Extracting Luttinger liquid parameter K based on U(1) symmetric infinite matrix product states

Abstract

We numerically calculate Luttinger liquid parameter <i>K</i> in the anisotropic spin XXZD models with spin <inline-formula><tex-math id="M15">\begin{document}$s = 1/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M15.png"/></alternatives></inline-formula>, 1, and 2. In order to obtain groundstate wavefunctions in Luttinger liquid phases, we employ the <inline-formula><tex-math id="M16">\begin{document}$U(1)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M16.png"/></alternatives></inline-formula> symmetric infinite matrix product states algorithm (iMPS). By using relation between the bipartite quantum fluctuations <i>F</i> and the so-called finite-entanglement scaling exponents <inline-formula><tex-math id="M17">\begin{document}$\kappa$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M17.png"/></alternatives></inline-formula>, the Luttinger liquid parameter <i>K</i> can be extracted. For <inline-formula><tex-math id="M18">\begin{document}$s = 1/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M18.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$D=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M19.png"/></alternatives></inline-formula>, the numerically extracted Luttinger liquid parameter <i>K</i> is shown to be good agreement with the exact value. On using the fact that the spin-1 XXZD Hamiltonian with <inline-formula><tex-math id="M20">\begin{document}$ D \leqslant - 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M20.png"/></alternatives></inline-formula> can be mapped to an effective spin-1/2 XXZ model, we calculate the Luttinger liquid parameter for the region of <inline-formula><tex-math id="M21">\begin{document}$ D \leqslant - 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M21.png"/></alternatives></inline-formula>. It is shown that our numerical value of the Luttinger liquid parameter agree well with the exact values, here, the relative error less than <inline-formula><tex-math id="M22">\begin{document}$1\%$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M22.png"/></alternatives></inline-formula>. Also, our Luttinger liquid parameter at <inline-formula><tex-math id="M23">\begin{document}$\Delta = - 0.5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M23.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M24">\begin{document}$ D = 0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M24.png"/></alternatives></inline-formula> is shown to be consistent with the result form the density matrix renormalization group (DMRG) method. These results suggest that the <inline-formula><tex-math id="M25">\begin{document}$U(1)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M25.png"/></alternatives></inline-formula> symmetric iMPS method can be applicable to calculate Luttinger liquid parameters if any system has a <inline-formula><tex-math id="M26">\begin{document}$U(1)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M26.png"/></alternatives></inline-formula> symmetry for gapless phases. For instance, we present our Luttinger liquid parameters for the first time for the spin-1 XXZD model under the other parameters and the spin-2 XXZD model with <inline-formula><tex-math id="M27">\begin{document}$D = 1.5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20190379_M27.png"/></alternatives></inline-formula>.