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arXiv:2308.08568

Gas in external fields: the weird case of the logarithmic trap

L. Ferrari

Published Aug 15, 2023• 0 citations
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Abstract

The effects of an attractive logarithmic potential $u_0\ln(r/r_0)$ on a gas of $N$ non interacting particles (Bosons or Fermions), in a box of volume $V_D$, are studied in $D=2,3$ dimensions. The unconventional behavior of the gas challenges the current notions of thermodynamic limit and size independence. When $V_D$ and $N$ diverge, with finite density $N/V_D<\infty$ and finite trap strength $u_0>0$, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at \emph{any} temperature. If, instead, $N/V_D\rightarrow0$, there exists a critical temperature $T_c$, such that the gas remains in the ground state at any $T

DOI
10.48550/arxiv.2308.08568

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The effects of an attractive logarithmic potential u0ln⁡(r/r0)u_0\ln(r/r_0)u0​ln(r/r0​) on a gas of NNN non interacting particles (Bosons or Fermions), in a box of volume VDV_DVD​, are studied in D=2,3D=2,3D=2,3 dimensions. The unconventional behavior of the gas challenges the current notions of thermodynamic limit and size independence. When VDV_DVD​ and NNN diverge, with finite density N/VD<∞N/V_D<\inftyN/VD​<∞ and finite trap strength u0>0u_0>0u0​>0, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at \emph{any} temperature. If, instead, N/VD→0N/V_D\rightarrow0N/VD​→0, there exists a critical temperature TcT_cTc​, such that the gas remains in the ground state at any $T
The effects of an attractive logarithmic potential u(r)=u0ln⁡(r/r0) (u0>0 being the trap strength and r0 an arbitrary length scale) on a gas of N non interacting particles (Bosons or Fermions), in a box of volume VD, are studied in D = 2, 3 dimensions. The unconventional behaviour of the gas challenges the current notions of thermodynamic limit and size independence. When VD and N diverge, with finite density N/VD<∞ and finite u0, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at any temperature. If, instead, N/VD→0, there exists a critical temperature Tc, such that the gas remains in the ground state at any T<Tc, and ‘evaporates’ above, in a non-equilibrium state of borderless diffusion. For the gas to exhibit a conventional Bose–Einstein condensation (BEC) or a finite Fermi level, the strength u0 must vanish with VD→∞, according to a complicated exponential relationship, as a consequence of the exponentially increasing density of states, specific of the logarithmic trap.
We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross-Pitaevskii scaling included, a less-than-100% condensation into a one-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.
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.
Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin-1∕21∕21∕2 fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where E0{E}^{0}E0 is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
We consider a gas mixture consisting of spinless fermions and bosons in one dimension interacting via a repulsive [Formula: see text]-function potential. Bosons and fermions are assumed to have equal masses and the interaction strength between bosons and among bosons and fermions is the same. Using the Bethe ansatz solution of the model, we study the ground state properties, the dressed energy potentials for the two bands of rapidities, the elementary particle and hole excitations, the thermodynamics, the finite size corrections to the ground state energy leading to the conformal towers, and the asymptotic behavior at large distances of some relevant correlation functions. The low-energy excitations of the system form a two-component Luttinger liquid. In an elongated optical trap the gas phase separates as a function of the distance from the center of the trap.