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Journal of Statistical Mechanics Theory and Experiment

Binding and stability of Coulomb systems

Walter E Thirring

Published May 28, 2010• 0 citations
Publisher

Abstract

For particles with Coulomb interactions, quantum mechanics predicts chemical binding and extensivity of the ground state energy (referred to as stability ). In contrast, classically the system has the tendency to crumble into binaries and singles. We study the problem with semi-classical density functionals and find that our simple systems can have either one of the two properties, but not both.

DOI
10.1088/1742-5468/2010/05/p05020

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For particles with Coulomb interactions, quantum mechanics predicts chemical binding and extensivity of the ground state energy (referred to as stability ). In contrast, classically the system has the tendency to crumble into binaries and singles. We study the problem with semi-classical density functionals and find that our simple systems can have either one of the two properties, but not both.
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Non-relativistic quantum chemical calculations of the particle mass, m±2, corresponding to the dissociation threshold in a range of Coulomb three-particle systems of the form m±1m±2m±3, are performed variationally using a series solution method with a Laguerre-based wavefunction. These masses are used to calculate an accurate stability boundary, i.e., the line that separates the stability domain from the instability domains, in a reciprocal mass fraction ternary diagram. This result is compared to a lower bound to the stability domain derived from symmetric systems and reveals the importance of the asymmetric (mass-symmetry breaking) terms in the Hamiltonian at dissociation. A functional fit to the stability boundary data provides a simple analytical expression for calculating the minimum mass of a third particle required for stable binding to a two-particle system, i.e., for predicting the bound state stability of any unit-charge three-particle system.
Abstract This chapter opens with the notion of stable interaction in quantum systems, somewhat different from classical systems. It is shown that an unstable classical system can become stable when treated quantum mechanically. The remaining of the chapter is devoted to a brief survey of the main results concerning the stability of Coulomb systems. Explicit evaluations of the total energy show that the exclusion principle is essential for their stability. The final part of the chapter deals with the thermodynamic limit of globally neutral Coulomb systems. The limit exists in spite of the long-range character of the interaction, and contrary to the stability property, the existence of the limit does not require quantum mechanics, being valid also in classical Coulomb systems.
Ordinary matter, composed of electrons and nuclei, is held together by Coulomb forces. These forces are potentially catastrophic for two reasons: the short range attraction could lead too a collapse while the long range repulsion and attraction could lead to an explosion or too strong interaction between macroscopic objects. The fact that neither of these things happens and that matter is essentially inert is a consequence of subtle properties of quantum mechanics and electrostatics outlined here.
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We study the quantum many-body ground states of electrons on the half-filled honeycomb lattice with short- and long-ranged density-density interactions as a model for graphene. To this end, we employ the recently developed truncated-unity functional renormalization group (TU-fRG) approach which allows for a high resolution of the interaction vertex' wave vector dependence. We connect to previous lattice quantum Monte Carlo (QMC) results which predict a stabilization of the semimetallic phase for realistic ab initio interaction parameters and confirm that the application of a finite biaxial strain can induce a quantum phase transition towards an ordered ground state. In contrast to lattice QMC simulations, the TU-fRG is not limited in the choice of tight-binding and interaction parameters to avoid the occurrence of a sign problem. Therefore, we also investigate a range of parameters relevant to the realistic graphene material which are not accessible by numerically exact methods. Although a plethora of charge-density waves arises under medium-range interactions, we find the antiferromagnetic spin-density wave to be the prevailing instability for long-range interactions. We further explore the impact of an extended tight-binding Hamiltonian with second-nearest-neighbor hopping and a finite chemical potential for a more accurate description of the band structure of graphene's pz{p}_{z}pz​ electrons.