A vertical diatomic artificial molecule in the intermediate coupling regime in a parallel and perpendicular magnetic field

We present experimental results for the ground state electrochemical potentials of a few electron semiconductor artificial molecule made by vertically coupling two quantum dots, in the intermediate coupling regime, in perpendicular and parallel magnetic fields up to 5 T. We perform a quantitative analysis based on local-spin density functional theory. The agreement between theoretical and experimental results is good, and the phase transitions are well reproduced.


I. INTRODUCTION
Semiconductor quantum dots (QD's) are widely regarded as artificial atoms with properties analogous to those of 'natural' atoms. Furthermore, systems composed of two QD's, 'artificial' quantum molecules (QM's), coupled either laterally or vertically, have recently been investigated experimentally 1 and theoretically. [2][3][4][5][6] Transistors incorporating QM's 7 made by vertically coupling two well defined and highly symmetric QD's 8 are ideally suited to study QM properties. We recently reported the addition energy spectra at zero magnetic field for such QM's as a function of interdot coupling strength. 9 In this work we present experimental and theoretical ground state electrochemical potentials for a diatomic QM in the intermediate coupling regime corresponding to an interdot distance, b = 3.2 nm, for magnetic fields (B) up to about 5 T. We assume here that the quantum mechanical coupling is sufficiently strong that the QM can be regarded as a symmetric 'homonuclear' diatomic QM. 9 We consider two different configurations, one corresponding to an applied magnetic field parallel (B ) to the drain current I d flowing through the constituent QD's, and another corresponding to an applied magnetic field perpendicular (B ⊥ ) to I d (see Fig. 1). The latter has received relatively little attention. 5,[10][11][12] We note that the QM physics we discuss in both magnetic field configurations is particularly relevant to the subject of solid-state quantum computing. 10 The interpretation of the experimental results here is based on the application of localspin density-functional theory (LSDFT). 6,13,14 In the B case it follows the development of the method thoroughly described in Ref. 6, which includes finite thickness effects of the dots, and uses a relaxation method to solve the partial differential equations arising from a high order discretization of the Kohn-Sham (KS) equations on a spatial mesh in cylindrical coordinates (axial symmetry is assumed). To describe the less-common B ⊥ case, a threedimensional (3D) LSDFT code has been developed to handle configurations without any spatial symmetry.
This work is organized as follows. In Sect. II we describe the experimental setup; in Sect. III we outline the method used to implement LSDFT in our axially symmetric QM system, and discuss the fully 3D configurations; and in Sec. IV we give the experimental and theoretical results. The interpretation of these results, and a short summary is presented in Sect. V.

II. EXPERIMENT
The molecules we study are formed by coupling, quantum mechanically and electrostatically, two QD's which individually display clear atomic-like features. 7,8 For the materials we typically use, the energy splitting between the bonding and anti-bonding sets of single particle (s.p.) molecular states, ∆ SAS , can be varied from about 4.5 meV (strong coupling) to about 0.1 meV (weak coupling). 6,7 In this paper, b, the thickness of the central barrier separating the two dots is fixed at 3.2 nm (∆ SAS ∼ 3 meV). Because this corresponds to intermediate coupling, we can reasonably neglect a small mismatch (of energy < ∆ SAS ) between the two dots, i.e. the QM is assumed to be symmetric 'homonuclear'. 9 Figure 1 shows (a) a schematic section of a sub-micron circular mesa, diameter D, containing two vertically coupled QD's, and (b) a scanning electron micrograph of a typical mesa after gate metal deposition. The starting material, a special triple barrier resonant tunneling structure, and the processing recipe are described elsewhere. 7,15,16 Drain current I d flows through the two QD's between the substrate contact and grounded top contact in response to voltage V d applied to the substrate, and gate voltage V g on the single surrounding gate. The one structure we describe here (D ∼ 0.5 microns) is cooled to about 300 mK or less.

III. THEORY
To analyze the experiments we have modeled the QM by two identical QD's stacked in the z direction (parallel to I d ). In this direction the QM is confined by two identical quantum wells of width w and depth V 0 separated by distance b= 3.2 nm. We have taken V 0 = 225 meV and w = 12 nm, which are also appropriate for the actual experimental device. To improve on the convergence of the 3D calculations, the somewhat ideal sharp double-well profile has been slightly rounded-off, as shown in Fig. 2. Given that the energy profiles of real structures are never abruptly sharp, this rounding-off is actually not unrealistic.
In the remaining two directions perpendicular to z the QM is confined by a harmonic oscillator potential V h = mω 2 (x 2 + y 2 )/2 of fixed strengthhω = 4.42 meV. This lateral confinement energy has been determined for N = 6 electrons using a general law 17 that quantitatively describes the phases of QM's in the strong, intermediate and weak coupling regimes as a function of B for a number of electrons, N, between 12 and 36. Lacking a better prescription at smaller N, 18 ω has been kept fixed for all N analyzed here, and in this sense we present a parameter-free calculation. In the following we will denote by V cf (x, y, z) the total confining potential obtained by adding the double well profile to the harmonic oscillator potential, V h . We stress that V cf (x, y, z) is axially symmetric around the z axis.
As it is well known, within LSDFT the ground state (g.s.) of the system is obtained by solving the Kohn-Sham (KS) equations. In the B case the problem is greatly simplified by explicitly using the axial symmetry of the system. The additional terms in the KS equations due to the presence of an arbitrary magnetic field are given below. The inclusion of these terms crucially does not break the axial symmetry of the KS Hamiltonian in the B case.
In the symmetric gauge the vector potential A( r ) corresponding to a constant magnetic field B is written as A = ( B ∧ r )/2, and its contribution to the KS Hamiltonian is where g * s is the effective gyromagnetic factor, L and S respectively are the orbital and spin angular momentum operators, and µ B is the Bohr magneton. Writing B = B(sinθ B , 0, cosθ B ) and introducing the cyclotron frequency ω c = eB/mc, it can be easily checked that H m = H m R + iH m I , with where η σ =+1(−1) for σ=↑(↓) with respect of the direction of the applied magnetic field. We have used effective atomic unitsh = e 2 /ǫ = m =1, where ǫ is the dielectric constant, and m the electron effective mass. In units of the bare electron mass m e , m = m * m e . In this system, the length unit is the effective Bohr radius a * 0 = a 0 ǫ/m * with a 0 =h 2 /m e e 2 , and the energy unit is the effective Hartree H * = Hm * /ǫ 2 . For a QD in GaAs, we take the following values: g * s = −0.44, ǫ = 12.4, and m * = 0.067. This yields a * 0 ∼ 97.94Å and H * ∼ 11.86 meV. From now on we will write the equations in these units.
Equation (2) reduces to the B case when θ B = 0, and to the B ⊥ case when θ B = π/2. In the former, since (x ∂ ∂y − y ∂ ∂x ) is proportional to L z , the problem remains axially symmetric. A detailed description of how the KS equations have been solved in this geometry can be found in Ref. 6.
In 3D the KS equations read The expression in the brackets is the KS Hamiltonian H KS , and V H (x, y, z) is the direct Coulomb potential. V xc = δE xc (n, m)/δn| g.s. , and W xc = δE xc (n, m)/δm| g.s. are, respectively, the variation of the exchange-correlation energy density E xc (n, m) in terms of the electron g.s. density n(x, y, z), and of the local spin magnetization m(x, y, z) ≡ n ↑ (x, y, z) − n ↓ (x, y, z).
The exchange-correlation energy has been taken from Perdew and Zunger, 13 and E xc (n, m) has been constructed as indicated in Ref. 6. It is worth noticing that if B = 0 then the s.p. wave functions Ψ σ (x, y, z) are complex, with their real and imaginary parts being coupled by H m . The KS and Poisson equations are solved on a 3D mesh after discretizing them using 7-point formulas, and using a two-grid version of the one-way multigrid method described in Ref. 19. The Poisson equation is solved using a first order relaxation scheme. 20 The required value of the Coulomb potential at the mesh boundary is obtained by a standard multipole expansion up to eighth order. The KS equations are solved using an imaginary time method, involving the third-order expansion of the forward solution of the imaginary time diffusion equation 20 i.e., where ǫ = Ψ(τ )|H KS |Ψ(τ ) . To further accelerate the self-consistent solution of both the KS and Poisson equations, we use the preconditioning smoothing operation described in Ref. 21. In the KS case, this means that ∆Ψ(τ ) has been smoothed as proposed in this reference.
The perfomance of the code has been further improved by adding a 'viscosity term', i.e., Eq. (5) has been changed into A viscosity term has also been included in the solution of the Poisson equation. We have used a 45 × 45 × 67 mesh with spatial steps ∆x = ∆y = 5.67 nm, and ∆z = 0.89 nm. The large asymmetry between the spatial meshes is motivated by the sharpness of the confining potential in the z direction. The heuristic viscosity parameter α V is fixed to a value of 0.8, and the time step δτ to the value of (∆z) 2 in effective atomic units. The stability of our results against the increase of the number of mesh points has been checked, and at B = 0 we have used the results of the axially symmetric code to test the results obtained with the 3D code. The accuracy of LSDFT for the B values of interest has been assessed by comparing the results for a single QD with those obtained using the current spin-density functional theory (CSDFT), [22][23][24][25][26] which in principle is better suited to high magnetic fields than LSDFT. Since CSDFT is a two dimensional (2D) theory, 22 we have also compared our LSDFT results with those obtained using the 2D-LSDFT which is implicit in any implementation of CSDFT, in particular see that of Ref. 23. The low and high field borders of the maximum-density droplet (MDD) phase using strictly 2D-LSDFT and CSDFT, as described in Ref. 23, have been obtained for QD's with N = 20, 28 and 36 electrons laterally confined by a harmonic oscillator potential of energyhω = 7.6 N −1/4 (meV). Such parameterization of the confining potential within LSDFT reproduces the experimental MDD. 27 The results are shown in Table I, together with the values obtained by using the present 3D-LSDFT for the same lateral confining potential. From Table I we can see that the overall agreement between the three calculations is clearly good, and thus we can confidently use LSDFT in the present calculations.

IV. RESULTS AND DISCUSSION
The experimental ground state electrochemical potentials for N = 3 to 6, as a function of B, are shown in Fig. 3 for (a) the parallel case, and (b) the perpendicular case. What is actually shown is the B-field dependence of the 3rd, 4th, 5th and 6th current (Coulomb oscillation) peaks measured by sweeping V g in the linear conductance regime for a small V d ≃ 0.1 mV. It is clearly evident that the dependencies for parallel and perpendicular cases are very different -in particular the former is stronger than the latter. We now attempt to explain the general appearance in both cases, and in particular the features marked by the different symbols, by using the computational methods described in the previous section.
We first address the B case. As in single QD's, at low B-fields, upward kinks (cusps) in the experimental N electron g.s. QM electrochemical potentials as a function of B are interpreted as changes in the N electron g.s. configuration of the QM which arise from s.p. level crossings. 8,27,28 We have plotted in Fig. 4 the calculated g.s. electrochemical potential µ(N), defined as where U is the total energy of the N electron QM g.s., as a function of B up to N = 6.
To label the g.s. configurations we have used the usual notation of molecular physics for s.p. electronic orbitals. 29 Upper case Greek letters are used for the total orbital angular momentum. We have also used the adapted version 3 of the ordinary spectroscopic notation 2S+1 L ± g,u with S being the total |S z |, and L being the total |L z |. The superscript +(−) refers to even (odd) states under reflection with respect to the z = 0 plane bisecting the QM. Even states are bonding (symmetric) states, and odd states are anti-bonding (anti-symmetric) states. The subscript g(u) refers to positive (negative) parity states. All these are good quantum numbers in the B case and can be used to label the different g.s.'s ('phases'). Following Refs. 4,26 we have also calculated the 'isospin' quantum number (the 'bonding number' in molecular physics) defined as I z = (N B −N AB )/2, with N B(AB) being the number of electrons in bonding (anti-bonding) s.p. states. This is an exact quantum number for homonuclear QM's in a parallel magnetic field. 30 Given the complexity of real vertical QM structures and the challenge in modelling them, 3,4,9,26 a comparison between Figs. 3(a) and 4 reveals a rather good agreement between theory and experiment. As a guide, and consistent with the calculated states and the observed B dependence, we indicate in Fig. 3(a) in simple box style 29 the dominant g.s. configurations at or near B = 0, and others at higher field which are stable over a relatively wide range of B . Up and down arrows indicate spin-up and spin-down electrons, and black (grey) arrows represent electrons in bonding (anti-bonding) s.p. states. 28,29 For N = 3, 5, 6, near B = 0, because the g.s.'s are close to each other, i.e. stable over a fairly narrow range, we show two configurations which in practice are hard to resolve. Some of these involve the population of the lowest anti-bonding state with a single electron, so isospin is non-maximal. Above B = 1 T, however, all the anti-bonding states are depopulated so isospin is maximal (I z = N/2), and filling of the QM resembles that of a single QD. The identifiable g.s. transitions in Figs. 3(a) are marked by black triangles. As expected, most appear as upward kinks. A couple, see first kinks for N = 5 and 6, appear as "downward" kinks because of the g.s. transitions which occur at almost the same B in N = 4 and 5 respectively.
Looking further at other details in Fig. 4, for N = 2, the singlet-triplet transition occurs at about 4.6 T which is close to the experimental value 16 of ∼ 4.2 T (not shown). We have found from the calculations an MDD configuration made of electrons filling just bonding s.p. states (MDD B ), which has a total angular momentum L z = N(N − 1)/2, and extends from ∼ 4.9 to ∼ 9.5 T for N = 3, from ∼ 5.1 to ∼ 9.0 T for N = 4, from ∼ 5.4 to ∼ 8.8 T for N = 5, and from ∼ 5.6 to ∼ 8.3 T for N = 6. These results are at variance with those of Ref. 26, where an MDD B g.s. was found for N = 3, but not for larger values of N. The reason of this discrepancy may be attributed either to the strictly 2D model used in their calculation to represent the constituent QD's, or more likely to their particular implementation of CSDFT. 31 A quantitative description of the spin phases of QM's in different coupling regimes, along with the experimental data, will be presented elsewhere. 17 In Fig. 4 we can see that for the larger N values studied here, the increase in angular momentum of the QM g.s. as it evolves from B = 0 towards the MDD B is accompanied by two isospin flips 26 caused by electrons jumping from anti-bonding to bonding states and vice-versa. Phase transitions from − to + g.s.'s involve ∆I z = 1 flips, whereas those from + to − g.s.'s involve ∆I z = −1 flips, and both are clearly seen in Fig. 4 for N = 5 and 6. Interestingly, they only happen for B < 2 T. We can see that after reaching the ν B = 2 g.s. (i.e., a filling factor two QM state made of just bonding s.p. states), which corresponds to the 1 ∆ + g phase for N = 4, to the 2 Γ + g phase for N = 5, and to the 1 I + g phase for N = 6, only bonding s.p. states are occupied, and as a consequence the QM reaches the MDD B state in a similar way to how a single QD reaches the MDD state, namely by populating bonding s.p. states of higher and higher s.p. orbital angular momentum l values. 32 In general, these isospin flips can produce a complex pattern in the s.p. spectrum as a function of B . As an example of this complexity, we present in Fig. 5  In the B ⊥ case, crucially, within LSDFT, the only good quantum number is the spin projection along the direction of the applied magnetic field, which we call s ⊥ , and the g.s. electrochemical potentials as a function of B ⊥ are expected to be much smoother than in the B case. This situation 5,10-12 unlike the B case, lacks an analytical solution even for the case of non-interacting electrons. We show in Fig. 6 the calculated non-interacting s.p. spectrum as a function of B ⊥ . At B = 0 the energy difference between bonding and antibonding l = 0 s.p. states is just ∆ SAS (likewise for the l = 1 states). The energy difference between l = 1 and 0 bonding (or anti-bonding) states is justh ω. Similar results have been reported elsewhere. 5,12 Note that the effective value of ∆ SAS is reduced as B ⊥ is increased because the tunneling electron trajectories are increased in the barriers. The small splitting between spin up and down states that originate from a common s.p. state with well defined orbital angular momentum at B = 0 are due to the Zeeman term. Increasing B ⊥ has also an effect on the quantum mechanical coupling of the QD's forming the QM, as the already sizeable electron density in the middle of the central barrier increases further (see n(z) in Fig. 2) which magnifies exchange-correlation effects between both QD's. The compression of n(z) in each well as B ⊥ increases shown in Fig. 2 can be understood from Eq. (2): note that when θ B = π/2, the first term in H m R becomes ω 2 c (y 2 + z 2 )/8. This also produces an upward shift of the s.p. spectrum of the QM as a whole (the horizontal dashed line in Fig.  2 lies above the horizontal solid line in the same figure).
The even (bonding) or odd (anti-bonding) character of the s.p. levels defining a QM state is strictly lost when a magnetic field perpendicular to I d is present. Intriguingly, however, the bonding/anti-bonding character present at B = 0 is sometimes retained to a large degree by the s.p. states at finite B ⊥ values. We have indeed found that the expectation value of the z → −z reflection operator is very close to ±1, as it should be for bona fide bonding/anti-bonding states, in many cases even for relatively large values of B ⊥ .
As an example of this, we show in Fig. 7 the energies of the occupied s.p. states as a function of B ⊥ for N = 5 and 6. Solid triangles represent 'quasi-bonding' states with Π z ≥ 0.95. Note that at 0 T for N = 5 the s.p. bonding state at ǫ ∼ 48.2 meV is two fold degenerate, and likewise for two of the N = 6 s.p. bonding states at ǫ ∼ 52 meV. Open triangles represent 'quasi-anti-bonding' states with Π z ≤ −0.95. Actually, there is only one such occupied anti-bonding s.p. state for N = 5 at B = 0, and none for N = 6. All other open symbols (circles and squares) correspond to s.p. states with negative Π z values larger than -0.95, i.e., cannot really be regarded even as 'quasi-anti-bonding' states.
The figure also shows that states that evolve from l = 0 s.p. states at B = 0 retain a quasi-bonding character up to quite high values of B ⊥ (at least up to 5 T), whereas other states, that at B = 0 are l = 1 s.p. states, do not. The quasi-bonding robustness of the lower-lying s.p. states may be due to the small effect that the applied magnetic field has on states that are l = 0 s.p. states at B = 0. The B ⊥ evolution of what at 0 T are the 2p-states is rather similar for N = 5 and 6 with a change from solid to open symbols near B ⊥ = 4 T. Interestingly, in spite of the lack of any spatial symmetry in the system when a perpendicular field is applied, the s.p. levels are still clearly distributed into shells as in the non-interacting case. 5,12 Notice also the different splitting between ↑ and ↓ states. For saturated (zero) spin (N = 6 case), this is essentially due to the small Zeeman term, whereas for non-saturated spin (N = 5 case) the splitting is mostly due 23 to the spin-dependent part of the exchange-correlation energy, W xc term in Eq. (3), and this effect is larger the higher the value of the g.s. spin. This explains the sizeable splitting between the two lower lying s.p. levels for N = 5 up to B ⊥ ∼ 0.5 T and the splitting of all the s.p. levels for N = 6 above B ⊥ ∼ 3.5 T (see also Fig. 8).
The calculated g.s. electrochemical potentials are shown in Fig. 8 as a function of B ⊥ . Comparing with Fig. 3(b) it can be seen that the agreement with experiment is good for 3 ≤ N ≤ 6. We have indicated the value of the total S ⊥ for all the relevant g.s. phases. In the B = 0 to 5 T range, there are some B ⊥ induced changes in S ⊥ , and these give rise to upward kinks (also marked in Fig. 3(b) by solid down triangles). Some downward kinks, identified by vertical arrows in the N = 5 and 6 g.s. electrochemical potentials do not correspond to changes in the N-electron S ⊥ . They are associated with s.p. level crossings between s.p. states of different Π z value of the N − 1 electron system. This is the case for N = 5 at B ⊥ ∼ 1.5 T and N = 6 at B ⊥ ∼ 0.75 T, as can be seen in Fig. 8 (also marked in Fig. 3(b) by solid circles). Because of the lack of spatial symmetry in the system, we do not, in general, attempt to identify the (dominant) g.s. configurations. The configurations shown in simple box style in Fig. 3 (b) are the dominant g.s. configurations at 0 T and they are expected to remain so for small values of B ⊥ . The singlet-triplet transition for N = 2 in Fig. 8 appears at ∼ 4.7 T, a value compatible with that found in the B case. The B ⊥ induced singlet-triplet transition in the experimental data is discussed elsewhere. 12 It can also be seen that for N = 4 Hund's first rule like filling occurs for B ⊥ < 1.5 T, even if the g.s. configuration is not strictly axially symmetric. Nonetheless, for N = 6 at B = 0 we have found an axially symmetric configuration corresponding to a 2D harmonic oscillator shell like filling.

V. SUMMARY
We have thoroughly discussed the ground state electrochemical potentials of a fewelectron semiconductor artificial QM in the intermediate coupling regime. A detailed com-parison between experimental data and LSDFT calculations shows overall a good agreement for both parallel and perpendicular magnetic fields. The agreement is even more remarkable since the frequency ω of the lateral confining potential has not been used here as a fitting parameter, but rather it has been derived from a law strictly valid for larger values of N. Had we used an even smaller value of ω, the agreement would have been even better.
Any sensible comparison with the results of other calculations 3,26 and with the experimental data should consider the strong influence of the confinement on the actual g.s. QM phases. Large ω values obviously favor the occupation of anti-bonding states. Consequently, decreasing ω might 'wash-out' phase transitions involving anti-bonding states. For example, see the 2 Π + u → 2 ∆ − u transition for N = 5 or the 1 Σ + g → 3 Π − g transition for N = 6 in Fig  4. We have checked that this is indeed the case when the N dependent confining potential hω = 5.78 N −1/4 (meV) of Refs. 3,26 is used. The same is true for the 2 Σ − u → 2 Π + u transition for N = 3 (these two states are practically degenerate at B = 0). Thus, a comparison between theory and experiment may help to find accurate and realistic values for the effective lateral confining potential.
Analysis of the B ⊥ case has shown that the N = 2 singlet-triplet transition sensibly occurs at a similar B value to that in the B case. In spite of the absence of the strong Landau quantization inherent to the B situation, the s.p. levels are, to a large extent, still distributed into shells. We have also found that spin effects arising from the spin dependence of the exchange-correlation energy can dominate those caused by the small Zeeman term as is also the case for a single QD.

ACKNOWLEDGMENTS
This work has been performed under grants PB98-1247 from DGESIC and 2001SGR-00064 from Generalitat of Catalunya, and by the Specially Promoted Research, Grant-in-aid for Scientific Research, and by DARPA-QUIST program (DAAD 19-01-1-0659). F.A. has been funded by CESCA-CEPBA Large Scale Facilities through the program "Improving the Human Potential", contract no. HPRI-1999-CT-00071. We are grateful for the assistance of T. Honda with processing the samples, and for useful discussions with A. Emperador, E. Lipparini, and Y. Tokura.
29 The simple schematic box notation in Fig. 3 that represents the molecular states is more intuitive than the full spectroscopic notation used in Fig. 4, although we do not use it in Fig. 4 because of lack of space. Nonetheless, it is easy to go from one notation to the other since each box represents a s.p. level characterized by a radial quantum number n and an angular momentum l. The black (grey) arrows denote the thrid component of spin for electrons in bonding (antibonding) states. At 0 T, for both bonding and antibonding states separately, the lowest box is the (n, l) = (0, 0) 1s state, and above that are two boxes, degenerate in energy, which are the (0, ±1) 2p states (see for example N = 5 in Fig. 3(a)). For the spectroscopic notation, the superscript − of states shown in Fig. 4 is associated with an electron in an antibonding box in Fig. 3(a), and the total S z and L z are readily obtained by adding the spin and orbital angular momentum of all the electrons. Parity (g or u) is obtained by multiplying the parities of the s.p. states. 30 We want to stress that in the B case we have explicitly imposed the s.p. wave functions to be axially symmetric, but have not forced them to be specularly symmetry about the z = 0 plane. Interestingly, in the case of a two-electron homonuclear QM in the weak coupling limit corresponding to fairly large interdot distances, we have found 9 that the LSDFT g.s. is a broken-symmetry state made of two s.p. wave functions each of them localized in one of the constituent dots, in a singlet spin state. The specularly symmetric configuration has a slightly higher energy and corresponds to a triplet spin state. 31 The authors of Ref. 26 have identified the vorticity V with the external magnetic field B. While this is indeed the case for a homogeneous 2D electron gas, and a fairly good approximation for large QD's, this may not be a good approximation for small systems, as in the present case. CSDFT implementations better suited for small systems are presented in Refs. 22-25, for example. 32 Due to the convention adopted for the s.p. wave functions, 6 the s.p. orbital angular momentum is actually −l.  The lines have been drawn as a to guide the eye. The vertical marks along the µ(N ) lines indicate phase boundaries. The value of S ⊥ in each phase is given. We have indicated by vertical arrows downward kinks arising from s.p. level crossings in the N -1 electron ground state that do not produce phase transitions in the N electron ground state.