On the (anisotropic) uniform metallic ground states of fermions interacting through arbitrary two-body potentials in d dimensions

0.5 MB·English

by  Behnam Farid

On the (anisotropic) uniform metallic ground states of fermions interacting through
arbitrary two-body potentials in d dimensions
Behnam Farid
Spinoza Institute, Department of Physics and Astronomy, University of Utrecht,
Leuvenlaan 4, 3584 CE Utrecht, The Netherlands ∗
(Received 16 April2003 and accepted 28 July 2003)
4 We demonstrate that the skeleton of the Fermi surface Sf;σ pertaining to a uniform metallic
0 ground state (corresponding to fermions with spin index σ) is determined by the Hartree-Fock
0 contribution Σhσf(k) to the dynamic self-energy Σσ(k;ε). That is to say, in order for k ∈Sf;σ, it is
2
necessary(butforanisotropicgroundstatesingeneralnotsufficient)thatthefollowing equationbe

n satisfied:
a
J εk+¯hΣhσf(k)=εf,

7
whereεkstandsfortheunderlyingnon-interactingenergydispersionandεffortheexactinteracting
] Fermi energy. The Fermi surface Sf;σ consists of theset of k points which in addition to satisfying
l theabove equation fulfil
e
-
r S (k)=0,
σ
t
s
. where
t
a
m Sσ(k):=π1 Z ∞ dεε Im[Σσ(k;εf+ε)+Σσ(k;εf−ε)].
- 0
d
The set of k points which satisfy the first of the above two equations but fail to satisfy the second
n
o constitutethepseudogapregionoftheputativeFermisurfaceoftheinteractingsystem. Weconsider
c the behaviour of the ground-state momentum-distribution function nσ(k) for k in the vicinity of
[ Sf;σ and show that, whereas for the uniform metallic ground states of theconventional single-band
3 Hubbard Hamiltonian, described in terms of an on-site interaction, nσ(k−f;σ)≥ 12 and nσ(k+f;σ)≤ 21
v (herek−f;σandk+f;σdenotevectorsinfinitesimallyclosetoSf;σ,locatedrespectivelyinsideandoutside
the underlying Fermi sea), for interactions of non-zero range these inequalities can be violated
0
5 (without thereby contravening the stability condition nσ(k−f;σ) ≥ nσ(k+f;σ)). This aspect is borne
3 out by the nσ(k) pertaining to the normal states of for instance liquid 3He (corresponding to a
4 range of applied pressure) as determined by means of quantum Monte Carlo calculations. We
0 furtherdemonstratethatforFermi-liquidmetallicstatesoffermionsinteractingthroughinteraction
3 potentialsofnon-zerorange(e.g. theCoulomb potential),thezero-temperaturelimit ofn (k) does
σ
t/0 nstoattenseeodf tthoebseineqgluea-blatond21Hfourbkba∈rdSfH;σa;mthilitsoniniasntr(iciftscuocnhtrsatsattewsitahttahlleeuxnisifto)r.mTFheisrmasi-plieqcutidshmoueltdalblice
a taken into account while analyzing the n (k) deduced from the angle-resolved photoemission data
m σ
concerning real materials. We discuss, in the light of the findings of the present work, the growing
- experimentalevidencewithregard tothe‘frustration’ ofthekineticenergyofthechargecarriersin
d
thenormal states of thecopper-oxide-based high-temperaturesuperconducting compounds.
n
o
c
:
v
i
X Preprint number: SPIN-2002/37
r
a
arXiv:cond-mat/0304350
Philosophical Magazine, 2004, Vol. 84, No. 2, 109-156
∗Electronic address: [email protected].
1
1. Introduction 2 defining Sf;σ, presented above, in this paper we deduce
the most general expression for n (k) in the close vicin-
σ
2. Preliminaries 4
ity of S and for specific cases deal with this expres-
f;σ
3. EquationsdeterminingtheFermisurfacesofuni- sion in considerable detail. Amongst others, we show
form metallicground states 7 that,incontrastwiththecaseoftheconventionalsingle-
band Hubbard Hamiltonian, n (k) can become smaller
4. Exposing a further close relationship between σ
Σσ(k;εf) and Sf;σ 8 (greater) than 21 for k inside (outside) the Fermi sea
and infinitesimally close to S , without infringing on
f;σ
5. Ground-state exchange and correlation poten- the condition n (k− )−n (k+ )≥0 implied by the as-
tials: a test case 10 σ f;σ σ f;σ
sumption of the stability of the GS of the system under
6. Behaviour of nσ(k) for k infinitesimally close to consideration. This aspect is significant in that in cases
Sf;σ 13 where n (k− ) < 1, by continuity n (k) is expected to
6.1. Thecasewhen0<γ<1, 0<τ <1 . . . . . . . 13 σ f;σ 2 σ
be more suppressed for k inside the Fermi sea (at least
6.1.1. 0<γ<τ <1. . . . . . . . . . . . . . . . . 13
6.1.2. 0<τ <γ<1. . . . . . . . . . . . . . . . . 14 in the vicinity of Sf;σ) in comparison with cases where
6.1.3. 0<τ =γ≤1 (includesFermiliquids) . . 14 n (k− ) ≥ 1, implying thus an increase in the contri-
σ f;σ 2
6.2. Thecasewhenγ=1, τ =1: general . . . . . . . 15 bution of the correlated kinetic energy ε n (k) to
66..22..21.. CCoonnssiiddeerriinnggnnσσ((kk+f−f;;σσ))≡≡ΛΛ+σ−σ//((11++ΛΛ+σ−σ)) .. .. .. 1176 t(hseeeGlaStetro).taIlnentherisgycoonfntehcetiosny,stnemoteutnhdaPetrkc,σonskidnσer(akt)ioins
6.2.2.1. Thecasewhen d+σ <0 . . . . . . . . . . 17 k,σ σ
6.2.2.2. Thecasewhen d+σ >0 . . . . . . . . . . 19 equal to the total number N of particles iPn the GS and
6.3. Thecasewhenγ=1, τ =1: Fermiliquids . . . 21 therefore independent of the magnitude of the coupling
constant of interaction.
7. Summary and concluding remarks 21
For a number of reasons of contemporary interest, a
Acknowledgements 24 reliable description of n (k) seems timely and desirable.
σ
Foremostamongstthese are investigations regardingthe
AppendixA:EstimationoferrorinSσ(k)within
determination of the mechanism of superconductivity in
the framework of a finite-order many-body per-
turbationtheory for Σσ(k;ε) 24 the copper-oxide-based high-temperature superconduct-
ingmaterials(hereafterreferredtoascupratesorcuprate
References 26
compounds);forrecentreviewssee[11–13]. Boththeear-
lier angle-resolvedphotoemission results [14](see in par-
ticular Fig. 3(F) herein) and the recent in-plane optical-
§ 1. INTRODUCTION
conductivity data concerning Bi Sr CaCu O [15,16]
2 2 2 8+δ
(see also [17]) reveal decrease in the in-plane kinetic
Theoretical investigation of anisotropic interacting
energy of the system upon entering into the supercon-
metallic systems is specifically demanding through the
ducting state, by an amount commensurate with the ex-
possibility,evenforisotropicinteractionpotentials,ofthe
pected superconducting condensation energy. 1 A simi-
deformation of the Fermi surface in these systems with
lartrendhasbeenobservedinthemeasuredout-of-plane
respect to the underlying non-interacting Fermi surface
(i.e. the c-axis)opticalconductivities ofTl Ba CuO ,
[1–3]. Forrecentapproachestothis problemwereferthe 2 2 6+x
La Sr CuO , and YBa Cu O [18]. 2 Assuming the
reader to [4,5] and the references herein. 2−x x 4 2 3 6.6
sufficiency of the BCS variational wavefunction to de-
Building on recent work [6] concerning the uniform
metallic ground states (GSs) of the conventional single-
band Hubbard Hamiltonian [7–10] for arbitrary spatial
dimensions d, in this work we consider similar states
1 This aspect has been explicitly quantified in [15–17].
for arbitrary two-body interaction potentials (provided
2 The ‘interlayer tunnelling’ theory by Anderson and co-
only that these possess Fourier transform) correspond-
workers [19,20] (see also [21]) ascribes the superconducting
ing to systems of fermions whose non-interacting energy
condensationenergywhollytothereductioninthekineticen-
dispersion ε can be both bounded (as is the case for
k ergyofthecupratecompoundsinconsequenceofthepossibil-
the single-band Hubbard Hamiltonian) and unbounded ityofcoherentinter-layertunnellingofthepairedelectronsin
from above (as in the case of the free-fermion model thesuperconductingstates,aprocessdeemedinfeasibleinthe
where εk = h¯2kkk2/[2m], in which m stands for the normalstatesof thesematerials (owingtotheincoherentna-
fermion bare mass). Our results, of which a number we ture of the underlying single-particle excitations). Although,
have presented in the abstract, make explicit some of among other things (see [22] and [20]), the measured c-axis
the fundamental differences between systems in which optical conductivity in Tl2Ba2CuO6+x, La2−xSrxCuO4, and
the interaction potential v(r − r′) is of contact type, YBa2Cu3O6.6 [18] may be interpreted within the framework
oftheinter-layertunnellingtheory[18](seehowever[23,24]),
of strictly bounded and of unbounded range. Follow-
some experiments [25,26] appear [27] (see also [28]) not to
ing the general considerations leading to the equations
support this theory; see however [29], [30] (and [31]). The
2
scribe the superconducting states of the cuprate com- relevantrangeof parameters. Of these, one concerns the
pounds (an assumption that has been widely antici- periodic(butincommensuratewithrespecttotheunder-
pated), 3 one readily observes that n (k− ) < 1, with lyinglattice)orderingofspinandcharge(alsoreferredto
σ f;σ 2
k ∈S , in combination with the assumption of con- asspin-chargestripes; forgeneralreviewssee[66,67]and
f;σ f;σ
tinuity of n (k) for k inside the Fermi sea and in the the referencesherein),whichappearsto competeagainst
σ
vicinityofS ,istantamounttoanexcesskineticenergy macroscopic phase separation [68]. This statement re-
f;σ
in the normal GS of the underlying system in compari- quires some qualification. Macroscopic phase separa-
son with the kinetic energy in the paired state (see the tionisbelievednotto concerntheHubbardHamiltonian
previous paragraph). [69,70],butsolelythet-J Hamiltonian. However,evenfor
The above-indicated experimental results on the one the t-J Hamiltonianthe actualsituationis notfirmly es-
hand and the feasibility of n (k− ) < 1 for the metal- tablished; on the one hand, Green-function Monte Carlo
σ f;σ 2
lic states offermions interactingthroughpotentials of at results (concerning finite systems) have been interpreted
least finite range on the other clearly signify the likeli- as implying phase separation at hole dopings and inter-
hood of the existence of a crucial relationship between actionstrengthsappropriatetothecupratesuperconduc-
the latter potentials and the unconventional properties tors[71],while,ontheotherhand,subsequentnumerical
ofthe metallic states ofthe cupratecompounds (see also calculations [70] based on ‘density-matrix renormaliza-
[37,38]). In this connection it is relevant to mention tion group’ (DMRG) method performed on larger sys-
that it has been rigorously shown [39] that for d = 2 tems have upheld the viewpoint that the ‘stripes’ state
the conventionalsingle-band Hubbard Hamiltonian does would have a lower energy than the phase-separated
not support d long-range pairing order at non-zero state. In spite of these opposing observations, it is gen-
x2−y2
temperatures,4 anaspectstronglyanticipatedby earlier erally believed that the sensitivity of the results corre-
Monte Carlo results (which incidentally also do not sup- sponding to the t-J Hamiltonian on such details as, for
port extended s-wave pairing) on finite systems [44–48]. instance, the shape and size of the clusters on which
Further,theviewisincreasinglygainingprominencethat computations are carried out does not warrant an a pri-
the long range of the Coulomb potential, such as taken ori neglect of the long range of the Coulomb interaction
account of by means of a variety of ‘extended’ Hubbard [61,63,64] (see also [72]). Later in this paper we shall
(and associated t-J) Hamiltonians [49–65], plays a cru- discuss and document the significance of a non-contact-
cial role in stabilizing one from amongst a number of type interaction in accurately describing the normal liq-
competing orders in the cuprate compounds over some uid state of 3He.
Using the experimental angle-resolved photoemission
dataitispossibletodeterminen (k)forkinthevicinity
σ
ofS ,ashasbeenreportedin[73]forBi Sr CaCu O
f;σ 2 2 2 8+δ
significance to pairing of the in-plane kinetic-energy frustra- (Bi2212)andBi2Sr2Cu1O6+δ (Bi2201),inboththeover-
tion inthenormalstates ofthecupratecompoundshasbeen dopedandtheunderdopedregime. In[73]theFermisur-
put forward and established by the model considerations of face has been asserted as being the locus of the k points
HirschandMarsiglio[32,60]andHirsch[33,52](seealso[34]). at which, firstly, the (experimental) single-particle spec-
3 According to this assumption, the zero-temperature BCS tralfunction A (k;ε) is peakedfor ε=ε and, secondly,
σ f
occupation fraction vk2 ≡ 1−u2k (see [35], Chapter 3 and in n (k) takes the value 1 (see [74]); 5 in [73] the abso-
particularFig. 3.1) playsthesamerole asn (k) doesinrela- σ 2
σ lute value of n (k) is determined by equating with unity
tion to the underlying normal GS. For completeness, within σ
the maximumvalue achievedbythe experimentaln (k),
the framework of the conventional BCS theory, and at zero σ
∀k, a procedure which evidently adds uncertainty to the
temperature, the n (k) pertaining to the normal GS coin-
σ
cides with the unit-step function, equal to unity for k inside experimentally determined nσ(k). As demonstrated in
the Fermi sea and to zero for k outside. Consequently, at [6], and in view of the findings in the present work, the
zero temperature the BCS state accommodates a higher ki-
neticenergythantheunderlyinguncorrelatednormalGS(for
a quantitative consideration see § 3.4 in [35]). Note in pass-
ing that Chester’s [36] findings do not apply to a pairing, or 5 Explicitly, in [73] the authors introduced three criteria to
reduced, Hamiltonian such as the one employed in the con- be obeyed “at a true FS [Fermi surface] crossing”, of which
ventional BCS theory. thesecondreads: “(2) n(k)[n (k)]shouldbenear50% ofits
σ
4 Thesuperconductingorderparameterinthecupratecom- maximal value, or equivalently the maximal gradient point”.
pounds is generally considered to have the dx2−y2 symme- Elsewheretheseworkersstated: “aFS[Fermisurface]crossing
try [11–13]. Note in passing that in practice the inter-planar shouldoccurwhenn(k)losesabouthalfofitsmaximumvalue
coupling of the electronic degrees of freedom is capable of (excludingthebackground)”. Asweindicateinthemaintext,
stabilizing a d-wavesuperconductingstate at finitetempera- theseworkersfixedthescaleoftheexperimentallydetermined
tures,suchasobtained within theframework oftheso-called n (k) by identifying with unity the maximum value of the
σ
‘clusterdynamicalmean-field’approximation[40,41](seealso latter function overall thek points probed experimentally.
[42,43]).
3
requirementn (k)= 1 fork∈S isnotjustifiedingen- tentialµ andcorrelationpotentialµ ,asencountered
σ 2 f;σ x;σ c;σ
eral; 6 firstly, within the framework of the conventional withintheframeworkoftheGSdensity-functionaltheory
single-band Hubbard model, the latter is solely a neces- [75–77]. Our expression for µ identically reproduces
x;σ
sary (and not sufficient) condition for metallic states to the exactµ correspondingto the Coulomb-interacting
x;σ
be Fermi-liquid(FL) and,secondly, formetallic statesof uniform-electron-gas system. Making use of nrpa(k), the
σ
fermionsinteractingthroughpotentialsofnon-zerorange random-phase approximation(RPA) to n (k), which for
σ
the condition n (k) = 1 for k ∈ S is not demanded systems of Coulomb-interacting fermions asymptotically
σ 2 f;σ
even for FL metallic states. In the light of these and approaches the exact n (k) in the high-density regime,
σ
neglecting for the moment the above-mentioned uncer- we calculate µ which in the latter regime up to an
c;σ
tainty concerning the absolute values of the experimen- unimportant additive constant reproduces the µ de-
c;σ
tally determined n (k), the results depicted in Fig. 2 of ducedfromquantumMonte-Carlocalculations. In§6we
σ
[73] have far-reaching implications. For instance, these consider the behaviour of n (k) for k in the vicinity of
σ
results clearly show that a contact-type interaction can- S ,bothinside andoutsidetheFermisea. Ratherthan
f;σ
not be capable of reliably describing Bi2212 and Bi2201 being exhaustive, in this Section we attempt to contrast
in the over-doped and the underdoped regimes (at no theimportantconsequencesthattherangeofinteraction
peak position of A (k;ε ) does n (k) acquire a value can have on the behaviour of n (k) in the mentioned re-
σ f σ σ
reasonably close to 1); a number of these results further gion of the k space. In § 7 we present a summary of
2
showthatthe pertinentsingle-particlespectralfunctions the main aspects of this paper. Here we also briefly re-
A (k;ε), corresponding to the systems considered, are late our theoretical findings with some available results
σ
most probablyfree fromquasi-particleδ-function contri- concerningcorrelatedfermionsystems,specificallyliquid
butions (establishing the breakdown of the FL state) at 3He in the normal state. Finally, in appendix A which
regionsofS neighbouringtheM pointsoftheunderly- follows the main body of this paper we present an anal-
f;σ
ing Brillouin zones (compare for instance Figs. 2(g) and ysis which exposes the nature of inaccuracy in S (k) as
σ
2(i) in [73] with the results in Eq. (109) below). calculated on the basis of a finite-order perturbation se-
The organization of this paper is as follows. In § 2 riesfor the self-energyoperator. Forreasonsthat willbe
we introduce the Hamiltonian on which the considera- clarified in appendix A, the defining expression for S
f;σ
tions in this paper are based; this Hamiltonian, which as introduced in this paper (i.e. Eq. (53) below) more
is expressed in terms of a single spin-degenerate energy reliably reproduces S than the conventional defining
f;σ
dispersion ε , is defined on the continuum; however, expression (i.e. Eq. (45)).
k
through some minor adjustments, which we explicitly
specify,thisreducestotheconventionalsingle-bandHub-
bardHamiltonian,definedonalattice. In§3wepresent § 2. PRELIMINARIES
thedetailsunderlyingtheequationsfromwhichS per-
f;σ
tainingtotheuniformGSsoftheHamiltonianintroduced In a recent work [6] we have in some detail consid-
in § 2 is determined. In § 4 we expose an intimate rela- ered the uniform metallic GSs of the single-band Hub-
tionship between S and Σhf(k) mediated through the bard Hamiltonian. Here we extend our investigations by
f;σ σ
single-particledensitymatrix̺ (r,r′)whichforuniform consideringuniformmetallicGSsofthefollowingHamil-
σ
GSs, where the latter function depends on r−r′, coin- tonian
cides with the Fourier transform of the GS momentum
distribution function nσ(k); the information with regard H =H0+gH1, (1)
toS ,throughthesingularnatureofn (k)fork∈S ,
f;σ σ f;σ where b b b
is shownto be already fully reflectedin the leading term
intheasymptoticseriesof̺σ(r,r′)forkr−r′k→∞. In H = ε cˆ† cˆ , (2)
§ 5 we examine the equation for S in terms of Σhf(k) 0 k k;σ k;σ
f;σ σ Xk,σ
by applying this equation to an isotropic metallic GS b
forwhichS is rotationallyinvariantandis fully deter-
f;σ
1
minedbytheconcentrationofthefermionsinthesystem. H = w¯(kqk)cˆ† cˆ† cˆ cˆ . (3)
By doing so we deduce expressions for the exchange po- 1 2ΩXσ,σ′ kX,p,q k+q;σ p−q;σ′ p;σ′ k;σ
b
In Eq.(1), g (which has the dimension of energy)stands
for the coupling-constant of interaction, ε for a spin-
k
6 For cases in which nσ(k) is discontinuous at k ∈ Sf;σ, in degenerate single-particle energy dispersion (which may
the present context the value assigned to nσ(k) for k ∈ Sf;σ be equated with h¯2kkk2/[2m], the energy dispersion of
is the arithmetic mean of nσ(k′) for k′ approaching k from non-interactingfreefermionswithmassm),cˆ†k;σ andcˆk;σ
inside and outside the Fermi sea. This mean value coincides for the canonical creation and annihilation operators re-
with thezero-temperature limit of nσ(k) for k∈Sf;σ [6]. spectively for fermions with spin index σ, gw¯(kqk) ≡
4
v¯(kqk) for the Fourier transform of the two-body in- where |Ψ i stands for the N-particle GS of H,
N;0
teraction potential v(r −r′) (assumed to be isotropic),
and Ω = Ld for the volume of the (macroscopic) d- nσ(k):=hΨN;0|cˆ†k;σcˆk;σ|ΨN;0i b (10)
dimensional hyper-cubic box occupied by the system.
The wave-vectorsums in Eqs.(2)and (3)are overa reg- for the GS momentum distribution function, and
ular lattice (the underlying lattice constant being equal
ϑ :=−hΨ | cˆ† ,[H ,cˆ ] |Ψ i, (11)
to2π/L)coveringinprincipletheentirereciprocalspace. k;σ N;0 k;σ 1 k;σ − + N;0
(cid:2) (cid:3)
TheHamiltonianinEq.(1)canbefruitfullyemployedto b
in which [, ] and [, ] denote commutation and anti-
investigate the consequences of interactionand interplay − +
commutation respectively. Making use of the canonical
between this and anisotropy (as implied by ε ) on the
k anti-commutationrelationsfor cˆ† andcˆ , one readily
observable quantities in many-particle systems. k;σ k;σ
obtains
On effecting the replacements
N 1
g⇀U, w¯(kqk)⇀ Ω , (4) ϑk;σ = Ω w¯(0)− Ω w¯(kpk)nσ(k+p)
N Xp
l
(for bounded w¯(0)). (12)
and restricting the wave-vector sums to N wavevec-
l
tors uniformly distributed over the first Brillouin zone
Note that N/Ω=:n is the total concentration of the
0
(1BZ) associated with the Bravais lattice spanned by
fermions in the N-particle GS of H. For the Hubbard
{R kj = 1,...,N }, the Hamiltonian in Eq. (1) goes
j l Hamiltonian, making use of the prescription in Eq. (4)
over into the conventional single-band Hubbard Hamil- b
and the fact that n (k+p)=N (recall that in the
tonian [7–10] H corresponding to the on-site interaction p σ σ
case of the HubbaPrd or an extended Hubbard Hamilto-
energy U; in cases where k+q and p−q on the right-
b nianthelattersumisrestrictedtopointsinsidethe1BZ),
hand side (RHS) of Eq. (3) lie outside the 1BZ, these
one obtains [6]
vectors are to be identified with the vectors inside the
1BZ obtained from k+q and p−q through Umklapp ϑ =n−n ≡n (Hubbard Hamiltonian), (13)
k;σ σ σ¯
processes. OnrelaxingthereplacementsinEq.(4),while
maintainingtheabove-mentionedrestrictionsconcerning where
the wave vectors, the Hamiltonian in Eq. (1) coincides
N
with an ‘extended’ Hubbard Hamiltonian. n := σ (14)
σ
N
Along the lines consideredin [6], it can be shown that l
for the Fermi surface Sf;σ, pertaining to the uniform stands for the number of fermions with spin index σ per
metallic GS of H, one has siteintheuniformN-particleGSoftheHubbardHamil-
tonian; thus n:=N/N . In Eq. (13), σ¯ denotes the spin
S ={kbk(ε −ε< )(ε+−ε> )=0}, (5) l
f;σ f k;σ f k;σ index complementary to σ, that is for σ =↑, σ¯ =↓ and
vice versa.
where ε+ =ε +0+ and
f f In cases where w¯(kqk) is unbounded for kqk = 0
> (such as is the case for the two-body Coulomb poten-
ε>< =ε +g βk<;σ . (6) tial), the expression in Eq. (12) has to be regularized;
k;σ k >
ν<(k) thisisachievedbyformallytakingintoaccounttheinter-
σ
action of the fermions with a positively chargeduniform
On the basis of similar arguments as presented in [6], it background through supplementing the Hamiltonian in
can be shown that (εf −ε<k;σ) = 0 implies that (ε+f − Eq. (1) with gH′, where
ε> ) = 0, and vice versa so that, for k ∈ S , ε<
akn;dσ ε>k;σ are up to an infinitesimal difference eqf;uσal. kI;nσ Hb′:=−n0w¯(κ)N, κ↓0, (15)
Eq. (6),
b b
in which N stands for the total number operator. Fol-
νσ<>(k):=(cid:26)1nσ−(kn)σ,(k), (7) lowing thisb, one obtains
′
1
ϑ =− w¯(kpk)n (k+p),
k;σ σ
β< = −1 w¯(kq′k) ΩXp
k;σ Ω Xσ′ pX′,q′ (for unbounded w¯(0)), (16)
×hΨN;0|cˆ†k;σcˆ†p′+q′;σ′cˆk+q′;σcˆp′;σ′|ΨN;0i, (8) where ′p leaves out p=0. For systems in the thermo-
dynamPic limit, sums of the form Ω−1 (or Ω−1 ′)
q q
β> =ϑ −β< , (9) can be replaced by (2π)−d ddq. In aPny of the abPove
k;σ k;σ k;σ
R
5
cases, whether one deals with the Hubbard Hamiltonian η ∼0 for k→S . (26)
k;σ f;σ
orwithamoregeneralHamiltonian,fromtheexpressions
in Eqs. (12) and (16) it is easily verified that Defining ζk;σ as follows
¯h 1
ϑk;σ ≡ g Σhσf(k), (17) ζk;σ:=ξk;σ− g (εf−εkf;σ), (27)
where Σhf(k) stands for the Hartree-Fock self-energy. on account of Eq. (22) we similarly have
σ
From Eqs. (6), (7), (9) and (17), one readily deduces
ζ ∼0 for k→S . (28)
that (cf. Eq. (56) in [6]) k;σ f;σ
Onthebasisoftheaboveexpressions,fromEqs.(20)and
n (k)ε< + 1−n (k) ε> =ε +h¯Σhf(k). (18)
σ k;σ σ k;σ k σ (21) for k→k ∈S we deduce
f;σ f;σ
(cid:0) (cid:1)
This result will prove useful in our later considerations
ε< ∼µ+a(k )·(k−k )+gζ , (29)
in this paper. k;σ f;σ f;σ k;σ
By introducing the decomposition
ε> ∼µ+a(k )·(k−k )−gΛ (k)(ζ −η ),
β< ≡n (k)ξ , (19) k;σ f;σ f;σ σ k;σ k;σ
k;σ σ k;σ
(30)
which implies no restriction so long as n (k) 6= 0, one
σ
canexpressε< , introducedin Eq.(6) above,asfollows: where
k;σ
ε<k;σ =εk+gξk;σ. (20) a(kf;σ):=∇kεk|k=kf;σ, (31)
From this, Eq. (6) and the results in Eqs. (18) and (17),
n (k)
one similarly has Λ (k):= σ . (32)
σ 1−n (k)
σ
ϑ −n (k)ξ
ε>k;σ =εk+g k;σ1−nσ (k) k;σ. (21) In Eqs. (29) and (30), µ stands for the ‘chemical poten-
σ
tial’ which for metallic states is infinitesimally greater
From the expression in Eq. (5) above and the ensuing than the Fermi energy εf (for details see [6,78]).
details(seethetextsubsequenttoEq.(6))oneinfersthe Making use of the equationof motion for the operator
following relationships from the expressions in Eqs. (20) cˆk;σ in the Heisenberg picture, that is
and (21) (for k ∈S )
f;σ f;σ
∂
i¯h cˆ (t)= cˆ (t),H , (33)
1 ∂t k;σ k;σ −
ξ ∼ (ε −ε ), k→k , (22) (cid:2) (cid:3)
k;σ g f kf;σ f;σ b
from the defining expression in Eq. (8), one obtains
1
ϑ ∼ (ε −ε ), k→k . (23)
k;σ g f kf;σ f;σ 1 ∂
β< = ¯h G (k;t−t′) −ε n (k) , (34)
k;σ g ∂t σ (cid:12) k σ
In other words, to the leading order in (k−k ), one n (cid:12)t′↓t o
f;σ (cid:12)
has where the single-particle Green(cid:12)function in the time do-
main is defined as follows:
ξ ∼ϑ for k→S . (24)
k;σ k;σ f;σ
G (k;t−t′):=−ihΨ |T cˆ (t)cˆ† (t′) |Ψ i, (35)
Note that here, as well as in Eqs. (22) and (23), S σ N;0 k;σ k;σ N;0
f;σ (cid:8) (cid:9)
is considered as given, so that the expressions in these inwhichT standsforthefermiontime-orderingoperator.
equationsreflectsomepropertiesofSf;σ ratherthannec- We have (below η ↓0)
essarily fully defining it.
In analogy with the expression in Eq. (19) above, we ∂ 1 ∞ dε
¯h G (k;t−t′) = eiεη/h¯εG (k;ε)
introduce the auxiliary function ηk;σ, satisfying ∂t σ (cid:12)(cid:12)t′↓t ¯hZ−∞ 2πi σ
nσ(k)ηk;σ:=ϑk;σ− g1(εf−εkf;σ). (25) (cid:12)(cid:12) ≡ ¯h1 Z−µ∞dεεAσ(k;ε), (36)
This expression is general for all k for which nσ(k) 6= 0. where Aσ(k;ε) stands for the single-particle spectral
In the light of Eq. (23), and since, for interacting GSs, function, defined in terms of Gσ(k;z), the analytic con-
n (k)6=0inaneighbourhoodofS (seelater),wemust tinuation of G (k;ε) = lim G (k;ε ± iη), ε>µ, as
σ f;σ σ η↓e0 σ <
have follows:
e
6
1
A (k;ε):= G (k;ε−iη)−G (k;ε+iη) , η ↓0. Sf;σ:={kkεk+h¯Σσ(k;εf)=εf}, (45)
σ σ σ
2πi
(cid:8) (cid:9)
e e (37) where, on general grounds, Im[Σσ(k;εf)] ≡ 0, ∀k
[81,6,78]. From Eqs. (17) and (23) we deduce
With (see Eq. (10) above)
ε +h¯Σhf(k)=ε for k∈S . (46)
k σ f f;σ
1 µ
n (k)= dεA (k;ε), (38)
σ ¯hZ σ Comparison of this expression with the defining expres-
−∞
sionfor S in Eq.(45)revealsthat to the leading order
f;σ
from Eqs. (19), (34), (36) and (38) we deduce that in (k−k ) we must have
f;σ
ξ ≡ βk<;σ = 1 −µ∞dεεAσ(k;ε) −ε , (39) Σσ(k;εf)∼Σhσf(k) as k→kf;σ ∈Sf;σ. (47)
k;σ νσ<(k) gnR −µ∞dεAσ(k;ε) ko The significance of this expression becomes more ap-
R
parent by considering the Kramers-Kr¨onig relation for
which in combination with Eq. (22) leads to
Re[Σ (k;ε)] in terms of Im[Σ (k;ε)]; since [82,78]
σ σ
µ dεεA (k;ε)
R−µ∞ dεA σ(k;ε) ∼εf for k→kf;σ ∈Sf;σ. (40) Σσ(k;ε)−Σhσf(k)=o(1) for |ε|→∞, (48)
−∞ σ
R the above-mentioned Kramers-Kr¨onigrelation reads
With
Re[Σ (k;ε)]=Σhf(k)
Z :=n (k− )−n (k+ ), (41) σ σ
kf;σ σ f;σ σ f;σ ∞ dε′ sgn(µ−ε′)Im[Σ (k;ε′)]
σ
−℘ . (49)
from Eqs. (38) and (40) we obtain Z π ε′−ε
−∞
1 µ From the leading term in the asymptotic series of the
Z = dεε A (k− ;ε)−A (k+ ;ε) , (42)
kf;σ ¯hεf Z−∞ (cid:2) σ f;σ σ f;σ (cid:3) integral on the RHS of Eq. (49) for ε → εf, making use
of the fact that Im[Σ (k;ε)] ∼ 0 as ε → ε , we obtain
σ f
which is alternative to the standard expression [82]
Zkf;σ = ¯h1 Z µ dε Aσ(k−f;σ;ε)−Aσ(k+f;σ;ε) , (43) Σσ(k;εf)=Σhσf(k)+Sσ(k), (50)
−∞ (cid:2) (cid:3)
where
directly obtained from the spectral representation of
n (k) in Eq. (38) above. The expression in Eq. (42) 1 ∞ dε
σ S (k):= Im[Σ (k;ε +ε)+Σ (k;ε −ε)].
σ σ f σ f
hasbeenexplicitlyderivedin[6]fortheuniformmetallic π Z ε
0
GSsofthesingle-bandHubbardHamiltonian. Hereasin
(51)
>
[6]thesingle-particleenergiesε< (associatedwithsome
k;σ
fictitious particles [6]) correspond to the single-particle Comparison of Eq. (47) with Eq. (50) reveals the neces-
spectral function A (k;ε), for which we have [6] sity of the condition
σ
A (k;ε)=h¯ ν<(k)δ(ε−ε< )+ν>(k)δ(ε−ε> ) . S (k)=0 (52)
σ σ k;σ σ k;σ σ
(cid:8) (cid:9)
(44) in order for k ∈ S . In the light of the results in
f;σ
Eqs. (47) and (50), we thus arrive at the following alter-
ItisreadilyverifiedthattheexactZ isobtainedbyre-
kf;σ native defining expression for S (cf. Eq. (45) above):
placingA (k;ε)inEqs.(42)and(43)byA (k;ε). Inthis f;σ
σ σ
>
connection note that, for stable (uniform) GSs, ε< >µ, S ={kkε +h¯Σhf(k)=ε ∧ S (k)=0}. (53)
k;σ< f;σ k σ f σ
∀k [6]. The GS total energy of the interacting system is
similarlyexactlyreproducedbyreplacingAσ(k;ε)inthe ThenecessityoftheadditionalconditionSσ(k)=0inthe
pertinent expression (see Eq. (51) in [6]) by Aσ(k;ε). defining expression for Sf;σ stems from the fact that the
equation in Eq. (46) applies for all k on a pre-supposed
S sothat,ingeneralEq.(46)cannotbesufficientfully
f;σ
§ 3. EQUATIONS DETERMINING THE FERMI todefine Sf;σ. The significanceofthisaspectcanbeeas-
SURFACES OF UNIFORM METALLIC GROUND iest appreciated by specializing Eq. (46) to the uniform
STATES metallic GSs of the single-band Hubbard Hamiltonian,
in which case Σhf(k) is independent of k and equal to
σ
TheFermisurfaceS isconventionallydefinedasfol- 1Un (see Eqs. (12), (17) and (4) above). The suffi-
f;σ h¯ σ¯
lows [79,80,6] ciency of ε +h¯Σhf(k) = ε to define S would in the
k σ f f;σ
7
presentcaseincorrectlyimplythatallsolutionsofthelat- since S (k)=0 applies for all k ∈S , in principle one
σ f;σ
ter equation, which constitute the entire S(0) (the Fermi has (see later)
f;σ
surface corresponding to ε and the partial number N
k σ ∇ S (k′) =λ (k∓)nˆ(k) for k∈S , (55)
of fermions with spin index σ in the exact interacting k′ σ k′=k∓ σ f;σ
GS),wereingeneralpointslocatedonS ;the uncondi- (cid:12)
f;σ where nˆ(k) s(cid:12)tands for the (outward) unit vector normal
tional equality Sf;σ = Sf(;0σ), rather than the relationship to S at k ∈ S ; here k− and k+ denote vectors in-
f;σ f;σ
Sf;σ ⊆ Sf(;0σ) deduced in [6], would rule out the feasibil- finitesimally close to k, with the former inside and the
ity that the pseudogap phenomenon is described in the latteroutsidetheFermisea,andλ (k)standsforascalar
σ
spacespannedby the uniformmetallic GSs ofthe single- function which cannot be identically vanishing. For con-
bandHubbardHamiltonian[6], contradictingthe results ciseness, in this work we shall denote the Fermi sea cor-
in [83–87]. Note that for isotropic GSs the condition respondingto fermions with spinindex σ by FS andits
σ
Sσ(k) = 0 is automatically satisfied at any k for which complementary part with respect to the entire available
Eq.(46)issatisfied,onaccountofthefactthatthemetal- reciprocal space, by FS [6].
σ
lic nature of a GS implies that Sσ(±k) = 0 for at least In writing the expression in Eq. (55) we have taken
one k (and all points related to this by symmetry) on account of the fact that S (k) may not be continuously
σ
Sf;σ, which by isotropy must remain valid for all k on differentiable in a neighbourhood of some or all points
Sf;σ. k ∈ Sf;σ (whence k± in λσ(k±)); however, in Eq. (55)
We note that along the lines of reasoning as in [6] it we have not taken account of the possibility that S (k)
σ
can be shown that nσ(k) is continuous (see Eq. (67) in maynotbedifferentiableinaneighbourhoodofk∈Sf;σ.
[6]) at all k which satisfy Eq. (46) but fail to satisfy The latter possibility in fact arises when, for instance,
Eq. (52). We further note that, since for k ∈ Sf;σ up fermions in d spatial dimensions interact through the
to infinitesimal corrections [6] we have ε<k;σ = ε>k;σ = εf, long-range Coulomb potential, in which case ∇kΣhσf(k)
Eq. (18) is seen to conform with the result in Eq. (46) is logarithmically divergent for k approaching any point
above. k on S (both from inside and from outside the
f;σ f;σ
It is interesting to rewrite the condition Sσ(k)=0 for Fermi sea). It follows that, for metallic states of sys-
k∈Sf;σ in the following appealing form: tems of Coulomb-interacting fermions for which quasi-
particles are well-defined in the neighbourhood of S ,
εfdε Im[Σσ(k;ε)] = ∞dε Im[Σσ(k;ε)], k∈Sf;σ. 8 the above-mentionedsingularity of ∇kΣhσf(k) mustf;bσe
Z−∞ εf−ε Zεf εf−ε fully cancelled by a singular contribution arising from
(54) ∇kSσ(k)(seeEq.(50)above),sothat,forthesemetallic
states, Eq. (55) cannot be meaningful.
The integrands on both sides of this expression are non-
negative over the pertinent ranges of integration. In the
light of this and of the inherent symmetry of these inte- § 4. EXPOSING A FURTHER CLOSE
grands for both |ε−ε | → 0 and |ε−ε | → ∞ (to the RELATIONSHIP BETWEEN Σσ(k;εF) AND SF;σ
f f
leadingordersin(ε−ε )and1/(ε−ε )respectively[78]),
f f
Eq. (54) can be viewed as implying some degree of simi- Thefactthat,fork→kf;σ ∈Sf;σ,Σσ(k;εf)∼Σhσf(k)
larity between |Im[Σσ(k;εf−ε)]| and |Im[Σσ(k;εf+ε)]| to leading order in (k−kf;σ) (see Eq. (47) above), de-
at intermediate values of ε (as measured with respect to serves some closer inspection. From Eqs. (12), (16) and
εf) when k ∈ Sf;σ; for k 6∈ Sf;σ, this similarity should (17) it is evident that Σhσf(k) is fully determined by
be, if not absent, less pronounced. 7 We point out that nσ(k). ForuniformGSsthisfunctionistheFouriertrans-
form of the single-particle density matrix ̺ (r−r′), or
σ
ddk
̺ (r)= eik·rn (k). (56)
7 We believe that the apparent violation of the result in σ Z (2π)d σ
Eq.(54)bythefirst-orderself-energyintermsofthescreened
Thesingularnatureofn (k)fork∈S hasfar-reaching
interaction function (as evident from the middle figures in σ f;σ
consequences for the behaviour of ̺ (r) for krk → ∞
Figs. 16 and 17 in [88], corresponding to k = 1.0kf) signifies σ
theshortcomingsofthisapproximateself-energy,asexplicitly [90,6] and consequently for that of Σhσf(k) for k in a
discussed in [78] (for instance, compare the first-order result
in Eq. (278) with the exact result in Eq. (209) and in the
light of the contrast between the two results, consider the
expressionconcerningIm[Σσ(k;ε)]inEq.(234a); inthiscon- 8 This implying the differentiability of Σσ(k;εf) and con-
nection,recallthatintheparamagneticstate,dealtwithhere, tinuity of ∇kΣσ(k;εf) in the infinitesimal neighbourhood of
n0;σ¯ −n0;σ as encountered in the latter equation is equal to Sf;σ [6,82].
zero).
8
neighbourhood of S . To illustrate this aspect, it is thatforthesesystemsourassumptionwithregardtothe
f;σ
convenient to introduce uniformity of the underlying GSs implies that we should
r only consider r ∈ {Rjkj = 1,...,Nl}) and for those
eˆr:=r where r:=krk. (57) defined on the continuum kz = ∞ so that nσ(kzeˆr) = 0;
for the former systems and for a given eˆ , n (k eˆ ) may
r σ z r
Withoutlossofgenerality,weexplicitlyconsiderthecase or may not be vanishing. In what follows, for simplic-
whend=2andassumethattheangleϕinthecylindrical ity we set n (k eˆ ) equal to zero. One observes that the
σ z r
coordinates (k,ϕ) of k is measured with respect to the full information concerning the shape and dimensions of
direction specified by eˆr. We thus re-write Eq. (56) as Sf;σ isalreadycontainedintheleading-orderterminthe
follows: large-krk asymptotic series of ̺ (r).
σ
1 ∞ For the Fock part Σfσ of the self-energy in the coordi-
̺ (r)= dk kI(k,r), (58) nate representation and for arbitrary GSs and two-body
σ (2π)2 Z
0 interaction functions v(r−r′) we have [78]
where
1
Σf(r,r′)=− v(r−r′)̺ (r′,r), (64)
2π σ ¯h σ
I (k,r):= dϕeikrcos(ϕ)n˜ (k,ϕ), (59)
σ Z0 σ where in the case of uniform GSs ̺σ(r′,r) is to be re-
placed by ̺ (r′ − r). It follows that the information
σ
in which ˜nσ(k,ϕ) ≡ nσ(k). Making use the stationary- concerning Sf;σ contained in ̺σ(r,r′) is similarly con-
phase method [89], we readily obtain tained in 9 Σf(r,r′). In particular, for any two-body
σ
potential other than those for which v(r − r′) ≡ 0
Iσ(k,r)∼2r2kπr nσ(keˆr)cos(kr−π/4), r →∞, (60) wkrhe−nrk′rk −→r∞′k ≥dirRect>ly0d,ettehrembinehesavtihoautrooffΣ̺σf((rr,,rr′′)) ffoorr
σ
kr−r′k→∞. For instance, for the uniform GSs of sys-
where we have made use of the fact that by the time-
temsofparticlesintwo-dimensionalspace(i.e. ford=2)
reversalsymmetry of the GS,
interactingthroughtheCoulombpotentialandforwhich
n˜σ(k,π)=n˜σ(k,0)≡nσ(keˆr). (61) nσ(keˆr) is discontinuous at k =kf;σ, from Eqs. (63) and
(64) we have (following our above convention, for uni-
Thus, from Eqs. (58) and (60) we have formGSs we use the notation Σf(r−r′) for what in the
σ
general case is denoted by Σf(r,r′))
2 ∞ σ
̺ (reˆ )∼ dk k1/2n (keˆ )
σ r (2π)3/2r1/2 Z0 σ r Σf(reˆ )∼−e2Zkf;σeˆrkf1;/σ2 sin(kf;σr−π/4), r →∞,
×cos(kr−π/4), r →∞. (62) σ r h (2π)3/2ǫ r5/2
0
(65)
Let k point in the direction of r, i.e. k = k eˆ .
f;σ f;σ f;σ r
The singular nature of nσ(keˆr) at k = kf;σ implies that where e2 stands for the particle-charge squared and ǫ0
for the further simplification of the expression on the for the vacuum permittivity (note that for −e the elec-
RHS of Eq. (62) it is necessary [78] to expresses the k tron charge, e2/h = 3.874...×10−5 S is the quantized
integral over [0,∞) in terms of at least two k integrals Hall conductance). 10 The fact that the oscillatory be-
overthedisjointintervals[0,k )and(k ,∞);thissub-
f;σ f;σ
division should be carried out for all k points at which
n (keˆ ) is singular; however, to avoid any complication
σ r
unnecessary to our present considerations, here we deal 9 This observation corroborates the validity of the alterna-
with the case where the most dominant singularity of tivedefinition for Sf;σ in Eq.(53) above.
nσ(keˆr) is located at k = kf;σ. Assuming nσ(keˆr) to 10 We note that the direct evaluation of the expression in
be discontinuous at k = kf;σ and continuous elsewhere, Eq.(65)fromtheFourierintegralrepresentationofΣfσ(r−r′)
through integration by parts, from Eq. (62) we obtain isnotstraightforward. ThisispartlybecauseΣfσ(keˆr)isnon-
[90,78] analytically singular at k = kf;σ (recall that in determining
the expression in Eq. (63) we have explicitly assumed that
̺σ(reˆr)∼ Z2k1f;/σ2eˆπr3k/f12;/σ2 sin(kf;rσ3r/2−π/4) taahtsyekml=epatdkofitn;iσcg)essixonptgahunalsati,roifntoyrotofhfΣendfσ(e(rtkeeˆ−rrm)rinc′)oa,ntoisoinsnetsohfoatfshaetoldairesgmceop-nkltoriyn−utriht′yke
σ
nσ(kzeˆr)kz1/2 sin(kzr−π/4) Laplace method [89] which requires analytic continuation of
+ 21/2π3/2 r3/2 , r →∞, (63) Σfσ(keˆr) into the complex k plane. In this connection we
pointoutthattheappearanceofsin(kf;σr−π/4)ontheRHS
whereforsystemsdefinedonlatticekzeˆr denotesthevec- ofEq.(65),ratherthancos(kf;σr−π/4),shouldbeareminder
tor from the originto the Brillouin zone boundary (note of the non-triviality of the result in Eq. (65).
9
haviour of the leading term in the asymptotic series of § 5. GROUND-STATE EXCHANGE AND
Σf(reˆ ) for r → ∞ is entirely determined by k in- CORRELATION POTENTIALS: A TEST CASE
σ r f;σ
dicates that, similar to n (k), Σhf(k) must be singular
σ σ
for all k on Sf;σ; 11 in the present case, where Σfσ(reˆr) A significant aspect of the result in Eq. (46) is made
decays with one power of 1/r faster than ̺σ(reˆr) for explicitasfollows. Forconcreteness,hereweconsiderthe
r →∞,theassumeddiscontinuityofnσ(k)atk=kf;σeˆr uniform GSs of systems defined on the continuum; these
implies singularity (explicitly, a logarithmic divergence uniformGSs arenecessarilyisotropic. Since latticemod-
12) in the behaviour of ∇kΣfσ(k) at k = kf;σeˆr. These els are excluded from our considerations in this Section
facts expose a fundamental distinction between systems (in a future work we explicitly deal with the more deli-
of fermions interacting through potentials of finite range cate case of lattice models), here we relax the definition
andpotentialsofinfiniterange. Notably,whereasforthe for n introduced in Eq. (14) above and introduce
σ
uniformmetallicGSsofthesingle-bandHubbardHamil-
tonian one has nσ(k)≥ 12 for k inside the Fermi sea and nσ:=Nσ. (66)
infinitesimally close to S [6], as we show later in this Ω
f;σ
paper(§6),thisisnotnecessarilythecasewhentheinter-
Withn:=n +n ,thetotalconcentrationofthefermions
σ σ¯
action potential is of longer range than the intra-atomic
intheN-particleuniformGSofthesystem,wedefinethe
potential encountered in the Hubbard Hamiltonian. In
spin magnetization fraction m as follows
this connection, recall that for the uniform metallic GSs
of the conventional single-band Hubbard Hamiltonian, n −n
σ σ¯
m:= . (67)
Σhσf(k) = h¯1Unσ¯, which is independent of k and thus n
regular for all k∈1BZ.
Denoting by E(N,M) the GS total energyof the system
Itisimportanttopointoutthatn (k)issingular(not
σ
necessarily discontinuous) at all k satisfying Eq. (46) describedbytheH inEq.(1),correspondingtothetotal
above. 13 However, since according to Eq. (53) the sat- number of particles N ≡Ωn and the total spin magneti-
b
isfaction of Eq. (46) at a given k is not sufficient for zation M ≡Ω(nσ−nσ¯), we define
k∈S ,itfollowsthatthekthatforagivendirectioneˆ
f;σ r 1
determinestheleadingterminthelarge-rasymptoticse- E(n,m):= E(N,M), (68)
riesof̺ (reˆ )(andofΣf(reˆ ))isnotnecessarilyapoint N
σ r σ r
of Sf;σ. Since such a k point satisfies Eq. (46) but fails which we assume to be a sufficiently smooth function of
to satisfy Eq. (52), it follows that at this point nσ(k) is n and m in the thermodynamic limit [91] (appendix B
continuous (see the paragraphpreceding that containing herein). In this limit, for a metallic GS we have
Eq. (54) above) whereby the leading term in the large-r
asymptotic series of ̺ (reˆ ) (and of Σf(reˆ )) acquires ∂
σ r σ r εf = nE(n,m) . (69)
an anomalousexponent, leading to a stronger power-law ∂n
(cid:0) (cid:1)
decay in the mentioned asymptotic term in comparison
From this we obtain
with the case corresponding to Zkeˆr 6= 0 for keˆr ∈ Sf;σ;
with reference to the expressions in Eqs. (63) and (65) d ∂E(n,m) σ,
nE(n,m) =ε −(m∓1) , α=
above (where k = kf;σ), the above-indicated anomalous dnα(cid:0) (cid:1) f ∂m nσ¯.
exponent would read 3/2+γ and 5/2+γ respectively,
(70)
with γ >0.
We now decompose E(n,m) as follows:
E(n,m)≡E (n,m)+E (n,m), (71)
k xc
11 This statement can also be explicitly verified from the
expression forΣhσf(k)≡(g/¯h)ϑk;σ (seeEq.(17)),makinguse where Ek(n,m), the ‘kinetic energy’ contribution to
of Eq. (16). E(n,m), corresponds to the GS of H in Eq. (80) be-
0
12 For some pertinent details we refer the reader to an ex-
low whose associated n and m are constrained to be the
tensive analysis of the large-r asymptotic behaviour of ̺ (r) b
σ same as those pertaining to the exact GS of H. Thus,
pertainingtouniformandisotropicGSsinappendixJof[78].
Eq. (71) defines the exchange-correlation energy contri-
From this analysis and the bounded result in Eq. (65) above b
bution E (n,m). By the assumption of the isotropy of
one can readily infer the nature of the singularity (i.e. loga- xc
rithmic divergence) of ∇kΣfσ(k) at k∈Sf;σ. theGSsunderconsiderationandinconsequenceofaLut-
13 The analysis underlying this statement is analogous to tingertheorem[92,80],accordingtowhichthenumberof
that presented in [6] for thecase of the single-band Hubbard k points enclosed by Sf;σ and Sf(;0σ) are equal, we have
Hamiltonian (see Eq.(68) and thesubsequent text in [6]). S = S(0), ∀σ. From this and in analogy with the re-
f;σ f;σ
sult in Eq. (70) we have
10