Finite-size analysis of the Fermi liquid properties of the homogeneous electron gas
We analyze the extrapolation to the thermodynamic limit of Fermi liquid properties of the homogeneous electron gas in two and three dimensions. Using field theory, we explicitly calculate finite-size effects of the total energy, the renormalization factor, and the effective mass at the Fermi surface within the random phase approximation (RPA) and discuss the validity for general metallic systems.
We consider the Fermi liquid phase of the (unpolarized) homogeneous electron gas in two and three dimensions () at zero temperature, . Quantum Monte Carlo (QMC) calculations [1, 2] have not only provided the most precise evaluations of the ground state energy, , as a function of density, , but also give access to spectral properties, e.g. the momentum distribution, , the effective mass, , of quasiparticle excitations, and the corresponding spectral weight, the renormalization factor [2, 3, 4, 5, 6, 7, 8, 9, 10]. Whereas QMC provides a powerful strategy to quantify correlations in Fermi systems, it still suffers from two fundamental limitations: the fixed-node error, and the extrapolation to the thermodynamic limit. As shown recently, calculations of the Fermi liquid parameters are drastically affected by size effects [4, 7], since typical QMC computations are done with electrons, but extrapolation schemes based on analytical knowledge of underlying properties significantly reduce the finite size error [11, 12, 4, 7]. Here, to provide further insight, we address the issue of extrapolating calculations of finite (small) systems to the infinite volume limit within common language of field theory. Strategies of improving the many-body wave function to reduce the fixed-node error in QMC are discussed elsewhere [13, 14, 15].
At high densities, , the random phase approximations (RPA) is expected to be valid, and we explicitly calculate leading order size corrections within this approximation, and discuss their general validity. Here, with is the mean inter particle distance, the Bohr radius, and and are the charge and mass of the electrons. Our results agree with the leading order corrections in [11, 4, 6, 7] where finite size corrections have been derived from a formulation directly based on the many-body wave function. In the following, we explicitly concentrate on three fundamental properties, the total energy, , the renormalization factor, , and the effective mass, , at the Fermi surface, .
From the dielectric function, , the ground state energy per particle, , inside a box of volume with periodic boundary conditions, and , can be obtained via coupling constant integration 
where is the Coulomb potential and
and the coupling constant integration can be done explicitly
Spectral properties are obtained from the single particle Green’s function, , which equals
where the self energy, accounts for interaction effects, and is the chemical potential which fixes the Fermi energy, , via . From a frequency integration over the Green’s function, the momentum distribution, , can be obtained, and the magnitude of the jump at the Fermi surface, , is directly related to the residual of at (,). In general, poles of the Green’s function give rise to quasi-particle excitations, whose energies are given in terms of an effective mass at the Fermi surface.
where is the unperturbed Fermi energy. From the self-energy, the renormalization factor, , at is given by
and the effective mass, , which characterizes the quasi-particle dispersion at the Fermi surface, is then obtained via
Formally, the RPA contains the leading order corrections beyond the independent particle Hartree-Fock approximation, and its validity is restricted to the high density region, . In this limit, it is consistent to expand Eq. (8). This leads to the so-called on-shell approximation (OSA) of the effective mass,
In the literature, there exists a long-standing controversy [19, 22, 23, 24, 25], whether Eq. (8) or Eq. (9) should be used to determine the effective mass. Strictly speaking, the validity of the RPA is confined to the high density region where both formulas essentially agree, , and the validity of RPA or RPA-like results at lower densities essentially relies on cancelation of errors. However, given the exact self-energy beyond RPA, only Eq. (8) provides the effective mass, and our finite-size analysis below is based on Eq. (8) , since, as we will see, our results are not restricted to the RPA.
3 Thermodynamic limit extrapolation of finite system results
All formulas of the previous section are valid for systems of finite size, , where wave vectors are discrete. Perturbative expressions for the Green’s function and the dielectric functions, as e.g. the RPA expressions, are based on the linked-cluster theorem, valid in a grand-canonical ensemble of fixed volume introducing small number fluctuations. Since the non-interacting Green’s function does not depend explicitly on the system size, the extrapolation from the finite to the infinite system corresponds to the conversions of all underlying discrete sums to integrals, , and finite size corrections are equivalent to discretization errors in quadrature. On this basis, the elimination of the size error in the second term on the rhs of the total energy expression, Eq. (1), is straightforward, leading to the Madelung constant
In the following, we show that a similar analysis of the quadrature error can be used to reduce the finite size error in other quantities.
If the integrand is regular, the discretization error is rapidly vanishing (of order or higher). The main contribution of order with comes from non-analytical points in the integrand. From the general structure of the perturbation expansion there are only certain discrete points where non-analyticities can develop: in the long wavelength limit, , where the Coulomb potential is singular, and for integer multiples of the Fermi wave-vector () due to the sharp Fermi surface of the underlying wave function.
Explicitly, the sharp Fermi surface at leads to well-known shell oscillations in the energy, already present in the energy of the ideal Fermi gas, Eq. (2). Twisted boundary conditions strongly reduce these oscillations , and grand-canonical twist averaging (GTABC) completely eliminates them . Within GTABC, we have
and . Similarly, GTABC also eliminates size effects of other non-interacting properties, e.g. in the ideal gas density response, , as can be seen from the spectral representation
although is still only given at discrete wave vectors ( with integer ). Therefore, twisted boundary conditions as well as GTABC do not modify the discrete sum in the exchange-correlation energy on the rhs of Eq. (1). However, GATBC lead to an accelerated convergence of the integrand towards the thermodynamic limit on all allowed wave vectors.
Within GTABC, we assume that size effects in the integrands of all expressions can be neglected, and focus on the presumably largest contributions from the Coulomb singularity at . For the total energy, Eq. (1), we expect that the leading order size corrections of the exchange-correlation energy, , is given by
Similarly, the effect of the thermodynamic limit extrapolation on the renormalization factor and the effective mass can be obtained from explicit (approximate) expressions of the self-energy, Eq. (6), together with Eq. (7) and Eq. (8).
The Lindhard function, , has the following limiting form in the long wave length limit
and we have
where the plasma frequency is given by .
4.1 Total energy
Inserting the limiting form of the dielectric function in Eq. (4), we obtain
and size corrections in the energy have a simple interpretation in terms of zero point energy of long wavelength plasmon modes. Explicitly, in three dimensions, we have
in agreement with Ref. , and
in two dimensions  where accounts for the cubic geometry.
4.2 Renormalization factor and effective mass
For the calculation of the renormalization factor and the effective mass, we need the derivatives of the self energy at the Fermi surface. Within the RPA, we have
Since for , the static dielectric function diverges in the long wave length limit,
providing perfect screening, and the dominant size correction, , of exactly equals that of , and is given by
and we have
in three dimension  with , and
in two dimensions  with . The leading order size corrections for the renormalization factor are then given by
and, using Eq. (8), the effective mass correction writes
Note, that, within the validity of the RPA, we may put on the rhs of Eq. (28). The slow decay of with increasing system size can lead to important qualitative and quantitative changes of spectral quantities when properly extrapolated to the thermodynamic limit with respect to calculations which rely on linear extrapolation schemes.
We have discussed size effects of the total energy, the renormalization factor, and the effective mass. Particularly, in two dimensions, and show an unexpected slowly decaying size effects . In general, we expect that these size effects are also present in other response quantities, e.g. the spin susceptibility, as they are connected to by relations based on Landau’s Fermi liquid theory. Our results on the finite size corrections are quite general and do not rely on the validity of the RPA, as they are based on the large wave length behavior of the dielectric function, Eq. (15), which applies to all metallic systems. Further, all results agree with the leading order corrections in [11, 4, 7] where finite size corrections have been derived from a formulation directly based on the many-body wave function.
This research was supported by CNRS-IDRIS, Project IDRIS 061801.
-  D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
-  B. Tanatar and D.M. Ceperley, Phys. Rev. B 39, 5005 (1989).
-  N. D. Drummond and R. J. Needs, Phys. Rev. B 79, 085414 (2009).
-  M. Holzmann, B. Bernu, V. Olevano, R. M. Martin, and D. M. Ceperley, Phys. Rev. B 79, 041308 (2009).
-  G. Ortiz and P. Ballone, Phys. Rev. B 50, 1391 (1994); G. Ortiz and P. Ballone, Phys. Rev. B 56, 9970 (1997).
-  S. Huotari, J. A. Soininen, T. Pylkkänen, K. Hämäläinen, A. Issolah, A. Titov, J. McMinis, J. Kim, K. Esler, D.M. Ceperley, M. Holzmann, and V. Olevano, Phys. Rev. Lett. 105, 086403 (2010).
-  M. Holzmann, B. Bernu, C. Pierleoni, J. McMinis, D.M. Ceperley, V. Olevano, and L. Delle Site, arXiv:1105.2338 (2011).
-  Y. Kwon, D.M. Ceperley, and R.M. Martin, Phys. Rev. B 50, 1684 (1994).
-  Y. Kwon, D.M. Ceperley, and R.M. Martin, Phys. Rev. B 53, 7376 (1996).
-  N. D. Drummond and R. J. Needs, Phys. Rev. B 80, 245104 (2009).
-  S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holzmann, Phys. Rev. Lett. 97, 076404 (2006).
-  N.D. Drummond, R.J. Needs, A. Sorouri, and W.M.C. Foulkes, Phys. Rev. B 78, 125106 (2008).
-  D.M. Ceperley and B.J. Alder, J. Chem. Phys. 81, 5833 (1984).
-  M. Holzmann, B. Bernu, and D.M. Ceperley, Phys. Rev. B 74, 104510 (2006).
-  M. Nava, E. Vitali, A. Motta, D.E. Galli, and S. Moroni, arXiv:1103.0915 (2011).
-  P. Nozières and D. Pines, Phys. Rev. 111, 442 (1958).
-  D. Bohm and D. Pines, Phys. Rev. 92, 609 (1953).
-  P. Nozières and D. Pines, The Theory of Quantum Liquids (Benjamin, New York,1966).
-  T.M. Rice, Ann. of Phys. 31, 100 (1965).
-  J.J. Quinn and R.A. Ferell, Phys. Rev. 112, 812 (1958).
-  H.-J. Schulze, P. Schuck, and N. Van Giai, Phys. Rev. B 61, 8026 (2000).
-  C. S. Ting, T. K. Lee, and J. J. Quinn, Phys. Rev. Lett. 34, 870 (1975); T. K. Lee, C. S. Ting, and J. J. Quinn, Phys. Rev. Lett. 35, 1048 (1975).
-  G.E. Santoro and G.F. Giuliani, Phys. Rev. B 39, 12818 (1989).
-  Y. Zhang and S. Das Sarma, Phys. Rev. B 71, 045322 (2005).
-  R. Asgari, B. Davoudi, M. Polini, G. F. Giuliani, M. P. Tosi, and G. Vignale, Phys. Rev. B 71, 045323 (2005).
-  C. Lin, F.-H. Zong, and D. M. Ceperley, Phys. Rev. E 64, 016702 (2001).