density is described as a sum over occupied bound and scattering wave functions of independent electrons moving in the ... of various problems related to the embedding of moving ions in a metallic ... at a nucleus with charge Z,9â11 is important in various fields ..... results could be helpful is the study of metallic hydrogen at.
PHYSICAL REVIEW B 72, 125113 共2005兲
Curvature of the total electron density at critical coupling A. Galindo,1,2 I. Nagy,2,3 R. Díez Muiño,2,4 and P. M. Echenique2,4,5 1Departamento
de Física Teórica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain International Physics Center DIPC, P. Manuel de Lardizabal 4, 20018 San Sebastián, Spain 3 Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, H-1521 Budapest, Hungary 4Unidad de Física de Materiales, Centro Mixto CSIC-UPV/EHU, P. Manuel de Lardizabal 3, 20018 San Sebastián, Spain 5Departamento de Física de Materiales, Facultad de Químicas UPV/EHU, Apartado 1072, 20080 San Sebastián, Spain 共Received 14 March 2005; revised manuscript received 28 July 2005; published 15 September 2005兲 2Donostia
A structural property of the spherically averaged total charge density 共r兲 of a degenerate electron gas distorted by the action of an attractive, screened potential V共r兲 is the subject of this study. The total charge density is described as a sum over occupied bound and scattering wave functions of independent electrons moving in the effective central field. It is shown that the second radial derivative ⬙ 共0兲 is a differentiable function of for any 艌 0. The smooth behavior implies that this curvature ⬙ 共0兲 does not provide a criterion for the bound-to-free transition in metallic environments. DOI: 10.1103/PhysRevB.72.125113
PACS number共s兲: 71.10.Ca, 71.55.⫺i
Screening is a fundamental property of an electron gas, the prototype many-electron model of metals. When embedding an external charge into it, the electrons redistribute themselves so as to completely shield the charge at large distances. The most accurate theoretical method currently applied to this fundamental problem is based on density functional theory 共DFT兲.1 The theory treats the screening action by using a grand-canonical ensemble for the electron gas with constant chemical potential and a variable number of constituents. In the Kohn-Sham 共KS兲 scheme,2 the physical groundstate density 共r兲 is constructed from one-particle wave functions by summing over occupied states.3 For a given density of the screening environment 0, and depending on the magnitude of the attractive external charge, the total density consists of bound 共bd兲 and scattering 共sc兲 states.4,5 Detailed knowledge of these wave functions is useful in the analysis of various problems related to the embedding of moving ions in a metallic environment.6–8 Theoretical information on the structural properties of the density 共r兲 other than its well-known behavior,
⬘共0兲 = − 2Z共0兲,
共1兲
at a nucleus with charge Z,9–11 is important in various fields of physics and chemistry. The above cusp condition does not depend on single-particle energies and also holds, separately, for bound and scattering states. Peierls12 investigated the possibility of discontinuity in the physical variable 共0兲 = bd共0兲 + sc共0兲 when a new bound state is created. He found a smooth behavior of 共0兲 at critical coupling. The smooth behavior was also demonstrated recently by an explicit calculation based on the Hulthén potential.13 Notice that it is the determinant form of the many-body wave function, imposed by the Pauli exclusion principle, the one that yields the smooth behavior of the averaged ground-state quantity.12 The analytical character of the density matrix in the many-body context14 has been also analyzed as a function of the coupling constant of a short1098-0121/2005/72共12兲/125113共5兲/$23.00
range potential V共r兲, by considering only s-type 共l = 0兲 states. For the particular case of a closed-shell atom without the presence of a screening environment, a closed expression for the second derivative of the electron density was deduced in Ref. 15 from the radial solutions of s and p共l = 1兲 occupied bound orbitals in the KS scheme. In a bare Coulomb potential, and after making the angular average of the probability density of scattering states gsc共r兲, the curvature at the origin has the following simple form:16
⬙ 共0兲 = 4Z2gsc共0兲. gsc
共2兲
This is, quite remarkably, independent of the scattering energy. Again, both s and p contributions were added up to obtain this result. In the same potential, if the probability density of bound states gbd共r兲 is described only with s-type states, one gets an energy-dependent expression
冉 冊
2 1 ⬙ 共0兲 = Z2 5 + 2 gbd共0兲, gbd 3 n
共3兲
where n is the principal quantum number. Both the gsc共0兲 and gbd共0兲 probability densities satisfy Eq. 共1兲. At vanishing one-particle binding energy 共n → ⬁兲, the prefactor of gbd共0兲 is 共10/ 3兲Z2. The difference between 4Z2 关Eq. 共2兲兴 and 共10/ 3兲Z2 关Eq. 共3兲兴 seems to indicate a discontinuous change in the curvature of the electronic density when approaching the zero-energy state from either the bound or the scattering side. This could, according to Ref. 16, provide a criterion for the bound-to-free transition in charged systems. Stimulated by this appealing possibility, we have focused on the study of the curvature in a different context, that of electrons moving in a central screened potential. To mark the difference, we shall distinguish in the following the electronic density obtained by summing over the occupied electronic states of a real screening problem 共bd and sc兲 from the probability densities 共gbd and gsc兲 obtained by using states of a given central field. The derivatives of g共r兲,
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a single-particle probability, may have relevance when dealing with the screening problem in metal-insulator transitions where the abrupt disappearance of the metallic character gives rise to the appearance of an atomic limit with a fixed number of electrons in the unit cell.17 The main goal of this work is thus a rigorous analysis of ⬙共0兲 and, more specifically, of its behavior as a function of the coupling constant . We will show that the angularaveraged charge density 共r兲 of a degenerate paramagnetic electron gas distorted by the action of a local short-range potential V共r兲, as well as the two radial derivatives ⬘ 共0兲 and ⬙ 共0兲, are differentiable functions of for any 艌 0. There are no discontinuities in that could be used as imprints for the appearance of bound states. In the following, we simply sketch the lines along which one can rigorously prove such differentiability. For the sake of legibility, only essential mathematical details are provided. Let V共r兲 be a central attractive potential of finite range and strength 0 ⬍ ⬍ ⬁. The potential V共r兲 is local and is intended to represent the complex many-body problem of the interacting electron gas in the presence of a singular potential in an approximate way. V共r兲 has the following series expansion around the origin: V共r兲 =
v−1 + r
Equations 共7兲 and 共9兲 illustrate the meaning of . The parameter includes both the effect of the external charge Z and the effect of the many-body electron screening through the ⌳ parameter. The bare Coulomb limit 关⌳ → 0 in Eq.共5兲兴 could be recovered by just making → ⬁, but then the rescaling of variables is not valid anymore. The Coulomb potential, which supports an infinite number of bound states, is out of the scope of this manuscript and will be treated elsewhere, together with extended proofs and additional analysis.18 The analysis and conclusions presented here do apply to other local short-range potentials that describe the screening of charges in electronic media, such as the Thomas-Fermi potential, for instance. In practical terms, the value of determines the number of bound states supported by the potential V. Now, let H ª − 21 ⌬ + V be the Hamiltonian, and let ,LM,E共r兲 = R,L,E共r兲Y LM 共⍀r兲 denote an eigenfunction of H for energy E 艋 0 and unit norm: 储,LM,E储 = 1. The radial wave function u,L,E共r兲 ª rR,L,E共r兲 is a regular solution of the reduced radial Schrödinger equation for L waves. For r →0
r→0
⬁
v jr j, 兺 j=0
v−1 艋 0.
共4兲
Some words may be needed to explain the particular notation used above. Any local analytic central potential with a Coulomb singularity at the origin can be described using the V共r兲 form. In order to clarify the meaning of the coupling constant in this context, let us take as an example the Hulthén potential VH共r兲,22,23 which will be later used for illustrative purposes: Z ⌳r , VH共r兲 = − r e⌳r − 1
,bd共0兲 = 2 兺 L,E
¯E = ⌳−2E,
¯共r ¯ 兲 = 共r兲.
⬙ 共0兲 = ,bd
共6兲
共7兲
The Schrödinger equation can then be rewritten as 1 ¯ 1¯¯ ¯ 兲 − ¯r ¯ 兲 = ¯E¯共r ¯ 兲. 共r 共r − ⌬ 2 e −1
共8兲
Hence, the form of the Hulthén potential in this new unit is the following: ¯兲 ª V共r ¯兲 = − V共r
1 ¯r
e −1
.
共9兲
共2L + 1兲 1 兩R,L,E共0兲兩2 = 兺 兩␣,0,E兩2 , 4 2 E
⬘ 共0兲 = 2v−1,bd共0兲, ,bd 1 2
冉兺 E
2 2 关52v−1 + 2共v0 − E兲兴兩␣,0,E兩2 3
冊
+ 6 兺 兩␣,1,E兩2 ,
Provided that ⌳ ⫽ 0, we can take 1 / ⌳ as a new length unit and make the following change of variables: = Z/⌳,
冊
where ␣,L,E is a normalization constant. From the small-r behavior of the radial wave functions u,L,E共r兲 关Eq. 共10兲兴, one can show that the contribution of the bound states to the angular-averaged density ,bd共r兲 and its radial derivatives ,bd ⬘ 共0兲 and ,bd ⬙ 共0兲 for r → 0 is
共5兲
1 Z ⌳r 共r兲 = E共r兲. − ⌬共r兲 − 2 r e⌳r − 1
v−1 r L+1
2 + 共L + 1兲共v0 − E兲 2 2v−1 r + O共r3兲 , 共10兲 共L + 1兲共2L + 3兲
+
where Z is the external charge and ⌳ a parameter describing the screening made by the electrons. VH共r兲 is expressed above in atomic units 共e = me = ប = 1兲. The Schrödinger equation with this potential is
¯r = ⌳r,
冉
u,L,E共r兲 ⬃ ␣,L,ErL+1 1 +
E
共11兲
where the sums run over all occupied bound states. Only those terms with L = 0 contribute to ,bd共0兲 and ,bd ⬘ 共0兲, whereas we emphasize that both L = 0 and L = 1 terms need to be considered for ,bd ⬙ 共0兲. A similar analysis can be made for the wave functions + + ,k 共r兲 of positive energy E = 21 k2 艌 0. Let ,k 共r兲 stand for an outgoing scattering state with incoming momentum k and + 共x兲 into normalized to the k scale. After expansion of ,k + partial waves, the radial function u,L,k共r兲 is the regular solution of the reduced radial Schrödinger equation, with the following behavior as r → 0:
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冢
+ + u,L,k 共r兲 ⬃ ␣,L,k rL+1 1 + r→0
just pay attention to the cases relevant in our problem: L = 0 and L = 1. From the behavior of the Jost functions for k → 0 and → L,n,
v−1 r L+1
冉
冊
冣
F,0共k兲 ⬃ 共− 1兲n关␣0,n共 − 0,n兲 + i␥0,nk + ¯ 兴,
1 2 2v−1 + 共L + 1兲 v0 − k2 2 r2 + O共r3兲 , + 共L + 1兲共2L + 3兲
F,L⬎0共k兲 ⬃ 共− 1兲n关␣L,n共 − L,n兲 + L,nk2 + i␥L,nk3 + ¯ 兴,
共12兲
where + ␣,L,k =
1 kL+1 . 共2L + 1兲!! F,L共k兲
共13兲
The function F,L共k兲 is the Jost function in the L channel.19 The contribution of the scattering states to the density ,sc共r兲 can be calculated adding up all scattering states up to the Fermi level. For r → 0,
,sc共0兲 = 2 =
冕 冕
d 3k
兩k兩艋kF
1 2
kF
0
dk
1 4
冕
S2
where ␣0,n , ␥0,n , ␣L,n , L,n , ␥L,n are strictly positive, one can prove the following propositions: Proposition 1: The appearance of a new bound state when crosses a critical value L,n accounts for a nonsmooth behavior of ,bd共0兲 , ,bd ⬙ 共0兲, as functions of around L,n. Up to a C1 function of , −3 4,bd共0兲 ⬃ 4␣0,n␥0,n 共 − 0,n兲共 − 0,n兲, →0,n
⬙ 共0兲 ⬃ 4,bd
→0,n
+ 兩,k 共r兲兩2d⍀r
k2 , 兩F,0共k兲兩2
⬙ 共0兲 ⬃ 4,bd
→1,n
2 2 2 ⬙ 共0兲 = 共52v−1 ,sc + 2v0兲,sc共0兲 + 3 32 ⫻关− 兩F,0共k兲兩 + 兩F,1共k兲兩 兴. −2
−2
8 2 2 3 关50,nv−1
−3 + 20,nv0兴␣0,n␥0,n
⫻ 共 − 0,n兲共 − 0,n兲, −4 1/2 −5/2 ␥1,n共 − 1,n兲 + 2␣1,n 1,n 兩 关4␣1,n1,n
−1 −1 ␥1,n兴共 − 1,n兲. − 1,n兩1/2 + 34 1,n
⬘ 共0兲 = 2v−1,sc共0兲, ,sc
冕
kF
共17兲
Proposition 2: In a neighborhood of = L,n, the quantities ,sc共0兲 , ,sc ⬙ 共0兲 behave as follows, up to a C1 function of : −3 兩共 − 0,n兲兩, 4,sc共0兲 ⬃ − 2␣0,n␥0,n
dkk4
→0,n
0
共14兲
As in the bound case, only s-wave terms contribute to ,sc共0兲 and ,sc ⬘ 共0兲, whereas both s and p waves are relevant for ,sc ⬙ 共0兲. Each individual 共either bound or scattering兲 term in the overall density presents discontinuities at the thresholds. Remarkably enough they cancel each other. To prove this cancellation, we need an auxiliary lemma to express the bound state contribution in terms of the Jost functions. We know that the negative bound energies are associated with the zeroes of the Jost functions in the positive imaginary axis. Let us denote ¯k = i¯, such that E = 21¯k2 = − 21 2 is a bound state ¯ 兲 = 0. If ␣ ¯ ª ␣ energy. Then, F,L共k ,L,i ,L,E, it can be proven by an extension of a well-known result for s waves20,21 that 兩␣,L,i¯兩2 = 共− 1兲L+1
共16兲
4 共¯兲2共L+1兲 . 2 共共2L + 1兲!!兲 关F,L共− i兲F,L共i兲兴=¯ 共15兲
Moreover, for real momenta F,L共k兲 vanishes only if k = 0 and = L,n, and thus Eqs. 共14兲 show that arbitrarily small momenta are fully responsible for the discontinuous behavior of ,sc共0兲 , ,sc ⬘ 共0兲 , ,sc ⬙ 共0兲 around critical strengths. Let us focus on the local behavior of the quantities 兩␣,L,i¯兩2 , 兩␣,L,k兩2 around the thresholds L,n for the appearance of the nth bound state of angular momentum L. We will
2 −3 2 ⬙ 共0兲 ⬃ − 34 共50,n + 20,nv0兲␣0,n␥0,n 4,sc v−1 →0,n
⫻ 兩共 − 0,n兲兩,
⬙ 共0兲 ⬃ 4,sc
→1,n
4 −1 −1 3 1,n␥1,n共−
1/2 −5/2 + 1,n兲 − 2␣1,n 1,n
−4 ⫻共 − 1,n兲1/2共 − 1,n兲 − 4␣1,n1,n ␥1,n兩
− 1,n兩共− + 1,n兲.
共18兲
Here, 共x兲 is the Heavyside step function. From these two propositions, one can easily prove the following theorem, which is the core of the present work: Theorem: 共1兲 The density 共0兲 at the origin is a C1 function of the coupling constant . 共2兲 The cusp function Cusp共兲 ª ⬘ 共0兲 / 共0兲 is a linear function: Cusp共兲 = 2v−1.
共19兲
共3兲 The second radial derivative ⬙ 共0兲 of the density at the origin is a C1 function of the coupling constant . The third point of the theorem, i.e., the differentiability of the total curvature ⬙ 共0兲 as a function of , is actually the most relevant for our purposes. It means that no discontinuities in ⬙ 共0兲 are expected whenever reaches the threshold for the appearance of a new bound state. In the following, we illustrate the smoothness of the elec-
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FIG. 1. 共Color online兲 Partial components of the electronic density at the origin ,bd共0兲 共upper panel兲 and ,sc共0兲 共middle panel兲 as a function of the coupling constant for a Hulthén potential V共r兲 = − / 共er − 1兲. The total density 共0兲 ª ,bd共0兲 + ,sc共0兲 is shown as well in the lower panel. The insets show the derivatives of these functions with respect to 关␦,bd共0兲 / ␦, ␦,sc共0兲 / ␦, and ␦共0兲 / ␦ in the upper, middle, and lower panels, respectively兴 in the vicinity of the threshold = 0,1 for the appearance of the first 共L = 0兲-bound state.
tronic density at the origin, as well as the smoothness of the curvature by making use of the Hulthén potential 关Eq. 共9兲兴. For this potential, explicit analytical expressions are available for the wave functions with L = 0.24 The corresponding eigenenergies of the bound states with L = 0 are En0 = −共 / n − n / 2兲2 / 2, with 1 艋 n ⬍ 冑2. Therefore there are threshold values 0,n ª 21 n2 , n 艌 1, marking the appearance of a new L = 0 eigenstate for any ⬎ 0,n. The normalization constants
FIG. 2. 共Color online兲 Partial components of the curvature at the origin ,bd ⬙ 共0兲 共upper panel兲 and ,sc ⬙ 共0兲 共middle panel兲 as a function of the coupling constant for a Hulthén potential V共r兲 = − / 共er − 1兲. The total curvature ⬙ 共0兲 ª ,bd ⬙ 共0兲 + ,sc ⬙ 共0兲 is shown as well in the lower panel. The insets show the same functions in the vicinity of the threshold = 1,1 for the appearance of the first 共L = 1兲-bound state.
兩␣,0,En0兩2 and the Jost function F,0 are also analytical. For bound eigenfunctions with L 艌 1, one has to resort to numerical computation. For instance, 1,n = 兵2.6530, 5.3626, 9.0504, . . . 其 for n = 1 , 2 , 3. . . The constants 兩␣,1,En1兩2 and the function F,1共k兲 also require numerical computation. Without loss of generality, we fix the Fermi level to kF = 21 for illustrative purposes. The smoothness does not depend on the particular value of kF. In Fig. 1 we show the behavior of the partial components of the density ,bd共0兲 and ,sc共0兲, as well as the total density 共0兲 ª ,bd共0兲
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+ ,sc共0兲 as functions of . Figure 1 shows clearly that the bound state contribution ,bd共0兲 and the scattering state contribution ,sc共0兲 present cusp points as functions of at the values in which a new bound state appears. The cusps are washed out when added up to obtain the all state contribution 共0兲. The lack of differentiability of ,bd共0兲 hardly leaps to the eye due to the dominance of the bound states contribution. To render it visible we show in the insets the behavior of the derivatives 共with respect to 兲 ␦,bd共0兲 / ␦, ␦,sc共0兲 / ␦, and ␦共0兲 / ␦, in a neighborhood of the first L = 0 threshold 0,1. The functions ,bd ⬙ 共0兲, ,sc ⬙ 共0兲, and ⬙ 共0兲 ª ,bd ⬙ 共0兲 + ,sc ⬙ 共0兲 are shown in Fig. 2. The discontinuities of ,bd ⬙ 共0兲 are again not apparent in this figure due to the scale used. A particular case 共the threshold for the appearance of the first bound state with L = 1兲 is shown in the inset. Similar insets are used in the plots of ,sc ⬙ 共0兲 and ⬙ 共0兲. In them, the lack of differentiability of ,bd ⬙ 共0兲 and ,sc ⬙ 共0兲 is clearly visible. Figure 2 shows the perfect cancellation of the discontinuities in the two partial components ,bd ⬙ 共0兲 and ,sc ⬙ 共0兲, which renders the function ⬙ 共0兲 a differentiable function of even at the values of for which new bound states arise. Summarizing, we have shown in a transparent, though rigorous, manner that 共r兲, ⬘ 共r兲, and ⬙ 共r兲 are differentiable functions of at r = 0, for any 艌 0, where is a coupling constant in an attractive, screened potential V共r兲, such as a Coulomb-like potential embedded in a polarizable medium. Particularly, the smooth behavior of ⬙ 共r兲 at r = 0 shows that the curvature does not provide a criterion for the bound-to-free transition in ground-state screening in para-
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 共1964兲. Kohn and L. J. Sham, Phys. Rev. 140, A1133 共1965兲. 3 W. Kohn, Rev. Mod. Phys. 71, 1253 共1998兲. 4 E. Zaremba, L. M. Sander, H. B. Shore, and J. H. Rose, J. Phys. F: Met. Phys. 7, 1763 共1977兲. 5 M. J. Puska and R. M. Nieminen, Phys. Rev. B 27, 6121 共1983兲. 6 P. M. Echenique, F. Flores, and R. H. Ritchie, in Solid State Physics Series, edited by H. Ehrenreich and D. Turnbull 共Academic Press, New York, 1990兲, Vol. 43, p. 229. 7 R. Díez Muiño, M. Alducin, and P. M. Echenique, Phys. Rev. B 67, 121101共R兲 共2003兲. 8 M. Alducin, A. Arnau, and I. Nagy, Phys. Rev. A 68, 014701 共2003兲. 9 T. Kato, Commun. Pure Appl. Math. 10, 151 共1957兲. 10 E. J. Steiner, J. Chem. Phys. 39, 2365 共1963兲. 11 J. C. Kimball, Phys. Rev. A 7, 1648 共1973兲. 12 R. Peierls, Surprises in Theoretical Physics 共Princeton University Press, Princeton, 1979兲, pp. 137–146. 13 I. Nagy and B. Apagyi, Adv. Quantum Chem. 46, 268 共2004兲. 14 W. Kohn and C. Majumdar, Phys. Rev. 138, A1617 共1965兲. 15 R. O. Esquivel, J. Chen, M. J. Stott, R. P. Sagar, and V. H. Smith, 1
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magnetic metallic environments. The case of a spin-polarized electron gas would require some minor modifications in our theoretical development, but should lead to the same conclusions with respect to the smoothness of the curvature for the spin-polarized partial electron densities. The smoothness of 共r兲 and its derivatives at the Coulomb cusp is relevant to any many-body electronic structure calculation based on trial wave functions. More specifically, it provides a strong test for those trying to describe transitions from localized to continuum states in a metallic environment. For instance, an exciting new field for which our results could be helpful is the study of metallic hydrogen at pressures beyond molecular dissociation.25,26 In this kind of problems, the value of the curvature at the origin is strongly related to sum rules for the partial structure factors.16 A temperature-dependent extension of the analysis, via modifications in the momentum distribution function, may also have applications in plasma physics. ACKNOWLEDGMENTS
The authors are thankful to Neil W. Ashcroft for his encouraging comments on this work. Financial support by the Basque Departamento de Educación, Universidades e Investigación, the University of the Basque Country UPV/EHU 共Grant No. 9/UPV 00206.215-13639/2001兲, the Spanish MEC 共Grant Nos. FIS2004-06490-C03-00 and FPA200402602兲, and the EU Network of Excellence NANOQUANTA 共Grant No. NMP4-CT-2004-500198兲 is acknowledged. The work of I.N. was partially supported by the Hungarian OTKA 共Grant Nos. T046868 and T049571兲.
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