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arXiv:2003.05904 · Physical review. B./Physical review. B

Direct determination of the Tomonaga-Luttinger parameter K in quasi-one-dimensional spin systems

M. Horvatić, M. Klanjšek, E. Orignac

Published Jun 9, 2020• 16 citations
arXivHTMLPDF

Abstract

We test the analytical formula for the enhancement of the nuclear magnetic resonance rate ${T}_{1}^{\ensuremath{-}1}$ by the critical spin fluctuations, over the simple power-law dependence predicted for a purely one-dimensional spin system, recently derived in the random phase approximation [M. Dupont et al., Phys. Rev. B 98, 094403 (2018)]. This prediction is experimentally confirmed by excellent fits to the published temperature dependence of ${T}_{1}^{\ensuremath{-}1}$ data in the two representative spin compounds, ${({\mathrm{C}}_{7}{\mathrm{H}}_{10}\mathrm{N})}_{2}{\mathrm{CuBr}}_{4}$ (DIMPY) and ${\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$, providing at the same time a direct and convenient experimental determination of the Tomonaga-Luttinger-liquid parameter $K$, very well in agreement with theoretical predictions.

DOI
10.1103/physrevb.101.220406

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We test the analytical formula for the enhancement of the nuclear magnetic resonance rate T1\ensuremath−1{T}_{1}^{\ensuremath{-}1}T1\ensuremath−1​ by the critical spin fluctuations, over the simple power-law dependence predicted for a purely one-dimensional spin system, recently derived in the random phase approximation [M. Dupont et al., Phys. Rev. B 98, 094403 (2018)]. This prediction is experimentally confirmed by excellent fits to the published temperature dependence of T1\ensuremath−1{T}_{1}^{\ensuremath{-}1}T1\ensuremath−1​ data in the two representative spin compounds, (C7H10N)2CuBr4{({\mathrm{C}}_{7}{\mathrm{H}}_{10}\mathrm{N})}_{2}{\mathrm{CuBr}}_{4}(C7​H10​N)2​CuBr4​ (DIMPY) and BaCo2V2O8{\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}BaCo2​V2​O8​, providing at the same time a direct and convenient experimental determination of the Tomonaga-Luttinger-liquid parameter KKK, very well in agreement with theoretical predictions.
We study the nuclear magnetic resonance (NMR) spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ in random one-dimensional spin chains as a function of the temperature and disorder strength. In the zero temperature limit, the system displays a disorder-induced quantum phase transition between a critical Tomonaga-Luttinger liquid (TLL) phase and a localized Bose glass phase. The 1/T11/{T}_{1}1/T1​ is investigated across this transition using large-scale simulations based on matrix product state techniques. We find that this quantity can detect the transition and probe the value of the dimensionless TLL parameter KKK. We also compute the NMR relaxation rate distributions for each temperature and disorder strength considered. In particular, we discuss the applicability of the stretched exponential fit to the return-to-equilibrium function to extract the 1/T11/{T}_{1}1/T1​ experimentally. The results presented here should provide valuable insights in regards of future NMR experiments in realistic disordered spin compounds.
Quasi-one-dimensional (1D) antiferromagnets are good candidates to realize a Tomonaga-Luttinger liquid (TLL), describing 1D interacting quantum systems, provided 3D coupling remains small as compared to temperature. In systems such as NiCl2{}_{2}2​-4SC(NH2{}_{2}2​)2{}_{2}2​ (DTN), made of weakly coupled spin S=1S=1S=1 chains, interchain coupling plays an important role at low temperature as it leads to antiferromagnetic long-range order below 111 K, corresponding to a Bose-Einstein condensate of magnons. Using time-dependent numerical simulations at finite temperature, the authors compute here the nuclear magnetic resonance (NMR) relaxation rate 1/T11/{T}_{1}1/T1​, a key experimental quantity, and address the question of the low-temperature crossover between TLL predictions and the higher-temperature regime in purely 111D quantum spins chains. They also comment on the experimental observation of the TLL regime, possibly masked by 333D long-range order in materials such as DTN.
A Rb-NMR study has been performed on the quasi-one-dimensional competing spin chain Rb2Cu2Mo3O12\mathrm{R}{\mathrm{b}}_{2}\mathrm{C}{\mathrm{u}}_{2}\mathrm{M}{\mathrm{o}}_{3}{\mathrm{O}}_{12}Rb2​Cu2​Mo3​O12​ with ferromagnetic and antiferromagnetic exchange interactions on nearest-neighboring and next-nearest neighboring spins, respectively. The system changes from a gapped ground state at zero field to a gapless state at ${H}_{\mathrm{C}}\ensuremath{\simeq}2\phantom{\rule{0.16em}{0ex}}\mathrm{T}$, where the existence of magnetic order below 1 K was demonstrated by a broadening of the NMR spectrum, associated with a critical divergence of 1/T11/{T}_{1}1/T1​. In the higher-temperature region, ${T}_{1}^{\ensuremath{-}1}$ showed a power-law-type temperature dependence, from which the field dependence of the Luttinger parameter KKK was obtained and compared with theoretical calculations based on the spin nematic Tomonaga-Luttinger liquid (TLL) state.
By means of nuclear spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}$, we follow the spin dynamics as a function of the applied magnetic field in two gapped quasi-one-dimensional quantum antiferromagnets: the anisotropic spin-chain system ${\mathrm{NiCl}}_{2}\mathrm{\text{\ensuremath{-}}}4\mathrm{SC}({\mathrm{NH}}_{2}{)}_{2}$ and the spin-ladder system (C5H12N)2CuBr4({\mathrm{C}}_{5}{\mathrm{H}}_{12}\mathrm{N}{)}_{2}\mathrm{Cu}{\mathrm{Br}}_{4}(C5​H12​N)2​CuBr4​. In both systems, spin excitations are confirmed to evolve from magnons in the gapped state to spinons in the gapless Tomonaga-Luttinger-liquid state. In between, ${T}_{1}^{\ensuremath{-}1}$ exhibits a pronounced, continuous variation, which is shown to scale in accordance with quantum criticality. We extract the critical exponent for ${T}_{1}^{\ensuremath{-}1}$, compare it to the theory, and show that this behavior is identical in both studied systems, thus demonstrating the universality of quantum-critical behavior.
It is generally difficult experimentally to distinguish magnetic multipolar orders in spin systems. Recently, it was proposed that the temperature dependence of the nuclear magnetic resonance relaxation rate 1/T11/{T}_{1}1/T1​ can involve an indirect but clear signature of the field-induced spin nematic or multipolar Tomonaga-Luttinger (TL) liquid phase [M. Sato, T. Momoi, and A. Furusaki, Phys. Rev. B 79, 060406(R) (2009)]. In this paper, we evaluate accurately the field and temperature dependence of 1/T11/{T}_{1}1/T1​ in spin-12\frac{1}{2}21​ frustrated J1{J}_{1}J1​-J2{J}_{2}J2​ chains combining field-theoretical techniques with numerical data. Our results demonstrate that isotherms of 1/T11/{T}_{1}1/T1​ as a function of magnetic field also exhibit distinctive nonmonotonic behavior in spin nematic TL liquid, in contrast with the standard TL liquid in the spin-12\frac{1}{2}21​ Heisenberg chain. The relevance of our results to quasi-one-dimensional edge-sharing cuprate magnets, such as LiCuVO4{\mathrm{LiCuVO}}_{4}LiCuVO4​, is discussed.
Motivated by recent studies on the quasi-one-dimensional (1D) antiferromagnet BaCo2V2O8, we investigate the Tomonaga-Luttinger-liquid properties of the 1D S=1/2 XXZ Heisenberg-Ising model in magnetic fields. By using the Bethe ansatz solution, thermodynamic quantities and the divergence exponent of the NMR relaxation rate are calculated. We observe a magnetization minimum as a function of temperature (T) close to the critical field. As the magnetic field approaches the critical field, the minimum temperature asymptotically approaches the universal relation in agreement with the recent results by Maeda et al. in Phys. Rev. Lett. 99 (2007) 057205. We observe T-linear specific heat below the temperature of the magnetization minimum. The field dependence of its coefficient agrees with the results based on conformal field theory. The field dependence of the divergence exponent of the NMR relaxation rate with decreasing temperature is obtained, indicating a change in the critical properties. The results are discussed in connection with experiments on BaCo2V2O8.