Dynamical Spectral Function of the Kagome Quantum Spin Liquid

TL;DR

The paper computes the full dynamical spin spectral function $S(\mathbf{k},\omega)$ of the $J_1$-$J_2$ kagome Heisenberg antiferromagnet directly in the infinite system using state-of-the-art iPEPS-based methods. By tracking $J_2/J_1$ across the magnetically ordered and quantum spin-liquid (QSL) phases, it reveals strong spin-wave renormalization, the emergence of a dense continuum at finite energies, and a gapless U(1) Dirac spin liquid characterized by vanishing gaps at multiple M points. The work introduces methodological improvements for efficient excitation spectra, analyzes bond-dimension and broadening effects, and provides spectral fingerprints—such as the low-energy continuum and M-point gap closure—that can guide experimental detection in kagome QSL materials. Overall, it offers a unified, high-resolution spectral picture of KHAF excitations across phases, bridging theory and potential INS/tunneling experiments.

Abstract

Quantum spin liquids (QSLs) host exotic fractionalized magnetic and gauge-field excitations whose microscopic origins and experimental verification remain frustratingly elusive. In the absence of static magnetic order, the spin excitation spectrum constitutes the crucial probe of QSL behavior, but its theoretical computation remains a serious challenge. Here we employ state-of-the-art tensor-network methods to obtain the full dynamical spectral function of the $J_1$-$J_2$ kagome Heisenberg model and benchmark our results by tracking their evolution across the magnetically ordered and QSL phases. Reducing $|J_2|/J_1$ causes increasingly strong spin-wave renormalization, flattening these modes then merging them into a continuum characteristic of deconfined spinons at all finite energies in the QSL. The low-energy continuum and the occurrence of gap closure at multiple high-symmetry points identify this gapless QSL as the U(1) Dirac spin liquid. These results establish a unified understanding of spin excitations in highly frustrated quantum magnets and provide clear spectral fingerprints for experimental detection in candidate kagome QSL materials.

Figures (13)

• Figure 1: Spin excitation spectra of the $J_1$-$J_2$ kagome Heisenberg antiferromagnet (KHAF). (a-f) Example spectra in the $q = 0$ magnetically ordered phase (a, b), the quantum spin-liquid (QSL) phase (c, d), and the $\sqrt{3}$$\times$$\sqrt{3}$ ordered phase (e, f). White lines show the energy levels obtained from iPEPS calculations. The inset in panel (e) shows the Brillouin zone and the high-symmetry path followed in our spectra; dashed and solid hexagons mark respectively the elementary and extended Brillouin zones. Below the spectra we represent the phase diagram determined by an iPEPS study of the ground states Liao2017.
• Figure 2: Spin structure factor of the KHAF at $J_2 = 0$. (a) Static structure factor shown over the entire Brillouin zone. (b)-(d) Dynamic structure factors shown for $\omega = 0.3J_1$ (b), $0.7J_1$ (c), and $1.0J_1$ (d).
• Figure 3: Properties of the low-energy spectrum. (a) Energy gaps at the M and M$^{\prime}$ points in the $J_2 = 0$ KHAF shown as functions of $1/D$. Error bars for $\Delta_{{\rm M}^{\prime}}$ represent the variation between M$^\prime$ points, as the Ansatz does not preserve the $C_3$ symmetry that makes these equivalent. Dashed lines are guides to the eye. (b) Energy gaps at M and M$^{\prime}$ as functions of $J_2$ for $D = 4$. (c) Lowest excitation energy shown around the high-symmetry path in the Brillouin zone for a range of $J_2$ values. (d) Spin excitation spectra at fixed energy $\omega = 0.3J_1$ shown on the same path for the same $J_2$ values.
• Figure S1: Comparison of the ground-state energy, $E_0$, and the staggered magnetization, $M_0$, of the $J_2 = 0$ KHAF computed using 3-PESS, 9-PESS, and iPEPS bases and optimized by simple-update (SU) and automatic differentiation (AD) methods.
• Figure S2: Comparison of the computation times required for 20 CTMRG steps in the calculation of the excitation spectrum for the $D = 4$ iPEPS on an NVIDIA A100 GPU, with and without employing the method of saving projectors. Results are shown as a function of the bond dimension, $\chi$, of CTMRG. (a) Computation time $T_1$ achieved by saving projectors. (b) Computation time $T_0$ without saving projectors and the speed-up factor, defined as $T_0/T_1$.