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

Abstract

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 admits 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 purely by quantum-statistical attraction and coined quantum-statistical condensate, which is 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.

Approaching off-diagonal long-range order for 1+1 -dimensional relativistic anyons

“…Moreover, it follows from Sec. II A that the fields in (17) change the charge in units of e in (13). This motivates interpreting e as anyonic charge.…”

“…The Hamiltonian of the model is obtained by making H 0 gauge invariant and adding gauge dynamics. The former task requires the usual substitution for gaugecovariant derivatives, i.e., i∂ x → i∂ x −eA(x), where e is the anyonic charge in (13). However, this is not enough due to the singular nature of the fields: One must also replace N[…”

Quantum entanglement in one-dimensional anyons

“…The problem can be solved by prescribing appropriate boundary conditions along the diagonal. We choose the free particle Hamiltonian for the system, also studied by Posske et al [21],…”

“…As already mentioned in the Introduction, we find it useful to recast the above results in the language of second quantization, as was done in [18]. We use the following generalised η-dependent algebra for the second quantized creation operator Ψ † (x), and annihilation operator Ψ(x) of the anyon fields…”

Bosonization study of a generalized statistics model with four Fermi points

“…The possibility of identical particles having generalized statistics in one dimension has been studied extensively over many years. Such generalizations can be introduced in many different ways, for instance, by modifying the conditions on the wave function and its derivative at the points when two of the particles have the same coordinate, modifying the commutation relations between the creation and annihilation operators in a second-quantized formalism, or modifying the form of the exclusion principle [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] . Several theoretical proposals have been made for realizing generalized statistics in one dimension [16][17][18][19][20] .…”