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arXiv:1302.0013 · Physical Review Letters

Ground-State Energies of Spinless Free Fermions and Hard-Core Bosons

Wenxing Nie, Hosho Katsura, Masaki Oshikawa

Published Sep 3, 2013• 17 citations
arXivHTMLPDF

Abstract

We compare the ground-state energies of bosons and fermions with the same form of the Hamiltonian. If both are noninteracting, the ground-state energy of bosons is always lower, owing to Bose-Einstein condensation. However, the comparison is nontrivial when bosons do interact. We first prove that, when the hopping is unfrustrated (all the hopping amplitudes are non-negative), hard-core bosons still must have a lower ground-state energy than fermions. If the hopping is frustrated, bosons can have a higher ground-state energy than fermions. We prove rigorously that this inversion indeed occurs in several examples.

DOI
10.1103/physrevlett.111.100402

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We compare the ground-state energies of bosons and fermions with the same form of the Hamiltonian. If both are noninteracting, the ground-state energy of bosons is always lower, owing to Bose-Einstein condensation. However, the comparison is nontrivial when bosons do interact. We first prove that, when the hopping is unfrustrated (all the hopping amplitudes are non-negative), hard-core bosons still must have a lower ground-state energy than fermions. If the hopping is frustrated, bosons can have a higher ground-state energy than fermions. We prove rigorously that this inversion indeed occurs in several examples.
We study the connections among particle statistics, frustration, and ground-state energy in quantum many-particle systems. In the absence of interaction, the influence of particle statistics on the ground-state energy is trivial: the ground-state energy of noninteracting bosons is lower than that of free fermions because of Bose-Einstein condensation and Pauli exclusion principle. In the presence of hard-core or other interaction, however, the comparison is not trivial. Nevertheless, the ground-state energy of hard-core bosons is proved to be lower than that of spinless fermions, if all the hopping amplitudes are non-negative. The condition can be understood as the absence of frustration among hoppings. By mapping the many-body Hamiltonian to a tight-binding model on a fictitious lattice, we show that the Fermi statistics of the original particles introduces an effective magnetic flux in the fictitious lattice. The latter can be effectively regarded as a kind of frustration since it leads to a destructive interference among different paths along which a single particle is propagating. If we introduce hopping frustration, the hopping frustration is expected to compete with the effective frustration due to the Fermi statistics, leading to the possibility that the ground-state energy of hard-core bosons can be higher than that of fermions. We present several examples, in which the ground-state energy of hard-core bosons is proved to be higher than that of fermions due to the hopping frustration. The basic ideas were reported in a previous paper [W.-X. Nie, H. Katsura, and M. Oshikawa, Phys. Rev. Lett. 111, 100402 (2013)]; more details and several extensions, including one to the spinful case, are discussed in this paper.
We study the connections among particle statistics, frustration, and groundstate energy, in many-particle systems.Statistics of identical particles is one of the most fundamental concepts, which pervades all of quantum mechanics.A consequence of the particle statistics appears in the ground-state energy.We compare the ground-state energies of bosons and fermions with the same Hamiltonian.In noninteracting systems, the influence of particle statistics on the ground-state energy is trivial: the ground-state energy of noninteracting bosons is lower than that of free fermions because of Bose-Einstein condensation (BEC) and Pauli exclusion principle.The relation that bosons have a lower ground-state energy than fermions is described as natural inequality in this thesis.However, the comparison of the ground-state energies of bosons and fermions is not trivial in the presence of interaction, because the simple argument based on the perfect BEC breaks down.In a system of interacting bosons, it is in fact already a nontrivial question whether the BEC actually takes place.In strongly correlated systems, the influence of particle statistics on the ground-state energy is still a relatively unexplored area.We have found a sufficient condition for the natural inequality to hold for spinless and spinful cases respectively, without relying on the occurrence of BEC.That is, if all the hopping amplitudes are nonnegative, the ground-state energy of hard-core bosons is still lower than that of fermions.The same argument implies that, once we relax the condition of nonnegative hopping amplitudes, it is possible to reverse the inequality so that the ground-state energy of bosons could be higher than that of fermions.By relaxing the condition, we indeed have found several concrete examples in which the ground-state energy of hard-core bosons is proved to be higher than that of fermions.Many of the examples are even proved rigorously in the thermodynamic limit.Our study leads to a novel physical understanding of the effects of particle statistics, in terms of frustration in quantal phase.This is more general than the picture based on the perfect BEC, and is indeed applicable to systems with interaction.We can map a quantum many-particle problem to a single-particle problem on a fictitious lattice in higher dimensions.When all the hopping amplitudes are nonnegative and the particles are bosons, the corresponding single-particle problem also has only nonnegative hopping amplitudes.In such a case, there is no frustration in the quantal phase of the wavefunction.On the other hand, Fermi statistics of the original particles gives an effective magnetic flux in the corre-sponding single-particle problem.This implies a frustration in the phase of the wavefunction, which can be regarded as "statistical frustration": it is induced by the Fermi statistics and it leads to a destructive interference among propagation along different paths.In the presence of a non-vanishing flux in the original many-body problem, we observe that there is also a magnetic flux in the corresponding single-particle problem, inducing a frustration among quantal phases, which we name hopping frustration.In the original many-particle problem, the statistical frustration appears rather differently from the hopping frustration.However, upon mapping to the single-particle problem on the fictitious lattice, both hopping frustration and statistical frustration are represented by non-vanishing flux in the fictitious lattice.This provides a unified understanding of frustration.Based on the unified understanding, we find the mechanism as to why the ground-state energy of hard-core bosons can be higher than that of fermions.For many-body bosonic systems, in which there is no statistical frustration, introduction of hopping frustration will not decrease the ground-state energy.This is known as Simon's universal diamagnetism of bosons.On the other hand, in many-body fermionic systems, where statistical frustration already exists, hopping frustration if introduced, is expected to compete with statistical frustration and sometimes partially cancel with each other, resulting in the reversed inequality between the ground-state energies of the hard-core bosons and fermions.
The one-particle density matrices for hard-core bosons in a one-dimensional harmonic trap are computed numerically for systems with up to 160 bosons. Diagonalization of the density matrix shows that the many-body ground state is not Bose-Einstein condensed. The ground-state occupation, the amplitude of the lowest natural orbital, and the zero momentum peak height scale as powers of the particle number, and the corresponding exponents are related to each other. Close to its diagonal, the density matrix for hard-core bosons is similar to that of noninteracting fermions.
We investigate the consequences of correlated hopping on the ground state properties of hard-core bosons on a square lattice as revealed by extensive exact diagonalizations and quantum Monte Carlo simulations. While for noninteracting hard-core bosons the effective attraction induced by the correlated hopping leads to phase separation at low density, we show that a modest nearest-neighbor repulsion suppresses phase separation, leading to a remarkable low-density pairing phase with no single particle Bose-Einstein condensation but long-range two-particle correlations, signaling a condensation of pairs. We also explain why the unusual properties of the pairing phase are a real challenge for standard one-worm quantum Monte Carlo simulations.
We investigate the energy spectrum of ground state and quasi-particle excitation spectrum of hard-core bosons, which behave very much like spinless noninteracting fermions, in optical lattices by means of the perturbation expansion and Bogoliubov approach. The results show that the energy spectrum has a single band structure, and the energy is lower near zero momentum; the excitation spectrum gives corresponding energy gap, and the system is in Mott-insulating state at Tonks limit. The analytic result of energy spectrum is in good agreement with that calculated in terms of Green's function at strong correlation limit.