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arXiv:2402.02206 · Journal of Physics A Mathematical and Theoretical

Preserving the Hermiticity of the one-body density matrix for a non-interacting Fermi gas

L. M. Farrell, D Eaton, P. Chitnelawong, K. Bencheikh, Brandon P. van Zyl

Published Aug 28, 2024• 0 citations
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Abstract

Abstract The one-body density matrix (ODM) for a zero temperature non-interacting Fermi gas can be approximately obtained in the semiclassical regime through different ℏ -expansion techniques. One would expect that each method of approximating the ODM should yield equivalent density matrices which are both Hermitian and idempotent to any order in ℏ . However, the Kirzhnits and Wigner–Kirkwood methods do not yield these properties, while the Grammaticos–Voros method does. Here we show explicitly, for arbitrary d-dimensions through an appropriate change into symmetric coordinates, that each method is indeed identical, Hermitian, and idempotent. This change of variables resolves the inconsistencies between the various methods, showing that the non-Hermitian and non-idempotent behavior of the Kirzhnits and Wigner–Kirkwood methods is an artifact of performing a non-symmetric truncation to the semiclassical ℏ -expansions. Our work also provides the first explicit derivation of the d-dimensional Grammaticos–Voros ODM, originally proposed by Redjati et al (2019 J. Phys. Chem. Solids 134 313–8) based on their d = 1 , 2 , 3 , 4 expressions.

DOI
10.1088/1751-8121/ad74be

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Abstract The one-body density matrix (ODM) for a zero temperature non-interacting Fermi gas can be approximately obtained in the semiclassical regime through different ℏ -expansion techniques. One would expect that each method of approximating the ODM should yield equivalent density matrices which are both Hermitian and idempotent to any order in ℏ . However, the Kirzhnits and Wigner–Kirkwood methods do not yield these properties, while the Grammaticos–Voros method does. Here we show explicitly, for arbitrary d-dimensions through an appropriate change into symmetric coordinates, that each method is indeed identical, Hermitian, and idempotent. This change of variables resolves the inconsistencies between the various methods, showing that the non-Hermitian and non-idempotent behavior of the Kirzhnits and Wigner–Kirkwood methods is an artifact of performing a non-symmetric truncation to the semiclassical ℏ -expansions. Our work also provides the first explicit derivation of the d-dimensional Grammaticos–Voros ODM, originally proposed by Redjati et al (2019 J. Phys. Chem. Solids 134 313–8) based on their d = 1 , 2 , 3 , 4 expressions.
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For an ideal D-dimensional Fermi gas under generic external confinement we derive the correcting coefficient (D−2)/3D(D-2)/3D(D−2)/3D of the von Weizsacker term in the kinetic energy density. To obtain this coefficient we use the Kirzhnits semiclassical expansion of the number operator up to the second order in the Planck constant ℏ\hbarℏ. Within this simple and direct approach we determine the differential equation of the density profile and the density functional of the Fermi gas. In the case D=2 we find that the Kirzhnits gradient corrections vanish to all order in ℏ\hbarℏ.
Forms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a one-electron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy V(r) is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution Vxc(r) to V(r) is expressed in terms of first- and second-order density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the first-order density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.
The equivalence in one-electron quantum baths between the practical implementation of density matrix embedding theory (DMET) and the more recent Householder-transformed density matrix functional embedding theory has been shown previously in the standard but special case where the reference full-size (one-electron reduced) density matrix, from which the bath is constructed, is idempotent [S. Yalouz et al., J. Chem. Phys. 157, 214112 (2022)]. We prove mathematically that the equivalence remains valid when the density matrix is not idempotent anymore, thus allowing for the construction of correlated (one-electron) quantum baths. A density-matrix functional exactification of DMET is derived within the present unified quantum embedding formalism. Numerical examples reveal that the embedding cluster can be quite sensitive to the level of density-matrix functional approximation used for computing the reference density matrix.
For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength λ\lambdaλ. This result permits one to generate an analogous λ\lambdaλ-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type exp⁡(−2x2/λ2)\exp(-2x^2/\lambda^2)exp(−2x2/λ2) (xxx denotes the distance from the wall boundary) which are non-analytic in λ\lambdaλ. Under some condition, the analyticity in λ\lambdaλ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of λ\lambdaλ, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the ν\nuν-dimensional (integer ν≥2\nu\ge 2ν≥2) space.
The single particle reduced density matrices for fermions in a linear potential in one and three dimensions are derived. An idempotent reduced density matrix for a nonuniform field with a single turning point is obtained. The density expression contains the contributions from the direct path and the path with one turning point. When applied to a potential well the corrections from the long-time paths are taken from the theory of Light and Yuan. Only classical quantities of momentum, time, and action enter in the final expression. The density distribution for the nonclassical region is more accurate than that given by the formula of Light and Yuan.