Quantum Fisher Information as a Thermal and Dynamical Probe in Frustrated Magnets: Insights from Quantum Spin Ice

TL;DR

The paper demonstrates that Quantum Fisher Information (QFI) can serve as a thermal and dynamical probe of frustrated quantum magnets, focusing on the pyrochlore quantum spin ice (QSI) model. It develops a multi-directed loop QMC framework (MDL) to compute QFI, complemented by exact diagonalization (ED) and gauge mean-field theory (GMFT), and connects theory to neutron-scattering observables for dipolar–octupolar DO materials such as Ce$_2$Zr$_2$O$_7$. The results show that QFI tracks thermal crossovers and phase boundaries, distinctly separating FM, CSI, QSI$_0$, and $ ext{QSI}_ obreak{ extstyle{rac{ ext{π}}{ ext{}}}}$ regimes, and reveals two finite-temperature crossovers with momentum-dependent signatures. The work establishes QFI as a powerful, experimentally relevant entanglement witness and a sensitive probe of finite-temperature dynamics in QSI, with direct implications for interpreting neutron scattering experiments in candidate QSL materials.

Abstract

Quantum Fisher information (QFI) is a novel measure of multipartite quantum entanglement that can be measured in inelastic neutron scattering experiments on quantum magnets. In this work, we demonstrate that the QFI can be used to understand the thermal and dynamical properties of quantum magnets by focusing on the pyrochlore lattice model of quantum spin ice (QSI), a three-dimensional quantum spin liquid that hosts fractionalized quasiparticles and emergent photons. We use the newly developed multi-directed loop update quantum Monte Carlo (QMC) algorithm and exact diagonalization (ED) to compute the QFI, which is further utilized to calibrate the gauge mean-field theory results. We show that the temperature and momentum dependence of the QFI can reveal characteristic energy scales of distinct phases and phase transitions in the global phase diagram. In particular, the QFI can clearly distinguish the ferromagnetic ordered phase, the thermal critical region above it, as well as two distinct QSI phases, namely zero-flux and $π$-flux QSI. Moreover, the QFI shows two crossover temperature scales, one from the trivial paramagnet to the classical spin ice regime and a lower temperature crossover to QSI. We discuss our results, especially for the $π$-flux QSI, in light of the ongoing experimental efforts on Cerium-based pyrochlore systems. Our results demonstrate that the QFI not only detects entanglement properties but can also be viewed as a sensitive thermal and dynamical probe in the investigation of quantum magnets.

Figures (11)

• Figure 1: Heat maps of the QFI as functions of temperature $T$ and $J_{\pm}$. Panels (a) and (b) show the QFI density $f_{Q}(S^{\pm}_\mathbf{q},T)$ in the $S^{\pm}$ channel at $\Gamma=(0,0,0)$, while panels (c,d) show it at $\Gamma^\prime=(4\pi,4\pi,0)$. Panel (a) and (c) are obtained from the ED calculation of 16-site cluster with $J_{\pm}$ ranging from $-0.045$ to $0.08$, and panels (b) and (d) are from the QMC simulation of $4\times L^3$ sizes ($L=4$) with $J_{\pm}$ ranging from $0.04$ to $0.10$. In panels (a) and (b), $f_{Q}(S^{\pm}_\Gamma,T)$ maps out the thermodynamic phase boundaries between the ferromagnetic phase (FM) and QSI$_0$. The temperature dependence of $f_{Q}(S^{\pm}_\Gamma,T)$ from QMC further discerns the crossover temperature scales between the high-temperature paramagnetic regime to classical spin ice and eventually to the QSI$_0$ regime (see also Fig. \ref{['fig:QFI_line']} (a)). In panels (c) and (d), $f_{Q}(S^{\pm}_{\Gamma^\prime},T)$ reflects the strength of the fluctuations in the thermal and quantum phase diagram, with strong QFI at the classical critical region above the FM phase and stronger QFI in the classical spin ice and QSI$_0$ region. Moreover, the strongest QFI signal, represented by the red region in panel (c) for $J_{\pm}<0$, reflects the QSI$_\pi$ regime (see also Fig. \ref{['fig:QFI_line']} (b)).
• Figure 2: Temperature evolution of the QFI in different phases obtained with various computational methods. The QFI $f_{Q}(S^{\pm}_\mathbf{q},T)$ are shown for (a) the $S^{\pm}$ channel at the $\Gamma$ point, (b) the $S^{\pm}$ channel at the $\Gamma^{\prime}$ point. The blue disk points represent results from ED calculations with $J_{\pm}=-0.3$ and $J_{\pm}=-0.125$ in the $\mathrm{QSI}_{\pi}$ regime, while the organge triangle correspond to QMC simulations with $J_{\pm}=0.045$ and $J_{\pm}=0.05$ in the $\mathrm{QSI}_{0}$ regime. The orange shaded areas accompanying the data highlight the crossover from paramagnetic regime to classical spin ice regime at $T\sim 1$ and that from the classical spin ice regime to QSI$_0$ at $T\sim |J^3_{\pm}|$. The red triangles indicate QMC results with $J_{\pm}$ ranging from $0.06$ to $0.08$ in the FM regime. The green points denote the GMFT calculation results at $J_{\pm}=0.045$ (QSI$_0$) and $J_{\pm}=-0.3$ (QSI$_\pi$) for different ground states at zero temperature. The QMC, ED, and GMFT results are consistent (see SM suppl for details).
• Figure 3: QFI in experimental coordinates for Cerium-based pyrochlore compounds. Panel (a) presents a heat map of the QMC computed QFI $f_{Q}(S^{\text{DO}}_{\Gamma'}, T)$, with $J_{\pm}$ ranging from $0.04$ to $0.10$. Panel (b) is the line cut of panel (a) from $J_{\pm}=0.045$ to $0.08$. The crossover temperature scales from the paramagnetic phase to CSI at $T\sim 1$ and from CSI to QSI$_0$ at $T\sim |J^3_{\pm}|$ are clearly manifest in the shaded data, highlighted in orange. Panel (c) shows ED results of $f_{Q}(S^{\text{DO}}_{\Gamma'}, T)$ for $J_{\pm}$ between $0.01$ and $0.08$. Panel (d) focuses $f_{Q}(S^{\text{DO}}_X, T)$ with $X=(0,0,2\pi)$ in the range from $-0.425$ to $0.00$.
• Figure S1: Schematic diagram of (a) a tetrahedron in the pyrochlore lattice and (b) the corresponding four leg operator vertex in the sampling space. (c) Illustration of the single loop update, (d) bi-loop update, and (e) tri-loop update in the multi-directed loop algorithm.
• Figure S2: The energy per site $e$ as a function of temperature $T$ at (a) $J_{\pm}=0.04$, (b) $J_{\pm}=0.045$, (c) $J_{\pm}=0.05$, and (d) $J_{\pm}$ ranging from $0.06$ to $0.10$ with an increment of 0.01. In (a-c), the blue points are obtained from SSE-QMC simulation with MDL update, and the orange lines are without MDL update. In (d), all points are obtained from SSE-QMC simulation without MDL update.