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A SEARCH FOR SUPERCONDUCTIVITY
IN CONJUGATED POLYMERS
by
Neil McCulloch
Submitted for the degree of Doctor of Philosophy
Heriot-Watt University
School of Engineering and Physical Sciences
Department of Physics
January 2010
The copyright in this thesis is owned by the author. Any quotation from the thesis or
use of any information contained in it must acknowledge this thesis as the source of
the quotation or information.
Abstract
The aim of this thesis is to search for a superconducting phase transition in a conjugated
polymer with implanted magnetic atoms. To carry out this investigation we have performed
a modified worldline quantum Monte Carlo (QMC) algorithm concentrating on the polymer
cis-polyacetylene at finite temperatures. The Hamiltonian used is that of the extended SuSchrieffer-Heeger discrete tight-binding electron-lattice model which incorporates on-site
and nearest-neighbour electron-electron Hubbard terms, Brazovskii-Kirova symmetry
breaking, and Kondo impurities.
We have found that through changing the conduction charge doping level real
space-charge pairs, bipolarons, can be stabilised in the presence of magnetic impurities
within certain parameter regimes. This has been established through direct observation of
the calculated lattice order parameter/staggered lattice distortion and net charge
distributions. Thus we have demonstrated that real space charge pairs and magnetic atoms
can coexist.
In order to search for a second-order phase transition, indicative of possible
superconducting behaviour, we have calculated the impurity magnetic staggered and
uniform susceptibilities. This was done at discrete temperatures in the 60K – 300K range in
the presence of stable hole bipolarons. The results show, at higher temperatures, an
enhancement of the staggered over the uniform susceptibility for next-nearest-neighbour
impurity separation. This is indicative of anti-ferromagnetic fluctuations arising from
indirect exchange.
In contrast, at lower temperatures the staggered and uniform
susceptibilities converge consistent with the Kondo effect and destruction of the indirect
exchange mechanism. When the impurity separation is increased to six lattice sites both
susceptibilities are qualitatively the same at all temperatures as the impurities fluctuate
independently. Thus the indirect exchange mechanism is short-ranged and the impurities
should be relatively close to each other to facilitate anti-ferromagnetic fluctuations in the
presence of bound hole pairs.
i
The susceptibility measurements showed no signature of a phase transition within
the aforementioned temperature range. The investigation is however inconclusive as lower
temperatures proved computationally prohibitive.
ii
This thesis is dedicated to my mum and dad (the answer was not 27!) och min lilla älskling
Karin.
iii
Acknowledgments
I wish to thank all the people who have made this thesis possible and who have supported
me throughout. In some sense this is the most important section of the thesis as without my
family and friends the scientific parts would not have been completed.
I thank my PhD supervisor Dr Eitan Abraham for his continued encouragement,
support and guidance. His general enthusiasm for both science and football has no doubt
made my time at Heriot-Watt much more enjoyable and fruitful.
I want to thank my mum and dad for everything they have done for me over the
years.
I thank the love of my life, Karin, for living in Scotland with me for nearly five
years even though everything is grey!
I must also give special thanks to: my brother Grant, Debbie and the babies Ayla
and Mirrin, Aunty Carol and Ash-Pash, my Grandpa Ramsay, the Linder family from
Borlänge, Luis and all the guys from DB2.40.
I apologise to any readers whom I have forgotten. It was not intentional!
iv
Publications
[1]
McCulloch, N. and E. Abraham, Bipolaron formation in the presence of magnetic
impurities. Physica C: Superconductivity, 2009. 469(21): p. 1966-1969
v
Contents
1. Introduction
1.1 Motivation
1
1.2 Genesis
2
1.3 Thesis structure and its original content
4
2. The Physics of Conjugated Polymers
2.1 Introduction
5
2.2 Microscopic models
10
2.2.1 SSH model for degenerate conjugated polymers
10
2.2.2 Modelling Coulomb interactions
12
2.2.3 Other models
13
2.2.4 Modelling non-degenerate conjugated polymers
15
2.2.5 Modelling magnetic impurity atoms
18
2.3 Conductivity in conjugated polymers
19
2.4 Experimental evidence for solitons in trans-polyacetylene
22
3. Quantum Monte Carlo (QMC)
3.1 Introduction to QMC
23
3.2 Monte Carlo in statistical mechanics
25
3.2.1 Importance sampling
26
3.2.2 The Metropolis algorithm
27
3.2.3 The heat-bath algorithm
29
3.3 QMC for a finite lattice
30
3.3.1 Trotter-Suzuki decomposition
3.4 Worldline QMC and the checkerboard break-up
30
31
3.4.1 Choice of trial worldline moves
35
3.4.2 Fermionic minus sign problem
36
4. Magnetic Impurities in cis-Polyacetylene
4.1 Introduction
38
vi
4.2 QMC algorithm
40
4.2.1 Measurements
47
4.3 Room-temperature bipolaron results
49
5. Search For A Superconducting Phase Transition
5.1 Introduction
58
5.2 Susceptibility v temperature results
60
6. Conclusions
65
Appendix A – Calculation of Matrix Elements Involving A Classical
Magnetic Impurity
69
Appendix B - Calculation of Matrix Elements Involving A Quantum
Magnetic Impurity
86
References
103
vii
Chapter 1
Introduction
1.1 Motivation
The achievement of high-temperature superconductivity in the copper oxides [2] is
a scientific breakthrough that defies full understanding to this day.
From a general
scientific perspective it is equally interesting that for several decades it was credo that
~25K was the maximum achievable critical temperature. Yet after the initial discovery in
1986 of Tc ~35K, the value went up to ~130K within a relatively short time. A bold
extrapolation is that there is nothing inherently prohibitive in nature to prevent the synthesis
of a room-temperature superconductor [3].
This latter observation prompted the
investigation in this thesis.
The present theoretical work is based on a conjecture put forward by Mourachkine
[4]. This author argues that a room-temperature superconductor should be possible to
synthesise by bringing together real-space bound charge pairs and a mechanism which
mediates coherence between such pairs.
In this thesis we have ‘synthesised’ such a
material via a suitable Hamiltonian. We then performed quantum Monte Carlo simulations
to study the formation of pairs, their stability, coherence and ultimately their synergy
leading to a possible phase transition.
The onset of superconductivity is characterised by a complex order parameter
whose amplitude embodies the binding of electrons or holes into Cooper pairs [5] and its
phase accounts for coherence among their wavefunctions to form a condensate Ψ. In
layered superconductors like the cuprates, the density of paired electrons Ns per layer
determines the phase stiffness, namely the energy cost to produce spatial variations in the
phase of Ψ within a layer. Superconductivity persists until the thermal energy kT becomes
comparable to the phase stiffness energy kTphase, which is proportional to Ns. At this point
phase coherence is destroyed and superconductivity obliterated as a result. In conventional
superconductors at low temperature, kTphase >> 2Δpair = kTpair, namely the pair binding
1
energy. In these materials as the temperature is increased, Cooper pair breaking drives the
transition to the normal state, and as it does, it reduces Ns until Tphase Tpair  Tc.
In the cuprates, the conventional ordering of the binding and phase stiffness
energies is reversed [6]. As Ns is modified away from the optimal value for the highest Tc
towards the Mott insulator (underdoping), reduced phase stiffness is accompanied by larger
binding energy [7, 8]. Thus, in underdoped cuprates, electrons may remain tightly bound
while long-range phase coherence vanishes at Tc [6, 9].
Furthermore, spectroscopic
evidence shows that pairing is found up to temperatures some 100K above Tc [7, 10, 11].
This fascinating observation opens up new frontiers for synthesising materials that can
sustain Cooper pairs at room temperature with an engineered phase coherence that
maximises Tphase. Pair creation and phase coherence are mediated by different physical
mechanisms and thus high temperature superconductivity requires that both scales be
elevated simultaneously.
Quasi-one-dimensional conjugated polymers are one group of materials for which
electron-phonon mediated real space bound charge pairs, bipolarons, are known to exist at
room temperature [12]. In contrast they are not superconducting at any temperature.
Mourachkine concludes that it is the lack of coherence between bipolarons which obstructs
superconductivity in such materials.
This poses the question as to how to engineer
coherence. In this thesis we investigate the speculative idea of mediating coherence
between bipolarons using the fluctuations of implanted magnetic atoms in cispolyacetylene.
As we are ultimately searching for a bipolaronic superconducting
mechanism this thesis may be relevant to readers interested both in the field of
superconductivity and/or the phenomenon of BCS-BEC crossover [13].
1.2 Genesis
Organic salts [14], heavy fermion compounds [15] and copper oxide (cuprates)
materials [2] are examples of superconductors in which BCS theory [16] breaks down. A
huge
number
of
unconventional,
non-weak-electron-phonon
mediated
BCS
superconducting materials have been fabricated since the discovery of the cuprates.
2
However, there is at present no high-Tc microscopic theory of superconductivity which can
make predictions about a wide group of materials as BCS can for conventional materials.
Yet there are numerous theories which can explain certain aspects of specific materials.
This poses the question: is there one microscopic theory which can explain all
superconducting materials or are there several different theories/mechanisms which can
give rise to a superconducting state? At this point in time it appears that the latter is true
although one can never rule out a unified theory beyond our present reach and
understanding.
The problem with unconventional superconductivity at the microscopic level is that
there are many different types of atoms, especially in the highest Tc compounds, and many
competing interactions. The complexity of these materials makes them extremely difficult
to model correctly and pinpoint the important factors contributing to the superconducting
state.
In this thesis our starting point is the non-degenerate conjugated polymer cispolyacetylene. We choose this material as a possible candidate for a new superconducting
material for several reasons. Firstly, through doping charges are naturally paired at certain
doping levels due to the relatively strong electron-phonon interaction, albeit in real space.
The formation of charge pairs is a known pre-requisite for any superconducting material.
In conjugated polymers these real-space pairs are called bipolarons and have been verified
by experiment at room temperature [12]. Secondly, conjugated polymers are quasi-onedimensional due to orbital overlap anisotropy. Several known superconducting materials
exhibit low-dimensional behaviour such as the cuprates [2] (quasi-2D), Bechgaard salts
[14] and carbon nanotubes [17] (both quasi-1D). Finally, superconductivity has already
been observed in carbon-based materials such as the fullerides C60 [18] as a further
example.
For these reasons we chose to investigate the possibility of turning cispolyacetylene into a superconductor. There are two main techniques used to create new
superconductors. Firstly, one can take a non-superconducting material and apply pressure.
Secondly, one can add new atoms. Sometimes a combination of both methods results in
3
success. Here we shall explore only the addition of new atoms. The immediate question
arising is which atoms should we add in order to coax cis-polyacetylene into a
superconducting state?
The coexistence between magnetism and superconductivity provides a criterion.
Long thought to be mutually exclusive [19], there are however materials for which
magnetism and superconductivity seem to be related. It is generally accepted that in the
heavy-fermion class of materials the electron pairing is magnetically mediated [20].
Cuprate materials such as YBCO [21] and NCCO [22] show superconductivity
neighbouring anti-ferromagnetic order on their respective phase diagrams. In addition in
the recently discovered Fe-based superconductors both magnetism and superconductivity
co-exist [23]. Then, given that bipolarons exist in cis-polyacetylene, can we engineer
coherence between them by implanting magnetic atoms thus leading to a new
superconductor?
1.3 Thesis structure and its original content
The layout of the thesis is as follows. Chapter 2 gives a description of the interesting
physics governing conjugated polymers. It then proceeds to detail the well defined models
and approximations used in explaining this class of material. Comparisons with previous
experiments are also made. Chapter 3 discusses the general quantum Monte Carlo (QMC)
algorithm used for extracting interacting many-body system properties at finite
temperature.
Chapter 4 discusses the extension of such an algorithm to include the
magnetic impurities and then presents the room-temperature bipolaron stability results.
Chapter 5 consists of the impurity susceptibility results at room temperature and below.
Chapter 6 is devoted to the conclusions. Chapters 1-3 are essentially introductory whilst
the original work comprises Chapters 4-6. For the sake of completeness we have included
appendices giving details of calculations. Their inclusion in the main body of the thesis
would have hindered the reader.
4
Chapter 2
The Physics of Conjugated Polymers
2.1 Introduction
Conjugated polymers are macroscopic molecules that have a backbone of carbon atoms
connected by alternating single and double covalent bonds. They have been the focus of
intense research since the 1970’s when the first samples of the topologically simplest
conjugated polymer trans-polyacetylene (PA) were synthesised and a corresponding
microscopic model was proposed [24-26]. There are several reasons for the great research
effort in the field of conjugated polymers. Firstly they can be reversibly doped through the
whole range from insulator to metal eventually reaching conductivities as high as 106 Scm1
[27]. Secondly they can exhibit a plethora of desired optical properties as well as retaining
the mechanical advantages of being plastics.
This has led to a large application of
conjugated polymers in the opto-electronic industry.
Thirdly, due to the quasi-one
dimensional nature of these polymers they exhibit a wealth of interesting physics.
To explain the physics of conjugated polymers we take the simplest and most
widely studied trans-PA. Although there exist many others, it is a useful example to use in
order to clarify several ideas for the reader. Trans-PA is made up of unit cells each
containing two carbon atoms and two hydrogen atoms as can be seen in Fig. 1. Each
carbon atom has four valence electrons, two of which form covalent bonds with
neighbouring carbons and one forms a covalent bond with the hydrogen. These three bonds
make an angle of 120 degrees with each other in the plane of the page and the
corresponding electrons commonly referred to as σ-electrons. The fourth valence electron
belonging to each carbon atom occupies the pz orbital perpendicular to the page. The
electrons occupying the pz orbitals are often termed π-electrons and are delocalised along
the chain direction due to the large pz orbital overlap of neighbouring carbon atoms. In
contrast it has been shown that the orbital overlap between neighbouring chains is minimal
and hence the motion of π- electrons perpendicular to the chain direction is restricted [12].
This is why conjugated polymers are termed quasi-one dimensional materials. In standard
5
tight-binding theory the corresponding electronic band structure would be that of a metal
with a finite density of states at the Fermi surface.
(a)
(b)
H
H
H
H
C
C
C
C
H
C
C
C
C
C
H
H
H
H
H
C
H
H
C
C
H
H
C
C
C
C
C
C
H
H
H
H
Fig. 1: trans-PA (a) A bond order phase (b) B bond order phase
However, as predicted theoretically by Peierls [28] as early as the 1950’s such one
dimensional metals should be unstable with respect to a reduction in symmetry. The basic
idea is as follows. Imagine two contributions to the total system energy of a single transPA chain. The first being the kinetic energy of the π-electrons and the second being the
potential energy due to the Coulomb interaction between localised electrons on
neighbouring C-H groups. Peierls showed that the minimum energy ground state was that
of a dimerised structure such that for small bond length displacements the increase in
potential energy was less than the reduction in the π-electron kinetic energy. Again, in
tight-binding theory, as a consequence of the dimerised bond structure a gap is present in
the π-electron band rendering neutral trans-PA a semi-conductor as it is now known to be.
This dimerisation or Peierls distortion, as it is commonly known, is a direct result of the
one-dimensional nature of trans-PA and all the other conjugated polymers; it would not be
present to such an extent in an equivalent 3-D counterpart.
We have just explained the existence of a dimerised ground state in trans-PA which
is due to the enhanced electron-phonon (lattice) interaction in one dimensional materials,
but there is more to it. As can be seen from Fig. 1 there exist two possibilities for the
bonding pattern in trans-PA: an A-phase (short-long-short) and a B-phase (long-shortlong). From first observation of the molecules symmetry one would expect both the A and
B-phases to have the same energy and that it would make no difference if one were to have
6
an A- or a B-phase present in the ground state. For this reason trans-PA is the simplest
example of a degenerate conjugated polymer. As we shall explain later, most conjugated
polymers in fact do not posses this degeneracy and the resulting physical differences are
significant. For now we will continue our review of the physics of trans-PA.
If for a moment one visualises a single chain of trans-PA with an odd number of
C-H groups (even number of bonds) it is not immediately clear what ground state bond
structure to expect. As we have just demonstrated there should be no reason for nature to
choose the A-phase over the B-phase bond order as both have the same energy. The
answer to this apparent problem was proposed in the early work by several authors [29-31].
In their work they demonstrated that under such conditions nature would not favour either
structure but instead choose both! The idea was that a chain starting in the A-phase on the
left of the chain would transform into the B-phase in the middle of the chain or vice versa.
Obviously for this situation to occur it was noted that at the transition point the bond
conjugation will be disrupted as shown in Fig. 2(a).
(a)
H
H
C
C
H
H
C
C
C
C
H
H
H
B-phase
.
C
C
H
H
C
C
C
H
Soliton
H
C
C
C
C
H
H
H
A-phase
(b)
LOP
X
Fig. 2: Ground state of trans-PA with an odd number of C-H groups (a) Electron is delocalised along the
length of the soliton (b) Lattice order parameter (LOP) corresponding to the neutral soliton.
7
In addition, it was hypothesised that this point of transition would not in fact be localised to
one site but extended over a region of several sites. Over the years several names have
been used to describe this region of bond order transition including: conjugational misfit,
Pople-Walmsley defect, domain wall, kink, dangling bond and the term which we will use
commonly throughout this work, namely soliton. It should be made clear that at this
junction the term soliton is used in these circumstances due to the non-linear shape of the
lattice order parameter (LOP) shown in Fig. 2(b).
_
The LOP is defined as
_
LOPi  ( y i  y i ) /  1 where y i   yi 1  2 yi  yi 1  / 4 and yi is the displacement of the ith
i
bond from the undimerised chain bond length. Unlike optical solitons which can pass
through each other unaltered the solitons discussed in the present work can be strongly
affected by each other and any disorder in the material.
As already mentioned, the perfectly dimerised structure of neutral trans-PA results
in a semiconductor in which all electronic states from the Fermi level are shifted to lower
energies. However, as the soliton region is not dimerised, a soliton electronic state is
situated within the energy gap [26].
For chains with an odd number of C-H groups it is
always energetically favourable to form a neutral soliton at gap-centre.
If one more
electron was added to this system it would also occupy this mid-gap state forming a
negative soliton. Alternatively an electron could be removed thus creating a positive
soliton.
Conduction
Band
Mid-Gap states
Valence
Band
(a)
(b)
(c)
Fig. 3: Soliton mid-gap states (a) charge neutral spin ½ soliton state (b) charge -e spin 0 soliton state (c)
charge +e spin 0 soliton state
8
One of the most controversial ideas of soliton physics can now be elucidated. In the case of
the neutral soliton, one electron occupies the mid-gap electronic state. The negative charge
of this electron cancels out the positive charge of the carbon nuclei hence the neutral
soliton. If one thinks of the soliton as a quasi-particle, as is justified, then it would appear
that the neutral soliton has spin one-half. Furthermore, charged solitons have no spin. This
counter-intuitive picture is the opposite from the standard particle picture of for example an
electron having both charge and spin simultaneously. It was these charge-spin relations
that at the time were regarded as extremely controversial and it was not until the much
celebrated microscopic model of Su et al was established that a detailed analysis of these
mysterious entities could be carried out [25, 26].
In summary, we have explained how the ground state of a single trans-PA chain
with an odd number of C-H groups consists of a soliton that separates perfect A- and Bbond-order phases. However, because polyacetylene is typically synthesised from carbon
pairs there exist many more even atom chains than odd atom chains thus resulting in a
minimal number of naturally occurring solitons (approximately 400 solitons per million
carbon atoms). It should be stated here, for what follows later, that for the chains with an
even number of sites the ground state is just the perfectly dimerised case. Additional
solitons can be created through hole or electron doping as well as photo-excitation
techniques. For example, suppose that an electron is injected into a neutral chain with an
even number of C-H groups. Then, instead of it entering the lowest energy state in the
conduction band (as in usual semiconductors), a new soliton state is created in the gap
which is energetically favourable. As the total number of electronic states is a constant the
creation of such a soliton state results in the depletion of states in both the valence and
conduction bands by half a state per spin. In this way doping occurs through soliton
formation in trans-PA.
9
2.2 Microscopic models
In this section we concentrate on the microscopic models used to explain the properties of
conjugated polymers. That a model is needed as opposed to an exact theory is the result of
the complexity of the quantum many-body problem for these systems.
In the next
subsection we will discuss the model Hamiltonian proposed by Su et al to describe transPA.
2.2.1 SSH model for degenerate conjugated polymers
In reality each atom in trans-PA has three degrees of freedom along an equal number of
axes. However, it was proposed by Su et al that only the normal mode predominantly
coupling to the π-electrons was important. Thus the only atomic co-ordinate in this model
is the projection of the atomic displacements along the chain axis. This first approximation
considerably simplifies any subsequent studies.
H
H
C
H
H
C
C
C
C
C
C
C
H
H
H
H
Fig. 4: Schematic of trans-polyacetylene. Only the movement of the C-H group in the chain direction is
modelled.
The second main approximation in constructing the SSH model is to treat the σelectrons adiabatically. I.e. they are confined to the nuclei and hence only move when the
nuclei move. In other words only the π-electron hopping is treated explicitly. It can be
seen from the band structure of trans-PA in Fig. 5 that the physics near the Fermi level is
governed by the π-electron bands.
10
Fig. 5: Band structure of trans-PA [32]. π-electron bands are circled.
Thus if we want to study the low energy physics of trans-PA this seems to be a valid
approximation. Given these two approximations the SSH Hamiltonian can be written as


H SSH   t 0   u i 1  u i  c i1, c i ,  H .C. 
i ,
K
2
 u
i
 ui  
2
i 1
1
2M
P
2
i
(1)
i
where t 0 is the mean transfer integral,  is the electron-phonon coupling parameter, u i is
the atomic displacement of the ith C-H group in the chain direction, ci, , ci ,
are the usual
fermionic creation/destruction operators creating and destroying π-electrons of spin  on
the ith site, H.C. is the Hermitian conjugate, K is the localised electron spring constant and
11
Pi is the conjugate momentum of the ith atom. It should be noted that this model neglects
inter-chain interactions. Using this model with parameters fitted to experimental data Su et
al calculated several properties of a soliton in trans-PA including its effective mass and
length and showed that it is a mobile quasi-particle due to its extended structure and low
effective mass. They were also able to predict the magnitude of bond dimerisation to be
approximately 0.075Ǻ which is in good agreement with subsequent experimental
investigations [26, 33, 34].
2.2.2 Modelling Coulomb interactions
To this day it is still unclear how to correctly deal with Coulomb interactions in conjugated
polymer models and to what degree they are already accounted for in the SSH model
parameter values [35, 36]. There have been many studies into the validity of the singleparticle model which would suggest that Coulomb terms need to be included in models of
trans-PA [37-53]. What is not clear from these studies is whether the explicit Coulomb
interactions stabilise or destabilise the dimerisation. This depends on the parameter values
used in the model and on the way the Coulomb terms are included i.e. Hubbard terms,
Hartree Fock, Coulomb perturbation.
One of the most common ways of dealing with Coulomb interactions in lattice
models is to include short range Hubbard type terms as follows:

H coul   Uni , ni ,  Vni ni 1

(2)
i
where U is the onsite Coulomb interaction and V is the nearest-neighbour Coulomb
interaction.
The
number
operators
are
defined
as
follows:
ni ,  ci, ci ,
and ni   ci, ci , . It should be noted that treating the Coulomb interactions within the

extended Hubbard model is a very rough approximation. As pointed out by Schrieffer et al
the large pz orbital overlap, which results in a large π-electron (transfer integral) bandwidth
in PA, also means that the charge is not localised on a site but can equally be found in
12
between sites (bond charge density). The effect of bond charge density on the dimerisation
of trans-PA has been studied and found to be significant [54].
2.2.3 Other models
The SSH model described in the previous section is the basis of the main work in this
thesis therefore we will only give a brief summary of other models used by others to study
conjugated polymers.
Several studies into conjugated polymers have been conducted using tight-binding
models like SSH but with additional approximations in order to simplify calculations.
Examples of these models are the following.
The Huckel model
This only models the hopping process of electrons along a chain. Electron interactions are
neglected and the geometry of the lattice is fixed.

H Huckel   Ei N i   t 1   i  ci1, ci ,  ci, ci 1,
i
i ,

(3)
where the first term is the potential energy of an electron on the ith orbital and the second
term represents an electron hopping from one atomic orbital to a nearest neighbour.  i is
the relative displacement of the ith bond from its equilibrium value.
The Peierls model
The Peierls model is just the SSH model in the limit of quasi-static atoms i.e. neglecting
atomic kinetic energy.


H Peierls   t 0   u i 1  u i  c i1, c i ,  H .C. 
i ,
13
K
2
 u
i
i 1
 ui 
2
(4)
with explicit electron-electron interactions neglected [28].
The Pariser-Parr-Pople (PPP) model
The PPP model is essentially the quantum chemistry equivalent of the extended Hubbard
model for modelling interacting electrons on a fixed lattice [55, 56].


1 
1 1

H PPP   t i ci, ci 1,  H .C.  U   N i ,   N i ,    Vi , j N i  1N j  1
2 
2  2 i j
i ,
i 
(5)
where the first term is the usual π-electron transfer integral, the second term is the on-site
Coulomb interaction between π-electrons and the third term is the nearest-neighbour
interaction between π-electrons. The reader can immediately see the similarity between the
PPP model and the extended Hubbard model by comparing the last two terms of Eqn. (5)
with the Hamiltonian of Eqn. (2).
Other studies have used combinations of the above described models including the
PPP-SSH model and the PPP-Peierls model to study conjugated polymers [57, 58].
The Continuum model
Although all of the models presented thus far are within the tight binding framework,
another important and significant model used to study conjugated polymers is known as the
continuum model. The continuum model can be derived from the SSH model in the
continuum limit: the electron-phonon interaction is weak and hence the magnitude of
dimerisation is small. In trans-PA the lattice constant is known to be approximately
a=1.22Ǻ compared to the magnitude of bond dimerisation u  0.075Ǻ and so this should be
a valid limit to take in the model. In this limit the electronic band structure can be linearly
approximated thus resulting in a field theory which can yield analytic solutions in closed
14
form. The continuum model Hamiltonian or TLM (Takayama, Liu, and Maki [59]) model
is given by
H TLM
 2




1



2


   dx  x    ivF  z
 x  x  x  
dx


x

 (6)
x
2  vF    w02






where  x  ,  x  are two-component fermionic field operators for which each
component represents a branch of the linearised band structure,  x and  z are Pauli spin

matrices, v F is the Fermi velocity,  is the lattice order parameter and  is the first
derivative of the order parameter with respect to time. The first two terms are the electron
kinetic energy and electron-lattice terms respectively. The second two terms represent the
lattice kinetic and potential energies.
This model has been extremely successful in
predicting properties of conjugated polymers. In particular it was the basis of several key
papers which predicted the existence of the polaron and bipolaron quasi-particles [60-62].
As expected, within the aforementioned limits, results such as the soliton formation energy
from the continuum model agree very well with those from the tight binding SSH model
[12].
2.2.4 Modelling non-degenerate conjugated polymers
We have already shown that in trans-PA, which has a degenerate ground state, non-linear
quasi-particles called solitons can exist. This is a direct consequence of the π-electronlattice coupling and the two possible degenerate bond order phases.
However, most
conjugated polymers are non-degenerate which means that the two bond-order phases are
energetically unequal. This can be seen in Fig. 6 where we use cis-PA as the prototypical
polymer. In cis-PA the difference in energy between the two bond-order phases arises
because of the orbital arrangement of the non-π electrons i.e. the σ-electrons localised in
bonds.
15
(a)
H
H
C
H
C
C
H
H
H
C
H
H
H
C
C
C
H
H
C
C
C
C
(b)
H
C
C
C
H
H
Fig. 6: cis-PA (a) High energy A-phase (b) Lower energy B-phase
The usual way to model this non-degeneracy is to add a symmetry-breaking term
into the SSH Hamiltonian. This term is the Brazovskii-Kirova (BK) term [60]. It should
be clear from the Hamiltonian below that even if the π-electron-phonon interaction could be
switched off there would still be dimerisation in the chain.


H BK  t BK   1 i ci, ci 1,  H .C.
i ,
(7)
where t BK is a phenomenological parameter that needs to be chosen consistently with
experiments on individual polymers. The prefactor of  1 i indicates that every odd (even)
numbered bond has an increased (decreased) orbital overlap relative to the undimerised
case.
When a BK term is introduced to the SSH model two solitons widely separated on a
linear chain will re-arrange themselves so that the region of high energy phase is
16
minimised. In other words, non-degeneracy leads to the confinement of solitons. The
resulting quasi-particle consisting of a confined pair of solitons has been named a polaron
or bipolaron depending on the individual charge of each soliton in the pair. A charged
polaron can be thought of as one charged soliton and one neutral soliton thus having charge
 e and spin ½. Similarly a charged bipolaron can be thought of as two charged solitons
thus having charge  2e and spin 0. Each polaron or bipolaron thus gives rise to two midgap states (one for each soliton component) as can be seen in Fig. 7. Within the BK model
solitons are always bound into polarons or bipolarons which means that even two like
charged quasi-particles can be confined to each other. This should not be surprising as the
BK model does not include any explicit Coulomb interactions which are the main cause of
bipolaron instability.
Conduction
Band
Mid-Gap states
Valence
Band
(a)
(b)
(c)
Fig. 7: Mid-gap states corresponding to (a) Positive (hole) polaron (b) Negative (electron) bipolaron (c)
Positive (hole) bipolaron
When do two separated solitons technically become confined and hence a polaron
or bipolaron? The answer is that if the wavefunctions of both solitons overlap in a nondegenerate chain, a bound state has been formed. Energy is then required in order to break
the polaron/bipolaron apart. Examples of a soliton and polaron/bipolaron can be seen in
Fig. 8. The bipolaron and polaron have similar LOPs. In general the bipolaron will have a
deeper lattice depression due to its double charge and consequent larger electron-phonon
interaction.
17
(a)
B phase
B phase
LOP
Soliton
Soliton
X
A phase
(b)
B phase
Bipolaron/Polaron region
B phase
LOP
X
Fig. 8: Lattice order parameter (LOP) for (a) two widely separated solitons each centred where the LOP
crosses the x-axis (b) Polaron or bipolaron centred at the chain centre.
2.2.5 Modelling magnetic impurity atoms
As the original work presented in this thesis is concerned with the effect of magnetic
impurities, we wish to give here a brief overview of how to model such entities placed into
host materials. The physical picture is as follows. Atoms containing d- or f-electron
orbitals can have unpaired electron spins and thus a magnetic moment. These impurity
orbitals have a weak hybridisation with the host material constituents (i.e. the conduction
band). The most celebrated theoretical models which describe these interactions are the
Anderson and the Kondo models [63, 64].
Here we shall only detail the Kondo
Hamiltonian as it is central to the later chapters. In addition to the usual conduction band
18
hopping term the Kondo Hamiltonian for each impurity is
H Kondo  J σ f  σ m 
(8)
where σ f and σ m are the second-quantised spin vector operators for the f electron of the
impurity atom and conduction electron at site m respectively.
We choose f-electron
impurity atoms in order to fix ideas although such a Kondo term can equally be applied to
d-electron impurities. In the limit of large f electron Coulomb repulsion the Kondo model
can actually be derived from the more general Anderson Hamiltonian [65]. It is because of
this large Coulomb repulsion that double occupancy is prohibited when simulating the
Kondo model i.e.
n


f
 1.
Such Hamiltonians have in the past proved successful in describing properties of the
heavy fermion superconducting family [66]. They can also explain resistance minima for
alloys containing magnetic impurities [64].
2.3 Conductivity in conjugated polymers
So far we have explained how quasi-particles can exist on single chains of conjugated
polymers. Charge injection through doping leads to the formation of solitons in trans-PA
whilst in non-degenerate polymers such as cis-PA, polarons and bipolarons can be created.
We will now attempt to address the question of how conductivity occurs in these materials
on a macroscopic level and the role that the aforementioned quasi-particles play in the
conduction process.
To try and answer these questions we will again confine our
discussion to the simplest and most studied trans-PA.
Throughout this chapter we present theoretical models of conjugated polymers as
idealistic single-chain molecules. As can be seen in Fig. 9 below the actual structure of
trans-PA consists of many intersecting fibrils each of approximately 200Ǻ diameter.
19
Fig. 9: Photograph of Shirikawa synthesised trans-PA [67]
In turn, each fibril is made up of between 100 and 1000 individual chains. Thus in reality
the conduction process in trans-PA will involve a combination of intra-chain, inter-chain
and inter-fibril charge transport processes.
To complicate matters even further the conductivity of trans-PA depends
significantly on the method of synthesis. The first trans-PA samples were created by
Shirikawa et al. [67] and can be doped right through the range from insulator to metal as
can be seen from Fig. 10 (a).
However the disappearance of conductivity at low
temperature is certainly not comparable to metallic materials (thermal phonons usually
decrease conductivity in metals due to electron-phonon scattering). It is believed that the
disorder present in Shirikawa trans-PA leads to the electronic wavefunctions being
localised. In turn the electrons can absorb phonons or emit energy to jump between the
localised states which results in charge transport [68]. As the temperature is lowered this
process can no longer take place and charge transport is hindered throughout the
macroscopic sample.
In the low doping regime, in which solitons are expected to be present, another
moderately successful theory of charge transport has been proposed for Shirikawa type
20
trans-PA [68]. The basic idea is that as charged solitons (created through doping) are
pinned to dopant ions they themselves cannot move freely. On the other hand neutral
solitons can move freely because of their neutrality and their low effective mass. If a
neutral soliton passes in the vicinity (approximately 7 lattice constants) of a charged soliton
they can effectively exchange positions. In this way charge transport could occur.
Trans-PA can also be synthesised using the Naarman method and contains much
fewer defects than the Shirikawa trans-PA [69]. The highly crystalline structure means that
the electronic wavefunctions are delocalised over large regions leading to soliton band
conductivity. Consequently at zero temperature, high conductivities are still achieved as
expected of metallic materials which can be seen in Fig. 10 (b).
Fig. 10: (a) Conductivity of iodine doped Shirikawa trans-PA. (b) Conductivity of iodine doped Naarman
trans-PA. The Shirikawa samples show a decrease in conductivity as the temperature tends to zero. In
contrast the higher purity Naarman sample has a relatively large conductivity as the temperature tends to zero
and phonons are frozen out. [70, 71]
Theoretical studies of inter-chain and inter-fibril interaction are extremely difficult
to undertake.
As we shall discuss in Chapter 3, fermionic lattice studies at finite
temperature are extremely restricted by the dimensionality and the size of the lattice. Due
21
to the complex topology of the fibrils and of the arrangement of chains within a single fibril
the conduction process in conjugated polymers is to this day not entirely understood.
2.4 Experimental evidence for solitons in trans-polyacetylene
As Chapter 4 is dedicated to polaron and bipolaron quasi-particles in a non-degenerate
system, here we will attempt to summarise the main experimental results pointing towards
the existence of solitons in trans-PA.
Firstly, soliton theory tells us that a neutral soliton has zero charge and spin ½.
Similarly, charged solitons have charge  e and spin 0. This spin-charge inversion has
been experimentally verified through studying the spin susceptibility as a function of
doping [72] and through electron spin resonance (ESR) [73-76]. Secondly the study of
doping induced infra-red active modes is consistent with charge doping through solitons
[77, 78]. Thirdly, theory tells us that the soliton is a highly mobile quasi-particle due to its
relatively low effective mass/extended structure. Evidence of mobile solitons has been
gathered experimentally using ESR and dynamic nuclear polarisation (DNP) techniques
[73-76, 79-84]. In addition, theory predicts the soliton half-width to be approximately 7
lattice constants in length. Good agreement has been found using electron-nuclear double
resonance studies (ENDOR) [85-87]. Finally, optical absorption experiments have given
direct evidence of the soliton mid-gap states [88-90].
22
Chapter 3
Quantum Monte Carlo (QMC)
3.1 Introduction to QMC
In general terms, a Monte Carlo simulation solves problems stochastically. A classic
review on the Monte Carlo technique is the highly cited reference by Metropolis and Ulam
[91] but since then a host number of publications is available. We are primarily interested
in the application to quantum statistical mechanics.
Monte Carlo simulations are particularly well suited to the interacting many-body
problems of statistical mechanics. They enable calculation of thermodynamic properties of
model systems/Hamiltonians which may elude an exact analytical answer. In addition the
method is quasi-exact in the sense that the statistical errors can be made extremely small by
simply making more independent measurements (longer run times/more CPU’s). For a full
review see Landau and Binder [92].
A quantum-mechanical approach to the many-body problem, at both zero and finite
temperatures, is precisely the scope of QMC techniques. The earliest such work dates back
to the nuclei and liquid helium ground-state simulations by Kalos [93, 94]. QMC was first
applied to spin lattice problems by Suzuki [95]. Since then the number of papers published
using QMC has been astronomical.
The invention of new QMC methods has also
progressed significantly over past years.
General classes of QMC algorithm include:
Variational MC (VMC), projector MC, Diffusion MC (DMC), Greens function MC and the
path integral QMC.
Each of these classes themselves has many variations and for
researchers starting to use QMC simulations the question of which type to use is
expediently model- and problem-dependent. Does such an algorithm allow studies of
arbitrary lattice dimension? What temperature range can be studied? If fermions are
simulated, is there a sign problem? Choosing the correct QMC algorithm, especially when
dealing with previously unstudied Hamiltonians, poses a challenging problem. A method
that works for one particular model may well not be suitable for another. Of course one
23
must use the literature as a guide in which a good starting point is the review of QMC
algorithms by Von der Linden [96].
Central to the original work of this thesis is the SSH Hamiltonian which is an
electron-lattice model. It is directly amenable to a QMC treatment. In saying this, the
number of QMC studies using the SSH model is rather small. Most QMC simulations
incorporating the SSH coupling term have been used to study what it was originally
invented for, namely conjugated polymers [46, 47, 97-103]. It has also been applied in a
QMC study of the fullerene C60 as well as to organic charge transfer salts [104, 105].
Other electron-phonon models widely studied with the use of QMC algorithms include the
Holstein and Frohlich coupling models. For a review of QMC applied to the Holstein
models see Ref.[106] and for the Frohlich models Ref.[107].
There are two main groups of QMC algorithm used to study electron-phonon
models which both belong to the path integral class, namely the determinantal and
worldline algorithms. The original determinantal algorithm was applied to the Holstein
model and consists of integrating out the fermionic degrees of freedom analytically thus
leaving a determinant which depends only on the bosonic/atomic fields [108, 109]. The
process of integrating out the fermionic degrees of freedom, leading to a determinant, can
be carried out if the fermionic Hamiltonian is either bi-linear or can be transformed into bilinear form using, for example, the Hubbard-Stratonovich transformation. In contrast to the
determinantal QMC, the worldline QMC [110] samples both bosonic and fermionic degrees
of freedom (as we shall explain). Whilst in general the worldline algorithms are faster than
the determinantal class they are usually restricted to one dimension by the fermionic sign
problem. This is discussed in more detail in § 3.4.2. The determinantal algorithms also
suffer from the sign problem which varies in magnitude with model, parameter regime, and
dimension. For a recent review of determinantal algorithms applied to fermion-only lattice
models see ref. [111].
In § 3.2 of this chapter we present a brief summary of the main components of
generic classical MC algorithms applied to statistical mechanics. These include: Markov
chain theory, importance sampling, the Metropolis algorithm, statistical error and
24
ergodicity. § 3.3 is concerned with the quantum many-body problem i.e. QMC. We shall
discuss the crossover from MC to QMC including the imaginary time path integral
representation and the Suzuki-Trotter decomposition. As a precursor to Chapter 4 we will
detail here the standard worldline QMC [110] algorithm for spinless fermions on a finite
lattice.
3.2 Monte Carlo in statistical mechanics
In this section we wish to discuss some general concepts of MC simulations including
Markov chains, importance sampling, the Metropolis algorithm, statistical errors and
ergodicity. To illustrate these points we will use the framework of statistical mechanics.
Statistical mechanics enables us to determine equilibrium thermodynamic properties
of large systems, e.g. of the order of Avogadro’s number, by means of the Boltzmann
distribution
1
P  e
Z
 E
kT
(9)
where Pµ is the probability of being in a state μ, E  is the energy of state μ, k is
Boltzmann’s constant, T is temperature and Z is the partition function which essentially
contains all the information about the system and is defined as
Z  e
 E
kT

(10)
Expectation values of observables are calculated according to
O   O P

(11)
where O is a general observable whose value is Oµ in the state µ and the summation is over
all possible system states. In theory one can use the partition function to subsequently
25
calculate all thermodynamic quantities such as internal energy, specific heat and free
energy. In reality the partition function cannot be directly written down due to the large
number of possible states involved in the problem. However, only a relatively small
fraction of possible states contribute significantly to Z and so if these “important states” can
be found a good estimate of observable quantities can be made.
3.2.1 Importance sampling
Let us now discuss how to find the important/most relevant states of the partition function.
Only this subset of states is used in calculating the thermal expectation value which will be
an approximation called an estimator. The subset of states is chosen by selecting them
according to some arbitrary probability distribution.
However, if this probability
distribution is indeed arbitrary it is highly likely that a large number of “important” states
will be omitted from the observable’s estimate whilst “unimportant” states will be included.
This of course will lead to an extremely bad estimate of the observable. The trick used in
statistical mechanics MC algorithms is to choose states  i according to the magnitude of
their corresponding Boltzmann factor P i . If for example there are ten thousand possible
states though only time to sample one of them, it is most likely that the state chosen will be
that of the state with the largest Boltzmann factor P i . Thus  i is the only state which
contributes to the observables average. Although the observable average will have a large
error due to sampling a single state, this error is significantly smaller than if the single state
had a corresponding small Boltzmann factor. The process continues by choosing a second
 i with probability P i and then a third etc. For a finite number of such samples an
estimate of Eqn. (11) can be written as
N
ON 
 O i e
i 1
N

e
  E i
  E
j
j 1
26
Pi1
P1j
(12)
where P is given above, N is the number of states selected from the probability distribution
P and  
1
. The factors P1 and P1 ensure that all states in the sample effectively
i
j
kT
contribute only once to the observable average as required by Eqn. (11). Cancellations in
the numerator and denominator, respectively, result in the following simple expression for
the estimator/thermal average of the observable O:
ON
1

N
N
 O
i 1
(13)
i
For every observable calculated using the latter method there will of course be an
associated statistical error due to the finite sample size. Given that each state is statistically
independent, the statistical standard error is given by
Error 
S
N
(14)
where S is the standard deviation of the observable estimate. Namely
S
O N2
 ON
2
(15)
3.2.2 The Metropolis algorithm
In the previous sub-section we have discussed the issue of importance sampling. Namely,
how to calculate a good estimate of an observables thermal average using a small fraction
of the total system states. As we have discussed, the solution is to select states according to
the Boltzmann probability distribution. In practice, it is common to use the Metropolis
algorithm to perform such a task. It can be summarised as follows:
27
1)
Choose an arbitrary initial system state i
2)
Change the system to state j and calculate the energy difference ΔE
3)
If ΔE ≤ 0 accept the new state (skip steps 4 and 5)
4)
If ΔE > 0 generate a random number 0 ≤ r < 1
5)
If r < exp( - β ΔE ) Accept the change
6)
Repeat steps 2-5
In repeating this procedure a number of times, eventually any new state will be
chosen according to the Boltzmann distribution. The time taken for this to happen is called
the thermalisation time or time taken for the system to reach thermal equilibrium. Thermal
averages of desired observables can then be calculated.
The above algorithm is an example of a Markov process for which the state of a
system is solely dependent on the directly preceding state and not on the systems history.
For such a process the transition probabilities Wi→j and Wj→i must be chosen to satisfy
detailed balance. Namely in equilibrium, on average, the system should go from state i to
state j the same number of times as going from state j to state i:
PiWi→j = PjWj→i.
(16)
When a thermodynamic system is in equilibrium we know that Pi and Pj are precisely the
Boltzmann probabilities. Given this, we are free to choose Wi→j and Wj→i according to Eqn.
(16). For a random initial system configuration, the iterative process of accepting/rejecting
new states according to these transition probabilities will eventually lead to equilibrium for
which
d Pi
 0 . The Metropolis algorithm uses just one choice of transition probabilities
dt
Wi→j = exp (-ΔE β)
ΔE > 0
Wi→j = 1
ΔE ≤ 0
(17)
28
In fact any choice of transition probabilities which satisfies the above detailed balance
condition and

Wi→j = 1 will also lead to the Boltzmann distribution.
j
In addition to the conditions above necessary for a Markov process there is also the
condition of ergodicity. In a nutshell this means that every state of a system can be reached
with a finite probability, no matter how small. Whether a simulation is ergodic or not will
depend on the precise way in which trial states are chosen. To illustrate a non-ergodic
algorithm consider the simple case of four coins on a table in a line. Say that we wish to
check each combination of heads and tails. To do so the easiest way would be to take an
initial configuration say H-H-H-H and flip single coins at random. If the process was
repeated a large number of times, each combination would appear with the probability P =
1/16. In contrast if we had chosen to flip only nearest neighbour same state (i.e. H-H → TT) coins together we would have missed out a large number of possible combinations.
Obviously this is an oversimplified example but a useful exercise to illustrate that the
choice of trial moves in MC simulations should be taken with caution.
3.2.3 The heat-bath algorithm
The heat-bath algorithm is another common importance sampling technique which is
similar to the Metropolis algorithm. For some models it can lead to faster thermalisation
times although there is a trade-off between this and longer CPU operations per spin
flip/trial change [112]. The general idea is to choose a new trial state at random and
accept/reject with probability proportional to the state’s Boltzmann weight. For the basic
Ising model in which each lattice site is either spin-up or spin-down the algorithm would be
as follows.
1) Choose a lattice site at random
2) Choose a random number between 0 ≤ r <1
3) If r < Pup =
exp   Eup 
exp   Eup   exp   E down 
4) If r < Pdown =
set spin to up
exp   E down 
set spin to down
exp   Eup   exp   E down 
29
Again this algorithm satisfies the necessary conditions for a Markov process and will thus
lead to a selection of states chosen according to the Boltzmann distribution.
3.3 QMC for a finite lattice
So far we have discussed general concepts and algorithms in relation to classical statistical
mechanics. In this section we shall confine our attention to finite lattice problems on the
quantum scale. The main difference is the promotion of the classical Hamiltonian to a
quantum mechanical operator:


exp   E    exp   Hˆ 
(18)
where Ĥ is a general Hamiltonian operator,  is a general quantum state and  is the
inverse temperature. By associating the inverse temperature  with imaginary time iτ,
statistical mechanics is directly related to the Feynman path integral formulation of
quantum mechanics through:
Z   D  exp  S  
(19)
where S   is the action in imaginary time and D  signifies a functional integral over
all possible system states. Now the underlying problem in making the transition from a ddimensional classical system to a d +1 (imaginary time dimension) quantum mechanical
system is the calculation of the required statistical weights given by Eqn. (18).
3.3.1 Trotter-Suzuki decomposition
One of the most common ways of calculating the desired statistical weights is to use the
Trotter-Suzuki approximation. The fundamental problem in calculating such a weight is
that the constituent parts of the Hamiltonian operator do not commute. For a general
30
Hamiltonian composed of two non-commuting operators, say H1 and H2, we can write [113,
114]:

exp   Hˆ

 
 exp   Hˆ 1  Hˆ 2
 


L

  2
   Hˆ 1 
   Hˆ 2  
 exp 
   O    
  exp 

 L 
 L  

 L 






 exp    Hˆ 1 exp    Hˆ 2

L

 O   
2
(20)

where     / L indicates that the inverse temperature/imaginary time has been divided
up into L time slices. It should also be noted that in using such an approximate expression
the error is of order    because the approximation is truncated at a level which omits
2
commutators [  Hˆ 1 ,  Hˆ 2 ] and above. The systematic error is therefore controlled by
the number of time slices used and is zero in the limit of infinite L. Furthermore, it should
be noted that the approximation can be truncated at higher order leading to a vast reduction
in systematic error for the same   or similarly the same systematic error using less
imaginary time slices. Although adding increased complexity to numerical coding, such
higher order Trotter-Suzuki approximants have displayed an improved convergence to the
exact answer [115].
3.4 Worldline QMC and the checkerboard break-up
The worldline QMC is a very powerful algorithm used to calculate thermodynamic
properties of a finite lattice system.
As already discussed, the importance sampling method used in QMC relies on the
ability to be able to calculate the statistical weights given by Eqn. (18). Furthermore, in
31
practice these weights can be calculated using the Trotter-Suzuki decomposition by
splitting the Hamiltonian up and thus introducing a controlled systematic error.
The key to the original worldline QMC algorithm is the way in which the
Hamiltonian is broken up known as the checkerboard break-up.
For Hamiltonians
including on-site and nearest-neighbour interactions, such as Eqn. (1), this involves the
separation of terms with i odd (e.g. involving atoms i=1 and i+1=2) and terms with i even
(e.g. involving atoms i=2 and i+1=3):
Hˆ  Hˆ 1  Hˆ 2   H i    H i 
i even
(21)
i odd
As the even sub-summation does not in general commute with the odd it is necessary to
apply the Trotter-Suzuki decomposition (see § 3.3.1):
exp   Hˆ exp   Hˆ 
1

 U 1U 2   O   
L
where

L
2
2

 O   
2

(22)


U 1  exp   Hˆ 1  U i 
i even


U 2  exp    Hˆ 2  U i 
i odd

U i   exp    Hˆ i 

(23)
The products in Eqn. (23) are justified as the individual U i  operators within each product
commute. Through insertion of a complete set of intermediate states between each U 1 and
U 2 the partition function can be written as
32
Z




 exp   H 
All states


1 .... 2 L

1 U 1 2 L 2 L U 2 2 L 1 ........... 3 U 1 2 2 U 2 1  O   
2

(24)
However the heat-bath and Metropolis importance sampling algorithms do not require the
calculation of Z. Instead the QMC algorithm only necessitates the calculation of the
following quantity:



2
 exp   Hˆ   1 U1 2 L 2 L U 2 2 L1 ... 3 U1 2 2 U 2 1  O   

(25)
A suitable choice of basis and the one chosen in the seminal Hirsch paper [110] is to
use that of occupation number:
1  n1 n2 n3 ..............nN
(26)
Thus for each imaginary time slice a set of occupation numbers is defined and for each 
interval there is one application of U 1 and one of U 2 . This can be visualised below where
the checkerboard nature of the Hamiltonian break-up manifests itself. By joining up the
occupations on successive time slices the “worldlines” are created. These worldlines are
not permitted to cross unshaded boxes as there would be no corresponding evolution
operator U 1 or U 2 . The sum over all states involved in the partition function is equivalent
to the sum over all possible worldline configurations.
33

Imag.
apply U1
Time

apply U2
apply U1
0
1
2
3
4
site
5
apply U2
6
Fig. 11: Checkerboard scheme for worldline quantum Monte Carlo. Shaded boxes correspond to two-site
matrix elements. Bold lines are typical fermionic paths (worldlines) evolving through imaginary time. The
fermion on the left of the board begins on site 2 then hops to site three within the first Δ τ interval before
hopping back to site 2 within the next Δ τ interval. There is one such lattice for each individual spin; namely
up and down.
Through choosing the occupation number basis the problem of calculating the
statistical weight reduces to that of a product of simple two-site matrix elements:
n1 ' n2 ' ....n N ' U 2 n1 n2 ....n N

 



 n1 ' n2 ' ...n N ' exp    Hˆ 1 exp    Hˆ 3 .... exp    Hˆ  N 1 n1 n2 ...n N




 n1 ' n2 ' exp    Hˆ 1 n1 n2 n3 ' n4 ' exp    Hˆ 3 n3 n4 .....
(27)
In summary the worldline QMC algorithm consists of the following steps.
1) Create an initial worldline configuration with one whole worldline corresponding to
each fermion in the system.
2) Make a change in the worldline making sure that the Hamiltonian symmetry is
obeyed and that worldlines do not hop across unshaded squares (see § 3.4.1).
3) Calculate the ratio of matrix elements corresponding to after and before the change.
34
4) Accept/reject the change according to the heat bath or Metropolis algorithm. (Others
may also be used)
5) Repeat steps 2-4 a number of times.
6) Once the system is in thermal equilibrium make measurements of thermodynamic
quantities.
If the model Hamiltonian under consideration includes both up and down fermion spins
then the above algorithm is applied to two space-time lattices, one for each spin.
3.4.1 Choice of trial worldline moves
As discussed previously any MC trial moves must be ergodic. The simplest such trial
move in the worldline algorithm is to attempt to move worldline segments across unshaded
squares as shown below. This, in fact, is the move originally proposed in the original paper
by Hirsch et al [110]. However, the simplest is not necessarily the best or most efficient.
The local nature of such a move means that such local worldline algorithms can take a large
number of trial moves to escape specific regions of phase space. Thus the number of trial
moves needed between statistically independent measurements can be extremely large
(autocorrelations). In turn this can lead to long run times and/or massive CPU requirements
to ensure all relevant regions of phase space are probed and accurate calculations of
observables performed.
An alternative to long run times could be to use one of the global update “loop”
algorithms [116-119].
Such algorithms essentially attempt to change many worldline
segments simultaneously in one trial move. Through implementing such global trial moves
the problem of autocorrelation can be reduced. Loop update QMC algorithms can be used
to simulate both spin and fermionic models on a finite lattice.
They also have the
advantage of being able to study the grand canonical ensemble which is problematic in the
standard local update schemes.
35
3.4.2 Fermionic minus sign problem
In performing QMC simulations involving fermionic degrees of freedom the statistical
weights used for importance sampling can become negative. This is a direct result of the
fermion statistics.
The importance sampling algorithms, however, treat the statistical
weights as probabilities and thus they must always remain positive. For any operator which
is diagonal, the expectation value can be calculated in the same way as classical MC. i.e.
Oˆ 

 O


 exp   Hˆ 


 exp   Hˆ 
(28)
The common trick to overcome negative configuration weights is to instead use the
absolute value [120]. The sign is then essentially absorbed into the observable as follows:
Oˆ 
 O S 
 S 


 exp   Hˆ 


 exp   Hˆ 


where S  is the sign of  exp   Ĥ 
and 
(29)
is an eigenstate of Ô . We can
thus write
Oˆ 
O S 
S
P
(30)
P
and the importance sampling algorithms are conducted with the following probability
distribution:
36
P 


 exp   Hˆ 



 exp   Hˆ 
(31)
This trick can be implemented as long as the average sign of the weights is well
above zero. If this is not the case then cancellations in the numerator of Eqn. (30) make
QMC simulations extremely difficult and practically impossible to perform without further
approximations such as fixed-node [121].
The fixed node approximation effectively
restrains fermions from hopping across a node of the wavefunction during the simulation
ensuring that QMC weights do not change sign. Of course the accuracy of such an
approximation relies upon that of the initially formed nodal structure which can be taken
from other quantum chemistry methods. The approximation is thus exact in the knowledge
of an exact nodal structure.
Before applying QMC to a new Hamiltonian there is no way of knowing the exact
magnitude of the sign problem, which is model-, algorithm- and parameter-dependent.
There are, however, general rules which can be used as a guide to its severity. Firstly, the
sign problem is more pronounced for fermionic hopping models in dimensions greater than
one. This is because fermions can exchange places easier. Secondly, the sign problem
generally increases when temperature is lowered. This is a result of the fermionic statistics
playing a more prominent role relative to temperature. Thirdly, smaller lattice sizes have a
corresponding smaller sign problem.
Finally, any fermionic spin Hamiltonian with
frustration, such as the triangular Heisenberg model will have an inherent severe problem.
Whilst most fermionic Hamiltonians will have a sign problem to some degree, there
are some exceptions: namely the 1-D t-J model [122], the t-J-V model [123] as well as both
the 1-D and 2-D bipartite Heisenberg models.
In summary, the sign problem can prevent investigations of certain fermionic
Hamiltonians within potentially interesting parameter regimes. Although there are ways
around this, as discussed above, there is to this day no complete solution and it is a topic of
continued research.
37
Chapter 4
Magnetic Impurities in cis-Polyacetylene
4.1 Introduction
The quasi-one-dimensional nature of conjugated polymers and inherent strong chargelattice interaction means that charge doping is fundamentally different from the standard
semiconductor picture. Charges introduced into neutral conjugated polymers result in a
distortion of the perfectly dimerised lattice. Subsequently, the charge can only propagate
along the chain if the surrounding distortion moves with it. The combination of charge and
lattice distortion acts like the charge has an increased mass and is therefore termed a quasiparticle [12, 68].
The first detailed theoretical studies into these quasi-particles were devoted to transPA, which has two degenerate bond-order phases. This degeneracy means that mobile
solitons are stable in such a material. In contrast to trans-PA all other conjugated polymers
do not possess this ground state degeneracy and therefore cannot support free soliton
solutions. They can however support bound pairs of solitons known as polarons and
bipolarons. Essentially, nature binds solitons into pairs in order to minimise the region of
the high-energy bond ordering.
The work of Brazovskii & Kirova [60] and Campbell & Bishop [61] laid the
foundations for a theoretical understanding of polarons and bipolarons in conjugated
polymers.
These initial investigations independently showed that for the continuum
version of the SSH model, the polaron solution existed in addition to the originally
discovered soliton solution.
For non-degenerate conjugated polymers, the Brazovskii-
Kirova model always favours bipolaron formation over two polarons when an even number
of charges are introduced [124, 125]. The validity of such a model of course depends on its
agreement with experiment.
38
From the work performed by Fesser et al it has been demonstrated that polarons and
bipolarons each have two mid-gap electronic states associated with their lattice distortions
[126]. The charged bipolarons, having both mid-gap states occupied or empty, have a
different optical absorption signature from polarons that have one fully occupied/empty
state and the other half occupied.
In addition, polarons having spin-1/2 manifest
themselves in spin detection experiments whilst bipolarons being spin compensated do not.
Optical absorption [127-130] and ESR (Electron Spin Resonance) [131] experiments on
polythiophene provide evidence that bipolarons are the stable quasi-particles at
intermediate doping levels whilst polarons can exist in the dilute doping regime. At high
doping levels a metallic-like state is found [127].
We mention results for polythiophene
as its carbon backbone is to good approximation the same as cis-PA. As bipolarons can
only be formed through the interaction of two polarons it is possible that the presence of
polarons in these materials is a result of disorder which could restrict the polarons from
recombining and forming a bipolaron [131].
Another explanation for polarons at low doping levels is that the explicit Coulomb
repulsion between charges (neglected in the BK single-particle model) is large enough to
prevent bipolaron formation.
The necessity for explicit Coulomb interactions has long been a matter of debate
amongst solid state physicists. More specifically, how strong are electron correlations in a
model system? To what extent are these correlations already accounted for in the other
parameters of the single particle models? Intuitively explicit Coulomb interactions should
be included if the competition between two like-charged polarons and a bipolaron is under
consideration. If one wishes to look at how easy it is to bring two same charges/charged
solitons together it makes no sense to omit the primary restrictive force i.e. Coulomb
repulsion. With this in mind, several studies have been conducted which include explicit
Coulomb interactions in the theoretical treatment [132-137]. Apart from the work of
Shimoi and Abe [135], the technical weaknesses of which been highlighted by Utz [136],
the general consensus is that bipolarons can exist for realistic values of the Coulomb
interaction terms.
Furthermore, through including these terms the polaron-bipolaron
transition at low doping levels can be explained. To summarise, if the explicit Coulomb
39
terms are included in any model of non-degenerate conjugated polymers their magnitude
should be such that (i) bipolarons are the generated quasi-particles upon even charge
doping and/or (ii) a doping-induced polaron-bipolaron phase transition is observed.
Given that bipolarons exist we investigate theoretically in this chapter how
implanted magnetic impurities affect such quasi-particles and vice-versa. We specifically
choose cis-PA as it is the topologically simplest non-degenerate conjugated polymer.
Furthermore, it can be viewed as a good approximation to other polymers such as
polythiophene when the effect of weakly interacting hetero-atoms, like sulfur, is neglected.
Questions naturally arising include the following: are bipolarons stable in the presence of
magnetic impurities and if so, how robust are they? How do these impurities affect the
competition between polarons and bipolarons? How are the impurities correlated in the
presence of quasi-particles? Is there interesting or unexplained behaviour of the impurity
susceptibility given their interaction with quasi-particles? Does such a system exhibit a
superconducting phase transition?
In §4.2, a description of the QMC implementation used to investigate these
questions is presented. It is essentially an extension of the electron-phonon worldline QMC
algorithms to incorporate magnetic impurities [98, 102]. Our room-temperature bipolaron
stability results comprise §4.3.
4.2 QMC algorithm
As the starting point for our QMC simulations we use the extended Hubbard SSH model
with Brazovskii-Kirova degeneracy lifting term. In addition there is a Kondo term
H Kondo  J σ f  σ m  for each magnetic impurity adjacent to site m (see §2.3.5):
N 1



H    t0   1 t1   xi 1  xi  ci1, ci ,  H .C. 
i 1,
K
2
i 1
N 1
 x
i 1
i 1
 xi 
2


1 N 2 N

Uni , ni ,  Vni ni 1   H Kondo
 Pi  
2M i 1
i 1
m
40
(32)
The quantity of interest enabling the QMC algorithm can be written as

K
exp   H   exp    
2

i 1
1 N 2 
 Pi   *
2 M i 1



N 1
2
 xi 1  xi  

 N 1

i 1

   t 0   1 t1    xi 1  xi  ci1, ci ,  H .C.  

 i 1,

exp     N


   Uni , ni ,  Vni ni 1   H Kondo
 


m
 i 1





(33)
 exp   H b  * exp   H fermion 
where t 0 is the mean transfer hopping integral, t1 is the phenomenological BrazovskiiKirova symmetry breaking term,  is the electron-phonon coupling parameter, xi is the
displacement for the ith CH group in the chain direction, ci, , ci ,
are the usual fermionic
creation/destruction operators creating and destroying pz electrons of spin  on the ith site,
K is the sigma-electron spring constant, Pi is the discretised momentum of the ith CH
group, U and V are, respectively, the on-site and nearest-neighbour Coulomb interactions.
The
number
operators
for
the
ith-site
are
defined
as
follows:
ni ,  ci, ci ,
ni ,  ci,ci , and ni  ni ,  ni , . We use cis-PA [47, 101] parameters: t 0 = 2.5 eV, t1 = 0.035
eV,  = 4.17 eVǺ-1, K = 27.8 eVǺ-2, M = 3145 eV-1Ǻ-2, U = 5 eV and V = 2.5 eV. In
what follows we will refer loosely to the CH groups as ‘atoms’.
The basis of the QMC algorithm is essentially that proposed by Hirsch [110] as
detailed in §3.4. The Hamiltonian in Eqn. (32) is substituted into Eqn. (21). Worldline
segments are moved across unshaded squares if the proposed move satisfies the
Hamiltonian symmetry and accepted/rejected according to the heat-bath algorithm. We
shall call this trial move Trial Move 1. The atomic co-ordinates are held constant whilst
41
updating these fermion configurations. Although the purely atomic terms are held constant
in this process, it should be noted that the calculated two-site matrix elements do depend on
the atomic configurations due to the electron-phonon coupling term.
In addition to Trial Move 1 we must extend the algorithm to include a move which
enables the possibility of spin flip due to the impurity presence.
The simplest move
allowing continuous worldlines is to attempt to flip the spins of a worldline segment
moving vertically on the impurity site and with an unshaded box to the right as shown in
Fig. 12. We call this move Trial Move 2. Thus the previously independent spin-up and
spin-down lattices are now coupled at the impurity locations. Furthermore, a conduction
electron can only flip its spin if there is not a conduction electron of opposite spin on the
same site. This is a consequence of the Pauli exclusion principle.
As a precursor to the fully quantum treatment, we have first (following Chen [138])
treated the magnetic impurity spin as a classical object with a fixed angle of orientation θ
with respect to the z-axis. In all results presented we shall make the clear distinction
between the use of classical and quantum impurities. For the classical impurity case the
impurity term H Kondo in Eqns. (32) and (33) becomes
H KondoClassical  J S imp  σ m 
(34)
where J is the bare exchange integral between the magnetic atom and the pz orbital at
some lattice site m, S imp is the classical impurity spin, and σ m is the second-quantised spin
operator for pz electrons at the site m. We assume that the impurity is closest to m. This
Kondo term can thus be written as
H KondoClassical 
J
  J sin   c  c

cos   c  c
 c c
 c c
m m 4
m m
 m m
 m m
4
(35)
where  is the angle between the classical impurity spin direction and the z-direction. The
classical treatment of the impurity spin means that θ remains constant throughout imaginary
42
time.
The full calculation of the desired two-site matrix elements for shaded boxes
including a classical impurity comprises Appendix A. Using these matrix elements new
fermionic configurations are again accepted/rejected using the heat-bath algorithm.
A fully quantum treatment of the Kondo impurities requires us to treat the impurity
spin as a quantum object as in Eqns. (32) and (33). Namely
H Kondo  J σ f  σ m 
(36)
where σ f and σ m are the second-quantised spin vector operators for the f electron of the
impurity atom and conduction electron at site m respectively. Note that to fix ideas we
have assumed magnetic f electrons and that throughout we impose the extra
condition  n f  1 . The Hilbert space must now be extended to include the quantum

impurity spin degrees of freedom. i.e.
nm, nm1, nm, nm1,
nm, nm1, nm, nm1, n f , n f ,
(37)
This is equivalent to introducing an impurity worldline at each impurity site. Such a
worldline must remain on the impurity site at all points in imaginary time as there is no felectron hopping term in the Hamiltonian. When modelling quantum impurities there is
one more restriction which must be adhered to. Namely, Trial Move 2 should only be
attempted if the associated f worldline segment can perform the opposite spin flip as in
Fig.12. This is because the fully quantum Kondo term commutes with the total spin
operator.
The full calculation of desired two-site matrix elements for shaded boxes
including a quantum impurity is detailed in Appendix B.
43
(a)
  
spin 
  
spin 
spin 
site 1
site 2
spin 
  0, 
  0, 
site 1
site 2
(b)
  
imp 
  
imp 
imp 
  0, 
site 1
site 2
  0, 
imp 
site 1
site 2
Fig.12: (a) Trial Move 2 of conduction electron worldlines. (b) Trial Move 2 of corresponding impurity
worldlines when quantum impurities are under consideration. Conduction electron worldline segments can
only be flipped from spin-up to spin-down if the impurity worldline segment can perform the opposite
movement. The impurity is located at site 1.
Whilst Trial Moves 1 and 2 are sufficient to sample all allowed fermionic
configurations, a third move is needed to update the ‘atomic’ degrees of freedom. The
purely atomic statistical weight can be written as (after associating the inverse temperature
with imaginary time)
44
exp   H b 

 K N 1
1 N 2 
2
 exp      xi1  xi  
 Pi  
2M i1  
 2 i1


K
M
2
  exp   b  xi1  j   xi  j  

2
2

i, j


(38)
 xi  j  1  xi  j  




b


2




This is equivalent to mapping the “atomic” co-ordinates on to a space-imaginary time
lattice. Trial Move 3 thus consists of selecting an atom at a particular imaginary time slice
and moving it by a random number rn in the interval –r ≤ rn ≤ r and accepting/rejecting the
new atomic position according to the Metropolis algorithm. Throughout, we have chosen
r  w / K where w  K / M in accordance with Hirsch’s original SSH work [110]. As
the atoms and fermions have different energy scales we have followed Takahashi [102] in
keeping a fixed atomic configuration over a finite number of fermionic time slices which
can be seen in Fig. 13. In other words the fermionic space-imaginary time lattice has Lf
time slices and the atomic/bosonic space-imaginary time lattice has Lb time slices. When
performing Trial Move 3 the quantity needed for the Metropolis algorithm thus consists of
the ratio of purely atomic weights multiplied by the ratio of fermionic two-site matrix
elements affected by the move.
exp   H b '  exp   H ' fermion  
*
exp   H b   exp   H fermion  
(39)
The primes indicate the respective Hamiltonians with proposed atomic configuration. As
Trial Move 3 is local on the atomic lattice, only nearest-neighbour (in both space and
imaginary time) purely atomic weights need to be calculated. In contrast, as L f is always
chosen several times larger than Lb , it turns out that the number of fermionic two-site matrix
elements affected by the move is also several times larger as shown in Fig. 13. Throughout
45

this chapter we have kept  f   / L f  0.0333 3 and  b   / Lb 1 . When varying the
inverse temperature we therefore vary L f and Lb . These chosen values have been shown to
be reasonable in similar studies [101, 102] whilst also meeting the accuracy criteria defined
by Galli [98].

rn
3 f
rn
Imaginary
Time
2
f

f
 b
0
1
2
4
site
3
5
6
Fig.13: Example of Trial Move 3. A move of atom 4 on the second atomic slice is depicted. Affected twosite fermionic matrix elements are shaded differently (left bottom corner to top right corner of plaquettes).
In summary, one can imagine two lattices such as Fig. 13 that have exactly the same
geometries. One lattice corresponding to spin-up conduction electrons and one for spindown. Both lattices are coupled through the Kondo interaction only at the sites m where
impurities reside. When considering quantum impurities there is an additional worldline
for each impurity. This is shown in Fig. 12. The initial fermionic worldlines are chosen at
random.
There is one worldline for each conduction electron.
The initial atomic
displacements are set identically to zero i.e. equal-width shaded squares in Fig. 13.
46
Trial Moves 1, 2 and 3 are then attempted at random to avoid any bias. One sweep is
defined as (on average) attempting to move every atomic space-time co-ordinate once (i.e.
Trial Move 3) and attempting trial moves 1 and 2 (remembering restrictions) on every
fermionic worldline segment once. Fixed end boundary conditions are used throughout.
Namely x1 ( )  xN ( )  0
 .
4.2.1 Measurements
After running the algorithm for a number of thermalisation sweeps, measurements of
observable quantities can be taken.
Atomic quantities and quantities with diagonal
fermionic operators can be calculated using Eqn. (30). To improve statistics the quantities
are also averaged over all time slices. The thermalisation time must be selected carefully in
order to disregard early observable measurements which have weights other than their
Boltzmann factors. Through inspecting a plot of observable value, such as the impurity
susceptibility, against the number of sweeps, an estimate of this initial transient can be
deduced. In practice, a single simulation consists of 6*105 equilibrium sweeps followed by
106 sweeps. For observables which can be calculated quickly such as the LOP and charge
density (see below) measurements are separated by five sweeps. For observables which
take longer to calculate, such as impurity susceptibilities, measurements are separated by
103 sweeps. If the number of sweeps between measurements is too small, consecutive
measurements will be statistically correlated and will not alter the observables average
leading to biased results. Thus, a considerable amount of computer time is spent on
calculating observables that do not contribute significantly. In contrast, if the number of
sweeps between measurements is too large, many uncorrelated measurements will be
omitted from the average. Due to the extended nature of the quasi-particle lattice distortion
and the local updating QMC procedure we found it necessary to perform several
simulations for each parameter set and take an average of the results. For lattice sizes N=70
and less, forty independent simulations were averaged to give the final results whilst eighty
were averaged for the larger lattice simulations; each with different initial random number
seeds and initial fermion configurations. Our computational resources were limited to a
maximum of 80 CPUs at any one instant. A typical simulation run time for an N = 60 site
47
chain with 4 magnetic impurities at an inverse temperature of β = 200 was approximately
one month.
The following quantities will be referred to throughout the remainder of this chapter
[100]: lattice order parameter (LOP), net charge density (CDi) on lattice site i and net spin
density (SDi) on lattice site i. The LOP is a measure of the magnitude and phase of the
lattice distortion and for a perfectly dimerised chain it would be uniform and finite. It is
defined as follows:
_
LOPi 
yi  yi
 1i
1 Lb
1 Lb
yi   xi 1  j    xi  j   xi 1  xi
Lb j 1
Lb j 1
_
yi 
(40)
1
 yi 1  2 yi  yi 1  .
4
The other observables are defined respectively as:
CDi  1  ni  1 
1
2L f
2Lf
 n ( j)
i
j 1
(41)
SDi  ni   ni  
1
2L f
48
n
2Lf
j 1
i
( j )  ni  ( j )

4.3 Room-temperature bipolaron results
We begin by removing one spin-up and one spin-down conduction electron from the half
filled band.
We set the inverse temperature to β = 40 eV-1 corresponding to room
temperature and work with a lattice of N = 100 sites. Under such circumstances, according
to polaron/bipolaron theory [60, 62], either two polarons or a single bipolaron should be
formed.
We show in Fig. 14 that the stable configuration is indeed the hole bipolaron
when there are no magnetic impurities involved. This is evidenced by the appearance of a
single dip in the LOP. The single dip is a consequence of both holes sharing the same
lattice distortion in order to minimise the system energy. In contrast, two polarons would
have been manifest by two distinct dips, one for each hole. Furthermore, this bipolaron is
robust enough to withstand both a single classical impurity and a pair of classical impurities
with J = -1.4eV and θ = 0 located near chain centre. It should be stated here that all values
of impurity exchange integral J selected throughout this work are of a suitable order of
magnitude for magnetic impurities in metals [139]. Although the bipolaron is stable in the
presence of such magnetic entities its shape is modified by their presence.
The net spin distribution in Fig. 14 (c) is consistent with the negative
(ferromagnetic) exchange integral value selected. The negative spin distribution in between
the two impurities is a reflection of the Pauli Exclusion Principle: as the impurities favour
a net up spin in this case, the electron hopping can only occur if there is an unoccupied
spin-up level on the atom in between.
49
0.1
(a)
0.09
Fig. 14: Variation of LOP, CD and SD with
0.08
lattice site for β = 40 eV-1 (a) LOP (b) Charge
LOP (Å)
0.07
density CD (c) Spin density SD. No impurity
0.06
(○), single classical impurity on 50th site with J
0.05
= -1.4eV θ = 0 (▲) and two classical impurities
0.04
on 50th and 52nd sites (▀). One spin-up and one
0.03
spin-down electron removed from half filled
band. The bipolaron is the stable entity in all
0.02
0
20
40
60
80
100
site i
cases although modified by the impurity
presence.
0.06
0.04
0.02
0
-0.02
CD
(b)
0.06
0.04
0.02
0
-0.02
With the aim of gaining confidence in
our results we have also performed
impurity-free
0.06
0.04
0.02
0
-0.02
degenerate
chain
simulations with the same parameters
as Takahashi [102]. The comparison
0
20
40
60
80
100
was extremely good.
In addition we
site i
have measured the statistical error in
0.12
(c)
the LOP to be  0.01 Ǻ. Whilst this
0.1
0.08
may seem extremely large we must
0.06
emphasise that this is a maximum error
SD
0.04
estimate.
0.02
We have treated each
0
independent simulation as a single
-0.02
measurement and thus the denominator
-0.04
-0.06
-0.08
0
20
40
60
80
of Eqn. (14) is of order
40 .
reality,
run-times
for
the
large
In
100
site i
performed, we expect the number of
independent measurements to be much larger resulting in a much smaller error. An attempt
was made to calculate the LOP autocorrelation function which would have given a better
50
estimate of the true error. Unfortunately the function was not exponentially decaying and a
good measure of the autocorrelation time could not be deduced. In hindsight, a statistical
binning analysis should have been employed.
0.1
LOP(Å)
(a)
0.09
Fig. 15: β = 40 eV-1 (a) LOP and (b) Spin density.
0.08
Single classical impurity at 50th site with J = -
0.07
5.4eV. θ = 0 (■), θ = π / 6 (▲), π / 3 (○) and π / 2
0.06
(×).
0.05
removed from half filled band. The LOP shows a
0.04
discontinuity at the impurity location. The spin
0.03
density varies consistently with varying classical
0.02
impurity angle θ.
One spin-up and one spin-down electron
0.01
0
0
20
40
60
80
100
site i
In order to study the effect of 
0.4
on the bipolaron stability and to test the
(b)
algorithm, we set the exchange integral to
0.3
J = -5.4eV for a single impurity on the
0.2
SD
50th site.
0.1
The resulting LOP and spin
distributions can be seen in Fig. 15 for
different angles. We observe that  has a
0
minimal effect on the LOP. Whilst there
-0.1
is still an attraction between holes, as
-0.2
0
20
40
60
80
100
evidenced by a non-negative LOP, there
site i
is now a discontinuity at the impurity site
due to the large value of the exchange integral J. This discontinuity will essentially
increase the effective mass of the bipolaron as a high bipolaron mobility relies on a slowly
varying LOP. When the classical impurity spin is aligned with the z-axis i.e.  =0 the
impurity acts as a localised magnetic field. The conduction electrons arrange themselves so
as to align with this localised field as can be seen from Fig. 15 (b). As the impurity spin is
rotated, the second term in Eqn. (35) (corresponding to a conduction electron spin flip)
increases reaching a maximum at    / 2 . In contrast the first term in Eqn. (35), which
51
breaks the spin-up spin-down degeneracy, becomes zero. The fact that neither spin is
favoured over the other in this example is clearly evident in Fig. 15(b) as expected. The
spin distributions for intermediate angles support this argument.
Through treating the impurity spins as classical objects we have thus introduced a
coarse-grained approximation which assumes the limit of large localised spins and makes
an analytical approach tractable [140].
Its validity in the present context requires a
comparison with a fully quantum treatment of the impurities. However, even if it is a bad
approximation (as we shall show) it nonetheless acts as a stepping stone to the more
complicated quantum impurity algorithm. This is what prompted this preliminary
investigation.
To allow a comparison with the semi-classical approach and to test the bipolaron
stability in the presence of quantum impurities we again remove one spin-up and one spindown conduction electrons from the half-filled band. We set the inverse temperature to β =
40 eV-1 corresponding to room temperature and work with a lattice of N = 100 sites as
before. Two quantum impurities are placed at the 50th and 52nd sites with the resulting LOP
and charge density displayed in Fig. 16.
We also include the same impurity-free
distributions to allow a straight comparison.
In contrast to the classical impurity case, it can be seen that the hole bipolaron is
unstable with respect to the formation of two asymmetric hole polarons for exchangeintegral values spanning a whole order of magnitude. In addition to the existence of two
polarons, there is also a noticeable dip in the LOP around the impurity sites. This is a result
of the interaction between the impurity and polaron spins.
52
0.1
(a)
0.09
LOP (Å)
0.08
0.07
0.06
0.05
0.04
0.03
0
20
40
60
80
100
60
80
100
site i
(b)
0.04
0.02
0
-0.02
CD
0.04
0.02
0
-0.02
0.04
0.02
0
-0.02
0
20
40
site i
Fig. 16: β = 40 eV-1 (a) LOP and (b) Charge density CD. J = 0 (♦), J = -0.14eV (○), J = -1.4eV (x). Quantum
impurities placed on 50th and 52nd sites. One spin-up and one spin-down electron removed from half-filled
band. A pair of quantum impurities restrict the hole bipolaron from forming over a whole order of magnitude
of impurity exchange integral J. Instead, two asymmetric hole polarons are centred at approximately the 20th
and 80th sites. The dip in LOP at chain centre arises from the polaron spins interacting with the impurity
spins.
53
By reducing the number of atoms in the chain and still removing two electrons from
the half-filled band we are essentially increasing the charge doping level. Fig. 17 shows the
LOP and charge density when we have reduced the chain length to 70 sites. Two quantum
impurity atoms are then located adjacent to the 34th and 36th sites. In this case, the hole
bipolaron is clearly the stable entity for all selected values of J . This is an extremely
important result as it shows that bipolarons can be stabilised by increasing the effective
charge doping level.
0.1
(a)
0.09
0.08
LOP (Å)
0.07
0.06
0.05
0.04
0.03
0.02
0
10
20
30
40
50
60
70
60
70
site i
(b) 0.06
CD
0.04
0.02
0
-0.02
0.06
0.04
0.02
0
-0.02
0.06
0.04
0.02
0
-0.02
0
20
40
site i
Fig. 17: (a) LOP and (b) Charge density CD. J = -1.0eV (○), J = -0.44eV (▲), J = -0.14eV (■). Impurities
placed on 34th and 36th sites. One spin-up and one spin-down electron removed from half-filled band. The
bipolaron is stable in all cases around the chain centre.
54
As we are ultimately interested in the engineering of coherence between bipolarons, it
is a natural progression to further remove one more spin-up and one more spin-down
electron. Thus in total four conduction electrons are removed from the half-filled band.
We again place two next-nearest-neighbour quantum impurities near the chain centre and
set the exchange integral J = - 0.14eV. We perform such a study on three different chain
lengths (doping level): namely N = 80 (5 %), N = 70 (5.71 %) and N = 50 (8 %).
From Fig. 18 it can be seen that two distinct bipolarons are formed for the N=50 and
N=70 chains. In contrast the N = 80 case gives three dips in the LOP which can be
explained as the overlap of a bipolaron and two polarons. In this case there are two
degenerate chain configurations. One configuration where the bipolaron is positioned on
the left of the chain (polarons at centre and right) and one where it is on the right (polarons
at centre and left). The bipolaron is never located at chain centre because of the reduction
in system energy when a polaron spin interacts with the impurity electrons.
Thus the presented LOP in Fig. 18(a) is the superposition of LOP’s for the two
degenerate configurations. The charge density in the top and middle frames of Fig. 18(b)
shows a total net charge of 2e for each half of the chain (bipolaron) as expected. The
bottom frame gives a non-integer net charge sum for each of the three corresponding lattice
distortions which is consistent with the interpretation of the LOP.
55
0.1
(a)
0.08
LOP (Å)
0.06
0.04
0.02
0
-0.02
(b)
0
10
20
30
40
site i
50
60
70
80
0.2
0.1
CD
0
0.2
0.1
0
0.2
0.1
0
0
20
40
site i
60
80
Fig. 18: (a) LOP and (b) Charge density CD. J = -0.14 eV. 5% doping (N=80) impurities on 40th and 42nd
sites (○), 5.71% doping (N=70) impurities on 34th and 36th sites (♦), 8% doping (N=50) impurities on 24th and
26th sites (+). Two spin-up and two spin-down electrons removed from half filled band. Two hole bipolarons
are formed on the N=50 and N=70 chains. A bipolaron and two hole polarons are formed on the N=80 chain.
So far we have shown that when two electrons are removed from a N = 100 site
chain of neutral cis-PA at room temperature, the stable configuration is the hole bipolaron.
This bipolaron is also stable in the presence of a pair of next-nearest-neighbour classical
magnetic impurities with exchange integral J = - 1.35eV, albeit modified by the impurities
presence. In contrast, by replacing the classical impurities with their quantum counterparts,
56
the bipolaron is no longer a stable entity. Instead, two asymmetric hole polarons are the
stable quasi-particles. The large discrepancies in these results cast severe doubt over the
validity of the classical spin approximation in this model.
Furthermore, we have
demonstrated that by holding the number of holes constant and reducing the chain length
(effectively increasing the charge doping level), hole bipolarons can be stabilised in the
presence of quantum magnetic impurities within certain parameter regimes.
57
Chapter 5
Search For A Superconducting Phase Transition
5.1 Introduction
In the previous chapter we showed that bipolarons could exist at room temperature in the
presence of magnetic impurities within certain regimes. Here we wish to explore the
possible implications. Motivated by the search for superconductivity we wish to study the
system’s behaviour as a function of temperature.
If the system resistivity could be calculated a jump from finite to zero resistance
would be evident if the material was cooled through its superconducting critical
temperature Tc.
However a calculation of the resistance within the current QMC
framework is non-trivial due to the problem of analytic continuation. As an alternative we
can resort to the following. As the normal-superconducting transition is known to be
second order, according to Landau’s definition of phase transitions, thermodynamic
quantities which are defined as a second derivative of free energy should display a
discontinuity at Tc in the thermodynamic limit.
As our system is far from the
thermodynamic limit we can only expect, at least in principle, a smoothed over
discontinuity as opposed to a sharp one. This is the aim of this chapter.
The magnetic susceptibility χ is defined as follows:

M  2F

H H 2
(42)
where M is magnetisation, H is applied magnetic field and F is the free energy of the
system.
58
The uniform impurity susceptibility χu and staggered impurity susceptibility χs
can also be written in terms of the fluctuations in magnetisation and staggered
magnetisation respectively [141]:
u 
1 
q 2

    .....    


0
z1
zq
z1
 .....  z q


   z1  .....  z q

2
(43)
s
1 
q 2
          .....    





0
z1
z2
   z1   z 2   z 3 .....  z q
z3
zq
z1
  z 2   z 3 .....  z q

2
where  z1 is the z-component of the first impurity spin, q is the number of impurities in the
system and  is the fermionic imaginary time step used in the QMC calculations. Eqn.
(43) is the definition used in all impurity susceptibility calculations presented in this work.
We choose to calculate the impurity magnetic susceptibility as supposed to the total
susceptibility as our QMC algorithm models the canonical ensemble with an equal number
of spin-up and spin-down electrons. Hence the fluctuation in total magnetisation and
therefore the total magnetic susceptibility would be zero.
From eqn. (43) it can be deduced that the larger the ferromagnetic
fluctuations the larger the uniform susceptibility, and the larger the anti-ferromagnetic
fluctuations the larger the staggered susceptibility. If
 u   s it means that the impurity
spins are fluctuating but in such a way as to preserve any ferromagnetic ordering, e.g. two
spin-up impurities fluctuate into two spin-down and so on. In contrast if  u   s the
impurity spins are fluctuating in such a way as to preserve any anti-ferromagnetic ordering.
If the impurity spins are fluctuating independently then  u   s .
In what follows we are essentially interested in finding out if the bipolaron-impurity
interplay results in a second-order phase transition as the temperature is lowered.
However, as T→ 0 the computational difficulties become virtually intractable for our
59
cluster of processors. As the inverse temperature β increases, so does the number of time
slices for a fixed imaginary time step entailing increasingly long simulation times. In
addition the number of impurities and length of polymer chain are restricted due to CPU
limitations.
5.2 Susceptibility v temperature results
The following results are for systems with two stable hole bipolarons. That the
bipolarons are stable, for the range of temperatures 58 ≤ T ≤ 292K considered in this study,
has been verified through LOP analysis similar to the one done before.
We begin by presenting the case of two magnetic impurities placed at the 34th and
36th sites of a N = 70 chain.
The resulting impurity susceptibility and staggered
susceptibility are plotted in Fig. 19. It can be seen from Fig. 19 that in general, for the J = 0.14eV impurity strength, the staggered susceptibility is always greater than the uniform
susceptibility. This can be attributed to collective anti-ferromagnetic impurity fluctuations
mediated by a RKKY [142] type indirect exchange interaction. The origin of this antiferromagnetic fluctuation enhancement must be of indirect nature as there are no direct
impurity-impurity interaction terms in the Hamiltonian.
The trend for the J = -0.8eV case is similar although at temperature 145K both
susceptibilities are convergent within statistical error. This convergence is evidence of the
Kondo effect in the current system [64], namely: the conduction electrons which act as
mediators between impurities are more localised around the impurity sites at low
temperature. Thus the impurities can no longer communicate and neither ferromagnetic nor
anti-ferromagnetic fluctuations are favoured, i.e. the impurities fluctuate independently of
each other. Furthermore, all susceptibilities in Fig. 19 can be seen to vary by a relatively
minimal amount over the wide range of temperatures studied. There is certainly no distinct
feature, within statistical errors, that would signify a phase transition. Qualitatively, both
uniform and staggered susceptibilities increase with increased exchange integral over the
whole temperature range.
60
6
Two Impurity Susceptibility
5
4
3
2
1
0
140
160
180
200
220
240
Temperature (K)
260
280
300
Fig. 19: Two impurity susceptibilities when impurities placed on sites 34 and 36 of N = 70 chain. Two hole
bipolarons are the stable quasi-particles. Staggered impurity susceptibility
impurity susceptibility
 u - continuous lines. □:–
s
- dashed lines. Uniform
impurity exchange integral J = -0.14eV.
o:- J = -
0.8eV. Lines are used only as a guide to the eye.
Next we added two more impurities into the system giving a total of four. In order for the
bipolarons to remain stable on the N = 70 chain, for all studied J, it was necessary to
position these four impurities at greater separations than before. Through trial and error, an
impurity separation of six sites is sufficient for bipolaron stability.
The impurity
susceptibility results are shown in Fig. 20.
All susceptibility curves in Fig. 20 are relatively constant over the whole
temperature range with no apparent signature of a phase transition. However, in contrast to
the two-impurity case, the four-impurity staggered and uniform susceptibilities are of
similar magnitude over the temperature interval. This could indicate that any indirect
exchange interaction contributing to the next-nearest-neighbour staggered susceptibility
enhancement (Fig. 19) is short-ranged in the current models. This is consistent with the
61
weak impurity-conduction electron coupling results from the two impurity Kondo work by
Fye and Hirsch [143].
4
Four Impurity Susceptibility
3.5
3
2.5
2
1.5
1
0.5
0
140
160
180
200
220
240
Temperature (K)
260
280
300
Fig. 20: Four impurity susceptibilities when impurities placed on sites 28, 34, 40 and 46 of N = 70 chain.
Two hole bipolarons are the stable quasi-particles. Staggered impurity susceptibility
Uniform impurity susceptibility
s
- dashed lines.
 u - continuous lines. □:– impurity exchange integral J = -0.14eV.
o:– J = -0.8eV. Lines are used only as a guide to the eye.
To probe further into the lower temperature regime the length of the chain was
reduced to N = 60. An advantage of this decreased chain length was that two bipolarons
were stabilised in the presence of four next-nearest-neighbour impurities for all J and
temperature studied.
The details of stabilising bipolarons by reducing chain length,
increasing charge doping, are explained in the previous chapter. The results are displayed
in Fig. 21.
62
4.5
Four Impurity Susceptibility
4
3.5
3
2.5
2
1.5
1
0.5
0
50
100
150
200
Temperature (K)
250
300
Fig. 21: Four impurity susceptibilities when impurities placed on sites 32, 34, 36 and 38 of N = 60 chain.
Two hole bipolarons are the stable quasi-particles. Staggered impurity susceptibility
Uniform impurity susceptibility
s
- dashed lines.
 u - continuous lines. □:– impurity exchange integral J = -0.14eV.
o:– J = -0.8eV. Lines are used only as a guide to the eye.
In this case, for both values of J, the staggered susceptibility is greater than the
uniform susceptibility at 290 K. This is consistent with an indirect exchange mediated
enhancement of the collective anti-ferromagnetic fluctuations of next-nearest-neighbour
impurities as in Fig. 19.
At lower temperatures the behaviour differs depending on
impurity strength. It is noticeable that for the stronger coupling J = -0.8eV case the Kondo
effect leads to equal uniform and staggered susceptibilities at 145K and below. In contrast
for the weaker coupling J = -0.14eV case the Kondo effect does not emerge until around
58K.
In summary, this chapter started with the premise that we are working with chains
of two stable hole bipolarons interacting with magnetic impurities. We have shown that,
63
for the exchange integral values selected, anti-ferromagnetic fluctuations can exist
alongside bipolarons within the current model at finite temperature.
However, these
collective fluctuations are short-ranged and for them to be achieved the magnetic atoms
must be placed on next-nearest-neighbour lattice sites. Furthermore, the material should be
well above its Kondo temperature.
64
Chapter 6
Conclusions
In this thesis we have performed a theoretical investigation into the effects of implanting
magnetic atoms into cis-polyacetylene. The model Hamiltonian used involves electronelectron Coulomb, electron kinetic energy, electron-phonon coupling and lattice energy
terms. The magnetic impurities have been modelled through the inclusion of a Kondo term
which couples conduction electrons with those localised on the impurity atoms. To gain
insight into such systems we have employed a modified worldline quantum Monte Carlo
algorithm in order to derive finite temperature properties. In what follows we summarise
our main findings and discuss possible continuations of the present research.
It has been known for several years that pairs of charges introduced into nondegenerate conjugated polymers, such as cis-polyacetylene, are bound together because of
the electron-phonon interaction.
This work has initially investigated the effect of having
spin-½ impurity atoms located in close proximity to such charge pairs/bipolarons. We have
found that although the impurity atoms can hinder bipolaron formation there are parameter
regimes in which these bipolarons and magnetic atoms can co-exist. We have found that a
key ingredient to bipolaron stability is the effective charge doping level or, equivalently,
the polymer chain length. A decrease in chain length has been demonstrated as a useful
tool in stabilising bipolarons. Calculation of the lattice order parameter (LOP)/staggered
lattice distortion and net charge distribution have allowed for a direct bipolaron stability
analysis. This analysis was performed for several values of Kondo impurity exchange
integral and chain length at room temperature.
We then continued by investigating the effect of lowering the system’s temperature.
Ensuring that two hole bipolarons were stable on the polymer chain we proceeded to
calculate the impurity uniform and staggered magnetic susceptibilities. In such systems,
short-ranged anti-ferromagnetic impurity spin fluctuations are observed for next-nearestneighbour impurities as long as the system is well above its Kondo temperature. If the
temperature is too low the Kondo effect restricts the conduction electrons from mediating
65
such collective fluctuations. In contrast, impurities placed a distance of six lattice sites
apart show no favoured uniform or staggered susceptibility at all temperatures and impurity
exchange values studied.
This is consistent with the magnetic impurities fluctuating
independently.
For all parameter regimes studied, no phase transition has been observed in the
impurity susceptibility calculations. This could be because of our finite system size being
far from the thermodynamic limit. Alternatively, it is likely that the magnetic impurity
strength would need to be just right in order to give the correct energy spin fluctuations to
mediate superconductivity.
Given that we have investigated only a small number of
discrete impurity strengths there is a large scope for further studies. Another possibility is
that our parameterisation for cis-PA is not quite right and in particular the on-site Coulomb
repulsion is underestimated. It would be interesting to investigate the effect on the system
of increasing this repulsion towards a Mott-insulator state which is known to be important
in the vicinity of the cuprate superconducting phase.
In addition it would have been useful to calculate different physical properties in
order to probe for a phase transition. A calculation of the current-current correlation
function, in principle, could have been used to calculate the systems resistance via the Kubo
formula. However due to the problem of analytically continuing the QMC data from
imaginary frequency to real frequency an accurate calculation of the resistance would have
proven troublesome.
Even though our work has not found a superconducting phase transition it could
have implications for superconductivity in such materials. In summary we have shown
that:
1.
Real space hole pairs can exist at room temperature in the presence of magnetic
impurities.
2.
Collective anti-ferromagnetic impurity spin fluctuations can occur.
66
It should be stated that the present work has been restricted by certain technical
problems. Firstly, simulation times are extremely long even though a large number of
processors have been deployed in order to reduce statistical errors. This has limited the
investigations to at most two bipolarons. Larger chain lengths allowing more bipolarons
were not feasible within our computational resources and timescale. In hindsight a nonlocal update loop worldline quantum Monte-Carlo algorithm [117] may have reduced
algorithm run time. However the benefits of such algorithms are usually only reaped when
simulating the grand canonical ensemble [118]. Thus several runs are needed for each
parameter set to ensure that the chemical potential gives exactly the desired hole doping
level. Secondly, the minus sign problem increases with the number of impurities [144] thus
again increasing simulation time to get a desired statistical uncertainty. The present thesis
thus focuses on systems with at most four impurities. Finally, our investigations are
restricted to relatively high temperatures ~ 58K.
Given that magnetically mediated
superconductivity in general occurs at < 3K in heavy fermion compounds it is perhaps not
surprising that we have not observed a phase transition. In saying this, such materials
charge pairs are bound through magnetic means [20]. In contrast in our current system the
charge pairing is electron-phonon mediated and it is only coherence that we seek from the
magnetic sources. In addition we have found that if the polymer is too cold, the Kondo
effect will dominate and magnetic fluctuations suppressed.
Further theoretical
investigations into magnetically doped conjugated polymers should address the above
issues in order to continue the search for superconductivity in these organic compounds.
From an experimental point of view what clues can be gained from this thesis into
how to coax a conjugated polymer into a superconducting state? These can be summarised
as follows:
1.
The charge doping level should be large in order to stabilise bipolarons in the
presence of impurities. Of course it cannot be too large otherwise disorder
will prevail and a metallic state entered [127].
2.
If magnetic fluctuations are important for superconductivity the impurities
should be located relatively close to each other, typically next-nearestneighbour apart. In addition, by contrast with conventional superconductors,
67
the polymer should most likely be at a relatively high temperature otherwise
the Kondo effect will dominate and hinder the possibility of magnetically
mediated coherence.
Although this thesis has not found evidence of superconductivity in conjugated
polymers, it has however clearly demonstrated that both charge-pairs and a possible
magnetic coherence mechanism can coexist at finite temperatures.
68
Appendix A
Calculation of Matrix Elements Involving a Classical Magnetic Impurity
In this Appendix we present the numerical details for the calculation of the matrix elements
involving the classical impurity Kondo term. After a change in either worldline (spin-up or
spin-down) is made, the QMC algorithm requires the calculation of the ratio of affected
matrix elements to determine whether or not to accept the new fermionic configuration. If
the worldline change affects the electron occupation on the impurity site then the following
matrix
elements
ni , ni 1, ni , ni 1, e
site version of H
fermio n

must
  H i fermion
,i 1

be
n'i , n'i 1, n'i , n'i 1,
calculated
. Where H i ,fermion
is the twoi 1
given by Eqn. 33 with H Kondo replaced with H KondoClassical from
Eqn. 34. The first step in calculating this matrix element is to apply the Hamiltonian to


each possible fermionic state i.e. H i ,fermion
n ' i , n ' i 1, n ' i , n ' i 1, . As the Hamiltonian under
i 1
consideration conserves particle number we can treat the different particle number
scenarios separately. We will now present the matrix element derivation for the single,
two-particle, three-particle and four-particle cases in turn. The single-particle Hamiltonian
can be written in matrix form as
1000
0100
0010
0001
1000
C
t
S
0
Hˆ 1 particle  0100
0010
t
S
0
0
0
C
0
t
0
0
t
0
0001
69
where C 
J
cos  , J is the exchange integral between conduction electrons on the
4
impurity site and the classical impurity,  is the angle between the classical impurity spin
direction and the z-direction, t is the hopping integral and S 
J
sin  . Now that we have
4
the Hamiltonian in matrix form we can attempt to calculate the desired matrix element. We
can write the following equation.

1



0
exp(  Hˆ 1 particle)  exp(  t 1 ) exp   S 2  exp    C 
0




0

0

1
Where  1  
0

0

1
0
0
0
0
0
0
1
0

0
  13   15   12 n 1

1

0 
And  12   14   12 n  I
Also note that  22   24   22 n
1

0

0

0

0
0
0
0
0
0
1
0
0

0
0

0 
And that
 2   23   22 n 1
0

0

1

0

0
0
0
0
1
0
0
0
0

0
0

0 
70
0 0 0

0 0 0
2
 O  


0 1 0

0 0 0  



The error O  
2
 is due to the Trotter-Suzuki decomposition [145] being applied once.
The Trotter-Suzuki decomposition is needed in this case because the three exponential
operators do not commute.
Now we wish to evaluate each of the three exponentials involved in the product in
turn. The first exponential can be written as the following approximate series which tends
to the exact exponential as the number of terms tends to infinity.
exp  t 1   I   t 1 
1
 t 2  12  1  t 3  13 .........................................
2!
3!


1

1
1
3
2
4
exp  t 1   I   t   t   ....... 1    t    t   ........ 12
3!
4!


 2!



 1

1
1
3
2
4
exp  t 1    t   t   ....... 1  1   t    t   ........ 12
3!
4!


 2!

exp  t 1   sinh  t  1  cosh  t  12
Finally we can write
0

1
exp  t 1   sinh  t 
0

0

1
0
0
0
0
0
0
1
0

0
 cosh  t  I
1

0 
We now evaluate the second exponential operator in the same way.
71
exp   S 2   I   S 2 
1
 S 2  22  1  S 3  23 .........................................
2!
3!


1

1
1
3
2
4
exp   S 2   I   S   S   ....... 2    S    S   ........ 22
3!
4!


 2!

0

0
exp   S 2    22  
0

0

0
1
0
0
0

0
1

1
2
4
 sinh  S  2    S    S   ........ 22

0
4!
 2!



1
0
0
0
0
0

0
exp   S 2   cosh  S  22  sinh  S  2  
0

0

0 0 0

1 0 0
0 0 0

0 0 1 
For the second exponential operator we get
1

0
exp   S 2   cosh  S 
0

0

0
0
0
0
0
0
1
0
0
0


0
0
 sinh(  S )

0
1


0
0 

For the full single-particle operator we can finally write
72
0
0
0
0
1
0
0
0
0  0
 
0  0

0  0
 
0   0
0
1
0
0
0
0
0
0
0

0
0

1 

exp   Hˆ 1 particle


1 0


cosh  S   0 0
0 0



0 0



0


1

 sinh  t  

0


0




1 0 0
1




0 0 0

0




cosh


t
I
*
exp



C

0

0 0 1




0

0 1 0 



0 0
0 0


0 0
0 0
 sinh(  S )

1 0
1 0




0 0
0 0
1 0 0
 
0 0 0

0 0 0
 
0 0   0
0

0 0 0
*
0 1 0

0 0 0  
0
0
0 0 0 

1 0 0 
0 0 0 

0 0 1 
Now applying this operator to the four possible single-particle states gives the desired
matrix elements as

exp   Hˆ 1 particle



J


sinh  t  cosh   4 sin   0100




J


 sinh  t sinh   sin   0001
4
J


1000  exp(  cos ) 

4
J


 cosh  t  cosh   sin   1000
4




J


 cosh  t sinh   sin   0010
4




exp   Hˆ 1 particle 0100  sinh(  t ) 1000  cosh  t  0100
73
















exp   Hˆ 1 particle

J


sinh  t  cosh   4 sin   0001




J


 sinh  t  sinh   sin   0100
4
J


0010  exp ( cos ) 

4
J


 cosh  t  cosh   sin   0010
4




J


 cosh  t  sinh   sin   1000
4















exp   Hˆ 1 particle 0001  sinh(  t ) 0010  cosh  t  0001
We now turn our attention to the two-particle matrix element derivation for sites involving
a classical impurity. We start by writing the two-particle Hamiltonian in matrix form
giving
0110
0110
1010
1100
1001
0101
0011
 C V
t
S
0
t
0
1010
ˆ
H 2 particle  1100
t
U
0
t
0
0
S
0
V C
0
0
0
1001
0101
0
t
t
0
0
0
C V
t
t
0
S
0
0011
0
0
0
S
0
V C
where V is the nearest neighbour Coulomb interaction and U is the on-site Coulomb
interaction. We can then write the following equation
74

exp   Hˆ 2 particle


V  C  


 

 U  

V  C  

I 
 exp  t 3  exp  t 4  exp   S 5  exp  S 6  exp   

V  C  

 0  


 
V  C  


 


 O  
2

where
 32   34   32 n
1

0
0

0
0

0

 42   44   42 n
1

0
0

0
0

0

0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0

0
0

0
0 
0 
 52   54   52 n
0

0
0

0
0

0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0

0
0

0
0 
1 
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0

0
0

0
0 
0 
75
 3   33   32 n1
0

1
0

0
0

0

1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0

0
0

0
0 
0 
 4   43   42 n1
0

0
0

0
1

0

0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0

0
0

0
0 
0 
 5   53   52 n1
0

0
0

0
0

0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0

0
0

1
0 
0 
 62   64   62 n
1

0
0

0
0

0

0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0 
0 
 6   63   62 n1
0

0
1

0
0

0

0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0 
0 
The Trotter-Suzuki decomposition is applied in this case as  3 and  4 do not commute
with  5 and  6 .
We now deal with each of the exponential terms in turn.
exp  t 3   I   t 3 
1
 t  2  32  1  t 3 33 .........................................
2!
3!
1

1
2
4
exp  t 3   I  sinh  t  3    t    t   ........ 32
4!
 2!

0

0
0
 1

1
2
4
exp  t 3   sinh  t  3  1   t    t   ........ 32  
4!
0
 2!

0

0

0

0
0
exp  t 3   sinh  t  3  cosh  t  32  
0
0

0

76
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0 
1 
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0 
1 
Using the exact same derivation we also obtain
0

0
0
exp  t 4   sinh  t  4  cosh  t  42  
0
0

0

0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0 
1 
For the third exponential we have
 S   53  ........
1
2
exp   S 5   I   S 5   S   52 
2!
3!
3
exp   S 5    sinh  S  5  I 
1
 S  2  52  .............
2!
1

0
0
2
exp   S 5    sinh  S  5  cosh  S  5  
0
0

0

And similarly for the fourth exponential,
77
0 0 0 0 0

1 0 0 0 0
0 1 0 0 0

0 0 0 0 0
0 0 0 1 0 
0 0 0 0 0 
0

0
0
exp  S 6   sinh  S  6  cosh  S  62  
0
0

0

0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0

0
0

0
0 
1 
Finally we can write for the two-particle operator


0 1 0 0 0 0
1




1
0
0
0
0
0


0



0

0 0 0 0 0 0







sinh


t

cosh


t

0 0 0 0 1 0
0

0 0 0 1 0 0
0





0 0 0 0 0 0
0





 0 0 0 0 0 0




 0 0 0 0 0 0

 0 0 1 0 0 0




 0 0 0 0 0 0

 0 0 0 0 0 0



0 0 0 0 0 1




exp   Hˆ 2 particle


0


0

0

sinh  t 
0

1



0


0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1


0
0

0
0
  cosh  t 
0
0

0
0

0
0 

78
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0  0
 
0  0
0  0

0  0
0   0
0   0
0 0 0 0 0 

1 0 0 0 0 
0 0 0 0 0 

0 0 1 0 0 
0 0 0 1 0 

0 0 0 0 0 
*









0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 

0 
0 
 *
0 
0 

1 

0


0

0

 sinh  S  
0

0



0


0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0


0
0

0
0
  cosh  S 
1
0

0
0

0
0 

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0

0


0

1

* sinh  S 
0

0



0


0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1


0
0

0
0
  cosh  S 
0
0

0
0

0
0 

0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 1
 
0  0
0  0

0  0
0   0
1   0
0  0
 
0  0
0  0

0  0
0   0
0   0
0
1
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0 

0 
0 
 *
0 
0 

0 
0 

0 
0 

0 
0 

1 

V  C  

 

U

 



V C 
) I 
* exp(  
V  C  

 0  




V  C  


 
Applying the two-particle operator to all possible two-particle states leads to the desired
matrix elements,



J


exp   Hˆ 2 particle 0110  exp    V  cos   *
4





J


2
sinh  t  cosh   4 sin   1001







J


 sinh  t  cosh  t  cosh   sin   0101 
4






J


 sinh  t  cosh  t  cosh   sin   1010 
4






J


2
 cosh  t  cosh   sin   0110

4






J


 sinh   sin   1100 

4




79
sinh 2  t  0101  cosh  t sinh  t  1001
ˆ
exp   H 2 particle 1010  exp   U 
2
 sinh  t  cosh  t  0110  cosh  t  1010





J


exp   Hˆ 2 particle 1100  exp    V  cos   *
4





J


2
sinh  t sinh   4 sin   1001







J


 sinh  t  cosh  t sinh   sin   1010 
4






J


 sinh  t  cosh  t sinh   sin   0101 
4






J


2
 cosh  t sinh   sin   0110

4






J


 cosh   sin   1100

4









J


exp   Hˆ 2 particle 1001  exp    V  cos   *
4





J


2
cosh  t  cosh   4 sin   1001







J


 sinh  t  cosh  t  cosh   sin   0101 
4






J


 sinh  t  cosh  t  cosh   sin   1010 
4






J


2
 sinh  t  cosh   sin   0110

4






J


 sinh   sin   0011 

4




exp   Hˆ 2 particle 0101  cosh 2  t  0101  sinh  t  cosh  t  1001
 cosh  t sinh  t  0110  sinh
80
2
 t  1010






J


exp   Hˆ 2 particle 0011  exp    V  cos   *
4





J


cosh   4 sin   0011







J


2
 sinh   sin   cosh  t  1001

4






J


 sinh   sin   sinh  t  cosh  t  0101 
4






J


 sinh   sin   sinh  t  cosh  t  1010 
4






J


 sinh 2  t sinh   sin   0110

4




We now turn our attention to the three-particle case and the corresponding matrix element
derivation. We follow the usual procedure and write the three-particle Hamiltonian in
matrix form.
1110
ˆ
H 3 particle  0111
1011
1101
1110
0111
1011
1101
U  2V
0
0
t
0
0
t
2V  C
t
t
U  2V
S
S
0
2V  C
0
81
We can then write the following equation

exp   Hˆ 3 particle


U  2V  


 

 2V  C  
2
 exp  t 7  exp  S 8  exp    
I   O  

U  2V


 
 2V  C  


 



Where
 7   73   72 n1
0

0

0

1

0
0
1
0
0
1
0
0
1

0
0

0 
 8   83   82 n1
0

0

0

0

0
0
0
1
0
0
0
0
0

1
0

0 
 72   74   72 n  I
 82   84   82 n
Firstly we deal with the first exponential.


exp  t 7  I   t 7 
1
 t  2  72  1  t  3  73 ............
2!
3!
1

1
2
4
exp  t 7   I  sinh  t  7    t    t   ........  72
4!
 2!

exp  t 7   sinh  t  7  cosh  t  I
82
0

0

0

0

0
1
0
0
0
0
0
0
0

0
0

1 
Secondly we deal with the second exponential.

0



0
exp  S 8   exp   S 
0


0



0 0 0

0 0 1
0 0 0

1 0 0  
 S   83  ........
1
2
exp  S 8   I   S 8   S   82 
2!
3!
3
1

0
exp  S 8   sinh  S  8   82  
0

0

0 0 0

0 0 0 1
1
2
4
  S   82   S   84  .......

0 1 0 2!
4!

0 0 0 
1

0
exp  S 8   sinh  S  8  cosh  S  82  
0

0

0 0 0

0 0 0
0 1 0

0 0 0 
Hence we can finally write the three-particle operator as
83

exp   Hˆ 3 particle

0


sinh  S  0
0



0




0


0
 sinh  t 

0


1



0 0 1


0 1 0
 cosh  t  I  *

1 0 0



0 0 0

0 0 0
0


0 0 1
0



cosh


S
0
0 0 0


0
1 0 0 

0 0 0 1
 
1 0 0 0

0 0 0 0
 
0 0 1   0
0 0 0 

0 0 0 
*
0 1 0 

0 0 0 

U  2V  


 

 2V  C  
exp    
I
U  2V  


 
 2V  C  


 

And now applying this operator to all possible three-particle states gives the desired threeparticle matrix elements.





J


cosh  t cosh   4 sin   0111 






J


 sinh  t cosh   sin   1011 
4

J





0111  exp     2V  cos    

4
J





  cosh  t sinh  sin   1101 




4






J


 sinh  t sinh   sin   1110 
4




exp   Hˆ 3 particle 1110  exp   U  2V cosh  t  1110  sinh  t  1101 
exp   Hˆ 3 particle



exp   Hˆ 3 particle 1011  exp   U  2V  cosh  t  1011  sinh  t  0111 
84

exp   Hˆ 3 particle



J


cosh  t cosh   4 sin   1101 






J


 sinh  t cosh   sin   1110 
4

J





1101  exp     2V  cos   

4
J

 



 cosh  t sinh   sin   0111 
4






J


 sinh  t sinh   sin   1011 
4




The Hamiltonian for the four-particle case is already diagonal and hence the corresponding
matrix element is trivially given by


exp   Hˆ 4 particle 1111  exp   U  4V  1111
85
Appendix B
Calculation of Matrix Elements Involving Quantum Impurities
In this Appendix we wish to present the full derivation of matrix elements involving
quantum impurities. This derivation is similar in detail to the case of classical impurity
matrix elements although the break-up of the Hamiltonian is different due to the extended
Hilbert space. A trial change in fermionic worldliness requires the calculation of two-site
matrix elements defined by
ni , ni 1, ni , ni 1, n f , n f , e

  H i fermion
, i 1

n ' i , n ' i 1, n ' i , n ' i 1, n ' f , n ' f ,
.
where H i ,fermion
is the two-site version of H fermion given by Eqn. 33. We have also imposed
i 1
the Kondo condition throughout that n f ,  n f ,  1. The first step in calculating such a
matrix element is to write the Hamiltonian in matrix form. As the Hamiltonian conserves
particle number we can treat the different total particle Hamiltonians individually.
We begin with the two-particle Hamiltonian which can be written as
010010
001010
000110
100001
010001
001001
000101
t
0
0
0
0
0
0
010010
100010
J
4
t
0
0
0
0
0
0
0
0
0
0
0
t
0
0
0
0
J
2
0
J

4
t
0
0
J
4
t
0
100010
001010
0
0
Hˆ 2 particle  000110
0
0
100001
0
0
010001
0
0
0
J

4
t
J
2
0
001001
0
0
0
0
0
0
000101
0
0
0
0
0
0
t
0
0
86
t
0
We can then write the following equation:



J


2
exp   Hˆ 2 particle  exp   t  1  exp    2  exp     3   O  
2 

0

1
0

0
where  1  
0
0

0
0


1 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 1 0 0 0 0`

0 1 0 0 0 0 0
  13   15   12 n1
0 0 0 0 1 0 0 
0 0 0 1 0 0 0

0 0 0 0 0 0 1
0 0 0 0 0 1 0 
And  12   14   12 n  I
 J 


 4 
 0 


 J
 
 4
 0 
I
2  
 J
 4 


 0 


 J 
 4 


 0 


 32   34   32 n
0

0
0

0

0
0

0
0

0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 1 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 1 0 0 0 
0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0 
 3   33   32 n1
0

0
0

0

0
0

0
0

0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 1 0 0 0

0 0 0 0 0 0 0
0 1 0 0 0 0 0 
0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0 
87
We now need to evaluate the exponential operators that are non-diagonal, namely those
involving  1 and  3 . We can write
exp  t 1   I   t 1 
1
 t  2  12  1  t  3  13 ............
2!
3!
1

1
2
4
exp  t 1   I  sinh  t  1    t    t   ........ 12
4!
 2!

exp  t 1   sinh  t  1  cosh  t  I
And in addition,
3
J 3

    3
2
J 
J
1
J
2

exp     3   I    3      32  
 ........
2 
2
2! 
2
3!

2
J 
J
1
J


exp     3    sinh    3  I      32  .............
2 
2
2! 
2


1

0
0

J 
J
J  2 0



exp     3    sinh     3  cosh    3  
2 
2
2



0
0

0
0

Finally we can write the full two-particle operator:
88
0 0 0 0 0 0 0

1 0 0 0 0 0 0
0 0 0 0 0 0 0

0 0 1 0 0 0 0
0 0 0 0 0 0 0 
0 0 0 0 1 0 0

0 0 0 0 0 1 0
0 0 0 0 0 0 1 

exp   Hˆ 2 particle



 J  


 

4


 


 0  



 


 J  


   


 4  


 0  


 I 
 sinh  t  1  cosh  t  I  * exp   

J

 


  4  



 


 0  



 

J


 




4  


 

0


 


 



1 0 0 0


0 1 0 0

0 0 0 0



0 0 0 1
J
J




*  sinh    3  cosh    32  

2
2


0 0 0 0

0 0 0 0



0 0 0 0

0 0 0 0


0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0 

0 
0 

0 
0 

0 

0 

1 
We then apply this two-particle operator to each possible two-particle state in turn to give
the desired matrix elements as follows:




J

exp   Hˆ 2 particle 100010  exp    sinh  t  010010  cosh  t  100010
4

exp   Hˆ 2 particle 010010  sinh  t  100010  cosh  t  010010
89




exp   Hˆ 2 particle 000110  sinh  t  001010  cosh  t  000110



J

sinh  t  cosh   2  000110




J

 sinh  t sinh    010001
2
J


001010  exp    
4 
J


 cosh  t  cosh    001010
2



J

 cosh  t sinh    100001
2






J

sinh  t cosh   2  010001 






J

 sinh  t sinh    000110 
2
J



100001  exp    


4
J



 cosh  t cosh    100001 
2





J


 cosh  t sinh    001010 
2










exp   Hˆ 2 particle
exp   Hˆ 2 particle












exp   Hˆ 2 particle 010001  sinh  t  100001  cosh  t  010001 
J

exp   Hˆ 2 particle 001001  exp    sinh  t  000101  cosh  t  001001 
4

exp   Hˆ 2 particle 000101  sinh  t  001001  cosh  t  000101 
90
We now turn our attention to the three-particle scenario. Following the same procedure as
above, we write the three-particle Hamiltonian in matrix form:
Hˆ 3 particle 
011010
101010
110010
011010
J
 V
4
t
101010
110010
t
U
0
J
V
4
0
100110
010110
001110
011001
101001
0
0
t
0
0
0
0
t
0
0
0
0
0
0
J
V
4
t
t
0
0
J
 V
4
100110
0
t
0
010110
t
0
0
001110
0
0
0
0
100101
010101
001101
0
0
0
0
110001
J

2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
J
2
0
0
0
0
0
t
0
t
0
0
0
0
0
t
0
0
0
J
V
4
0
0
0
t
0
0
0
0
0
0
0
0
J
2
0
0
0
0
0
J
V
4
t
0
0
0
0
0
0
0
100101
0
0
0
0
0
010101
0
0
0
0
001101
0
0
0
0
011001
101001
110001

U

0
J
V
4
0
t
0
0
J
2
0
t
0
0
0
0
0
0
0

J
V
4
t
0
0
We can then write the following equation:



J
J




2
exp   Hˆ 3 particle  exp   t  4  exp  t  5  exp     6  exp    7  exp    8   O  
2
2





Where
 4   43   42 n1
0

1
0

0
0

0

0
0

0
0

0
0

1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
91
0

0
0

0
0 
0

0
0

0
0

0
0 
 42   44   42 n
1
 
1
 0
 
1
1
 
 0
  I
1
1
 
 0
1
 
1
 0
 
 5   53   52 n1
 6   63   62 n1
0

0
0

0
1

0

0
0

0
0

0
0

0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
0

0

0
0

0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
92
0

0
0

0
0 
0

0
0

0
0

0
0 
0

0
0

0
0 
0

0
0

0
0

0
0 
 52   54   52 n
1
 
1
0
 
1
1
 
0
  I
1
1
 
0
1
 
1
0
 
 62   64   62 n
 7   73   72 n1
 0
 
 0
 0
 
 0
 0
 
1
  I
 0
 0
 
 0
1
 
 0
 0
 
0

0
0

0
0

0

0
0

1
0

0
0

 72   74   72 n
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
93
0

0
0

0
0 
0

0
0

0
0

0
0 
1
 
 0
 0
 
 0
 0
 
 0
  I
 0
 0
 
1
 0
 
 0
 0
 
 J

 V 
 4

 U 


 J

 V 
 4

 J V 
 4



 0 
 J

 V 

8   4
 J V 
 4



 U 


 J V 
 4

 J

 V 
 4

 0 


 J

 V 
 4

In following exactly the same steps as the two-particle scenario for breaking up the
exponentials we obtain the full three-particle operator. It can be written as
94

exp   Hˆ 3 particle


 0 
  

 0 

1 

  

 0 

 0 

  

1 

 sinh  t  4  cosh  t  42    I  *
 0 

 0 

  

1 

 0 

  

 0 

1 

  


 0  
1 
   
  

 0  
1 



1 

1  
   
  

 0  
1 



1 

 
 0  
  

1  

  J 
  J  2  0  
2
sinh  t  5  cosh  t  5   0  I  *  sinh  2  6  cosh  2  6   1  I  *




   
  



1 
0  



  

 
1  
1 



 0 

0  
   
  

 0  
1 



1 

 1   
  


 0 
  

1 

1 

  

1 

1 

  


  J 
  J  2  1  
sinh  2  7  cosh  2  7   1  I  * exp    8  I 




  

1 

  

 0 

1 

  

1 

1 

  

95
Now applying the full three-particle operator to all possible three-particle states gives the
matrix elements required.



 J

exp   Hˆ 3 particle 011010  exp       V   *
 4




J

sinh   2  110001 







J


2
cosh  t  cosh    011010 

2





J

cosh  t sinh  t cosh    010110  
2





J

cosh  t sinh  t cosh    101010  
2





J


2
sinh  t cosh    100110

2



cosh 2  t  101010  cosh  t sinh  t  100110  
ˆ
exp   H 3 particle 101010  exp   U 

2
cosh  t sinh  t  011010  sinh  t  010110 





J

exp   Hˆ 3 particle 110010  exp      V   110010
4





J

exp   Hˆ 3 particle 100110  exp      V   *
4


cosh 2  t  100110  cosh  t sinh  t  101010  


2
cosh  t sinh  t  010110  sinh  t  011010 
cosh 2  t  010110  cosh  t sinh  t  011010  
ˆ
exp   H 3 particle 010110  

2
cosh  t sinh  t  100110  sinh  t  101010 


96

exp   Hˆ 3 particle

exp   Hˆ 3 particle



J

2
 cosh  t sinh   2  100101 







J

cosh  t sinh  t sinh    101001  
2





J
 J
 

001110  exp       V   cosh  t sinh  t sinh    010101 
2
 4
 





J


2
sinh  t sinh    011001 

2





J


cosh    001110

2




cosh 2  t  011001 




J
  cosh  t sinh  t  010101  
011001  exp      V   

4
  cosh  t sinh  t  101001  

sinh 2  t  100101



cosh 2  t  101001  cosh  t sinh  t  100101
ˆ
exp   H 3 particle 101001  exp   U  
2
cosh  t sinh  t  011001  sinh  t  010101



exp   Hˆ 3 particle






J

2
cosh  t sinh   2  011010 







J

cosh  t sinh  t sinh    010110  
2







J
J




110001  exp       V   cosh  t sinh  t sinh    101010  
2
 4
 





J

2
sinh  t sinh    100110 

2





J

cosh    110001

2



97

exp   Hˆ 3 particle



J

 sinh   2  001110 







J

2
cosh  t  cosh    100101 

2





J
 J
 

100101  exp       V   cosh  t sinh  t  cosh    101001  
2
 4
 





J

cosh  t sinh  t  cosh    010101  
2





J

2
sinh  t  cosh    011001

2



cosh 2  t  010101  cosh  t sinh  t  011001  
exp   Hˆ 3 particle 010101  

2
cosh  t sinh  t  100101  sinh  t  101001 





J

exp   Hˆ 3 particle 001101  exp      V   001101
4


We finally derive the four-particle matrix elements involving a quantum impurity site. Our
starting point is to write the four-particle Hamiltonian in matrix form.
98
111010
111010
U  2V
011110
0
101110
0
011110
0
J
  2V
4
t
101110
0
110110
t
111001
0
011101
0
101101
0
t
0
0
0
0
U  2V
0
0
0
110101
0
J

2
0
Hˆ 4 particle  110110
111001
t
0
0
0
0
0
0
J
 2V
4
0
0
0
0
0
U  2V
011101
0
0
0
0
0
101101
0
0
0
0
110101
0
0
J

2
0
J
 2V
4
t
0
t
t
0
U  2V
0
0
J
  2V
4
0
0
t
0
which can be written as


 U  2V 


  J  2V 
 4



 U  2V 


 J  2V 
J
I
Hˆ 4 particle  t  9   10   4


2
 U  2V 
 J

  2V 
 4

 U  2V 


 J

   2V 
 4

where
 9   93   92 n1
0

0
0

1

0
0

0
0

0 0 1 0 0 0 0

0 1 0 0 0 0 0
1 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 1 
0 0 0 0 0 1 0

0 0 0 0 1 0 0
0 0 0 1 0 0 0 
99
 92   94   92 n  I
 10   103   102 n1
 102   104   102 n
0

0
0

0

0
0

0
0

0

0
0

0

0
0

0
0

0 0 0 0 0 0 0

0 0 0 0 0 0 1
0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0 
0 0 0 0 0 0 0

0 0 0 0 0 0 0
1 0 0 0 0 0 0 
0 0 0 0 0 0 0

1 0 0 0 0 0 0
0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0 
0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 1 
By following the usual breakup procedure we can write the four-particle operator as
100


exp   Hˆ 4 particle  sinh  t  9  cosh  t  I  *

1 0 0 0


0 0 0 0

0 0 1 0



sinh   J   cosh   J  2   0 0 0 1

 10

 10 

 2 
 2 
0 0 0 0

0 0 0 0



0 0 0 0

0 0 0 0


0 0 0 0 

0 0 0 0 
0 0 0 0 

0 0 0 0 
*
1 0 0 0 

0 1 0 0 

0 0 1 0 

0 0 0 0 



 

 U  2V  



 

J



  2V  


 
4



 


 U  2V  



 


 J  2V  

 
exp     4




U  2V  





 J
 


  2V  


 4
 


 U  2V  



 


 J
 



2
V


 

 4
 


Then applying this operator to every possible four-particle state gives the matrix elements.


exp   Hˆ 4 particle 111010  exp   U  2V  cosh  t  111010  sinh  t  110110

exp   Hˆ 4 particle




J

cosh  t  cosh   2  011110  






J

cosh  t sinh    110101  
2


 J


011110  exp       2V   


4
J





sinh  t  cosh    101110  
2





J

sinh  t sinh    111001

2



101










exp   Hˆ 4 particle 101110  exp   U  2V cosh  t  101110  sinh  t  011110


J

exp   Hˆ 4 particle 110110  exp      2V  cosh  t  110110  sinh  t  111010
4



exp   Hˆ 4 particle 111001  exp   U  2V cosh  t  111001  sinh  t  110101 

J

exp   Hˆ 4 particle 011101  exp      2V   cosh  t  011101  sinh  t  101101 
4


exp   Hˆ 4 particle 101101  exp   U  2V cosh  t  101101  sinh  t  011101 

exp   Hˆ 4 particle



J

cosh  t cosh   2  110101  






J


cosh  t sinh    011110  
2


 J


110101  exp       2V   

J
 4
 


sinh  t cosh    111001  
2





J

sinh  t sinh    101110 
2



102
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