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arXiv:2408.11163 · PNAS Nexus

Tomonaga–Luttinger liquid and quantum criticality in spin-12 antiferromagnetic Heisenberg chain C14H18CuN4O10 via Wilson ratio

Sharath Kumar Channarayappa, Sankalp Kumar, N. S. Vidhyadhiraja, Sumiran Pujari, M. P. Saravanan, Amal Sebastian, Eun Sang Choi, Shalinee Chikara, Dolly Nambi, Athira B. Suresh, Siddhartha Lal, D. Jaiswal‐Nagar

Published Aug 23, 2024• 6 citations

Abstract

Abstract The ground state of a one-dimensional spin-12 uniform antiferromagnetic Heisenberg chain (AfHc) is a Tomonaga–Luttinger liquid which is quantum-critical with respect to applied magnetic fields up to a saturation field μ0Hs beyond which it transforms to a fully polarized state. Wilson ratio has been predicted to be a good indicator for demarcating these phases [Phys. Rev. B 96, 220401 (2017)]. From detailed temperature and magnetic field-dependent magnetization, magnetic susceptibility and specific heat measurements in a metalorganic complex and comparisons with field theory and quantum transfer matrix method calculations, the complex was found to be a very good realization of a spin-12 AfHc. Wilson ratio obtained from experimentally obtained magnetic susceptibility and magnetic contribution of specific heat values was used to map the magnetic phase diagram of the uniform spin-12 AfHc over large regions of phase space demarcating Tomonaga–Luttinger liquid, saturation field quantum critical, and fully polarized states. Luttinger parameter and spinon velocity were found to match very well with the values predicted from conformal field theory.

DOI: 10.1093/pnasnexus/pgae363

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Abstract The ground state of a one-dimensional spin-12 uniform antiferromagnetic Heisenberg chain (AfHc) is a Tomonaga–Luttinger liquid which is quantum-critical with respect to applied magnetic fields up to a saturation field μ0Hs beyond which it transforms to a fully polarized state. Wilson ratio has been predicted to be a good indicator for demarcating these phases [Phys. Rev. B 96, 220401 (2017)]. From detailed temperature and magnetic field-dependent magnetization, magnetic susceptibility and specific heat measurements in a metalorganic complex and comparisons with field theory and quantum transfer matrix method calculations, the complex was found to be a very good realization of a spin-12 AfHc. Wilson ratio obtained from experimentally obtained magnetic susceptibility and magnetic contribution of specific heat values was used to map the magnetic phase diagram of the uniform spin-12 AfHc over large regions of phase space demarcating Tomonaga–Luttinger liquid, saturation field quantum critical, and fully polarized states. Luttinger parameter and spinon velocity were found to match very well with the values predicted from conformal field theory.

Magnon bound states emerging in one-dimensional (1D) spin chains still lack a rigorous understanding. In this Rapid Communication we show that the length-1 spin strings significantly dominate the critical properties of spinons, magnons, and free fermions in the 1D antiferromagnetic spin-1/2 chain. Using the Bethe ansatz solution, we analytically calculate the scaling functions of the thermal and magnetic properties of the model, providing a rigorous understanding of the quantum criticality of spinons. It turns out that the double maxima in specific heat elegantly mark two crossover temperatures fanning out from the critical point, indicating three quantum phases: the Tomonaga-Luttinger liquid (TLL), the quantum critical, and fully polarized ferromagnetic phases. For the TLL phase, the Wilson ratio

{R}_{W}=4{K}_{s}

remains almost temperature independent, where

{K}_{s}

is the Luttinger parameter. Furthermore, by applying our results, we precisely determine the quantum scalings and critical exponents of all magnetic properties in the ideal 1D spin-1/2 antiferromagnet

\mathrm{Cu}(\mathrm{C}{}_{4}\mathrm{H}{}_{4}\mathrm{N}{}_{2})(\mathrm{NO}{}_{3}){}_{2}

, recently studied by Kono et al. [Phys. Rev. Lett. 114, 037202 (2015)]. We further find that the magnetization maximum used in experiments is not a good quantity to map out the finite-temperature TLL phase boundary.

Using micromechanical force magnetometry, we have measured the magnetization of the strong-leg spin-1/2 ladder compound (C(7)H(10)N)(2)CuBr(2) at temperatures down to 45 mK. Low-temperature magnetic susceptibility as a function of field exhibits a maximum near the critical field H(c) at which the magnon gap vanishes, as expected for a gapped one-dimensional antiferromagnet. Above H(c) a clear minimum appears in the magnetization as a function of temperature, as predicted by theory. In this field region, the susceptibility in conjunction with our specific-heat data yields the Wilson ratio R(W). The result supports the relation R(W)=4K, where K is the Tomonaga-Luttinger-liquid parameter.

High-precision dc magnetization measurements have been made on Cu(C4H4N2) (NO3)2 in magnetic fields up to 14.7 T, slightly above the saturation field Hs=13.97 T, in the temperature range from 0.08 to 15 K. The magnetization curve and differential susceptibility at the lowest temperature show excellent agreement with exact theoretical results for the spin-1/2 Heisenberg antiferromagnet in one dimension. A broad peak is observed in magnetization measured as a function of temperature, signaling a crossover to a low-temperature Tomonaga-Luttinger-liquid regime. With an increasing field, the peak moves gradually to lower temperatures, compressing the regime, and, at Hs, the magnetization exhibits a strong upturn. This quantum critical behavior of the magnetization and that of the specific heat withstand quantitative tests against theory, demonstrating that the material is a practically perfect one-dimensional spin-1/2 Heisenberg antiferromagnet.

In the present work, we provide a comprehensive numerical investigation of the magnetic properties and phase spectra of three types of spin-1/2 branched chains consisting of one, two and three side spins per unit block with intra-chain interaction and a uniform inter-chain interaction in the presence of an external magnetic field. In a specific magnetic field interval, the low-temperature magnetization of these chains shows a step-like behavior with a pronounced plateau depending on the strength and the type of intra-chain interaction being ferromagnetic or antiferromagnetic. We demonstrate that when inter-chain interaction J1 is antiferromagnetic and intra-chain interaction J2 is ferromagnetic, the magnetization of the models manifests a smooth increase without a plateau, which is evidence of the existence of a Luttinger-like spin liquid phase before reaching its saturation value. On the other hand, when J1 is ferromagnetic and J2 is antiferromagnetic, the low-temperature magnetization of the chain with two branches shows an intermediate plateau at one-half of the saturation magnetization that breaks a quantum spin liquid phase into two regions. The magnetization of the chain with three branches exhibits two intermediate plateaus and two regions of a quantum spin liquid. We demonstrate that the chains with more than one side spin illustrate in their ground-state phase diagram a Kosterlitz-Thouless transition from a gapful phase to a gapless spin liquid phase.

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.