Suppressed HF Behaviour in the Periodic Anderson Model
European Scientific Journal April 2018 edition Vol.14
Suppressed HF Behaviour in the Periodic Anderson Model
Okunzuwa I. S. Arthur I. I. Ejere 0
0 Department of Physics, Faculty of Physical Sciences, University of Benin , Benin city , Nigeria
The periodic Anderson model is applied to 4 electrons on 4 sites with periodic boundary conditions. We applied magnetic field to the localized forbitals, Eσf. The number of electrons is taken to be one per site and the interactions between different sites is restricted to nearest neighbors. The many body eigenvalues are calculated exactly using exact diagonalization technique. We find that the specific heat is suppressed by the variation of the band energy of the localized f-orbitals as mediated by the application of the magnetic field, H, under various hybridization energy. A continuous suppression of the specific heat reduces the heavy fermion behavior in the system.
Average energy; Ferromagnetism; Ground State energy; partition function; specific heat
Many rare–earth and actinide compounds have been known over the
decade to exhibit mixed valence phenomena [Batista et al, 2003]. The valence
fluctuation in such systems are highly correlated, they are also found to exhibit
Fermi-liquid phenomena with a heavy mass at low temperature. There is also
increasing interest in these heavy fermion materials. The common feature of
these heavy fermion systems is that at low temperatures they exhibit large
values of specific heat coefficient [Batista et al, 2003].
There has been extensive theoretical work on mixed valence (MV) and
Kondo systems [Batista et al, 2003, Chabe et al, 2008]. However, the Kondo
problem and the MV regime are well understood. The Anderson Model has
been used to extensively study the thermodynamic, transport and excitation
properties of these systems using some powerful techniques [Herin et al,
However, the lattice problem in these regimes is neither well
understood nor solved. There is need to also explain the wide range of physical
A Fermi liquid theory which discards quasi particle interactions has
been developed [Jarrell, 1995]. It is found that the large effective mass of HF
systems does not arise primarily due to band renormalization and hence
contradict many theories [Chabe et al, 2008]. From the foregoing remarks, it
is obvious that there remain a need for greater understanding of the MV and
by extension the HF systems.
Though, sometimes small clusters may not represent bulk materials.
However, the large number of states obtained even in small clusters
statistically may give results which fairly represent a large system over a
reasonable temperature range [Sordi et al, 2007]. In the present work, the
periodic Anderson Model is applied to a chain of four electrons on four sites
with period boundary conditions. There would be a systematic study of the
effect of varying energy of the localized f-orbital mediated by the application
of magnetic field to the localized orbital only. One extended orbital would be
considered per site per spin with an inter-atomic transfer integral t. There is
also one localized f-orbital per site per spin having energy Ef with coulombic
repulsion u between two electrons in the f orbitals on the same site. An
hybridization term v, between the localized and extended orbitals is also
present in the model. One electron per site would be allowed, and it is
restricted to nearest neighbours. An exact diagonalization technique would be
applied to generate the matrix. The matrix would be diagonalized using the
mathematical 9 wolfram program.
The ground state is a singlet (S = 0) while the other states are excited
(S > 0) and are nearly degenerated with the ground state. The specific heat
increases sharply at low temperatures which is comparable to the heavy
fermion behaviour. The systematic variation of the energy of the f-orbital and
its corresponding effect on the specific heat of the heavy fermion system
would be carried out.
We use the Anderson Lattice model (AL) [Yu et al, 2008a] which is a
translationally invariant generalization of the Anderson impurity model which
has been successfully used to explain the Kondo effect.
The Anderson Lattice (AL) Hamiltonian in second quantization is
= ∑ + + ∑ + + ⁄ ∑ ( + + H.C) + ∑ .
interaction between the conducting band electrons and the localized
felectrons. The onsite coulomb repulsion is U and Eσf is the eigenvalue for a
single f-electron. The local spin does not flip in the Anderson Model. Because
of the hybridization term, a local spin that is down can become a conduction
state with spin down, and then wander away [Sordi et al, 2007]. Later, a
different conduction electron with spin up can come and reside in the local
orbital.[ Hywel ,2008] This process gives the appearance of the local spin
having flipped from up to down.
Furthermore, Eσf = Ef – σh, h = μBH and H denotes the magnetic field.
The ground state energy is obtained using the exact diagonalization
method and the model (i.e. eqn. 1) is studied under cyclic boundary conditions.
This case has the advantage of eliminating edge effect. We use the Wolfram
Mathematica 9 Software [Wolfram Research, Inc (2007] which is effective in
diagonalizing large matrix sizes. With these tools, we were able to study
systems of 4 electrons on 4 sites for which the Hamiltonian matrix size is 70
x 70. Computing the ground state energy, using
E=⟨ | | ⟩ (
⟨ | ⟩
Where we have E = 1[u+√16 2 +
+ 2ϴ ]
Calculating the properties of the system of particular interest is the
partition function Z of the electrons. When a grand-canonical ensemble is
considered we can deduce a number of static thermodynamic properties. The
partition function is the trace of an operator Z [Zhu. and Zhu 2001].
Z=Tr 〈 |exp(− )| 〉 (
Where = 1/
and H is Hamiltonian matrix.
Hence, the partition function of the system under consideration is given by
Z=2exp(-ϐ∝ )+4exp(-ϐϴ) (
We define the expectation value < H > = E as the operator for H. That is
E=〈 〉 = 1 〈 | exp(− )| 〉 (
) describes the thermal average of the quantity H. This can be used
to characterize the electron correlation [Galpin et al, 2008]].
Therefore E=〈 〉=( + )exp(− ( + ))+2 exp(− ) (
exp(− ( + ))+2exp(− )
The temperature dependence of the thermodynamic, quantity (specific
heat) is calculated from the many body eigenvalue (the ground state energy)
by using the standard statistical mechanics procedure for a canonical
ensemble. The expression for specific heat is given by equation 10. Our result
is expressed in units of J/mol-k, where it is understood that ‘mole’ means N
ϒ=[∝exp(-∝ϐ)+2ϴexp(-ϴϐ)][∝exp(-∝ϐ) + 2 exp(-ϴϐ)]
=exp(-2∝ϐ) + 4exp(-2ϴϐ) + 4exp(-∝ +ϴ)ϐ
(2 2/ )exp(-ϴϐ)]
Result and discussion
We discuss the temperature dependence of the specific heat for a
solution of values of Eσf and v/t/ chosen from a large set of available results to
represent interesting cases. We found that the specific heat curves are
qualitatively similarly for different values of Eσf. . Figures 1 and 2 are plotted
on a common linear scale of temperature T/t/, hybridization v, respectively. If
t is set to 1ev, then the scale is in Kelvin.
Firstly, we consider the symmetric case, Ef = -u/2. We fix t = 1, u =
1.0, H = 0.01 and v vary from 0.1 to 1.0 (all energies are in units of t). When
the hybridization is relatively large (v = 0.1 to 1.0) no peak structure is
obtained in the temperature range of interest to us. This indicate the absence
of the Schottky type behavior as there is no peak observed for Eσf = -5.0 and v
increases from 0.1 to 1.0. This indicates the absence of degeneracy .
Fig 1: Cv vs v for various εσf for t=1, H=0.01, u=1
From the ranges of values obtained from Eσf ranging from -0.5 to -0.4
shows that the Specific Heat (SH) resembles that of a Kondo system. Under
very low temperature (T = 1k) the SH shows a near linear increase as the
hybridization increases which is indicated by Cv (T1) in Fig. 1. It is also
observed that this trend changes slightly as temperature is gradually increased
until a broad peak is reached at T = 5k as indicated by Cv(T5) . This behavior
can be explained in the following way: when the hybridization v, gradually
increases for Eσf = -0.5, the ‘f’ level are well defined and essentially full (nf ~
1) at low temperatures. Excitations occur giving rise to a low temperature
linear behavior of SH.
Secondly, our results indicate that the SH increases very rapidly at low
temperatures. The unusual rise mimics the onset of the HF behavior. In order
to study this in more details, we present a typical example of a tetrahedron for
t = 1, u = 1, H = 1.0, v = 0.1 and plot in Fig. 2, Cv/T against T for Eσf ranging
from -0.5 to -0.4.
We notice that the rate of increase of Cv/T (at very low temperatures)
is maximum for Eσf = -0.4. Again, for each Eσf, the ground state increases.
The ground state is a singlet. The next higher energy states are well separated
from this ground state. The low–temperature rise in SH is determined by the
separation of these levels. As Eσf increases, the separation between the ground
state and other excited states increases, and the rise in C/T correspondingly
increases. Thus, we notice that the HF behaviour is obtained when the many
body ground state is a singlet. The electrical resistance of a pure metal drops
as temperature is lowered. However, the resistance saturates as the temperature
is lowered below about 10k. This behaviour changes dramatically when
magnetic atoms such as Cobalt, are added. Rather than saturating, the
electrical resistance increases as the temperature is lowered further. The
socalled Kondo temperature-roughly speaking the temperature at which the
resistance starts to increase again –completely determines the low temperature
electronic properties of the material [Kubo, 2015]. This HF behaviour,
however is affected by the application of magnetic field (H). The resistivity
of the PAM with magnetic field applied to the localized orbitals is obtained
from eqn. 14 below.
+ ] -2/kT [2 (
) + 2
√ /(6 √ + 6 + √ + √ (15)
The graph of ρ plotted for various for H=1.0, u=1.0, v=1.0 is given
in fig. 3 below.
From fig. 3 and with H only acting on the localized orbital results in a
downturn of the resistivity, ρ at temperatures below 5k. This downturn
indicates a suppressed HF behaviour.
Summary and conclusion
We have calculated the thermodynamic property (specific heat) for a
range of parameters U, V and Eσf and the resistivity for various values of
localized band energies by using the Periodic Anderson Model. Our most
important conclusion is that calculation are capable of reproducing the types
of SH behavior observed in Heavy Fermions and Kondo systems. An
important parameter in determining the SH property it was found that as Eof
is increased, there was a broad peak observed at low temperature T = 1k. A
striking feature of our results is the occurrence of HF behavior. We find that
when Eσf is large and negative for very low values of v, the many body ground
state is a singlet. The other excited states have much higher energy. The
specific heat rises sharply at low temperatures and the system exhibits HF
We conclude that at low temperatures, the SH deeps as v reduces. With
the application of magnetic field of H =1.0, the energy of the localized band
Eσf is varied. This affects the SH and the resistivity as they are suppressed by
the activities of the ‘f’ electrons. It is therefore concluded, that as Eσf increases
further, the system would leave the HF regime.
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