Robustness of a local Fermi liquid against ferromagnetism and phase separation.

Comment on " Robustness of a Local Fermi Liquid against Ferromagnetism and Phase Separation " In a Letter [1], on which ongoing research is based [2], Engelbrecht and Bedell (EB) studied the properties of Fermi liquids with a local (i.e., k independent) irreducible self-energy. Two of their main results were that such local Fermi liquids are robust against ferromagnetism and phase separation. In this Comment we point out that the conclusions of [1] are not generally valid for Fermi liquids with a local self-energy. We show that the conclusions of EB are not valid for lattice models in high spatial dimensions ͑d ! `͒, which are the only systems known to date for which the self-energy is purely local. We argue that the authors' statement that " the dynamical mean-field theories, which become exact in infinite d, should lead to results that are compatible with ours. .. , " does not hold. First we recall some of the properties of lattice Fermi systems in d ෇ `. In high spatial dimensions the self-energy S͑k͒ is local [3]. Thus it can be seen as a functional of the local Green function: S ෇ S͓G loc ͑v͔͒, where G loc ͑v͒ ෇ N 21 P k G͑k͒ and N is the number of lattice sites. However, the irreducible [4] vertex functions are not local [5,6]. Their momentum dependence enters through the parameter x͑k͒ ෇ d 21 P d i෇1 cos͑k i ͒ [7], e.g., the three contributions to the irreducible ver-tex function in second order in a local interaction are for d ! ` dependent on x͑p 2 p 0 ͒ (bubble), x͑p 2 p 0 ͒ (parallel interaction lines), or x͑p 1 p 0 1 q͒ (crossing interaction lines). EB assume that a local self-energy S ෇ S͓G loc ͑v͔͒ implies a local irreducible vertex function: G IR ෇ dS͑v͒͞dG loc ͑v 0 ͒ [below Eq. (4) in [1] ]. But even in the limit d ! `, where the assumption of a local self-energy is best, the locality of the self-energy does not imply the locality of the irreducible vertex function; see above. This means that functional derivation and limit process do not commute. By extension, an almost local self-energy in finite dimensions does not imply an almost local irreducible vertex function. Based on this crucial point, we argue that EB's concept of a strictly local Fermi liquid is not …