Many-body localization of a one-dimensional anyon Stark model

<sec> In this work, a one-dimensional interacting anyon model with a Stark potential in the finite size is studied. Using the fractional Jordan Wigner transformation, the anyons in the one-dimensional system are mapped onto bosons, which are described by the following Hamiltonian: </sec><sec>　　　　　　　　<inline-formula><tex-math id="M1">\begin{document}$ \begin{aligned} \hat{H}^{\text{boson}}=-J\sum_{j=1}^{L-1}\left( \hat{b}_{j}^{\dagger}\hat{b}_{j+1}{\mathrm{e}}^{{\mathrm{i}}\theta \hat{n}_{j}}+{\mathrm{h.c.}}\right)+\frac{U}{2}\sum_{j=1}^{L}\hat{n}_{j}\left( \hat{n}_{j}-1\right)+\sum_{j=1}^{L}{h}_{j}\hat{n}_{j},\;\;\;\;\;\;\;\;\;\end{aligned}$\end{document}</tex-math></inline-formula>where <i>θ</i> is the statistical angle, and the on-site potential is <inline-formula><tex-math id="M2">\begin{document}$h_{j}=-\gamma\left(j-1\right) +\alpha\Big( \dfrac{j-1}{L-1}\Big)^{2}$\end{document}</tex-math></inline-formula> with <i>γ</i> representing the strength of the Stark linear potential and <i>α</i> denoting the strength of the nonlinear part. Using the exact diagonalization method, the spectral statistics, half-chain entanglement entropy and particle imbalance are numerically analyzed to investigate the onset of many-body localization (MBL) in this interacting anyon system, induced by increasing the linear potential strength. As the Stark linear potential strength increases, the spectral statistics transforms from a Gaussian ensemble into a Poisson ensemble. In the ergodic phase, except for <i>θ</i> = 0 and π, where the average value of the gap-ratio parameter <inline-formula><tex-math id="M4">\begin{document}$\left\langle r\right\rangle\approx 0.53$\end{document}</tex-math></inline-formula>, due to the destruction of time reversal symmetry, the Hamiltonian matrix becomes a complex Hermit matrix and <inline-formula><tex-math id="M5">\begin{document}$\left\langle r\right\rangle\approx 0.6$\end{document}</tex-math></inline-formula>. In the MBL phase, <inline-formula><tex-math id="M6">\begin{document}$\left\langle r\right\rangle\approx 0.39$\end{document}</tex-math></inline-formula>, which is independent of <i>θ</i>. However, in the intermediate <i>γ</i> regime, the value of <inline-formula><tex-math id="M7">\begin{document}$\left\langle r\right\rangle$\end{document}</tex-math></inline-formula> strongly depends on the choice of <i>θ</i>. The average of the half-chain entanglement entropy transforms from a volume law into an area law, which allows us to construct a <i>θ</i>-dependent MBL phase diagram. In the ergodic phase, the entanglement entropy <i>S</i>(<i>t</i>) of the half chain increases linearly with time. In the MBL phase, <i>S</i>(<i>t</i>) grows logarithmically with time, reaching a stable value that depends on the anyon statistical angle. The localization of particles in a quench dynamics can provide the evidence for the breakdown of ergodicity and is experimentally observable. It is observed that with the increase of <i>γ</i>, the even-odd particle imbalance changes from zero to non-zero values in the long-time limit. In the MBL phase, the long-time average value of the imbalance is dependent on the anyon statistical angle <i>θ</i>. From the Hamiltonian <inline-formula><tex-math id="M10">\begin{document}$\hat{H}^{\text{boson}}$\end{document}</tex-math></inline-formula>, it can be inferred that the statistical behavior of anyon system equally changes the hopping interactions in boson system, which is a many-body effect. By changing the statistical angle <i>θ</i>, the many-body interactions in the system are correspondingly changed. And the change of the many-body interaction strength affects the occurrence of the MBL transition, which is also the reason for MBL transition changing with the anyon statistical angle <i>θ</i>. Our results provide new insights into the study of MBL in anyon systems and whether such phenomena persist in the thermodynamic limit needs further discussing in the future.</sec>