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arXiv:1508.04779 · Physical Review B

Lieb-Liniger model with exponentially decaying interactions: A continuous matrix product state study

Julián Rincón, Martin Ganahl, Guifré Vidal

Published Sep 2, 2015• 29 citations
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Abstract

The Lieb-Liniger model describes one-dimensional bosons interacting through a repulsive contact potential. In this work, we introduce an extended version of this model by replacing the contact potential with a decaying exponential. Using the recently developed continuous matrix product states techniques, we explore the ground-state phase diagram of this model by examining the superfluid and density correlation functions. At weak coupling superfluidity governs the ground state, in a similar way as in the Lieb-Liniger model. However, at strong coupling quasicrystal and super--Tonks-Girardeau regimes are also found, which are not present in the original Lieb-Liniger case. Therefore the presence of the exponentially decaying potential leads to a superfluid/super--Tonks-Girardeau/quasicrystal crossover, when tuning the coupling strength from weak to strong interactions. This corresponds to a Luttinger liquid parameter in the range $K\ensuremath{\in}(0,\ensuremath{\infty})$, in contrast with the Lieb-Liniger model, where $K\ensuremath{\in}[1,\ensuremath{\infty})$, and the screened long-range potential, where $K\ensuremath{\in}(0,1]$.

DOI
10.1103/physrevb.92.115107

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The Lieb-Liniger model describes one-dimensional bosons interacting through a repulsive contact potential. In this work, we introduce an extended version of this model by replacing the contact potential with a decaying exponential. Using the recently developed continuous matrix product states techniques, we explore the ground-state phase diagram of this model by examining the superfluid and density correlation functions. At weak coupling superfluidity governs the ground state, in a similar way as in the Lieb-Liniger model. However, at strong coupling quasicrystal and super--Tonks-Girardeau regimes are also found, which are not present in the original Lieb-Liniger case. Therefore the presence of the exponentially decaying potential leads to a superfluid/super--Tonks-Girardeau/quasicrystal crossover, when tuning the coupling strength from weak to strong interactions. This corresponds to a Luttinger liquid parameter in the range K\ensuremath∈(0,\ensuremath∞)K\ensuremath{\in}(0,\ensuremath{\infty})K\ensuremath∈(0,\ensuremath∞), in contrast with the Lieb-Liniger model, where K\ensuremath∈[1,\ensuremath∞)K\ensuremath{\in}[1,\ensuremath{\infty})K\ensuremath∈[1,\ensuremath∞), and the screened long-range potential, where K\ensuremath∈(0,1]K\ensuremath{\in}(0,1]K\ensuremath∈(0,1].
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
The large-distance behavior of the density-density correlation function in the Lieb-Liniger model at finite temperature is investigated by means of the recently derived nonlinear integral equations characterizing the correlation lengths. We present extensive numerical results covering all the physical regimes from weak to strong interaction and all temperatures. We find that the leading term of the asymptotic expansion becomes oscillatory at a critical temperature which decreases with the strength of the interaction. As we approach the Tonks-Girardeau limit the asymptotic behavior becomes more complex with a double crossover of the largest and next-largest correlation lengths. The crossovers exist only for intermediate couplings and vanish for $\ensuremath{\gamma}=0$ and $\ensuremath{\infty}$.
We investigate the ground-state properties of the quasi-one-dimensional dipolar gases using continuous matrix product states techniques. Making use of the first- and second-order correlation functions, we find that the system supports the superfluid, super-Tonks-Girardeau, and quasicrystal phases according to the Luttinger liquid theory. We also map out the phase diagram on the parameter plane consisting the contact and dipolar interaction strengths. Furthermore, we compute the Luttinger parameter, the structure factor, and the momentum distribution of the system. Finally, we show that the predicted dipolar effect can potentially be observed in quasi-one-dimensional gases of polar molecules.
We investigate the ground-state properties of the quasi-one-dimensional dipolar gases using continuous matrix product states techniques. Making use of the first- and second-order correlation functions, we find that the system supports the superfluid, super-Tonks-Girardeau, and quasicrystal phases according to the Luttinger liquid theory. We also map out the phase diagram on the parameter plane consisting the contact and dipolar interaction strengths. Furthermore, we compute the Luttinger parameter, the structure factor, and the momentum distribution of the system. Finally, we show that the predicted dipolar effect can potentially be observed in quasi-one-dimensional gases of polar molecules.
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δδδ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length aaa and the radius rrr of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g∼a/r2g \sim a/r^2g∼a/r2 is derived. Our bounds are uniform in ggg in the whole parameter range 0≤g≤∞0\leq g\leq \infty0≤g≤∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼r−2\sim r^{-2}∼r−2 above the ground state energy.