Haldane phase, field-induced magnetic ordering and Tomonaga-Luttinger liquid behavior in a spin-one chain compound NiC$_2$O$_4$$\cdot$2NH$_3$

We performed single-crystal magnetic susceptibility and $^1$H NMR measurements on a quasi-1D, spin-1 antiferromagnet NiC$_2$O$_4$$\cdot$2NH$_3$, with temperature down to 100 mK and with field up to 26 T. With field applied along the chain direction (crystalline $b$ direction), a spin gap is determined at low fields. Our susceptibility and spin-lattice relaxation measurements reveal a Haldane phase at low field, with an intrachain exchange coupling $J$ $\approx$ 35 K and an easy-plane single-ion anisotropy of 17 K. A field-induced antiferromagnetic (AFM) ordering emerges at fields of 2.1 T, which sets a three-dimensional (3D) quantum critical point (QCP). The high-temperature spin-lattice relaxation rates $1/T_1$ resolves an onset of Tomonaga-Luttinger liquid behavior at field above $3.5$ T, which characterizes a hidden 1D QCP.