by restudying the Bethe lattice with different
bandwidth, i.e. W2 = 2W1, where the orbital-selective
Mott transition can happen8.
DMFT equation with the high-frequency supplemented
self-energy function, instead of using Eq. (20) in ref. 9.
Our self-energy data in Fig. 1 is identical to those in Fig.
a,σ(iωn) = −1
12;34δ12δ34 is re-
We directly solved the
βt1= 50,U/t1= 4,J/U = 0.25,t2= 2t1
narrow-band, dual expansion
wide-band, dual expansion
FIG. 1: Benchmark: imaginary part of the impurity self-
energies for βt1 = 50,U/t1 = 4,J/U = 0.25 in unit of t1.
Both the narrow and wide bands are metallic. The dual ex-
pansion gives two different asymptotic behaviours of the self-
energy for two different bands, as expected. However, the
atomic self-energy does not have such a resolution.
12 of ref. 9, meaning that the dual expansion method is
reliable to produce the high-frequency tail of the self-
energy and can be used in the CT-HYB for solving im-
purity problems. To see the performance of the dual ex-
pansion method for a finite spatial-dimension problem, in
Fig. 2, we show the comparison of the self-energy func-
tion calculated for a 2-orbital Hubbard model at two-
dimension (see the Hamiltonian in Eq. (1) ). The im-
provement from the dual expansion is clearly seen from
the agreement between the CT-HYB and the dual expan-
sion results. Increasing the hybridization strength, this
agreement becomes even better. Thus, a smaller num-
ber of Matsubara frequencies is required to simulate in
such a case. However, the atomic self-energy has a larger
deviation from the CT-HYB results for smaller ωn.
Similar ideas were used to formulate effective impurity
solvers19,21for the DMFT. We use it here to get the cor-
rect high-frequency tail of the impurity self-energy, while
still keeping the low frequency self-energy function sim-
ulated from the QMC. This method only needs the hy-
bridization function at each DMFT iteration. The dual-
expansion can be carried out independent of the CT-HYB
simulation. Thus, it does not introduce additional nu-
merical cost to the CT-HYB, which is another essential
difference with respect to previous works12–16.
FIG. 2: The comparison of the impurity self-energy calcu-
lated from the CT-HYB, atomic Hubbard model and the dual
expansion method. The parameter sets for the 2-orbital Hub-
bard model are βt = 50,U/t = 4.0,J/U = 0.25,V/t = 1.0.
See the text for more details.
B.4-point correlation function χ12;34
The dual-expansion, discussed in the above section, re-
quires the knowledge of the atomic 4-point correlation
12;34. In the multi-orbital case, such calcula-
tion can be hard since the large-dimensional matrix mul-
tiplication is time-consuming. In this case, one can again
use the block diagonal form of the Hamiltonian matrix
and employ the “look-up” routine as we did in the trace
calculation. Here, we want to further simplify the cal-
culation by employing the symmetry of χat
12;34. Such a
symmetry turns out to be also very useful in the simula-
tion of the impurity 4-point correlation function χimp
Thus, in this section we try to keep our discussion gen-
eral. We start from the simulation of the χimp
CT-HYB and discuss the symmetries of it afterwards.
The same symmetry requirements are satisfied by χat
Although in the CT-HYB, Wick’s theorem apparently
is not supported by the impurity action, the 4-point cor-
relation function can be simulated by removing two rows
and two columns from the determinant matrix, which re-
sults in an expression analogous to those for the CT-INT
and the CT-AUX. Effectively, one can still simulate the
4-point correlation function as if Wick’s theorem holds.
Here, we use the following notation to symbolically rep-
resent this expression:
= g12(ω1,ω2)g34(ω3,ω4) − g14(ω1,ω4)g32(ω3,ω2) (17)
where labels 12;34 represent “orbitals, sites, spins” et al.
In the CT-HYB, the two-frequency dependent propaga-
tors gαβ(ω,ω?) is given as
4? − ?c1c∗
gαβ(ω1,ω2) = −1
It has the following symmetry in Matsubara frequency
gαβ(ω1,ω2) = g∗
which reduces the numerical effort by a factor of two. A
similar symmetry is also satisfied by χ.
12;34(ω,ω?) = χ−Ω,∗
In what follows, we denote ω = ω1, ω+Ω = ω2, ω?+Ω =
ω3, ω?= ω4. Eq. (20) says, only for Ω > 0, χ needs to
Symmetry (20) relates the positive to the correspond-
ing negative frequencies of χ. It is also possible to find
symmetries which connect different ω, ω?in the same Ω-
sector. This can be achieved via the fact that, χ12;34is
anti-symmetric under the exchange 1 ⇔ 3 and 2 ⇔ 4:
χ34;12(ω?+ Ω,ω + Ω;−Ω) = χ12;34(ω,ω?;Ω).
Combining Eq. (21) with Eq. (20), we have
χ34;12(−ω?− Ω,−ω − Ω;Ω) = χ∗
Given the spin configurations of different χ channels, we
12;34satisfies both symmetries in Eq. (20) and
Eq. (22). However, χσσ;¯ σ¯ σ
12;34only satisfies the symmetry
shown in Eq. (20) and the following relation.
χσσ;¯ σ¯ σ
12;34= χ¯ σ¯ σ;σσ
One can implement the symmetries in Eqns. (20) and
(22) as follows: 1). χ↑↑;↑↑, χ↓↓;↓↓, and χ↑↑;↓↓are simu-
lated only for Ω > 0. 2). For each specific Ω considered,
Eq. (22) is further applied to χ↑↑;↑↑and χ↓↓;↓↓. Only for
one part of the frequency points in this Ω-sector, χ↑↑;↑↑
and χ↓↓;↓↓need to be simulated. 3). At the end of the
calculation, Ω < 0 components are calculated through
Eq. (21). 4). χ↓↓;↑↑is calculated by Eq. (23). In ad-
dition to the symmetries shown in Eqns. (20) and (22),
it is possible to find more symmetries to relate different
Before finishing this section, we want to note the 4-
point correlation function is useful not only for the physi-
cal response function and the dual-expansion scheme, but
also relates closely with the extension of the DMFT. In
the DF method18and the dynamical vertex approxima-
tion (DΓA)20, non-local self-energy is constructed from
the impurity two-particle vertices.
As a typical application, we consider here a 2-orbital
Hubbard model (see the Hamiltonian in Eq. (1)), with
rotationally invariant interactions, i.e. U?= U−2J,U??=
U?− J. To make a link with realistic material systems,
this multi-orbital Hubbard model can be viewed as an ef-
fective model for the eg-orbital systems. The rotational
invariance of the interaction term is not obligatory in
the CT-HYB solver, here we use it only as one possible
situation. By making use of the DMFT, the 2-orbital
Hubbard model has been studied by many groups9,22–29.
These calculations are either based on semicircular den-
sity of states, which corresponds to the Bethe lattice, or
they employ an impurity solver with certain limitations
in temperature or interaction strength. Here, we solve
the DMFT equation at finite dimension and tempera-
tures. In these cases, the DMFT loop cannot be closed
by a simple relation in the imaginary-time space like on
the Bethe lattice. Thus, our dual-expansion method dis-
cussed in Sec. IIA, turns to be a decisive tool. Our
calculations are mainly performed on ordinary desktop
Compared to the single-orbital case, two issues in a
multi-orbital model are of obvious interests:
(1). What is the effect of the orbital fluctuations?
general believe is, that it is competitive to the Coulomb
interaction. As a result, the metallic state can be stabi-
lized up to a large interaction value30,31.
(2). How does the Hund’s coupling modify the transition
from the metal to Mott insulator (MIT)?
that the 2-orbital Hubbard model behaves quite differ-
ently with and without J22,23.
The phase diagrams of the 2-orbital Hubbard model can
be found in ref. 23,28. Here we study, in particular, the
coexistence region for different values of J in Fig. 3 (a),
which indicates the MIT is of first order. Compared to
the phase diagrams for the Bethe lattice23,28, the reduc-
It is known
(a) Phase Diagram
-20 -15 -10 -5 0 5 10 15 20
-20 -15 -10-5 0 5 10 15 20
FIG. 3: The phase diagram of the 2-orbital Hubbard model
at half filling. The MIT at βt = 50 for two values of J/U
are shown as histograms.The two local density of states
on the right hand side correspond to the two solutions for
U/t = 8.4,J/U = 0.1.
tion of the spatial dimension does not change significantly
the critical Coulomb interaction value of Uc when it is
normalized by the full bandwidth. However, Ucbecomes
larger compared to the single-orbital model, which con-
firms that the orbital fluctuation stabilize the metallic
phase. With the increase of the Hund’s rule coupling J,
we found the coexistence region to become smaller. For
the two values of J/U in our calculations, the reduction
is about 0.2 eV. On the other hand, Bulla, et al.32found,
for J/U > 0.25, the transition to be of second order. At
J/U = 0.25, our results show that the coexistence re-
gion still has a reasonably large width. Thus, we believe
that even for J/U > 0.25, the MIT remains first order.
Whether, with the increase of J/U, the coexistence re-
gion completely disappear for sufficient large Hund’s rule
coupling J deserves more investigations.
On the right hand side of Fig. 3, two different solu-
tions of the local density of state, i.e. A(ω), are displayed
for U/t = 8.4. They correspond to the metallic, see Fig.
3 (b), and insulating states, see Fig. 3 (c), in the coex-
istence region. A(ω) is obtained by using the stochastic
analytical continuation directly on the Matsubara data
In Fig. 4, the typical behavior of the metal to band-
insulator transition is shown by calculating the impu-
rity Green’s function and the corresponding self-energy
as a function of the hybridization. Increasing the hy-
bridization V/t tends to open a band gap. Furthermore,
with the increase of V/t, the impurity Green’s function
at the lowest Matsubara frequency becomes smaller and
finally approaches zero, see Fig. 4 (a). The metal to
band-insulator transition happens somewhere between
V/t = 2.5 and 3.0 for U/t = 4. This transition is not
visible from the self-energy plot, where Σimp(iωn) be-
haves similarly for different values of V/t. The slope, i.e.
∂Σimp(ω)/∂ω|ω0, remains negative for all hybridization
FIG. 4: Behavior of the impurity self-energy and Green’s func-
tion around the metal to band-insulator transition as func-
tions of the hybridization strength V/t.
strengths, see Fig. 4 (b). In contrast, the slope of the lo-
cal Green’s function around ω0has different signs before
and after the metal-insulator(band) transition.
Increasing further the value of U/t strengthens both
the intra- and inter-orbital interactions. Finally, for val-
ues of U/t of the order of the non-interacting bandwidth,
the metal to band-insulator transition is replaced by the
Mott-insulator to band-insulator crossover as a function
of the hybridization strength ∆/t. This behavior is dis-
played in Fig.5.In contrast to the metal to band-
insulator transition shown in Fig.
a choice of U/t = 9, we notice that the local Green’s
4, in Fig.5 with
FIG. 5: Similar to figure 4, but with different interaction
strength U/t = 9, where a Mott insulating state is found
at ∆/t = 0,0.125,0.25.The increase of the hybridization
between two orbitals greatly changes the behavior of the self
energy, while leaves the one-particle Green’s function nearly
2 4 6
-6 -4-2 0
2 4 6
FIG. 6: The inter- and intra-orbital components of the re-
ducible impurity two-particle susceptibility for βt = 20,U/t =
6,J/U = 0.25.
function stays nearly unchanged with modifying the hy-
bridization strength V/t, i.e. Gimpshows an insulating
behavior for all values of V/t. However, for different val-
ues of V/t, the insulating nature is indeed different. This
can be seen from the variation of the self-energy func-
tion shown in the right-hand side of Fig. 5. Increas-
ing V/t results in increasing of ∂Σ(ω)/∂ω|ω0for any fi-
nite V/t, indicating the crossover from Mott-insulator to
By applying the symmetries presented in Sec. IIB,
we show the inter-orbital and intra-orbital reducible spin
susceptibilities in Fig. 6 for βt = 20,U/t = 6,J/U = 0.25
and V/t = 0, with a,b the orbital indices.
ab,Ω− ˜ χσσ,¯ σ¯ σ
the subtraction of the impurity bubble susceptibilities.
They are plotted as functions of the two fermionic fre-
component is given, the implementation discussed in Sec.
IIB works for any value of Ω. Fig. 6 (a) and (b) refer
to the 3D plots of ˜ χspin,ab
top-view plots are shown in Fig. 6 (c) and (d). Based
on the CT-HYB, the 4-point correlation functions were
recently also calculated for the effective one and four-
orbital systems13,35for different problems. Another effi-
cient and stable, but approximate algorithm can be found
in Ref. 36.
n) are the impurity susceptibilities with
nfor fixed Ω = 0. While only the Ω = 0
n), the corresponding 2D
From Fig. 6, we see that the reducible two-particle sus-
ceptibility ˜ χspin,ab
(ωn,ωn?) decays rather fast as a func-
tion of ωn and ωn?. The dominant contribution comes
from the elements with ωn= 0, or ωn? = 0, or ωn= ωn?.
For our parameter set, the inter-orbital spin susceptibil-
ity shows a sharper structure than the intra-orbital one,
which can be viewed as a precursor of the possible orbital
In this paper, we showed how the high-frequency tail of
the self-energy can be calculated in a controlled manner
from the dual transformation in CT-HYB. This scheme
provides an efficient recipe for finite dimension DMFT
studies when taking the CT-HYB as an impurity solver.
Our procedure is based on a Matsubara frequency space
simulation and produces more moments from the dual
expansion. Thus, it generates a better high frequency
self-energy behavior. Most importantly, it does not in-
troduce any numerical cost to the runtime simulation.
We also simulated the 4-point correlation function for
different spin configurations in the particle-hole channel.
To this end, we implemented different symmetries to re-
duce the memory and CPU requirements without loosing
As a first application, we demonstrated the usefulness
of our method for a 2-orbital model with a general on-
site interaction. From this study, we deduced a substan-
tial influence of the Hund’s rule coupling on the metal-
insulator transition phase diagram, especially on the co-
existence region. In particular, we find that for any finite
value of J/t, the MIT stays first order.
Our scheme is also of particular use for connecting
the dual fermion method, which many be viewed as a
non-local extension of the DMFT, with a priori DFT
techniques. A multi-orbital dual fermion calculation will
be especially interesting and rewarding for the DFT +
dual fermion study of material systems. In such study,
the CT-HYB effectively works on an impurity problem
with the DFT dispersions as input.
good control on the ”minus-sign” problem. The high-
momentum resolution, provided by the dual fermion al-
gorithm, makes the result ready to be compared with
experiments, such as ARPES data.
Thus, one has a
One of us (G.L.) acknowledges the valuable discus-
sions with Philipp Werner, Xi Dai and Zhong Fang and
is grateful for the hospitality of Institute of Physics, Chi-
nese Academy of Science. We thank Fakher Assaad for
providing us the initial stochastic analytical continua-
tion code, from which the extension to Matsubara fre-
quency space was made. This work was supported by
the DFG Grants No. Ha 1537/23-1 within the Forscher-gruppe FOR 1162.
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