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Particle-hole ring diagrams for fermions in two dimensions

Norbert Kaiser, Physik-department T Technische Universität München

Published Aug 17, 2016• 2 citations
OpenAlex

Abstract

The set of particle-hole ring diagrams for a many-fermion system in two dimensions is studied. The complex-valued polarization function is derived in detail and shown to be expressible in terms of square-root functions. For a contact-interaction the perturbative contributions to the energy per particle $\bar E(k_f)$ are calculated in closed analytical form from third up to twelfth order. The resummation of the particle-hole ring diagrams to all orders is studied and a pronounced dependence on the dimensionless coupling parameter $\alpha$ is found. There is a substantial difference between the complete ring-sum with all exchange-type diagrams included and the standard resummation of the leading $n$-ring diagrams only. The spin factor $S_n(g)$ associated to the $n$-th order ring diagrams is derived for arbitrary spin-degeneracy $g$.

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The set of particle-hole ring diagrams for a many-fermion system in two dimensions is studied. The complex-valued polarization function is derived in detail and shown to be expressible in terms of square-root functions. For a contact-interaction the perturbative contributions to the energy per particle Eˉ(kf)\bar E(k_f)Eˉ(kf​) are calculated in closed analytical form from third up to twelfth order. The resummation of the particle-hole ring diagrams to all orders is studied and a pronounced dependence on the dimensionless coupling parameter α\alphaα is found. There is a substantial difference between the complete ring-sum with all exchange-type diagrams included and the standard resummation of the leading nnn-ring diagrams only. The spin factor Sn(g)S_n(g)Sn​(g) associated to the nnn-th order ring diagrams is derived for arbitrary spin-degeneracy ggg.
For a system of fermions with a three-body contact interaction the second-order contributions to the energy per particle $\ensuremath \bar E(k_f)$ are calculated exactly. The three-particle scattering amplitude in the medium is derived in closed analytical form from the corresponding two-loop rescattering diagram. We compare the (genuine) second-order three-body contribution to $\ensuremath \bar E(k_f)\sim k_f^{10}$ with the second-order term due to the density-dependent effective two-body interaction, and find that the latter term dominates. The results of the present study are of interest for nuclear many-body calculations where chiral three-nucleon forces are treated beyond leading order via a density-dependent effective two-body interaction.
Exact two-dimensional analytic wave functions for an arbitrary number N of contact-interacting lowest-Landau-level (LLL) spinful fermions are derived with the use of combined numerical and symbolic computational approaches via analysis of exact Hamiltonian numerical diagonalization data.Closed-form analytic expressions are presented for two families of zero-interaction-energy states at given total angular momentum and total spin 0 S N/2 in the neighborhood of the ν = 1 filling, covering the range from the maximum density droplet to the first quasihole.Our theoretical predictions for higher-order spatial and momentum correlations reveal intrinsic polygonal, multi-ring crystalline-type structures, which can be tested with ultracold-atom experiments in rapidly rotating traps, simulating quantum Hall physics (including quantum LLL skyrmions).
We calculate the fermion Green function and particle-hole susceptibilities for a degenerate two-dimensional fermion system with a singular gauge interaction. We show that this is a strong-coupling problem, with no small parameter other than the fermion spin degeneracy N. We consider two interactions, one arising in the context of the t-J model and the other in the theory of half-filled Landau level. For the fermion self-energy we show that the qualitative behavior found in the leading order of perturbation theory is preserved to all orders in the interaction. The susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathit{Q}}$ at a general wave vector Q\ensuremath{\ne}2pF{\mathbf{p}}_{\mathbf{F}}pF​ retains the Fermi-liquid form. However, the 2pF{\mathit{p}}_{\mathit{F}}pF​ susceptibility ${\mathrm{\ensuremath{\chi}}}_{2\mathit{p}\mathit{F}}$ either diverges as T\ensuremath{\rightarrow}0 or remains finite but with nonanalytic wave-vector, frequency, and temperature dependence. We express our results in the language of recently discussed scaling theories, give the fixed-point action, and show that at this fixed point the fermion-gauge-field interaction is marginal in d=2, but irrelevant at low energies in d\ensuremath{\ge}2.
The correlation energy per particle (expressed in rydbergs) of a two-dimensional electron gas is calculated by summing the ring diagrams. For high densities, this is found to be of the form C+Drslnrs+O(rs)C+D{r}_{s}\mathrm{ln}{r}_{s}+O({r}_{s})C+Drs​lnrs​+O(rs​), where rs{r}_{s}rs​ is a dimensionless parameter such that ${r}_{s}^{\ensuremath{-}2}$ is proportional to the number of electrons per unit area. The value of DDD is -0.172 and that of CCC is -0.38 \ifmmode\pm\else\textpm\fi{} 0.04, the latter involves a numerical evaluation. By the same methods, we have also calculated the difference in total energy of the nonmagnetic and ferromagnetic states and we find for rs>2.3{r}_{s}>2.3rs​>2.3 the system may be ferromagnetic. We have also obtained some exact relationships between the pair correlation function at zero separation and the asymptotic behavior of the structure factor and the momentum distribution function. These are of interest in relation to scattering experiments.
We develop the parquet-diagram summation method for neutron matter interacting via potentials that include spin, tensor, and spin-orbit components. For that purpose, we derive an exact expression for the sum of all ring diagrams in terms of effective local particle-hole interactions involving the above four operators. The parquet equations are closed by deriving the spin-orbit contribution to those particle-hole interactions. We show that many-body correlations screen the bare spin-orbit potential considerably, and the corrections of that screened spin-orbit potential to the other three interaction channels are quite small. We apply our method to the calculation of the response of neutron matter to density and both longitudinal and transverse spin-dependent external fields.