Download Novel Results for Condensed Matter Systems with Time Reversal Symmetry

Novel Results for Condensed Matter Systems With Time
Reversal Symmetry
Alexander A Baytin
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Frederick D Haldane
June, 2009
c Copyright 2009 by Alexander A Baytin.
°
All rights reserved.
Abstract
The first half of the Thesis is dedicated to the study of the Spin Hall Effect. Contrary to
Quantum Hall Effect that requires broken time inversion symmetry, the Spin Hall current
may exist in “ordinary” systems due to antisymmetric behavior of spin under time inversion.
Obviously such current may only exist when the system couples spatial coordinates of
electrons with their spin coordinates, which naturally leads to investigation of how the spin
orbit coupling may lead to existence of the Spin Hall Current.
First we present a method of computing Spin Hall Current based on Streda formula. We
then show an elegant derivation of the absence of Spin Hall Current in conductors with SOC
of Rashba type. Next we show in detail how the spin current emerges in semiconductor
systems, providing intuitive explanation for its Z2 nature by looking at the edge states.
We then demonstrate by a direct numerical computation that it is possible to distinguish
between the Spin Hall insulator and an ordinary insulator by comparing their response to
an adiabatic pump of a magnetic flux into the system.
The second half of the Thesis is dedicated to studying the effects of interactions that
lead to formation of superconducting state in metallic systems that are too small to be
considered superconductors. This happens when the single energy level spacing becomes
comparable to the bulk superconducting gap. Even though such small systems do not carry
superconducting current they exhibit peculiar correlation effects in the crossover region.
Two different approaches are used to tackle exact solutions of superconducting (or pairing) Hamiltonian and to compute quantities of interest, such as superconducting gap, excitation energies and parity effects. Since the exact solution are in fact systems of equations
iii
that cannot be solved analytically, we focus on its various expansions. The first approach
amounts to a systematic expansion of exact solution in the inverse values of coupling constant. The second approach is an expansion of solutions in the inverse number of electron
pairs. Having these expansions allows getting intuition for all the regimes of the pairing
Hamiltonian.
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Acknowledgements
I would like to start with expressing deepest gratitude to my advisor Duncan Haldane. For
all the time he invested in me, sharing his knowledge and wisdom. For his endless patience
and for making this happen.
I thank my parents for making me understand importance of education from an early
age and for helping me not to stray from this road. And for everything else.
I thank my wife Alexandra for all her support, patience and care.
I am very grateful to my undergraduate advisor Alexander Andreev for igniting my
interest in Condensed Matter Physics and for his guidance and support.
I thank Boris Altshuler and Emil Yuzbashyan for all the good times we had when working
on mesoscopic superconductors. I am very grateful to Rosario Fazio and Luigi Amico for
organizing my visit to the University of Catania that I have immensely enjoyed. I would
like to kindly thank Chiara Nappi for all the help.
And last but not nearly the least I thank my friends for making the years spent in
Princeton so great : Pedro Goldbaum, Dmitry and Tania Gordeev, Subroto Mukerjee,
Alexey Makarov, Akakii Melikidze, Sergey Nadtochiy, Julia Pachos, Srinivas Raghu, Kumar
Raman, Slava Rychkov, Dmitry Sarkisov, Michael Shefter and Alexei Tchouvikov.
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Contents
Abstract
iii
Acknowledgements
v
Contents
vi
1 Introduction
1
1.1
Spin Hall Effect in Conductors and Topological Insulators . . . . . . . . . .
1
1.2
Exact results for BCS Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Spin Hall Effect
9
2.1
Introduction to Spin Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Streda Formula and Spin Hall Effect . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Spin Hall Insulators and Adiabatic Z2 Pump . . . . . . . . . . . . . . . . .
18
2.3.1
Berry Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.2
Topological Classification of Spinless Electronic Bands . . . . . . . .
21
2.3.3
Quantum Hall Effect in the absence of magnetic field - a case of broken
Time Reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3.4
Quantum Spin Hall Effect in Graphene . . . . . . . . . . . . . . . .
32
2.3.5
Edge States, Z2 Nature of Insulating States and Spin Currents . . .
34
2.3.6
Z2 Pump and Topological Insulators . . . . . . . . . . . . . . . . . .
39
vi
2.4
Appendix - Numerical Method For Finding Edge States . . . . . . . . . . .
3 Strong Coupling Expansion of BCS Hamiltonian
43
45
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.2
The strong coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.3
The Strong Coupling Expansion
. . . . . . . . . . . . . . . . . . . . . . . .
54
3.3.1
The ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3.2
Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.5
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4 Large N Expansion of BCS Hamiltonian
66
4.1
Review of Richardson’s 1/m expansion . . . . . . . . . . . . . . . . . . . . .
69
4.2
Ground state energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.3
Comparison to previous studies . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.4
Excitation energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.5
Matveev-Larkin parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.7
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.8
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.9
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.10 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
References
87
vii
Chapter 1
Introduction
1.1
Spin Hall Effect in Conductors and Topological Insulators
It is a common belief that most exciting effects in condensed matter physics are associated with breaking of one symmetry or another. Notable examples are considered to be
superfluidity, which results from spontaneous breaking of particle number conservation and
superconductivity which results from spontaneous breaking of charge conservation. One of
the most studied effects in modern condensed matter physics - the Quantum Hall Effect
(QHE) is a nontrivial response of a system to an applied magnetic field which fundamentally stems from a fact that time inversion symmetry is broken. QHE may exist even when
there is zero net magnetic flux through the system. Nevertheless, in this work we consider
several systems that present considerable interest even though no fundamental symmetries
are broken (with possible exception of inversion symmetry). The first are two dimensional
conductors and insulators with spin orbit coupling. Such systems have received much attention in the past several years in connection with the Spin Hall Effect. In a Spin Hall
system a spin current flows orthogonal to the applied electric field. Contrary to a regular
Hall effect, such spin current response can exist without breaking of time reversal symmetry.
Apart from the fact that existence of such novel effect is highly interesting conceptually,
1
2
being able to control spin currents turned out to be a very important problem for the applied field of spintronics [1]. Much of early research on Spin Hall Systems was focused on
two dimensional conductors with spin orbit interaction of Rashba type. In early works it
was found that the spin hall conductance has a universal value of
e
2π
which turned out to
be quite a controversial result, as it was later found that accurately taking into account
all relevant terms of perturbation expansion in Kubo formula the Spin Hall conductance
vanishes. This controversy has been amplified by the fact that in the presence of spin orbit
interaction the total spin of the system is not conserved and therefore the spin current is
not a well defined quantity. In the second chapter we describe an elegant method based on
the Streda formula to derive the Hall conductance and show that it indeed vanishes for a
system with Rashba interaction. The vanishing of spin hall conductance turned out to be a
non-universal result, as it was found in [2] that in a system with different symmetries and
spin orbit interactions the value of hall conductance can be non zero.
An attention to insulators with spin orbit interactions was drawn by works of Kane
and Mele [3, 4] where it was shown that under circumstances such insulators may exhibit
non-dissipative Spin Hall effect. Moreover, they showed that system could be driven into
spin hall state by tuning strengths of spin orbit interaction, thus leading to a discovery of
an interesting new class of the so called Spin Hall Insulators. The authors were motivated
by original work of Haldane on zero-field Hall Effect [5]. In that pioneering paper Haldane
considered electrons in graphene with time inversion symmetry being broken by means of
staggered magnetic flux. The total magnetic flux through the system was zero and the system possessed translation symmetry. The system has two Bloch bands, each band carrying
nontrivial winding number and therefore exhibiting Hall effect, as follows from results of
Thouless et al [6]. Kane and Mele made an observation that for electrons on honeycomb
lattice the spin orbit interaction that preserves the component of spin perpendicular to the
plane is equivalent to two systems of spinless electrons, first system corresponding to up
spins and the other corresponds to down spins. The systems experience opposite in signs
staggered magnetic fields with net zero flux thus making them equivalent to models studied
3
by Haldane in [5]. Total charge Hall current would therefore cancel for such a system but
the spin currents would add up, as the bands carry opposite spins, thus exhibiting a robust
non-dissipative Spin Hall Effect.
Since the existence of Spin Hall current in insulators has been shown to have a deep
connection with that of a charge Hall Effect, an essential question of existence of an invariant
characterizing the Spin Hall State has emerged. In the same paper [3] Kane and Mele
observed that contrary to the ordinary Hall Effect, total Chern number of all occupied
bands in an insulator is not representative of a spin hall insulator for a one simple reason
that it is identically zero. This fundamental result is due is due to a general property
of Chern number being antisymmetric under time reversal operation. In the absence of
time reversal breaking each band would have its Kramers partner with an opposite Chern
number, and therefore the total Chern number of all occupied bands should vanish.
At first sight, in the case of the model considered by Kane and Mele a quantity that could
indicate the existence of spin hall effect could be a spin - weighted sum of Chern numbers
of the occupied bands. Nevertheless such characterization would not generic enough, since
for an arbitrary spin orbit interaction none of the spin components are conserved. Yet, it
was numerically observed that the edge states characteristic for Spin Hall effect persist even
when one turns on the interaction of Rashba type which breaks the conservation of spin
component. Therefore such naive generalization of Chern invariant to spin bands must be
discarded.
In [4] it was observed that the fundamental difference between topological and ordinary
insulating phases can be found when looking at the states localized at the edge of the
system. Any state adiabatically connected to the trivial insulator has an even number of
Kramers pairs at the edge while the topological insulator has an odd number of such states.
Such distinction implies that there exists a Z2 - type invariant which differentiates between
the phases. It was found that the number of edge pairs is related to the number of zeroes of
the Pfaffian of electron Bloch functions in the momentum space. The authors also provided
an invariant which counts number of zeroes and thus provided a characterization of the
4
insulating phases.
Later in [7] Fu and Kane by considering a toy one dimensional model with spin orbit
interaction and time reversal symmetry demonstrated that it is possible to pump spin and
charge into the system by means of an adiabatic cycle. They related the possibility of such
adiabatic pump to the existence of the Z2 invariant for topological insulators by placing
the insulator on the cylinder and performing the thought experiment done originally by
Laughlin [8] in the context of the Quantum Hall Effect.
Eventually the topological arguments leading to existence of invariants for topological
insulators similar to the arguments given in original TKNN [6] paper were presented by
Moore and Balents [9]. In original TKNN [6] paper it was argued that in the absence
of time reversal symmetry the Brillouin zone is equivalent to a torus and therefore every
Hamiltonian for non-interacting electrons can be described as a continuous mapping of a
torus to a space of hermitian matrices. Such mappings then could be classified by standard homotopy arguments leading to existence of integer invariants which could then be
ultimately related to the Hall conductance of the system. For topological insulators the
existence of time reversal symmetry entails Kramers degeneracies and therefore original
TKNN arguments can not be applied due to degeneracies. Essentially, a whole Brillouin
zone is too large of an object to consider for such mappings since every point on a Brillouin
zone has its Kramers partner - a point with an opposite momentum. This lead Moore
and Balents to considering continuous mappings from the Effective Brillouin Zone which
is essentially a half Brillouin zone glued such that Kramers degeneracies are accounted for.
Topological analysis of all such mappings allowed to rigorously demonstrate existence of Z2
invariants in two dimensional systems by purely homotopic arguments. For more details we
refer the reader to the original paper [9].
1.2
Exact results for BCS Hamiltonian
Since it has been discovered in 1911 by Kammerlingh Onnes, superconductivity has become
one of the most studied phenomena in condensed matter physics. Microscopic explanation
5
of superconductivity by Bardeen, Cooper and Schrieffer is one of the landmark achievements
of theoretical physics. The key notion in the BCS theory is that of pairing of electronic states
related by Kramers symmetry. Namely, the famous BCS wavefunction of superconductor
ground state is written in terms of pair creation operator
|ΨG >=
Y
(1 + ∆k Dk† )|vac >,
Dk† = a†k a†−k
(1.1)
k
Important thing about wavefunction [1.1] is that it explicitly breaks particle number conservation for a Hamiltonian which actually conserves number of particles. This feature of BCS
wavefunction, which later become known as spontaneous symmetry breaking, has turned to
be responsible for plethora of exciting effects in theoretical physics.
Breaking of particle number conservation can be simply seen as a tool to elegantly
derive properties of superconductors and it holds strictly only when the number of particles
is infinite and therefore is valid only for macroscopic systems. On the other hand it is well
known that in the realm of mesoscopic physics effects associated with finite system size
become crucial for explaining system behavior, such as in the case of conductivity in the
regime of Coulomb Blockade [10].
The fact that superconductivity is associated with breaking of particle number conservation hints at the idea that in mesoscopic systems superconductivity can quite different
from that of bulk systems. In this work we focus on extreme case of the so-called zerodimensional mesoscopic system, also known as artificial atoms. These systems are so small
that any spatial dynamics leads to excitation energies that are comparable to or much larger
than other characteristic energy scales present. As a result such systems exhibit peculiar
response to an applied gate voltage, as discussed below.
It has been noticed by Anderson as early as in 1958 [11] that some kind of transition
from superconductor to insulator must happen when the energy spacing due to system
size becomes comparable with bulk superconducting gap ∆, since creating superconducting
excitation will be too expensive. At the time the issue raised by Anderson presented purely
theoretical interest, until in mid 1990s Ralph, Black and Tinkham succeeded in producing
ultrasmall Al grains with radii of order of 5nm [12]. Using methods of single electron
6
tunneling spectroscopy authors were able to extract discrete excitation spectrum of Al
grains. Namely, the setup of experiment is shown in Figure 1.1. The grain was attached via
oxide tunnel barriers to two leads, thus forming a typical single electron transistor setup
[10]. Generally, conductance of such transistor shows peaks at values of gate voltage that
correspond to energy levels of the system. Thus it is possible to obtain the excitation
energies from the patterns of conductance peaks as long as the peaks are well resolved.
Figure 1.1: a) shows design of Al single electron transistor, used in RBT experiments. b)
shows equivalent the transistor equivalent circuit.
Even though ultrasmall grains do not exhibit superconducting current, being zero dimensional ”by construction”, it was found that when the bulk gap is larger than level
spacing nontrivial correlations exist between electrons which leads to interesting effects,
such as a parity effect. Namely, a grain with an even number of electrons had a distinct
spectroscopic gap, larger than level spacing while an odd grain did not have such a large
gap. This was a clear indication of pair correlations in the grains and has been studied
using self consistent BCS-like theory in [13, 14]. Still, a rigorous analysis of the spectra was
missing due to many-body nature of the problem.
One of the most peculiar developments in the study of ultrasmall superconducting grains
was the fact that the hamiltonian of discrete electron levels used for describing grains in
fact has an exact solution, originally discovered by Richardson [15] and independently by
Gaudin [16]. This fact was pointed out to condensed matter community by Richardson
himself, who published his work in the context of nuclear physics, using the discrete level
model to describe parity effects in nuclei. Having obtained the exact solution, Richardson
7
explored its properties in a series of papers [15]-[17]. The fact that existence of such solution
has evaded attention of condensed matter community was quite remarkable, especially given
that it was invaluable for studying the region of grains where bulk gap was comparable to the
level spacing and therefore no small parameters could be introduced. Once the existence
of exact solution became known, a number of works emerged where it was interpreted
in the light of modern theory of quantum integrable systems. Namely it was shown that
connections exist between Richardson’s exact solution and Bethe Ansatz [18, 19], Conformal
Field Theory [20]-[21] and Chern Simons Theory [22].
1.3
Summary of Thesis
In Chapter 2 we present our results on Spin Hall Effect. In Section 2.1 we give an introduction to the physics of Spin Hall effect in conductors with spin orbit interactions and
point out contradictions involved when trying to compute spin hall conductivity. In the
Section 2.2 we show an elegant way to compute Spin Hall conductivity by using Streda
formula similar to the way it was used to compute the Quantum Hall Effect. We show that
in the two dimensional system with Rashba-type interaction the Spin Hall Effect vanishes,
thus providing an intuitive explanation to the absence of Spin Current in the system with
Rashba interaction and can also be applied to tackle more complicated system, such as
three dimensional systems with spin orbit interaction considered in [23].
Section 2.3 is dedicated to the physics of topological insulators. First we give a brief
introduction into the topic of topological classification of Quantum Hall Insulators. We show
how nontrivial winding numbers of electron Berry phases result in existence of Quantum
Hall Effect through Kubo formula, as was originally shown by Thouless et al [6]. We
then describe how same results can be explained in the framework of homotopy theory
which is a powerful tool for classification of topological phases of quantum systems. After
informal introduction of relevant results of homotopy theory we show, following [24] how two
dimensional spinless conductors can be classified, in accordance with results by Thouless et
al. We then show a specific example of Hamiltonian with broken time reversal and inversion
8
symmetries, originally considered by Haldane [5], where transition to Hall Insulator state
can be achieved by tuning parameters of symmetry breaking terms.
We then show, following pioneering works by Kane and Mele that Spin Hall Effect can
exist for system electrons in graphene with spin orbit interaction by mapping such system
onto two copies of Haldane’s model. If the electrons spin along the axis perpendicular to
the graphene plane is conserved, the system exhibits quantized spin hall conductance. On
the other hand, in a general system the spin hall conductance will not be conserved, but
will be protected against disorder by Kramers symmetry.
We then present our main result on the Spin Hall Insulators which describes a physically
observable method for probing topological insulators. Namely, we show that by pumping
a quantum of orbital magnetic flux into the topological insulator results in a simultaneous
pump of a unit charge into the system at the location of the flux injection. We explain
relation of this approach to the common method of analyzing the edge states in cylinder
geometry and show explicit numerical calculations that demonstrate the effect of the charge
pump.
Chapters 3 and 4 are dedicated to various exact results on BCS Hamiltonian. This work
has been done in collaboration with Boris Altshuler and Emil Yuzbashyan and results have
been published in [25, 26].
In Chapter 3 we show how to use Richardson’s exact solution of BCS Hamiltonian
to perform systematic expansion of Hamiltonian’s ground state and excitation energies in
the limit of strong coupling constant. In Chapter 4 we use Richardson’s exact solution to
perform expansion of BCS Hamiltonian energies in the limit of large number of particles.
Starting from original work by Richardson [17] where the connection between the exact
solution and the “classical” solution by Bardeen et al [27] is shown. We then proceed to
obtain the finite size corrections to that solution by expanding the Richardson’s solution in
the inverse number of particles.
Chapter 2
Spin Hall Effect
2.1
Introduction to Spin Hall Effect
In a Spin Hall system a spin current flows in response to an applied electric field. Contrary
to Quantum Hall Effect and Anomalous Hall Effect where dissipation-less electric current
exists as a consequence of broken time reversal symmetry [28], Spin Hall effect may exist in
a system where time reversal symmetry is conserved. This will be explicitly demonstrated
in the section 2.3.4.
It is also intuitively clear that emergence of spin current due to applied electric field
should be a consequence of spin orbit interactions in the system, as there must exist a way
of transferring the perturbations in spatial dynamics of electrons due to applied electric
field into dynamics of its spin. In other words, if electron spacial and spin coordinates were
not coupled there would be no resulting spin dynamics, except possibly for dynamics purely
in spin space, caused by spin-spin interactions.
This simple reasoning motivated a vast body of research aimed at studying two and three
dimensional electron systems with spin orbit interaction. One of the most studied models
in two dimensions was a model of electrons with Rashba spin orbit interaction [29]. The
Rashba spin orbit interaction is typical for electron systems where the two dimensional state
is obtained by applying electric field to the two-dimensional systems. The two dimensional
9
10
system obtained that way is shown in Figure 2.1. The applied perpendicular electric field
induces spin orbit interactions which can be described by an effective Hamiltonian
H ef f = ²(k − kR ẑ × s)
(2.1)
where ²(k) is the band Hamiltonian and s is the electron spin. Expanding Hamiltonian near
its minimum
k2
2m
produces the Rashba spin orbit interaction
H=
k2
+ 2λR k × s
2m
(2.2)
kR
. Experimentally, Rashba spin-orbit coupling strength can be varied over
where λR = − 4m
a wide range by tuning a gate field, with typical values being of order of 0.1²F /kF [30, 31].
The energy levels of this system are given by
(0)
ε± =
p2
∓ λR |p|
2m
(2.3)
Figure 2.1: Energy in Rashba model. Rashba coupling produces system with two
Fermi momenta pF ± . There exist singly occupied states for pF − < p < pF + where spin is
perpendicular to the electron momentum.
11
In one of the early papers on Spin Hall Effect [23] a classical argument was given to
obtain a non-zero Spin Hall conductivity for the model [2.2]. Assuming that EF > 0 there
are two Fermi surfaces with Fermi momenta pF ± where 0 < pF − < pF + and
pF + − pF − = 2m|λR |
(2.4)
Figure 2.1 shows energies in the p space. Spin direction is perpendicular to velocity, and
therefore tangential to the constant energy surfaces (which can be seen directly from the
Hamiltonian [2.2]). When p < pF − both states are occupied and their total spin vanishes.
The states with pF − < p < pF + are singly occupied and individually carry non-zero spin.
The spin density is obtained by summing over all individual states and therefore vanishes
due to inversion symmetry p → −p. When p > pF + states are not occupied and the spin
vanishes trivially. When an electric field accelerates the Fermi sea so it is no longer centered
at p = 0, the total spin density no longer cancels, and the spin density grows as the Fermi
sea is boosted, corresponding to an intrinsic 2D torque density
τ=
eλR m
(−Ey , Ex , 0)
4πh̄2
(2.5)
which conserves the ẑ component of spin normal to the 2D plane. Summing up all the states
the non-zero conductivity σ =
e
8π
was obtained. This intrinsic value was also obtained by
direct computation using Kubo formula and Landauer-Buttiker formalism (see discussion
and references in [32]). Also, in [33, 34] effects of disorder were taken into account and the
same universal value for Spin Hall conductivity was produced in the clean limit. This value
for intrinsic Spin Hall conductivity was considered to be agreed upon until it was realized
that not all vertex corrections were previously accounted for and the surprising result of
vanishing of Spin Hall conductivity in the clean limit was obtained [35, 36, 37].
Here we provide a more intuitive way of computing intrinsic spin hall conductivity that
avoids using disorder averaging techniques. The method is an extension of the Streda
formula [38] that was originally used to compute Quantum Hall effect. First we describe
the original method and then use a simple calculation to show that the intrinsic Spin Hall
conductivity indeed vanishes for the Rashba Hamiltonian.
12
2.2
Streda Formula and Spin Hall Effect
As it has been discussed in the previous section one of the ambiguities in determining the
spin current is that in the presence of spin orbit interactions the spin is not conserved. Here
we propose a a general ambiguity-free method for calculating the intrinsic dissipationless
current response of a system to an applied electric field, which is known to give correct
results in the cases of the Anomalous Hall Effect [28] and Quantum Hall effect [39] and
appears to be generally consistent even in the absence of a local conservation law.
The method is based on applying the Streda formula [2.8] which relates the linear response of a medium to a uniform electric field to its linear response to a uniform magnetic
flux density, with the chemical potential µ held fixed. This method allows elegant derivation of zero Spin Hall conductivity previously done in [35, 36, 37] using straightforward
perturbation expansion.
In [38] Streda showed that the current response of a two dimensional quantum system
to the in-plane electric field E
xy y
σH
E
(2.6)
xy x
J y = − σH
E
(2.7)
Jx =
is given by
µ
xy
σH
=
∂ρ
∂B
¶
(2.8)
µ
Originally this formula was applied by Streda to investigate Quantum Hall conductivity.
Later it was applied by Haldane to the Anomalous Hall Effect [28].
Here we provide a heuristic explanation for the Streda formula. First consider a two
dimensional system without applied electric field. Suppose that the system has a charge
density response to the applied orbital magnetic field:
xy
δρ = σH
δB z
(2.9)
13
Let us boost the system in the x direction with velocity vx . Due to present magnetic field
the boost induces electric field in the moving frame
E y = −v x B z
(2.10)
Also, due to charge density induced by magnetic field [2.9] the system carries electric current
xy x z
xy y
J x = −δρv x = −σH
v B = σH
E
(2.11)
When viewed from the boosted system, according to the last term in equation [2.11]
there exists a response to electric field and the conductivity turns out to be precisely equal
to the charge density response to magnetic field. The same result can be strictly derived
using Kubo formula, see [38].
Figure 2.2: From magnetically induced charge to electrically induced non-dissipative current
Fundamentally the existence of Streda formula provides a connection between Faraday’s and charge conservation laws. Indeed, if we substitute [2.9] into charge conservation
equation
∂ρ
+ ∇J = 0
∂t
(2.12)
14
we immediately obtain Faraday’s law
µ
xy
σH
¶
∂B z
+ ∂x E y − ∂y E x = 0
∂t
(2.13)
More generally, if there exists a charge density response of the form
¯
lim
B→0
∂ρ ¯¯
= χa (µ)
∂B a ¯µ
(2.14)
the dissipationless current will be given by
J a = σab E b ,
σa b = ²abc χc (µ)
(2.15)
This relation between the intrinsic responses to electric and magnetic fields is known
to be obeyed when ρ is the (conserved) electronic charge density, and µ is the electronic
chemical potential. The intrinsic Hall conductivity is completely determined by electronic
states at the Fermi level. In the case of the QHE, where the bulk system in the clean limit
has no states at the Fermi level, the states at the Fermi level are the chiral edge states of
the QHE. In the case of the AHE, where the bulk system is metallic, the non-quantized
intrinsic Hall conductivity is completely determined by the Berry phases of quasiparticles
moving on the Fermi surface [28]. Since the charge density is even under time reversal and
magnetic field is odd, the susceptibility must also be odd under time-reversal. It follows
therefore that charge Hall Effect can not exist unless time reversal symmetry is broken.
We now apply similar reasoning to derive spin current response to an applied electric
field. In a system with spin orbit interactions the spin density obeys an equation which is
an analog of [2.12]
∂t si + ∇a Jsia = τ i
(2.16)
where s is the spin density vector. Since spin orbit interactions do not conserve the spin
of the system the right hand side of [2.12] contains an intrinsic torque term τ which equals
to the rate of generation of spin density [23]. Here i labels spin component and a labels
the current direction. Time reversal symmetry implies Kramers degeneracy where the spin
of single-particle states with wavenumber k is balanced by those with wavenumber −k. If
15
inversion symmetry is absent, acceleration of the Fermi sea of occupied electronic states by
an applied electric field will generate a (time-dependent) local spin density. The response
of the spin density to electromagnetic fields then takes the form
si = χia B a + τ ia Ea t
(2.17)
In the 3D case, B = (Bx, By, Bz); in 2D, only the response to the flux density component
B z is relevant. Here χia is even under time reversal (and spatial inversion), so can in principle
take a non-zero value in a system with time reversal symmetry, with or without inversion
symmetry; τ ia is even under time reversal, but odd under inversion, so vanishes if inversion
symmetry is present. By assumption, if a Spin Hall effect is present,
J ia = σsiab Eb
(2.18)
The continuity equation for spin density now takes the form
τ ia Ea + (σsiab − ²abc χic )∇a Eb = τ i
(2.19)
the assumption that the intrinsic torque depends only on the local electric field strength,
and not on its gradient, again implies the Streda-type relation
σsiab = ²abc χic
(2.20)
Assuming the validity of these somewhat-heuristic arguments, we are now equipped with
an unambiguous method for computing σsiab by calculating the spin response χia to an
(orbitally-coupled) magnetic flux density B by diagonalizing the one-particle Hamiltonian
H(π, r, S), where
[πa , πb ] = ieh̄²abc B c
(2.21)
We now apply this method to demonstrate the absence of Spin Hall conductivity in
Rasba model [2.2]. It is most convenient to use the method of second quantization to tackle
the problem. The raising operator in this case is given by
l
a† = −i √ (πx − i(eB)πy )
2h̄
(2.22)
16
where l = (h̄/|eB|)1/2 is the magnetic length. For eB > 0 the Hamiltonian can be written
as
H=
√ h̄λR + †
1
h̄2 †
(a
a
+
)
(S a + S − a)
2
+
ml2
2
l
(2.23)
It turns out that Rashba model possesses a symmetry operator. Consider the following
operator
n = a† a +
1
− Sz
2
(2.24)
It can be easily verified that this operator commutes with [2.23] and takes the integer values
from zero to infinity. This operator mixes the orbital and spin degrees of freedom and can be
viewed as a “generalized Landau index”. In order to find the spectrum of the hamiltonian
we consider states with the same values of operator [2.24], namely |a† a = N, S z =↓> and
|a† a = N + 1, S z =↑> and diagonalize [2.23] on this subspace. Note that the state |0, ↑>
has no partner and contains uncompensated spin. This observation is crucial, as it will be
shown below that the spin response of this single unpaired level will cancel contributions
from the rest of the levels. Simple computation gives the energy levels of the hamiltonian
[2.23]
µ
En± =
E0 =
1
p2n
∓ ( h̄ωc )2 + (λR pn )2
2m
2
1
h̄ωc
2
¶1/2
,
n = 1, 2, . . .
(2.25)
(2.26)
where ωc = eB/m is the cyclotron frequency and pn = (2n)1/2 h̄/l is the momentum associated with the generalized index value of n. Figure 2.2 shows how the states with S z = ±1/2
mix while leaving one state unpaired.
In order to calculate Spin Hall conductivity according to spin-Streda formula [2.20] we
must now compute the spin density. Since the unpaired state carries fixed spin S z = 1/2 its
contribution to spin density comes purely from the change in the Landau level occupancy
eB
4π .
On the other hand states with nonzero n carry spins
Ã
z
Sn±
( 21 h̄ωc )2
= ±sgn(eB) 1
( 2 h̄ωc )2 + (λR pn )2
!1/2
(2.27)
17
Figure 2.3: Spectrum of Rashba Hamiltonian is obtained through hybridization of states
with same value of generalized Landau index. One state has no partner and exactly cancels
spin response due to paired levels.
We should sum over all occupied state that do not have corresponding counterpart. When
magnetic field goes to zero this amounts to summing over all the states with momenta
between p− and p+ . As a reminder, the following relation holds
X
n
m
=
eB
Z
dε
(2.28)
where ε is the electron energy. Using this correspondence we find the spin density including
the unpaired level to be
n
z
Stot
Ã
+
X
( 21 h̄ωc )2
eB
(1 −
=
4π
( 21 h̄ωc )2 + (λR pn )2
n−
!1/2
1
eB
(1 −
)=
4π
2m|λR |
Z p+
p−
dp) = 0
(2.29)
where we used [2.28]. Therefore there is no spin response to the orbital magnetic field in
the Rashba system and therefore the Spin Hall conductivity vanishes.
18
2.3
2.3.1
Spin Hall Insulators and Adiabatic Z2 Pump
Berry Phases
Early works on Spin Hall Effect focused on conductors with spin orbit coupling until Kane
and Mele showed in elegant work [3] that insulators can also exhibit Spin Hall effect. They
considered a model of electrons on honeycomb lattice with spin orbit interactions which
turned out to be equivalent to two copies of model previously considered by Haldane [5]
where he showed that Quantum Hall Effect can exist in a system without net magnetic flux.
Haldane shown that in order to produce the Hall current it is sufficient to break time reversal
symmetry (which can done by applying staggered magnetic flux to the system). Since the net
magnetic flux was zero, the system possessed translation symmetry and therefore powerful
results of band structure theory could be applied. In particular the existence of Quantum
Hall Effect could be related to electronic bands having nontrivial Chern numbers, as was
shown in a pioneering work by Thouless et al. Here we describe these results in more detail,
as they are crucial for understanding of the original results presented later.
We start by looking at spinless electrons moving on a lattice and looking at their response
to applied electric field. Such a general system was considered in detail by Thouless et al.
[6] who related existence of dissipationless Hall currents in such systems with nontrivial
topological Chern numbers of electronic bands. A concept of Chern number is intimately
related to the notion of electronic Berry phases in the system (see [40] for introduction).
If we consider general quantum system with Hamiltonian depending on (in general, multidimensional) parameter g, the eigenvalues and eigenvectors of such Hamiltonian will also
be the functions of parameters vector:
Ĥ(g)|ψn (g) >= En (g)|ψn (g) >
(2.30)
Obviously the choice of eigenvector is not unique - if the state |ψn > is not degenerate any
other state multiplied by a phase
|ψn0 >= eiχn (g) |ψn >
(2.31)
19
is also an eigenstate. In other words, non-degenerate eigenstate has a g - local, abelian
symmetry. The fact that the state n is not degenerate is essential as otherwise the symmetry
would be described by arbitrary non-abelian SU (N ) transformation where N is the degree
of the degeneracy of the state.
It is possible to quantify the amount of phase change of eigenstate as the parameters
vector g changes by an infinitesimal amount δg - simple calculation gives
< ψn (g)|ψn (g + δg) >= | < ψn (g)|ψn (g + δg) > |eiAn (g)δg
(2.32)
where, assuming that the state |ψn > is normalized we introduced a vector potential also
known as Berry Connection
A(g) = −i < ψ(g)|∇g × |ψ(g) >
(2.33)
(it is assumed here that the eigenstate is normalized). Note that under phase transformation
[2.31] the Berry Connection gets transformed as follows
A(g) → A(g) + ∇χ(g)
(2.34)
which is exactly analogous to the transformation of the vector potential in electrodynamics.
That is why the transformation [2.31] is in fact a gauge transformation of an eigenstate.
Continuing analogy with electrodynamics an object equivalent to electromagnetic tensor
can be introduced can be introduced
Fαβ (g) = ∇gα Aβ (g) − ∇gβ Aα (g)
(2.35)
Imagine that the parameters vector is changed along the closed contour C, i.e. such that it
eventually returns to its initial state. Then, according to [2.32] the change of the phase of
the wavefunction is given by
I
δC φ =
C
Z
Adg =
SC
dg α ∧ dg β Fαβ
(2.36)
where SC is any surface for which the contour C is a boundary. The last equation is a
consequence of Stokes theorem. Now, since returning to the same parameter value simply
20
gives the overlap the state with itself, the phase factor should be equal to one. This simple
logic leads us to a nontrivial “topological quantization” identity which holds for an arbitrary
surface SC
Z
SC
dg α ∧ dg β Fαβ = 2πK
(2.37)
where K is an integer. This identity can be interpreted as the fact that the number of Berry
flux es should always be an integer number.
We can now apply this interesting result to the case of interest - the electrons in a
periodic potential. In this case all the states of the Hamiltonian are in fact Bloch states
parameterized by a Bloch vector k
|ψn (k) >= eikr |un (k) >
(2.38)
We can identify the Bloch vector k with the parameters vector g of the periodic Hamiltonian.
This becomes apparent when we apply a unitary transformation to the periodic Hamiltonian
Ĥ(k) = e−ikr Ĥeikr
(2.39)
It is easy to see that the Bloch functions |un (k) > become eigenstates of the transformed
Hamiltonian
Ĥ(k)|un (k) >= En (k)|un (k) >
(2.40)
Also, the state |un (k) > spans an electronic band as the Bloch vector changes within the
Brillouin zone.
Let us first consider a two dimensional system, where results are particularly illustrative
and intuitive. In this case the Berry curvature [2.35] is a pseudoscalar
n
Fxy
= −i < ∇gx un (k)|∇gy un (k) >
(2.41)
In a seminal paper [6] Thouless et al. observed that transverse conductivity in the two
dimensional system obtained by direct application of Kubo formula is given by
σxy
e2
=
h
Z
BZ
Fxy dkx dky =
e2 X (1)
C
h n n
(2.42)
21
(1)
where Cn is an integer K from [2.37] which corresponds to an integral of Berry flux over the
whole Brillouin zone (the choice for notation will be explained later) for the n-th electronic
(1)
band. Alternatively, Cn is a winding number of a phase of a Bloch function |un (k) > when
the Bloch vector k goes around the Brillouin zone.
Remarkably, the quantization rule for Hall conductance [2.42] has been obtained purely
by considering phase changes of electron wavefunctions with changing Bloch vector. This
may look as a coincidence but a more fundamental mathematical analysis shows that two
(1)
dimensional periodic systems indeed can be classified according to the integer numbers Cn .
Moreover, for the case of infinite number of bands (as opposed to various tight-binding
models) the set of integers that characterize each band provides a unique classification of
the Hamiltonian in the sense that any two Hamiltonians with identical sets of these integer
numbers can be continuously deformed one into another. To see how this happens we have
to make brief informal digression into the Homotopy theory and how it applies to condensed
matter physics. The ideas described below were pioneered in the articles by Avron et al [24]
where the connection between the invariants found by Thouless et al and the topological
classification of Hamiltonian systems using methods of Homotopy was presented.
2.3.2
Topological Classification of Spinless Electronic Bands
Topology studies classifications of various mappings of topological spaces. In particular,
homotopy theory studies classification of mappings of n - dimensional sphere S n into topological spaces X. Namely, consider two mappings f and g from S n into topological space
X. These mappings are called homotopic if there exists a continuous parametric map
F (t) : S n × [0, 1] → X such that F (0) = f , F (1) = g. In other words, the two mappings
f and g are homotopic if they can be continuously deformed one into another. Moreover,
since gluing two spheres as shown in Figure 2.4 is topologically equivalent to a sphere, it is
possible to define a product of two maps h = f ∗ g as the map from the new sphere into
X. Also, a constant map is a map f0 of S n into one point in x0 ∈ X. Maps f homotopic
to f (x0 ) are such maps that can be contracted to a point. Note that it is necessary to
22
Figure 2.4: Two glued spheres are topologically equivalent to a sphere.
specify the target point x0 only in the case when the target space X contains disjoint sets.
Otherwise all the constant maps are homotopic as they can be deformed one into another by
continuously moving the target point in X. In the future we assume that the target space
X is connected and will not specify the target point x0 . It can be shown that equivalence
classes of maps from S n described above together with a product operator (f ∗ g) and a unit
element (constant map) form a group called n-th homotopy group of X, which is denoted as
πn (X). In essence, understanding the structure of homotopy groups πn (X) is the subject of
Homotopy theory. Of special importance is the group π1 (X) which is called a fundamental
group of the manifold. Since a mapping of S 1 (which is a circle) to X is essentially a loop
in X, fundamental group provides classification for all possible loops that exist in X.
Let’s consider a few examples of homotopy groups for various spaces X which will
illustrate ideas above and also will be useful for understanding results that will follow.
• Fundamental group of a circle : X = S 1 - in this case π1 (S 1 ) classifies mappings of
circle to a circle. It is convenient to study this mappings by looking at mappings from
23
one unit circle eiφ on a complex plane onto another eiψ . In other words, we want to
classify all mappings
φ → ψ,
φ ∈ [0, 2π]
(2.43)
In addition, the map to the target S 1 must return to its starting point, which defines
the terminal value of ψ to be of 2πn where n is an integer. Intuitively it is clear that
maps with different values of n can not be continuously contracted to one another
and vice versa - paths with same values of n can be contracted to one another and
are therefore homotopic. Therefore n is the only ”topological” characteristic of a
mapping from S 1 to S 1 . Therefore the fundamental group of a circle is a group of
integer numbers Z. The integer elements of the group have the meaning of the winding
numbers of the mapping S 1 → S 1 - they indicate how many times the image of the
first circle wraps around the target circle, see Figures 2.5-2.6.
• Higher homotopy groups of a circle: it can be shown that all higher homotopy groups
of a circle are zero - that is, every higher dimensional loop in S 1 can be contracted to
a point. Therefore the only nontrivial homotopy group of circle is π1 (S 1 ) = Z
Figure 2.5: Mapping of S 1 → S 1 with winding number n = 1.
• Homotopy groups of N -torus : X = T N . Next we study homotopy groups of an
N -dimensional torus, which is nothing but a Cartesian product of N circles, i.e.
1
TN = S
× S1 ×
. . . × S 1}
{z
|
N times
(2.44)
24
A map from S n to T N is described by N independent maps of S n to different circles
which comprise the torus according to [2.44], each map being some member of πn (S 1 )
which was described above. We therefore conclude that the fundamental group of a
torus is a direct sum of integer numbers
π1 (T N ) = Z
|
πn (T N ) = 0,
M
Z
M
{z
...
M
Z
}
N times
n = 2, 3, . . .
(2.45)
In other words, all one dimensional loops on N -torus are characterized by a set of N
integer numbers which are essentially the winding numbers for all the circles comprising the torus. All higher loops on the torus can be contracted to zero.
Figure 2.6: Mapping of S 1 → S 1 with winding number n = 2.
We shall show now that it is precisely because of this property of the torus that all two
dimensional periodic Hamiltonians can be characterized by a set of integer numbers, as was
originally found by Thouless et al. First we realize that any periodic Hamiltonian [2.39] can
be viewed as an element of space of continuous mappings from a 2-torus T 2 (parameterized
by Bloch vector k) onto a space of Hermitian matrices. Moreover we focus on a subspace of
all Hermitian non degenerate matrices and consider mappings to this space which consists
of matrices such that for all values of k no bands cross each other. Denote that space as
M . Note that the results of the homotopy theory can not be directly applied to classify
mappings from T 2 to an arbitrary space X, since T 2 is not a sphere. Intuitively though,
it is clear that such mappings must depend on two elements of π1 (X) which describe how
25
main circles of the torus map onto X and a “leftover” two-dimensional mapping π2 (X). For
a general N-torus T N the mapping T N → X will be classified by N elements elements of
π1 (X), N (N − 1)/2 elements of π2 (X) and so on.
It turns out that classification of mappings T N → M according to the scheme above is
particularly simple, since M has a very simple homotopy group structure. Namely,
πk (M ) = 0,
k 6= 2
π2 (M ) = Z ∞
(2.46)
where Z ∞ is an infinite set of integers. To show that it is indeed a case we consider a
space which consists of Hamiltonians with eigenvalues 1, 2, 3, . . . (the choice of the values
is not important, as long as it is nondegenerate). Denote this space as N . It is clear that
every element of a mapping T N → M can be continuously deformed to an element of a
mapping T N → N and vice versa by changing the k-depenedent eigenvalues and keeping
eigenvectors fixed. We therefore conclude that πk (M ) = πk (N ) for arbitrary k. We can
write an arbitrary element of N as H = U DiagU −1 where U is a unitary operator and
Diag is a diagonal matrix with elements 1, 2, 3, . . .. Such identification of element from N
with some unitary matrix is not unique though - any matrix Ũ = U D where D is a diagonal
unitary matrix will produce the same matrix H as can be seen from
Ũ Diag Ũ −1 = U DDiagD−1 U −1 = U DiagU −1 = H
(2.47)
Thus the space N is a quotient space
N = U/DU
(2.48)
where U is a space of unitary matrices and DU is the space of diagonal unitary matrices.
It can be shown that all homotopy groups of space U are zero and therefore the following
identity holds
πk (N ) = πk−1 (DU ) = πk−1 (T ∞ )
(2.49)
The second equality follows from the fact that unitary diagonal matrices are isomorphic
to a torus (each diagonal element is a complex number with unit absolute value and is
26
therefore a circle). Thus, according to [2.46] all the homotopy groups of N are zero except
for π2 (N ) = Z ∞ . This means that each band Hamiltonian can be characterized by a set of
integer numbers, each number having a meaning of a winding number for a corresponding
band. This is nothing but a set of integer numbers first discussed by Thouless et all.
The results so far were shown for infinite-dimensional band Hamiltonians. Note that for
a finite number of bands the analysis is more complicated, since in the formula [2.48] the
homotopy groups of p-dimensional unitary matrices U (p) are no longer zero. This produces
additional global invariants of a mapping and also puts constraints on integer numbers
which characterize the torus - namely the sum of band winding numbers must be zero.
This follows from the fact that π1 (U (p)) = Z. Still, important homotopy results hold in a
weaker form - if two band Hamiltonians are characterized by two sets of winding numbers
(n1 , . . . , np ) and (n01 , . . . , n0p ), they can be continuously transformed one into another only
if all these numbers coincide.
Another consideration proves to be useful for understanding of emergence of nontrivial
winding numbers from topologically trivial Hamiltonians. It often happens that a transition
to a topologically nontrivial state happens when some parameter θ of Hamiltonian is driven
through some critical value. For example, we shall see that a transition to a nontrivial
topological insulator state is driven by a staggered lattice potential (Boron-Nitride term).
It is clear that at the critical value of a parameter some bands must touch, as this is the
only way for them to change their winding numbers. For simplicity we assume that it only
happens to two bands at one time. In other words, as we sweep the parameter θ through its
critical value the bands which had winding numbers n1 and n2 touch at a critical value θ0
and then split again, but with different winding numbers - n01 and n02 . What is interesting
though, is that we can view these two neighboring bands as one ”superband” and allow its
sub-bands to be degenerate. Yet, the winding number of the band must be conserved at all
times, as the superband itself is not degenerate with any other band. When the bands do
not touch the winding number of the superband is just a sum of the two winding numbers
27
of each band. We therefore conclude that a following conservation law must hold
n1 + n2 = n01 + n02
(2.50)
This logic can be generalized to a larger number of bands. As an interesting example we
can view the whole Hamiltonian as one uberband. Since we can do anything now with its
sub-bands (which comprise the whole Hamiltonian) we can transform it to a Hamiltonian
with all bands being non-degenerate and having zero winding numbers. By conservation law
[2.50] we conclude that the total winding number of the whole Hamiltonian must be zero.
This leads to the second homotopy group of m-dimensional non-degenerate Hamiltonians
π2 (N m ) to be Z m−1 as has already been mentioned.
2.3.3
Quantum Hall Effect in the absence of magnetic field - a case of
broken Time Reversal symmetry
We have shown how topological considerations lead to existence of TKNN integers that
characterize band Hamiltonians and whose existence results in Quantum Hall Effect. The
emergence of the QHE is customarily associated exclusively with nonzero magnetic field
passing through the two dimensional electron system. On the other hand a uniform magnetic field is a rather awkward object to work with since it explicitly breaks translation
invariance and only at the values of the field that are commensurate with the underlying
lattice a periodicity is restored and the results [6] can be applied. This was precisely a
setup considered originally by TKNN authors where they shown that for an arbitrary rational magnetic flux φ = pq φ0 the winding numbers for each resulting band can be obtained
by solving a certain equations with integer variables (known as Diophantine equations).
It turns out that there exist an easier way to create QHE without subjecting system to
a uniform magnetic field as was originally shown by Haldane in [5]. He proved that in order
to create nontrivial winding band numbers it is sufficient to break time reversal symmetry
only locally, such as by applying staggered magnetic field. As a specific demonstration of
these ideas Haldane considered a model of electrons in graphene. Since to the end of this
28
chapter we shall be working with this model exclusively, here it is worthwhile to review the
results obtained in [5] in more detail.
In graphene, electrons move on a honeycomb lattice in two dimensions, as shown in
Figure 2.17. The lattice consists of two sublattices, which are shown as black and white
circles. The Hamiltonian of the system is given by
H = −t1
X
(c†i cj + hc) − t2
<ij>
X
(c†i cj + hc) +
<<ij>>
X
α
²αi cα†
i ci
(2.51)
iα
where the sum < ij > runs over all nearest neighbor sites while the sum << ij >> runs
over nearest neighbor sites. Also, ²αi denotes the site energy with α being the index of
sublattice (α = 1, 2). With such notation an α - independent term ²i has a meaning of a
smooth potential while a strongly α-dependent ²α indicates a staggered potential, which in
its most dramatic realization looks like
²αi = (−1)α Ubn
(2.52)
Such term is present in the Boron-Nitride (BN) which is a binary chemical compound,
consisting of equal proportions of boron and nitrogen. Boron Nitride is isoelectronic (has
the same valence structure) to carbon and can be used to form Boron Nitride nanotubes
similar to carbon nanotubes. The term [2.52] which is not present in carbon, differentiates
between two different atom types in BN.
It is worth mentioning at this point that if the system [2.51] has a nontrivial topological
state, transition to such state should be driven by the BN term [2.52], since it is obvious
that at very large values of that term the system will consist of two almost flat bands,
each centered around the value of Ubn . By taking Ubn to infinity, it is clear that the bands
eventually become absolutely flat and therefore trivial. We conclude that there should exist
∗ which governs the transition to the topologically nontrivial insulating
a critical value Ubn
state.
In order to break time reversal invariance the hopping t2 should be made a complex
t2 eiφij where the phase φij is positive when going clockwise in sublattice A and negative
when going clockwise in sublattice B. The total flux through each elementary cell is thus
29
made zero by construction, as there is no phase change for an electron going around any
loop enclosing full elementary cell. On the other hand the flux is non-zero through every
half elementary cell and therefore the time reversal invariance is explicitly broken here.
The Hamiltonian [2.51] should be first brought to Bloch representation [2.40] by transforming wavefunctions |ψ(r) > according to a discrete version of [2.38]
|ψ(rα ) >=
X
eikrα uα (k),
α = 1, 2
(2.53)
k
where the index α labels sublattices A and B and k runs over the Brillouin zone. The
resulting k - dependent Hamiltonian acts on two dimensional spinor uα (k)
H(k) = h1 (k) + h2 (k) + h3 (k)
h1 (k) = t1
3
X
(2.54)
cos(kai )σ1 + sin(kai )σ2
(2.55)
i=1
h2 (k) = Ubn σ3
h3 (k) = 2t2 cos(φ)
Ã
X
!
cos(kbi ) σ0 − 2t2 sin(φ)
i
here σi ,
Ã
X
!
sin(kbi ) σ3
(2.56)
(2.57)
i
i = 1 − 3 are the Pauli matrices, σ0 denotes the unit matrix and φ = 2πΦ/Φ0
where Φ0 is the flux quantum.
Effectively Hamiltonian [2.54] describes a spin 1/2 in magnetic field that is parameterized by k. A more convenient parametrization of such Hamiltonian is a set of spherical
coordinates
Ĥ = h(r(k))n(θ(k), φ(k))σ̂
(2.58)
where n is a unit vector and r denotes a combination of parameters that do not change the
direction of effective magnetic field. One of the two eigenstates of [2.58] is given by
|ψ(k)) >= U (n(k))| ↑>
(2.59)
where U (n) is the operator of unitary rotation (defined up to a U (1) phase). The wavefunction [2.59] therefore does not depend on r and is just a function of two spherical angles.
We can therefore easily compute Berry curvature [2.35] with respect to these angles, which
is its natural parametrization and then convert it to the curvature in Bloch space according
30
to
Ã
Fk = Fθ,φ D
θ, φ
kx , k y
!
(2.60)
where D(. . .) denotes a Jacobian of coordinate transformation. Since the rotation matrix
U (n) is explicitly given by
σ̂n
2
(2.61)
Fθφ = sin(θ)
(2.62)
U (n) = ei
the Berry curvature is given by
which corresponds to a constant effective magnetic field
1
2
perpendicular to the sur-
face of the sphere. This can be seen by comparing expressions for infinitesimal fluxes
Fθφ dθdφ = B(θ, φ) sin(θ)dθdφ. In fact, the effective field produced by Hamiltonian [2.58]
exactly corresponds to a magnetic field of Dirac Monopole
Bef f =
n
2r2
(2.63)
Therefore according to [2.37] the phase change of the wavefunction is just the area of the
surface that is subtended by unit vector n - a well known result. Viewed in this light, an
arbitrary band Hamiltonian represents a mapping of a torus on a two dimensional sphere
and therefore
R
Fxy dkx dky is proportional to the number of times the sphere is “covered”
by such mapping. According to general homotopy results this number therefore provides
a unique topological classification of an arbitrary two-band Hamiltonian. Note also that
Berry curvatures of two bands are opposite to one another and as the result the surfaces on
the sphere that they cover. This is again is consistent with a more general result that the
sum of topological invariants for all the bands must be zero.
For a particular case of graphene Hamiltonian the energies are given by
´
³
√
√
3ky
x
cos
t2 ±
E± (k) = 2 cos φ cos 3kx + 2 cos 3k
2
2
r³
´
´2
´
³
³
√
√
√
√
3ky
3ky
3kx
2
x
t2 + Ubn
2 cos 3kx + 4 cos 3k
2 cos 2 + 3 t1 + 2 sin φ sin 3kx − 2 cos 2 sin 2
(2.64)
31
Figure 2.7: Two bands of graphene Hamiltonian in the absence of time reversal and inversion
symmetry breaking terms. Brillouin zone is a hexagonal lattice with two inequivalent corner
points touching at Dirac points.
In the absence of symmetry breaking terms Ubn and φ the two bands touch in two points
that are determined by equation
X
exp(ikF ai ) = 0
(2.65)
i
These points completely determine the dissipationless dynamics of electrons. Dynamics of
electrons near these points is governed by linear relativistic-type dispersion relation, hence
these points are called Dirac points of graphene. The t2 term breaks particle-hole symmetry
but does not eliminate Dirac points, as can be seen in Figure 2.7.
On the other hand the terms Ubn and φ open gap between two bands. Interestingly, the
effect of two perturbations is in a way opposite to each other, namely when
√
Ubn = ±3 3t2 sin φ
(2.66)
one of the Dirac points closes again (both points can not close simultaneously since inversion
symmetry is broken), see [2.3.3]. This degeneracy has important effect on the winding
number of the Hamiltonian and as a result leads to quantum hall effect.
32
Figure
√ 2.8: Symmetry breaking terms open gap between Dirac points which closes at Ubn =
±3 3t2 sin φ at one of the Dirac points (depending on the sign of Ubn
In particular, Berry curvature is given by [2.62]. Notably, when neither inversion symmetry nor time reversal symmetry are unbroken, i.e. Ubn = 0 and φ = 0 the effective
magnetic field is always in xy plane for any values of k and therefore the berry curvature
is always zero. For general values of symmetry breaking terms the Berry curvature is not
zero, but it does not mean that the system is in topologically nontrivial state. As was
mentioned previously, at large values of Ubn the graphene system essentially consists of two
independent bands pierced by magnetic fluxes of opposite signs. Since the values of the
fluxes are identical the systems will exhibit zero quantum hall effect. On the other hand
when Ubn crosses degeneracy point [2.66] the bands “exchange” nontrivial winding numbers
(opposite in sign, according to [2.50]).
2.3.4
Quantum Spin Hall Effect in Graphene
Having shown the connection between topological properties of Bloch bands and the existence of quantum hall current we can now demonstrate, following Kane and Mele [3] how
Spin Hall effect may exist in a system of electrons with spin on graphene lattice. Consider
33
a model with spin orbit interaction
H = −t1
X
(c†i cj + hc) − t1
<ij>
X
X
(c†i cj + hc) − Uz
<<ij>>
i(ai × aj )sαβ c†iα cj β (2.67)
<<ij>>αβ
where the in the second sum ai × aj denotes a vector product of two nearest neighbor
vectors that provide the shortest path between two next-nearest points i and j. The second
hopping spin orbit coupling term is invariant under time reversal since both the spin and
the imaginary unity change sign simultaneously under its operation.
Comparing Hamiltonian [2.67] with [2.51] we can see that the spin orbit Hamiltonian
considered by Kane and Mele [2.67] is just a two copies of Haldane’s Hamiltonian [2.51]
for up and down spins. The two Hamiltonians differ in values of magnetic flux φ = ± π2 .
According to [2.54] corresponding bands will have effective magnetic fields pointing out of
the xy plane in opposite directions and as a result, equal and opposite winding numbers.
As a result, charge hall currents from two planes would exactly cancel each other while
the Spin Hall currents, which contains an extra spin factor, would add up to produce a
“universal” value
σSH = 2
e
h̄ e2
=
2e h
2π
(2.68)
which coincidentally is the Spin Hall conductivity which was originally reported for Rashba
model [23].
Moreover, it is possible to provide a transition from such spin conducting state to a trivial
insulating state by turning on inversion breaking potential Ubn and gradually increasing it
√
until one of the Dirac points closes at Ubn = 3 3Uz , after which the system becomes a
simple insulator.
This important result indeed looks so universal that it is very tempting to speculate that
the Spin Hall conductivity is indeed quantized and its value is given by a spin-weighted sum
of band winding numbers. The problem with such reasoning is that the z component of spin
in general is not conserved, as can be easily seen by turning on lattice version of Rashba
spin orbit interaction
VR = iUR
X †
ci (σ × dij )z cj
αβ
(2.69)
34
where the vector dij connects nearest neighbor sites.
Rashba term completely breaks the nice picture of quantized Spin Hall conductance and
conserved spin current and one has to face same dilemma as in the case of conductor with
spin orbit interaction considered in the previous section. Nevertheless it turns out that
there still exist more subtle characteristics of the insulating state which distinguish it from
ordinary insulator.
Figure 2.9: Schematics of Laughlin experiment. Electric field is applied by means of adiabatic time-dependent orbital magnetic flux. Such flux can be viewed conveniently as a
twisted boundary conditions. Adiabatic pump of a flux quantum is equivalent to changing
boundary conditions by 2π and therefore returns the spectrum to its initial state. Existence
of any type of current is therefore contingent on the system having edge states connecting
Bloch bands. (Drawing by Duncan Haldane)
2.3.5
Edge States, Z2 Nature of Insulating States and Spin Currents
Even though it might be problematic to define spin current when spin is not conserved, it
is still possible to study accumulation of spin on the edges of the system when the electric
field is applied to the system. A very insightful way of doing that, due to Laughlin, was
originally applied to the problem of Quantum Hall effect [8]
Consider the system we study in a cylinder geometry, as shown in Figure 2.9. A
small electric field can be viewed as an adiabatic application of a time-dependent orbital
magnetic flux through the system, i.e. E =
1 dΦ
c dt .
This flux can be interpreted sim-
35
Figure 2.10: t1 = 1, t2 = 0.1, Ubn = UR = Uz = 0
A singular case of spin orbit Hamiltonian - both Dirac points are closed and there are four
degenerate “zero” states connecting them.
ply as a twisted boundary conditions on the electron wavefunctions, ∆φ = 2π ΦΦ0 where
Φ0 is the flux quantum. Total spin accumulated at one of the cylinder boundaries is
< Sz > (t) = σSH
R
Edt = σSH Φ(t), assuming that Φ(0) = 0. When flux equals to Φ0
the Hamiltonian returns to its original and so its eigenstates.
Assuming that lower band is completely filled, the only way to have charge or spin
transported is by means of a state connecting two bands. Since we know well the spectrum
of system in the bulk, such state may exist only on the boundary of the system. In other
words, dissipationless transport in insulators occurs by means of the edge states - a well
known result. In the case of electrons with spin, edge state will have a pair of edge states,
related by Kramers symmetry. They move in different directions along the edge and we
shall call them ”right” and ”left” movers. After pumping one flux quantum through the
cylinder the right moving edge will bring an electron into the empty band while the left
mover will bring the hole into the occupied band, therefore the amount of spin accumulated
36
Figure 2.11: t1 = 1, t2 = 0.1, Ubn = 0.2, UR = 0.1, Uz = 0.1
Spin Hall Insulating regime - two pairs of edge states connect two bands.
at the boundary after one such cycle is < SZ >L − < SZ >R , from which follows that the
Spin Hall conductivity is given by
σSH =
e
(< Sz >L − < Sz >R )|EF
h
(2.70)
the result originally obtained in [4]. Here, the average value of spin is taken for the edge
states at Fermi energy. In case Sz is conserved, right and left movers have equal and opposite
spins of
1
2
and we therefore obtain the “universal” result [2.68]. Yet, for a general spin orbit
interaction, spin conductivity is not quantized.
Figures [2.10-2.13] show results of numerical diagonalization of spin orbit Hamiltonian.
The system is placed on a cylinder and is therefore has a conserved momentum k along
the cylinder circumference. Once k is fixed the Hamiltonian becomes one dimensional
along the cylinder axis. For each value of k the diagonalization of resulting one-dimensional
Hamiltonian is performed for the cylinder length Lx = 30. The resulting spectrum is plotted
as a function of k, with k changing from 0 to 2π when the system returns to its initial state.
Importantly, the value k = π is special since it is equivalent to its time-reversed value of
37
Figure 2.12: t1 = 1, t2 = 0.1, Ubn = 0.3, UR = 0.03, Uz = 0.05
When the inversion breaking term Ubn increases, the system eventually ends up in the trivial
insulating regime where the edge states do not connect bands anymore and therefore cannot
transfer spin. The parameters of Hamiltonian are chosen such that the system is very close
to the transition point.
−π. It is therefore at this value that we expect to see crossings of all Kramers pairs. From
the spectrum figures we can see how by increasing inversion breaking term Ubn one starts
from Spin Hall state (Figure 2.11) and eventually “disconnects” bands from each other
(Figure 2.12) and eventually leads to two simple separated bands (Figure 2.13).
According to [2.70] it is not the quantization of Spin Hall conductivity that distinguishes
the Spin Hall insulators but rather a mere existence of spin current as well as the existence
of edge states. More specifically, Kane and Mele noticed that for a general system with time
reversal symmetry a robust spin current may exist only if the number of right-left partners
at the edge is an odd number. If the number of edge state pairs is even, all right movers
will hybridize with left movers with exception of their partners, see [2.14] and there will be
no transport. On the other hand if the number of pairs is odd, after hybridization there
will still be an edge state connecting the bands, see example [2.15] where there are three
38
Figure 2.13: t1 = 1, t2 = 0.1, Ubn = 1, UR = 0.03, Uz = 0.05
When Ubn gets very large the system essentially splits in two independent trivial bands.
Kramers pairs at the edge.
The general argument for existence of robust hall current goes as follows. Consider some
energy value and look at the number of Kramers pairs P (E) that cross that energy. States
may hybridize with any other states other than their Kramers partners hence various small
perturbations of Hamiltonian may only change P (E) by two. If we start with an even P (E)
it is possible by continuously changing Hamiltonian to make it zero. On the other hand if
P (E) is odd, it is impossible to make it zero by continuously changing the Hamiltonian and
therefore at every energy level there will be at least one Kramers pair. This means that
there will always be a Kramers pair which connects two bands - its existence is protected
by Z2 nature of P (E).
It therefore becomes clear that time reversal invariant insulators are characterized by
Z2 number which is of topological nature, since it is preserved by continuous change of the
Hamiltonian. Simple sum of spin-weighted band Chern numbers is not sufficient since a)
conservation of spin is broken by Rashba-type interactions and b) such sum is not a Z2
39
Figure 2.14: When edge carries even number of Kramer pairs the right movers hybridize
with non-partner left movers and the system will not carry spin current.
number since Chern number can be an arbitrary integer. Kane and Mele found in [4] that
Z2 feature of Spin Hall insulator can be extracted by counting number of zeroes of Bloch
wavefunctions. Topological insulators have odd number of zero pairs.
In the next section we demonstrate that nontrivial Z2 phases exhibit an interesting
observable effect. Namely, for such phases pumping flux quantum into the system results
in a simultaneous pumping of one electron. On the other hand trivial Z2 phase shows no
such effect - the number of electrons pumped into the system is zero. Such effect clearly
has a Z2 nature and therefore can serve as an alternative measure for classification of Z2
phases of topological insulators.
2.3.6
Z2 Pump and Topological Insulators
Here we show that inserting orbital magnetic flux produces states localized at the point
of flux insertion. For an insulator in nontrivial Z2 phase this bound state connects the
two bands, just as the edge states connect bands in the case of Laughlin experiment and
40
Figure 2.15: When edge carries odd number of Kramers pairs right movers hybridize with
non-partner left movers, but hybridization will not block spin current - there will still remain
two partner states connecting upper and lower bands.
therefore adiabatic change of flux would pump the charge into the system. In fact, a system
with inserted flux is topologically equivalent to a cylinder and therefore there is a very close
relation of the flux pumping effect and the results discussed in the previous section. The
difference between two setups is more of a quantitative nature, since the case of a cylinder
is very convenient for numerical and analytical analysis of effects of magnetic field because
the dimensionality can be reduced to one, while the case where the flux is directly inserted
into the system represents a directly observable effect, although analysis of the problem is
more complicated, since we are dealing with a full two dimensional problem.
To insert local magnetic flux into the system we apply twisted periodic boundary conditions to the Hamiltonian [2.67] with a Rashba term [2.69] in a special way. Figure 2.17
shows the honeycomb lattice with its elementary cells. By choosing nonuniform boundary
conditions it is possible to simulate insertion of two fluxes of opposite signs into the system.
Positions of two inserted fluxes labeled as A and B with their equivalent positions are shown
41
Figure 2.16: Flux insertion into hexagonal lattice. A hexagonal lattice is shown with
inequivalent sites pictured as white and black circles. The dashed lines show elementary
cells. Two fluxes labeled as A and B are inserted at positions indicated by large circles.
Electrons experience phase change when they cross the boundary, with the value of phase
change dependent of the position of the hopping. We set φ11 = φ/2, φ12 = −φ/2, φ2 = 0.
If we move electron around the position A clockwise the phase change of an electron is φ,
while if we move the electron around the position B the phase change of the electron is −φ.
with large circles. Consider choosing phase changes of electrons for crossing boundaries to
be φ11 if electron crosses x-boundary between points A and B, φ12 when electron crosses
x-boundary after point B and φ2 when electron crosses the y-boundary. With such choice
of phases φ2 simply represents regular twisted boundary conditions while other two phases
describe two opposite fluxes φ11 − φ12 located at points A and B (this can be easily seen
by going around these points and observing the change of phase) and an overall twisted
boundary condition of (φ11 + φ12 )/2.
For present discussion we set the values of phases to be as follows: φ11 = φ/2, φ12 =
−φ/2, φ2 = 0 which precisely describes two fluxes of opposite values φ. We then see how
spectrum evolves as a function of flux φ. Moreover, since we want to study insertion of
a single flux, we need to isolate two fluxes from each other. This should be done in two
42
Figure 2.17: Spectrum of the topological insulating phase with a changing magnetic flux.
One pair of localized states is elevated above another by applying local potential around
the flux position.
ways. First, the system has to be taken as large as possible to prevent the bound states
localized at different flux insertion points from influencing each other. Secondly, we need to
be able to separate spectra from different fluxes. If we simply consider Hamiltonian [2.51]
in twisted boundary conditions environment the spectra of bound states will coincide. We
can differentiate between two flux insertion positions by elevating energies of one of them
by switching a local uniform potential around one of the fluxes.
The results of diagonalization are presented in Figure 2.17. Indeed, we can see two sets
of Kramers pairs connecting electron bands, each pair corresponding to its flux insertion
point. Such spectrum implies that we can create a charge in the system by adiabatically
changing magnetic flux. Figure 2.18 shows wavefunctions of two localized states at the
intermediate value of flux φ = φ0 /2. The wavefunctions are well separated, as required.
We have therefore explicitly demonstrated that in topological ly insulated phase insertion of a flux creates bound states localized at the flux (see [2.18]), such that when the
43
Figure 2.18: Wavefunctions of states localized at flux insertion points. Horizontal axes label
lattice sites. Vertical axis shows the electron density function.
value of flux is adiabatically changed by a quantum φ0 , an electron will be pumped into a
system. Conversely, when the system is not in the topological insulator phase, results of
diagonalization show that even though localized wavefunctions still exist at the location of
the flux, the corresponding bound energies do not connect the bands as the value of the flux
changes, therefore adiabatic pump will not exist in this case. Thus, we have shown that its
existence is one of the striking indications of a novel Z2 phase.
2.4
Appendix - Numerical Method For Finding Edge States
Here we describe an efficient approach that has been used here to find the edge states of
the SOC Hamiltonian. First, let us formulate the problem: for sufficiently small systems
where Hamiltonian matrix has dimension less than 10000 a straightforward diagonalization
is possible, using standard linear algebra packages, such as LAPACK. When the size of
the matrix is much larger, if Hamiltonian possesses any symmetries, it is in sometimes
44
possible to split the full matrix into sub-matrices corresponding to the eigenstates of a
symmetry operator. If no such symmetries exist or if the resulting matrices are too big,
straightforward diagonalization is not possible and one has to be more sophisticated when
finding the eigenstates of interest (or any eigenstates for that matter).
A very popular method for finding eigenstates of very large sparse matrices is the Lanczos
algorithm (or more generally, Arnoldi’s algorithm). It does not require the matrix to be
stored in memory but rather all that needs to be defined is the multiplication operation
of the Hamiltonian, ie a function of a vector argument x that returns vector y such that
y = Ĥx. Lanczos algorithm is an advanced version of a Power method which states that
starting with a random vector x0 and iteratively multiplying it with Hamiltonian the result
converges to an eigenstate with a largest absolute value.
Since energies of edge states typically lie between the band energies, direct application
of Lanczos method is impractical. Let us assume that it is known that a typical energy
that lies in the band gap is E0 . The idea is to use Lanczos method to find eigenstates of a
matrix M̂ =
1
Ĥ−E0 Iˆ
where Iˆ is the unit matrix. Obviously, matrix M̂ has same eigenstates
as Ĥ, eigenvalues of Ĥ are trivially computed from those of M̂ and Lanczos method can be
directly applied to M̂ . The computational price to pay for such modification of the problem
is that multiplication operation for matrix M̂ is not readily available. Rather, instead of a
ˆ = x.
multiplication operation one has to use a solver to find vector y such that (Ĥ − E0 I)y
Obviously, this is a costlier operation than multiplication but robust solvers are available
that greatly facilitate the task.
We have used a Fortran package ARPACK that provides functionality for such Shift and
Invert eigenstate finding. It worked very efficiently for Hamiltonians of dimensions 105 and
higher.
Chapter 3
Strong Coupling Expansion of BCS
Hamiltonian
Emil A. Yuzbashyan1,2 , Alexander A. Baytin1,2 , and Boris L. Altshuler1,2
1
2
Physics Department, Princeton University, Princeton, NJ 08544
NEC Research Institute, 4 Independence Way, Princeton, NJ 08540
Abstract
The paper is devoted to the effects of superconducting pairing in small metallic grains.
It turns out that at strong superconducting coupling and in the limit of large Thouless
conductance one can explicitly determine the low energy spectrum of the problem. We
start with the strong coupling limit and develop a systematic expansion in powers of the
inverse coupling constant for the many-particle spectrum of the system. The strong coupling
expansion is based on the formal exact solution of the Richardson model and converges for
realistic values of the coupling constant. We use this expansion to study the low energy
excitations of the system, in particular energy and spin gaps in the many-body spectrum.
45
46
3.1
Introduction
Since mid 1990’s, when Ralph, Black, and Tinkham succeeded in resolving the discrete
excitation spectrum of nanoscale superconducting metallic grains [12], there has been considerable effort to describe theoretically superconducting correlations in such grains (see e.g.
[41] for a review). However, very few explicit analytical results relevant for the low energy
physics of superconducting grains have been obtained, since, in contrast to bulk materials,
the discreetness of single electron levels plays an important role. In this paper we address
this problem in the regime of well developed superconducting correlations.
The electron–electron interactions in weakly disordered grains with negligible spin–orbit
interaction are described by a simple Hamiltonian [42]
Huniv. = HBCS − JS(S + 1)
HBCS =
X
²i c†iσ ciσ − λd
i,σ
N
X
† †
ci↓ ci↑ cj↑ cj↓
(3.1)
(3.2)
i,j=1
where ²i are single electron energy levels, d is the mean level spacing, c†iσ and ciσ are
creation and annihilation operators for an electron on level i, S and N are the total spin
and number of levels respectively. There are only two sample–dependent coupling constants:
λ and J that correspond to superconducting correlations and spin–exchange interactions
respectively. Throughout the present paper, for the sake of brevity, we consider only the
less trivial case of ferromagnetic exchange, J > 0.
Although Hamiltonian (3.1) is integrable [43, 44] and solvable by Bethe’s Ansatz, the
exact solution [45] yields a complicated set of coupled polynomial equations (see Eq. (3.3)
below). As a consequence, very few explicit results have been derived and most studies
resorted to numerics based on the exact solution. The purpose of the present paper is to
remedy this situation and to build a simple and intuitive picture of the low energy physics
of isolated grains in the superconducting phase.
It is well known that physical observables of a superconductor are nonanalytic in the
coupling constant λ at λ = 0. On the other hand, the opposite limit of large λ turns out
47
to be regular and relatively simple. Here we use the exact solution to obtain an explicit
expansion in powers of 1/λ for the ground state and low lying excitation energies.
We will distinguish between two types of excitations: ones that preserve the number of
Cooper pairs (the number of doubly occupied orbitals) and ones that do not. Only the latter
excitations are capable of carrying nonzero spin. It turns out that for J = 0 to the lowest
order in 1/λ both types of excitations are gaped with the same gap λN d. We compute
explicitly the two gaps to the next nonzero order in 1/λ and find the gap for pair–breaking
excitations to be larger. The difference between the two gaps turns out to be of the order
of d2 /∆, where d is the mean single particle level spacing and ∆ is the BCS energy gap, i.e.
the difference vanishes in the thermodynamical limit. We were not able to determine the
convergence criteria for the strong coupling expansion exactly, however we present evidence
that the expansion converges up to realistic values of λ between λc1 ≈ 1 and λc2 ≈ 1/π.
Hamiltonian (3.2) was studied extensively in 1960’s in the context of pair correlations
in nuclear matter (see e.g. [46]). A straightforward but important observation was that
singly occupied levels do not participate in pair scattering [47]. Hence, the labels of these
levels are good quantum numbers and their contribution to the total energy is only through
the kinetic and the spin–exchange terms in (3.1). Due to this “blocking effect” the problem of diagonalizing the full Hamiltonian (3.1) reduces to finding the spectrum of the BCS
Hamiltonian (3.2) on the subspace of either empty or doubly occupied – “unblocked” orbitals. The latter problem turns out to be solvable [45] by Bethe’s Ansatz. The spectrum
is obtained from the following set of algebraic equations for unknown parameters Ei :
−
m
n
X
X
0
2
1
1
+
=
λd j=1 Ei − Ej
E − 2²k
k=1 i
i = 1, . . . , m
(3.3)
where m is the number of pairs and n is the number of unblocked orbitals ²k . Bethe’s
Ansatz equations (3.3) for the BCS Hamiltonian (3.2) are commonly referred to as Richardson’s equations. The eigenvalues of the full Hamiltonian (3.1) are known to be related to
Richardson parameters, Ei , via
E=
m
X
i=1
Ei +
X
B
²B − JS(S + 1)
(3.4)
48
where
P
B ²B
is a sum over singly occupied – “blocked” orbitals and S is the total spin of
blocked orbitals (i.e. the total spin of the system).
BCS results [27] for the energy gap, condensation energy, excitation spectrum, etc. are
recovered from exact solution (3.3) in the thermodynamical limit [17]. The proper limit
is obtained by taking the number of levels, N , to infinity, so that N d → 2D = const,
m = n/2 = N/2, where D is an ultraviolet cutoff usually identified with Debye energy. In
particular, for equally spaced levels ²i , the energy gap ∆ and the ground state energy in
the thermodynamical limit are
∆(λ) =
D
sinh(1/λ)
BCS
Egr
= −Dm coth 1/λ
(3.5)
Since the BCS Hamiltonian (3.2) contains only three energy scales: D, ∆, and d, there
are only two independent dimensionless parameters: N , and λ. The perturbation theory in
small λ breaks down in the superconducting state as is already suggested by BCS formulas
(3.5). Thus, it is natural to consider the opposite limit of large λ and treat the kinetic term
in Hamiltonian (3.1) as a perturbation.
The paper is organized as follows. In Section 2 we consider the limit λ → ∞, which is
the zeroth order of our expansion. In this limit one can determine the spectrum straightforwardly by representing the BCS Hamiltonian (3.2) in terms of Anderson pseudospin
operators [11]. In particular, one finds that at J = 0 excitations with nonzero spin to the
lowest order in 1/λ have the same gap (the spin gap) as spinless excitations. Next, we
rederive the same results from Richardson’s equations (3.3) and also show that in the limit
λ → ∞ the roots of Richardson’s equations are zeroes of Laguerre polynomials.
In Section 3 Bethe’s Ansatz equations (3.3) are used to expand the ground state and low–
lying excitation energies in series in 1/λ. We write down several lowest orders explicitly and
give recurrence relations that relate the kth order term to preceding terms. These relations
can be used to readily expand up to any reasonably high order in 1/λ. Finally, we compute
the spin gap to the next nontrivial order in 1/λ and demonstrate that at J = 0 the first
excited state always have zero spin.
49
3.2
The strong coupling limit
In this section we analyze the lowest order of the strong coupling expansion.
As the strength of the coupling constant λ increases, the spectrum of the BCS Hamiltonian 3.2 undergoes dramatic changes as compared to the spectrum of noninteracting
Hamiltonian HBCS (λ = 0). First, there is a region of small λ where the superconducting
coupling causes only small perturbations in the electronic system. This region shrinks to
zero in the thermodynamical limit and is roughly determined by the condition ∆(λ) ≤ d
[11], where ∆(λ) is given by (3.5). For larger λ the perturbation theory in λ breaks down
[48] and strong superconducting correlations develop in the system. A representative energy
level diagram is shown on Fig 1. In the crossover regime the spectrum displays numerous
level crossings which reflect the break down of perturbation theory in λ. The fact that
the crossings occur for random single electron levels ²i , i.e. in the absence of any spatial
symmetry, is a characteristic feature of quantum integrability [49].
The lowest order of the strong coupling expansion is obtained by neglecting the kinetic
energy term in the BCS Hamiltonian (3.2). This limit can in principle be realized in a grain
of an ideal regular shape [50]. In this case the single electron levels are highly degenerate
and if the energy distance between degenerate many-body levels is much larger than λd,
only the partially filled Fermi level is relevant. Then, the kinetic term in (3.2) is simply a
constant proportional to the total number of particles and can be set to zero.
An efficient way to obtain the spectrum of Hamiltonian (3.1) in the strong coupling
limit is by representing the interaction term in the BCS Hamiltonian in terms of Anderson
pseudospin-1/2 operators [11].
Kiz
=
c†i↑ ci↑ + c†i↓ ci↓ − 1
2
Ki+ = (Ki+ )† = c†i↑ ci↓
(3.6)
The pseudospin is defined only on unblocked levels, where it has all properties of spin-1/2,
~ 2 = 3/4.
i.e. proper commutation relations and definite value of K
i
~ i.
The interaction term in the BCS Hamiltonian (3.2) takes a simple form in terms of K
50
40
Energy
20
0
-20
-40
-60
-80
20
15
1
0
1.5
2
4
6
8
10
λ
Figure 3.1: Results of exact numerical diagonalization. Energies of BCS Hamiltonian (3.2)
for m = 4 pairs and n = 8 unblocked single particle levels ²i versus coupling constant λ.
All energies are measured in units of the mean level spacing d. The single particle levels ²i
are computer generated random numbers. As the strength of the coupling λ increases, the
levels coalesce into narrow well separated rays (bands). The width of these bands vanishes
in the limit λ → ∞ (see Eq. (3.47) and the discussion around it). Slopes of the rays and
the number of states in each ray are given by Eq. (3.7, 3.15, 3.12). The ground state is
nondegenerate, while the first group of excited states contains n − 1 = 7 states. Note also
the level crossings for λ ∼ 1 (see the insert on the above graph).
51
h
∞
HBCS
= −λd K + K − = −λd K(K + 1) − (K z )2 + K z
i
(3.7)
~ =P K
~
where K
i i is the total pseudospin of the unblocked levels. The z-projection of the
total pseudospin according to (3.6) is K z = m − n/2, where m and n are the total number
of pairs and unblocked (either doubly occupied or empty) levels respectively. It is simple to
check that replacing a doubly occupied level with two singly occupied ones does not affect
the difference m − n/2. As a result,
Kz = m −
N
n
=M−
2
2
(3.8)
where M is the maximum possible number of pairs and N is the total number of levels
respectively. Hence, the last two terms in (3.7) yield a constant independent of the number
of blocked levels. This constant can be set to zero by an overall shift of all energies.
Therefore, the full Hamiltonian (3.1) in the strong coupling limit is
∞
Huniv.
= −λdK(K + 1) − JS(S + 1)
(3.9)
Since there are n pseudospin-1/2s, the total pseudospin K takes values between |K z | and
n/2,
n
n
≥ K ≥ |m − |,
2
2
(3.10)
while the total spin S ranges from 0 (1/2) to M − m (M − m + 1/2) for even (odd) total
number of electrons. For the sake of brevity, let us from now on consider only the case
of even total number of electrons. Then, the sum of the total spin and pseudospin is
constrained by
K +S ≤
N
2
(3.11)
The degree of degeneracy D(K, S, n) of each level is [51]
D(K, S) =
( n2
(N − n)!(2S + 1)
n!(2K + 1)
n
N
−n
+ K + 1)!( 2 − K)! ( 2 + S + 1)!( N 2−n − S)!
(3.12)
The ground state of Hamiltonian (3.9) has the maximum possible pseudospin, K = N/2,
and minimal possible spin, S = 0, provided that λd > J (recall that we consider only positive
values of the exchange coupling J).
52
There are two ways to create an elementary excitation. First, one can decrease the total
pseudospin K while keeping the total number of pairs M unchanged. The second type of
excitations corresponds to breaking pairs and blocking some of the single electron levels.
These excitations can contribute to the total spin of the grain S. They also affect the
pseudospin since its maximal value Kmax = n/2 is determined by the number of unblocked
levels. The lowest–lying excitations correspond to K = N/2 − 1, which can be achieved
both with and without breaking a single Cooper pair. Therefore, we find from (3.9) that
the pair–conserving excitations are separated by a gap ∆pair = N λd while pair–breaking
excitations can lower their energy by having nonzero spin S. Since the maximum value of
S for two unpaired electrons is S = 1, we get ∆spin = N λd − 2J.
In the opposite case J > λd, K = 0 and the total spin has the maximum possible value
S = M in the ground state, i.e. J = λd is the threshold of Stoner instability in the strong
coupling limit.
The above results can be obtained directly from exact solution (3.3). Moreover, individual parameters Ei can also be determined and, since eigenstates of the BCS Hamiltonian
(3.2) are given in terms of Ei (see [45]), this can be used to calculate various correlation
functions in the strong coupling limit.
The value of the total pseudospin K turns out to be related to the number, r, of those
roots of equations (3.3) which diverge in the limit λ → ∞ (see below). To the lowest order
in 1/λ we can neglect single electron levels ²i in Eqs. (3.3) for these roots
0
r
X
2
n0
1
+
=
−
λd j=1 Ei − Ej
Ei
i = 1, . . . , r
(3.13)
where n0 = n + 2r − 2m and summation excludes j = i. For the remaining m − r roots we
have
n
X
1
= 0 i = r + 1, . . . , m − r
E − 2²k
k=1 i
(3.14)
Multiplying each equation in (3.13) by Ei and adding all Eqs. (3.13), we obtain the eigenenergies of the BCS Hamiltonian (3.2) for n unblocked levels and m pairs
E = −λd r(n − 2m + r + 1)
(3.15)
53
Comparing this to (3.7) and (3.8), we find the relationship between r and K
r = K + m − n/2
(3.16)
Since the total pseudospin, K, is constrained by (3.10), the number, r, of diverging Richardson parameters, Ei , is also constrained
2m − n ≤ r ≤ m if n < 2m
(3.17)
0≤r≤m
if n ≥ 2m
Bellow in this Section we show that Eqs. (3.13) have a unique solution. As a result,
the degeneracy of energy levels (3.12) is equal to the number of solutions of Eqs. (3.14) for
the remaining Ei . This number can be computed [16, 52] directly from (3.14) and indeed
coincides with (3.12).
Finally, Eqs. (3.13) can be solved to determine parameters Ei to the lowest order in 1/λ
(see also [53, 54]). To this end it is convenient to introduce a polynomial f (x) of order r
with zeroes at x = xi = Ei /(λd)
f (x) =
r
Y
(x − xi ),
(3.18)
i=1
Using
lim
x→xi
2
f 00 (x) X
=
f 0 (x)
x
−
xj
j6=i i
one can rewrite Eqs. (3.13) as
F (xi ) = 0
where F (x) = xf 0 (x) − xf 00 (x) + n0 f 0 (x)
(3.19)
Since F (x) and f (x) are two polynomials of the same degree r with the same roots xi , they
are proportional to each other. The coefficient of proportionality is the ratio of coefficients
at xr and, according to (3.19), is equal to r. Therefore, F (x) = rf (x), or equivalently
xf 00 − (x + n0 )f 0 + rf = 0
(3.20)
54
0
The only polynomial solution to this equation is the Laguerre polynomial Lr−1−n . Thus, to
the order λ the nonvanishing roots of Richardson’s equations (3.3) in the strong coupling
limit are determined by
0
L−1−n
r
µ
Ei
λd
¶
= 0 n0 = n + 2r − 2m
(3.21)
where r is the number of nonvanishing roots to the order λ. This number and the total
pseudospin are related by (3.16). The ground state has r = m, the first degenerate group
of excited states corresponds to r = m − 1, etc. The constraint r ≥ 2m − n in (3.17) follows
from the requirement that the roots of (3.21) be nonvanishing [53]. Moreover, it can be
shown [53] using conditions (3.17) that all Richardson parameters Ei are complex for even
values of r, while for odd r there is a single real (negative) root. The fact that the roots
of (3.13) are generally complex was also noted in [52] on the basis of numerical solution of
Richardson’s equations.
3.3
The Strong Coupling Expansion
Now we turn to the expansion in powers of 1/λ around the strong coupling limit. The evolution of energy levels with λ can be viewed as a motion of one–dimensional particles whose
positions are the energies of the BCS Hamiltonian (3.2) (see e.g. [55, 49]). Then, single electron levels ²i determine the initial conditions at λ = 0. As the coupling λ increases beyond
the crossover between the weakly perturbed Fermi gas and the regime of strong superconducting correlations, the particles gradually loose the memory of their initial positions and
eventually the spectrum becomes independent of ²i . In this limit, the excited levels coalesce
into highly degenerate rays with a universal slope (see Fig. 1 and Eq. (3.15)). In the strong
coupling expansion the system of one–dimensional particles evolves from larger to smaller
λ. One expects this evolution to be nonsingular until we come close to the level crossings
(see the beginning of the previous section), i.e. the crossover region, where both expansions
in λ and in 1/λ break down.
A quantitative estimate of the convergence of 1/λ expansion can be obtained by con-
55
sidering various limiting cases. In the thermodynamical limit the ground state energy is
given by BCS expression (3.5). This limit is equivalent to keeping only the terms of order
N in the 1/λ expansion. We observe from BCS expressions (3.5) that the expansion in 1/λ
converges for λ > 1/π. In the opposite case of one pair and two levels, 2M = N = 2, the
ground state energy can be computed exactly by e.g. solving Eqs. (3.3) with the result
2
Egr
= −d(λ +
p
1 + λ2 )
(3.22)
In this case the expansion of the ground state energy (3.22) in 1/λ converges for λ > 1. In
general, we believe that strong coupling expansion yields convergent rather than asymptotic
series with the radius of convergence between λc1 ≈ 1 and λc2 ≈ 1/π.
Bellow in this Section we develop an efficient algorithm for calculating the low energy
spectrum to any order in 1/λ. While the pseudospin representation detailed in the previous
Section provides a simple and intuitive description of the strong coupling limit, the usual
perturbation theory becomes unmanageable beyond the first two orders in 1/λ. An approach
based on Bethe’s Ansatz equations, on the other hand, turns out to be well suited for the
purposes of systematic expansion.
3.3.1
The ground state
Here we expand the ground state energy in 1/λ. Richardson’s equations (3.3) lead to
recurrence relations for the coefficients of the expansion. From these relations the ground
state energy can be computed to any reasonably high order in 1/λ, e.g. we write down the
energy up to 1/λ7 . As it was mentioned above we take the number of electrons to be even
and consider only the case when λd > J. As we have seen in the previous Section, this
inequality ensures that in the ground state all levels are unblocked and all electrons are
paired, i.e Richardson’s equations (3.3) should be solved at
m=M
n=N
We begin by introducing a convenient set of variables
sp ≡
N
X
(2²k )p
k=1
σp ≡
M
X
1
i=1
Eip
(3.23)
56
Variables σp can be expanded into series in the inverse coupling constant λ.
σp =
∞
X
akp λ−k−p
(3.24)
k=0
Next, we rewrite Richardson’s Eqs. (3.3) in a form more suitable for our purpose. We divide
the equation for Ei by Eip with p ≥ −1 and add all M equations for each p. Expanding
1/(1 − 2²k /Ei ) in 2²k /Ei and using an identity
X
i>j
2
Ei − Ej
Ã
1
1
p −
Ei
Ejp
!
= pσp+1 −
p
X
σp−k+1 σk
k=1
we obtain
Egr (M, N, sp ) =
M
X
Ei =
i=1
−M (N − M + 1)λd −
∞
X
sk σk = −M (N − M + 1)λd − d
∞
X


j=0
k=1
j+1
X
k=1
(3.25)
p
∞
X
σp X
−
σp−k+1 σk = (N − p)σp+1 +
−
sj σj+p+1
λd k=1
j=1
Now plugging σp =

 λ−j
sk aj−k+1
k
P∞
k −k−p
k=0 ap λ
p≥0
(3.26)
into the last equation and setting the coefficient at
λ−h−p−1 to zero, we obtain
p X
h
h
X
ahp X
s
h
sk ah−k
+
ah−s
a
+
p−k+1 k
p+k+1 = −(N − p)ap+1
d
k=1 s=0
k=1
(3.27)
Note that from σ0 = M it follows a00 = M and ak0 = 0 for k ≥ 1. The values of ak0 serve as
boundary conditions for recurrence relations (3.27). Note also that according to (3.27) the
coefficients aph do not depend on λ as expected from their definition (3.24). Coefficients a0p
determine σp for the ground state to the lowest nonvanishing order in 1/λ and therefore can
be expressed in terms of zeroes of Laguerre polynomial (3.21) with r = M . Using (3.21),
we obtain
a0p dp = (−1)p
¯
dp
¯
−1−N
log
L
(x)
¯
M
x=0
dxp
According to Eq. (3.25) in order to determine the ground state energy to order 1/λj one
has to calculate the first j − p + 2 coefficients akp in the expansion of σp . To do this, we first
compute a0p for p ≤ j + 1, then a1p for p ≤ j, then a2p for p ≤ j − 1, etc. In other words, we
57
start from a01 element of matrix ahp and use recurrence relations (3.27) to move down the
first column of this matrix until a0j+1 , then to move down the second column from a11 to a1j
etc.
While we were not able to express akp in terms of p and k explicitly, the above procedure
allows for an efficient calculation, e.g. using Mathematica, of the ground state energy to
any given order. For example, the ground state energy to order 1/λ2 is
µ
¶
s2 M (N − M )
s1 M
− s2 − 1
Egr (M, N, sp ) = −M (N − M + 1)λd +
N
N N 2 (N − 1)λd
¶
µ
µ
¶
M (N − M )(N − 2M )
s1 s2 M (N − M )(N − 2M )
s2
+ s3 −
− s2 − 1 s1
4
2
N
N (N − 1)(λd)
N N 3 (N − 1)(N − 2)(λd)2
(3.28)
From Eq. (3.28) one can make several observations.
1. For N = M the first two terms give the exact energy. This is seen by noting that
N = M means that all levels are doubly occupied, i.e. there is only one state. Averaging
Hamiltonian (3.1) over this state gives the exact energy of the system which turns out to
be equal to the first two terms in (3.28). Therefore, the remaining terms in the 1/λ series
for the ground state energy are proportional to N − M .
2. When N = 2M , all terms with even nonzero powers of 1/λ vanish. This can be
demonstrated, e.g., by writing the kinetic term in the BCS Hamiltonian (3.2) in terms of
pseudospin operators (3.6)
H(λ = 0) =
N
X
i=1
2²i Kiz +
s1
s1
≡ H0 +
2
2
(3.29)
and noting that N = 2M correspond to zero z-projection of the total pseudospin. In this
case, by Wigner-Eckart’s theorem [56], Kiz has nonzero matrix elements only for transitions
K → K ± 1, while matrix elements for transitions K → K are equal to zero. The terms
with even nonzero powers of 1/λ vanish because they contain at least one matrix element
of H0 from (3.29) between states with the same K. These terms are therefore proportional
to N − 2M . Even terms also vanish when ²i are distributed symmetrically with respect to
zero. Hence, they reflect an asymmetry in the distribution of ²i . For example, the ground
58
state energy for N = 2M and equidistant single electron levels distributed symmetrically
between ±D = ±(m − 1/2)d is
"
E02m
= −Dm λ
128m3
2m + 1
16m2 + 22m + 7
2m + 2
+
−
+
2m − 1 3(2m − 1)λ
180(2m − 1)2 λ3
380m2
(3.30)
#
+
+ 344m + 93
+ O(1/λ7 )
7560(2m − 1)3 λ5
One can check that in the limit m → ∞ this expression reproduces the BCS result (3.5)
for the ground state energy up to terms of order 1/λ7 , while for m = 1 we recover (3.22).
Note also that the case p = N in Eq. (3.27) does not seem to be problematic as at p = N
the factors of 1/(N − p) in Eqs. (3.28, 3.30) are always compensated by a factor of (N − p)
in the numerator of the corresponding term.
3. Richardson’s equations (3.3) remain invariant if single electron levels ²k are shifted by
δ and parameters Ei are shifted by 2δ. The total energy E =
PM
i=1 Ei
then shifts by 2M δ.
Note that this shift is entirely contained in the second term of expansion (3.28). Thus, the
remaining combinations of sk at each power of 1/λ are “shiftless”. For example,
s2 −
3.3.2
s2
s21
→ s2 + 2δs1 + N δ 2 − (s21 + 2N δs1 + N 2 δ 2 )/N = s2 − 1
N
N
Excited states
Let us now expand energies of low–lying excitations in 1/λ. These expansions turn out
to be analogous to that for the ground state energy. We begin with the excitations that
conserve the number of pairs and then turn to the simpler case of pair–breaking excitations.
It was demonstrated in Section 2 that for λd > J lowest pair–conserving excitations
correspond to total pseudospin K = N/2 − 1 and total spin S = 0, where N is the total
number of single particle levels. The number of such states according to degeneracy formula
(3.12) is N − 1 and their energy is −λdK(K + 1) according to (3.9). We also know from
Section 2 that for these states one of parameters Ei (say EM ) remains finite as λ → ∞,
while all others diverge in this limit.
59
To distinguish EM from the rest of parameters Ei , we denote it by η. Richardson’s
equations (3.3) read
M
−1
N
X
X
0
2
1
2
1
+
=
−
−
λd
E
−
E
E
−
2²
E
i
j
i−η
k
j=1
k=1 i
−
M
−1
N
X
X
2
1
1
−
=
λd
Ej − η k=1 η − 2²k
j=1
i<M
i=M
(3.31)
(3.32)
Expanding the LHS of Eqs. (3.31) in 2²k /Ei and η/Ei and performing the same manipulations that lead to Eqs. (3.25, 3.26) for the ground state, we obtain
M
−1
X
Ei = −(M − 1)(N − M )λd −
i=1
∞
X
(sk − 2η k )σk
(3.33)
k=1
p
−
∞
X
σp X
−
σp−k+1 σk = (N − p − 2)σp+1 +
(sj − 2η j )σj+p+1
λd k=1
j=1
where now σp =
PM −1
i=1
p≥0
(3.34)
1/Eip . We see that replacements
M →M −1
N →N −2
sp → sp − 2η p
(3.35)
transform Eqs. (3.33, 3.34) into Eqs. (3.25, 3.26) for the ground state. Thus, energies of
first N − 1 excited states are
Epair =
M
−1
X
Ei + EM = Egr (M − 1, N − 2, sp − 2η p ) + η
(3.36)
i=1
Let us also rewrite Eq. (3.32) for η as
N
X
1
1
=−
− 2σ1 − 2ησ2 − 2η 2 σ3 − . . .
η
−
2²
λd
k
k=1
(3.37)
One can see (by e.g. sketching the LHS of Eq. (3.37) ) that this equation has N − 1 roots
with the kth root lying between 2²k and 2²k+1 . To the lowest order in 1/λ this equation
reads
N
X
1
=0
η − 2²k
k=1 0
(3.38)
Eqs. (3.34) and (3.37) are to be solved iteratively order by order in 1/λ. The procedure
is similar to that for the ground state, e.g., recurrent relations analogous to (3.27) can also
60
be derived. The only difference is that the coefficients at powers of 1/λ now depend also
on η0 , which has to be obtained from (3.38). For example, the excitation energies (3.36) to
the first two orders in 1/λ are
Epair = −(M − 1)(N − M )λd +
(s1 − 2η0 )(M − 1)
+ η0
N −2
(3.39)
Epair − Egr = N λd + η0 (1 − 2f )
(3.40)
f = (M − 1)/(N − 1) ≈ M/N
(3.41)
where
is the filling ratio.
Energies of higher excitations can be computed in the same way by solving 2, 3, 4,
. . . coupled equations of the type of (3.38). For instance, energies of the next group of
excited levels to the first two orders in 1/λ are determined by solutions of the system
n
X
2
1
=
η
−
2²
η
−
η2
1
k
k=1 1
n
X
2
1
=−
η − 2²k
η1 − η2
k=1 2
Now let us consider pair–breaking excitations. For λd > J low energy excitations of this
sort correspond to breaking a single pair of electrons thereby decreasing the number of pairs
by 1 and the number of unblocked levels by 2. Let the single electron levels occupied by two
unpaired electrons have energies ²a and ²b . Since the lowest energy is achieved by having
the unpaired electrons in a triplet state (recall that J > 0 corresponds to the ferromagnetic
exchange), the energy of lowest pair–breaking excitations according to (3.4) is
Espin = ²a + ²b − 2J + Egr (N − 2, M − 1, sp − (2²a )p − (2²b )p )
(3.42)
Note that, unlike η in (3.36), single electron energies ²a and ²b do not depend on λ. Therefore, to compute the energy of pair–breaking excitations we need only recursion relations
(3.27) for the ground state with N 0 = N − 2, M 0 = M − 1, and s0p = sp − ²pa − ²pb . In
61
particular, to the first two orders in 1/λ we get from (3.28)
Espin = −(M − 1)(N − M )g +
(s1 − 2²a − 2²b )(M − 1)
+ ²a + ²b − 2J
N −2
Espin − Egr ≈ N λd + (²a + ²b )(1 − 2f ) − 2J
(3.43)
It is instructive to compare the above results with the BCS theory [27]. For this purpose
let us write down the energies of the pair–conserving excitations for large M and N up to
the order 1/λ.
Epair − Egr ≈ 2Dλ + ηk (1 − 2f ) + ηk2
f (1 − f )
Dλ
(3.44)
where D = N d, f is the filling ratio (3.41), and ηk is the kth root of Eq. (3.37). In
deriving the above equation from (3.40) and (3.28) we shifted the single electron levels so
that ²̄i =
³P
N
i=1 ²i
´
/N = 0. In BCS theory (i.e. in the limit N, M → ∞) pair–conserving
excitation energies are [17]
q
2 (²k − µ)2 + ∆2
(3.45)
where µ is the chemical potential and ∆ is the gap. In the strong coupling regime both µ
and ∆ are of order λ. Expanding the square root in expression (3.45) in small ²k up to ²2k ,
we see that (3.45) and (3.44) coincide to this order if we identify
q
∆ = Dλ 4f (1 − f ) µ = (2f − 1)Dλ
ηk = 2²k
The first of these equations follows from (3.37) in the limit of large N , while the remaining
two can be derived from the BCS equation for the gap and chemical potential (see e.g. [17]).
Similarly one can check that pair–breaking excitations (3.42) correspond to two Bogoliubov
quasi-particles with total energy
q
q
(²a − µ)2 + ∆2 +
(²b − µ)2 + ∆2
(3.46)
Note that in the BCS limit the difference between pair–breaking and pair–conserving excitations disappears and expression (3.45) simply corresponds to two quasi-particles in a
singlet state each having the energy
p
(²k − µ)2 + ∆2 .
62
We have seen in Section 2 (see also Fig. 1) that in the strong coupling limit manyparticle energy levels of the BCS Hamiltonian (3.2) coalesce into narrow well separated
bands. Expression (3.44) can be used to estimate the ratio of the width of the first band,
W1 , to the single particle bandwidth D = N d.
W1
f (1 − f )
≈ 2(1 − 2f ) +
D
λ
(3.47)
where W1 is the width. Note that at half filling, f = 1/2, the width of the first band goes to
zero as λ → ∞. In general, it follows from Wigner-Eckart’s theorem [56] (see the discussion
in item 2 under the ground state formula (3.28)) that at half filling widths of higher bands
also vanish as λ → ∞.
According to the BCS equations for the excitation energies (3.44) and (3.46) the gaps
h
∆spin = Espin − Egr
i
min
h
and ∆pair = Epair − Egr
i
min
for the two types of excitations
coincide in the thermodynamical limit. We have also seen in Section 2 (see the discussion
below degeneracy formula (3.12)) that when 0 < J < λd and J/(λd) remains finite as
λ → ∞, spin-1 excitations have lower energy as compared to pair–conserving excitations
. If, however, J ∼ d or smaller, keeping J to the lowest order in 1/λ in excitation energy
(3.42) is not justified. In this case the two gaps are the same to this order. Therefore, it is
interesting to set J = 0 and evaluate the gaps to the next nonzero order.
Depending on the filling ratio f (see Eq. (3.41)) we can distinguish two different cases.
1. f 6= 1/2. Lowest lying excitations correspond to smallest or largest possible values of
η0 and ²a + ²b depending on the sign of (1 − 2f ). To determine the maximal and minimal
η0 , note that the kth root of Eq. (3.38) lies between 2²k and 2²k+1 . If N is large and
²k − ²k+1 → 0 as N → ∞, the smallest and largest solutions of (3.38) are η0min ≈ 2²1 and
η0max ≈ 2²n respectively. We have from (3.40, 3.43)
∆spin − ∆pair = d|1 − 2f | > 0
(3.48)
where we have used ²n − ²n−1 ≈ ²2 − ²1 ≈ d and d is the mean level spacing.
2. f = 1/2. To the first two orders in 1/λ: ∆spin − ∆pair = 0. In the next order we
63
obtain from (3.28, 3.36, and 3.42)
∆pair − ∆spin =
η02 − 2²2a − 2²2b
2N λd
where we shifted single electron levels so that ²̄i =
³P
N
i=1 ²i
(3.49)
´
/N = 0. We show in the
Appendix using Eq. (3.38) for η0 that the minimal value of η02 is always smaller than that
of 2(²2a + ²2b ). Therefore, ∆spin > ∆pair .
Thus, at J = 0 the pair–breaking excitations always have a larger gap in the strong
coupling limit. Note that for λ = 0 the situation is opposite as it always costs less energy
to move one of the two electrons on the highest occupied single electron levels to the next
available level. Since according to BCS expression (3.5) the energy gap in the strong coupling
limit is 2∆ ≈ 2Dλ = N λd, we see from (3.49) that at the half–filling ∆pair −∆spin ≈ d2 /∆,
i.e. the difference between the two gaps vanishes in the thermodynamical limit.
3.4
Conclusion
We determined the spectrum of the Universal Hamiltonian (3.1) in the strong superconducting coupling (λ ≥ 1) limit (3.9, 3.12, 3.21) and developed a systematic expansion in
1/λ around this limit (3.27, 3.28, 3.36, 3.42) for the ground state and low–lying excitation
energies. We detailed an algorithm by which these energies can be explicitly evaluated up
to arbitrary high order in 1/λ and estimated that the expansion converges for λ > λc where
λc lies between λc1 ≈ 1 and λc2 ≈ 1/π. Technically, this expansion is based on the existence
of the exact solution [45] of the BCS Hamiltonian (3.2). We found that in the strong coupling limit Richardson parameters are zeroes of appropriate Laguerre polynomials (3.21)
and analyzed their behavior at large enough but finite λ .
We found that it is important to distinguish between two types of excitatione in the
problem: those that conserve the total number of paired electrons and those that do not. We
determined the energy gaps for both types and found that at zero spin–exchange constant,
J = 0, in contrast to the weak superconducting coupling limit, the gap for pair–breaking
excitations is always larger (3.48, 3.49).
64
We believe there are two physically motivated questions within the scope of validity
(see [42]) of the Universal Hamiltonian (3.1) that still need further clarification. The first
problem is to develop a quantitative description of the crossover between a perturbed Fermi
gas and the region of strong superconducting correlations (see [48] and the discussion in the
beginning of Sections 2 and 3). The second problem is to study analytically the interplay
between superconducting correlations and spin–exchange (see e.g. [57]).
3.5
Appendix
We show here using Eq. (3.38) for η0 that the minimal value of η02 is always smaller than
that of 2(²2a + ²2b ), i.e.
x20 < 2(a2 + b2 )
(3.50)
where x0 is the smallest in absolute value solution of (3.38), a and b are the two smallest
in absolute value single electron levels ²i , and |a| ≤ |b|. Indeed, consider a function
g(x) =
N
X
1
x − 2²k
k=1
(3.51)
To prove (3.50) we need to show that g(x) has a zero on the interval (−c, c), where
q
c=
2(a2 + b2 )
For N = 2 there is only one zero, x0 = ²1 + ²2 , and (3.50) clearly holds. Consider N > 2.
First, note that g(x) has a single pole at x = a on this interval from −c to c, and
g(a+) > 0, while g(a−) < 0. Hence, there is a zero between c and −c iff either g(c) < 0 or
g(−c) > 0. To show that this is the case it is sufficient to demonstrate that g(c)−g(−c) < 0.
We have
g(c) − g(−c) =
N
X
i=1
X
2c
2c
=
2
2
2
c − 4²i
c − 4²2i
² 6=a,b
i
which is indeed negative since c2 < 4²2i for all ²i except ²i = a, b.
65
g(x)
x
-c
2a
+c
P
1
Figure 3.2: A schematic plot of the function g(x) = N
k=1 x−2²k on the interval from −c
p
to c, where c = 2(a2 + b2 ), a and b are the two smallest in absolute value single electron
levels ²i , and |a| ≤ |b|. Note that since 2|b| > c there is only one pole on this interval. In
the vicinity of 2a we have g(x) ≈ 1/(x − 2a) and therefore g(x) is positive on the immediate
right of x = 2a and negative on the left.
Chapter 4
Large N Expansion of BCS
Hamiltonian
Since mid 1990’s, when Ralph, Black, and Tinkham succeeded in resolving the discrete
excitation spectrum of nanoscale superconducting metallic grains [12], there has been considerable effort to describe theoretically superconducting correlations in such grains (see e.g.
Ref. [41] for a review). A key question in any such description is how results of the BCS
theory are modified in finite systems. In this paper we address this problem by developing
a systematic expansion in the inverse number of electrons on the grain for the low energy
spectrum of the problem.
In 1977, Richardson used exact solution (3.3) to outline [17] a method for expanding the
low energy spectrum in powers of the inverse number of pairs, 1/m. Richardson showed that
BCS results [27] for the energy gap, condensation energy, excitation spectrum etc. are recovered from exact solution (3.3) in the thermodynamical limit. The proper limit is obtained
by taking the number of levels, n, to infinity, so that nd → 2D = const, m = n/2, where
D is an ultraviolet cutoff usually identified with Debye energy. In particular, for equally
spaced levels ²i , the energy gap ∆ and the ground state energy in the thermodynamical
limit are
∆0 (λ) =
D
sinh(1/λ)
BCS
Eg.s.
(λ) = −Dm coth 1/λ
66
(4.1)
67
In the present paper we show that the ground state and excitation energies of BCS
Hamiltonian (3.2) can be evaluated explicitly to any order in d/∆0 ∼ 1/m in terms of
the BCS gap ∆0 , chemical potential µ, mean level spacing d, ultraviolet cutoff D, and the
thermodynamic density of states ν(²). In the physical limit ∆0 /D → 0, the expansion is
applicable for ∆0 ≥ d. In fact, we believe that in this limit the expansion is in powers of
d/∆0 with a convergence radius d/∆0 ∼ 1.
BCS Hamiltonian (3.2) supports two types of low energy excitations. Excitations of the
first type preserve the number of pairs (pair-preserving excitations). The second type of
low lying excitations (pair-breaking excitations) is obtained by breaking a single electron
pair. In the thermodynamical limit both types of excitations are gapped with the same
gap, ∆p = ∆b = 2∆0 , where ∆p and ∆b are the energy gaps for pair-preserving and pairbreaking excitations respectively. In Section 4.4, we evaluate leading finite size corrections
(of order 1/m) to the gaps ∆p and ∆b . Interestingly, it turns out that these corrections
coincide, even though the two gaps are not identical in higher orders in 1/m. In the limit
∆0 /D → 0, our result yields ∆p = ∆b = 2∆0 − d. We also show that the energy levels of
lowest excitations of two types cross at certain value of the coupling constant λ.
Another measure of the low energy properties of BCS model (3.2) is the parity parameter[58]
introduced by Matveev and Larkin. This parameter is defined as
2m+1
∆M L = Eg.s.
−
´
1 ³ 2m+2
2m
Eg.s. + Eg.s.
2
(4.2)
l
where Eg.s.
is the ground state energy of BCS Hamiltonian (3.2) with l electrons. Matveev
and Larkin evaluated ∆M L in the physical limit ∆0 /D → 0 in two different regimes: ∆0 À d
and ∆0 ¿ d. They found that in the first regime the leading finite size correction to the parity parameter (4.2) comes entirely from the stationary point (mean field) expression for the
ground state energy of BCS Hamiltonian (3.2). Here we use our method to calculate ∆M L
in the regime ∆0 > d for an arbitrary ratio ∆0 /D. We show that the contribution of quantum fluctuations to the leading finite size correction to ∆M L behaves as (∆0 /D) ln(∆0 /D)
for small ∆0 /D.
The ground state energy of pairing Hamiltonian (3.2) has been discussed recently in a
68
number of papers. Numerical fits for finite size corrections to the ground state energy in the
weak coupling regime, λ ¿ 1, have been proposed [48, 52]. Here we evaluate the leading
finite size correction exactly and find a complete agreement with numerical results [48, 52]
in the weak coupling regime.
In Ref. [48], authors studied the condensation energy, defined as the difference between
the ground state energy and the expectation value of BCS Hamiltonian (3.2) in the Fermi
ground state. This difference was calculated in second order perturbation theory in λ
BCS (λ) − E BCS (0). The authors found that the two
and compared to BCS expression Eg.s.
g.s.
√
expressions become of the same order when d ≤ ∆0 ≤ Dd and interpreted this as a new,
”intermediate”, regime of pairing correlations in metallic grains. We argue below that,
although the finite size correction to the condensation energy indeed becomes of the same
√
order as the BCS result for d ≤ ∆0 ≤ Dd, this fact does not indicate a new physical regime,
but is rather an artifact of the model. Main contribution to the finite size correction to the
condensation energy comes from energies close to the ultraviolet cutoff D and therefore is
beyond limits of applicability of BCS Hamiltonian (3.2). Effects coming from this range of
energies can be properly accounted for [59] within the Eliashberg theory [60].
The paper is organized as follows. Section 4.1 is devoted to the review of a general
method [17] of 1/m expansion due to Richardson. In Section 4.2, we show that Richardson’s results can be used to evaluate ground state and excitation energies of BCS Hamiltonian (3.2) to any order in 1/m and explicitly calculate the leading correction to the ground
state energy. In Section 4.3, we discuss various limits of our results and make a comparison
with previous work. Results for the excitation spectrum and Matveev Larkin parameter
are collected in Sections 4.4 and 4.5 respectively, where we also determine the gaps for
pair-breaking and pair-preserving excitations and discuss the range of applicability of the
1/m expansion.
69
4.1
Review of Richardson’s 1/m expansion
Here we briefly review Richardson’s 1/m expansion [17] for the ground state and excitation
energies of pairing Hamiltonian (3.2). The details can be found in the original work [17].
In subsequent sections we will use Richardson’s results to explicitly evaluate finite size
corrections to the low energy spectrum of BCS Hamiltonian (3.2).
Richardson’s 1/m expansion is based on an electrostatic analogy to equations (3.3).
In this analogy, the roots Ei of equations (3.3) are interpreted as locations of m twodimensional free charges of unit strength in the complex plane. The free charges are subject
to a uniform external field −1/(λd) and the field of n fixed charges of strength 1/2 located
at the points ²k on the real axis. The total electrostatic field at a point z associated with
the charge distribution is given by
F (z) =
m
X
i=1
n
1X
1
1
1
−
−
z − Ei 2 k=1 z − ²k
λd
(4.3)
The field F (z) contains complete information about the spectrum of BCS Hamiltonian
(3.2). For example, the energy spectrum is related to the quadrupole momentum of F (z).
Indeed, defining multipole moments of F (z) by
F (z) =
∞
X
F (m) z −m
(4.4)
m=0
and expanding equation (4.3) in 1/z, we obtain
E=2
m
X
Ei = 2F (2) +
i=1
n
X
²k
(4.5)
k=1
1
= F (0)
λd
1
m − = F (1)
2
(4.6)
−
(4.7)
The 1/m expansion is facilitated by the following field equation that can be derived from
equations (3.3) and (4.3):
Ã
1X
1
1 X 1
2
dF
+ F2 =
+
+
dz
2 k (z − ²k )2 4 k z − ²k
λd
!2
−
X
k
Hk
z − ²k
(4.8)
70
where Hk is the field at the location of the fixed charge ²k due to the free charges
Hk =
X
2
²k − Ei
i
(4.9)
Equation (4.8) can be solved by expanding the field F (z) in powers of 1/m
F (z) =
∞
X
Fr (z)
(4.10)
r=0
where Fr (z) is of order m1−r . It turns out [17] that the lowest order in (4.10), F0 (z),
together with field equation (4.8) completely determine the field F (z) to higher orders in
1/m. Moreover, to obtain higher orders, Fr (z) for r ≥ 1, from F0 (z) one needs to solve only
algebraic equations.
Different states of the system are described by different F0 (z). For example, one can
show that the BCS ground state corresponds to
F0 (z) = −
X
k
p
(z − µ)2 + ∆2
p
2(z − ²k ) (²k − µ)2 + ∆2
(4.11)
The parameters ∆ and µ correspond to the BCS gap and chemical potential respectively.
Equations for ∆ and µ can be derived by substituting F0 (z) into equations (4.6) and (4.7)
X
1
2
p
=
λd
(²k − µ)2 + ∆2
k
n − 2m =
X
k
²k − µ
(²k − µ)2 + ∆2
p
(4.12)
(4.13)
There are no higher order corrections to equations (4.12) and (4.13), since by construction
F0 (z) yields exact monopole and dipole moments of F (z), F (0) (z) and F (1) (z).
Note that, according to equations (4.3) and (4.11), F0 (z) also describes the fixed charges
exactly, since
lim (z − ²k )F0 (z) = −
z→²k
1
2
(4.14)
Higher order corrections to the field F (z) can be expressed only in terms of ²k , ∆, µ and
finite zeroes of F0 (z)
n
X
1
=0
(xl − ²k ) (²k − µ)2 + ∆2
k=1
p
(4.15)
71
For example,
Ã
X z + ²k − 2µ
X z + xl − 2µ
1
z−µ
F1 (z) =
−
−
2Z(z) k Z(z) + Z(²k )
Z(z) + Z(xl )
Z(z)
l
where
!
(4.16)
q
(z − µ)2 + ∆2
Z(z) =
One can show (by e.g. sketching the LHS of equation (4.15)) that there are n − 1 finite
solutions to equation (4.15), each of them lying between two consecutive single electron
levels ²k .
The ground state energy to the first two orders in 1/m, i.e. to the order m0 , can be
obtained from F0 (z) and F1 (z) using equation (4.5).
E = E0 + E1
E0 =
X
k
∆2 X q
−
(²k − µ)2 + ∆2
²k − µ(n − 2m) +
λd
k
E1 = −mλd +
n−1
X ·q
(xl − µ)2 + ∆2 −
l=1
Nl
Pl
(4.17)
¸
(4.18)
where
Nl =
X
k
1
(xl − ²k )2
Pl =
X
k
(xl − ²k
)2
1
(²k − µ)2 + ∆2
p
To calculate excitation energies one needs to appropriately modify F0 (z), the lowest order
in 1/m of the electrostatic field F (z). Here we simply write down excitation energies to the
first two nonzero orders in 1/m referring readers interested in the detailed derivation to the
original work.[17]
e(l) = e1 (l) + e2 (l)
l = 1, . . . , n − 1
q
e1 (l) = 2 (xl − µ)2 + ∆2
e2 (l) = 2
X 1 ·
m6=l
Pl
(F10 )2 − (F1 )2 +
where
d
2F10
(F10 − F1 ) +
dz
xm − xl
(4.19)
¸
(4.20)
z=xm
p
F10 (z)
= F1 (z) +
(xl − µ)2 + ∆2
1
p
−
2
2
z − xl
(z − xl ) (z − µ) + ∆
and e(l) is the excitation energy relative to the ground state.
(4.21)
72
Finally, we note that the lowest nonzero order of 1/m expansion, E0 and e1 (l) for the
ground state and excitation energies, reproduces the mean field (BCS) results for pairing
Hamiltonian (3.2). Therefore, the mean field for pairing Hamiltonian (3.2) is exact in the
thermodynamical limit, while contributions E1 and e2 (l), equations (4.18) and (4.20), are
leading finite size corrections to the thermodynamical limit.
4.2
Ground state energy
Here we evaluate the leading finite size correction to the ground state energy of BCS Hamiltonian (3.2).
First, we note that, as shown in Appendix A, expression (4.18) for the finite size correction E1 can be cast into a simpler form
µ
E1 = λd
¶
n−1
n q
Xq
X
n
−m +
(xl − µ)2 + ∆2 −
(²k − µ)2 + ∆2
2
l=1
k=1
(4.22)
To facilitate comparison to the mean field BCS result (4.1), we assume below n = 2m equally
spaced single electron levels ²k = (k − m − 1/2)d with energies ranging from D = (m − 1/2)d
to −D. It should be emphasized, however, that explicit results in terms of ∆, µ, and the
density of states ν(²) can be equally well obtained for arbitrary continuous ν(²).
Since n = 2m and ²k are distributed symmetrically with respect to zero, equation (4.13)
yields µ = 0, while equations (4.12), (4.17), and (4.22) become
2m
X
1
2
q
=
λd k=1 ²2 + ∆2
(4.23)
2m q
∆2 X
−
²2 + ∆2
E0 =
λd k=1 k
(4.24)
k
E1 =
2m−1
X q
x2l
+
∆2
2m q
X
²2k + ∆2
−
l=1
(4.25)
k=1
Equation (4.15) for xl now reads
f (xl ) =
2m
X
k=1
1
q
(xl − ²k ) ²2k + ∆2
=0
(4.26)
73
Since for each ²k there is ²k0 = −²k , f (z) is an odd function of z. Therefore, xl = 0 is a
solution of equation (4.26), while the remaining n − 2 = 2m − 2 nonzero solutions come
in pairs of xl and −xl . Let us label m − 1 positive roots xl with l = 1, . . . , m − 1 and
relabel m positive single electron energies ²k with k = 0, 1 . . . , m − 1. Then, we can rewrite
equation (4.25) as
s
E1 = ∆ − 2
m−1
m−1
Xq
Xq
d2
+ ∆2 + 2
x2l + ∆2 −
²2k + ∆2
4
l=1
k=1
(4.27)
where we have separated contributions to the summations of xl = 0 and ²k = ±d/2.
Because xl is located between ²l and ²l−1 = ²l − d, we can write it as xl = ²l − αl d, where
q
0 < αl < 1. Expanding
x2l + ∆2 in xl in the vicinity of xl = ²l and bearing in mind that
d ≈ D/m is of order 1/m, we obtain
E1 = −∆ − 2
m−1
X
l=1
q
αl d
²2l + ∆2
(4.28)
where we neglected terms of order 1/m. With the same accuracy, we can replace the
summation over k with an integration
E1 = −∆ − 2
Z D
0
²α(²)
d² √
²2 + ∆2
(4.29)
Note that E1 is indeed of order m0 as it should be. The function α(²) is evaluated in
Appendix B. The result, up to terms of order 1/m, is
" √
#
√
1
D ²2 + ∆2 − ² D2 + ∆2
1
√
√
α(²) = − arccot ln
π
π
D ²2 + ∆2 + ² D2 + ∆2
Introducing a new variable
" √
#
√
D ²2 + ∆2 − ² D2 + ∆2
1
√
√
x = ln
π
D ²2 + ∆2 + ² D2 + ∆2
D∆ sinh(πx/2)
² = −q
,
∆2 cosh2 (πx/2) + D2
we can cast expression (4.29) into a more convenient form
√
Z ∞
∆ ∆2 + D2
dx
q
E1 = −2
π (1 + x2 ) ∆2 + D2 (cosh(πx/2))−2
0
(4.30)
(4.31)
(4.32)
74
To complete the evaluation of the ground state energy to order m0 , we also need to calculate
the leading term, E0 with the same accuracy. The first step is to replace summation in
equations (4.12) and (4.24) with integrations according to the following formula:
d
n2
X
f (jd) =
Z n2 d
n1 d
j=n1
dxf (x) +
d
[f (n1 d) + f (n2 d)] + o(1/m)
2
Equations (4.12) and (4.24) now read
2
=
λ
E0 =
Z D
−D
∆2 1
−
λd
d
d²
d
√
+√
²2 + ∆2
∆2 + D2
Z D
−D
p
d² ²2 + ∆2 −
(4.33)
p
∆2 + D2
(4.34)
The solution of equation (4.33) for ∆ to order m0 is obtained by dropping the second term
on the RHS. Evaluating the integral, we obtain ∆0 = D/[sinh(1/λ)] in agreement with
equation (4.1). To compute the correction of order 1/m to ∆, we substitute ∆ = ∆0 + δ∆
into equation (4.33) and expand in δ∆. Keeping only terms of order 1/m, we find
∆ = ∆0 + d
q
Plugging ∆ into equation (4.34) and using
of order 1/m
∆0
2D
(4.35)
∆20 + D2 = D coth(1/λ), we obtain up to terms
¶
µ
1
D coth(1/λ)
E0 = − m +
2
(4.36)
Note also that ∆ in expression (4.32) for E1 can be replaced by ∆0 up to terms of order
1/m. Thus, the ground state energy of BCS Hamiltonian (3.2) for m pairs and n = 2m
equally spaced levels is
·
Eg.s.
where
φ(λ) = 2
Z ∞
0
¸
1
= −D coth(1/λ) m + + φ(λ)
2
(4.37)
dx
cosh(πx/2)
q
π(1 + x2 ) cosh2 (πx/2) + sinh2 (1/λ)
(4.38)
The plot of function φ(λ) is shown on Fig. 1.
The finite size correction to the mean field BCS result (4.1) is
·
Eg.s. =
BCS
Eg.s.
+ Ef.s.
Ef.s.
1
+ φ(λ)
= −D coth(1/λ)
2
¸
(4.39)
75
1
φ(λ)
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
λ
2
2.5
3
Figure 4.1: The plot of function φ(λ) defined by equation (4.38). This function appears
in leading finite size corrections to ground state (4.37) and excitation (4.52, 4.55) energies
of BCS Hamiltonian (3.2) and to Matveev-Larkin parameter (4.57). Note the asymptotics
φ(λ) → 0 and φ(λ) → 1 for λ → 0 and λ → ∞ respectively.
Note that Ef.s. is different from E1 given by equation (4.32) due to contribution of order
m0 from E0 .
Higher order corrections to the ground state energy can also be evaluated explicitly. The
first step is to express them in terms of ∆ and xl following prescriptions of Ref. [17]. Then,
∆ and xl have to be calculated to appropriate order in 1/m using methods of this section
and Appendix B. Final results for higher order corrections will involve multiple integrations
similar to the integration in equation (4.38). For example, the expression for the correction
of order 1/m contains a triple integral.
The general case when the distribution of single electron levels in the limit m, n → ∞,
m/n = fixed is described by a continuous density of states ν(²) can be treated similarly.
Final expressions for corrections will now be in terms of ∆, µ, and ν(²). For example, the
correction of order m0 will be again given by the integral in equation (4.29) where the limits
of integration should now be −D and D, ² has to be replaced with ² − µ, and the integrand
has to be multiplied by ν(²) . The function α(²) will still be given by equation (B.61) where
76
now ν(²) has to be included under the integral.
4.3
Comparison to previous studies
Here we analyze our result and compare it to previous results. First, we check whether
equation (4.39) reproduces the results of 1/λ expansion [25] around λ = ∞. Expanding
the integrand in equation (4.38) in 1/λ, evaluating the resulting integrals, and expanding
coth(1/λ) in 1/λ, we obtain
µ
·
Ef.s.
1
19
143
1
3
−
+
+O
= −D λ +
2
3λ 360λ2 15120λ5
λ7
¶¸
Comparing this expression with terms of order m0 in 1/λ expansion [25] for the ground
state energy (see equation (30) of Ref. [25]), we find that the two results coincide.
Now let us consider the limit of small λ. The asymptotic behavior of φ(λ) for small λ
is worked out in Appendix C. Here we write down the first two terms
φ(λ) = λ + ln 2 · λ2 + O(λ3 )
(4.40)
q
Expanding coth(1/λ) =
1 + ∆20 /D2 in ∆0 /D and using D = (m − 1/2)d, we obtain from
equation (4.37)
µ
Eg.s.
1
= −D m +
2
¶
−
∆20
− Dλ − ln 2 · Dλ2 + O(λ3 )
2d
(4.41)
The first term in equation (4.41) is the energy of noninteracting Fermi ground state to order
m0 . The second term is the nonperturbative mean field (BCS) contribution to the ground
state energy. The first two terms are extensive and survive the thermodynamical limit.
The last two terms give the correction to the ground state energy that one would obtain in
the second order of ordinary perturbation theory in λ around noninteracting Fermi ground
state.
We see that our result (4.39) yields the leading finite size correction to the thermodynamical limit for all values of λ. In particular, there is no breakdown in the regime of
ultrasmall grains, i.e. for d > ∆0 . As we will see in subsequent sections, this is not a generic
77
feature of our approach, but is specific to the ground state energy and is probably related
to the ultraviolet nature (see below) of the finite size correction calculated above.
A frequently discussed quantity [41, 52, 48] is the difference between the ground state
energy and the expectation value of BCS Hamiltonian (3.2) in the unperturbed Fermi
ground state, |F.g.s.i, i.e. a state where single particle levels below the Fermi level, ²k < 0,
are doubly occupied, while the remaining levels are empty. This difference is often called
condensation energy, even though this name is misleading for the reasons detailed below.
However, to facilitate a comparison with results of Ref. [52] and [48], we will use the same
terminology in this section. We have
µ
Econd. = hF.g.s.|HBCS |F.g.s.i − Eg.s. = −D m +
1
2
¶
− 2λmd − Eg.s.
Using D = (m − 1/2)d and equation (4.41), we obtain
Econd. =
∆20
+ ln 2 · Dλ2 + O(λ3 )
2d
(4.42)
Comparison shows that the exact result (4.42) for Econd. to order m0 is in complete agreement with fits to numerical data.[48, 52]
Finally, note that the second term in expression (4.42) is ultraviolet divergent, since it
depends explicitly on the ultraviolet cutoff D. For pairing by phonons the ultraviolet cutoff
D can be identified with the Debye energy ωD . To properly take into account any effect
that comes from energies of the order of ωD , one needs to go beyond the BCS theory which
is appropriate only at energies much lower than ωD . The contribution from these energies
to finite size corrections can be adequately treated [59] within the Eliashberg theory [60].
In particular, the hard cutoff at D = ωD has to be replaced by a soft effective cutoff due
to the 1/ω 2 decay of the phonon propagator for frequencies ω À ωD . Therefore, even
though the contribution of the finite size correction in equation (4.42) becomes important
√
for ∆0 ≤ Dd, the conclusion of Ref. [48] that this is an indication of any new physical
regime is not justified.
78
4.4
Excitation energies
In this section we evaluate leading finite size corrections to lowest excitation energies. As
we will see below, the results of this section are accurate only in the regime of relatively
large grains, ∆0 > d, i.e. within terms of order o(d/∆0 ). These higher order corrections
can also be straightforwardly calculated using methods of Sections 4.2. However, we will
only evaluate corrections of order d/∆0 here.
As in Section 4.2, we will perform calculations for the case of 2m electrons and n = 2m
equally spaced levels ²k = (k − m − 1/2)d with energies ranging from D = (m − 1/2)d to
−D. In this case, equation (4.13) implies µ = 0. A more general case when the single
electron levels are distributed with a smooth density of states can be treated similarly (see
the discussion below equation (4.39)).
Note that Hamiltonian (3.2) conserves the number of paired electrons. Therefore, the
excitations can be grouped into two types: those that preserve the number of pairs and those
that break pairs. Energies of low lying pair-preserving excitations in the thermodynamical
limit are given by equation (4.19) with µ = 0
q
ep1 = 2 x2l + ∆20
(4.43)
where xl are the roots of equation (4.15). Low lying pair-breaking excitations are obtained
by breaking a single pair and placing the two unpaired electrons on two single electron levels
²a and ²b . The energy of this excitation according to equation (3.4) is
eb = ²a + ²b + Eg.s. (²a , ²b ) − Eg.s.
(4.44)
where Eg.s. (²a , ²b ) is the ground state energy of BCS Hamiltonian (3.2) with levels ²a and
²b blocked. In the thermodynamical limit, using equation (4.17), we obtain
eb1 =
q
²2a + ∆20 +
q
²2b + ∆20
(4.45)
Therefore, in the thermodynamical limit both types of excitations are gapped with the same
gap 2∆0 , i.e.
∆p1 = ∆b1 = 2∆0
(4.46)
79
Since pair-breaking excitations are capable of carrying spin-1, ∆b can also be called the
spin gap. To calculate corrections to ∆p1 and ∆b1 , one needs to go beyond mean field
approximation.
First, let us determine the energy of lowest lying pair-breaking excitations to order
1/m. Breaking a pair changes both the number of pairs to m0 = m − 1 and also the number
of unblocked levels to n0 = 2m − 2 = 2m0 . The lowest energy is archived by blocking
levels ²a = d/2 and ²b = −d/2. Since this leaves the distribution of single particle levels
symmetric with respect to zero, the chemical potential µ in equation (4.13) remains equal to
zero, µ0 = µ = 0. However, the blocking affects the gap ∆0 , since now terms corresponding
to ²k = ±d/2 have to be excluded from gap equation (4.12). Using equation (4.12), we
obtain
X
k
where
∆0
q
1
²2k + ∆02
X
1
2
q
+
=p 2
2
d /4 + ∆2
² + ∆2
k
(4.47)
k
is the value of the gap with levels ±d/2 blocked. Expanding the LHS of equation
(4.47) in δ∆ = ∆0 − ∆ and using gap equation (4.12), we obtain
s
δ∆ = −d 1 +
∆2
D2
(4.48)
According to equation (4.44), to order 1/m the lowest lying pair-breaking excitations have
the following energy:
∆b = E00 (∆0 ) − E0 (∆) + E10 (∆0 , x0l ) − E1 (∆, xl )
(4.49)
where E0 (∆) and E1 (∆, xl ) are given by equations (4.24) and (4.25) respectively and primes
denote quantities for the ground state with levels ±d/2 blocked. Equations (4.24), (4.25),
(4.48), and (4.35) imply
E00 (∆0 ) − E0 (∆) = 2∆0 +
X
k
s
δ∆∆
d∆0
∆20
−
d
1
=
2∆
+
+
0
D
D2
4(²2k + ∆2 )3/2
E10 (∆0 , x0l ) − E1 (∆, xl ) =
X
xl δxl
∂E1 (∆)
q
δ∆ +
∂∆
x2 + ∆2
l
(4.50)
(4.51)
l
where E1 (∆) is given by equation (4.32) and δxl is the change in xl due to blocking levels
±d/2.
80
We see from equation (4.26) that the effect of removing levels ²k = ±d/2 from the
summation in equation (4.15) is strongest for the roots closest to the blocked levels ±d/2.
For these roots δxl ∼ d. On the other hand, due to an additional factor of xl in front of δxl
in equation (4.51), the contribution of each of these xl to the RHS of equation (4.51) is of
order d2 /∆. By splitting the sum in equation (4.26) into two sums as in Appendix B, one
can show that the contribution of all these roots to the sum in equation (4.51) is of order
o(1/m). For the remaining roots, δxl /xl is of order 1/m and each term in equation (4.26)
can be expanded into δxl /(xl − ²k ). We have
X
²k 6=±d/2
q
1
²2k + ∆2 (x0l − ²k )
=
2m
X
1
q
²2k + ∆2 (xl + δxl − ²k )
k=1
−
2m
X
1
2
q
=
xl ∆ k=1 ²2 + ∆2 (x − ² )
l
k
k
Expanding into δxl , we obtain
δxl
X
k
1
q
²2k + ∆2 (xl − ²k )2
=−
2
xl ∆
The summation here can be evaluated in the same way as the first sum in equation (B.60)
of Appendix B is evaluated. Recall that roots of equation (4.26) xl and therefore δxl are
distributed symmetrically with respect to zero. Using the notation introduced in the text
following equations (4.26) and (4.27), we have for xl > 0
q
δxl xl = −
2d2 ²2l + ∆2 sin2 πα(²l )
∆
π2
where α(²l ) is given by equation (4.30). Substituting δxl xl into equation (4.51) and using
equations (4.50), (4.49), (4.30), and (4.32), we obtain
s
∆b = 2∆0 − d 1 +
∆20 d∆0
+
[1 + φ(λ)]
D2
D
(4.52)
where we used the change of variables (4.31) and φ(λ) is defined by equation (4.38). Expression (4.52) yields the energy of lowest lying pair-breaking excitations up to terms of
order o(d/(min[D, ∆0 ])).
In the physical limit of weak coupling, ∆0 /D → 0, according to equation (4.40), expression (4.52) becomes
∆b = 2∆0 − d + o(d/∆0 )
(4.53)
81
Next, we turn to excitations that preserve the number of pairs. Energies of these excitations to order 1/m are given by equations (4.19) and (4.20). Equation (4.43) shows
that the lowest lying excitation corresponds to xl = 0. We have, up to terms of order
o(d/(min[D, ∆0 ]))
∆p = 2∆ + 2
X 1 ·
xm 6=0
Pl
(F10 )2 − (F1 )2 +
d
2F 0
(F10 − F1 ) + 1
dz
xm
¸
(4.54)
z=xm
where F1 (z) and F10 (z) are defined by equations (4.16) and (4.21). Taking into account that
both ²k and xl are distributed symmetrically with respect to zero and µ = 0, we can rewrite
these equations as


X
X
z
1
1
1


q
q
F1 (z) = √
−
−√
√
√
2
2
2
2
2
2
2 + ∆2 +
2 + ∆2 +
2
2
2 z +∆
z
+
∆
z
²
+
z
x
+
∆
∆
k
l
k
l
q
F10 (z) = F1 (z) +
x2l + ∆2
1
√
−
2
2
z
−
xl
(z − xl ) z + ∆
Summations in F1 (z) and in equation (4.54) can be evaluated in the same way as sums in
equations (4.51) and (4.27) have been evaluated. Even though this calculation looks rather
different from the one that lead to equation (4.52), it yields an identical result, i.e.
∆p = ∆b + o(d/(min[D, ∆0 ]))
(4.55)
Thus, both gaps coincide up to terms of order o(1/m). However, this coincidence is not
preserved in higher orders. Indeed, it was shown in Ref. [25] that in the strong coupling
limit, λ À 1, the gap for pair-breaking excitations is larger ∆b − ∆p ' d2 /∆0 > 0. On
the other hand, at λ = 0 the gap for pair-preserving excitations is larger, ∆b − ∆p = −d.
Therefore, the lowest energy levels of the two types of excitations cross at certain value of
∆0 . Equation (4.55) shows that the distance between the two levels is reduced from d at ∆0
to o(d/∆0 ) d even when d ¿ ∆0 ¿ D. However, the knowledge of higher order corrections
to the gaps ∆b and ∆p is needed to determine whether the crossing occurs in the physical
regime ∆0 /D → 0, i.e. at ∆0 ' d.
82
4.5
Matveev-Larkin parameter
Finally, let us evaluate the Matveev-Larkin parameter [58]. This parameter is a measure of
a parity effect in the grain and is defined as follows:
´
1 ³ 2m+2
2m
Eg.s. + Eg.s.
2
2m+1
∆M L = Eg.s.
−
(4.56)
l
where Eg.s.
is the ground state energy of BCS Hamiltonian (3.2) with l electrons.
The calculation of ∆M L is similar to the one that lead to equation (4.52), only now we
also have to take into account the change in the chemical potential
q
µ2m+2 − µ2m = 2(µ2m+1 − µ2m ) = −2(∆2m+2 − ∆2m ) = d 1 +
∆20
D2
∆2m+2 − ∆2m = O(d2 /∆0 )
The calculation results in
∆M L
∆b
d
= ∆0 −
=
2
2
s
1+
∆20 d∆0
+
[1 + φ(λ)]
D2
2D
(4.57)
where φ(λ) is defined by equation (4.38). As before, this expression is accurate up to terms
of order o(d/(min[∆0 , D])). In the physical limit ∆0 /D → 0, expression (4.57), according
to equation (4.40), reduces to the one obtained in Ref. [58]
∆M L = ∆0 −
d
+ o(d/∆0 )
2
(4.58)
The first three terms on the RHS of equation (4.57) come from the mean field (stationary
point) approximation (4.17) for the ground state energy. The last term in equation (4.57)
represents the contribution of order 1/m of quantum fluctuations around the stationary
point. The asymptotic behavior of this term in the physical limit ∆0 /D → 0 is given by
equation (4.40). In terms of d, ∆0 , and D it reads d ln(∆0 /D)∆0 /D. In this limit quantum
fluctuations will contribute to higher orders in d/∆0 as evidenced by the result[58] for ∆M L
in the regime d ¿ ∆0 . Therefore, it is of certain interest to use methods of Section 4.2 to
evaluate further corrections to ∆M L .
83
We conclude this section with a comment on the range of applicability of 1/m expansion
detailed in this paper. It is clear from equations (4.53) and (4.58) that the expansion is
applicable in the regime ∆0 ≥ d. In fact, results of Ref. [25] and [17] (see also Section 4.1)
suggest that the expansion is in powers of d/∆0 with a convergence radius d/∆0 ' 1.
4.6
Conclusion
In this paper we have shown that finite size corrections to the thermodynamical limit for
pairing Hamiltonian (3.2) can be evaluated explicitly in terms of the BCS gap ∆0 , chemical
potential µ, mean level spacing d, ultraviolet cutoff D, and the thermodynamic density of
states ν(²) to any order in d/∆0 ∼ 1/m.
We evaluated leading corrections to the ground state and lowest excitation energies, and
to Matveev-Larkin parameter (equations (4.39, 4.52, 4.53, 4.55, 4.57, 4.58)). Our results for
the ground state energy are in agreement with previous numerical studies. We showed that
the finite size correction to the condensation energy is ultraviolet divergent and therefore
comparing it to the BCS result is not justified.
We found that the gaps for pair-breaking and pair-conserving excitations of pairing
Hamiltonian (3.2) coincide up to terms of order o(1/m), where m is the number of electron
pairs on the grain. In higher orders in 1/m the two gaps are different, the difference being
of order d2 /∆0 , where d is the mean level spacing and ∆0 is the BCS gap (4.1). We showed
that the energy levels of the lowest excitations of two types cross at a certain value of the
coupling constant λ.
The range of applicability of 1/m expansion detailed in the present paper is ∆0 ≥ d.
In fact, we believe that in the physical limit ∆0 /D → 0 the expansion is a power series in
d/∆0 with a convergence radius of order one.
Note that our results significantly simplify in the physical limit ∆0 /D → 0 (e.g. compare
equations (4.52) and (4.53)). An interesting open problem is to take this limit directly in
Richardson’s equations (3.3) and to develop a simplified version of the 1/m expansion for
this case. In particular, this might help to address the problem of the crossover between
84
the fluctuation dominated (d À ∆0 ) and the bulk (d ¿ ∆0 ) regimes.
4.7
Acknowledgements
We are grateful to Akaki Melikidze for showing to us how expression (4.18) can be simplified
to equation (4.22). We thank Igor Aleiner for useful discussions. One of the authors, B. L.
A., also acknowledges the support of EPSRC under the grant GR/S29386.
4.8
Appendix A
Here we show that expression (4.18) for the correction to the ground state energy can be
simplified to equation (4.22). Indeed, define
f (z) =
n
X
1
where dk = p
(²k − µ)2 + ∆2
dk
z − ²k
k=1
Equation (4.15) now reads f (xl ) = 0. The function f (z) has n − 1 finite zeroes at z = xl
and also a zero at z = ∞. Its dual function, g(z) = 1/f (z), has n − 1 poles at z = xl and
Pn
also a pole at z = ∞ with a residue (
g(z) =
n−1
X
l=1
where we have used
P
k
k=1 dk )
−1 .
Therefore, it can be represented as
n−1
X ml
z
λdz
ml
+P
+
=
z − xl
z − xl
2
k dk
l=1
dk = 2/(λd) in accordance with gap equation (4.12). The following
equations for the residues of g(z) and f (z) are helpful:
"
X
1
dk
z − xl
= 0
=−
ml = lim [(z − xl )g(z)] = lim
z→xl
z→xl f (z)
f (xl )
(xl − ²k )2
k
#−1
=−
1
Pl
X
1
λd
1
ml
=
= g 0 (²k ) = −
+
2
dk
limz→²k [(z − ²k )f (z)]
(x
−
²
)
2
l
k
l
where the prime denotes the derivative with respect to z. Using these equations, we obtain
n−1
X
l=1
µ
n−1
n
n
XX
X
ml
1
λd
Nl
=−
=
−
2
Pl
(x
−
²
)
d
2
l
k
k
l=1 k=1
k=1
¶
n q
λdn X
=−
+
(²k − µ)2 + ∆2 (A.59)
2
k=1
Finally, substituting equation (A.59) into expression (4.18), we obtain equation (4.22).
85
4.9
Appendix B
In this Appendix we solve equation (4.26) for xl . As was discussed below equation (4.16),
each solution xl lies between two consecutive single electron levels ²k . Consider the solution
x(²) that lies between ² − d and ², where we dropped subscripts for simplicity.
Now let us multiply equation (4.26) by d and rewrite it as
X
d
|²k −²|≤Jd
(x(²) − ²k ) ²2k + ∆2
q
X
d
|²k −²|>Jd
(x(²) − ²k ) ²2k + ∆2
+
q
=0
(B.60)
√
where 1 ¿ J ¿ Λ = min[∆, D]/d. For example, one can choose J = Λ. In the first
q
√
summation in equation (B.60), ²2k + ∆2 can be replaced by ²2 + ∆2 with a relative error
of order Jd/∆. We obtain
·
X
d
|²k −²|≤Jd
(x(²) − ²k ) ²2k + ∆2
q
= 1+O
µ
Jd
∆
¶¸
¸
J ·
X
1
1
1
√
−
²2 + ∆2 p=0 p + 1 − α(²) p + α(²)
where α(²) is defined by x(²) = ² − α(²)d. To determine α(²) to the leading (m0 ) order
√
in 1/m, we can now take the limit m → ∞. With a suitable choice of J (e.g. J = Λ),
J → ∞ and (Jd)/D → 0 in this limit, while the second sum in equation (B.60) becomes a
principal value integral. Using,
∞ ·
X
p=0
¸
1
1
= −π cot(πα(²))
−
p + 1 − α(²) p + α(²)
we obtain
π cot(πα(²)) =D
−D
d²0
√
(² − ²0 ) ²02 + ∆2
(B.61)
Finally, evaluating the integral, we arrive at equation (4.30).
Corrections δα(²) to α(²) of order 1/m and higher can also be evaluated explicitly by
expanding equation (4.26) in δα(²). These corrections contribute to terms of order 1/m
and higher in the ground state energy.
86
4.10
Appendix C
Here we determine the asymptotic behavior for small λ of the integral
φ(λ) = 2
Z ∞
0
dx
cosh(πx/2)
q
2
π(1 + x ) cosh2 (πx/2) + sinh2 (1/λ)
First, we note that up to terms of order e−1/λ , one can rewrite this integral as
φ(λ) = 2
Z ∞
−∞
dx
√
π[1 + (x + x0 )2 ] 1 + e−πx
where x0 =
2
πλ
Let us divide the domain of integration into three intervals: (−∞, −a), (−a, a), and (a, ∞),
where 1 ¿ a ¿ x0 , and denote the corresponding integrals by I3 , I2 , and I1 respectively.
Each of the integrals Ik can be expanded into its own small parameter that depends on a.
The dependence on a will cancel out when the results are added together. We have
I3 = 2
Z ∞
a
dx
√
=2
π[1 + (x − x0 )2 ] 1 + eπx
I2 = 2
Z a
2
x30
I1 = 2
Z ∞
a
−a
Z ∞
dx e−πx/2 − e−3πx/2 + . . .
a
π
1 + (x − x0 )2
dx
2
√
= 2
2
−πx
x0
π[1 + (x0 + x) ] 1 + e
Z a
−a
Z a
−a
dx
√
−
π 1 + e−πx
xdx
2πa + 4 ln 2
√
+O
+ ... =
−πx
π 2 x20
π 1+e
dx
√
=2
π[1 + (x + x0 )2 ] 1 + eπx
Ã
Z ∞
dx 1 − e−πx /2 + . . .
a
π
= O(e−πa/2 )
1 + (x + x0 )2
a2
x30
!
2a
2
−
=
+O
πx0 πx20
Ã
a2
x30
!
Adding I1 , I2 , and I3 , we obtain equation (4.41). Higher order terms can also be calculated
by the same method.
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