Interactions, superconducting $T_c$, and fluctuation magnetization for
  two coupled dots in the crossover between the Gaussian Orthogonal and Unitary
  ensembles
Untitled
Interactions, superconducting Tc, and fluctuation magnetization
for two coupled dots in the crossover between the Gaussian
Orthogonal and Unitary ensembles
Oleksandr Zelyak∗ and Ganpathy Murthy†
Department of Physics and Astronomy,
University of Kentucky, Lexington, Kentucky 40506, USA
Igor Rozhkov‡
Department of Physics, University of Dayton,
300 College Park, Dayton, OH 45469
(Dated: November 25, 2018)
Abstract
We study a system of two quantum dots connected by a hopping bridge. Both the dots and
connecting region are assumed to be in universal crossover regimes between Gaussian Orthogonal
and Unitary ensembles. Using a diagrammatic approach appropriate for energy separations much
larger than the level spacing we obtain the ensemble-averaged one- and two-particle Green’s func-
tions. It turns out that the diffuson and cooperon parts of the two-particle Green’s function can
be described by separate scaling functions. We then use this information to investigate a model
interacting system in which one dot has an attractive s-wave reduced Bardeen-Cooper-Schrieffer
interaction, while the other is noninteracting but subject to an orbital magnetic field. We find that
the critical temperature is nonmonotonic in the flux through the second dot in a certain regime of
interdot coupling. Likewise, the fluctuation magnetization above the critical temperature is also
nonmonotonic in this regime, can be either diamagnetic or paramagnetic, and can be deduced from
the cooperon scaling function.
PACS numbers: 73.21.La, 05.40.-a, 73.50.Jt
Keywords: quantum dot, scaling function, crossover, quantum criticality
http://arxiv.org/abs/0704.0919v1
I. INTRODUCTION
The idea of describing a physical system by a random matrix Hamiltonian to explain
its spectral properties goes back to Wigner1,2. It was further developed by Dyson, Mehta
and others, and became the basis for Random Matrix Theory (RMT)3. First introduced in
nuclear physics, RMT has been used with great success in other branches of physics and
mathematics. A notable example was a conjecture by Gorkov and Eliashberg4 that the
single-particle spectrum of a diffusive metallic grain is controlled by RMT. This conjecture
was proved by Altshuler and Shklovskii5 who used diagrammatic methods and by Efetov
who used the supersymmetry method6. In 1984 Bohigas, Giannoni and Schmit7 conjectured
that RMT could also be employed in the study of ballistic quantum systems whose dynam-
ics is chaotic in the classical limit. Their conjecture broadened the area of applicability
of RMT enormously and was supported by numerous ensuing experiments and numerical
simulations7–10. The crucial energy scale for the applicability of RMT is the Thouless energy
ET = ~/τerg, where τerg is the time for a wave packet to spread over the entire system. For
a diffusive system of size L, we have ET ≃ ~D/L2, while for a ballistic/chaotic system we
have ET ≃ ~vF/L, where vF is the Fermi velocity.
In this paper we consider a system of two quantum dots/nanoparticles which are coupled
by a hopping bridge. The motion of electrons inside each dot can be either ballistic or
diffusive. In the case of ballistic dots we assume that the dots have irregular shapes leading
to classically chaotic motion, so that RMT is applicable.
RMT Hamiltonians fall into three main ensembles3. These are the Gaussian Orthogonal
Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble
(GSE). They are classified according to their properties time-reversal (TR). The Hamilto-
nians invariant with respect to TR belong to the GOE. An example of GOE is a quantum
dot which has no spin-orbit coupling and is not subject to an external magnetic field. GUE
Hamiltonians, on the contrary, are not invariant with respect to TR and describe motion
in an orbital magnetic field, with or without spin-orbit coupling. Hamiltonians from GSE
group describe systems of particles with Kramers degeneracy that are TR invariant but
have no spatial symmetries, and correspond to systems with spin-orbit coupling but with
no orbital magnetic flux. In our paper we only deal with the first two classes.
For weak magnetic flux the spectral properties of the system deviate from those predicted
by either the GOE or the GUE11. In such cases the system is said to be in a crossover3. For
these systems the Hamiltonian can be decomposed into real symmetric and real antisym-
metric matrices:
HS + iXHA√
1 +X2
, (1)
where X is the crossover parameter12 which is equal, up to factors of order unity, to Φ/Φ0,
where Φ is the magnetic flux through the dot, and Φ0 = h/e is the quantum unit of magnetic
flux. Note that the gaussian orthogonal and unitary ensembles are limiting cases of X → 0
and X → 1 respectively.
To understand the meaning of the crossover parameter consider the Aharonov-Bohm
phase shift picked up by a ballistic electron in a single orbit in the dot:
∆φ = 2π
. (2)
For one turn the flux enclosed by the trajectory is proportional to Φ = BL2, where L is the
size of the dot. After N turns the total flux is Φtotal =
NΦ, where factor
N originates
from the fact that electron has equal probability to make clockwise or counterclockwise
orbits, and thus does a random walk in the total flux enclosed. The minimal phase shift
for the electron to notice the presence of the magnetic flux is of the order 2π, and thus
the minimal cumulative flux enclosed by the orbit should be Φ0 =
NΦ. This leads to
N = (Φ0/Φ)
2, while the time to make N turns is τ = LN/vf (for a ballistic/chaotic dot).
From the Heisenberg uncertainty principle the associated energy scale is:
Ecross ≈
, (3)
where ET is the ballistic Thouless energy
13. For a diffusive dot it should be substituted by
the diffusive Thouless energy ET ∼= ~D/L2. One can see that when Φ is equal to Φ0, EX is
equal to ET which means that energy levels are fully crossed over.
In this paper the reader will encounter many crossover parameters, and thus many
crossover energy scales. By a line of argument similar to that leading to Eq. (3), it
can be shown that to every crossover parameter Xi there is a corresponding energy scale
EXi ≃ X2i ET .
Breaking the time reversal symmetry of system changes the two-particle Green’s function.
While the two-particle Green’s function can in general depend separately on ET , EX , and
the measurement frequency ω, it turns out that in the universal limit ω, EX ≪ ET , it
becomes a universal scaling function of the ratio EX/ω. The scaling function describes the
modification of 〈GR(E+ω)GA(E)〉 as one moves away from the “critical” point ω = 0. The
limits of the scaling function can be understood as follows: If the measurement frequency
ω is large (small) compared to the crossover energy scale EX , the 〈GR(E +ω)GA(E)〉 takes
the form of the GOE (GUE) ensemble correlation function. If ω ∼ EX , the Green’s function
describes the system in crossover regime.
The one-particle Green’s function 〈GR(E)〉 is not critical as ω → 0, although it gets
modified by the interdot coupling. The two-particle Green’s function 〈GR(E + ω)GA(E)〉
always has a diffuson mode14, that diverges for small ω in our large-N approximation,
which means that our results are valid on scales much larger than mean level spacing. This
divergence is not physical and will be cut off by vanishing level correlations for ω ≪ δ
in a more exact calculation15. On the other hand, the energy scale ω should be smaller
than Thouless energy of the system for RMT to be applicable. These limitations hold for
the crossover energy EX as well. In what follows we study the regime corresponding to
δ ≪ ω,EX ≤ ET .
The other term that appears in the two-particle Green’s function is a cooperon mode. In
general the cooperon term is gapped if at least one of the crossover parameters is different
from zero. In the case when the total Hamiltonian of the system is time reversal invariant, all
the crossover parameters are zero and the cooperon, just like the diffuson, becomes gapless.
Finally, when each part of compound system belongs to the GUE (the case when all crossover
parameters are much larger than ω) the cooperon term disappears.
Our study has a two-fold motivation. The first part comes from works on cou-
pled structures with noninteracting particles in acoustic and electronic systems16–18, and
crossovers11,19–22. We focus on a complete description of the crossover regimes in all three
regions (the two dots and the bridge) and define scaling functions for the diffuson and
cooperon parts of the two-particle Green’s function. Using parameters analogous to EX we
describe crossover regimes in dots 1 and 2 and the effects of the tunable hopping between
them. Varying these parameters allows us to obtain results for various physical realizations,
when different parts of the compound system behave as pure GOE, GUE, or belong to the
crossover ensemble. In electronic systems it is easy to break time-reversal by turning on an
external orbital magnetic flux. In acoustic systems one can break time-reversal by rotating
the system or a part thereof. As mentioned before, the system of two dots coupled by hop-
ping has been investigated before using supersymmetry methods18. However, the authors
considered only the GUE, whereas here we are interested in the full crossover. In fact, the
crossover is essential to the second aspect of our work, as will become clear immediately.
The second part of our motivation is the possibility of using the information gained in
noninteracting systems to predict the behavior of interacting systems23–26. We consider
interacting systems controlled by the Universal Hamiltonian27–30, which is known to be the
interacting low-energy effective theory31–33 deep within the Thouless band |ε− εF | ≪ ET in
the renormalization group34,35 sense for weak-coupling when the kinetic energy is described
by RMT and the Thouless number g = ET/δ ≫ 1. For the GOE the Universal Hamiltonian
HU has the form
27–30
α,scα,s +
N̂2 − JS2 + λT †T (4)
where N̂ is the total particle number, S is the total spin, and T =
cβ,↓cβ,↑. In addition
to the charging energy, HU has a Stoner exchange energy J and a reduced superconducting
coupling λ. This last term is absent in the GUE, while the exchange term disappears in the
In this paper we concentrate on the reduced Bardeen-Cooper-Schrieffer (BCS) coupling
λ which leads to a mean-field superconducting state when λ < 0. Previous work by one
of us26 sets the context for our investigation. We consider an interacting system which has
a single-particle symmetry and a quantum phase transition in the limit ET /δ → ∞. An
example relevant to us is a superconducting nanoparticle originally in the GOE. It has the
reduced BCS interaction and time-reversal symmetry, and the (mean-field) quantum phase
transition is between the normal and superconducting states and occurs at λ = 0. Now con-
sider the situation when the symmetry is softly broken, so that the single-particle dynamics
is described by a crossover RMT ensemble. It can be shown26 that this step allows us to
tune into the many-body quantum critical regime36–38 of the interacting system. Thus, the
scaling functions of the noninteracting crossover are transmuted into scaling functions of
the interacting system in the many-body quantum critical regime. In our example, the or-
bital magnetic flux breaks the time-reversal symmetry which is crucial to superconductivity.
FIG. 1: The system of two vertically coupled quantum dots.
When the orbital flux increases to a critical value, it destroys the mean-field superconduct-
ing state. Above the critical field, or more generically above the critical temperature, the
system is in the quantum critical regime.
To be more specific, we consider two vertically coupled quantum dots, the first of which
has an attractive reduced BCS coupling, while the second has no BCS coupling. Fig. 1
shows the geometry, the reason for which will become clear soon. We apply an orbital
magnetic flux only through (a part of) the second dot, and observe the effect on the coupled
system. Our main results are for the mean-field critical temperature Tc of the system, and
its magnetization in the normal state (above Tc) as a function of the flux in the normal
nanoparticle. Such a system could be realized physically without too much difficulty, by,
for example, growing a thin film of normal metal (such as Au) on an insulating substrate,
then a layer of insulator which could serve as the hopping bridge, and finally a thin film of
superconductor(such as Al, which has a mean-field superconducting transition temperature
of around 2.6K). The orbital flux can be applied selectively to the Au layer as shown in
Fig. 1 by a close pair of oppositely oriented current carrying wires close to the Au quantum
dot, but far from the Al quantum dot.
The reason for this geometry is that we want to disregard interdot charging effects entirely
and concentrate on the BCS coupling. The Hamiltonian for the coupled interacting system
contains charging energies for the two dots and an interdot Coulomb interaction24.
N21 +
N22 + U12N1N2 (5)
Defining the total number of particles as N = N1 +N2, and the difference in the number as
n = N1 −N2 the interaction can also be written as
U1 + U2 + 2U12
U1 + U2 − 2U12
U1 − U2
nN (6)
We see that there is an energy cost to transfer an electron from one dot to the other. This
interaction is irrelevant in the RG sense24, but vanishes only asymptotically deep within an
energy scale defined by the hopping. Our geometry is chosen so as to make U1 = U2 = U12 as
nearly as possible, which can be achieved by making the dots the same thickness and area,
and by making sure that their vertical separation is much smaller than their lateral linear
size. In this case, since N is constant, we can ignore charging effects entirely. Charging effects
and charge quantization in finite systems can be taken into account using the formalism
developed by Kamenev and Gefen39, and futher elaborated by Efetov and co-workers40,41.
Since our primary goal is to investigate quantum critical effects associated with the BCS
pairing interaction, we will assume the abovementioned geometry and ignore charging effects
in what follows.
After including the effect of the BCS interaction, we find the surprising result that in
certain regimes of interparticle hopping strength, the mean-field transition temperature of
the system can increase as the flux through the second quantum dot increases. Indeed, its
behavior can be monotonic increasing, monotonic decreasing, or nonmonotonic as the flux
is increased. We can qualitatively understand these effects by the following considerations.
In the absence of orbital flux, hopping between the dots reduces Tc since it “dilutes” the
effect of the attractive BCS coupling present only in the first dot. The application of an
orbital flux through the second dot has two effects: (i) To raise the energy of Cooper pairs
there, thus tending to localize the pairs in the first dot and raise the Tc. (ii) To cause time-
reversal breaking in the first dot, and reduce Tc. The nonmonotonicity of Tc arises from the
competition between these two effects.
Another quantity of interest above the mean-field Tc is the fluctuation magnetization
which corresponds to gapped superconducting pairs forming and responding to the external
orbital flux. In contrast to the case of a single quantum dot subjected to an orbital flux,
we find that the fluctuation magnetization42 can be either diamagnetic (the usual case) or
paramagnetic. A paramagnetic magnetization results from a free energy which decreases as
the flux increases. The origin of this effect is the interplay between the localizing effect of
high temperature or the orbital flux in the second dot on the one hand, and the reduced
BCS interaction on the other.
The regimes we describe should be distinguished from other superconducting single-
particle RMT ensembles discovered in the past decade43,44, which apply to a normal meso-
scopic system in contact with two superconductors with a phase difference of π between
their order parameters43 (so that there is no gap in the mesoscopic system despite Andreev
reflection), or to a mesoscopic d-wave superconducting system44. In our case, the symmetry
of the superconducting interaction is s-wave. However, the most important difference is that
we focus on quantum critical fluctuations, which are inherently many-body, while the RMT
classes described previously are single-particle ensembles43,44.
This paper is organized as follows. In Sec. II we review the basic steps of calculating
the one-particle and two-particle Green’s functions for a single dot. Then in Sec. III we
present the system of Dyson equations for the one-particle Green’s function in the case of
two coupled dots and solve it in the limit of weak coupling. In addition, we set up and solve
the system of four Bethe-Salpeter equations for the two-particle Green’s function. In Sec.
IV we apply our results to the system of superconducting quantum dot weakly coupled to
other quantum dot made from a normal metal. We end with our conclusions, some caveats,
and future directions in Sec. V.
II. REVIEW OF RESULTS FOR A SINGLE DOT.
Our goal in this section is to calculate the statistics of one and two-particle Green’s
functions for an uncoupled dot in a GOE→ GUE crossover (see appendix A, and12 for more
details), starting from the series expansion of Green’s function:
〈β|GR(E)|α〉 = GRαβ(E) =
E+ −H
(E+)2
(E+)3
+ . . . . (7)
We are interested in averaging this expansion over the appropriate random matrix ensemble.
The corresponding Dyson equation for averaged Green’s function is:
The bold line denotes the averaged propagator 〈GR(E)〉 and regular solid line defines the
bare propagator 1/E+ with E+ = E + iη, where η is infinitely small positive number. Here
Σ stands for self-energy and is a sum of all topologically different diagrams.
One can solve Dyson equation approximating self-energy only by first leading term and
find:
−E2, (9)
where δ is the mean level spacing. This approximation works only for E ≫ δ. As E gets
comparable with δ, other terms in expansion for Σ should be taken into account.
Then, the average of the one-particle Green’s function is given by:
〈GRαβ(E)〉 =
)2 −E2
Next, we repeat the procedure for the averaged two-particle Green’s function, which can be
represented by the series:
where two bold lines on the left hand side denote 〈GR(E + ω)GA(E)〉. The leading contri-
bution comes from ladder and maximally crossed diagrams. The sum of these diagrams can
be conveniently represented by Bethe-Salpeter equation. For example, the contribution of
all the ladder diagrams can be expressed in closed form by:
where ΠD is a self-energy. For maximally crossed diagrams we have similar equation:
where ΠD and ΠC are related to the connected part of two-particle Green’s function as:
In the limit of ω being much smaller than bandwidth (ω ≪ Nδ), the two-particle Green’s
function (connected part) is expressed as:
〈GRαγ(E + ω)GAδβ(E)〉 =
δαβδγδ
δαδδγβ
1 + iEX
The second term is a contribution of maximally crossed diagrams. EX is a crossover energy
scale, connected to the crossover parameter as EX = 4X
2Nδ/π.
Depending on values of EX one can speak of different types of averaging. If EX ≪ ω, we
get average over GOE ensemble, if EX is of order ω, averaging is performed over ensemble
being in crossover, and, if EX ≫ ω, contribution of maximally crossed diagrams can be
disregarded, thus going to the limit of the GUE ensemble.
III. TWO COUPLED DOTS.
Next we discuss general framework of our calculation and calculate correlation functions
for our system of interest, which is two weakly coupled quantum dots (see appendix B for
more technical details). The Hamiltonian for this system can be represented as:
Htot =
V † 0
V † H2
 . (16)
where H1 andH2 are the Hamiltonians of uncoupled dots 1 and 2. The coupling is realized by
a matrix V . The elements of H1, H2, and V are statistically independent random variables.
We assume that both dots and the hopping bridge are in crossover regimes, characterized
by parameters X1, X2, and Γ respectively.
In the crossover matrices Hi and V are given by:
HSi + iXiH
1 +X2i
, i = 1, 2; V =
V R + iΓV I√
1 + Γ2
, (17)
where H
i is a symmetric (antisymmetric) part of Hi, and V
R,I is real (imaginary) matrix.
In what follows we assume that the bandwidths in dot 1 and dot 2 are the same. That
is, N1δ1 = N2δ2. This should not make any difference in the universal limit N → ∞.
In addition we introduce the parameter ξ – the ratio of mean level spacing in two dots:
ξ = δ1/δ2. For each realization of matrix elements of the Hamiltonian Htot, the Green’s
function of this system can be computed as follows:
G = (I ⊗ E −H)−1 =
E −H1 −V
−V † E −H2
G11 G12
G21 G22
 . (18)
Each element of G has the meaning of a specific Green’s function. For example, G11 and
G22 are the Green’s functions that describe particle propagation in dots 1 and 2 respectively.
On the other hand, G12 and G21 are the Green’s functions representing travel from one dot
to another.
Calculating (I ⊗E −H)−1 one finds the components of G. For example,
G11 =
(E −H1)− V (E −H2)−1V †
= G1 +G1V G2V
†G1 +G1V G2V
†G1V G2V
†G1 + . . .
where G1 and G2 are bare propagators in dot 1 and dot 2 defined by G1 = (E −H1)−1 and
G2 = (E −H2)−1.
To find the ensemble average of G11 one needs to average the whole expansion (19) term
by term. For coupled dots Gij interrelated and in large N approximation can be found from
the following system of equations:
The bold straight and wavy lines with arrows represent averaged Green’s functions
〈Gαβ,1(E)〉 and 〈Gij,2(E)〉 respectively, while regular solid lines are bare propagators in
dots 1 and 2. The dotted line describes pairing between hopping matrix elements V , and
the dashed (wavy) line denotes pairing between matrix elements of H1 (H2).
The system (20) accounts for all possible diagrams without line crossing. Diagrams
containing crossed lines of any type are higher order in 1/N and can be neglected when
N → ∞. If the hopping between dots is zero, this system decouples into two separate
Dyson equations for each dot. In the case of weak coupling (U ≪ 1), where U is a parameter
controlling the strength of coupling between dots, this system can be readily solved. As zero
approximation, we use results for a single dot.
In this approximation one-particle Green’s function for dot 1 and dot 2 are calculated as
follows:
〈GRαβ,1(E)〉 =
〈GRαβ,0(E)〉
E−2Σ0
1− ǫ2
1 + U
1 + i ǫ√
〈GRij,2(E)〉 =
〈GRij,0(E)〉
1− U√
E−2Σ0
1− ǫ2
1 + U
1 + i ǫ√
where ǫ is a dimensionless energy ǫ = πE/2Nδ. We used subindex 0 in Σ0 and 〈GR0 (E)〉 to
denote solutions for one uncoupled dot.
In the large N approximation the contribution to the two-particle Green’s function comes
from ladder diagrams and maximally crossed diagrams. It is convenient to sum them sepa-
rately. The ladder diagram contribution can be found from the following system of equations:
where ΠDij with proper external lines denote various two-particle Green’s functions. As in
the case of the one-particle Green’s function equations, if the inter-dot coupling is zero, the
system reduces to two Bethe-Salpeter equations for uncoupled dots.
The system of four equations (22) can be broken into two systems of two equations to
〈GRαγ,1(E + ω)GAδβ,1(E)〉D1 =
N21 δ1
δαβδγδ
〈GRil,2(E + ω)GAkj,2(E)〉D2 =
N22 δ2
δijδlk
where gD are the scaling functions of diffusion terms in dot 1 and dot 2 defined by:
gD1 =
1 + i√
1 + i(
ξ + 1√
gD2 =
1 + i
1 + i(
ξ + 1√
Here EU = 2UNδ/π is the interdot coupling energy scale. These dimensionless functions
show how diffusion part is modified due to the coupling to another dot.
Next, for the maximally crossed diagrams the system of equations we have:
The subsequent solution of this system produces:
〈GRαγ,1(E + ω)GAδβ,1(E)〉C1 =
N21 δ1
δαδδγβ
〈GRil,2(E + ω)GAkj,2(E)〉C2 =
N22 δ2
δikδlj
where gC are the scaling functions for cooperon term defined according to:
gC1 =
1 + i√
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
gC2 =
1 + i
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
Here EX1,2 = 4X
1,2Nδ/π, and EΓ = 4Γ
2EU/(
ξ + 1√
) are the crossover energy scales,
describing transition from GOE to GUE ensemble in dot 1 and dot 2, as well as in hopping
bridge V .
As we determined how the scaling function gC modifies cooperon part of two-particle
Green’s function and depends on the crossover energy scales defined above, we are ready to
proceed with write up the connected part of the total two-particle Green’s function, which
is a sum of diffuson and cooperon parts:
〈GRαγ,1(E + ω)GAδβ,1(E)〉 =
N21 δ1
δαβδγδ
1 + i√
1 + i(
ξ + 1√
N21 δ1
δαδδγβ
1 + i√
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
〈GRil,2(E + ω)GAkj,2(E)〉 =
N22 δ2
δijδlk
1 + i
1 + i(
ξ + 1√
N22 δ2
δikδlj
1 + i
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
In general, the coupling between dots changes the bandwidth of each dot. Corrections to
the bandwidth are of the order of U and can be neglected for weak coupling. Calculating
approximations to the second order in U one can be ensure that one-particle and two-particle
Green’s functions can be treated perturbatively.
Diagrams on Fig.2 show the typical behavior of absolute value and phase of scaling
functions gD and gC in dot 1. All energy parameters are measured in units of EU .
Next we analyze the temporal behavior of the computed statistical characteristics. The
Fourier transform of the two-particle Green’s function shows the time evolution of the density
matrix of the system. One can observe that the diffuson part of 〈GRGA〉 diverges for small
ω. To get the correct behavior we replace 1/ω with ω/(ω2 + η2), and take η to zero in the
final result. As for the cooperon term, it stays regular in the small ω limit if at least one of
the crossover parameters differs from zero.
First of all, we look at the Fourier transform of 〈GRGA〉 in the first dot. We have
FIG. 2: Absolute value and phase of diffuson (a,b) and cooperon (c,d) scaling functions in dot
1. Frequency ω is measured in units of EU . For these graphs the crossover parameters are:
EX1/EU = EX2/EU = 1, EΓ/EU = 0.8, ξ = 1.
〈GRαγ,1(t)GAδβ,1(t)〉 = δαβδγδ
N1(1 + ξ)
ξ+ 1√
)EU t
+ δαδδγβ
EX2 +
a+ − a−
e−ta− − e−ta+
, (29)
where a± depend on the crossover parameters (see Eq. (C11) in appendix C)
Then, for the corresponding quantity in the second dot the Fourier transform produces:
〈GRil,2(t)GAkj,2(t)〉 = δijδlk
N2(1 + ξ)
ξ+ 1√
)EU t
+ δikδlj
EX1 +
a+ − a−
e−ta− − e−ta+
. (30)
IV. TWO COUPLED METALLIC QUANTUM DOTS
In this section we apply the results obtained in the previous sections to an interacting
system. We consider two vertically coupled metallic quantum dots, as shown in Fig. 1, the
first of which is superconducting and the second noninteracting. For simplicity the quantum
dots are assumed to have the same level spacing (ξ = 1). The calculations presented
in this section can be extended to the case ξ 6= 1 in a straightforward way. The first
(superconducting) quantum dot and the hopping bridge belong to the GOE ensemble. A
nonzero orbital magnetic flux penetrating the second (noninteracting) quantum dot drives
it into the GOE to GUE crossover described by the crossover energy scale EX2 . The other
crossover energy scale EU describes the hopping between the quantum dots. Because of this
hopping one can observe a nonzero magnetization in the first particle caused by a magnetic
flux through the second particle. Roughly speaking, when the electrons in the first dot travel
to the second and return they bring back information about the orbital flux.
We wish to compute the magnetization as a function of orbital flux, as well as the mean-
field critical temperature. It should be noted that since the quantum dot is a finite system,
there cannot be any true spontaneous symmetry breaking. However, when the mean-field
superconducting gap ∆BCS ≫ δ, the mean-field description is a very good one45–47. Recent
numerical calculations have investigated the regime ∆BCS ≃ δ where quantum fluctuations
are strong48. We will focus on the quantum critical regime of the system above the mean-field
critical temperature/field, so we do not have to worry about symmetry-breaking.
We start with BCS crossover Hamiltonian for the double-dot system including the inter-
actions in the first dot and the hopping between the dots26:
HBCSX2 =
µ0ν0c
cν0s − λT †T +
i0j0s
i0j0s
ci0s +
Vµ0i0(c
ci0s + h.c.)
µ,scµ,s − δλ̃T †T, (31)
where H(2) contains the effect of the orbital flux through the second quantum dot. Here
T, T † are the operators which appear in the Universal Hamiltonian, and are most simply
expressed in terms of electron creation/annihilation operators in the original GOE basis of
the first dot (which we call µ0, ν0) as
cµ0,↓cµ0,↑ (32)
Now we need to express the operators cµ0,s in terms of the eigenoperators of the combined
single-particle Hamiltonian of the system of two coupled dots. The result is
Mµνcν,↓cµ,↑, Mµν =
ψµ(µ0)ψν(µ0), (33)
where ǫµ denotes the eigenvalues of the total system, cµ,s operator annihilates electron in the
orbital state µ with spin s, ψµ(µ0) is the eigenvector of the compound system, δ is the mean
level spacing of a single isolated dot, λ̃ > 0 is the attractive dimensionless BCS coupling
valid in region of width 2ωD around the Fermi energy. Note that while the indices µ, ν
enumerate the states of the total system, the index µ0 goes only over the states of the first
dot, since the superconducting interaction is present only in the first dot.
To study the magnetization of the first quantum dot in the crossover we follow previous
work by one of us26: We start with the partition function Z = Tr(exp−βH) where β = 1/T
is the inverse temperature. We convert the partition function into an imaginary time path
integral and use the Hubbard-Stratanovich identity to decompose the interaction, leading
to the imaginary time Lagrangian
c̄µ,s(∂τ − ǫµ)cµ,s + σT̄ + σ̄T (34)
where σ, σ̄ are the bosonic Hubbard-Stratanovich fields representing the BCS order parame-
ter and c̄, c are Grassman fields representing fermions. The fermions are integrated out, and
as long as the system does not have a mean-field BCS gap, the resulting action for σ, σ̄ can
be expanded to second order to obtain
Seff ≈
|σ(iωn)|2( 1λ̃ − fn(β, EX , ωD)) (35)
fn(β, EX , ωD) = δ
|Mµν |2 1−NF (ǫµ)−NF (ǫν)ǫµ+ǫν−iωn (36)
where ωn = 2πn/β, and the sums are restricted to |ǫµ|, |ǫν| < ~ωD. We see that the
correlations between different states µ, ν play an important role. Deep in the crossover (for
EX ≫ δ) we can replace |Mµν |2 by its ensemble average26. We will also henceforth replace
the summations over energy eigenstates by energy integrations with the appropriate cutoffs.
In previous work26 the statistics23–25 of |Mµν |2 was used to obtain analytical results for this
expression.
The (interacting part of the) free energy of the system in the quantum critical regime is
given by26:
ln(1− λ̃f(iωn, β, EX2)), (37)
where f is the scaling function given by expression:
f(iωn, β, EX2) = δ
|Mµν |2
1− nµ(β)− nν(β)
ǫµ + ǫν − iωn
, (38)
nν(β) = (1 + exp(βǫν))
−1 is the Fermi-Dirac distribution. We have shifted the energy so
that the chemical potential is 0.
Converting this double sum into integral and substituting |Mµν |2 by its ensemble average
(see Appendices D and E), we get:
dǫ1dǫ2
(ǫ1 − ǫ2)2 + EX2EU + E2X2
((ǫ1 − ǫ2)2 − EX2EU)2 + (EX2 + 2EU)2(ǫ1 − ǫ2)2
tanh(βǫ1
) + tanh(βǫ2
ǫ1 + ǫ2 − iωn
where ωD is the Debye frequency, and β = 1/kBT is the inverse temperature.
One can decompose the ratio in the first part of integrand into two Lorentzians to get26:
E2X2 + EUEX2 − E
E22 − E21
4(~ωD)
2 + ω2n
C ′/β2 + (E1 + |ωn|)2
E22 −E2X2 −EUEX2
E22 −E21
4(~ωD)
2 + ω2n
C ′/β2 + (E2 + |ωn|)2
. (40)
Here C ′ ≈ 3.08 and E1,2 depend on crossover energy scales as follows:
E21,2 =
(EX2 + 2EU)
2 − 2EUEX2 ∓
(EX2 + 2EU)
2(E2X2 + 4E
. (41)
The magnetization can then be obtained from the free energy:
M = −
=Mnonint +
1− λ̃fn
, (42)
where Mnonint is the contribution from noninteracting electrons
49. We will be interested in
the second term, which is the fluctuation magnetization42.
For illustrative purposes, we use the parameters for Al in all our numerical calculations,
with ωD = 34meV and λ̃ = 0.193. This leads to a mean-field transition temperature
Tc0 = 0.218meV = 2.6K for an isolated Al quantum dot in the absence of magnetic flux.
In all our calculations we evaluate Matsubara sums with a cutoff exp−|ωn|/ωD. We have
verified that changing the cutoff does not qualitatively affect our results, but only produces
small numerical changes.
It will be informative to compare the two-dot system with a single dot subject to an
orbital magnetic flux26 (see Fig. 3). We draw the reader’s attention to two important
features. Firstly, the critical temperature Tc decreases monotonically with EX , resulting from
the fact that time-reversal breaking disfavors superconductivity. Secondly, the fluctuation
magnetization is always negative, or diamagnetic, resulting from the fact that the free energy
monotonically increases as the orbital flux increases.
Now let us turn to our system of two quantum dots coupled by hopping. Before we carry
out a detailed analysis, it is illuminating to inspect the behavior of E1,2 and the coefficients
of the two logarithms in Eq. (40) (which we call A1,2) as a function of EX2 . This is shown
in Fig. 4. E1 tends to EX2/2 for EX2 ≪ EU , and to EU in the opposite limit EX2 ≫ EU .
E2 tends to EU for EX2 ≪ EU , while in the opposite limit EX2 ≫ EU E2 → EX2 . Both
coefficients A1,2 start at
for small EX2. For EX2 ≫ EU A1 → 1, while A2 → 0.
FIG. 3: Magnetization (per unit volume) in a single dot system as a function of temperature
for different values of crossover parameters EX . Panel (d) shows the dependence of the critical
temperature on EX .
The asymptotic regimes T,EX2 ≪ EU and T,EX2 ≫ EU can be understood simply. In
the first regime, EU is the largest energy scale, and far below it the spatial information that
there are two distinct quantum dots is lost. The system behaves like a single large dot with
a smaller “diluted” superconducting coupling. On the other hand, when T,EX2 ≫ EU , A2
is vanishingly small, and the system resembles the isolated first dot with a superconducting
coupling λ̃ but with a crossover energy EU . Note that the approach of the energies to the
asymptotes is slow, so for a particular value of EU it may happen that one cannot realistically
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6
FIG. 4: The behavior of Log coefficients in Eq. (40) and E1,E2 as functions of the ratio EX2/EU .
approach the asymptotic regime without running into either δ at the lower end or ωD at the
higher end. Finally, one can envisage situations in which EX2 ≪ EU but T ≥ EU , for which
there are no simple pictures.
The temperature dependence of magnetization per unit volume for different values of
crossover parameters EX2 and EU (excluding the part due to noninteracting electrons) is
shown in Fig. 5.
In the range where magnetization changes significantly, the fluctuation magnetization
shows both diamagnetic and paramagnetic behavior. This is in contrast to the case of a
single superconducting quantum dot subjected to an orbital flux where the fluctuation mag-
netization is always diamagnetic (Fig. 3). Close to T = 0 an increase in temperature makes
the fluctuation magnetization more diamagnetic. A further temperature increase changes
the fluctuation magnetization from diamagnetic to paramagnetic. For large values of temper-
ature the fluctuation magnetization is paramagnetic and decreasing as T increases. Another
set of diagrams, Fig. 6, demonstrates the dependence of the fluctuation magnetization in
the first dot on crossover parameter EX2 in the second dot. Generically, we find that at low
T the fluctuation magnetization is diamagnetic while at high T it is paramagnetic.
The variation of crossover energy scales EX2 and EU does not change the qualitative
behavior of the fluctuation magnetization as a function of T or EX2 . A paramagnetic
magnetization is counterintuitive in superconducting system, because one believes that “an
orbital flux is the enemy of superconductivity”, and therefore that the free energy must
FIG. 5: Magnetization (per unit volume) as a function of temperature for different values of
crossover parameters EX2 and EU . The fluctuation magnetization is diamagnetic for low T and
paramagnetic for high T .
always increase as the orbital flux increases. This assumption is false for our system. The
explanation is fairly simple, as we will see immediately after the results for Tc have been
presented.
The mean-field critical temperature Tc of transition between normal and superconducting
state strongly depends on EX2 and EU . As one can see from Fig. 7, for very strong
hopping (EU ≫ Tc0) between quantum dots Tc is monotonically decreasing as EX2 increases.
On the other hand, for intermediate hopping Tc has a maximum as a function of orbital
flux, which means that for small values of orbital magnetic flux Tc increases as the orbital
flux increases. Finally, when EU is very weak, Tc monotonically increases as a function of
FIG. 6: Fluctuation magnetization in the first dot vs crossover parameter EX2 in the second dot
for different values of temperature. The fluctuation magnetization is diamagnetic for low T and
paramagnetic for high T .
orbital flux through the second quantum dot. This is in contrast to the behavior of a single
superconducting quantum dot for which Tc decreases monotonically as a function of orbital
flux.
These counterintuitive phenomena can be understood in terms of the following cartoon
picture. One can think of the two dots as two sites, each capable of containing a large
number of bosons (the fluctuating pairs). The BCS pairing interaction occurs only on the
first site. When there is no magnetic flux, hopping delocalizes the bosons between the two
sites, leading to a “dilution” of the BCS attraction and a low critical temperature. The effect
FIG. 7: Critical temperature as a function of EX2 for several intermediate to strong values (com-
pared to Tc0) of the hopping parameter EU . For larger values of EX2 (not shown on graphs) critical
temperature is equal to zero.
of the magnetic flux on the second dot is twofold: (i) Firstly, it gaps the cooperon of the
second dot, which we think of as raising the energy for the bosons to be in the second dot.
(ii) Secondly, by virtue of the interdot hopping, a small time-reversal symmetry breaking
is produced in the first dot, thereby raising the energy of the bosons there as well. As the
flux through the second dot rises, the bosons prefer to be in the first dot since they have
lower energy there. The more localized the cooper pairs are in the first dot due to effect (i),
the more “undiluted” will be the effect of the BCS attraction λ, and the more favored will
FIG. 8: Behavior of critical temperature TC as a function of EX2 for small to intermediate values
(compared to Tc0) of EU .
be the superconducting state. However, effect (ii) produces a time-reversal breaking in the
first dot, thus disfavoring the superconducting state. These two competing effects lead to
the varying behaviors of Tc and the fluctuation magnetization versus the orbital flux in the
second quantum dot. When the hopping between the quantum dots is weak (EU < Tc0), the
first effect dominates, and Tc increases with EX2 . When the hopping is stronger (EU ≃ Tc0)
the first effect dominates at small orbital flux, and the second at large orbital flux. Finally,
at very large hopping (EU ≫ Tc0), effect (ii) is always dominant.
When considering the magnetization one must take into account the temperature as
well, so the picture is more complex. The general feature is that effect (i) which tends
to localize the pairs in the first dot also tends to decrease the interacting free energy of
the system, which leads to a paramagnetic fluctuation magnetization. Effect (ii), which
breaks time-reversal in the first dot, increases the free energy of the system and thus leads
to a diamagnetic fluctuation magnetization. Based on our results we infer that at high
temperature the coherence of pair hopping is destroyed leading to more localization in the
first quantum dot. The consequences of high T are thus similar to that of the effect (i): A
lowering of the interacting free energy and a paramagnetic fluctuation magnetization.
We can make this picture a bit more quantitative for the behavior of Tc with respect to
EX . Consider once more the scaling function of Eq. (40), which we reproduce here for the
reader’s convenience
fn(EX2 , EU , T ) =
E2X2 + EUEX2 − E
E22 − E21
4(~ωD)
2 + ω2n
C ′/β2 + (E1 + |ωn|)2
E22 −E2X2 −EUEX2
E22 −E21
4(~ωD)
2 + ω2n
C ′/β2 + (E2 + |ωn|)2
. (43)
It is straightforward to show that fn reaches its maximum value for ωn = 0. The condition
for Tc is then
λ̃f0(EX2 , EU , Tc) = 1 (44)
Let us first set EX2 = 0. Let us also call the mean-field critical temperature of the isolated
first dot in the absence of a magnetic flux Tc0 (recall that for the parameters pertinent to Al,
Tc0 = 0.218meV = 2.6K). Now there are two possible limits, either EU ≪ Tc0 or EU ≫ Tc0.
In the first case we obtain
Tc(EU) ≃ Tc0
λ̃C ′T 2c0
+ · · ·
In the second case, EU ≫ Tc0, we obtain
Tc(EU) ≃ Tc0
e−1/λ̃ (46)
Note that this can be much smaller than Tc0 and is an illustration of the “dilution” of the
BCS attraction due to the second dot mentioned earlier. Of course, there will be a smooth
crossover between the expressions of Eq. (45) and Eq. (46), so that Tc is always smaller
than Tc0.
FIG. 9: The behavior of E∗X2 vs EU for numerical simulation and analytical approximation.
Now under the assumption EX2 , Tc ≪ EU we can solve analytically for Tc to obtain
T 2c (EX2 , EU) ≃ −
C ′2E2U
e−4/λ̃e
−ln ωD
One can further find the maximum of this expression. It turns out that EU has to be larger
than a critical value E∗U for there to be a maximum.
E∗U = ωDe
For our values of the parameters ωD = 34meV , λ̃ = 0.193, we find E
U = 0.245meV . The
position of the maximum can now be estimated asymptotically for EU > E
E∗X2 ≃ 16e
−1E∗U
Fig.9 compares the dependence of E∗X2 vs EU in case of numerical simulation and the one
described by Eq. (49). For large values of EU compared to E
U the numerically computed
curve matches the analytical approximation.
V. CONCLUSION AND DISCUSSION
In writing this paper we began with two objectives. We intended to compute noninteract-
ing scaling functions in the GOE→GUE crossover in a system of two dots coupled by hop-
ping, and to use this information to investigate the properties of an interacting system23–26
in the many-body quantum critical regime36–38.
We have considered a system of two coupled quantum dots, each of which could have
its own time-reversal breaking parameter, coupled by a bridge which could also have time-
reversal breaking. For each crossover parameter, there is a corresponding crossover energy
scale, which represents the inverse of the time needed for the electron to “notice” the presence
of that coupling in the Hamiltonian. We have computed the two-particle Green’s functions in
the coupled system in a large-N approximation12, valid when all energies of interest are much
greater than the mean level spacing. This allows us to compute the correlations of products
of four wavefunctions belonging to two different energy levels (which have been previously
calculated for a single dot for the pure ensembles by Mirlin using supersymmetry methods50,
and for the Orthogonal to Unitary crossover by Adam et al23). The two-particle Green’s
function splits naturally into a diffuson part and a cooperon part. Each of these parts can
be represented as 1−iω times a scaling function, where ω represents the frequency at which
the measurement is being performed. For example, when we use the two-particle Green’s
function to find the ensemble average of four wavefunctions belonging to two energies, ω is
the energy difference between the two states. The “scaling” nature of the scaling function
is represented by the fact that it depends only on the ratio of ω to certain crossover energy
scales. For the diffuson part the crossover energy EU is controlled solely by the strength of
the hopping between the two dots, while the scaling function for the cooperon part depends
sensitively on the time-reversal breaking in all three parts of the system.
In the second part of the paper, we consider the case when one of the dots has an
attractive BCS interaction, implying that it would be superconducting in the mean-field
limit at zero temperature if it were isolated, and the other dot has no electron interactions
but is penetrated by an orbital magnetic flux. The BCS interaction is one part of the
Universal Hamiltonian27–30, known to be the correct low-energy effective theory31–33 in the
renormalization group34,35 sense for weak-coupling and deep within the Thouless band |ε−
εF | ≪ ET . In order to eliminate complications arising from the charging energy, we consider
a particular geometry with the dots being vertically coupled and very close together in the
vertical direction, as shown in Fig. 1. Our focus is on the quantum critical regime36–38,
achieved by increasing either the temperature or the orbital flux through the second dot.
The first dot is coupled by spin-conserving hopping to a second dot on which the electrons are
noninteracting. This coupling always reduces the critical temperature, due to the “diluting”
effect of the second dot, that is, due to the fact that the electrons can now roam over both
dots, while only one of them has a BCS attraction. Thus, the mean-field critical temperature
Tc of the coupled system is always less than that of the isolated single superconducting dot
Tc0. This part of the phenomenology is intuitively obvious.
However, when the hopping crossover energy EU is either weak or of intermediate strength
compared to Tc0, turning on an orbital flux in the second dot can lead to a counterintuitive
increase in the mean-field critical temperature of the entire system. For very weak hopping,
the mean-field Tc monotonically increases with orbital flux through the second dot, reaching
its maximum when the second dot is fully time-reversal broken. For intermediate hopping
strength, the mean-field Tc initially increases with increasing orbital flux to a maximum.
Eventually, as the orbital flux, and therefore the crossover energy corresponding to time-
reversal breaking in the second dot increases, the critical temperature once again decreases.
For strong hopping EU ≫ Tc0, Tc monotonically decreases as a function of the orbital flux
in the second quantum dot.
We have obtained the detailed dependence of the fluctuation magnetization in the quan-
tum critical regime as a function of the dimensionless parameters T/EX2 and EX2/EU . Once
again, the coupled dot system behaves qualitatively differently from the single dot in having
a paramagnetic fluctuation magnetization in broad regimes of T , EX2 , and EU .
We understand these phenomena qualitatively as the result of two competing effects of
the flux through the second dot. The first effect is to raise the energy for Cooper pairs
in the second dot, thereby tending to localize the pairs in the first dot, and thus reducing
the “diluting” effect of the second dot. This first effect tends to lower the interacting free
energy (as a function of orbital flux) and raise the critical temperature. The second effect is
that as the electrons hop into the second dot and return they carry information about time-
reversal breaking into the first dot, which tends to increase the free energy (as a function of
orbital flux) decrease the critical temperature. The first effect dominates for weak hopping
and/or high T , while the second dominates for strong hopping and/or low T . Intermediate
regimes are more complex, and display nonmonotonic behavior of Tc and the fluctuation
magnetization.
It should be emphasized that the quantum critical regime we focus on is qualitatively
different from other single-particle random matrix ensembles applicable to a normal meso-
scopic system which is gapless despite being in contact with one or more superconducting
regions43,44, either because the two superconductors have a phase difference of π in their
order parameters43, or because they are d-wave gapless superconductors44. The main dif-
ference is that we investigate and describe an interacting regime, not a single-particle one.
Without the interactions there would be no fluctuation magnetization.
Let us consider some of the limitations of our work. The biggest limitation of the nonin-
teracting part of the work is that we have used the large-N approximation, which means that
we cannot trust our results when the energy scales and/or the frequency of the measurement
becomes comparable to the mean level spacing. When ω ≃ δ the wavefunctions and levels
acquire correlations in the crossover which we have neglected. Another limitation is that we
have used a particular model for the interdot hopping which is analytically tractable, and is
modelled by a Gaussian distribution of hopping amplitudes. This might be a realistic model
in vertically coupled quantum dots, or where the bridge has a large number of channels, but
will probably fail if the bridge has only a few channels. These limitations could conceivably
be overcome by using supersymmetric methods14,18.
Coming now to the part of our work which deals with interactions, we have restricted
ourselves to the quantum critical regime of the system, that is, when there is no mean-field
BCS gap. Of course, a finite system cannot undergo spontaneous symmetry-breaking. How-
ever, in mean-field, one still finds a static BCS gap. The paradox is resolved by considering
phase fluctuations of the order parameter which restore the broken symmetry48. To system-
atically investigate this issue one needs to analyze the case when the bosonic auxiliary field
σ in the coupled-dot system acquires a mean-field expectation value and quantize its phase
fluctuations.
We have also chosen a geometry in which interdot charging effects can be ignored. How-
ever, most experimental systems with superconducting nanoparticles deal with almost spher-
ical particles. For two such nanoparticles coupled by hopping, one cannot ignore charging
effects24,39–41. We expect these to have a nontrivial effect on the mean-field Tc and fluctuation
magnetization of the combined system. We defer this analysis to future work.
There are several other future directions in which this work could be extended. New
symmetry classes51,52 have been discovered recently for two-dimensional disordered/ballistic-
chaotic systems subject to spin-orbit coupling53,54. In one of these classes, the spin-orbit
coupling is unitarily equivalent to an orbital flux acting oppositely51,52 on the two eigen-
states of a single-particle quantum number algebraically identical to σz . Due to the unitary
transformation, this quantum number has no simple interpretation in the original (Orthog-
onal) basis. However, it is clear that the results of this paper could be applied, mutatis
mutandis, to two coupled two-dimensional quantum dots subject to spin-orbit couplings. In
particular, consider the situation where one quantum dot has no spin-orbit coupling, but
does have a Stoner exchange interaction, while the other dot is noninteracting, but is made
of a different material and has a strong spin-orbit coupling. Work by one of us has shown26
that by tuning the spin-orbit coupling one can access the quantum critical regime, which is
dominated by many-body quantum fluctuations. The above configuration offers a way to
continuously tune the spin-orbit coupling in the first dot by changing the strength of the
hopping between the dots.
In general, one can imagine a wide range of circumstances where changing a crossover
parameter in one (noninteracting) dot allows one to softly and tunably break a symmetry
in the another (interacting) dot, thereby allowing one access to a quantum critical regime.
We hope the present work will be useful in exploring such phenomena.
Acknowledgments
The authors would like to thank National Science Foundation for partial support under
DMR-0311761, and Yoram Alhassid for comments on the manuscript. OZ wishes to thank
the College of Arts and Sciences and the Department of Physics at the University of Kentucky
for partial support. The authors are grateful to O. Korneta for technical help with graphics.
APPENDIX A: ONE UNCOUPLED DOT
In this Appendix we calculate one-particle and two-particle Green’s functions for a single
dot undergoing the crossover. The strength of magnetic field inside the dot is controlled by
crossover parameter X . The Hamiltonian of the system in crossover is:
HS + iXHA√
1 +X2
, (A1)
where HS,A are symmetric and antisymmetric real random matrices with the same variance
for matrix elements. Normalization (1 + X2)−1/2 keeps the mean level spacing δ fixed as
magnetic field changes inside the dot.
We define the retarded one-particle Green’s function as follows:
GRαβ(E) =
E+ −H
(E+)2
+ . . .
(E+)2
(E+)3
+ . . . ,
Here H is a Hamiltonian, and E+ is the energy with infinitely small positive imaginary part
E+ = E + iη.
This series has nice graphical representation:
GR(E) = , (A3)
where straight solid line represents 1/E+ and dashed line stands for Hamiltonian.
= , H = (A4)
Just as in disordered conductor or quantum field theory the target is not the Green’s
function itself, but rather its mean and mean square. We take on random matrix ensemble
average of Gαβ . Such averaging assumes knowledge of 〈Hn〉, where angular brackets stand
for gaussian ensemble averaging, and n = 1,∞. For n = 1 we have 〈H〉 = 0, while for n = 2
the second moment reads:
〈HαγHδβ〉 =
〈HsαγHsδβ〉 −X2〈HaαγHaδβ〉
1 +X2
δαβδγδ +
1 +X2
δαδδγβ . (A5)
All higher moments of H can be computed using Wick’s theorem55. Thus, the ensemble
averaging leaves only the terms containing even moments of H . Introducing the notation
for 〈HH〉 = , we obtain, for the averaged GR series:
〈GR(E)〉 = (A6)
Then, the expansion (A6) can be written in a compact form of Dyson equation:
The bold line denotes the full one-particle Green’s function averaged over Gaussian en-
semble, and Σ is a self-energy, representing the sum of all topologically different diagrams.
The corresponding algebraic expression for the Dyson equation can be easily extracted from
Eq. (A7) producing:
Gαβ =
GανΣνµ
, (A8)
where Gαβ means 〈GRαβ(E)〉. Now, using the fact that Gαβ = Gαδαβ and Σαβ = Σαδαβ (no
summation over α implied), one can solve this equation and obtain:
Gαβ =
E+ − Σ
. (A9)
Next we approximate self-energy by the first term in large N approximation:
Σαβ = = G
〈HαγHγβ〉 ≈
E+ − Σ
. (A10)
Solving Eq. (A10) for the self-energy we determine:
−E2. (A11)
Consequently, the ensemble average of one-particle Green’s function is given by:
〈GRαβ(E)〉 =
)2 − E2
; 〈GAαβ(E)〉 = 〈GRβα(E)〉∗. (A12)
Next, to study the two-particle Green’s function we notice that the main contributions
come from ladder and maximally crossed diagrams:
(A13)
Two bold lines on the left side stand for the average two-particle Green’s function 〈GR(E+
ω)GA(E)〉. The sum of ladder diagrams is described by Bethe-Salpeter equation:
(A14)
δαδδβγ +
F [E, ω]
, (A15)
where ΠD is a ladder approximation of diffuson part of two-particle Green’s function. Here
F [E, ω] is a product of two inversed averaged one-particle Green’s functions and in the limit
ω ≪ Nδ is:
F [E, ω] = 〈GR(E + ω)〉−1〈GA(E)〉−1 ≈ −
. (A16)
One can solve this equation taking into account Π
δγ = Π
Dδαδδβγ:
F [E, ω]
F [E, ω]−
. (A17)
Multiplying ΠD by F 2[E, ω] we arrive at the following expression for the diffuson term:
〈GRαγ(E + ω)GAδβ(E)〉D =
δαβδγδ
. (A18)
Then, we turn our attention to the equation for maximally crossed diagrams. We have
(A19)
and ΠC is expressed in terms of F [E, ω] again:
1 +X2
F [E, ω]
F [E, ω]− 1−X2
. (A20)
Assuming X to be small compared to unity (weak crossover), we evaluate the contribution
of maximally crossed diagrams to Green’s function to get:
〈GRαγ(E + ω)GAδβ(E)〉C =
δαδδγβ
1 + iEX
, (A21)
where EX = 4X
2Nδ/π is a crossover energy scale. Final expression for the connected
part of the two-particle Green’s function is:
〈GRαγ(E + ω)GAδβ(E)〉 =
δαβδγδ
δαδδγβ
1 + iEX
. (A22)
APPENDIX B: TWO COUPLED DOTS
This Appendix contains details of the derivation for statistical properties of the Green’s
functions for the two coupled dots connected to each other via hopping bridge V . Coupling
between dots is weak and characterized by dimensionless parameter U . For the system of
uncoupled dots the Hilbert space is a direct sum of spaces for dot 1 and dot 2. Hopping V
mixes the states from two spaces. The Hamiltonian of the system can be represented as:
Htot =
V † H2
 . (B1)
For H1,2 and V we have:
HSn + iXnH
1 +X2n
, i = 1, 2; V =
V R + iΓV I√
1 + Γ2
. (B2)
Here S (A) stands for symmetric (antisymmetric), and R (I) means real (imaginary).
Below we use Greek indices for dot 1, and Latin indices for dot 2. We also found it convenient
to keep bandwidth of both dots the same; that is, N1δ1 = N2δ2 with ξ = δ1/δ2.
The following averaged products of matrix elements of H can be obtained:
〈HαγHδβ〉 =
δαβδγδ +
1−X21
1 +X21
δαδδγβ
〈HilHkj〉 =
δijδlk +
1−X22
1 +X22
δikδlj ,
where X1 and X2 are the crossover parameters in dot 1 and 2. Pairings between V matrix
elements are:
〈VαiVβj〉 = 〈V †iαV
jβ〉 =
1− Γ2
1 + Γ2
N1N2δ1δ2U
δαβδij
〈VαiV †jβ〉 =
N1N2δ1δ2U
δαβδij,
with Γ a crossover parameter in hopping bridge. Normalization for V pairing is chosen to
coincide with that of 〈HH〉 when ξ = 1.
To determine one-particle Green’s function we use the system listed in Eq. (20). The
straight and wavy bold lines with arrows represent averaged functions 〈GR1 (E)〉, 〈GR2 (E)〉
in dot 1 and 2, regular lines represent bare propagators, and the rest of the lines describe
pairings between Htot matrix elements. We have:
= 〈GR1 (E)〉; =
= 〈GR2 (E)〉; =
= 〈H1H1〉 = 〈H2H2〉 = 〈V V †〉.
The corresponding analytical expressions of this system of equations are:
Σ11G1
Σ12G1
Σ22G2
Σ21G2
with G1 and G2 connected to Green’s functions via: 〈GRαγ,1(E)〉 = G1δαγ , 〈GRil,2(E)〉 = G2δil
The self-energies Σnm are to be determined using standard procedure14.
We observe, that the system of two linear equations (B) has a solution:
E+ − Σ11 − Σ12
, G2 =
E+ − Σ22 − Σ21
Here we approximated self-energies by the first term in large N expansion again. In this
approximation evaluation of Σnm yields:
Σ11αβ = Σ
11δαβ = = G1
〈HαγHγβ〉 =
E+ − Σ11 − Σ12
Σ12αβ = Σ
12δαβ = = G2
〈VαiV †iβ〉 =
N1N2N2δ1δ2U
E+ − Σ22 − Σ21
Σ22ij = =
E+ − Σ22 − Σ21
Σ21ij = =
N1N2N1δ1δ2U
E+ − Σ11 − Σ12
Thus, to find all Σnm one needs to solve the following system of equations:
E+ − Σ11 − Σ12
E+ − Σ22 − Σ21
N1N2N2δ1δ2U
E+ − Σ22 − Σ21
E+ − Σ11 − Σ12
N1N2N1δ1δ2U
Observing that Σ21 = UΣ11/
ξ and Σ12 = U
ξΣ11 we decouple the system given in Eq.
(B6). For example, the pair of first and third equations can be rewritten as:
(Σ11)2 − EΣ11 + U
ξΣ11Σ22 = −
(Σ22)2 − EΣ22 +
Σ11Σ22 = −
For weak coupling the solution can be found by expanding self-energies Σ11 and Σ22 in
series in U . Taking the solution for single dot as zero approximation (below all the solutions
for the uncoupled dot will be marked with subscript 0) we get
Σ11 = Σ110 + UΣ
1 (B8)
Σ22 = Σ220 + UΣ
1 . (B9)
Note that N1δ1 = N2δ2, and Σ
0 = Σ
0 ≡ Σ0.
Plugging into the right hand side of Eq. (B9) in system (B7) we arrive at:
Σ11 = Σ0
1 + U
E+ − 2Σ0
Σ22 = Σ0
E+ − 2Σ0
Σ21 =
Σ12 = U
ξΣ22.
(B10)
Neglecting the higher powers in U for one-particle Green’s functions we finally arrive at
the following expressions for the single particle Green’s functions:
〈GRαβ,1(E)〉 =
〈GRαβ,0(E)〉
E−2Σ0
1− ǫ2
1 + U
1 + i ǫ√
〈GRij,2(E)〉 =
〈GRij,0(E)〉
1− U√
E−2Σ0
1− ǫ2
1 + U
1 + i ǫ√
where ǫ = πE/2Nδ.
Now we switch our attention to the calculational procedure for the average of the two-
particle Green’s functions 〈GRαγ,1(E + ω)GAδβ,1(E)〉 and 〈GRil,2(E + ω)GAkj,2(E)〉. In the limit
of large N1 and N2 ladder and maximally crossed diagrams contribute the most. For ladder
diagrams we obtain the system of Bethe-Salpeter equations (see Eq. (22)). Here we used
the following notation:
= 〈GR11(E + ω)GA11(E)〉, = 〈GR22(E + ω)GA22(E)〉
= 〈GR12(E + ω)GA21(E)〉, = 〈GR21(E + ω)GA12(E)〉
(B11)
For the diffuson ΠDnm the system of algebraic equations reeds:
ΠD11 =
F1[E, ω]
N1N2δ1δ2U
F2[E, ω]
ΠD22 =
F2[E, ω]
N1N2δ1δ2U
F1[E, ω]
ΠD12 =
N1N2δ1δ2U
F1[E, ω]
N1N2δ1δ2U
F2[E, ω]
ΠD21 =
N1N2δ1δ2U
F2[E, ω]
N1N2δ1δ2U
F1[E, ω]
(B12)
where F1[E, ω] and F2[E, ω] are defined as products of inverse averaged one-particle Green’s
functions in the first and second dots respectively. For small values of U and ω these
functions can be approximated as follows:
F1[E, ω] = 〈GR1 (E + ω)〉−1〈GA1 (E)〉−1 ≈
ξU − iω̃
F2[E, ω] = 〈GR2 (E + ω)〉−1〈GA2 (E)〉−1 ≈
− iω̃
(B13)
where ω̃ = πω/2Nδ. The system of four equations given by the Eq. (B12) can be decoupled
into the two systems of two equations each. To determine ΠD11 one solves the system of the
first and the last equations of Eq. (B12) to get:
F1(E, ω)
ΠD11 −
N1N2N2δ1δ2U
F2(E, ω)
F2(E, ω)
ΠD21 −
N1N2N1δ1δ2U
F1(E, ω)
N1N2δ1δ2U
(B14)
Then, solving the resulting system (Eq. (B14)) and attaching external lines one obtains
expression for the two-particle Green’s function in dot 1:
〈GRαγ,1(E + ω)GAδβ,1(E)〉D =
N21 δ1
δαβδγδ
1 + i√
1 + i(
ξ + 1√
. (B15)
The corresponding correlator for dot 2 is readily obtained as well:
〈GRil,2(E + ω)GAkj,2(E)〉D =
N22 δ2
δijδlk
1 + i
1 + i(
ξ + 1√
. (B16)
For the second part of the Green’s function (which is the sum of maximally crossed
diagrams) the system of equations is described by Eq. (24). Transforming this graphical
system into the algebraic one, we get:
ΠC11 =
1−X21
1 +X21
1−X21
1 +X21
F1[E, ω]
1− Γ2
1 + Γ2
N1N2δ1δ2U
F2[E, ω]
ΠC22 =
1−X22
1 +X22
1−X22
1 +X22
F2[E, ω]
1− Γ2
1 + Γ2
N1N2δ1δ2U
F1[E, ω]
ΠC12 =
1− Γ2
1 + Γ2
N1N2δ1δ2U
1− Γ2
1 + Γ2
N1N2δ1δ2U
F2[E, ω]
1−X21
1 +X21
F1[E, ω]
ΠC21 =
1− Γ2
1 + Γ2
N1N2δ1δ2U
1− Γ2
1 + Γ2
N1N2δ1δ2U
F1[E, ω]
1−X22
1 +X22
F2[E, ω]
(B17)
Once again, the system at hand breaks into systems of two equations each. We proceed
by combining the first and the last equations to obtain:
1−X21
1 +X21
F1[E, ω]
ΠC11 −
1− Γ2
1 + Γ2
N1N2N2δ1δ2U
F2[E, ω]
1−X21
1 +X21
1− Γ2
1 + Γ2
N1N2N1δ1δ2U
F1[E, ω]
1−X22
1 +X22
F2[E, ω]
ΠC21 =
1− Γ2
1 + Γ2
N1N2δ1δ2U
(B18)
Now we can construct approximations for the expressions, containing crossover parame-
ters. For example, for small values of X and Γ the solution for ΠC11 is expressed as follows:
ΠC11 =
(1− 2X21 )( U√ξ − iω̃ + 2X
2 ) + (1− 4Γ2)U2
ξU − iω̃ + 2X21 )( U√ξ − iω̃ + 2X
2 )− (1− 4Γ2)U2
. (B19)
Next, introducing crossover energy scales:
EX = 4X
EU = 2U
4Γ2EU√
ξ + 1√
(B20)
we obtain the solution for ΠC11 in the following form:
ΠC11 =
1− EU√
− EX2
1− EX1+EX2
EX1EX2
(iω)2
EX1EU√
ξ(iω)2
ξEX2EU
(iω)2
ξ + 1√
(B21)
Then, adding external lines to ΠC11 for Green’s function we get:
〈GRαγ,1(E + ω)GAδβ,1(E)〉C =
N21 δ1
δαδδγβ
1 + i√
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
(B22)
Similar manipulations for the corresponding correlator of Green’s functions for the second
room result in:
〈GRil,2(E + ω)GAkj,2(E)〉C =
N22 δ2
δikδlj
1 + i
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
(B23)
Finally, the connected part of the total two-particle Green’s function is obtained as a sum
of diffuson and cooperon parts, yielding:
〈GRαγ,1(E + ω)GAδβ,1(E)〉 =
N21 δ1
δαβδγδ
1 + i√
1 + i(
ξ + 1√
N21 δ1
δαδδγβ
1 + i√
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
(B24)
〈GRil,2(E + ω)GAkj,2(E)〉 =
N22 δ2
δijδlk
1 + i
1 + i(
ξ + 1√
N22 δ2
δikδlj
1 + i
1 + i
EX1+EX2
− EX1EX2
− EX1EU√
ξEX2EU
ξ + 1√
1 + iEΓ
(B25)
APPENDIX C: FOURIER TRANSFORM OF TWO-PARTICLE GREEN’S
FUNCTION
To be able to study temporal behavior of electrons in the rmt system we introduce the
Fourier transform of two-particle Green’s function. We define it via the following integral:
〈GRαγ(t)GAδβ(t)〉 =
(2π)2
exp−iωt〈GRαγ(E + ω)GAδβ(E)〉dωdE. (C1)
To get the correct behavior of the diffuson part for small ω, we replace 1/ω by ω/(ω2+η2),
where η is infinitesimal positive number. Now we introduce for dot 1:
fD(ω) =
N21 δ1
δαβδγδ
1 + i√
1 + i
ξ + 1√
→ δαβδγδ
N21 δ1
ω2 + η2
ω + i√
ω + i
ξ + 1√
= δαβδγδ
N21 δ1
ω + i√
(ω − iη)(ω + iη)(ω + i(
ξ + 1√
. (C2)
The Fourier transform of this diffuson term gives:
fD(t) =
exp(−iωt)fD(ω)dω. (C3)
Next steps are the standard steps of integration in complex plane. For t > 0 one closes
contour in lowerhalf plane. One root is located in upper half plane and two more are located
in lower half plane. The integration yields:
fD(t) = δαβδγδ
N21 δ1
− η)e−ηt
ξ + 1√
)EU − η
ξ + 1√
ξE2Ue
ξ+ 1√
)EU t
ξ + 1√
)2E2U − η2
 . (C4)
As η approaches zero, fD(t) becomes:
fD(t) = δαβδγδ
N21 δ1
1 + ξ
ξ+ 1√
)EU t
. (C5)
The full Fourier transformation includes integration over E as well. In current approx-
imation, when E is close to the center of the band, 〈GR1GA1 〉 is independent of E. It will
depend on E if we integrate over the whole bandwidth. The exact dependence of 〈GR1GA1 〉
on E far from the center of the band is not known. To get correct expression we assume
that integration over E adds to 〈GR1GA1 〉 multiplicative factor N1δ1 along with normalization
coefficient A. Also, for index pairing α = β and γ = δ, GRαγG
δβ becomes transition proba-
bility density P (t)α→γ. Using equipartition theorem, for t→ ∞ summation of P (t)α→γ over
α one can get total probability to stay in dot 1. It is equal to N1/(N1 +N2). That is,
dEfD(t) =
N1 +N2
1 + ξ
. (C6)
Integration over E and summation over α gives the factor of AN21 δ1. We identify the
normalization constant as A = 1/π. Note, that we did not use cooperon part fC(t) to
determine normalization constant A. The reason for that is chosen index pairing. After
the summation over α cooperon part contribution is of the order 1/N1 compared with the
diffuson part. After integration over E with proper normalization fD(t) becomes:
fD(t) = δαβδγδ
N1(1 + ξ)
ξ+ 1√
)EU t
. (C7)
Then we perform the Fourier transform of the cooperon part:
fC(ω) =
N21 δ1
δαδδγβ
1− EX2
1− EX1+EX2
EX1EX2
(iω)2
EX1EU√
ξ(iω)2
ξEX2EU
(iω)2
ξ + 1√
The fC(ω) is a regular function when ω approaches limiting values, provided at least one
of the crossover energy scales EX1 , EX2 , or EΓ differs from zero.
To make fC(ω) more suitable for the Fourier transform we manipulate Eq. (C8) into:
fC(ω) = −δαδδγβ
N21 δ1
iω − EX2 −
(iω)2 −
(EX1 +
ξEU ) + (EX2 +
EX1EX2 +
EX1EU√
+ EX2EU
ξ + (
)EUEΓ
and observe that the poles of fC(ω) are given by
iω± =
(EX1 +
ξEU) + (EX2 +
(C10)
with D = ((EX1 +
ξEU)− (EX2 +EU/
ξ))2 + 4E2U(1− 4Γ2). The parameter D is always
positive and ω± are imaginary complex numbers.
It can be proved that (EX1 +
ξEU)+(EX2 +EU/
D for all values of parameters,
which means that the poles are pure imaginary numbers in lower half complex plane:
ω± = −i
(EX1 +
ξEU) + (EX2 +
= −ia±, a+ > a− > 0. (C11)
The function fC(ω) now reads:
fC(ω) = −δαδδγβ
N21 δ1
iω − EX2 − EU√ξ
(iω − a−)(iω − a+)
. (C12)
We perform the Fourier transform and use the normalization factor to obtain:
fC(t) =
exp(−iωt)fC(ω)dωdE = δαδδγβ
EX2 +
a+ − a−
e−ta− − e−ta+
(C13)
Hence, the full expression for the Fourier transform for the two-particle Green’s function
in the dot are given by:
〈GRαγ(t)GAδβ(t)〉11 = δαβδδγ
N1(1 + ξ)
ξ+ 1√
)EU t
+ δαδδγβ
EX2 +
a+ − a−
e−ta− − e−ta+
(C14)
〈GRik(t)GAlj(t)〉22 = δijδkl
N2(1 + ξ)
ξ+ 1√
)EU t
+ δilδkj
EX1 +
a+ − a−
e−ta− − e−ta+
, (C15)
where a± is defined through Eq. (C11).
APPENDIX D: CORRELATION OF FOUR WAVE FUNCTIONS
In this appendix we obtain correlation of four wave functions 〈ψn(α)ψ∗n(γ)ψm(δ)ψ∗m(β)〉
for the system of two coupled dots. This has been obtained in a single dot for the pure
ensembles by supersymmetry methods by Mirlin50, and for the GOE→GUE crossover by
Adam et al23. We consider ensemble average of the following product:
GRαγ(E + ω)−GAαγ(E + ω)
GRδβ(E)−GAδβ(E)
− 〈GRαγ(E + ω)GAδβ(E)− 〈GAαγ(E + ω)GRδβ(E)〉 = −2(δαβδγδRe[D1] + δαδδγβRe[C1]), (D1)
where D1 and C1 are the diffuson and cooperon expressions from Eq. (27). Here we used
the fact that ensemble average of GRGR and GAGA are smaller than GRGA and GAGR.
On the other hand, we have:
GRαγ(E)−GAαγ(E) = −2πi
ψn(α)ψ
n(γ)δ(E −En), (D2)
GRαγ(E + ω)−GAαγ(E + ω)
GRδβ(E)−GAδβ(E)
− 4π2〈
ψn(α)ψ
n(γ)ψm(δ)ψ
m(β)δ(E + ω − En)δ(E − Em)〉. (D3)
We know that in the crossover components of eigenvalues and eigenvectors are correlated
with each other. This correlation is small already on the distances of a few δ and can be
neglected in the limit ω ≫ δ, so Eq. (D3) can be approximated by:
−4π2〈ψn̄(α)ψ∗n̄(γ)ψm̄(δ)ψ∗n̄(β)〉〈
δ(E + ω −En)〉〈
δ(E − Em)〉.
where n̄ and m̄ mark energy levels close to E + ω and E respectively.
The average of the sum is a density of states ρ(E) = 〈
n δ(E − En)〉 = 1/δ. Then, we
GRαγ(E + ω)−GAαγ(E + ω)
GRδβ(E)−GAδβ(E)
〈ψn(α)ψ∗n(γ)ψm(δ)ψ∗m(β)〉. (D4)
For the two coupled dots we have:
Re[D1] =
ω2 + (
ξ + 1√
)2E2U
In order to calculate Re[C1] from Eq. (27) we are going to assume that magnetic field is
zero in the first dot and in the hopping region (EX1 = EΓ = 0), and the second dot is in
GOE to GUE crossover (EX2 ∼ ω). Then,
Re[C1] =
N21 δ1
2 + (EU +
ξEX2)EUEX2
(ω2 −
ξEUEX2)
2 + (EX2 + (
ξ + 1√
)EU)2ω2
The relation between the mean level spacing δ for the system of coupled dots and the
mean level spacing in the first uncoupled dot δ1 is as follows. The averaged density of states
in coupled system is going to be the sum of densities of each dot: 〈ρ〉 = 〈ρ1〉 + 〈ρ2〉, or
δ−1 = δ−11 + δ
2 . Thus, we conclude that δ = δ1/(1 + ξ).
Finally, we set Eq. (D1) and Eq. (D4) equal and obtain correlation of for the wave
functions:
〈ψn(α)ψ∗n(γ)ψm(δ)ψ∗m(β)〉 = δαβδγδ
π(1 + ξ)2N21
ω2 + (
ξ + 1√
)2E2U
+ δαδδγβ
π(1 + ξ)2N21
ξω2 + (
ξE2X2 + EUEX2)
(ω2 −
ξEUEX2)
2 + (EX2 + (
ξ + 1√
)EU)2ω2
. (D7)
APPENDIX E: SUM RULE FOR DOUBLE DOT SYSTEM
To verify the expressions we have obtained for the averaged Green’s functions we use a
sum rule.
The pair annihilation (creation) operator T (T †) in the basis of two uncoupled dots is a
sum of two terms belonging to each dot:
cα0,↓cα0,↑ +
ci0,↓ci0,↑,
T † =
α0,↑c
α0,↓ +
i0,↑c
i0,↓.
Greek indices go over the states in the first dot, and Latin indices go over the states in
the second dot. The subindex 0 denotes the basis of two uncoupled dots.
Our first goal is to calculate the commutator [T †, T ]. As operators from different dots
anticommute, one gets:
[T †, T ] =
α0,β0
α0,↑c
α0,↓, cβ0,↓cβ0,↑] +
i0,j0
i0,↑c
i0,↓, cj0,↓cj0,↑] = N̂1e + N̂2e −N1 −N2, (E2)
where N̂1e, N̂2e are the operators of total number of electrons in dot 1 and dot 2, and N1, N2
are the total number of levels in dot 1 and dot 2.
The expectation value of [T †, T ] in ground state at zero temperature is:
[T †, T ] = 〈Ω|[T †, T ]|Ω〉 = Ne −N. (E3)
Ne and N are the total number of electrons and levels in both dots. This number is conserved
when going to another basis.
Now we choose the basis of the system of coupled dots. In this basis cα0,s =
m ψm(α0)cm,s, and ci0,s =
m ψm(i0)cm,s, where cm,s is annihilation operator in new basis.
Using this transformation, we rewrite pair destruction operator as follows:
cα0,↓cα0,↑ +
ci0,↓ci0,↑ =
m1,m2
Dm1m2cm1,↓cm2,↑, (E4)
where Dm1m2 is defined by the following expression:
Dm1m2 =
m1,m2
ψm1(α0)ψm2(α0) +
ψm1(i0)ψm2(i0)
cm1,↓cm2,↑
ψm1(p0)ψm2(p0). (E5)
The index p0 runs over all states in the first and second dots for the basis of uncoupled dots.
In the new basis the T, T † operators look like this:
m1,m2
Dm1m2cm1,↓cm2,↑,
T † =
m1,m2
D∗m1m2c
m2,↑c
m1,↓.
Consequently, in the new basis,
[T †, T ] =
m1,m2
m3,m4
D∗m1m2Dm3m4 [c
m2,↑c
m1,↓, cm3,↓cm4,↑]
m2,m4
D∗m1m2Dm1m4
m2,↑cm4,↑ −
m1,m3
D∗m1m2Dm3m2
cm3,↓c
m1,↓. (E7)
One can go further and use completeness condition
m(p0)ψm(n0) = δp0n0 to show
that in the new basis the value of commutator is N̂e−N . Our next goal, however, is to take
the disorder average of the vacuum expectation value and to prove the invariance of [T †, T ].
Taking into account that 〈Ω|c†m1,↑cm2,↑|Ω〉 = δm1m2Θ(µ − Em1) and 〈Ω|cm2,↓c
m1,↓|Ω〉 =
δm1m2(1−Θ(µ− Em1)), the ground state expectation value for the commutator is:
[T †, T ] = 〈Ω|[T †, T ]|Ω〉 =
m1,m2
|Dm1m2 |2[2Θ(µ− Em1)− 1], (E8)
where Θ(x) is a step function.
Averaging over disorder gives:
〈[T †, T ]〉 = 2
m1,m2
Θ(µ− Em1)〈|Dm1m2 |2〉 −
m1,m2
〈|Dm1m2 |2〉 (E9)
Converting this into integral, we get:
〈[T †, T ]〉 = 2
dE1dE2ρ(E1)ρ(E2)〈|D(E1, E2)|2〉
dE1dE2ρ(E1)ρ(E2)〈|D(E1, E2)|2〉 (E10)
The density of states ρ(E) is the Winger’s semicircle law:
ρ(E) =
W 2 − E2,
where 2W is the bandwidth and N is the number of states in the system.
To proceed we need to find the ensemble average of the following object:
〈|Dm1m2 |2〉 =
p0,n0
〈ψ∗m1(p0)ψ
(p0)ψm1(n0)ψm2(n0)〉. (E11)
Using results of appendix D one can obtain expression for the correlation of four wave
functions in the form:
〈ψ∗m1(p0)ψ
(p0)ψm1(n0)ψm2(n0)〉 =
2π2ρ(E1)ρ(E2)
〈GRn0p0(E2)G
(E1)〉
〈GRn0p0(E2)G
(E1)〉
. (E12)
Note, that to get the correct answer for the sum rule one should keep 〈GRGR〉 term as
well. Summation in Eq. (E12) is performed over the states in both dots.
When the dots have equal mean level spacing δ1 = δ2 = δ0, one particle Green’s function
can be found exactly from the system (B7) without approximation in U :
(E)〉 =
W 2 −E2
(E)〉 =
W 2 − E2
e−iφ,
(E13)
where W = 2N0δ0
1 + U/π is the half bandwidth and sin φ = E/W . Here both indices
p0 and p
0 belong either to the first or to the second dot.
The sum in Eq. (E12) can be broken into four sums, when the indices p0, n0 belong either
to the first dot, or to the second dot, or one of the indices go over the states in the first dot,
and the other one goes over the states in the second dot.
For example, for 〈GRGA〉 part we have the following expression:
〈GRn0p0(E2)G
(E1)〉 =
(1 + U)
(1 + U)e−iφ21 − ζ
[(1 + U)e−iφ21 − 1][(1 + U)e−iφ21 − ζ ]− U2
(1 + U)
(1 + U)e−iφ21 − 1
[(1 + U)e−iφ21 − 1][(1 + U)e−iφ21 − ζ ]− U2
(1 + U)
[(1 + U)e−iφ21 − 1][(1 + U)e−iφ21 − ζ ]− U2
(E14)
Here φ21 = φ2 − φ1, and ζ = (1−X22 )/(1 +X22 )
The first term in Eq. (E14) is the contribution of 〈GR〉〈GA〉 plus the cooperon part of
two particle Green’s function in the first dot. The second term describes contribution of
free term and cooperon part in the second dot. The last term is a sum of transition parts
from dot 1 to dot 2 and vice versa. It appears that these transition terms are equal, which
explains coefficient 2 in front of the last term in Eq. (E14).
Summation of the 〈GRGR〉 gives similar result:
〈GRn0p0(E2)G
(E1)〉 =
(1 + U)
(1 + U)e−iψ21 + ζ
[(1 + U)e−iψ21 + 1][(1 + U)e−iψ21 + ζ ]− U2
(1 + U)
(1 + U)e−iψ21 + 1
[(1 + U)e−iψ21 + 1][(1 + U)e−iψ21 + ζ ]− U2
(1 + U)
[(1 + U)e−iψ21 + 1][(1 + U)e−iψ21 + ζ ]− U2
(E15)
where ψ21 = φ2 + φ1.
In principle, there should be terms corresponding to diffusons in dot 1 and dot 2. However,
these terms after summation over p0, n0 are 1/N0 smaller than the others and in the large
N0 limit can be neglected.
Although one can use Eq. (E10) to verify the sum rule, it is more convenient to work
with derivative of Eq. (E10) over µ at µ = 0.
It gives:
〈[T †, T ]〉µ=0 = 2ρ(0)
dE2ρ(E2)〈|D(E1 = 0, E2)|2〉. (E16)
On the other hand, this expression should be equal to:
(Ne −N) =
ρ(E)dE −N
= 2ρ(µ). (E17)
Comparison of Eq. (E16) and (E17) at µ = 0 results in the following condition for the
sum rule:
dE2ρ(E2)〈|D(E1 = 0, E2)|2〉 = 1. (E18)
The integral in Eq. (E18) was computed numerically and matched the unity with high
accuracy.
∗ Electronic address: zelyak@pa.uky.edu
† Electronic address: murthy@pa.uky.edu
‡ Electronic address: rozhkois@notes.udayton.edu
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