Phase Boundaries of the Pseudogap Anderson Impurity Model
Aaron S. Mohammed, 1* Tathagata Chowdhury, 2 Kevin Ingersent 2
1
Department of Applied Physics, University of South Florida, 4202 East Fowler Avenue, Tampa, FL
33620, USA
2
Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611, USA
August 2014
As the temperature of metals that contain low concentrations of magnetic impurities reach low
temperatures, a phenomenon known as the Kondo effect takes place in which tunnel-coupling between
the electrons of the magnetic impurity and the electrons of the conduction band dominate, causing there to
be an increase in resistance. Although impurity models have explained this effect since the 1960’s,
nanotechnology such as quantum dots and scanning tunneling microscopes have rekindled curiosity since
they allow us to study this effect within controlled settings. In this paper, we focus on testing the accuracy
of scaling approximations that predict the shape of the phase boundaries of the pseudogap Anderson
impurity model.
I. INTRODUCTION
In a many-body system, quantum phase
transitions (QPTs) take place when the ground
state changes at the absolute zero of temperature.
Whereas classical phase transitions are due to
changes in temperature, QPTs occur due to
changes in a non-thermal parameter such as
pressure or chemical composition. This paper
focuses on QPTs in models describing impurities
in metals. During the 1960s, the Anderson and
Kondo models were developed to explain a unique
phenomenon called the Kondo effect. Lately, this
effect has been of great interest due to the
possibility of studying it in controlled settings
created by modern nanotechnologies, e.g.,
quantum
dots
and
scanning
tunneling
microscopes.1 Because the Kondo effect involves
strong interactions between electrons, gaining a
better understanding of it may help in
understanding other strongly interacting systems
such as heavy-fermion materials and high
temperature superconductors.1
As the temperature of a pure metal decreases,
its resistance also decreases. Depending on the
metal, one of two different effects can occur once
the temperature becomes very low: either the
1
resistance obtains a constant minimum value or it
completely disappears, changing the metal’s phase
Figure 1: A plot of resistance vs. temperature is shown
for a metal that contains a certain concentration of
magnetic impurities. Tk represents the metal’s Kondo
temperature.
to superconducting. As shown in Fig. 1, if the
metal contains a very low concentration of
magnetic impurities (as low as a few parts per
million), it can undergo the Kondo effect where its
resistance will actually begin to increase again
before finally flattening once the temperature
drops far below a characteristic value called the
Kondo temperature. The increase in the metal’s
resistance arises from the spin-exchange processes
that occur between the electrons of the metal and
the electrons of the magnetic impurity. 1 Due to
quantum fluctuations, an electron in the magnetic
impurity can tunnel into an unoccupied bulk state
of the metal. Within a very small timeframe, an
electron from the bulk state can tunnel into the
impurity and take the original electron’s place. If
the replacement electron has a different spin value
than the original, then the total spin of the impurity
will change. These spin-exchange processes
increase the resistance because they lead to an
increase in scattering, and they eventually lead at
absolute zero to the complete screening of the
impurity spin by the conduction electrons. 1 The
Anderson model is made up of a single magnetic
impurity tunnel-coupled to the conduction band of
the metal. In order to determine the properties of
this system, we need to use the eigenstates and
eigenvalues of the entire system. These can be
found by using a Hamiltonian that describes the
Anderson model. Because this many-body system
involves a very large number of terms, second
quantized notation is used to represent the
Hamiltonian. Instead of representing each
individual particle’s state with wave functions, as
is the case with first quantized notation, second
quantized notation uses occupation numbers that
represent the number of particles (𝑛) in a certain
state (𝒅). The Hamiltonian for this model is
𝐻𝐴 = 𝐻𝑖𝑚𝑝 + 𝐻𝑏𝑎𝑛𝑑 + 𝐻𝑖𝑚𝑝−𝑏𝑎𝑛𝑑 .
(2)
In this equation
†
𝐻𝑏𝑎𝑛𝑑 = ∑ 𝜀𝒌 𝑐𝒌σ
𝑐𝒌σ
(3a)
𝐤,𝛔
describes the conduction band of the host metal,
†
where 𝑐𝒌σ
and 𝑐𝒌σ represents the creation and
annihilation operators for a conduction band
electron with energy 𝜀𝒌 and spin z-component 𝜎
(which can be either ↑ or ↓ ). When these
operators are together, they count the number of
such electrons;
𝐻𝑖𝑚𝑝 = U𝑛𝒅↑ 𝑛𝒅↓ + 𝜀𝒅 𝑛𝒅
(3b)
represents the magnetic impurity, where 𝑈 is the
Coulombic repulsion between two electrons on the
2
impurity level, 𝑑σ† and 𝑑σ are the creation and
annihilation operators for an electron with energy
𝜀𝒅 and spin z-component 𝜎 in the impurity level,
and 𝑛𝒅 = 𝑛𝒅↑ + 𝑛𝒅↓;
𝐻𝑖𝑚𝑝−𝑏𝑎𝑛𝑑 =
V
√𝑁𝑘
∑(𝑑σ† 𝑐𝒌σ + H. c. )
(3c)
𝐤,𝛔
accounts for the tunnel-coupling between the
impurity and the conduction band electrons, with
𝑁𝑘 being the number of unit cells in the metal and
V the hybridization matrix element.
(a)
(b)
Figure 2: (a) The density of states for a
pure metal. (b) The density of states for a
semi-metal.
In order to represent many-body interactions, such
as these, the electronic states of the system would
have to be considered. The number of electronic
states within a certain range of energy levels can
be determined by calculating the density of states.
The number of electronic states (𝜌(𝜀)) for a
specific energy level (𝜀) is given by
𝜀 r
𝜌(𝜀) = 𝜌𝑜 | | ,
𝐷
(1)
where 𝐷 is equal to half the bandwidth of the
conduction band, which in this paper is 2, and r is
a constant2. All of the energy states below the
Fermi energy of the metal are occupied, while all
of the states above are vacant. For a normal metal,
where r = 0, the number of electronic states are
the same between any given range of energies. For
a value of r > 0 , however, the number of states
decreases as the lower energies approach the
Fermi energy and then begin increasing again as
the energy continues to increase; this is called a
pseudogap and is found in semi-metals. The major
difference between metals and semi-metals, within
the context of the Kondo effect, is that semimetals undergo QPTs while pure metals do not.
The two different quantum phases for a semimetal are the local moment (LM) and strong
coupling (SC) phases. The magnetic impurity’s
spin degree of freedom is free in the LM phase,
while it is extinguished in the SC phase due to the
Kondo effect, this basically means that the
impurity’s magnetic moment is completely
screened out by the surrounding electrons.
Phase diagrams for the pseudogaped
Anderson model have been approximated using a
scaling method called poor-man’s scaling analysis.
These calculations are only approximations,
however, and are not as accurate as numerical
calculations. In this project, we decided to test the
accuracy of the predictions of the behavior of the
pseudogap model phase diagrams developed from
the results of the scaling approximations. We did
this by using the numerical renormalization group
method to make more accurate calculations of
quantum critical points and analyzed them to see
how well the predictions matched the results.
However, the Kondo effect is notoriously difficult
to capture using algebraic methods, so the
3
reliability of the predicted phase boundaries is
unclear. The goal of this project was to compared
the phase boundaries predicted by poor man’s
scaling with those obtained via a highly reliable
but computationally intensive technique called the
numerical renormalization group. Section II
briefly describes this method, while Section III
presents and discusses our numerical results.
knowing what combination of parameters would
cause the impurity to be in the LM phase and what
combination would cause it to be in the SC phase.
At absolute zero, if the impurity is in the LM
phase, its magnetic susceptibility, Tχ, would be
1
equal to 4. If the impurity were in the SC phase, Tχ
𝑟
8
would be equal to when the system is in particlehole symmetry or 0 otherwise. In order to solve for
Tχ, equation (2) must be diagonalized to obtain all
of the eigenstates and eigenvalues of the system.
However, this would mean having to consider all
of the different electronic states across the
bandwidth. As one can imagine, if we do this, our
matrix will end up with such a large
dimensionality that even a computer would have a
hard time solving it. In order to solve this problem,
we use the numerical renormalization group
(NRG) method that was developed be Kenneth
Wilson during the 1970s. What this method
essentially does is logarithmically discretize the
𝜀
continuous spectrum of the bandwidth [-1 < < 1]
(a)
D
(b)
into
positive
and
negative
intervals D+
n =
−n
−(n+1)
[Λ−(n+1) , Λ−n ] and D−
] ,
n = [−Λ , −Λ
where n = 0, 1, 2 … If Λ is taken to be equal to 1,
then the problem is unchanged and will take an
infinite amount of time to be solved.
−Λ0
Figure 3: (a) The phase diagram of a semimetal with 0 < r < ½ (b) The phase
diagram of a semi-metal with r > ½
II. METHODS
In order to determine where the critical points
of the phase boundaries were, we needed a way of
−Λ−1 −Λ−2
Λ−2 Λ−1
Λ0
If Λ is set to 2 or 3, however, these intervals get
smaller and smaller as the energies gets closer to
the Fermi energy. The reason for this
discretization is so that only one electronic state is
represented in each of these intervals or so called
bins. So instead of having an infinite set of states
over the entire bandwidth, we only sample one
4
state per bin with an energy that is equal to the
average over the bin, weighted by the density of
states
𝜀n±
=
∫
±,n
∫
𝑑𝜀 𝜌(𝜀)𝜀
±,n
𝑑𝜀 𝜌(𝜀)
.
(4)
by iterative diagonalization. After each iteration,
only the eigenstates with the lowest energies are
kept. These solutions describe the physics in a
temperature range equal to
n
DΛ−2
Tn =
,
𝑘B
The energy of each state is now equal to
1
𝜀n± = ± Λ−n (1 + Λ−1 ).
2
(5)
The discretized conduction band is mapped via the
Lanczos transformation onto a semi-infinite chain
form. The Anderson impurity model is now solved
III. RESULTS
where 𝑘B is Boltzmann’s constant. In order to
make these numerical calculations, we used an
existing NRG code to repeatedly solve the
Anderson impurity model and track Tχ of the
impurity
vs.
T
automatically.
𝜀𝑑𝑐 ≈ −
A. 𝜺𝐝 (𝐫, 𝐔, 𝚪), r > 1
Log (−
The first scaling approximations tested
predicted the behavior of thehighlighted portions
of the phase boundary for r > 1 shown in Fig. 5.
(6)
ΓU
(𝑟 − 1)𝜋
𝜀𝑑𝑐
) = Log(U) − Log((𝑟 − 1)𝜋)
Γ
𝜀𝑑𝑐 ≈ −
Log (−
Γ
r𝜋
𝜀𝑑𝑐
) = Log(r𝜋)
Γ
r
𝜋𝑟 (1 + ) U 1−𝑟
2
Γc ≈ ≈
( )
r
2
𝑟(1+ )
2
4𝑒
Figure 5: The red highlighted area shows the
portion of the boundary predicted by equation (7).
The blue highlighted area shows the portion of the
boundary predicted by equation (9).
Log(Γc ) = Log (
r
𝜋𝑟 (1 + 2)
r
𝑟(1+ )
2
4𝑒
U
) + (1 − r) Log ( )
2
5
-8
-6
-4
-2
0
2
4
0
-1
Log (dc/)
-2
-3
-4
r = 1.1
r = 1.5
r=2
-5
-6
-7
 = 10-3
-8
Log (U)
6
R
Γ
10-3
1.1
10-2
10-1
10-3
1.5
10-2
10-1
10-3
2
10-2
10-1
Slope
Y-Intercept
Predicted
YIntercept
9.36E-01
-1.15E-01
5.03E-01
1.17E-02
-6.05E-01
-5.39E-01
9.38E-01
-1.17E-01
5.03E-01
1.17E-02
-6.05E-01
-5.39E-01
9.53E-01
-1.45E-01
5.03E-01
1.13E-02
-6.04E-01
-5.39E-01
9.90E-01
-3.59E-01
-1.96E-01
1.33E-02
-7.41E-01
-6.73E-01
9.90E-01
-3.62E-01
-1.96E-01
1.33E-02
-7.41E-01
-6.73E-01
9.91E-01
-3.91E-01
-1.96E-01
1.30E-02
-7.40E-01
-6.73E-01
9.97E-01
1.47E-02
9.97E-01
-6.02E-01
-8.66E-01
-6.04E-01
-4.97E-01
-7.98E-01
-4.97E-01
1.47E-02
-8.66E-01
-7.98E-01
9.98E-01
-6.22E-01
-4.97E-01
1.45E-02
-8.65E-01
-7.98E-01
7
B. 𝚪𝐜 (𝐫, 𝐔), r <
𝟏
𝟐
𝐔
, 𝜺𝐝 = − 𝟐
-14 -13
-12
-11 -10
-9
-8
-7
-6
Log (c)
-15
-5
-4
r = 0.05
r = 0.1
r = 0.15
r = 0.2
r = 0.3
r = 0.4
1
2
 d U
hhhh
-3
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
Log (U)
r
Slope
Predicted
Predicted
Y-Intercept
Slope
Y-Intercept
0.05 0.950012
0.95
-3.74
-3.92
0.1
0.900033
0.9
-2.95
-3.22
0.15 0.850064
0.85
-2.44
-2.82
0.2
0.80011
0.8
-2.04
-2.53
0.3
0.70026
0.7
-1.39
-2.14
0.4
0.600461
0.6
-0.766
-1.87
8
-6.5
-2
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
Log [c(-U/2)-c(Ed)]
-3
-4
-5
r = 0.1
r = 0.2
r = 0.3
-6
-7
-8
-9
U = 10-3
-10
Log [Ed + U/2]
9
-3.5
0
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Log [c(-U/2)-c(Ed)]
-1
-2
-3
r = 0.1
r = 0.2
r = 0.3
-4
-5
-6
U=1
-7
Log [Ed + U/2]
U
10^-3
1
r
Slope
0.1
2.015
0.2
0.3
2.000
1.966
0.1
0.2
2.000
2.026
0.3
2.005
CONCLUSION
Although there are substantial deviations of our prefactors between numerical results and
approximations, these approximations accurately predict the behavior of the phase boundaries of the
Anderson impurity model. Therefore, these approximations are accurate qualitatively.
IV.
V.
REFERENCES
* Electronic address: amohammed2@mail.usf.edu
1. L. Kouwenhoven and L. Glazman, Revival of the Kondo effect. Physics World 14, 33 –38 Jan. 2001
10