Low energy excitations of double quantum dots in the lowest Landau level regime

We study the spectrum and magnetic properties of double quantum dots in the lowest Landau level for different values of the hopping and Zeeman parameters by means of exact diagonalization techniques in systems of N=6 and N=7 electrons and filling factor close to 2. We compare our results with those obtained in double quantum layers and single quantum dots. The Kohn theorem is also discussed.


I. Introduction
The understanding of the structure and properties of double quantum dots (DQD's) grown on the perpendicular direction to their plain, and subject to an external constant magnetic field B, has attracted special interest due to the wealth of new quantum states that are actually realized. This wealth does not only refer to the ground states (GS's), which provide a quite intricate phase diagram 1,2 , but also to the variety of different excited states. This allows to have a large quantity of different sets of properties of the DQD system that can model for instance, point contacts within a device built for electron transport.
Special interest has the regime for which the magnetic field is so strong that only the lowest Landau level (LLL) is occupied but not strong enough that prevent any spin polarization. We will always assume that the second Landau level is far enough that we can ignore any mixture between Landau levels. In the symmetric gauge, the projection of the DQD Hamiltonian to the LLL is given by 3 and β =h 2 ω c = eB cm * is the cyclotron frequency, m * being the effective electron mass in the semiconductor host and e and c are the electron charge and the speed of light in vacuum respectively. The frequency associated with the parabolic confining potential in both dots is given by ω 0 , M is the total angular momentum and N is the total number of electrons. The Zeeman coupling is given by ∆ z = gµ B B with g being the Landé g-factor and µ B being the Bohr magneton ( for the free electron mass, µ B = eh/2mc). The single particle energy gap between symmetric (s) and antisymmetric (a) states, combinations of |r and |l single particle states of electrons confined in the right and left dot respectively (see below) is given by 2∆ t and X = N s − N a is the balance between symmetric and antisymmetric single particle states. Finally, H int is the Coulomb interaction term.
Hereafter all distances will be given in units of the magnetic length defined as: and all energies in units of e 2 /(ǫl B ) being ǫ the dielectric constant of the semiconductor host. The Zeeman energy, the tunneling term scaled by ∆ t , the kinetic contribution given by α and β and the Coulomb interaction, provide the ingredients of the system.
The applied magnetic field can be directed in any direction in space in such a way that its action has different effect in the kinetic contribution in which only the component along the Z-direction plays a role and in the Zeeman effect in which the total magnetic field contributes. The eigenstates of the Hamiltonian are characterized by the total angular momentum M, the total spin along the B-direction given by S z and the parity P 1,4 . These parameters are related with the invariance of the Hamiltonian under space rotations along the Z-direction, rotations of spin and specular reflection in the plane inter-dots respectively 1 . We will denote by (M, S z , P ) the configuration that determines each separated subspace of eigenstates.
Previous studies in DQD's based on exact diagonalizations on one hand 1,2 and the large experience extracted from mean field (MF) approximations and effective field the- A simultaneous change of spin and parity occurs for the GS in each phase transition whereas the total angular momentum remains unchanged 1 .
No direct information about the order in plane has been obtained so far from the exact solutions due to the fact that they have all the quantum numbers (M, S z , P ) well defined. No information can directly be obtained from order parameters that depend on single operators of the type S R x or S L x since their expectation values vanish. However, indirect information about the order in plane can be obtained from the properties of the excited states as we will show in the discussion of Figs.(1) and (2) below. Within the great number of possible low energy excitations that provide information about the properties of the system, we will draw our attention to those related to three main different points: First, we search for particle-like excitations independent of the electron-electron interaction and so independent of the number of electrons. We discuss when the Kohn theorem 10 holds in a parabolic DQD system ( as it is always the case for a single parabolic QD) Second, we look for the types of excitations that soften as they come close to a GS transition that takes place with the variation of some input parameter. These excitations provide a clear and easy way to map the GS phase diagram.
Third, in the last point we concentrate in the dispersion relation ω(l) of two different type of excitations, one with and the other without a simultaneous spin and isospin flip (see below). In a single QD the Coulomb contribution to the dispersion relation of excitations over a ferromagnetic GS decreases with angular momentum due to expansion. This is the general behavior, except at the "magic" values of l * 11 for which the system increases in angular momentum from l * to l * + 1 but does not increase in size and hence the Coulomb contribution remains constant. These magic values of l are related to the incompressible states. Our aim is to see if there is a similar behavior in the DQD system. This paper is organized as follows: In Section II we make a brief account of the exact diagonalization used in our calculations. In Section III we present the outstanding features of some excited states of different multipolarity over different GS's, paying special attention to the different behavior of even and odd systems. We will follow the three main points previously mentioned. Finally in Section IV we draw our conclusions.

II. Exact diagonalization
Exact diagonalization can be performed in separate boxes for different configurations characterized by (M, S z , P ). Each configuration determines a finite-eigenstate subspace.
Within the LLL regime, we will use in all expansions the non-interacting single particle wave functions which do not have nodes along the radius and are given by where m is the single particle angular momentum, Λ = r, l, σ = ↑, ↓, r = (x, y), and φ m ( r) are the Fock Darwin wave functions given by 11 Along the Z-direction we assume delta charge distributions separated by d, the distance between the dots (we consider d = 1 in all numerical performances).
Within our calculations we will use symmetric |s and antisymmetric |a single particle states, related by |r and |l by and we will use the term isospin referred to this degree of freedom (sometimes referred as pseudospin in literature, see Ref. 5 ). None of the parameters r and l or s and a are well defined quantum numbers since the Coulomb interaction mixes s and a and the tunneling process mixes r and l. The well established restriction is that parity must be preserved, this means that the change in symmetry ( s → a or a → s) due to electron interaction must take place by pairs of electrons, and never by one alone (see below).
We will consider excited states over the three possible types of GS's and choose the parameters in such a way that for an even number of electrons in the ferromagnetic GS we have filling factor ν = 2. We proceed as follows. Once the input parameters are fixed, we determine first the finite number of Slater determinants built up from single particle wave functions of the type given by Eq.(6) which provide a bases for each subspace configuration (M, S z , P ). Then, the diagonalization of the Hamiltonian given by Eq.(1) is straightforward except for the Coulomb term. Although the Coulomb interaction does not mix |r and |l single particle states, some manipulations must be done in order to operate over |s and |a wave functions. In second quantization, the interaction Hamiltonian is given by where V is the Coulomb interaction and the sub-indexes denote angular momentum m = 0, 1, 2, .., spin= ↑, ↓ and isospin τ = a, s. Taking into account that the single particle wave functions are related by Eq. (8) and (9) and that the right and left single states are δ-distributions of the type δ(z) and and for each electron. As a consequence, there are only three possibilities for the expectation values of V : (i) the interacting electrons do not change their isospin as in (ii) only one electron changes its isospin as in, and (iii) both electrons change their isospin index as in In the brackets on the right hand side of these equations, the integral over the Zcoordinate has been performed and the potentials are given by with r =| r 1 − r 2 |, 1 and 2 being the two interacting electrons. That is to say, the Coulomb interaction either leaves unchanged the isospin of the electrons, if it operates or changes the isospin of two electrons, if it operates with in such a way that parity P , given by P = (−1) X/2 is preserved (X = N s − N a ) 1 . The change of isospin of a single electron is forbidden. Taking into account Eqs. (8)- (9) and the expression of the exciting potential (asso- it is easy to see that The system jumps from a 1-dim space configuration to a 2-dim space. The result is the excitation of the CM by α leaving the internal Coulomb energy unchanged. This is the well known intra-Landau level dipole excitation whose energy decreases when the magnetic field increases that is, the ω − = α far-infrared resonance (FIR) 11 . No inter-Landau level transition of energy given by is possible within the LLL considered in our calculation. Other Coulomb-independent interaction excitations are possible in the N=7 system. If we also allow changes in parity, that is to say, if we consider, the CM is excited with an energy α + ∆ t (it is a CM excitation since the same result would be obtained for N=1). The previous case (of energy α) would correspond to the excitation made by a nearly constant electric field ( we are assuming dipole approximation) directed along the X direction (E ) and the last one (of energy α + ∆ t ) would correspond to an electric field with an additional non-vanishing component along the Z direction (let us call it E xz ). Incidentally, this result can be used to determine experimentally the relative orientation of the DQD respect to the incident beam. The system absorbs a photon of energy α + ∆ t with maximum probability when the angle between the direction of the incident electric field and the normal to the plane of the DQD is θ = π/4.
For both excitations, with and without parity change, the system goes from a 1-dim space (|FM ) to a 2-dim space of excited states. The CM excitation leaves the system in the higher energy state within the excited 2-dim configuration in such a way that, and where the eigenenergies E i have been ordered from lower to higher energy within each subspace. See Table ( In all the cases the angular momentum is preserved and a simultaneous spin and parity flip takes place. That is to say, where E 1 (M, S z , P ) is the GS energy at ∆ t and E 1 (M, S z ± 1, −P ) is the lowest eigenenergy within the excited configuration at the same ∆ t . These excitations provide the lowest value ( ∆M = l = 0 ) of the multipole dispersion relation ω(l) given below.
The increase of angular momentum in one unit increases the kinetic energy of the system by α (α = 0.2). This increase is partially compensated by the decrease of Coulomb energy due to expansion. For both systems, the Coulomb contribution is numerically, for the particular values of the parameters that we have taken, about 2α/3 in such a way that the total amount of energy gained in each step is about α/3. This gives a quite monotonous behavior.
In contrast, unexpected features were obtained for excitations of the type (solid line): where spin and parity flip takes place simultaneously. For low values of l, the contribution of the Coulomb energy to ω is given by a more or less constant amount of, numerically, α/2. However, after several steps, 4 for N=6 and 6 for N=7 the system suffers a sudden expansion that reduces drastically the Coulomb interaction. In the N=7 case, due to the presence of an unpaired electron, the amount of the Coulomb contribution is so large that the total energy decreases. Looking at the occupancies of the single particle In the DQD most of the absorbed energy is transformed into internal energy releasing the electrons from their interaction.

IV Discussion and Conclusions
We have analyzed several types of low energy excitations over the three possible GS of a DQD confined by parabolic potentials in each plane and separated by a distance d.
The LLL regime was considered and the input parameters were chosen in such a way that the filling factor of the ground states and some of the excited states is close to ν = 2. Due to the extra degree of freedom (as compared with a single dot or a single layer), represented by the isospin states and due to the interplay between tunneling and Coulomb interaction, the energy ω has a Coulomb contribution. This is in contrast to the case of a single layer for which, in the limit k → 0 the non-interacting contribution given by ∆ z is recovered.
The softening of the ω modes with the variation of ∆ t shows phase instabilities previously detected in the determination of the GS phase diagram from exact diagonalization 1 .
There are some similarities and some differences between our results and those reported by Das Sarma et al. 7 within a HF calculation for a double layer system. We will follow the arguments given by these authors to analyze our results. Within the |FM and |SYM phases, the structure of the curves ω/∆ t is similar, in both cases the ω mode softens as it approaches the phase transition from |FM to |C and from |SYM to |C respectively. In addition, the increase of ω as it moves away from the boundary is larger in the symmetric phase in both calculations. However, within the canted state we obtain finite values of ω although much smaller than those in the |FM or |SYM phases, in contrast to the results obtained in the DL for which ω = 0 over the full canted phase.
Aside from the previous outcomings, there is another main difference: the canted state in our calculation is an eigenstate of the S z operator whereas this is not the case for the HF canted phase. At the boundaries separating different phases, however, due to energy degeneracy, the superposition of states of different well defined spin gives rise to states which are not eigenstates of S z . Das Sarma et al. 7 have proved that the existence of a gapless mode is directly due to the canted antiferromagnetic spin ordering. As a consequence, even though the canted states in a DQD may be interpreted as having antiferromagnetic order in the plain of the dots 1 , the order is not complete, producing gapped excitations probably due to edge effects. The exceptions are at the boundaries separating different phases where due to degeneracy, the gapless mode is recovered. Finally, in the limit l → ∞ the dispersion relation of the SDW does not approach asymptotically a constant value as it is the case for a single layer (SL) or DL systems due to the fact that at this limit the excitation energy always recovers the single particle value.
For SL or DL this energy is given by ω ′ + V ex where ω ′ is the non-interacting excitation value ( a combination of ∆ z and ∆ t ) and V ex is the exchange single particle energy of an electron in the GS, independent of the linear momentum k due to the degeneracy existing in extended systems. In contrast, in a DQD the parabolic potential breaks the degeneracy producing an increase of energy with increasing angular momentum. This gives a nearly linear curve as l → ∞ typical of a single particle.
No total spin or space correlations in the ground states have been investigated through the density-correlation functions which may signal additional symmetry breaking effects.  15 . This is left for future investigations.

Acknowledgements
We gratefully acknowledge C. Tejedor

I. APPENDIX
For simplicity we neglect spin indices since they are irrelevant for the following discussion. We characterize the first quantized wave function of the N particle system as Ψ a 1 ,...,a N (x 1 , ...., x N ), where a i = 1, 2 are layer indexes indicating the dot in which the i-th electron sits. Hence the first quantized Hamiltonian is a matrix in layer-index space An in-plane homogeneous electric field clearly produces a contribution proportional to the identity matrix in the layer space, and hence the interaction only depends on the center of mass coordinates, namely the Kohn theorem applies. A homogeneous electric field perpendicular to the plane E ⊥ produces a contribution diagonal in layer space (but not proportional to the identity matrix) which reads where d is the distance between the two dots.
The hopping term is not diagonal in layer space, where s 11 = s 22 = 0 and s 12 = s 21 = 1. Then it is easy to check that where ǫ b i a i is the antisymmetric tensor. Hence a homogeneous electric field perpendicular to the plane changes the dynamics in layer space in a non-trivial way. Since the latter is entangled with the relative motion through the Coulomb term we conclude that a homogeneous electric field perpendicular to the plane may produce transitions between different states with the same center of mass quantum numbers, namely the Kohn theorem does not apply.   Table 1: Eigenenergies obtained from di erent con gurations. We considered: t = 0:07, and t = 0:06, for N = 7 and N = 6 respectively, in both cases = 0:2, = 1:4 and z = 0:02 and the GS is ferromagnetic. A is the excitation due to an electric eld given byẼ = E kî , and B due toẼ =Ẽ xz beingî the unitary vector along the X-axis. C is due to an electric eld expanded up to the quadrupolar approximation and directed along the X-axis. N=6 GS A B C (6,3,0) (