Tomonaga-Luttinger Liquid Spin Dynamics in the Quasi-One-Dimensional Ising-Like Antiferromagnet ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$

Combining inelastic neutron scattering and numerical simulations, we study the quasi-one-dimensional Ising anisotropic quantum antiferromagnet ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in a longitudinal magnetic field. This material shows a quantum phase transition from a Néel ordered phase at zero field to a longitudinal incommensurate spin density wave at a critical magnetic field of 3.8 T. Concomitantly, the excitation gap almost closes and a fundamental reconfiguration of the spin dynamics occurs. These experimental results are well described by the universal Tomonaga-Luttinger liquid theory developed for interacting spinless fermions in one dimension. We especially observe the rise of mainly longitudinal excitations, a hallmark of the unconventional low-field regime in Ising-like quantum antiferromagnetic chains.

Figure 1

(a) The ground-state phase diagram of the spin-$1/2$ $XXZ$ chain under a longitudinal field with Hamiltonian (1). The Heisenberg case corresponds to $\mathrm{\Delta }=1$ and ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ to $\mathrm{\Delta }=1.9$. The gray-shaded area ($H^{*}<H<H_{\mathrm{sat}}$) is dominated by transverse spin-spin correlations and the red-shaded area ($H_c<H<H^{*}$) by longitudinal correlations. (b),(c) Magnetic structure (blue arrows) of a single ${\mathrm{Co}}^{2+}$ screw chain of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ (blue and red spheres are Co and O, respectively) at (b) $H=0$ in the gapped Néel phase ($H<H_c$) and at (c) ${\mu }_{0}H=6\mathrm{T}$ in the low-field regime of the TLL phase ($H_c<H<H^{*}$) with only the antiferromagnetic component shown. The amplitude of the magnetic moments in (c) is multiplied by 5 for clarity.

Figure 2

Field dependence of low-energy magnetic excitations in ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ across the quantum phase transition occurring for $\mathbf{H}\parallel \mathbf{c}$. Open and closed symbols correspond, respectively, to antiferromagnetic and zone center positions, and their surrounding. Circles correspond to AFM $\mathbf{Q}=(2,0,1)$ and ZC $\mathbf{Q}=(3,0,1)$ positions. Triangles denote the same AFM and ZC positions in the IC phase. Diamonds correspond to the associated satellites $\mathbf{Q}=(2,0,1+\delta )$ and $\mathbf{Q}=(3,0,1+\delta )$. The critical field is indicated by the dashed black line.

Figure 3

Inelastic scattering intensity maps showing the intrachain dispersion of the magnetic excitations around the AFM point $\mathbf{Q}=(2,0,1)$ in a longitudinal field of (a) 4.2 T and (c) 6 T, obtained experimentally from a series of constant-$Q_L$ energy scans. They are compared with numerically calculated scattering cross sections $S_n$ (Supplemental Material [24]) at (b) 4.2 T and (d) 6 T, which are the superposition of (e) transverse $S_{xx}$ and (f) longitudinal $S_{zz}$ dynamical structure factors.

Figure 4

Spin dynamics in the LSDW phase of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in the vicinity of a ZC position at ${\mu }_{0}H=4.2\mathrm{T}$ and $T=150\mathrm{mK}$. Energy scans at three different scattering vectors (b) the ZC position (3,0,1), (c) its satellite $(3,0,1+\delta )$ with $\delta =0.082$, and (d) further away along $c^{*}$. (a) Low-energy part of these scans. (e) Inelastic scattering intensity map obtained from constant-$Q_L$ energy scans as (a)–(d), which shows the dispersion of the excitations along the $c^{\star }$ direction, and is enlarged in (f). (g),(h) Numerically calculated intensity color maps to be compared with experimental maps (e),(f). In both the experimental and theoretical maps, the color scale is saturated in order to emphasize the weaker modes. (i) Transverse $S_{xx}$ and (j) longitudinal $S_{zz}$ components of the numerically calculated intensity color map (g).