Low-dimensional quantum magnets, particularly highly frustrated ones, exhibit strong spin fluctuations, which can be probed via nuclear magnetic resonance (NMR). However, numerical simulations of NMR relaxation rates, like the spin-lattice relaxation rate $1/T_1$, remain challenging for the real-time evolution or analytical continuation. Here we develop an imaginary-time approach for computing $1/T_1$ using thermal tensor networks (thermoTNs). We demonstrate the accuracy and versatility of the method by applying it to one-dimensional chain and two-dimensional lattice models. Our results show that the critical exponents $\eta$ and $z\nu$ can be extracted from the low-temperature scaling behaviors of simulated $1/T_1$ data near the quantum critical points. We subsequently employ this method to investigate three representative quantum magnets: the Ising chain compound ${\mathrm{CoNb}}_2{\mathrm{O}}_6$, and the triangular-lattice quantum antiferromagnets ${\mathrm{Ba}}_8{\mathrm{CoNb}}_6{\mathrm{O}}_{24}$ and the ${A\mathrm{YbSe}}_2$ series ($A=$ Na, K, Cs). On triangular lattice, the simulated $1/T_1$ data clearly track the transition from spin-ordered states to quantum spin liquids (QSLs), highlighting NMR relaxation rate as a sensitive probe for QSL in these actively studied systems. Moreover, the approach can be extended to transport properties like the thermal conductivity, by substituting relevant operators. We show that thermoTN, equipped with imaginary-time proxies, serves as a powerful and sensitive tool for simulating equilibrium and nonequilibrium phenomena in frustrated quantum magnets and correlated systems in general.

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• Exotic phases of matter
• Frustrated magnetism
• Quantum criticality
• Spin dynamics
• Spin fluctuations
• Spin liquid
• 1-dimensional spin chains
• Antiferromagnets
• Triangular lattice
• Density matrix renormalization group
• Nuclear magnetic resonance
• Nuclear magnetic resonance relaxation rate
• Tensor network renormalization