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Journal of Physics Materials

Characteristic features of strong correlation: lessons from a 3-fermion one-dimensional harmonic trap

Victor Caliva, Johanna I. Fuks

Published Jul 1, 2024• 0 citations
Publisher

Abstract

Abstract Many quantum phenomena responsible for key applications in material science and quantum chemistry arise in the strongly correlated regime. This is at the same time, a costly regime for computer simulations. In the limit of strong correlation analytic solutions exist, but as we move away from this limit numerical simulation are needed, and accurate quantum solutions do not scale well with the number of interacting particles. In this work we propose to use few-particle harmonic traps in combination with twisted light as a quantum emulator to investigate the transition into a strongly-correlated regime. Using both analytic derivations and numerical simulations we generalize previous findings on 2 Coulomb interacting fermions trapped in a one-dimensional harmonic trap to the case of 3 fermions. The 4 signatures of strong correlation we have identified in the one-dimensional harmonic trap are: (i) the ground state density is highly localized around N equilibrium positions, where N is the number of particles, (ii) the symmetric and antisymmetric ground state wavefunctions become degenerate, (iii) the von Neumann entropy grows, (iv) the energy spectrum is fully characterized by N normal modes or less. Our findings describe the low-energy behavior of electrons in quantum wires and ions in Paul traps. Similar features have also been reported for cold atoms in optical lattices.

DOI
10.1088/2515-7639/ad63cb

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Abstract Many quantum phenomena responsible for key applications in material science and quantum chemistry arise in the strongly correlated regime. This is at the same time, a costly regime for computer simulations. In the limit of strong correlation analytic solutions exist, but as we move away from this limit numerical simulation are needed, and accurate quantum solutions do not scale well with the number of interacting particles. In this work we propose to use few-particle harmonic traps in combination with twisted light as a quantum emulator to investigate the transition into a strongly-correlated regime. Using both analytic derivations and numerical simulations we generalize previous findings on 2 Coulomb interacting fermions trapped in a one-dimensional harmonic trap to the case of 3 fermions. The 4 signatures of strong correlation we have identified in the one-dimensional harmonic trap are: (i) the ground state density is highly localized around N equilibrium positions, where N is the number of particles, (ii) the symmetric and antisymmetric ground state wavefunctions become degenerate, (iii) the von Neumann entropy grows, (iv) the energy spectrum is fully characterized by N normal modes or less. Our findings describe the low-energy behavior of electrons in quantum wires and ions in Paul traps. Similar features have also been reported for cold atoms in optical lattices.
The transition into a strongly-correlated regime of 3 fermions trapped in a one-dimensional harmonic potential is investigated. This interesting, but little-studied system, allows us to identify characteristic features of the regime, some of which are also present in strongly-correlated materials relevant to the industry. Furthermore, our findings describe the behavior of electrons in quantum dots, ions in Paul traps, and even fermionic atoms in one-dimensional optical lattices. Near the ground state, all these platforms can be described as fermions trapped in a harmonic potential. The correlation regime can be controlled by varying the natural frequency of the trapping potential, and to probe it, we propose to use twisted light. We identify 4 signatures of strong correlation in the one-dimensional 3-fermion trap, which are likely to be present for any number N of trapped fermions: i) the ground state density is strongly localized with N maximally separated peaks (Wigner Crystal) ii) the symmetric and antisymmetric ground state wavefunctions become degenerate (bosonization) iii) the von Neumann entropy grows, iv) the energy spectrum is fully characterized by N normal modes or less.
We conduct a theoretical study of SU(N\mathrm{SU}(NSU(N) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the SU(N\mathrm{SU}(NSU(N) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these NNN-component systems. Lastly, we consider balanced SU(N\mathrm{SU}(NSU(N) Fermi gases that have an equal number of particles in each spin state for N=2,3,4N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4N=2,3,4. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of NNN-component fermions may be accurately inferred from the corresponding few-body systems of NNN distinguishable particles.
We explore the potential of twisted light as a tool to unveil many-body effects in parabolically confined systems. According to the Generalized Kohn Theorem, the dipole response of such a multi-particle system to a spatially homogeneous probe is indistinguishable from the response of a system of non-interacting particles. Twisted light however can excite internal degrees of freedom, resulting in the appearance of new peaks in the even multipole spectrum which are not present when the probe is a plane wave. We also demonstrate the ability of the proposed twisted light probe to capture the transition of interacting fermions into a strongly correlated regime in a one-dimensional harmonic trap. We report that by suitable choice of the probe's parameters, the transition into a strongly correlated phase manifests itself as an approach and ultimate superposition of peaks in the second order quadrupole response. These features, observed in exact calculations for two electrons, are reproduced in adiabatic Time Dependent Density Functional Theory simulations.
A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.
A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.
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.