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arXiv:1604.02466 · Physical Review A

Critical points of the anyon-Hubbard model

J. Arcila-Forero, Roberto Franco, Jereson Silva-Valencia

Published Jul 19, 2016• 30 citations
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Abstract

Anyons are particles with fractional statistics that exhibit a nontrivial change in the wave function under an exchange of particles. Anyons can be considered to be a general category of particles that interpolate between fermions and bosons. We determined the position of the critical points of the one-dimensional anyon-Hubbard model, which was mapped to a modified Bose-Hubbard model where the tunneling depends on the local density and the interchange angle. We studied the latter model by using the density-matrix renormalization-group method and observed that gapped (Mott insulator) and gapless (superfluid) phases characterized the phase diagram, regardless of the value of the statistical angle. The phase diagram for higher densities was calculated and showed that the Mott lobes increase (decrease) as a function of the statistical angle (global density). The position of the critical point separating the gapped and gapless phases was found using quantum information tools, namely the block von Neumann entropy. We also studied the evolution of the critical point with the global density and the statistical angle and showed that the anyon-Hubbard model with a statistical angle $\ensuremath{\theta}=\ensuremath{\pi}/4$ is in the same universality class as the Bose-Hubbard model with two-body interactions.

DOI
10.1103/physreva.94.013611

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Anyons are particles with fractional statistics that exhibit a nontrivial change in the wave function under an exchange of particles. Anyons can be considered to be a general category of particles that interpolate between fermions and bosons. We determined the position of the critical points of the one-dimensional anyon-Hubbard model, which was mapped to a modified Bose-Hubbard model where the tunneling depends on the local density and the interchange angle. We studied the latter model by using the density-matrix renormalization-group method and observed that gapped (Mott insulator) and gapless (superfluid) phases characterized the phase diagram, regardless of the value of the statistical angle. The phase diagram for higher densities was calculated and showed that the Mott lobes increase (decrease) as a function of the statistical angle (global density). The position of the critical point separating the gapped and gapless phases was found using quantum information tools, namely the block von Neumann entropy. We also studied the evolution of the critical point with the global density and the statistical angle and showed that the anyon-Hubbard model with a statistical angle \ensuremathθ=\ensuremathπ/4\ensuremath{\theta}=\ensuremath{\pi}/4\ensuremathθ=\ensuremathπ/4 is in the same universality class as the Bose-Hubbard model with two-body interactions.
Recently optical lattices allow us to observe phase transition without the uncertainty posed by complex materials, and the simulations of these systems are an excellent bridge between materials-based condensed matter physics and cold atoms. In this way, the computational physics related to many-body problems have increased in importance. Using the density matrix renormalization group method, we studied a Hubbard model for anyons, which is an equivalent to a variant of the Bose-Hubbard model in which the bosonic hopping depends on the local density. This is an exact mapping between anyons and bosons in one dimension. The anyons interlope between bosons and fermions. For two anyons under particle exchange, the wave function acquires a fractional phase e iθ . We conclude that this system exhibits two phases: Mott-insulator and superfluid. We present the phase diagram for some angles. The Mott lobe increases with an increase of the statistical. We observed a reentrance phase transition for all lobes. We showed that the model studied is in the same universality class as the Bose-Hubbard model with two-body interactions.
We investigated a one-dimensional system of anyons that interact with each other under a local three-body term. Using a fractional Jordan-Wigner transformation, we arrived at a modified Bose-Hubbard model, which exhibits gapped and gapless phases. We built the phase diagram of the system fixing the hopping parameter or the statistics, showing the evolution of the critical points, which were estimated with von Neumann block entropy. A superfluid to Mott insulator quantum phase transition with one particle per site can be driven by the statistics or the interaction. Specifically, we show that for larger angles there is a finite critical value of the interaction at which the Mott phase appears. Also, we found that the critical angles increase with the hopping. Diverse gapless phases were observed away from the pseudofermion limit.
We study one-dimensional lattice anyons with extended Hubbard interactions at unit filling using bosonization and numerical simulations. The behavior can be continuously tuned from bosonic to fermionic statistics by adjusting the topological exchange angle θ, which leads to a competition of different instabilities. We present the bosonization theory in the presence of dynamic gauge fields, which predicts a phase diagram of four different gapped phases with distinct dominant correlations. Advanced numerical simulations determine and analyze the exact phase transitions between Mott insulator, charge density wave, dimerized state, and Haldane insulator, all of which meet at a multicritical line in the parameter space of anyonic angle θ, on-site interaction U, and nearest neighbor repulsion V. Superfluid and pair-superfluid phases are stable in a region of small V.
Using the density matrix renormalization group method, we determine the phase diagram of the Bose-Hubbard model with local two- and three-body interactions, describing polar molecules in one-dimensional optical lattices. The difference in the block von Neumann entropy with different system sizes was used to establish the critical points. We found that the quantum critical point position increases with the three-body interaction. We show that the model studied is in the same universality class as the model with pure two-body interactions.
We theoretically investigate topological properties of the one-dimensional superlattice anyon-Hubbard model, which can be mapped to a superlattice bose-Hubbard model with an occupation-dependent phase factor by fractional Jordan-Wigner transformation. The topological anyon-Mott insulator is identified by topological invariant and edge modes using exact diagonalization and the density matrix renormalization group algorithm. When only the statistical angle is varied and all other parameters are fixed, a statistically induced topological phase transition can be realized, which provides insights into the topological phase transitions. What's more, we give an explanation of the statistically induced topological phase transition. The topological anyon-Mott phases can also appear in a variety of superlattice anyon-Hubbard models.
Using the Anyon–Hubbard Hamiltonian, we analyze the ground-state properties of anyons in a one-dimensional lattice. To this end we map the hopping dynamics of correlated anyons to an occupation-dependent hopping Bose–Hubbard model using the fractional Jordan–Wigner transformation. In particular, we calculate the quasi-momentum distribution of anyons, which interpolates between Bose–Einstein and Fermi–Dirac statistics. Analytically, we apply a modified Gutzwiller mean-field approach, which goes beyond a classical one by including the influence of the fractional phase of anyons within the many-body wavefunction. Numerically, we use the density-matrix renormalization group by relying on the ansatz of matrix product states. As a result it turns out that the anyonic quasi-momentum distribution reveals both a peak-shift and an asymmetry which mainly originates from the nonlocal string property. In addition, we determine the corresponding quasi-momentum distribution of the Jordan–Wigner transformed bosons, where, in contrast to the hard-core case, we also observe an asymmetry for the soft-core case, which strongly depends on the particle number density.