Accurate Dynamic Structure Factor of
the Homogeneous Electron Liquid
The electron liquid, an assembly of N electrons
embedded in a uniform positive background, has beenstudied to clarify the nature of electron correlation inmetals, putting aside the influence of the periodic ion
potential. This simplified model poses us a toughproblem, but after the struggles lasting longer thanhalf a century, accurate knowledge of its almost all
static properties is now acquired by a number ofsophisticated methods including quantum Monte
Carlo simulations over the entire region of metallic
densities 1.88<rs<5.6 where rs is the conventional
density parameter.
As for dynamical properties, on the other hand, ourknowledge is still not enough in spite of all previous
efforts to go beyond the random-phase approximation(RPA) in an attempt to explain the double-peakstructure in the dynamical structure factor S(q, )observed in light metals like Al. Among
approximation schemes, the Baym-Kadanoff (BK)
conserving one formulated in terms of the full Green'sfunction G is most suitable for the evaluation ofS(q, ). Accuracy of the result depends critically on
the choice of the energy functional [G], but theresult will never become exact, since no algorithm isknown to give the exact [G].
A conceptually different scheme is developed toobtain the exact S(q, ) [1]; instead of pursuing [G],we pay attention to the exact functional relations
between the self-energy and the vertex function ,obeying the microscopic conservation law. In
particular,
is determined by the Bethe-Salpeter
equation with an irreducible electron-hole interaction
given by the functional derivative,/ G. Startingfrom an arbitrary input, we iteratively revise bothandsimultaneously towards self-consistency
through the relations, whereby the number of terms
representing
generated in this iterative process
rapidly increases, eventually covering all terms
derivable from the exact[G] when the
self-consistency is achieved.
In a numerical algorithm, however, the functional
differentiation / G is not feasible. Thus we need toinvent an alternative scheme to revise on a computer
accurately and efficiently. Recently a physicallymotivated and accurate enough prescription is given
for [2], enabling us to obtain the result of S(q, ) [3]that is most accurate among all existing ones.
Throughout the metallic-density region, qualitative
features of S(q, ) remain the same; a typical example
at rs=5 is given in Fig. 1. Besides a broad peak b
located at almost the same position in the RPA, thereappears a clear shoulder a accompanied by asteepened slope of the linear term in S(q, ) for small. The peak b, being identified as “the center ofgravity” of one-pair excitations, is transformed
continuously into the plasmon peak as |q| decreases.

The shoulder a is well developed particularly for 1.4 <|q|/pF <2 and the slope is surprisingly enhanced overthe RPA value accordingly. These features imply
inseparable coupling between one- and multi-pair
excitations in the one-pair excitation region, attractive
electron-hole multiple scattering (excitonic effect) on
the low- side of the one-pair region, and extra
contributions from multi-pair excitations outside of it.

Fig. 1. Dynamical structure factor of the electron liquid
at rs=5. The value indicated by p is the plasmon energy
at q=0. The excitonic effect manifests itself in thestructure indicated by a.

Since accurate knowledge of S(q, ) of the electron
liquid is now available, we have compared it with theexperimental one for Al, a metal regarded as mostelectron-liquid-like with rs=2.08, in order to see how
the band effects modifies the actual structure of S(q, ).
The result of comparison is shown in Fig. 2, whichleads us to conclude that the electron-liquid model
applies well to Al on the whole and especially so for<EF ( 11eV). The discrepancy from the model athigher is ascribed to the strong tight-binding natureof unoccupied 3d and 4f bands in Al, leading to thestructure c in Fig. 2.

Fig. 2. Comparison between S(q, ) of the electron liquid
at rs=2.08 and the experimental one for Al metal.

References[1]Y. Takada, Phys. Rev. B 52, 12708 (1995).
[2]Y. Takada, Phys. Rev. Lett. 87, 226402 (2001).
[3]Y. Takada and H. Yasuhara, Phys. Rev. Lett. 89,216402 (2002).

Authors

Y. Takada and H. Yasuharaa
aInstitute for Materials Research, Tohoku University.