Frequency sum rules are derived in extended quantum systems of non relativistic fermions from a minimal set of assumptions on dynamics in infinite volume, for ground and thermal states invariant under space translations or a lattice subgroup. For the jellium Coulomb model, they imply the one point result for the plasmon energy spectrum in the zero momentum limit. In general, the density waves energy spectrum is shown to converge, in the limit of large wavelenght, to a point measure at zero frequency, for any number of fermion fields and potentials with integrable second derivatives. For low momentum, ~ k^2 for potentials V with r^2 d_i d_j V integrable, ~ k^{a-d+2} for potentials decaying at infinity as 1/r^a, d-2 < a < d, d the space dimensions. For one component models with short range interactions, the fourth momentum of the frequency is expressed, at lowest order in k, purely in terms of the three point correlation function of the density.
Frequency sum rules are derived in extended quantum systems of non relativistic fermions from a minimal set of assumptions on dynamics in infinite volume, for ground and thermal states invariant under space translations or a lattice subgroup. For the jellium Coulomb model, they imply the one point result for the plasmon energy spectrum in the zero momentum limit. In general, the density waves energy spectrum is shown to converge, in the limit of large wavelenght, to a point measure at zero frequency, for any number of fermion fields and potentials with integrable second derivatives. For low momentum, ~ k^2 for potentials V with r^2 d_i d_j V integrable, ~ k^{a-d+2} for potentials decaying at infinity as 1/r^a, d-2 < a < d, d the space dimensions. For one component models with short range interactions, the fourth momentum of the frequency is expressed, at lowest order in k, purely in terms of the three point correlation function of the density.
Frequency sum rules are derived in extended quantum systems of nonrelativistic fermions from a minimal set of assumptions on dynamics in infinite volume, for ground and thermal states invariant under space translations or a lattice subgroup. For the jellium Coulomb model, they imply the one point result for the plasmon energy spectrum in the zero momentum limit. In the presence of a uniform magnetic field, they allow for a rigorous derivation of the gyroplasmon spectrum, as a pure point spectrum, in the same limit. For all potentials with integrable second derivatives, the density wave energy spectrum is shown to converge, in the limit of large wavelength, to a point measure at zero frequency, for any number of fermion fields. ⟨ω2(k)⟩∼k2 for potentials with r2∂i∂jV integrable, ⟨ω2(k)⟩∼kα−d+2 for potentials decaying at infinity as 1/rα, d−2<α<d, d the space dimensions. For one component models with short range interactions, the fourth moment of the frequency is derived and expressed, at lowest order in k, in terms of the three point correlation function of the density.
The momentum density, n(k) of interacting many-body Fermionic systems is studied (for k>kF) using examples of several well-known two-body interaction models. It is shown that n(k) can be approximated by a zero-range model for momenta k less than about 0.1/re, where re the effective range. If the scattering length is large and one includes the effects of a fixed value of re≠0, n(k) is almost universal for momenta k up to about 2/re. However, n(k) can not be approximated by a zero-range model for momenta k greater than about 1/(are2)1/3, where a is the scattering length, and if one wishes to maintain a sum rule that relates the energy of a two component Fermi-gas to an integral involving the density. We also show that the short separation distance, s, behavior of the pair density varies as s6.
We derive explicit results from an exact expression for fermion loops with an arbitrary number of density vertices in two dimensions at zero temperature, which has been obtained recently by Feldman et al. [in Singularities, edited by G. M. Greuel (Birkhaeuser, Basel, 1998)]. The 3-loop is an elementary function of the three external momenta and frequencies, and the N-loop can be expressed as a linear combination of 3-loops with coefficients that are rational functions of momenta and frequencies. We show that the divergencies of single loops for low energy and small momenta cancel each other when loops with permutated external variables are summed. The symmetrized N-loop, i.e., the connected N-point density-correlation function of the Fermi gas, does not diverge for low energies and small momenta. In the dynamical limit, where momenta scale to zero at fixed finite energy variables, the symmetrized N-loop vanishes as the $(2N\ensuremath{-}2)\mathrm{th}$ power of the scale parameter.
We derive exact general relations between various observables for N spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range re in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading 1/k6 tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-1/a and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.
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.
Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box.We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles.Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps).In non-interacting systems, the characteristic length scale of the oscillations is set by (2k F ) -1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wignercrystal oscillations of period (4k F ) -1 .Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix.While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number.We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface.In addition, we determine the spatially varying on-site density-density correlation, which in turn yields Tan's contact density and, upon integration, Tan's contact.As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.