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Register a new userExistence of a Density Functional for an Intrinsic State
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A generalization of the Hohenberg-Kohn theorem proves the existence of a density functional for an intrinsic state, symmetry violating, out of which a physical state with good quantum numbers can be projected.
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In this work we consider the existence and uniqueness of the ground state of the regularized Hamiltonian of the Supermembrane in dimensions $D= 4,,5,,7$ and 11, or equivalently the $SU(N)$ Matrix Model. That is, the 0+1 reduction of the 10-dimensional $SU(N)$ Super Yang-Mills Hamiltonian. This ground state problem is associated with the solutions of the inner and outer Dirichlet problems for this operator, and their subsequent smooth patching (glueing) into a single state. We have discussed properties of the inner problem in a previous work, therefore we now investigate the outer Dirichlet problem for the Hamiltonian operator. We establish existence and uniqueness on unbounded valleys defined in terms of the bosonic potential. These are precisely those regions where the bosonic part of the potential is less than a given value $V_0$, which we set to be arbitrary. The problem is well posed, since these valleys are preserved by the action of the $SU(N)$ constraint. We first show that their Lebesgue measure is finite, subject to restrictions on $D$ in terms of $N$. We then use this analysis to determine a bound on the fermionic potential which yields the coercive property of the energy form. It is from this, that we derive the existence and uniqueness of the solution. As a by-product of our argumentation, we show that the Hamiltonian, restricted to the valleys, has spectrum purely discrete with finite multiplicity. Remarkably, this is in contrast to the case of the unrestricted space, where it is well known that the spectrum comprises a continuous segment. We discuss the relation of our work with the general ground state problem and the question of confinement in models with strong interactions.
Based on Generalized Bloch equation the trans-series expansion for the phase (exponent) of the ground state density for double-well potential is constructed. It is shown that the leading and next-to-leading semiclassical terms are still defined by the flucton trajectory (its classical action) and quadratic fluctuations (the determinant), respectively, while the the next-to-next-to-leading correction (at large distances) is of non-perturbative nature. It comes from the fact that all flucton plus multi-instanton, instanton-anti-instanton classical trajectories lead to the same classical action behavior at large distances! This correction is proportional to sum of all leading instanton contributions to energy gap.
Nuclear structure models built from phenomenological mean fields, the effective nucleon-nucleon interactions (or Lagrangians), and the realistic bare nucleon-nucleon interactions are reviewed. The success of covariant density functional theory (CDFT) to describe nuclear properties and its influence on Brueckner theory within the relativistic framework are focused upon. The challenges and ambiguities of predictions for unstable nuclei without data or for high-density nuclear matter, arising from relativistic density functionals, are discussed. The basic ideas in building an ab initio relativistic density functional for nuclear structure from ab initio calculations with realistic nucleon-nucleon interactions for both nuclear matter and finite nuclei are presented. The current status of fully self-consistent relativistic Brueckner-Hartree-Fock (RBHF) calculations for finite nuclei or neutron drops (ideal systems composed of a finite number of neutrons and confined within an external field) is reviewed. The guidance and perspectives towards an ab initio covariant density functional theory for nuclear structure derived from the RBHF results are provided.
The fission process is a fascinating phenomenon in which the atomic nucleus, a compact self-bound mesoscopic system, undergoes a spontaneous or induced quantum transition into two or more fragments. A predictive, accurate and precise description of nuclear fission, rooted in a fundamental quantum many-body theory, is one of the biggest challenges in science. Current approaches assume adiabatic motion of the system with internal degrees of freedom at thermal equilibrium. With parameters adjusted to data, such modelling works well in describing fission lifetimes, fragment mass distributions, or their total kinetic energies. However, are the assumptions valid? For the fission occurring at higher energies and/or shorter times, the process is bound to be non-adiabatic and/or non-thermal. The vision of this project is to go beyond these approximations, and to obtain a unified description of nuclear fission at varying excitation energies. The key elements of this project are the use of nuclear density functional theory with novel, nonlocal density functionals and innovative high-performance computing techniques. Altogether, the project aims at better understanding of nuclear fission, where slow, collective, and semi-classical effects are intertwined with fast, microscopic, quantum evolution.
The nonrelativistic reduction of the self-consistent covariant density functional theory is realized for the first time with the similarity renormalization group (SRG) method. The reduced nonrelativistic Hamiltonian and densities are calculated by solving the corresponding flow equations with a novel expansion in terms of the inverse of the Dirac effective mass. The efficiency and accuracy of this newly proposed framework have been demonstrated for several typical spherical nuclei. It is found that the exact solutions of the total energies, traces of vector and scalar densities, and the root-mean-square radii are reproduced quite well for all nuclei. This allows one to directly compare and bridge the relativistic and nonrelativistic nuclear energy density functional theories in the future.
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