For one-dimensional many-body systems interacting via the \textit{Coulomb force} and with \textit{arbitrary} external potential energy, we derive (\textit{i}) the \textit{node coalescence condition} for the wave function. This condition rigorously proves the following: (\textit{ii}) that the particles satisfy \textit{only} a node coalescence condition; (\textit{iii}) that irrespective of their charge or statistics, the particles cannot coalesce; (\textit{iv}) that the particles cannot cross each other, and must be ordered; (\textit{v}) the particles are therefore distinguishable; (\textit{vi}) as such their statistics are not significant; (\textit{vii}) conclusions similar to those of the spin-statistics theorem of quantum field theory are arrived at via non-relativistic quantum mechanics; (\textit{viii}) the noninteracting system \textit{cannot} be employed as the lowest-order in a perturbation theory of the interacting system. (\textit{ix}) Finally, the coalescence condition for particles with the short-ranged delta-function interaction and \textit{arbitrary} external potential energy, is also derived. These particles can coalesce and cross each other.
For one-dimensional many-body systems interacting via the \textit{Coulomb force} and with \textit{arbitrary} external potential energy, we derive (\textit{i}) the \textit{node coalescence condition} for the wave function. This condition rigorously proves the following: (\textit{ii}) that the particles satisfy \textit{only} a node coalescence condition; (\textit{iii}) that irrespective of their charge or statistics, the particles cannot coalesce; (\textit{iv}) that the particles cannot cross each other, and must be ordered; (\textit{v}) the particles are therefore distinguishable; (\textit{vi}) as such their statistics are not significant; (\textit{vii}) conclusions similar to those of the spin-statistics theorem of quantum field theory are arrived at via non-relativistic quantum mechanics; (\textit{viii}) the noninteracting system \textit{cannot} be employed as the lowest-order in a perturbation theory of the interacting system. (\textit{ix}) Finally, the coalescence condition for particles with the short-ranged delta-function interaction and \textit{arbitrary} external potential energy, is also derived. These particles can coalesce and cross each other.
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.
In this work, we study the statistics of a generic noninteracting many-body fermionic system whose single-particle counterpart has a single-particle mobility edge (SPME). We first prove that the spectrum and the extensive conserved quantities follow the multivariate normal distribution with a vanishing standard deviation $\ensuremath{\sim}O(1/\sqrt{L})$ in the thermodynamic limit, regardless of SPME. Consequently, the theorem rules out an infinite-temperature or high-temperature many-body mobility edge (MBME) for generic noninteracting fermionic systems in any dimension. Further, we also prove that the spectrum of a one-dimensional fermionic many-body system with short-range interactions is qualitatively similar to that of a noninteracting many-body system up to the third-order moment. These results partially explain why neither short-range [Phys. Rev. B 107, 035129 (2023)] nor long-range interacting systems exhibit an infinite-temperature MBME in a one-dimensional system.
Interactions in many-body systems bring about collective phenomena such as superconductivity and magnetism. In many cases, simple mean-field theory provides a basic understanding of these phenomena. In fermion systems in one dimension, however, neither the mean-field theory nor perturbation theory works if it starts from the non-interacting fermions. This is because the interaction effects in one dimension are much stronger than those in higher dimensions. Intuitively speaking, two particles cannot avoid collision in a single-way track in contrast with two and three dimensions. Thus the interaction effects appear in a drastic way in one dimension. Another aspect in one dimension, which overcompensates the difficulty of perturbation and mean-field theories, is that a complete account of interaction effects is possible under certain conditions. The class of systems satisfying such conditions is referred to as exactly solvable. Soon after the establishment of quantum mechanics, Bethe solved exactly the Heisenberg spin model in one dimension [28]. The basic idea of the solution is now called the Bethe ansatz. Since then, theoretical physics in one dimension has developed into a magnificent edifice, including sophisticated mathematical techniques. In many cases, the eigenfunctions derived by the Bethe ansatz consist of plane waves that are defined stepwise for each spatial configuration of particles. Since the coefficients of plane waves depend on the configuration, the property of the wave function cannot be made explicit without detailed knowledge of these coefficients. We mention some of the recent monographs on the Bethe ansatz and its extensions [54, 118, 179]. A comprehensive account on exactly solvable models has recently been given by Sutherland [178].
Time-dependent quantal density-functional theory (Q-DFT) is a description of the s-system of noninteracting fermions with electronic density equivalent to that of Schr\"odinger theory, in terms of fields whose sources are quantum-mechanical expectations of Hermitian operators. The theory delineates and defines the contribution of each type of electron correlation to the local electron-interaction potential ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ of the s system. These correlations are due to the Pauli exclusion principle, Coulomb repulsion, correlation-kinetic, and correlation-current-density effects, the latter two resulting, respectively, from the difference in kinetic energy and current density between the interacting Schr\"odinger and noninteracting systems. We employ Q-DFT to prove the following sum rules and properties of the s system: (i) the components of the potential due to these correlations separately exert no net force on the system; (ii) the torque of the potential is finite and due solely to correlation-current-density effects; (iii) two sum rules involving the curl of the dynamic electron-interaction kernel defined as the functional derivative of ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ are derived and shown to depend on the frequency dependent correlation-current-density effect. Furthermore, via adiabatic coupling constant (\ensuremath{\lambda}) perturbation theory, we prove: (iv) the exchange potential ${\ensuremath{\nu}}_{x}(\mathbf{r},t)$ is the work done in a conservative field representative of Pauli correlations and lowest-order $O(\ensuremath{\lambda})$ correlation-kinetic and correlation-current-density effects; (v) the correlation potential ${\ensuremath{\nu}}_{c}(\mathbf{r},t)$ commences in $O({\ensuremath{\lambda}}^{2}),$ and, at each order, it is the work done in a conservative field representative of Coulomb correlations and correlation-kinetic and correlation-current-density effects; (vi) we derive the integral virial theorem relating ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ to the electron-interaction and correlation-kinetic energy for arbitrary coupling constant strength \ensuremath{\lambda}, and show there are no explicit correlation-current-density contributions to the energy. From this integral virial theorem we (vii) obtain the fully interacting $(\ensuremath{\lambda}=1)$ and exchange-only $(\ensuremath{\lambda}=0)$ integral virial theorems as special cases, the latter showing there is no explicit correlation-kinetic contribution to the exchange energy; and (viii) write expressions for the electron-interaction and correlation-kinetic actions for arbitrary coupling constant \ensuremath{\lambda} in terms of the corresponding fields.
We review one dimensional matrix theory and its variations, collective field theory and quantum phase space description. In the planar limit, these theories become classical and can be easily analyzed. With these descriptions, one dimensional interacting many-body system can be solved exactly when the particle number goes to infinity. As an example, bosonic and two-component fermionic systems with a δ-function interaction are analyzed in detail.