Critical points of the anyon-Hubbard model

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 $\theta =\pi /4$ is in the same universality class as the Bose-Hubbard model with two-body interactions.

Figure 1

System size dependence of the chemical potential of anyons in 1D with statistical angle $\theta =\pi /4$ and $\rho =2$. The upper set of data in each panel corresponds to the particle excitation energy and the lower one to the hole excitation energy. In the left panel ($t/U=0.05$), we show a state with a finite difference at the thermodynamic limit, while this difference vanishes in the right panel ($t/U=0.25$).

Figure 2

Density $\rho$ vs chemical potential $\mu$ at the thermodynamic limit for $t/U=0.1$ and a statistical angle $\theta =\pi /4$.

Figure 3

Phase diagram of the anyon-Hubbard model with statistical angle $\theta =\pi /4$ for the densities $\rho =1,2$, and 3 using DMRG (blue line-circle) and comparison with the mean-field solution (gray squares) for the same densities (mean-field data were taken from [29]). Inset: display sequence from Mott insulator to supefluid and back to Mott insulator for $\theta =\pi /4$ and $\rho =1$ at fixed $\mu =0.13$.

Figure 4

von Neumann block entropy $S_L\left(l\right)$ as a function of $l$ for a system with size $L=512,\rho =1$, and $\theta =\pi /4$. Here we consider two different values of the hopping parameter, $t/U=0.2$ and $t/U=0.6$. In the inset, the von Neumann block entropy $S_L\left(l\right)$ as function of the logarithmic conformal distance $\lambda$ is shown, revealing a linear behavior for the critical state. Otherwise, the noncritical state does not exhibit linear behavior, because of the short correlation length. We found that the central charge is $c=0.97$.

Figure 5

Estimator $\mathrm{\Delta }{S}_{LK}$ as a function of the hopping parameter $t/U$ for $\theta =0$ and $\theta =\pi /4$. Here, we fixed $L=256$ and $\rho =1$. In the inset, the estimator $\mathrm{\Delta }{S}_{LK}$ vs $t/U$ for $\theta =\pi /4$, and different system lengths $L=64,128,256$, and 512. The lines are visual guides.

Figure 6

Energy gap as a function of $t_c-t$ for $\rho =1$ and $\theta =\pi /4$. In the inset, $\text{ln}\mathrm{\Delta }\mu$ vs $1/\sqrt{t_c-t}$. Here, the points are DMRG results, and the fits to the Kosterlitz-Thouless formula are shown by lines.

Figure 7

Quantum critical point position ${(t/U)}_c$ as a function of the density for the anyon-Hubbard model with $\theta =\pi /4$ and $\theta =0$. The dashed lines represent the best fit of the numerical data with the function of Eq. (9). The numerical constants obtained for $\theta =\pi /4$ are $(\alpha ,\beta ,\gamma )=(-0.037,0.45,-0.7)$ Inset: $\mathrm{\Delta }{t}_c$ between $\theta =0$ and $\theta =\pi /4$ as a function of the density $\rho$ (the data of $\theta =0$ were taken from [50]).

Figure 8

Estimator $\mathrm{\Delta }{S}_{LK}$ as a function of the hopping parameter $t/U$ for various statistical angles $\theta =0.25\pi ,0.3\pi$, and $0.35\pi$. Here, we fixed $L=256$ and $\rho =1$.

Figure 9

Critical point evolution with statistical angle $\theta$ for the anyon-Hubbard model. The dashed lines represent the best fit of the numerical data. The stars represent the critical points found by Keilmann et al. We cannot explore larger values of $\theta$ due to the dramatic increase in the number of states that must be maintained in order to achieve the limit $\left(\text{ln}2\right)/6$.