Second quantization of Leinaas-Myrheim anyons in one dimension and their relation to the Lieb-Liniger model

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.

Figure 1

Examples of composite anyons described by clusters $\mathbf{\mu }$, called strings in the context of the Lieb-Liniger model. These are the fundamental building blocks of the anyonic wave functions and can be conceived as individual particles. For clusters of more than one anyon, $\eta <0$ is implicit. (a) Single anyon; (b) two-anyon bound state; (c) maximally bound cluster of $n$ anyons: the quantum-statistical condensate.

Figure 2

Observables for confined anyons. The anyonic properties for $0<\eta <\infty$ continuously interpolate between bosons ($\eta =0$) and fermions ($\eta =\infty$). The statistical condensate forms at $\eta <0$. (a) Discrete energy spectrum of two anyons. To depict the full range of $\eta$, we plot against ${\varphi }_{\eta }(1/L)$. The two-anyon bound state and its excitations emerge for negative $\eta$. (b) Momentum density at zero temperature in the limit of infinitely many particles (numerical calculations for $n=512$ anyons, where the curves almost converge to the limit $n\to \infty$). (c) Finite-size oscillations of the particle density $\rho$ for four anyons.

Figure 3

Geometric interpretation of the statistical phase of two clusters, each consisting of two anyons. The radius of the circle denotes the relative momentum between clusters $K_2-K_1$. The statistical angle is obtained by adding up three summands. Two of these summands are the normal statistical angle ${\varphi }_{\eta }$, and the third summand is the statistical angle of the doubled statistical parameter ${\varphi }_{2\eta }$. The statistical parameter $\eta$ appears in the lengths of the drawn tangential segments.