Supplement: the Lieb–Liniger model

Fabio Franchini

Friedrich Hübner, Benjamin Doyon

In conventional fluids, it is well known that Euler-scale equations are plagued by ambiguiti es and instabilities. In one dimension smooth initial conditions may develop shocks, and weak solutions, such as for domain wall initial conditions (the paradigmatic Riemann problem of hydrodynamics), are not unique (and must be supplemented by physically motivated conditions such as non-negative entropy production). The absence of shock formation experimentally observed in quasi-one-dimensional cold-atomic gases, which are described by the Lieb-Liniger model, provides perhaps the strongest pointer to a modification of the hydrodynamic equation due to integrability. Generalized hydrodynamics (GHD) is the required hydrodynamic theory, taking into account the infinite number of conserved quantities afforded by integrability. We provide a new quadrature for the GHD equation—a solution in terms of a Banach fixed-point problem where time has been explicitly integrated. The quadrature is an efficient numerical solution tool; and it allows us, in the Lieb-Liniger model, to rigorously show that no shock may appear at all times, and, when combined with recent hydrodynamic fluctuation theories, to obtain new expressions for correlations in nonstationary states, establishing for the first time the presence of discontinuities that are fundamental to the nonequilibrium physics of integrable systems.

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

Pieter Naaijkens

Thore Posske, Björn Trauzettel, Michael Thorwart

In one spatial dimension, anyons in the original description of Leinaas and Myrheim are formally equivalent to locally interacting bosons described by the Lieb-Liniger model. This allows an interesting reinterpretation of interacting bosons in the context of anyons. We elaborate on this parallel, particularly including the many-body bound states from the attractive Lieb-Liniger model. In the anyonic context these bound states are created solely by quantum-statistical attraction and coined the quantum-statistical condensate, which is shown to be more robust than the Bose-Einstein condensate. We introduce the second quantization formalism for the present anyons and construct the generalized Jordan-Wigner transformation that connects them to the bosons of the Lieb-Liniger model.

Anna Vershynina, Élliott H. Lieb