Ground-State Static Correlation Functions of the Lieb–Liniger Model

Jean-Sébastien Caux, Pasquale Calabrese

Julián Rincón, Martin Ganahl, Guifré Vidal

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 $K\ensuremath{\in}(0,1]$.

Charles Radin

E. J. K. P. Nandani, Xi‐Wen Guan

The Lieb–Liniger model is a prototypical integrable model and has been turned into the benchmark physics in theoretical and numerical investigations of low-dimensional quantum systems. In this note, we present various methods for calculating local and nonlocal M -particle correlation functions, momentum distribution, and static structure factor. In particular, using the Bethe ansatz wave function of the strong coupling Lieb–Liniger model, we analytically calculate the two-point correlation function, the large moment tail of the momentum distribution, and the static structure factor of the model in terms of the fractional statistical parameter α = 1−2/ γ , where γ is the dimensionless interaction strength. We also discuss the Tan’s adiabatic relation and other universal relations for the strongly repulsive Lieb–Liniger model in terms of the fractional statistical parameter.

Jacopo De Nardis, Miłosz Panfil

We study the density-density correlation function of the 1D Lieb–Liniger model and obtain an exact expression for the small momentum limit of the static correlator in the thermodynamic limit. We achieve this by summing exactly over the relevant form factors of the density operator in the small momentum limit. The result is valid for any eigenstate, including thermal and non-thermal states. We also show that the small momentum limit of the dynamic structure factors obeys a generalized detailed balance relation valid for any equilibrium state.