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arXiv:2403.11895 · Physical review. B./Physical review. B

Thermal tensor network approach for spin-lattice relaxation in quantum magnets

Ning Xi, Yuan Gao, Huihang Lin, Qiaoyi Li, Chengchen Li, Enze Lv, Shuang Liang, Lei Wang, Rong Yu, Xiaoqun Wang, Wei Li

Published Mar 4, 2026• 0 citations
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Abstract

Low-dimensional quantum magnets, particularly those with strong spin frustration, are characterized by their notable spin fluctuations. Nuclear magnetic resonance (NMR) serves as a sensitive probe of low-energy fluctuations that offers valuable insight into rich magnetic phases and emergent phenomena in quantum magnets. Although experimentally accessible, the numerical simulation of NMR relaxation rates, specifically the spin-lattice relaxation rate $1/T_1$, remains a significant challenge. Analytical continuation based on Monte Carlo calculations are hampered by the notorious negative sign for frustrated systems, and the real-time simulations incur significant costs to capture low-energy fluctuations. Here we propose computing the relaxation rate using thermal tensor networks (TTNs), which provides a streamlined approach by calculating its imaginary-time proxy. We showcase the accuracy and versatility of our methodology by applying it to one-dimensional spin chains and two-dimensional lattices, where we find that the critical exponents $η$ and $zν$ can be extracted from the low-temperature scalings of the simulated $1/T_1$ near quantum critical points. Our results also provide insights into the low-dimensional and frustrated magnetic materials, elucidating universal scaling behaviors in the Ising chain compound CoNb$_2$O$_6$ and revealing the renormalized classical behaviors in the triangular-lattice antiferromagnet Ba$_8$CoNb$_6$O$_{24}$. We apply the approach to effective model of the family of frustrated magnets AYbCh$_2$ (A = Na, K, Cs, and Ch = O, S, Se), and find dramatic changes from spin ordered to the proposed quantum spin liquid phase. Overall, with high reliability and accuracy, the TTN methodology offers a systematic strategy for studying the intricate dynamics observed across a broad spectrum of quantum magnets and related fields.

DOI
10.1103/q6dt-kzgc

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Low-dimensional quantum magnets, particularly those with strong spin frustration, are characterized by their notable spin fluctuations. Nuclear magnetic resonance (NMR) serves as a sensitive probe of low-energy fluctuations that offers valuable insight into rich magnetic phases and emergent phenomena in quantum magnets. Although experimentally accessible, the numerical simulation of NMR relaxation rates, specifically the spin-lattice relaxation rate 1/T11/T_11/T1​, remains a significant challenge. Analytical continuation based on Monte Carlo calculations are hampered by the notorious negative sign for frustrated systems, and the real-time simulations incur significant costs to capture low-energy fluctuations. Here we propose computing the relaxation rate using thermal tensor networks (TTNs), which provides a streamlined approach by calculating its imaginary-time proxy. We showcase the accuracy and versatility of our methodology by applying it to one-dimensional spin chains and two-dimensional lattices, where we find that the critical exponents ηηη and zνzνzν can be extracted from the low-temperature scalings of the simulated 1/T11/T_11/T1​ near quantum critical points. Our results also provide insights into the low-dimensional and frustrated magnetic materials, elucidating universal scaling behaviors in the Ising chain compound CoNb2_22​O6_66​ and revealing the renormalized classical behaviors in the triangular-lattice antiferromagnet Ba8_88​CoNb6_66​O24_{24}24​. We apply the approach to effective model of the family of frustrated magnets AYbCh2_22​ (A = Na, K, Cs, and Ch = O, S, Se), and find dramatic changes from spin ordered to the proposed quantum spin liquid phase. Overall, with high reliability and accuracy, the TTN methodology offers a systematic strategy for studying the intricate dynamics observed across a broad spectrum of quantum magnets and related fields.
Abstract Motivated by recent progress on the experimental realization of proximate deconfined quantum critical point in a frustrated quantum magnet, we study the low-energy spin dynamics of a related checkerboard J-Q model by using quantum Monte Carlo simulations. The ground state of this model undergoes a weakly first-order quantum phase transition with an emergent O(4) symmetry between an antiferromagnetic state and a plaquette valence bond solid. The calculated spin lattice relaxation rate of nuclear magnetic resonance, 1 / T 1 , exhibits distinct low-temperature behaviors depending on the ground states. With decreasing the temperature, 1 / T 1 rises up on the antiferromagnetic side, characterizing a crossover to the renormalized classical regime, whereas 1 / T 1 drops exponentially on the side of valence bond solid, reflecting the gap opening in the plaquette ordered phase. The extracted spin gap scales with the distance to the transition point as a power-law with an exponent φ ≈ 0.3, consistent with the scaling ansatz ϕ = ν z with ν ≈ 0.3 and z = 1. Near the quantum phase transition, the temperature dependent 1 / T 1 shows a broad crossover regime where a universal scaling 1 / T 1 ∼ T η with η ≈ 0.6 is found. Our results suggest a quantum scaling regime associated with the emergent enhanced symmetry near this first-order quantum phase transition.
We have a strong interest in the interplay between geometrical frustrations and quantum fluctuations in the strongly correlated matter because it causes the emergence of new exotic quantum phases. For example, in the past decade, we have found a number of new Mott insulator materials on a triangular layer: Cs2CuCl4 [1] and organic charge transfer salts [2], such as κ-(BEDT-TTF)2 X and β-Z[Pd(dmit)2]2. There are geometrical frustrations and quantum fluctuations in these materials. Probably, the interplay has an important role for the disordered behaviors at very low temperatures in some materials. In order to study the low temperature property of strongly correlated matter in low-dimensional systems, we need a theoretical tool beyond the mean-field theory. The numerical approach is a candidate. Unfortunately, there are serious drawbacks in the conventional numerical methods. For example, quantum Monte Carlo (QMC) methods are powerful tools because they are unbiased. However, the weight of QMC samples can be negative in quantum frustrated magnets, leading to a cancellation in sign, and the accuracy of simulations fatally decreases (this is the so-called negative sign problem.) Exact diagonalization gives us the detail of a ground state, but it can only be applied to small systems. On the other hand, the variational method is flexible, and it is powerful if we choose the appropriate variational wave function. In the past decade, based on the study of entanglement, we discussed the new types of wave functions for quantum many-body systems. We define the probability amplitudes of these wave functions by the tensor contractions of small tensors. Since the tensor contractions can be drawn as a network, it is called a tensor network state. In this paper, we will report the performance of tensor network state for the variational study of quantum frustrated magnets. There are two major tensor network states: projected entangled pair state(PEPS) [3] and multiscale entanglement renormalization ansatz(MERA) [4]. In the following, we will focus on the MERA, and we will report the performance for S = 1/2 spatially anisotropic triangular antiferromagnets [5]. We organize this paper as follows: In Sec. 2, we will briefly introduce the tensor network. In addition, we will explain a MERA tensor network for triangular lattice models in detail. In Sec. 3, we will show the performance of MERA for the S = 1/2 antiferromagnetic Heisenberg model on a
The quantum Ising chain in a transverse magnetic field is $t\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}e$ paradigm of a quantum phase transition, displaying universal quantum criticality at low temperatures. Insights from this model have informed studies of quantum criticality in numerous modern correlated-electron materials. There have been a number of experimental studies of the quantum Ising chain, most notably in the material CoNb2{}_{2}2​O6{}_{6}6​. It has recently become clear that there was a surprising discrepancy between the observations: the energy gap in the quantum disordered regime as measured by nuclear magnetic resonance was approximately twice that measured by other methods. Here, the authors show via careful analysis of the low-temperature excitations that this is not a discrepancy, rather, it is in fact to be expected from the subtle interactions between the excitations. Although the underlying spin excitations are bosonic, their interactions transmute them into fermions, and this is a key ingredient in understanding the experiments.
We have extended the canonical tree tensor network (TTN) method, which was initially introduced to simulate the zero-temperature properties of quantum lattice models on the Bethe lattice, to finite temperature simulations. By representing the thermal density matrix with a canonicalized tree tensor product operator, we optimize the TTN and compute accurately the thermodynamic quantities, including the internal energy, specific heat, spontaneous magnetization, etc., at various temperatures. By varying the anisotropic coupling constant $\mathrm{\ensuremath{\Delta}}$, we obtain the phase diagram of the spin-1/2 Heisenberg XXZ model on the Bethe lattice, where three magnetic ordered phases, namely the ferromagnetic, XY, and antiferromagnetic phases, are found in low temperatures and separated from the high-TTT paramagnetic phase by a continuous thermal phase transition at Tc{T}_{c}Tc​. The XY phase is separated from the other two phases by two first-order phase transition lines at the symmetric coupling points $\mathrm{\ensuremath{\Delta}}=\ifmmode\pm\else\textpm\fi{}1$. We have also carried out a linear spin wave calculation on the Bethe lattice, showing that the low-energy magnetic excitations are always gapped, and find the obtained magnon gaps in very good agreement with those estimated from the TTN simulations. Despite the gapped excitation spectrum, Goldstone-like transverse fluctuation modes, as a manifestation of spontaneous continuous symmetry breaking, are observed in the ordered magnetic phases with $|\mathrm{\ensuremath{\Delta}}|\ensuremath{\le}1$. One remarkable feature there is that the transverse correlation length reaches ${\ensuremath{\xi}}_{c}=1/ln(z\ensuremath{-}1)$ for $T\ensuremath{\le}{T}_{c}$, the maximal value allowed on a zzz-coordinated Bethe lattice.
The frustrated square-lattice J1-J2 Ising model exhibits rich magnetic phases, yet its phase transitions remain difficult to characterize due to critical slowing down in conventional Monte Carlo and truncation errors inherent to tensor renormalization group methods. We develop a tensor-network Monte Carlo (TNMC) algorithm for this frustrated model, which supports both open and periodic boundary conditions. For periodic lattices, we use a sweeping block-update strategy that updates spin blocks while fixing surrounding spins. To boost efficiency, we couple our sampler with histogram reweighting, and test each technique standalone to validate these independent optimizations. We perform extensive finite-size scaling analyses at g=0.55 and g=1.0, extracting apparent critical temperatures and effective thermal exponents from Binder-cumulant data. Our method offers an efficient, nearly bias-free path for large-scale simulations of J1-J2-type frustrated magnets under periodic boundaries and generalizes readily to other frustrated spin lattices.
This thesis is composed by three projects in which we investigate thermal effects in low dimensional quantum spin systems. Thanks to recent developments in DMRG/MPS techniques, it is possible to simulate for such systems the dynamics at finite temperature. This allows us in the first two parts to study the temperature dependence of the NMR spin-lattice relaxation rate and of the INS spectra. Numerical calculations are performed for a wide range of temperatures. This represents a bridge between analytical results available for very low and very high temperatures. Interesting deviations from bosonization results valid at low energy, are seen at temperatures comparable to the energy scale of the system. In the third part we study the temperature and magnetic field dependence of the Grüneisen parameter, a valuable quantity for the detection and investigation of quantum phase transitions. Using an effective theory valid close to criticality, we show deviations from the critical behavior.