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arXiv:cond-mat/0410683 · Physical Review Letters

Fermionization in an Expanding 1D Gas of Hard-Core Bosons

Marcos Rigol, Alejandro Muramatsu

Published Jun 22, 2005• 141 citations
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Abstract

We show by means of an exact numerical approach that the momentum distribution of a free expanding gas of hard-core bosons on a one-dimensional lattice approaches that of noninteracting fermions, acquiring a Fermi edge. Yet there is a power-law decay of the one-particle density matrix ${\ensuremath{\rho}}_{x}\ensuremath{\sim}1/\sqrt{x}$, as usual for hard-core bosons in the ground state, which accounts for a large occupation of the lowest natural orbitals for all expansion times. The fermionization of the momentum distribution function, which is not observed in equilibrium, is analyzed in detail.

DOI
10.1103/physrevlett.94.240403

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We show by means of an exact numerical approach that the momentum distribution of a free expanding gas of hard-core bosons on a one-dimensional lattice approaches that of noninteracting fermions, acquiring a Fermi edge. Yet there is a power-law decay of the one-particle density matrix \ensuremathρx\ensuremath∼1/x{\ensuremath{\rho}}_{x}\ensuremath{\sim}1/\sqrt{x}\ensuremathρx​\ensuremath∼1/x​, as usual for hard-core bosons in the ground state, which accounts for a large occupation of the lowest natural orbitals for all expansion times. The fermionization of the momentum distribution function, which is not observed in equilibrium, is analyzed in detail.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
We develop an exact approach to study the quench dynamics of hard-core bosons initially in thermal equilibrium in one-dimensional lattices. This approach is used to study the sudden expansion of thermal states after confining potentials are switched off. We find that a dynamical fermionization of the momentum distribution occurs at all temperatures. This phenomenon is studied for low initial site occupations, for which the expansion of the cloud is self-similar. In this regime, the occupation of the natural orbitals allows one to distinguish hard-core bosons from noninteracting fermions. We also study the free expansion of initial Mott insulating domains at finite temperature and show that the emergence of off-diagonal one-body correlations is suppressed gradually with increasing temperature. Surprisingly, the melting of the Mott domain is accompanied by an effective cooling of the system. We explain this phenomenon analytically using an equilibrium description based on an emergent local Hamiltonian.
The momentum distribution of an expanding cloud of one-dimensional hard-core anyons is studied by an exact numerical approach, and shown to become indistinguishable from that of a noninteracting spin-polarized Fermi gas for large enough times (dynamical fermionization). We also consider the expansion of one-dimensional anyons with strongly attractive short-range interactions suddenly released from a parabolic external potential, and find that the momentum distribution approaches that of its dual system, the ideal Bose gas (dynamical bosonization). For both processes the characteristic time scales are identified, and the effect of the initial confinement is analyzed comparing the dynamics associated with both harmonic and hard-wall traps.
Based on an exact treatment of hard-core bosons confined on one-dimensional lattices, we obtain the large distance behavior of the one-particle density matrix, and show how it determines the occupation of the lowest natural orbital in the thermodynamic limit. We also study the occupation ${\ensuremath{\lambda}}_{\ensuremath{\eta}}$ of the natural orbitals for large-$\ensuremath{\eta}$ at low densities. Both quantities show universal behavior independently of the confining potential. Finite-size corrections and the momentum distribution function for finite systems are also analyzed.
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.
Abstract Quasicondensation in one dimension is known to occur for equilibrium systems of hard-core bosons (HCBs) at zero temperature. This phenomenon arises due to the off-diagonal long-range order in the ground state, characterized by a power-law decay of the one-particle density matrix g 1 ( x , y ) ∼ | x − y | − 1 / 2 —a well-known outcome of Luttinger liquid theory. Remarkably, HCBs, when allowed to freely expand from an initial product state (i.e. characterized by initial zero correlation), exhibit quasicondensation and demonstrate the emergence of off-diagonal long-range order during nonequilibrium dynamics. This phenomenon has been substantiated by numerical and experimental investigations in the early 2000s. In this work, we revisit the dynamical quasicondensation of HCBs, providing a fully analytical treatment of the issue. In particular, we derive an exact asymptotic formula for the equal-time one-particle density matrix by borrowing ideas from the framework of quantum Generalized Hydrodynamics. Our findings elucidate the phenomenology of quasicondensation and of dynamical fermionization occurring at different stages of the time evolution, as well as the crossover between the two.