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arXiv:1801.00144

Ground-state energy of one-dimensional free Fermi gases in the thermodynamic limit

Peter Otte, Spitzer, Wolfgang

Published Dec 30, 2017• 1 citation
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Abstract

We study the ground-state energy of one-dimensional, non-interacting fermions subject to an external potential in the thermodynamic limit. To this end, we fix some (Fermi) energy $ν>0$, confine fermions with total energy below $ν$ inside the interval $[-L,L]$ and study the shift of the ground-state energy due to the potential $V$ in the thermodynamic limit $L\to\infty$. We show that the difference $\mathcal{E}_L(ν)$ of the two ground-state energies with and without potential can be decomposed into a term of order one (leading to the Fumi-term) and a term of order $1/L$, which yields the so-called finite size energy. We compute both terms for all possible boundary conditions explicitly and express them through the scattering data of the one-particle Schrödinger operator $-Δ+V$ on $L^2(\mathbb R)$.

DOI
10.48550/arxiv.1801.00144

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We study the ground-state energy of one-dimensional, non-interacting fermions subject to an external potential in the thermodynamic limit. To this end, we fix some (Fermi) energy ν>0ν>0ν>0, confine fermions with total energy below ννν inside the interval [−L,L][-L,L][−L,L] and study the shift of the ground-state energy due to the potential VVV in the thermodynamic limit L→∞L\to\inftyL→∞. We show that the difference EL(ν)\mathcal{E}_L(ν)EL​(ν) of the two ground-state energies with and without potential can be decomposed into a term of order one (leading to the Fumi-term) and a term of order 1/L1/L1/L, which yields the so-called finite size energy. We compute both terms for all possible boundary conditions explicitly and express them through the scattering data of the one-particle Schrödinger operator −Δ+V-Δ+V−Δ+V on L2(R)L^2(\mathbb R)L2(R).
The isotropic scattering phase shift is calculated for nonrelativistic bosons interacting at low energies via an arbitrary finite-range potential in ddd space-time dimensions. Scattering on a $(d\ensuremath{-}1)$-dimensional torus is then considered, and the eigenvalue equation relating the energy levels on the torus to the scattering phase shift is derived. With this technology in hand, and focusing on the case of two spatial dimensions, a perturbative expansion is developed for the ground-state energy of NNN identical bosons which interact via an arbitrary finite-range potential in a finite area. The leading nonuniversal effects due to range corrections and three-body forces are included. It is then shown that the thermodynamic limit of the ground-state energy in a finite area can be taken in closed form to obtain the energy per particle in the low-density expansion by explicitly summing the parts of the finite-area energy that diverge with powers of NNN. The leading and subleading finite-size corrections to the thermodynamic limit equation of state are also computed. Closed-form results---some well known, others perhaps not---for two-dimensional lattice sums are included in an Appendix.
When noninteracting fermions are confined in a DDD-dimensional region of volume O(LD)O({L}^{D})O(LD) and subjected to a continuous (or piecewise-continuous) potential VVV which decays sufficiently fast with distance, in the thermodynamic limit, the ground-state energy of the system does not depend on VVV. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in DDD dimensions. We find that this universality property holds under a quite mild condition on VVV, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.
We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e., in dilute regimes, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size $\ensuremath{\ell}$ keeping the particle number NNN fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-NNN limit keeping the ratio $N/\ensuremath{\ell}$ fixed corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-$\ensuremath{\ell}$ limit. When the trapped system is in the metallic phase, corrections at finite $\ensuremath{\ell}$ are $O({\ensuremath{\ell}}^{\ensuremath{-}1})$ and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as $O({\ensuremath{\ell}}^{\ensuremath{-}1/3})$ only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.
We study the ground state energy of a system of N fermions with two spin states in the large N limit. The particles are placed in an inhomogeneous trapping potential and interact via scaled interactions. We study a dilute limit where the range of the interaction potential is much smaller than the typical inter-particle distance. We show that the energy per particle converges to the Thomas-Fermi energy of the system, with a perturbative term corresponding tot he interaction and exhibiting the scattering length of the potential. The proof is decomposed into two bounds. First, we construct an appropriate test-state to prove the upper bound. Then, we prove the lower bound by the Dyson lemma, which allows us to regularize the interaction potential, and several semi-classical tools.
We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in d -dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting N -Fermion systems. We interpret the decay exponent \gamma in terms of scattering theory and find \gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\mathrm {HS}}^2 , where T_E is the transition matrix at the Fermi energy E . This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352–359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.