Rigorous lower and upper bounds for the slope parameter

It is proved within the framework of axiomatic field theory that the logarithmic derivative of the absorptive part of the scattering amplitude with respect to momentum transfer is bounded from above by $\text{(15}\text{log}\text{s)}\text{[4√t(2 ? √t)]}$ for a sequence of s→+∞, and from below either in the s-channel by const. × s^?5 log^?4s or in the u-channel by const. × u^?5 log^?4u for at least one sequence of s or u →+∞, respectively. In the particular case of the s?u even-symmetric amplitude, a stronger lower bound is obtained; namely, const. × s^?5 log^?4s for at least one sequence of s→+∞. Here s, t, and u are the usual Mandelstam variables, and all bounds are obtained in the forward and the unphysical regions: 0?t<4 (in units of pion mass).It is observed that the Regge amplitude β(t)s^α(t) of high-energy scattering gives the same energy dependence as the above upper bound, and, furthermore, that the slope of the Regge trajectory is bounded from above by $\text{15}\text{[4√t(2 ? √t)]}\text{for}\text{0 < t < 4}$.     

Upper bounds for the ground-state energy of the exciton-phonon system

Upper bounds for the ground-state energy of the exciton-phonon system are calculated variationally for all ratios of the polaron radii to the exciton radius, whereby the constituents of the exciton are no longer treated as two independent polarons. The numerical results compare satisfactory with experimental data even for the thallous halides.     

“Optimal” bounds on the ground-state energy ofN-body systems of bosons and fermions interacting by attractive forces

We present inequalities on the ground state energy ofN-body systems which reduce, for bosons and fermions, to the exact solution in the limit where forces approach harmonic oscillator forces.     

Phase diagram of strongly correlated Fermi systems

Phase transitions caused by the redistribution of quasiparticle occupation numbers n(p) in homogeneous Fermi systems with particle repulsion are analyzed. The phase diagram of a strongly correlated Fermi system, when drawn in the coordinates “density ρ-dimensionless coupling constant η,” resembles a Washington pie for a rather broad class of interactions. Its upper part is “filled” with Fermi condensate, and the bottom part is filled with normal Fermi liquid. Both parts are separated by a narrow interlayer of Lifshitz phase with a multiply connected Fermi surface.     

Converging upper and lower bounds for ground-state energies of atomic nuclei

The production of K⁺-mesons in proton-nucleus collisions from 1.0 to 2.3 GeV is analyzed with respect to one-step nucleon-nucleon (NN NYK⁺) and two-step -nucleon (N K⁺YN) or pion-nucleon (N K⁺Y) production channels on the basis of a coupled-channel (CBUU) transport approach including the kaon final-state-interactions (FSI). Momentum-dependent potentials for the nucleon, hyperon and kaon in the final state are included as well as K⁺ elastic rescattering in the target nucleus. The transport calculations are compared to the experimental K⁺ spectra taken at COSY-Jülich. Our systematic analysis of K⁺ spectra from ¹²C, ⁶³Cu, ¹⁰⁷Ag and ¹⁹⁷Au targets as well as their momentum differential ratios gives a repulsive K⁺ potential of 20±5 MeV at normal nuclear matter density.     

Low-temperature collapse of the Fermi surface and phase transitions in correlated Fermi systems

A topological crossover, associated with the collapse of the Fermi surface in strongly correlated Fermi systems, is examined. It is demonstrated that in these systems, the temperature domain where standard Ferrai liquid results hold dramatically narrows, because the Landau regime is replaced by a classical one. The impact of the collapse of the Fermi surface on pairing correlations is analyzed. In the domain of the Lifshitz phase diagram where the Fermi surface collapses, splitting of the BCS superconducting phase transition into two different ones of the same symmetry is shown to occur.     

The upper and lower bounds of energy for nuclear and coulomb few-body systems

The upper and lower bounds of energy are found for three-, four-, and five-particle nuclear and Coulomb systems in the framework variational method with the trial functions of exponential and Gaussian types. The two-sided estimates of energy not only allow one to fix the limits for the exact value of energy but also provide an additional opportunity for extrapolation of the variational estimates to the exact value of energy. This allows one to reduce the volume of calculations by shortening the number of trial functions without the loss of accuracy.     

Fermi surface reconstruction in strongly correlated Fermi systems as a first-order phase transition

A quantum phase transition in strongly correlated Fermi systems beyond the topological quantum critical point has been studied using the Fermi liquid approach. The transition takes place between topologically equivalent states with three Fermi surface sheets, but one of them is characterized by a quasiparticle halo in the quasiparticle momentum distribution n(p), and the other one is characterized by a hole pocket. It has been found that the transition between these states is a first-order phase transition for the interaction constant g and temperature T. The phase diagram in the vicinity of this transition has been constructed.     

Adaptation of the Landau-Migdal quasiparticle pattern to strongly correlated Fermi systems

A quasiparticle pattern advanced in Landau’s first article on Fermi-liquid theory is adapted to elucidate the properties of a class of strongly correlated Fermi systems characterized by a Lifshitz phase diagram featuring a quantum critical point (QCP) where the density of states diverges. The necessary condition for stability of the Landau Fermi-Liquid state is shown to break down in such systems, triggering a cascade of topological phase transitions that lead, without symmetry violation, to states with multi-connected Fermi surfaces. The end point of this evolution is found to be an exceptional state whose spectrum of single-particle excitations exhibits a completely flat portion at zero temperature. Analysis of the evolution of the temperature dependence of the single-particle spectrum yields results that provide a natural explanation of classical behavior of this class of Fermi systems in the QCP region.     

Universal behavior of the collision rate in strongly correlated Fermi systems

Low temperature transport phenomena in strongly collision rate τ(T) is evaluated in closed form and applied to the calculation of the resistivity ρ(T). The ratio of ρ(T) to the square of the specific heat C _V(T) is shown to be a nearly universal function independent of the effective interaction between particles in medium. The result is found to be close to the well known Kadowaki-Woods ratio deduced from available experimental data on heavy fermion systems.     

Strongly correlated Fermi systems as a new state of matter

The aim of this review paper is to expose a new state of matter exhibited by strongly correlated Fermi systems represented by various heavy-fermion (HF) metals, two-dimensional liquids like ³He, compounds with quantum spin liquids, quasicrystals, and systems with one-dimensional quantum spin liquid. We name these various systems HF compounds, since they exhibit the behavior typical of HF metals. In HF compounds at zero temperature the unique phase transition, dubbed throughout as the fermion condensation quantum phase transition (FCQPT) can occur; this FCQPT creates flat bands which in turn lead to the specific state, known as the fermion condensate. Unlimited increase of the effective mass of quasiparticles signifies FCQPT; these quasiparticles determine the thermodynamic, transport and relaxation properties of HF compounds. Our discussion of numerous salient experimental data within the framework of FCQPT resolves the mystery of the new state of matter. Thus, FCQPT and the fermion condensation can be considered as the universal reason for the non-Fermi liquid behavior observed in various HF compounds. We show analytically and using arguments based completely on the experimental grounds that these systems exhibit universal scaling behavior of their thermodynamic, transport and relaxation properties. Therefore, the quantum physics of different HF compounds is universal, and emerges regardless of the microscopic structure of the compounds. This uniform behavior allows us to view it as the main characteristic of a new state of matter exhibited by HF compounds.     

Upper bounds on the free energy of random spin systems

The sequence of upper bounds on the quenched free energy of a random spin system considered by Goulart Rosa recently is rederived by an alternative method, and a similar sequence is constructed for the annealed free energy. The dependence of the free energy on interactions between spins is discussed.     

Rigorous bounds on the storage capacity of the dilute Hopfield model

We study a neural network model consisting ofN neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1-p. For the Hopfield Hamiltonian on such a network, we prove that ifp c[(lnN)/N]^1/2, the model can store at leastm= _cpN patterns, where _c 0.027 ifc 3 and decreases proportional to 1/(–lnc) forc small. This generalizes the results of Newman for the standard Hopfield model.