We describe a simple model for symmetry breaking of electronic structure configurations of one-electron systems. This model involves generalizing the problem to D-dimensional space and finding the solution at , a semiclassical limit which can be solved exactly. The large-D limit model reduces the problem to a variational calculation which is equivalent to mean-field theories of critical phenomena in statistical mechanics. We show that symmetry breaking of electronic structure configurations can be described as standard phase transitions. Rich phase diagrams with multicritical points are reported for both linear and planar one-electron systems.
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.
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
Quantum criticality of metal-insulator transitions in correlated electron systems is shown to belong to an unconventional universality class with violation of the Ginzburg-Landau-Wilson (GLW) scheme formulated for symmetry breaking transitions. This unconventionality arises from an emergent character of the quantum critical point, which appears at the marginal point between the Ising-type symmetry breaking at nonzero temperatures and the topological transition of the Fermi surface at zero temperature. We show that Hartree-Fock approximations of an extended Hubbard model on square lattices are capable of such metal-insulator transitions with unusual criticality under a preexisting symmetry breaking. The obtained universality is consistent with the scaling theory formulated for Mott transitions and with a number of numerical results beyond the mean-field level, implying that preexisting symmetry breaking is not necessarily required for the emergence of this unconventional universality. Examinations of fluctuation effects indicate that the obtained critical exponents remain essentially exact beyond the mean-field level. It further clarifies the whole structure of singularities by a unified treatment of the bandwidth-control and filling-control transitions. Detailed analyses of the criticality, containing diverging carrier density fluctuations around the marginal quantum critical point, are presented from microscopic calculations and reveal the nature as quantum critical ``opalescence.'' The mechanism of emerging marginal quantum critical point is ascribed to a positive feedback and interplay between the preexisting gap formation present even in metals and kinetic energy gain (loss) of the metallic carrier. Analyses of crossovers between GLW type at nonzero temperature and topological type at zero temperature show that the critical exponents observed in
(V,Cr)2O3 and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.
We explore in detail properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large
D limit, or as disordered models. Both models have a mass parameter
m and the transition from the perturbative large
m region to the strongly coupled ``black-hole'' small
m region is associated with several interesting phenomena. One model, with
U(n)2 symmetry and equivalent to complex Sachdev-Ye-Kitaev, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having nontrivial, nonmean-field critical exponents for standard thermodynamical quantities. Quasinormal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced
U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass
m∗. For
0<m<m∗, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{+}$ and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{-}}$ for the Euclidean two-point function when $t\ensuremath{\rightarrow}+\ensuremath{\infty}$ and $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ respectively. We provide several detailed and pedagogical derivations, including rigorous proofs or simplified arguments for some results that were already known in the literature.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.