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arXiv:2411.19538 · Physical review. B./Physical review. B

Haldane phase, field-induced magnetic ordering, and Tomonaga-Luttinger liquid behavior in the spin-one chain compound NiC 2 O 4 · 2 NH 3

Shuo Li, Zhanlong Wu, Yanhong Wang, Jun Luo, Kefan Du, Xiaoyu Xu, Ze Hu, Ying Chen, Jie Yang, Zheng-Xin Liu, Rong Yu, Yi Cui, Rui Zhou, Hongcheng Lu, Weiqiang Yu

Published May 29, 2025• 1 citation
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Abstract

The Haldane phase is a celebrated example of a magnetic topological phase but rarely reported in materials. The authors show here an NMR study on the spin-1 chain magnet NiC${}_{2}$O${}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}2NH${}_{3}$, which determines its ground state to be a Haldane phase at zero field. An applied magnetic field induces a magnetic ordering at low temperatures and a Tomonaga-Luttinger liquid phase above the ordering temperature with attractive fermionic interactions.

DOI
10.1103/physrevb.111.195164

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The Haldane phase is a celebrated example of a magnetic topological phase but rarely reported in materials. The authors show here an NMR study on the spin-1 chain magnet NiC2{}_{2}2​O4{}_{4}4​\ifmmode\cdot\else\textperiodcentered\fi{}2NH3{}_{3}3​, which determines its ground state to be a Haldane phase at zero field. An applied magnetic field induces a magnetic ordering at low temperatures and a Tomonaga-Luttinger liquid phase above the ordering temperature with attractive fermionic interactions.
The spin-1/21/21/2 ferromagnetic XXZ chain is a prototypical many-body quantum model, exactly solvable via the integrable Bethe ansatz method, hosting a Tomonaga-Luttinger spin liquid. However, its clear experimental realizations remain absent. Here, we present a thorough investigation of the magnetism of the structurally disorder-free compound LuCu(OH)3SO4\mathrm{LuCu}{(\mathrm{OH})}_{3}{\mathrm{SO}}_{4}LuCu(OH)3​SO4​. By conducting magnetization and electron-spin-resonance measurements on the single-crystal sample, we establish that the title compound approximates the spin-1/21/21/2 ferromagnetic XXZ chain model with a nearest-neighbor exchange strength of ${J}_{1}\ensuremath{\sim}65\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and an easy-plane anisotropy of $\ensuremath{\sim}0.994$. The specific heat demonstrates a distinctive power-law behavior at low magnetic fields (with energy scales $\ensuremath{\le}0.02{J}_{1}$) and low temperatures $(T\ensuremath{\le}0.03{J}_{1})$. This behavior is consistent with the expectations of the ideal spin-1/21/21/2 ferromagnetic XXZ chain model, thereby supporting the formation of a gapless Tomonaga-Luttinger spin liquid in LuCu(OH)3SO4\mathrm{LuCu}{(\mathrm{OH})}_{3}{\mathrm{SO}}_{4}LuCu(OH)3​SO4​.
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.
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.
Critical properties of field-induced magnetic phases are investigated in the one-dimensional S=1 Haldane system with planar anisotropy, using the finite-temperature density-matrix renormalization group method. We point out that the underlying physics of the properties differs significantly between cases of different field directions: The magnetic phase is characterized as the gapless Tomonaga-Luttinger regime in one case and as the gapful soliton regime in the other. In connection with the information obtained for the one-dimensional spin chain, possible implications for the field-induced magnetic long-range ordering are discussed.
A bosonized-field-theory representation of spin operators in the uniform-field-induced Tomonaga–Luttinger-liquid (TLL) phase in spin-1 Haldane chains is formulated by means of the non-Abelian (Tsvelik's Majorana fermion theory) and the Abelian bosonizations and Furusaki–Zhang technique. It contains massive magnon fields as well as massless boson fields. From it, asymptotic forms of spin correlation functions in the TLL phase are completely determined. Applying the formula, we further discuss the effects of perturbations (bond alternation, single-ion anisotropy terms, staggered fields, etc) for the TLL state, and the string order parameter. Throughout the paper, we often consider in some detail how the symmetry operations of the Haldane chains are translated in the low-energy effective theory.