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Physical Review A

Quantum criticality and stability of three-body Coulomb systems

Sabre Kais, Qicun Shi

Published Nov 15, 2000• 50 citations
Publisher

Abstract

We present quantum phase transitions and critical phenomena of three-body Coulomb systems with charges $(Q,$ q, $Q)$ and masses $(M,$ m, $M).$ Full numerical results, using the finite-size scaling method, for an arbitrary mass ratio $0<~\ensuremath{\kappa}{=(1+m/M)}^{\ensuremath{-}1}<~1$ over the range $1<~\ensuremath{\lambda}=|Q/q|<~1.25,$ show that there exists a transition curve ${\ensuremath{\lambda}}_{c}(\ensuremath{\kappa})$ through which all systems undergo a first-order phase transition from stable to unstable. Particularly, ${\ensuremath{\lambda}}_{c}(\ensuremath{\kappa})$ has a minimum at ${\ensuremath{\kappa}}_{m}=0.35,$ which leads to a new proposed classification of the three-body Coulomb systems: moleculelike systems, $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{m},$ such as ${\mathrm{Ps}}^{\ensuremath{-}}$ $(\ensuremath{\kappa}=0.5)$ and atomlike systems, $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{m},$ such as $\overline{p}\overline{p}d$ $(\ensuremath{\kappa}=0.33).$

DOI
10.1103/physreva.62.060502

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Abstract We present quantum phase transitions and critical phenomena at the large‐dimension (D) limit for three‐body ABA Coulomb systems with charges (Q,q,Q) and masses (M,m,M). The Hamiltonian depends linearly on two parameters λ=∣Q/q∣ and κ=[1+(m/M)]−1. The system exhibits critical points with mean field critical exponents (α=0, β=½, δ=3, γ=1). We calculate the critical curve λc(κ) through which all systems undergo a continuous‐phase transition from the symmetrical configuration, the two like particles have the same distance from the reference particle, to the unsymmetrical phase. The critical curve at D→∞ limit is a convex function of κ and very similar to the one obtained at D=3 with variational calculations. We also calculated the line of zero angular correlation in the mass polarization term, which separates the symmetrical phase to an atom‐like region and a molecule‐like region. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
In this review we discuss quantum phase transitions and the mapping between symmetry breaking of electronic structure configurations at the large-dimension limit and mean-field theory of phase transitions. We show that the finite size scaling method can be used for the calculations of the critical parameters of the few-body Schrodinger equation. In this approach, the finite size corresponds to the number of elements in a complete basis set used to expand the exact eigenfunction of a given Hamiltonian. The critical parameters such as the critical nuclear charges will be used to explain and predict the stability of atomic and molecular negative ions. For N-electron atoms with 2 N 86, results show that, at most, only one electron can be added to a free atom in the gas phase. However, doubly charged atomic negative ions might exist in a strong magnetic field.
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Quantum phase transitions are strongly relevant in a number of fields, ranging from condensed matter to cold atom physics and quantum field theory. This book, now in its second edition, approaches the problem of quantum phase transitions from a new and unifying perspective. Topics addressed include the concepts of scale and time invariance and their significance for quantum criticality, as well as brand new chapters on superfluid and superconductor quantum critical points, and quantum first order transitions. The renormalisation group in real and momentum space is also established as the proper language to describe the behaviour of systems close to a quantum phase transition. These phenomena introduce a number of theoretical challenges which are of major importance for driving new experiments. Being strongly motivated and oriented towards understanding experimental results, this is an excellent text for graduates, as well as theorists, experimentalists and those with an interest in quantum criticality.
In three research articles we have studied the critical properties of effective lattice models for strongly correlated electron systems by Monte Carlo simulations. A similar model is used in a fourth article for investigating thermal fluctuations of vortices in a rotating Bose–Einstein condensate. In the first part of this thesis we review the necessary background and introduce the models one by one. The last part is a collection of the papers.\n\nPaper I [1]: We consider the scaling of the mean square dipole moment in a plasma with logarithmic interactions in a two- and three-dimensional system. In both cases, we establish the existence of a low-temperature regime where the mean square dipole moment does not scale with system size and a hightemperature regime does scale with system size. Thus, there is a nonanalytic change in the polarizability of the system as a function of temperature, and hence a metal-insulator transition in both cases. The relevance of this transition in three dimensions to quantum phase transitions in 2 + 1-dimensional systems is briefly discussed.\n\nPaper II [2]: The existence of a discontinuity in the inverse dielectric constant of the two-dimensional Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a threedimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.\n\nPaper III [3]: We perform large-scale Monte Carlo simulations on an effective gauge theory for an easy plane quantum anti-ferromagnet, including a Berry phase term that projects out the S = 1/2 sector. Without a Berry phase term, the model exhibits a phase transition in the 3DXY universality class associated with proliferation of gauge-charge neutral U(1) vortices. The instantons that eliminate the phase transition in the gauge-charged sector are cancelled by the Berry phases. The result is a first order phase transition. This gauge theory therefore does not exhibit deconfined criticality.\n\nPaper IV [4]: We perform Monte Carlo studies of vortices in three dimensions in a cylindrical confinement, with uniform and nonuniform density. The former is relevant to rotating 4He, the latter is relevant to a rotating trapped Bose condensate. In the former case we find dominant angular thermal vortex fluctuations close to the cylinder wall. For the latter case, a novel effect is that at low temperatures the vortex solid close to the center of the trap crosses directly over to a tension-less vortex tangle near the edge of the trap. At higher temperatures an intermediate tensionful vortex liquid located between the vortex solid and the vortex tangle, may exist.
The Hamiltonian for an arbitrary Coulomb three-body system is written in closed form in terms of nine generators of three independent o(2,1) algebras and its six physical parameters (i.e., three masses and three charges). Moreover, it is shown that such a Hamiltonian can also be written in the second quantized form by using six pairs of creation and annihilation (boson) operators. The obtained finite-term representation of the Coulomb three-body Hamiltonian in the second quantized form is exact, i.e., it is not based on any approximation. In particular, this means that the original Coulomb three-body problem can be reduced to the equivalent problem of three interacting, complex (i.e., charged) boson fields. The developed procedure opens new avenues in obtaining analytical and semianalytical (or highly accurate) solutions for various Coulomb three-body problems.