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Journal of the Physical Society of Japan

Boundary Effect on NMR Relaxation Rate for Tomonaga–Luttinger Model

源太 脇田, Yoshikazu Suzumura

Published Oct 10, 2007• 2 citations
Publisher

Abstract

The NMR relaxation rate, 1/ T 1 , in one-dimensional Tomonaga–Luttinger model with an open boundary has been investigated to comprehend the electronic state, which depends on x , the distance from the boundary. Based on the bosonization scheme, we examine the spin response function, R ( x , T ) [∝1/( T 1 T )], as the function of x , temperature ( T ) and interaction. It is shown that, with increasing x , R ( x , T ) increases from zero with an oscillation and reduces to a finite value after taking a maximum of the peak height. The amplitude of the oscillation is enhanced by interaction but decreases exponentially due to thermal fluctuation. We demonstrate properties associated with the Tomonaga–Luttinger liquid from the view point of the interplay of boundary, interaction and temperature.

DOI
10.1143/jpsj.76.104709

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The NMR relaxation rate, 1/ T 1 , in one-dimensional Tomonaga–Luttinger model with an open boundary has been investigated to comprehend the electronic state, which depends on x , the distance from the boundary. Based on the bosonization scheme, we examine the spin response function, R ( x , T ) [∝1/( T 1 T )], as the function of x , temperature ( T ) and interaction. It is shown that, with increasing x , R ( x , T ) increases from zero with an oscillation and reduces to a finite value after taking a maximum of the peak height. The amplitude of the oscillation is enhanced by interaction but decreases exponentially due to thermal fluctuation. We demonstrate properties associated with the Tomonaga–Luttinger liquid from the view point of the interplay of boundary, interaction and temperature.
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.
Recently, it has been shown that spin nematic (quadrupolar) or higher multipolar correlation functions exhibit a quasi-long-range order in the wide region of the field-induced Tomonaga-Luttinger-liquid (TLL) phase in spin-12\frac{1}{2}21​ zigzag chains. In this Rapid Communication, we point out that the temperature dependence of the NMR relaxation rate 1/T11/{T}_{1}1/T1​ in these multipolar TLLs is qualitatively different from that in more conventional TLLs of one-dimensional quantum magnets (e.g., the spin-12\frac{1}{2}21​ Heisenberg chain); 1/T11/{T}_{1}1/T1​ decreases with lowering temperature in multipolar TLL. We also discuss low-energy features in spin dynamical structure factors which are characteristic of the multipolar TLL phases.
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.
PACS. 05.30.-d – Quantum statistical mechanics. PACS. 67.40.Db – Quantum statistical theory, ground state, elementary excitations. PACS. 05.30.Jp – Boson systems. Abstract. – The ground state properties and low-lying excitations of a (quasi) one-dimensional system of longitudinally confined interacting bosons are studied. This is achieved by extending Haldane’s harmonic-fluid description to open boundary conditions. The boson density, oneparticle density matrix, and momentum distribution are obtained accounting for finite-size and boundary effects. Friedel oscillations are found in the density. Finite-size scaling of the momentum distribution at zero momentum is proposed as a method to obtain from the experiment the exponent that governs phase correlations. The strong correlations between bosons induced by reduced dimensionality and interactions are displayed by a Bijl-Jastrow wave function for the ground state, which is also derived. The names of Tomonaga and Luttinger are commonly associated with interacting electron systems in one-dimension (1d). The so-called Tomonaga-Luttinger liquids (TLL’s) have the remarkable property that their low-energy spectrum is completely exhausted by gapless
We present NMR measurements of a strong-leg spin-1/2 Heisenberg antiferromagnetic ladder compound (C7H10N)2CuBr4 under magnetic fields up to 15 T in the temperature range from 1.2 K down to 50 mK. From the splitting of NMR lines, we determine the phase boundary and the order parameter of the low-temperature (three-dimensional) long-range-ordered phase. In the Tomonaga-Luttinger regime above the ordered phase, NMR relaxation reflects characteristic power-law decay of spin correlation functions as 1/T1∝T(1/2K-1), which allows us to determine the interaction parameter K as a function of field. We find that field-dependent K varies within the 1<K<2 range, which signifies attractive interaction between the spinless fermions in the Tomonaga-Luttinger liquid.