Diagnosing energy gap in quantum spin liquids via polarization amplitude

TL;DR

This work introduces a gap-diagnostic method based on the polarization amplitude derived from a twist operator, evaluated via iDMRG, to distinguish gapped from gapless phases in quantum spin systems. By defining $|z^q|=\lim_{L_{\mathrm{tw}}\to\infty} |\langle U^q\rangle|$ with $U=\exp\left(\frac{2\pi i}{L_{\mathrm{tw}}}\sum_j j S_j^z\right)$, the authors show that $|z^q|$ approaches unity in gapped phases and vanishes in gapless ones, while carefully addressing the infinite-system quantization subtleties. They benchmark the method on the spin-$1/2$ XXZ chain, where $|z^2|$ signals the TLL-to-Néel transition, and apply it to the infinite-cylinder XY--$J_\chi$ model to detect a transition from a gapless XY phase to a gapped chiral spin liquid, with $|z|$ approaching unity in the CSL and decaying in the XY phase. The results demonstrate that polarization amplitudes constitute a practical, ground-state diagnostic tool for gap analysis in quasi-one-dimensional and quasi-two-dimensional quantum magnets, including spin liquids, within the iDMRG framework.

Abstract

Identifying whether a many-body ground state is gapped or gapless is a fundamental yet challenging problem, especially in quantum spin liquids. In this work, we develop a gap-diagnostic scheme based on the polarization amplitude defined via a twist operator, evaluated within the infinite density-matrix renormalization group (iDMRG) framework. As a benchmark, analysis of the spin-$1/2$ XXZ chain demonstrates that the polarization amplitude clearly distinguishes the gapless Tomonaga-Luttinger liquid from the gapped Néel phase. We then extend this framework to infinite cylinders of the spin-$1/2$ XY-$J_χ$ model on the square lattice. We find that the polarization amplitude sharply detects the transition between the gapless XY phase and the gapped chiral spin liquid phase. These results show that polarization amplitudes provide a strong energy-gap diagnostic in two-dimensional frustrated quantum magnets, including quantum spin liquids.

Figures (7)

• Figure 1: Schematic illustration of the spatial region on which the twist operator acts. (a) For an infinite one-dimensional chain in the thermodynamic limit ($L_x \to \infty$), the twist operator $U$ is applied to a finite region of length $L_{\mathrm{tw}}$. (b) For an infinite cylinder with circumference $L_y$ (here $L_y = 4$) and infinite axial length ($L_x \to \infty$), the twist operator $U$ is applied to a finite region of length $L_{\mathrm{tw}}$ along the axial direction.
• Figure 2: Spin polarization amplitude $|z^2|$ of the XXZ chain as a function of the anisotropy $\Delta$. Blue circles, green squares, and red triangles correspond to $L_{\mathrm{tw}}=20,\,100,\,1000$, respectively. Here $L_{\mathrm{tw}}$ denotes the region length on which the twist operator is applied and should not be confused with the system size.
• Figure 3: In-plane spin correlations in the square-lattice XY--$J_\chi$ model on infinite cylinders. We plot the equal-time correlation function $\bigl|\langle S^{+}_{0} S^{-}_{j_r}\rangle\bigr|$ as a function of the distance $r$ along the cylinder axis for circumferences $L_y=4$ (circles) and $L_y=6$ (crosses). (a) XY phase ($J_\chi=0.0$--$0.8$): the correlations show a slow, approximately algebraic decay, consistent with quasi-long-range order on cylinders. (b) Chiral spin liquid phase ($J_\chi=0.9,\,1.0$): the correlations decay much more rapidly over distance, indicating a finite correlation length and consistent with a non-zero gap.
• Figure 4: Ground-state diagnostics for the square-lattice $S=1/2$ XY--$J_\chi$ model on infinite cylinders. (a) In-plane Néel correlation $m_{xy}^2$ within the $L_y\times L_y$ region (Eq.\ref{['eq:neel']}) as a function of $J_\chi$ for $L_y=4$ and $6$. (c) Scalar spin chirality $\langle \chi\rangle$ (Eq. \ref{['eq:chiral']}) as a function of $J_\chi$ for $L_y=4$ and $L_y = 6$, where. (a,b) Polarization amplitude $|z|$ as a function of $J_\chi$ for (a) circumference $L_y=4$ and (b) $L_y=6$. Different symbols correspond to different twist lengths $L_{\mathrm{tw}}$ (length along the cylinder axis) on which the twist operator is applied. The polarization amplitude changes from nearly zero to a finite value across the transition.
• Figure 5: Scaling of the spin polarization amplitude with the inverse twist length in the square-lattice XY--$J_\chi$ model on infinite cylinders. The vertical axis shows the spin polarization amplitude $|z|$, while the horizontal axis represents the twist length $L_{\mathrm{tw}}$. Circle and cross symbols correspond to cylinders with circumferences $L_y = 4$ and $L_y = 6$, respectively. Within each set, the curves are ordered from bottom to top as $J_\chi = 0.0, 0.2, 0.4$, and $0.6$. (a) XY phase ($J_\chi = 0.0$--$0.6$): $|z|$ shows as a power-law decay and vanishes in the $L_{\mathrm{tw}} \to \infty$ limit, with dashed lines indicating power-law fits of the form $|z| \propto 1/L_{\mathrm{tw}}^{\alpha}$. (b) Chiral spin-liquid phase ($J_\chi = 0.9,\,1.0$): $|z|$ extrapolates toward values close to unity as $L_{\mathrm{tw}} \to \infty$, consistent with a gapped chiral spin-liquid regime. Dashed lines indicate exponential fits of the form $|z|= |z_{\infty}| e^{-\lambda/L_{\mathrm{tw}}}$, where $|z_{\infty}|,\lambda$ are fitting parameters.