M2 Internship report rare-earth nickelates

Internship Report
ICFP master M2, quantum physics track
École Normale Supérieure
Charge density waves in rare-earth
nickelates: a mean-field approach
Author:
Yiteng Dang
Supervisors:
Dr Michel Ferrero
Prof Antoine Georges
April 2015

Contents
Preface 2
1 Theoretical background 3
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Numerical calculations of the DoS . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Nearly free electron model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 An application of mean-field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Anti-ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.3 Analysis of the gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.1 Properties of imaginary-time Green’s functions . . . . . . . . . . . . . . . . . . . . 12
1.5.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Nickelates 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Low-energy two-orbital Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Mean-field Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Self-consistency equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Half-filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6.1 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Quarter-filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 Comparison of Quarter-filling results and half-filling results . . . . . . . . . . . . . . . . . 24
Conclusion 25
Appendix A: Green’s Functions 26
Appendix B: Derivation of the self-consistency equation for the two-band model 32
References 35
1

Preface
In this M2 internship, we apply mean-field theory to study the presence of charge density waves in
rare-earth nickelates. We start by recalling basic concepts and techniques of solid state physics. I had
limited background in this domain by the start of the internship and gained a better understanding of the
basics through exercises, which form a substantial part of my internship. Hence Chapter 1 these basic
concepts and especially the application of numerical methods. This includes the density of states for the
tight-binding model in different dimensions, the nearly free electron model, mean-field theory applied to
the Hubbard Model and imaginary time Green’s functions. The first two sections are typically taught in
a basic solid state physics course and should be very accessible. The Hubbard Model has been discussed
in the M2 course Condensed Matter Theory of the ICFP Master, while Green’s functions appeared in the
M2 course Strongly Correlated Fermions and Bosons, which started after the full-time internship period
had finished. These concepts are discussed in more detail and various numerical methods are applied to
show how to solve typical problems in condensed matter theory. Chapter 2 deals with nickelates and
forms the core of the internship . We introduce nickelates and its main properties and then discuss the
mean-field approach for a model Hamiltonian which captures the essential physics. In the end a phase
diagram for the zero-temperature case is given.
Acknowledgements
I would like to thank Prof Antoine Georges and Dr Michel Ferrero for giving me this wonderful oppor-
tunity to do my M2 internship in their group. They have been very knowledgeable not only in their
domain but also in setting up a schedule that has permitted me to get on track with the knowledge
required to do research in their area. My thanks also goes to my (changing) roommates Sophie, Loucas
and Alice at Collège de France, Priyanka and Wei at École Polytechnique, to Pascal, Thomas and the
rest of the group, both for useful discussions as well as for keeping the spirit high.
2

1 Theoretical background
1.1 Preliminaries
As prepration for the internship, I studied basic solid-state physics from the books by Alloul [1] and
Ashcroft and Mermin [2]. This was complemented by problem class (PC) exercises from the M1 course
PHY552A Quantum physics of electrons in solids (introduction to condensed matter physics) given
by Prof Antoine Georges at École Polytechnique. Some of the problems studied included the one-
dimensional electron gas in a weak periodic potential and the dispersion relation of graphene.
1.2 Density of states
I then moved on to study numerical methods to compute the density of states (DoS) for the tight-binding
model on a hypercubic lattice in different dimensions. Exact results are known only in d = 1, 2, ∞. In all
other cases the DoS can only be computed numerically. We focus on three different numerical methods:
(1) numerical integration, (2) Monte Carlo direct sampling and the (3) Metropolis algorithm.
The DoS is defined as
ρ( ) =
1
N k
δ( − k). (1)
We suppose that the spectrum k is known. The normalisation factor 1/N ensures that ρ( )d = 1.
In the continuum limit (N → ∞)
ρ( ) →
FBZ
dd
k
(2π)d
δ( − k) (2)
where the integral is now over the First Brillouin Zone (FBZ).
For the tight-binding model in d dimensions the spectrum is given by
k = −2t
d
i=1
cos(ki a). (3)
In one dimension we can compute the integral directly to obtain the analytical result
ρ( ) =
1
2π|t|
1
1 − (2t
)2
(Tight-binding DoS in 1D). (4)
In 2 dimensions it is also possible to find an analytic solution, in terms of elliptic functions. In the
large d limit, we can treat the cosine contribution from each of the dimensions as a random variable
with mean 0 and variance σ2
= 2t2
(this can be computed using the exact DoS for d = 1 as probability
distribution). The central limit theorem states that the resulting distribution must be Gaussian with
mean 0 and variance 2t2
d. This means that the distribution in arbitrary d would look like
ρd( ) =
1
t
√
πd
exp −
2
4t2d
. (5)
Hence the DoS vanishes as d → ∞, unless we rescale the hopping t → t√
d
. By doing this, we find a
standard deviation of σ =
√
2t, independent of d.
3

1.2.1 Numerical calculations of the DoS
We have mentioned three different ways to numerically compute the DoS. To begin with, for numerical
integration, we take k-points on a d-dimensional grid in the first Brillouin Zone and evaluate their
energies. We divide the total range of the energies into different bins and for each evaluated energy, add
one to the bin it belong to. In 1D we can compare the result with the analytic result and find good
agreement. We also find interesting results for 2D and 3D: in both cases we find singularities, which
are referred to as Van Hove singularities in the literature. In 2D the DoS diverges at k = 0, which in
reciprocal space corresponds to the square formed by joining the midpoints of the Brillouin Zone edges.
The divergence is logarithmic, as verified in Figure 2a, where we fit a function of the form a log | |+b to
the data, with a < 0 and b small. In 3D the DoS is continuous, but its derivative has a discontinuities
on both sides of the energy spectrum.
The downside of the previous method is that for higher dimensions the algorithm quickly becomes too
time-consuming. A better method would then be to use Monte Carlo algorithms. In particular, we
do random sampling of the first Brillouin Zone. This speeds up considerably at higher dimensions, so
that one easily obtains results for d = 10 for example. From intermediate results we already see that
the distribution tends towards a gaussian. For d = 100 the result fits perfectly with a gaussian with
standard deviation of
√
2 (t = 1 in all our computations).
The third method is to apply a Metropolis algorithm. In this case, we need a weight function w(k) so
that the integral
dd
k ρ(k) = dd
k
ρ(k)
w(k)
w(k) (6)
can be evaluated a similar way, but with an acceptance probability for the move kold → knew:
min(1,
w(knew)
w(kold)
). (7)
In general it is not obvious which function to use as weight function, but for d = 2, the logarithmic
fitting function is a good candidate. The results obtained with this function are shown in Figure (2).
We also implement the algorithm with C++ and note that the convergence for this code is much faster.
1.3 Nearly free electron model
We study the problem of an electron in a weak periodic potential. Physically, such a situation occurs
when the electron is placed in a crystalline lattice where the potential due to the ions is weak. Such
situations occur with certain metals in groups I, II, III and IV of the periodic table. Of course, the model
does not correspond perfectly to reality since it does not take into account impurities and deviations
from perfect periodicity that are present in any material. Nevertheless, we shall see that interesting
results can be obtained from this model.
We assume that the system is periodic and obeys the Born-von Karman boundary conditions (cf. [2]).
We can then expand the wave function in terms of plane waves
ψ(r) =
q
cqeiq·r
, (8)
where the sum is over wave vectors q allowed under the boundary conditions.
We consider a periodic potential V (r), i.e. V (r + R) = V (r) for all vectors R of the crystal Bravais
lattice. Expanding the potential in the plane wave basis, we see that only wave vectors of the reciprocal
lattice (denoted by K) are allowed:
V (r) =
K
VKeiK·r
. (9)
4

(a) d = 1 (b) d = 2
(c) d = 3 (d) d = 10
(e) d = 100
Figure 1: DoS for d = 1, d = 2, d = 3, d = 10 and d = 100 using numerical integration, Monte Carlo
direct sampling or the Metrpolis algorithm.
5

(a) The divergence at = 0 in d = 2. (b) Metropolis algorithm for d = 2.
(c) Results with the C++ code (plotted with
GNUPlot).
Figure 2: Some specific results for d = 2.
Since the potential is real, we must have
VK = V ∗
−K. (10)
The solution of the wave function is given by the Schrödinger equation, which using our plane wave
expansion can be rewritten as
q
eiq·r
2
2m
q2
− E cq +
K
VKcq−K = 0. (11)
Because the plane waves form an orthogonal basis for our wave function, the term inside the brackets
must vanish. Introducing q =
2q2
2m
, we find
( q − E) cq +
K
VKcq−K. (12)
In matrix form the expression reads








...
...
...
... ...
. . . q−K VK V2K . . .
. . . VK q VK . . .
. . . V2K VK q+K . . .
... ...
...
...
...
















...
ck−q
ck
ck+q
...








= E








...
ck−q
ck
ck+q
...








(13)
6

We see that the coefficients for a given wave vector q are only coupled to those which differ by a
vector of the reciprocal lattice. This is still an infinite set, but if the potential decays quickly enough,
that is to say if VK become smaller for increasing K sufficiently fast, then we can try to find an approx-
imate solution by considering a finite submatrix of the infinite matrix.
In the case when two levels are close to each other in energy, but far away from all other levels, we can
consider a 2 × 2 submatrix, whose wave vectors we denote as q and q − K. By solving for the spectrum
of this system, we see that the periodic potential induces changes from the free energy spectrum only
when q is near a Bragg plane (determined by the reciprocal vector K)1
.
More concretely, the effect becomes clear in one dimension, where Bragg planes are single points of the
form 1
2
K where K is a reciprocal vector. In this case, band gaps of size 2VK open up near these points.
By folding the bands into the Reduced Brillouin Zone, we find that gaps of size 2VK with K = 2π
a
n open
between the n-th and (n+1)-th energy bands.
We numerically verify these results. If we take an n × n matrix, then we will find n eigenvalues
{E
(k)
q |k = 1, . . . , n} with E
(1)
q ≤ E
(2)
q ≤ . . . ≤ E
(n)
q . By doing this for different q inside the first Brillouin
Zone, we construct the energy bands of the system. One can do this for a variety of potential profiles
(some examples are given in Figure 3). For each of these potentials we find gaps near the edges of
the Brillouin Zone as expected. However, the overall band picture looks very similar for all different
potential profiles we studied (see Figure 4).
1.4 An application of mean-field theory
In this section, we discuss an example of an application of mean-field theory. This is used to study
ferromagnetic and anti-ferromagnetic ordering in the Hubbard Model. The example is illustrative of the
usefulness of mean-field theory to predict qualitatively interesting phenomena.
1.4.1 Ferromagnetism
The Hubbard model is given by
H = −t
i,j ,σ
c†
i,σcj,σ + U
i
ni,↑ni,↓ − µ
i,σ
ni,σ, (14)
where c†
i,σ and ci,σ are spin σ fermionic creation and annihilation operators and ni,σ = c†
i,σci,σ. The
hopping parameter is given by t, the interaction energy by U and the chemical potential by µ. Under
the mean-field approximation, we replace the quadratic term by
ni,↑ni,↓ → ni,↑ ni,↓ + ni,↑ ni,↓ − ni,↑ ni,↓ . (15)
Let us now assume that there is long-range order in the system. Also assume that ni,σ = nσ is
independent of the site. Then we can define the average occupation number n and magnetization m by
n = n↑ + n↓
m = n↑ − n↓ .
(16)
Equivalently, we can write nσ = n+σm
2
. In momentum basis the operators take the form
ck,σ =
1
√
N i
ci,σe−ik·ri
. (17)
1
A derivation is given in Chapter 9 of [2].
7

(a) V (K) = −1
n2 . (b) V (K) = −1
n
(c) Asymmetric potential, V (K) = −1
n2 + i−0.5
n2.5
(d) Lorentzian profile. One can show that V (K) ∝
e−π|K|
∞
n=0
cos (4π2Kn). In practice we take a finite
sum over n.
Figure 3: Some potential profiles used to determine the band picture of nearly free electrons in one
dimension. Note that K = 2π
a
n, with n ∈ Z.
Let us denote nk,σ = c†
k,σck,σ. Under this Fourier transform, combined with the expressions of nσ in
terms of n and m, the Hamiltonian reads
HMF =
k,σ
k − µ + U
n + σm
2
nk,σ, (18)
where k = −2t
d
i=1
cos (ki a). Define the Fermi-Dirac function as
f (x) =
1
1 + eβx
. (19)
The expectation value of nk,σ is then
nk,σ = f k − µ + U
n + σm
2
. (20)
8

(a) V (K) = −1
n2 (b) V (K) = −1
n
Figure 4: Band pictures for the nearly free electron model, for two of the potentials shown above. Small
gaps open near the borders and the centre of the Reduced Brillouin Zone. The qualitative behaviour is
hardly different for different potential profiles.
Hence by going over to an integral representation we can write our self-consistency equations as
m = dω D(ω) f ω − µ + U(
n + m
2
) − f ω − µ + U(
n − m
2
)
n = dω D(ω) f ω − µ + U(
n + m
2
) + f ω − µ + U(
n − m
2
) . (21)
At T = 0, we find
m = dωD(ω + µ −
Un
2
) θ(ω −
Um
2
) − θ(ω +
Um
2
) = UmD(µ −
Un
2
) + O(m3
). (22)
9

Recall that m = ∂F
∂B
where F is the free energy and B the magnetic field, and from Landau theory we
know that F can be expanded in terms of m. This gives
m = αm + βm3
+ O(m5
). (23)
If β is negative, the condition of having a non-zero solution is α > 1. This translates to the condition
UD(µ − Un
2
) > 1.
Similarly, if we expand the equation for n in powers of m, we obtain
n = 2 dωD(ω)f(ω + µ −
Un
2
) + O(m) = 2
µ−Un
2
−∞
dωD(ω) + O(m). (24)
From this we see that in terms of the chemical potential in the absence of magnetic ordering µ0 = µ− Un
2
,
the chemical potential for the interacting system is given by
µ = µ0 +
Un
2
. (25)
Hence we find the Stoner Criterion
UCD(µ0) = 1. (26)
Consider flipping the spin of one electron in a configuration with equal number of spin up and spin
down electrons. There is a competition between the loss of potential energy resulting from placing the
electron in an unoccupied orbit and the gain of kinetic energy resulting from moving to a higher energy
level. By energy considerations one can then derive that we require U > UC for ferromagnetism (cf.
[9]). In the Landau theory picture this thus corresponds to a negative β.
1.4.2 Anti-ferromagnetism
Anti-ferromagnetism in the Hubbard model can be studied by a similar mean-field approach. We define
sublattices lattices A and B, such that for each pair of neighbouring one site is in A and the other in
B. We then expect the average magnetization, known as the staggered magnetization, to take opposite
values on A and B:
ni,σ = nσ =
1
2
(n + ξσm). (27)
Here the factor ξ equals +1 on A and −1 on B. For a (hyper)cubic lattice, with lattice spacing a, we
can write this as
ξ = eiq·ri
, q = (
π
a
, . . . ,
π
a
). (28)
The interaction term under mean-field theory becomes
HU =
k,σ
Un
2
nk,σ +
Um
2
σc†
k,σck+q,σ . (29)
At half-filling, we have µ = U
2
and the mean-field Hamiltonian can be expressed in the form
HMF =
k,σ
k≤0
c†
k,σ c†
k+q,σ
k
Umσ
4
Umσ
4
− k
ck,σ
ck+q,σ
. (30)
The spectrum of this mean-field Hamiltonian is given by
±Ek = ± 2
k + ∆2, (31)
10

where ∆ = Um
2
is the gap. The Hamiltonian is diagonalized by a Bogoliubov transform
ak,σ
bk,σ
=
cos θ −σ sin θ
σ sin θ cos θ
ck,σ
ck+q,σ
, (32)
where the angle θ obeys the property sin 2θ = − Um
2Ek
. We note that ak,σ and bk,σ also obey Fermi-Dirac
statistics. This finally gives the result
HMF =
k,σ
Ek b†
k,σbk,σ − a†
k,σak,σ (33)
1.4.3 Analysis of the gap
The next step is to consider the self-consistency equation for m. We can write
m = ( ni,↑ − ni,↓ )eiq·ri
. (34)
With a few steps of calculation one can show that
m =
Um
2 k
(f(−Ek) − f(Ek))
2
k + ∆2
(35)
In the zero-temperature limit, f(Ek) = θ(µ − Ek) = 0 and f(−E) = 1 − f(E) = 0, and we obtain
1 =
U
2 k
1
2
k + ∆2
. (36)
This equation has the same form as the BCS gap equation. In terms of the DoS, the zero-temperature
self-consistency equation writes
1 =
U
2
dω
D(ω)
√
ω2 + ∆2
. (37)
We now analyze this equation in the large U and small U limits. The question each time is whether there
is a solution for ∆ for arbitary values of U. This would imply the existence of an anti-ferromagnetic
phase, since it would give a non-trivial solution for m. To be fully rigorous one would need to check
that this solution has a lower energy than the configuration with m = 0, which is always a solution, but
this goes beyond the work performed here.
In the large U limit, we can write the gap equation as
1 = F(∆)U, F(∆) =
1
2
dω
D(ω)
√
ω2 + ∆2
. (38)
If U 1, then F(∆) must be small, implying ∆ must be large. Therefore, to first order
F(∆) =
1
2∆
dω
D(ω)
1 + ( ω
∆
)2
≈
1
2∆
dωD(ω) =
1
2∆
. (39)
Hence we see that in the large U limit, F(∆) ∼ 1/∆, so the gap is of order U.
In the small U case, the situation is more subtle. When ∆ goes to zero, either F(∆) diverges or it
does not. If it remains bounded, then it attains a maximum value which would imply that U cannot be
arbitrarily small. On the contrary, if it diverges such a value is not attained and therefore we may find
a solution for arbitrary U > 0. We immediately check that for ∆ = 0, we have
F(0) =
1
2
dω
D(ω)
ω
(40)
11

This function is likely to diverge near ω = 0. It would converges if D(ω) has a power law behaviour
near 0 (i.e. D(ω) ∝ ωα
for some α > 0), but such behaviour is highly unlikely. Hence we conclude that
at zero temperature, there is anti-ferromagnetic ordering for arbitrarily small U.
At finite temperature, we have the equation
1 =
U
2
dωD(ω)
f(−Ek) − f(Ek)
√
Ek
(41)
The maximum of the function F(∆) is obtained at
F(0) =
1
2
dωD(ω)
f(−|ω|) − f(|ω|)
|ω|
, (42)
which converges for arbitrary T > 0. Therefore, we see that at arbitrarily small finite T, there exists a
critical U below which there is no solution for ∆. This UC is determined by
UC =
1
F(0)
. (43)
1.5 Green’s functions
In this section we study properties of Green’s functions and spectral functions and verify such properties
by numerical computation. Calculating Green’s functions is an important task in condensed matter
physics and there exists a variety of intricate tools for doing such computations, including diagrammatic
expansions and dynamical mean-field theory. In this section we shall only touch upon basic properties
which can be directly derived from the definitions. We present the relevant properties below but refer
to Appendix A for a full derivation of these properties.
1.5.1 Properties of imaginary-time Green’s functions
The imaginary time Green’s function for a single fermionic particle is defined as
G(τ − τ ) = − Tτ c(τ)c†
(τ ) , (44)
where Tτ is the time-ordering operator. G(τ) is defined for 0 < τ < β but satisfies the property
G(τ + β) = −G(τ). (45)
The function is not well-defined for τ = 0 and τ = β, but the limiting values obey the relation
G(0+
) + G(β−
) = G(0+
) − G(0−
) = −1. (46)
Because G(τ) is defined on a finite interval, the Fourier transform G(iωn) is defined for discrete ωn =
(2n+1)π
β
which are called Matsubara frequencies.
There exists a function G(z) which is analytic everywhere on the complex plane except for the real axis
and matches G(iωn) as defined before on the given points of the imaginary axis. The retarded Green’s
function can be then obtained from analytic continuation by the substitution
GR
(ω) = G(iωn → ω + i0+
). (47)
Recall that the spectral function is defined as
A(ω) = −
1
π
ImGR
(ω). (48)
12

In the particular case when the spectral function is even - this may be the case for certain metals and
semi-conductors - the Green’s functions has a few more properties. In particular, the real part of G(iωn)
is zero in this case, while the imaginary part is an odd function of iωn. Taking the limit iωn → 0 gives
lim
iωn→0
Im G(iωn) = −πA(ω = 0). (49)
Finally, it can be shown that the Green’s function is symmetric on [0, β], i.e. G(β − τ) = G(τ).
Conversely, the Green’s function can be obtained from the spectral function through a Hilbert transform.
Furthermore, it can also be computed from a kernel expression (for 0 < τ < β) given by
G(τ) =
∞
−∞
dω K(τ, ω )A(ω ), (50)
with
K(τ, ω) = −
e−τω
1 + e−βω
. (51)
From this relation we can derive an approximate relation
G(β/2) ≈ −
π
β
A(ω = 0). (52)
1.5.2 Numerical results
For the free particle at energy k and chemical potential µ, one can obtain the exact results
G(τ) = −
e−τ( k−µ)
1 + e−β( k−µ)
G(iωn) =
1
iωn − ( k − µ)
. (53)
In general, the integral in equation (50) might not yield any analytic solution, but we can numerically
perform this integral by discretizing time and frequency. Taking meshes ω ∈ {ωi : ωi+1 − ωi = ∆ω ∀i}
and τ ∈ {τj : τj+1 − τj = ∆τ ∀j} we write this as
G(τi) =
j
∆ωK(τi, ωj)A(ωj). (54)
In matrix form we get
−→
G = K ·
−→
A, (55)
where we absorb ∆ω into
−→
A. Note that by inverting the matrix K we obtain a method for getting A(ωj)
from G(τ). However, as discussed below, this is not a very eﬀective method for obtaining the spectral
function since it quickly becomes unreliable as the matrix size increases and in the presence of noise.
Let us now use this procedure to compute the Green’s function for diﬀerent spectral functions. We
model a free particle, a metal and an insulator by choosing spectral functions of the following form:
1. for the free particle A(ω) = δ(ω − E).
2. for a metal a semi-circle,
A(ω) =
2
πR2
√
R2 − ω2 if − R < ω < R
0 elsewhere
. (56)
13

3. for the insulator two semicircles with a gap in between,
A(ω) =



1
πR2 R2 − (ω − L)2 if R − L < ω < R + L
1
πR2 R2 − (ω + L)2 if − (R + L) < ω < −(R − L)
0 elsewhere
(57)
The exact result for the free particle and the numerical results for the metal and insulator are shown in
Figure 5. Note that we can easily verify many of the properties derived above. We see that the bounds
G(0+
) and G(β−
) indeed add up to −1 in all our calculations, as required by equation (46). We see
that the value at β/2 is also as expected from equation (52) for the metal and insulator: it takes ﬁnite
value −π
β
A(ω = 0) = −2
β
= −0.04 for the metal and value 0 for the insulator. This approximation is
not valid for the free particle due to the fact that A(ω) is a delta function. For the Fourier components,
we immediately note that the real part is zero for the metal and insulator. Finally, the value as n → 0
of G(iωn) is also equal to −πA(ω = 0) = −2 in the case of the metal while it goes to 0 for the insulator,
as expected from equation (49).
Finally, we invert (54) to obtain the spectral function from the imaginary time Green’s function.
We only do this for the semi-circular A(ω) of the metal, to show that already in this case there are
severe limits. Starting from the Green’s function G(τ) obtained from the A(ω) through applying (54),
we invert the matrix K to re-obtain A(ω). Discretization errors enter in this step and these quickly
lead to wrong results when the matrix size is increased. For instance, our results in Figure (6) show
that while the result is still fairly accurate for lattices of size 15, it is no longer so for lattices of size
20. In addition, if we add a noise to the matrix elements of K−1
(for each matrix element we add a
small random number), the result also quickly becomes unreliable. This is related to the fact that the
determinant of K is very small, of the order of more than 10−36
for the plots below, so that the matrix
is in practice ’hardly invertible’.
14

(a) Free particle (β = 1) (b) Free particle (β = 10)
(c) Metal (d) Metal
(e) Insulator (f) Insulator
Figure 5: Green’s functions in time and frequency domain for diﬀerent systems. In all cases we take
R = 1 and L = 1.
15

(a) Grid size 15 × 15, no noise (b) Grid size 20 × 20, no noise
(c) Grid size 15 × 15, noise of order 10−9
(d) Grid size 15 × 15, noise of order 10−8
Figure 6: The spectral function recovered from the G(τ) by inverting the matrix equation (54) for the
metal we modelled. We assumed a semi-circular A(ω) to begin with and obtained G(τ) by the same
equation (54). The inﬂuence of grid size and noise is dramatic.
16

2 Nickelates
2.1 Introduction
It is well-known that certain compounds containing copper and oxygen - called cuprates - exhibit high
TC superconductivity and have rich and complicated phase diagrams. Typically, the crystal structure of
these cuprates is (close to) perovskite, with possible lattice distortions. In this report we study a different
class of materials - usually referred to as rare-earth nickelates - which does not exhibit superconductivity,
but exhibits an interesting phase transition that is not yet completely understood (cf. e.g. [4], [6], [7],
[8]).
The chemical formula for rare-earth nickelates is RNiO3, where R is a rare earth element (typically
studies focus on the fifteen Lathanides, the series from La to Lu in the periodic table). These materials
have a perovskite structure, typically with NiO6 octahedra sitting on an orthorhombic or square lattice
at the centre of which sits the rare earth cation. It is known that oxygen has valency −2 and rare
earth elements have valency +3, hence nickel in these compounds takes the form of Ni3+
. The electronic
configuration of Ni is 3d8
4s2
(recall that the 4s band is filled before the 3d band). The ionized Ni3+
has
the configuration is 3d7
. Due to the crystal structure the five bands of the 3d shells are split into a set
of three (called the t2g manifold) and a set of two (the eg manifold). The filling order of the bands is
dictated by the Aufbau principle and Hund’s Rules. In the case the splitting is large, the electrons fill
the t2g manifold completely before entering the eg manifold (such configurations are called low-spin).
We can then construct a low-energy description by focussing only on the eg bands, as detailed below.
The rare-earth nickelates undergo an interesting phase transition as temperature is lowered, which
involves electronic, structural and magnetic changes. Such a transition has been observed for nearly
all rare-earth elements and the temperature of this phase transition varies with the rare-earth element.
More specifically, a bad-metal [7] to insulator transition has been shown to take place simultaneously
with a structural transition from a orthorhombic structure to a monoclinic structure. In the latter case,
one differentiates between compressed octahedra with short Ni-O bonds (SB) and stretched octahedra
with long Ni-O bonds (LB). In addition, there is a transition from a paramagnetic state to an anti-
ferromagnetic state which takes place close to or slightly below the temperature of the metal-insulator
transition.
The exact nature of this transition is not well-understood, but several mechanisms are known to play a
role. To begin with, one can ask how the electrons in the eg band are placed for our d7
configuration.
The Jahn-Teller effect predicts that the degeneracy of the eg bands must be lifted and as a result the
octahedra become tilted, but this has not been observed in all compounds. Another mechanism that
appears to play a role is charge disproportionation, by which we mean that inequivalent Nickel sites will
get charge Ni3 + δ
(on SB sites) and Ni3 − δ
(on LB sites). In particular, it has been proposed that such
a charge disproportionation takes place when the Hund’s coupling J is sufficiently large [8]. Whether
this takes place depends on the competition between the on-site repulsion U and Hund’s coupling J.
We study the presence of such charge density waves through a mean-field approach, based on a model
Hamiltonian first introduced in [4]. To simplify things, we study the half-filled case for which there are
n = 2 electrons per site. We show that in the presence of large J and when the difference in energy
between inequivalent sites ∆s is large, charge ordering indeed takes place and construct a tentative phase
diagram at T = 0.
Earlier studies have led to a phase diagram of nickelates showing the dependence on the rare earth
element R and on temperature, as shown in Figure 7. This shows the concurrent existence of CDWs
and insulating properties, leading us to interpret CDWs as a sign of the insulating phase. The ultimate
goal would be to determine the link between the model Hamiltonian we present below and the actual
physics of nickelates as established by experimental results.
17

Figure 7: Phase diagram for nickelates (from [10]). CO, PM and AFM stands for charge ordered,
paramagnetic and antiferromagnetic respectively. The tolerance factor t is a measure of the deviation
from a perfect cubic perovskite lattice (corresponding to t = 1).
2.2 Low-energy two-orbital Hamiltonian
The Hamiltonian under study is (cf. [4])
H =
m=1,2 σ=↓,↑

−t
i,j
c†
mσicmσj + h.c. −
Λs
2 i∈A
c†
mσicmσi +
Λs
2 i∈B
c†
mσicmσi − µ
i
c†
mσicmσi

 + Hint.
(58)
Hint =
i
Hint(i) is the Kanamori Hamiltonian, with on each site a Hamiltonian of the form (dropping
the i on the right hand side)
Hint(i) = U
m
ˆnm↑ˆnm↓ + (U − 2J)
m =m
ˆnm↑ˆnm ↓ + (U − 3J)
m<m ,σ
ˆnmσ ˆnm σ
+J
m=m
c†
m↑c†
m↓cm ↓cm ↑ − J
m=m
c†
m↑cm↓c†
m ↓cm ↑. (59)
This is a two-band Hamiltonian, corresponding to the description of electrons in the eg manifold. Note
that alone the kinetic term, which we will denote as Ht, would constitute the tight-binding Hamiltonian.
The on-site interaction term can be interpreted as follows: the first three terms correspond to energy
costs for placing electrons in different configurations. The first term describes the repulsive interaction
of two electrons with opposite spin on the same band, with interaction energy U. In the Hubbard model
this is the only interaction term, but in our model we take into account interactions between electrons
in different bands. Hence there is a Hund’s coupling J and interactions between different bands between
electrons with opposite spin (with energy U − 2J) and parallel spin (with energy U − 3J). This is in
accordance with Hund’s Rules, which favours parallel spin alignments over anti-parallel alignments and
18

placing electrons in different bands over placing them in the same band. The last two terms are called
spin-flip and pair-hopping terms, but as we shall see they play no role at mean-field level. Finally, the
existence of distorted SB and LB octahedra induces an energy difference ∆s between neighbouring sites
of sublattice A (corresponding to LB sites) and sublattice B (corresponding to SB sites). In the fully
disproportionated case, we would expect two electrons per site on the LB sites and none on the SB sites.
2.3 Mean-field Hamiltonian
Under mean-field theory, the Hamiltonian reads (cf. Appendix B for details):
HMF = −t
m,σ, i,j
c†
mσicmσj + h.c.
+
m,σ,i
U ˆnmσ i + (U − 2J) ˆnm σ i + (U − 3J) ˆnm σ i − (−1)i ∆s
2
− µ ˆnmσi (60)
2.4 Order parameter
We are looking for charge density waves, which are periodic fluctuations of the charge density on the
lattice. For simplicity we only look for configurations where nearest neighbours have different charge
density. Hence we divide the lattice into sublattices A and B such that for any pair of neighbouring
sites one site belongs to A and the other to B. We then assume the existence of a long-range order
determined by
ˆnmσi =
n + (−1)i
δ
4
, (61)
such that for i ∈ A (i even) we expect a higher occupancy than for i ∈ B. We assume that because
of band degeneracy and spin symmetry, the average occupancy per site ˆni =
m,σ
ˆnmσi = 4 ˆnmσi is
independent of m and σ. Hence if we take i ∈ A, j ∈ B,
n =
m,σ
ˆnmσi + ˆnmσj
2
=
ˆni + ˆnj
2
(62)
is the average occupation per site, while the order parameter is related to the difference in occupation
of neighbouring sites:
δ =
m,σ
ˆnmσi − ˆnmσj
2
=
ˆni − ˆnj
2
. (63)
We can now write
HU =
m,σ,i
˜U + (−1)i ˜∆ ˆnmσi, (64)
with
˜U = (3U − 5J)n − µ
˜∆ = (3U − 5J)δ −
∆s
2
. (65)
A more physical picture is obtained by defining Ueff
= ˜U and
∆eff
s = ∆s − 2δ(3U − 5J). (66)
19

In terms of these definitions the mean-field Hamiltonian reads
HMF = −t
m,σ, i,j
c†
mσicmσj + h.c. +
m,σ,i
Ueff
− (−1)i ∆eff
s
2
ˆnmσi. (67)
From this expression we see the interpretation of Ueff
as effective repulsive energy and ∆eff
2 as effective
disproportionation energy. However, the following we continue working with ˜U and ˜∆s, for convenience
of notation. We note that in the half-filled case, n = 2 and full disproportionation would correspond to
δ = 2, which amounts to placing all electrons on lattice A and none on B. For the quarter filled case,
n = 1, and in the case of full charge disproportionation would correspond to δ = 1.
2.5 Self-consistency equation
After diagonalizing the Hamiltonian, we find two bands in k ≤ 0 region, with spectra given by
Ek,± = ˜U ± 2
k + ˜∆2. (68)
By performing a Bogoliubov transform as in the anti-ferromagnetic case for the Hubbard Model, we
obtain the self-consistency equation
δ =
4
Ns
k≤0
−(3U − 5J)δ +
∆s
2
[f(Ek,−) − f(Ek,+)]
2
k + ˜∆2
. (69)
Details of the derivation are given in Appendix B.
2.6 Half-filling
In nickelates the Ni3+
configuration is d7
, which corresponds to quarter-filling (n = 1) of the eg bands
(this has also been confirmed by Density Functional Theory calculations, see [4]). However, we start by
studying the half-filled case, for which the chemical potential is known and hence results are more easily
obtained.
Recall that for the Hubbard model at half-filling there is a particle-hole symmetry, from which we
can derive that the chemical potential is µ = U
2
. This follows directly from invariance under the
transformation ciσ → (−1)i
c†
iσ. Similarly, we can deduce the condition for half-filling in our two-band
model by requiring invariance under the transformation
cmσi → (−1)i
c†
mσi. (70)
The kinetic term remains invariant, while
ˆnmσi → 1 − ˆnmσi. (71)
Hence the chemical potential changes sign, and in the interaction term we need encounter terms of the
form (1 − ˆnmσi)(1 − ˆnm σ i) which also give contributions to the chemical potential. Adding up these
terms gives
µ
m,σ,i
ˆnmσi − U
m,i
(ˆnm↑i + ˆnm↓i) − (U − 2J)
m=m ,i
(ˆnm↑i + ˆnm ↓i) − (U − 3J)
m=m ,i
(ˆnmσi + ˆnm σi)
= [µ − 2 (3U − 5J)]
m,σ,i
ˆnm,σ,i.
(72)
20

One can check that the spin-flip and pair-hopping terms are invariant under this transformation. Hence
requiring this term to be equal to the original chemical potential term −µ
m,σ,i
ˆnmσi gives µ = 3U − 5J.
Finally, the terms involving ∆s get a minus sign, but this simply corresponds to a change of A and B
sites.
2.6.1 Limiting cases
In the limit of large ∆s, we expect that all electrons will move to A sites while the B sites will be empty.
Hence we would expect δ = 2. We check that
1
2
k + ˜∆2
≈
1
(∆s
2
)2
=
2
∆s
. (73)
At T = 0, at half-filling all the states in the lower band are filled while those of the upper band are
empty, giving f(Ek,+) = 0 and f(Ek,−) = 1. Therefore, to good approximation we have (since the region
k ≤ 0 has Ns/2 points)
δ =
4
Ns
k≤0
[f(Ek,−) − f(Ek,+)] = 2. (74)
Similarly, in the limit of large J, (U − 3J) becomes negative and therefore it becomes favourable to
place electrons in different bands on the same site. We then also expect full disproportionation. Indeed,
in the limit |J| ∆s we can neglect the term with ∆s
2
, giving
1
2
k + ˜∆2
≈
1
((5J − 3U)δ)2
= −
1
(3U − 5J)δ
. (75)
From this we obtain the same equation as (74).
Finally, for ∆s = 0 we find a solution δ = 0 as well as the equation
1 =
1
Ns
k
(5J − 3U)
2
k + ˜∆2
[f(Ek,−) − f(Ek,+)] . (76)
This is very similar to the BCS gap equation we encountered before when studying anti-ferromagnetism
in the Hubbard model. However, in general Ek,− = −Ek,+ and therefore we might expect slightly
different results.
2.6.2 Numerical results
We study the self-consistency equation (69) numerically. This is done by performing the sum over a grid
representing the Reduced Brillouin Zone [−π
2
, π
2
]3
(with 27000 points) and a grid of δ (with 100 points).
We find the zeros by fitting a 10th order polynomial function to the data and looking for its roots. The
quality of the fit and the results below suggest that this accuracy is sufficient for our purposes.
We first show δ as a function of the coupling 3U − 5J for different values of ∆s in Figure (9a). Note
that indeed δ → 2 as J becomes large. Furthermore, a higher energy difference ∆s induces a higher
disproportionation for all values of the coupling as expected. It should be remarked that for ∆s = 0,
δ tends to 0 for large values of 3U − 5J but does not reach zero in the limited regime we studied. By
contrast, for ∆s = 0.0 a sharp onset of charge order can be identified, close to 3U − 5J = −4.0. Note
that this is different than expected from the analogy with the BCS equation, for which we expect charge
order for any 3U − 5J < 0. It remains to be investigated whether this is due to numerical inaccuracy.
21

One might expect this to be a finite temperature effect, but Figures 8(a), 8(b) and 8(c) suggest that
at low temperatures such effects are small. We see that the onset indeed shifts to higher values of the
coupling as temperature is decreased, but from 8(c) we see that for β ≈ 1.0 onwards lowering the tem-
perature further hardly affects the results. Hence we expect our results at β = 5.0 to be representative
for the low temperature, T → 0 limit.
Finally, we check that δ → 2 as ∆s becomes large (Figure 8(d)), in line with our analytic predictions.
We construct a low temperature phase diagram from the data of Figure (9a). The results are shown
in Figure (10): we identify in the (3U − 5J, ∆s) plane different regions with varying degree of charge
ordering (as determined by the size of δ). Qualitatively the results agree to expectations: higher values
of ∆s favour charge ordering and stronger Hund’s coupling J also increases the degree of charge ordering.
(a) Temperature dependence of δ(3U−5J) at ∆s =
0.0. The onset of order occurs at lower J when
temperature is decreased.
(b) Temperature dependence of δ(3U − 5J) at
∆s = 1.0. Above β = 1.0 temperature effects are
small.
(c) δ(β) quickly saturates above β = 1.
(d) δ(∆s) → 2.0 as ∆s becomes large, with
(U, J, β) fixed.
Figure 8: Various numerical results showing the dependence of δ on the parameters of the problem.
2.7 Quarter-filling
At quarter-filling, the chemical potential cannot be easily determined and one has to resort to numerical
methods to find an approximate value. This can be done by tuning the chemical potential in such a way
22

that n = 1, where the filling n can be determined from the sum
n =
4
Ns
k≤0
−(3U − 5J)δ +
∆s
2
[f(Ek,−) + f(Ek,+)]
2
k + ˜∆2
. (77)
(a) The gap δ as a function of the only coupling
in the problem (3U − 5J), for different values of
∆s, at β = 5.0. The ∆s = 0.0 case is plotted at
β = 1.0, but it can be checked that in this case
results hardly change when β = 5.0. The sudden
onset of charge order occurs near 3U −5J = −4.0.
(b) Quarter-filling: the gap δ as a function of the
only coupling in the problem (3U −5J), for differ-
ent values of ∆s, at β = 5.0.
(a) Half-filling (b) Quarter-filling
Figure 10: Low temperature phase diagram for the half-filled and quarter-filled cases (β = 5). Darker
colours correspond to higher values of δ.
23

2.8 Comparison of Quarter-filling results and half-filling results
(a) ∆s = 0.0 (b) ∆s = 1.0
(c) ∆s = 2.0 (d) ∆s = 3.0
Figure 11: Quarter-filling versus half-filling
24

Conclusion
In this internship, we studied charge density waves in rare earth nickelates. This was done by a mean-
field approach for a model Hamiltonian, in which we restrict to charge density order. The presence of
such order is associated with the insulating phase of nickelates and it would be interesting to compare
results to the known phase diagrams from experimental results. This would allow us to identify the role
of the coupling constants U and J and how they are related to measurable physical quantities. The
mean-field approach can also be extended to include spin density order, which is known to be present
in nickelates [10]. However, one should keep in mind that the validity of mean-field theory is typically
restricted to small values of the coupling constants.
It has been suggested that the model provides an accurate description of nickelates when the coupling
(3U − J) ≤ ∆s becomes small or negative ([4]). At mean-field level, the results that can be obtained
are limited. To gain a full understanding of the system would typically require Dynamical Mean-Field
Theory (DMFT) calculations, which would be the logical extension to this internship.
We have also applied various numerical methods to condensed matter systems. The tools sometimes
boil down to simple matrix diagonalisation (nearly free electrons) and numerical integration (DoS for
the tight-binding model, nickelates self-consistency equation and obtaining Green’s functions from the
spectral function). We have seen that choices made in the implementation can have dramatic results,
such as in the case of introducing noise in the Green’s function computation and varying grid size.
However, when the right choices are made, excellent results can be obtained which sometimes can be
shown to be in excellent agreement with analytic results (1d DoS of the tight-binding model).
25

Appendix A: Green’s Functions
In this appendix we give an overview of properties of Green’s functions, both in real-time and imaginary
time. For details, the reader can refer to [3], especially chapters 9 and 11. However, not all properties
we present are to be found in this reference and therefore a complete presentation of the subject is given
here.
Real-time Green’s functions
In the most general sense, Green’s functions refer to solutions to ordinary and partial differential equa-
tions. For instance, in classical electromagnetism the Poisson equation is solved by introducing a Green’s
function for the 2
. In many-body quantum physics, the Green’s functions are also solutions to dif-
ferential equations in the equation of motion theory. However, this link is not obvious and generally
one does not need to use this property to define them. Instead, let us give a fully general definition of
Green’s functions which can be used to extract physical information about a system.
In the context of condensed matter systems, Green’s functions are correlation functions of observables.
For instance, if A and B are time-dependent observables for a fermionic system we can define
CAB(t − t ) = −iθ(t − t ) {A(t), B(t )} . (78)
For the bosonic case the anti-commutators have to be replaced by commutators, but we shall consider
only fermionic systems below.
In particular, we are interested in studying Green’s functions for single particles. The most commonly
used is the retarded Green’s function
GR
(ν, t; ν , t ) = −iθ(t − t ) {aν(t), a†
ν (t )} , (79)
where aν and a†
ν are creation and annihilation operators in a basis {|ν }. Here retarded refers to the
fact that causality is implied by the time ordering t > t . We also define the advanced Green’s function
GA
(ν, t; ν , t ) = iθ(t − t) {aν(t), a†
ν (t )} . (80)
In position space the retarded Green’s function takes the form
GR
(r, t; r , t ) = −iθ(t − t ) {Ψ(r, t), Ψ†
(r , t )} . (81)
In momentum space we have
GR
(k, t; k , t ) = −iθ(t − t ) {ak(t), a†
k (t )} . (82)
The correlators can be evaluated by expanding in an eigenbasis of the Hamiltonian |n . This gives
the so-called Lehmann representation. For free particles, for which the Hamiltonian is diagonal, the
frequency space Green’s function can be expressed as
GR
(ν, ω) =
1
Z n,n
| n|cν|n |2
ω + En − En + iη
(e−βEn
+ eβEn ), (83)
where η = 0+
is a positive infinitesimal. From this representation we define the spectral function
A(ν, ω) = −
1
π
Im G(ν, ω). (84)
The spectral function can be regarded as a generalization of the DoS in free systems. It is normalized
to 1 with our convention and the average occupation in state ν can be expressed as
nν = cνc†
ν = dωA(ν, ω)nF (ω). (85)
The spectral function can be directly measured using techniques such as ARPES (angular-resolved
photoemission spectoscopy).
26

Free fermions
Consider a system of free fermions described by the Hamiltonian
H =
k
kc†
kck. (86)
The time-evolution of the annihilation operator is given by
ck(t) = cke−i kt
, (87)
and that for the creation operator is given by its hermitian conjugate. Hence the retarded Green’s
function is given by
GR
(k, t − t ) = −iθ(t − t ) {ck, c†
k} e−i k(t−t )
= −iθ(t − t )e−i k(t−t )
(88)
In frequency space we obtain
GR
(k, ω) =
1
ω − k + iη
. (89)
The spectral function is therefore
A(k, ω) = δ(ω − k). (90)
The spectral function is sharply peaked at the energy values that can be taken by the free particle system.
GR
(k, t − t ) describes the evolution of an excitation of momentum k in the system. Since this is an
oscillatory function which does not decay in time, we say that the lifetime of the excitation is inﬁnite.
This is of course due to the absence of interactions. If we add interactions to the system, the lifetime
becomes ﬁnite and the spectral function broadens. In more general cases, one of which will be discussed
below, the term iη in the denominator of GR
(k, ω) is replaced by a term −Σ(k, ω). Let us assume for
simplicity that it does not depend on k, ω. Writing Σ = Σ − iΣ , we can deduce that
GR
(k, t) = −iθ(t)e−i( k+Σ )t
e−Σ t
. (91)
Hence we see that the imaginary part of Σ gives the inverse lifetime of an excitation. Furthermore, the
spectral function becomes a Lorentzian of the form
A(k, ω) =
Σ
(ω − k − Σ )2 + Σ 2
. (92)
Single fermion level coupled to a bath
We now study a single fermion level coupled to a bath. The Hamiltonian of the full system is H =
H0 + Hbath + Hhyb, where
H0 = 0c†
0c0 (93)
describes the level,
Hbath =
ν=0
νc†
νcν (94)
describes the bath and
Hhyb =
ν=0
Vν(c†
νc0 + c†
0cν) (95)
27

is the hybridisation term. We are interested in the retarded Green’s function of the single fermion
GR
00(ω) = dteiωt
GR
00(t) where GR
00(t) = −iθ(t) {c0(t), c†
0} . In order to compute this function, let us
first consider a fully generic quadratic Hamiltonian
H =
νν
hνν c†
νcν . (96)
If H is real, then the matrix ˆh is hermitian and therefore there exists a unitary transformation P such
that PˆhP−1
= ˆ where ˆ is a diagonal matrix. We define new creation and annihilation operators
dµ =
ν
Pµνcν, d†
µ =
ν
P∗
µνcν (97)
and it can be checked that in the new basis the Hamiltonian takes the form
µ
µd†
µdµ. (98)
The Green’s functions of the c operators and the d operators are related by a similar transformation
Gd
µ(ω) =
ν,ν
PµνGc
νν (ω)P−1
ν µ. (99)
Since Gd
µ(ω) = 1
ω− µ+iη
, if we invert the previous relation we find an expression for Gc
ν,ν . In matrix
notation (where we introduce ˆgµν = Gc
µν) this reads
ˆg = (ω + iη) − ˆh
−1
. (100)
In order to find the inverse, we use the matrix identity
A B
C D
=
A − BD−1
C BD−1
0 Î
Î 0
C D
. (101)
Inverting this equation gives
A B
C D
−1
=
I 0
−CD−1
D−1
(A − BD−1
C)−1
(BD−1
C − A)−1
BD−1
0 Î
. (102)
Applied to our Hamiltonian we first remark that ˆh takes the form
ˆh =






0 . . . Vν . . .
...
Vν
...
...
... ...
. . . ν . . .
... ...
...






(103)
Note that the bottom right submatrix can principle be infinite dimensional, containing all eigenvalues
of the bath Hamiltonian. Using the above expressions we find
GR
00(ω) = (A − BD−1
C)−1
=



(ω + iη − 0) − . . . Vν . . .




...
... ...
. . . (ω + iη − ν)−1
. . .
... ...
...








...
Vν
...








−1
=
1
ω + iη − 0 − ∆(ω)
(104)
28

where
∆(ω) =
ν
V 2
ν
ω + iη − ν
. (105)
We can make the approximation ∆(ω) = −i∆0, where ∆0 is a constant, at low temperatures and if the
DoS of the bath is not too singular. In this case we find for the spectral function
A(ω) = 2π
2∆0
(ω − 0)2 + ∆2
0
. (106)
By transforming back to the time variable, we find for the retarded Green’s function
GR
00(t) = −iθ(t)eiωt
e−∆0t
. (107)
Hence we see that the lifetime ∆0 is related to the width of the spectral function A(ω).
Imaginary time Green’s functions
We have already discussed the imaginary time Green’s functions in the main text. In this section we
will derive the results we have stated in the main text. Recall that the imaginary time Green’s function
is defined as
G(τ − τ ) = − Tτ c(τ)c†
(τ ) , (108)
where more explicitly, for fermions we have
Tτ c(τ)c†
(τ ) = θ(τ − τ )c(τ)c†
(τ ) − θ(τ − τ)c†
(τ )c(τ). (109)
The time-ordering operation is not defined for equal times, but we can still take the limit τ → 0+
to get
G(0+
) = −(1 − c(0+
)c†
(0) ) = n − 1, (110)
where n = c(0)c†
(0) (assume the change at time 0±
is negligible). Similarly, we have
G(0−
) = −G(β−
) = c†
(0−
)c(0) ) = n (111)
for β−
= β + 0−
. This leads to our previous relation for the Green’s function at the borders of the
interval on which it is defined.
G(0+
) + G(β−
) = G(0+
) − G(0−1
) = −1. (112)
More precisely, the Fourier transform is defined as
G(iωn) =
β
0
dτG(τ)eiωnτ
, (113)
where ωn = (2n+1)π
β
. Conversely, the inverse Fourier transform is obtained from
G(τ) =
1
β n
G(iωn)e−iωnτ
. (114)
The relation between the real-time retarded Green’s function and the Matsubara Green’s function in
frequency space can be best seen from the Lehmann representation (cf. Bruus and Flensberg Ch. 11).
In this representation it is immediately clear that they are related by the transformation
GR
(ω) = G(iωn → ω + i0+
), GA
(ω) = G(iωn → ω + i0−
). (115)
29

We also note the identity
G(ω + i0+
) − G(ω + i0−
) = GR
(ω) − GA
(ω) = −2iπA(ω). (116)
For the last identity we have used GA
(ω) = [GR
(ω)]∗
, which can be shown using the formula we will
now give.
Explicitly, the Hilbert transform that relates G(z) to A(ω) is
G(z) =
∞
−∞
A(ω )
z − ω
dω . (117)
We now have
G(ω ± i0+
) =
∞
−∞
A(ω)
ω − ω ± i0+
, (118)
from which we see that indeed GA
(ω) = [GR
(ω)]∗
.
Consider now the case where A(ω) is even. We have
G(iωn) =
∞
−∞
A(ω )
iωn − ω
= −
∞
−∞
A(ω )ω
ω2
n + (ω )2
− i
∞
−∞
A(ω )ωn
ω2
n + (ω )2
. (119)
The real part of G(iωn) is zero, while the imaginary part is odd in ωn. Furthermore, the Lorentzian
ωn
ω2
n+(ω )2 in the imaginary part goes to πδ(ω ) in the limit ωn → 0. Hence we see that
lim
iωn→0
Im G(iωn) = −πA(ω = 0). (120)
The fact that G(τ) is symmetric on [0, τ] follows from
G(β − τ) =
1
β n
eiωn(β−τ)
G(iωn)
=
1
β n
−e−iωnτ
G(iωn)
=
1
β n
− sin (ωnτ) Im G(iωn)
=
1
β n
eiωnτ
G(iωn)
= G(τ). (121)
Finally, let us elaborate on how we obtained the kernel expression which relates G(τ) to A(ω). We
30

start by noting the following identity (for 0 < τ < β):
G(τ) =
1
β n
G(iωn)e−iωnτ
=
1
β n
∞
−∞
dω
A(ω )
iωn − ω
e−iωnτ
=
1
β
∞
−∞
dω
n
e−iωnτ
iωn − ω
A(ω )
=
∞
−∞
dω K(τ, ω )A(ω ), (122)
with
K(τ, ω) =
1
β n
e−iωnτ
iωn − ω
. (123)
This is precisely the imaginary time Green’s function for a free particle at energy ω, since G(iωn) = 1
iωn−ω
for this system. However, from the deﬁnition of the Green’s function we also know (still assuming τ > 0)
that for free particles with energy E we have
G(τ) = − Tτ c(τ)c†
(0)
= −(1 − nF (ω))e−Eτ
= −
e−τE
1 + e−βE
(124)
Therefore we indeed have, as stated before,
K(τ, ω) = −
e−τω
1 + e−βω
. (125)
Finally, equation (52) follows from
K(τ, ω)|τ=β/2 = −
1
2 cosh(βω
2
)
. (126)
For large β, this function decreases quickly with ω. Using the property
∞
−∞
1
cosh( βω
2
)
= 2π
β
, we obtain the
approximate formula
G(β/2) ≈ −
π
β
A(ω = 0). (127)
31

Appendix B: Derivation of the self-consistency equation for the
two-band model
In this appendix we give a derivation of the self-consistency equation (69). Let us recall the full Hamil-
tonian
H =
m=1,2 σ=↓,↑

−t
i,j
c†
mσicmσj + h.c. −
Λs
2 i∈A
c†
mσicmσi +
Λs
2 i∈B
c†
mσicmσi − µ
i
c†
mσicmσi

 + Hint,
Hint(i) = U
m
ˆnm↑ˆnm↓ + (U − 2J)
m =m
ˆnm↑ˆnm ↓ + (U − 3J)
m<m ,σ
ˆnmσ ˆnm σ
+ J
m=m
c†
m↑c†
m↓cm ↓cm ↑ − J
m=m
c†
m↑cm↓c†
m ↓cm ↑. (128)
Mean-field theory
Under mean-field theory, the first term expands as
ˆnm↑ˆnm↓ →ˆnm↑ ˆnm↓ + ˆnm↑ ˆnm↓ + ˆnm↑ ˆnm↓
−c†
m↑cm↓ c†
m↓cm↑ − c†
m↑cm↓ c†
m↓cm↑ − c†
m↑cm↓ c†
m↓cm↑ (129)
The next two terms have similar expansions. The terms involving ˆnmσ add up to an interaction term
which can be simplified to
HU =
m,σ,i
U ˆnmσ i + (U − 2J) ˆnm σi + (U − 3J) ˆnm σ i − (−1)i ∆s
2
− µ ˆnmσi, (130)
where (−1)i
=
1 if i ∈ A
−1 if i ∈ B
.
The remaining contributions from the first three terms add up to
H1 = −U
m,σ=σ
c†
mσcmσ c†
mσ cmσ − (U − 2J)
m=m ,σ=σ
c†
mσcm σ c†
m σ cmσ
−(U − 3J)
m=m ,σ
c†
mσcm σ c†
m σcmσ . (131)
A similar calculation for the last two terms of the interaction Hamiltonian leads to
H2 = J
m=m σ=σ
c†
mσ cm σ + c†
m σ cmσ c†
mσcm σ − c†
mσ cmσ c†
mσcm σ − c†
mσcmσ c†
m σ cm σ. (132)
Adding up and simplifying the results, we finally arrive at the mean-field Hamiltonian
HMF = Ht + HU + H1 + H2
= Ht + HU − U
i,m σ=σ
c†
mσicmσ i c†
mσ icmσi
+
i,m=m ,σ=σ
J c†
mσ icm σ i + J c†
m σ icmσ i − (U − 3J) c†
m σicmσi c†
mσicm σi
+
i,m=m ,σ=σ
−J c†
mσicmσ i c†
m σ icm σi − (U − J) c†
mσicm σ i c†
m σ icmσi (133)
32

This is the full mean-field Hamiltonian, where we have made no assumptions about the averages. To
simplify things, we first note that the two bands are degenerate in energy. This is manifest through the
symmetry m ↔ m of the original Hamiltonian. As a result, we expect terms such as c†
mσcm σ and
c†
mσcm σ to be zero, since they are related to the hopping between bands. This leaves us with
H1 → −U
m σ=σ
c†
mσ cmσ c†
mσcm σ
H2 → −J
m=m σ=σ
c†
mσ cmσ c†
m σcmσ + c†
mσcmσ c†
m σ cm σ (134)
We also assume the absence of spin order, since the Hamiltonian symmetric under exchange of up and
down spin. Hence we take c†
mσcmσ = 0. This leads to H1 = H2 = 0. Hence our final mean-field
Hamiltonian takes the form
HMF = −t
m,σ, i,j
c†
mσicmσj + h.c.
+
m,σ,i
U ˆnmσ i + (U − 2J) ˆnm σ i + (U − 3J) ˆnm σ i − (−1)i ∆s
2
− µ ˆnmσi (135)
Diagonalisation
We now go to momentum space to derive the self-consistency equation for δ. We suppress the indices
m, σ and corresponding sums for the moment and focus on the position and momentum indices. Recall
that for a cubic lattice in d dimensions with lattice spacing a
Ht =
k
kc†
kck, (136)
with k = −2t
d
i=1
cos kia. Let us write
(−1)i
= eiq·Ri
, q = (
π
a
, . . . ,
π
a
). (137)
where Ri is the position vector of site i. Using these properties we can rewrite the mean-field Hamiltonian
as
HMF =
k∈FBZ
k + ˜U c†
kck + ˜∆c†
kck+q , (138)
where the sum is over the First Brillouin Zone (FBZ). Because k = − k+q, we can rewrite this as a
sum over the region where k ≤ 0,
HMF =
k≤0
c†
k c†
k+q
k + ˜U ˜∆
˜∆ − k + ˜U
ck
ck+q
. (139)
The matrix is diagonalized in the same way as in the matrix we found for anti-ferromagnetism in the
Hubbard Model. Hence we introduce operators
ak
bk
=
cos θ − sin θ
sin θ cos θ
ck
ck+q
. (140)
33

The diagonal Hamiltonian then takes the form
HMF =
k≤0
Ek,−a†
kak + Ek,+b†
kbk, (141)
where the spectrum is now given by
Ek,± = ˜U ± 2
k + ˜∆2. (142)
Self-consistency equation
We give the full derivation of the self-consistency equation (69).
Let Ns be the total number of lattice sites (in real space). The self-consistency equation for δ reads
δ =
1
2
ˆni∈A − ˆnj∈B
=
1
Ns
i∈A
ˆni −
j∈B
ˆnj
=
1
Ns i
ˆni eiq·Ri
=
4
Ns i
ˆnimσ eiq·Ri
=
4
Ns i
1
Ns
k,k’∈FBZ
c†
kck’ ei(k −k+q)·Ri
=
4
Ns
k∈FBZ
c†
kck-q
=
4
Ns
k∈FBZ
c†
kck+q
=
4
Ns
k≤0
c†
kck+q + c†
k+qck
= −
4
Ns
k≤0
2 a†
kak sin θ cos θ − b†
kbk sin θ cos θ
= −
4
Ns
k≤0
sin 2θ [f(Ek,−) − f(Ek,+)]
=
4
Ns
k≤0
− ˜∆
2
k + ˜∆2
[f(Ek,−) − f(Ek,+)] . (143)
Here
i
denotes the sum over all lattice sites. We have suppressed m, σ when switching to k-space.
Substituting for ˜∆ in the numerator, we obtain the self-consistency equation (69). Given that the
equation is invariant under k → − k we may also write this as
δ =
8
Ns
k∈FBZ
−(3U − 5J)δ +
∆s
2
[f(Ek,−) − f(Ek,+)]
2
k + ˜∆2
. (144)
34

References
[1] Henri Alloul, Introduction to the Physics of Electrons in Solids (Graduate Texts in Physics).
Springer, 2011 edition. Translated by Stephen Lyle.
[2] Neil W. Ashcroft and N. David Mermin, Solid State Physics, Cengage Learning, 1976,
[3] Henrik Bruus and Karsten Flensberg, Many-body Quantum Theory in Condensed Matter Physics,
Oxford University Press, 2004.
[4] Alaska Subedi, Oleg E. Peil and Antoine Georges, A low energy description of the metal-insulator
transition in the rare-earth nickelates, Phys. Rev. B 91, 075128 (2014).
[5] Antoine Georges, Luca de Medici, and Jernej Mravlje, Strong Correlations from Hunds Coupling,
Annu. Rev. Condens.Matter Phys. 4, 137 (2013).
[6] M. L. Medarde, Structural, magnetic and electronic properties of RNiO3 perovskites (R = rare
earth), J. Phys.: Condese. Matter 9, 1679 (1997).
[7] R. Jaramillo, S. D. Ha, D. M. Silevitch and Shiram Ramanathan, Origins of bad metal conductivity
and the insulator-metal transition in the rare-earth nickelates, Nature Physics 10, 304 (2014).
[8] I. I. Mazin, D. I. Khomskii, R. Lengsdorf, J. A. Alonso, W. G. Marshall, R. M. Ibberson, A.
Podlesnyak, M. J. Martnez-Lope, and M. M. Abd-Elmeguid, Charge Ordering as Alternative to
Jahn-Teller Distortion, Phys. Rev. Lett. 98, 176406 (2007).
[9] A. Thomas et al., Elementary Introduction to the Hubbard Model, QUEST documentation, retrieved
April 2015 from: http://quest.ucdavis.edu/tutorial/hubbard7.pdf.
[10] R. Scherwitzel. Metal-insulator transitions in nickelate heterostructures. Th`ese de doctorat: Univ.
Gen`eve, 2012, no. Sc. 4427.
35

M2 Internship report rare-earth nickelates

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Viewers also liked

Viewers also liked (20)

Similar to M2 Internship report rare-earth nickelates

Similar to M2 Internship report rare-earth nickelates (20)

Recently uploaded

Recently uploaded (20)

M2 Internship report rare-earth nickelates