Merging of single-particle levels and non-Fermi-liquid behavior of finite Fermi systems

0.13 MB·English

by  V. A. Khodel

Merging of single-particle levels and non-Fermi-liquid behavior of finite Fermi systems
V. A. Khodel,1,2 J. W. Clark,2 Haochen Li,2 and M. V. Zverev1
1 Russian Research Centre Kurchatov Institute, Moscow, 123182, Russia
2 McDonnell Center for the Space Sciences and Department of Physics,
Washington University, St. Louis, MO 63130, USA
(Dated: February 6, 2008)
7 We examine the problem of finite Fermi systems having a degenerate single-particle spectrum
0 and show that the Landau approach, applied to such a system, admits the possibility of merging
0 single-particle levels. It is demonstrated that the opportunity for this behavior is widespread in
2 quantummany-bodysystems. Thesalient feature of the phenomenon is theoccurrence of noninte-
gral quasiparticle occupation numbers,leading to a radical alteration of thestandard quasiparticle
n
picture. Implications of this alteration are considered for nuclear, atomic, and solid-state systems.
a
J
PACSnumbers: 71.10.Hf,21.10.-k,36.20.Kd,21.60.-n
5

]
l Landau Fermi-liquid theory [1] (FL) is recognized as kind in the nuclear interior and for electron-electron in-
e
one of the foundation stones of our understanding of the teractions in atoms.
-
r properties of condensed matter. When adapted to fi- Webegintheanalysisofthisphenomenonbyconsider-
t
s nite Fermi systems in nuclear and atomic physics, the ing two neutron levels in an open shell of of a schematic
.
t standardFL quasiparticlepicture prescribesthatthe to- model of a spherical nucleus of mass number A and ra-
a
m tal angular momenta of the ground states of odd nuclei dius R=r0A1/3. The levels are denoted by 0 and +, in
must be carried by a single quasiparticle, and that the order of increasing energy. As usual, the sp energies ε
- λ
d electronic configurations of ions of elements of the pe- and wave functions ϕλ(r)=Rnl(r)Φjlm(n) are solutions
n riodic table must repeat those of preceding atoms. Its of equation [p2/2M + Σ(r,p)]ϕ (r) = ǫ ϕ (r), where
λ λ λ
o iconic stature notwithstanding, the standard FL picture Σ stands for the quasiparticle self-energy. In a spheri-
c
sometimesfails. Currently,theoriginofnon-Fermi-liquid cal nucleus with even numbers of neutrons and protons,
[
(NFL) behavior of homogeneous Fermi systems is one of which has zero total angular momentum in its ground

1
the central issues of condensed-matter physics. Inhomo- state due to pairing correlations, the sp energies ǫ are
λ
v
geneous, finite Fermi systems, although unburdened by independent of the magnetic quantum number m. We
7
the damping of single-particle (sp) excitations cited as suppose that the orbital angular momenta of levels 0
1
1 the source of failure of the Landau quasiparticle picture and + obey l0 6= l+ ≫ 1 and follow the variation of
1 [2],arealsoknowntoexhibitviolationsofFLtheory. For the distance between these levels, as N ≫ 1 neutrons
0 example,thetotalangularmomentaofthegroundstates are added to the level 0, changing the density ρ(r) by
7
of many odd-A nuclear isotopes in the transition region δρ(r) = NR2 (r)/4π. We neglect self-interaction cor-
0 n0l0
cannotbeattributedtoaspstate,andtheelectroniccon- rections and retain only a major, spin- and momentum-
/
t
a figurationsof elements not belongingto principalgroups independent part V(r) of the self-energy Σ and a pri-
m differ from those expected in FL theory. mary,δ(r)-like portionofthe Landau-Migdalinteraction
- Here we focus attentiononfinite Fermisystems whose function f. In this case, the FL relation between the
d self-energy and the density ρ that is responsible for the
sp spectrum ǫ has a degeneracy and show that, rather
n λ
variation of ǫ simplifies to [3, 4]
o surprisingly, NFL behavior can arise within the Landau λ
c approach[1] itself, with the mergingof splevels lying on δV(r)=f[ρ(r)]δρ(r) . (1)
:
v oppositesidesoftheFermisurface. Thisphenomenon,al-
i mostneveraddressedincondensed-matterphysics,stems The interaction matrix elements
X
from the variationof spenergies under change of the oc- r2dr
ar cupationofthelastunfilledsplevelduetotheinteraction fkk′ =Z Rk2(r)f[ρ(r)]Rk2′(r) 4π (2)
betweenquasiparticles. Inthesystemstobestudied,the
energeticdistance betweena splevelbeing filled andthe are assigned values f00 = u, f++ = v, and f0+ =
nearest empty level shrinks progressively as the former w. In a semiclassical approximation where Rk(r) ∼
level is filled, leading to a crossing of the levels in cases r−1R−1/2cos pk(r)dr, one has u≃v ≃3w/2.
where standard FL theory is obeyed. However, in the Based on thRese assumptions, the dimensionless energy
case of interest, the levels do not cross one another; in- shifts ξk(N)=[ǫk(N)−ǫk(0)]/D are given by
stead, they merge. As will be seen, a primary condition
ξ (N)=n U and ξ (N)=n W , (3)
0 0 + 0
for merging to occur is that the Landau-Migdal interac-
tion function f is repulsive in coordinate space, which where n = N /(2j +1) is the occupation number of
k k k
holds for the interactions between particles of the same level k, D is the initial distance between levels + and 0,
2
U = u(2j +1)/D, and W = w(2j +1)/D. We have of merging has the familiar form
0 0
neglected second-order corrections that have little effect
1−n n
on the results since they are almost independent of the G(r,r′,ǫ)= k + k ϕ (r)ϕ∗(r′) ,
sp quantum numbers. k=X0,+(cid:18)ǫ−ǫk+iδ ǫ−ǫk−iδ(cid:19) k k
According to Eqs. (3), the dimensionless distance (7)
d(N)=[ǫ (N)−ǫ (N)]/D =1+ξ (N)−ξ (N)changes but the occupation numbers n of the merging sp levels
+ 0 + 0 k
sign at N = D/(u−w). This always occurs before fill- become fractional.
c
ing of the level 0 is complete if the distance D is rather
small. For N above some critical value N , Eqs. (3) be- (cid:19) [ (cid:20) (cid:19) [ (cid:20)
c
comeinapplicable,sincethepatternoforbitalfillingmust (cid:20)
(cid:21)
change. In the standard FL picture, which allows only
for crossing of sp levels, all the added N > N quasi-
c G(cid:20) G
particles must settle into the empty sp level +, which (cid:3)
entails a change of the relevant sp energies. The sign of (cid:19)
(cid:19)
the difference ǫ (N) = δǫ (N)−δǫ (N) of the shifts of
r + 0
the sp energies ǫ and ǫ due to total migration of the
+ 0 (cid:20) (cid:20)
quasiparticles from level 0 into level + is crucial. With
the help of Eq. (1), it is seen that
Q(cid:3)(cid:15)(cid:3)Q(cid:19)(cid:14) Q(cid:19) Q(cid:14) (cid:3) Q(cid:19) Q(cid:14) Q(cid:3)(cid:15)(cid:3)Q(cid:3)(cid:19)(cid:14)
ǫ (N)=N(u−2w+v) . (4)
r
(cid:19) (cid:19)
In our model, ǫ (N)≃2Nu/3is positive. The positivity
r (cid:19) [ (cid:20) (cid:19) [ (cid:20)
of this key quantity means that level + remains above
level 0 upon the migration. The standard FL scenario FIG. 1: Top panels: Dimensionless distance d=(ǫ+−ǫ0)/D
must then encounter a catastrophe at N > Nc: on the betweenlevels+and0asafunctionoftheratiox=N/(2j0+
onehand, quasiparticlesmustleavelevel0;onthe other, 2j++2). Lower panels: Occupation numbers nk for levels 0
their total migration into level + is prohibited. To end and +. Input parameters: U = V = 3,W = 1. For the left
the deadlock, both levels must be partially occupied, in column,theratior≡(2j0+1)/(2j++1)=2/3;fortheright,
r=3.
contradistinction to FL theory where one and only one
levelcanbepartiallyoccupied. Suchdualpartialoccupa-
Results from numerical calculations are plotted in
tion is possible only if the sp energies ǫ and ǫ coincide
0 +
Fig. 1, which consists of two columns, each made up of
withthe chemicalpotentialµ. Ifso,atN >N ,Eqs.(3)
c
twoplots. Theupperpanelsshowthedimensionlessratio
becomes
d(x)=[ǫ (x)−ǫ (x)]/D versus x=N/(2j +2j +2),
+ 0 0 +
which runs from 0 to 1. The lower panels give the oc-
ǫ (0)+N u+N w =ǫ (0)+N w+N v, (5)
0 0 + + 0 +
cupation numbers n (x) and n (x), which, in the range
+ 0
of x where d(x) = 0, behave as n (x) = x(1+r)/2−
+
which has the solution
r/(2(U−W))andn (x)=x(1+r)/(2r)+1/(2(U−W)),
0
with r =(2j +1)/(2j +1).
N =(N −N )(u−w)(u+v−2w)−1 . (6) 0 +
+ c In the variable x, the model exhibits three distinct
regimes. In two of them, d 6= 0 and the standard
Significantly, a resolution of the dilemma has been FL picture holds. It fails in the third region, where
found within the Landauapproachitself. To understand d=0 and integrationof Eq.(7) overǫ yields the density
the consistency of this resolution, recall that the occu- ρ(r) = n ϕ (r)ϕ∗(r) for the added quasiparti-
k=0,+ k k k
pation numbers of Landau quasiparticles are given by cles. ThPisresultcannotbeattributedtoasinglesplevel,
[1] n (T) = [1+e(ǫλ−µ)/T]−1. At T = 0, this formula implying that well-defined sp excitations in the familiar
λ
guaranteesthatoccupationnumbersarerestrictedtothe Landausenseno longer exist. Passagethroughthe three
values 0 and 1, but only for those sp levels with energies regimes can be regarded as a second-order phase tran-
ǫ 6=µ;otherwisetheindex ofthe exponentisuncertain. sition, with the occupation number n (x) treated as an
λ +
As seen above, the merging of two (or more) sp levels order parameter.
drives the levels exactly to the Fermi surface. Solution The two sp levels remain merged until one of them is
of equations of merging such as (5) removes the residual completely filled. If the level 0 fills first, as in the left
uncertainty at the expense of introducing fractional oc- column of Fig. 1, the episode of merging is ended by
cupationnumbersn ,violatingwhatwouldappeartobe repulsion of the two levels, as if they possess the same
λ
an elementary truth of FL theory, but in fact maintain- symmetry – despite the fact that in the open shell, they
ing consistency. The sp Green function in the presence alwayshave different symmetries. In another case where
3
level + becomes fully occupied before level 0, as in the paredto exchange[8],sothatf takesthe Hartree-Fock
ee
right column, the distance d(x) becomes negative, and form. Carrying out the same operations that led to the
the two levels just cross each other at this point. necessary condition (4), we readily arrive at the corre-
In the nuclear many-body problem, both types of sp sponding condition
level degeneracy – either initially present or arising in
the scenario described above – are lifted when pairing fnnll+fnn′′ll′′ −2fnnl′l′ >0 (8)
correlations are explicitly involved [3, 4, 5]. In atomic
nuclei, realistic pairing forces are so weak that in nu- for the merging of two electron sp levels with quantum
cleiwithclosedshells,pairingcorrelationsarecompletely numbers n,l and n′,l′. Introducing Rnl(r) = rRnl(r),
suppressed, with the effect that the nuclear pairing en- the interaction matrix elements are constructed as
ergy becomes a part of a shell correction to the Bethe-
Weisz¨acker liquid-drop formula. A model calculation [6] fnnl′l′ =e2Z [R2nl(r1)R2n′l′(r2)
shows that merging of even two sp levels approximately
doubles the gap value and drastically increases the pair-
ing energy. More realistic calculations are needed to de- 1
termine whether this enhancementis enoughto promote −(2j+1)−1Rnl(r1)Rn′l′(r1)Rnl(r2)Rn′l′(r2)]r dr1dr2,
>
the existence of a new minimum in the ground-state en- (9)
ergy functional of superheavy nuclei. where r is the greater of r , r . In obtaining Eq. (9),
> 1 2
We turn now to the merging of sp levels in atoms. In we have neglected insignificant contributions to the ex-
fact,itwasinatomicphysicsthatoneofthefirstmodels change part of f coming from multipole moments cre-
ee
of NFL behavior involving nonintegral occupation num- atedbyelectronsmovinginthe openshell. Semiclassical
bers was developed by Slater et al. [7]. This model is estimatesalongthesamelinesasbeforeconfirmthatthe
based the observation that results of Hartree-Fock (HF) nondiagonal matrix elements of f are of much smaller
ee
calculations for atoms of intermediate groups are often size than the diagonal ones. We may then conclude that
improvedif,inanextendedHFenergyfunctional,contri- the difference on the l.h.s. of inequality (8) is positive
butions of two leading Slater configurations are included independently of the quantum numbers, so that the nec-
additively with factors 0 < x < 1 and 1−x. The best essary condition for merging of the sp levels is met.
choice for x is found from a minimization procedure, It is instructive to compare the values of the basic pa-
yielding equations similar to Eqs. (5), which naturally rameterN thatgovernsthemergingphenomenoninnu-
c
entail fractional occupation numbers n . As we have al- clear and atomic systems. In the nuclear problem, the
λ
readyseenwithinthe frameworkofthe non-perturbative neutron-neutron interaction is characterized by the di-
Landau approach, this feature is not a prerogative or mensionless constant [3] F = f p M/π2 ≃ 1; from
NN NN F
artifactof the HF method: nonintegraloccupation num- Eq. (2) one then obtains u ≃ǫ0/A, where ǫ0 = p2/2M
F F F
bersmaylegitimatelycomeintoplayprovidedthesignof is the Fermi energy. On the other hand, the distance D
the keyquantity δǫ fromEq.(4)is positive,and,hence, between sp levels in spherical nuclei with closed shells is
r
the criterion for merging of sp levels is met. It therefore of order of ǫ0/A2/3. Consequently, the critical particle
F
comes as no surprise that the conditions for merging of numberN =D/(u−w)mustbe oforderofA1/3,which
c
splevelsaresatisfiedincertainstronglycorrelatedFermi in turn means that not every pair of nuclear sp levels
systems [9, 10] for which the HF method is inapplicable. adjacent to the Fermi surface is susceptible to merging.
Two circumstances complicate the analysis of merg- The situation in heavy atoms is quite different: the
ing electron sp levels in atoms. First, the sp energies diagonal matrix elements of the electron-electron inter-
ǫ cease to be m-independent due to the absence of actionintheopenshellareoforderseveraleV,markedly
njlm
Cooper pairing. Difficulties stemming from this fact can enhanced in the event f-orbital collapse, as occurs in
be avoided if, following Ref. 7, one tracks the center-of- rare-earth and transuranic elements [11, 12, 13, 14, 15].
gravity ǫo = ǫ /(2j +1) energies of levels, rather The distance D between the sp levels adjacent to the
k m km k
than individuPal m-levels or a band. Then one only has Fermi surface in atoms with closed electron shells is
to deal with the spherically symmetric part of δρ(r), as known to be some 1–2 eV as well, so the critical num-
in the nuclear problem. Second, the self-energy Σ has ber N in atoms is of order of unity. Consequently, in
c
a nonlocal character due to the presence of long-range elements withnuclearchargeZ >20,the spGreenfunc-
Coulombinteractions. In another respect, the treatment tion has the form (7) – except for elements of principal
of merging sp levels in atomic systems is much simpler groups, where the standard FL picture still holds. Fur-
than in the nuclear case, because the spacing parame- thermore, when N substantially exceeds N , as is the
c
ter r = r /a is less than unity (r being the radius of case ofmany rare-earthandtransuranicelements, merg-
s 0 B 0
the volume per electron and a , the Bohr radius). This ing of levels triggered by collapse of the f orbitals in-
B
implies that correlation contributions to the electron- escapablyinvolvesmostofthesplevelsintheopenshell.
electron interaction function f are rather small com- Fractional occupation numbers become a necessity, with
ee
4
the result that these elements lose their chemical indi- latter, sp levels can merge even if f is momentum-
viduality, a well-known feature of the sequence of rare- independent. The reason for this difference is simple.
earth elements. The veracity of these inferences can be In the homogeneous case, the matrix elements u, v, and
testedbymeansofprecisemeasurementsofthedifference w areequaltoeachother,implying zeroenergygaindue
σ(Z +1)−σ(Z) between cross sections for elastic scat- totherearrangementwhenvelocity-dependentforcesare
tering of charged particles by rare-earth or transuranic absent. The sameisseenfromEq.(10); the groupveloc-
atomswithatomicnumbersdifferingbyunity. Inthethe- ity, whose sign determines whether the Landau FL state
oryofGoeppert-Mayer[11],thisdifferenceisdirectlyex- is stable, is unaffected by the momentum-independent
pressed in terms of the single 4f wave function, whereas part of f.
if merging occurs, the density change involves all the The merging of sp levels quite often violates a sym-
merged sp levels. metry inherent in the initial ground state. In nuclei, for
According to the above argument, in systems devoid example,themembersofpairsofsplevelswithquantum
of pairing the centers of merged sp levels “get stuck” at numbersn,l andn±1,l±2lie quite closeto eachother.
the Fermi surface. We observe that this could provide As the lowerofthe twolevelsis being filled, the distance
a simple mechanism for pinning narrow bands in solids from its neighbor shrinks, resulting in an increase of the
to the Fermi surface. To exemplify this point, consider nuclear quadrupole moment. If merging occurs, spheri-
a model in whichthe electronspspectrum, calculatedin cal symmetry is broken. This mechanism is presumably
local-densityapproximation(LDA),isexhaustedby(i)a responsible for the occurrence of islands of nuclear de-
wide bandthat dispersesthroughthe Fermi surface,and formation beyond the mainland of rare-earth elements.
(ii) a narrow band, placed below the Fermi surface at a Merging can also promote the enhancement of parity vi-
distance D . Turning on the electron-electron interac- olation effects in nuclei and atoms. These issues will be
n
tions produces a change of the sp energies in accordance addressed in a future article.
with the Landau equation In conclusion, our exploration of the mergence of
single-particle levels in finite Fermi systems exhibiting
ǫ(p)=ǫ (p)+ f(p,p )n(ǫ(p ))d3p /(2π)3 . (10) degeneracy reveals that if the merging of two or more
LDA 1 1 1
Z
sp levels occurs, the ground state experiences a rear-
To proceed, we assume that only matrix elements rangementthatintroducesaamultitude ofquasiparticle
f(n)(p,p ) of the interaction function f referring to the terms, endowing it with a more complex character,as in
1
narrow band are significant, the others being negligible. the comparison of a chorus with a dominant soloist.
Iftheshiftδǫ(n) inthelocationofthenarrowbanddueto We thank K.Kikoin,E.Saperstein,andV.Yakovenko
switching on the intraband interactions exceeds the dis- for fruitful discussions. This research was supported by
tanceD ,thenthestandardFLscenariocallsforthenar- Grant No. NS-8756.2006.2 from the Russian Ministry of
n
row band to be completely emptied; but then δǫ must Educationand Science andby the McDonnell Center for
n
vanish,andquasiparticlesareobligedtoreturn. Toelim- the Space Sciences.
inatethismismatch,onlyafractionofthe particlesleave
the narrowband, in just the rightproportionto equalize
the band chemical potentials. The feedback mechanism
we have described positions the narrow band exactly at
[1] L.D.Landau,JETP30,1058,(1956); Sov.Phys.JETP,
the Fermi surface, resolving a long-standing problem in
3, 920, (1957).
the LDA scheme.
[2] D. Pines and P. Nozieres, Theory of Quantum Liquids,
It is worth noting that the bare narrow-band group Vol. 1 (W. A. Benjamin, NewYork-Amsterdam,1966).
velocity, proportional to the corresponding bandwidth [3] A. B. Migdal, Theory of Finite Fermi Systems and Ap-
W(n), is rather small, whereas the matrix elements plications to Atomic Nuclei (Wiley,New York,1967).
f(n)(p,p ),whichdonotcontainthisfactor,arenotsup- [4] P.RingandP.Schuck,TheNuclearMany-Body Problem
1
(Springer-Verlag, Berlin, 1980).
pressed. Consequently, the group velocity might change
[5] S. T. Belyaev, Mat. Fys. Medd.Dan. 31, No.11 (1959).
its sign when the interaction correction is taken into ac-
[6] V. A. Khodel, J. W. Clark, H. Li, and M. V. Zverev,
count, giving rise to the previously studied phenomenon
/cond-mat 0609581.
known as fermion condensation, which involves whole- [7] J.C.Slater,J.R.Mann,T.M.Wilson,andJ.H.Wood,
sale mergence of sp levels in homogeneous Fermi fluids Phys. Rev.184, 672 (1969); and references therein.
[9, 10, 16]. In spite of evident commonalities, there is a [8] M.Ya.AmusiaandV.R.Shaginyan,J.Phys.B25,L345
crucial difference between the conditions for the “level- (1992), and references cited therein.
[9] V. A. Khodel and V. R. Shaginyan, JETP Letters 51,
mergence” phenomenon in homogeneous Fermi liquids
553 (1990); Condensed Matter Theories 12, 221 (1997);
and in finite Fermi systems with degenerate sp levels.
[10] V. A. Khodel, V. V. Khodel, and V. R. Shaginyan,
In the former, the presence of a significant velocity- Phys. Rep. 249, 1 (1994); V. A. Khodel, M. V. Zverev,
dependent component in the interaction function f is and V. M. Yakovenko, Phys. Rev. Lett. 95, 236402
needed to promote fermion condensation, while in the (2005).
5
[11] M. Goeppert Mayer, Phys. Rev. 60, 184 (1941). [14] I.M.Band,K.A.Kikoin,M.B.Trzhaskovskaya,andD.
[12] D. G. Griffin and K. L. Andrew, Phys. Rev. 177 62, I. Khomskii, Sov. Phys.JETP 67, 1561 (1988).
(1969). [15] L. D. Landau and E. M. Lifshitz, Quantum Mechan-
[13] V.A.Shaburov,I.M.Band,A.I.Grushko,T.B.Mezent- ics, Non-relativistic theory(Pergamon,NewYork,1965),
seva, E. V. Petrovich, Yu. P. Smirnov, A. E. Sovestnov, p. 290.
O.I.Sumbaev,M.B.Trzhaskovskaya,andI.A.Markova, [16] P. Nozi`eres, J. Phys.I France 2, 443 (1992).
Sov.Phys. JETP 38, 573 (1974).