Extracting Luttinger liquid parameter <i>K</i> based on <i>U</i>(1) symmetric infinite matrix product states
Authors: Wang Xiu-Juan , Li Sheng-Hao ,   State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing 400044, China,   Chongqing Vocational Institute of Engineering, Chongqing 400037, China,   Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, China
Published: December 2020

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>.