Renormalization Group Approach to Interacting Fermions

The stability of nonrelativistic fermionic systems to interactions is studied within the Renormalization Group framework. A brief introduction to $\phi^4$ theory in four dimensions and the path integral formulation for fermions is given. The strategy is as follows. First, the modes on either side of the Fermi surface within a cut-off $\Lambda$ are chosen and a path integral is written to describe them. An RG transformation which eliminates a part of these modes, but preserves the action of the noninteracting system is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A $d=1$ warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, the RG correctly yields a scale invariant system (Luttinger liquid) In $d=2$ and 3, for rotationally invariant Fermi surfaces, {\em automatically} leads to Landau's Fermi liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function $F$, with the only relevant perturbations being of the superconducting (BCS) type The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally non-invariant (but time-reversal invariant) Fermi surfaces also, A study of a nested Fermi surface shows an additional relevant flow leading to charge density wave formation. For small $\Lambda / K_F$, a 1/N expansion emerges, with $N = K_F/ \Lambda$, which explains why one is able to solve the narrow cut-off theory. The search for non-Fermi liquids in $d=2$Comment: 191 pages, Latex, Yale, July 1993; 18 EPS figures appende

Ecological autocatalysis:A central principle in ecosystem organization?

Ecosystems comprise flows of energy and materials, structured by organisms and their interactions. Important generalizations have emerged in recent decades about conversions by organisms of energy (metabolic theory of ecology) and materials (ecological stoichiometry). However, these new insights leave a key question about ecosystems inadequately addressed: are there basic organizational principles that explain how the interaction structure among species in ecosystems arises? Here we integrate recent contributions to the understanding of how ecosystem organization emerges through ecological autocatalysis (EA), in which species mutually benefit through self-reinforcing circular interaction structures. We seek to generalize the concept of EA by integrating principles from community and ecosystem ecology. We discuss evidence suggesting that ecological autocatalysis is facilitated by resource competition and natural selection, both central principles in community ecology. Furthermore, we suggest that pre-emptive resource competition by consumers and plant resource diversity drive the emergence of autocatalytic loops at the ecosystem level. Subsequently, we describe how interactions between such autocatalytic loops can explain pattern and processes observed at the ecosystem scale, and summarize efforts to model different aspect of the phenomenon. We conclude that EA is a central principle that forms the backbone of the organization in systems ecology, analogous to autocatalytic loops in systems chemistry.</p

Growth Rate Hypothesis does not apply across colimiting conditions: cholesterol limitation affects phosphorus homoeostasis of an aquatic herbivore

1. Herbivores show stronger control of element homoeostasis than primary producers, which can lead to constraints in carbon and nutrient transfer efficiencies from plants to animals. Insufficient dietary phosphorus (P) availability can cause reduced body P contents along with lower growth rates of animals, leading to a positive relationship between growth and body P. 2. We examined how a second limiting food component in combination with dietary P limitation influences growth and P homoeostasis of a herbivore and how this colimitation influences the hypothesized positive correlation between body P content and growth rates. Therefore, we investigated the responses in somatic growth and P stoichiometry of Daphnia magna raised on a range of diets with different amounts of P and the sterol cholesterol. 3. Somatic growth rates of D. magna increased asymptotically with increasing P as well as with increasing cholesterol availability. The body P content increased with increasing dietary P and stabilized at high dietary P availability. The observed plasticity in D. magna's P stoichiometry became stronger with increasing cholesterol availability, i.e. with decreasing colimitation by cholesterol. 4. At P-limiting conditions, the positive correlation between body P content and growth rate, as predicted by the growth rate hypothesis (GRH) applied to the within-species level, declined with increasing cholesterol limitation and disappeared entirely when cholesterol was not supplied. Thus, even when Daphnia shows no growth response owing to strong limitation by the colimiting nutrient, the body P content may vary substantially, calling into question the unconditional use of herbivores' P content as predictor of a potential P limitation in nature. 5. The observed interaction between dietary P and cholesterol on Daphnia's growth and stoichiometry can be used as a conceptual framework of how colimiting essential nutrients affect herbivore homoeostasis, and provide further insights into the applicability of the GRH within a consumer species