Fermionic long-range correlations realized by particles obeying deformed statistics

Abstract

Deformed exchange statistics is realized in terms of electronic operators. This is employed to rewrite Hubbard type lattice models for particles obeying deformed statistics (we refer to them as deformed models) as lattice models for electrons. The resulting models show up gauge-like modulations in the hopping processes, which induce long-range correlations in the lattice. The conditions for the Bethe ansatz solvability of the latter are interpreted as restrictions imposed on the statistics to be compatible with the Bethe ansatz solvability of the deformed models. It is found that solvable deformed models are not unitarily equivalent to fermionic models if the exchange of particles with the same spin-orientations is deformed. Statistics deformations, where the exchange relation of two particles is influenced by the presence of other particles, cannot be realized by fermionic operators.The statistics of degrees of freedom drastically affects the physical properties of a many-particle system. Besides bosonic and fermionic statistics, a continuous family of intermediate statistics serves to explain important effects involved in two or one dimensional physics. Remarkably, in D = 2, excitations in the fractional quantum Hall effect can be described as anyons 1,2 . Onedimensional systems can occur either because only onedimensional dynamics is allowed in the system (even if the system lives in higher dimensions) or because the samples are indeed one-dimensional (like quantum wires, carbon nanotubes, systems with charge density wave order, etc). In the present paper we shall focus on deformed statistics in one dimension. Defining arbitrary statistics in one dimension exhibits several peculiarities 3 . In particular, imposing the statistics in one dimension can be interpreted as a "continuity condition" on the wave function (arising from the set of coordinates such that two or more particles coincide), fixing its symmetry. One dimensional fractional statistics arises since this constraint on the wave function can be imposed arbitrarily. Explicit realizations of 1D fractional statistics quasi particles, formulated in "first quantization", are the eigenstates of Calogero-Sutherland models.In Refs. 4 and 5, the notion of Deformed Exchange Statistics (DES) was defined as a specific deformation of electronic commutation rules (in second quantization). Mathematical aspects of this type of statistics have been also studied in Refs. 6,7. We applied DES to investigate how robust is the solvability by Coordinate Bethe Ansatz (CBA) 8,9 of the XXZ and the Hubbard model with respect to such a modification of the particle content (we have called those DES preserving CBA solvability Solvable DES). In the present paper we show that the solvable deformed Hubbard model (without Schulz-Shastry type correlations) is equivalent to adding correlations similar to those discussed by Schulz and Shastry 10 to the undeformed Hubbard model. We prove this realizing DES operators by composites of electronic operators. Using the results of Ref. 11 we ...

Ulrich Eckern's 60th birthday

“…For example, the role of randomness on the ground state properties of integrable models were studied successfully with the help of the density matrix renormalization group technique [9]. Particles obeying fractional statistics were treated with the Bethe ansatz [10]. In order to investigate electronic and magnetic properties of real materials, Ulli and his group employed density functional theory (DFT) [11].…”

Extraspecial Two-Groups, Generalized Yang-Baxter Equations and Braiding Quantum Gates

“…Note Added: Deformations with phase parameters similar to those appearing in eq. (3.12) have been used in the context of deformed exchange statistics in 1D systems, leading to a deformed solution of the Yang-Baxter equation [37]. Also the matrix (3.11) recently appears in the context of an SL(2) invariant extension of the entanglement measure concurrence to higher (half-integer) spins in ArXiv.…”

Linear-optical dynamics of one-dimensional anyons

“…Some arise as dimensional reduction of twodimensional anyon states [17,18], but most are obtained as free-particle descriptions of exactly solvable models with local two-body interactions in one dimension [19][20][21][22][23]. The systems we consider are anyons defined via deformed commutation relations [24][25][26][27][28][29][30][31], where the ±1 bosonic/fermionic exchange phase is replaced by a nontrivial complex phase. These models play a role in onedimensional many-body systems with three-body inter-actions [32][33][34][35][36][37][38] studied in optical lattice implementations [39][40][41][42], and can be related to standard fermionic and bosonic systems via generalized Jordan-Wigner transformations [43][44][45].…”