Effect of magnetic order on longitudinal Tomonaga-Luttinger liquid spin dynamics in weakly coupled spin-$\frac{1}{2}$ chains

The quantum many-body interactions in one-dimensional spin-$\frac{1}{2}$ systems are subject to Tomonaga-Luttinger liquid (TLL) physics, which predict an array of multiparticle excitations that form continua in momentum-energy space. Here we use inelastic neutron spectroscopy to study the TLL spin dynamics in ${\mathrm{SrCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$, a compound which contains weakly coupled spin-$\frac{1}{2}$ chains of Co atoms, at 0.05 K and in a longitudinal magnetic field up to 9.0 T. The measurements were performed above 3.9 T, where the ground-state Néel antiferromagnetic (AFM) order is completely suppressed and the multiparticle excitations are exclusively of the TLL type. In this region and below 7.0 T, the longitudinal TLL mode—psinon/antipsinon (P/AP)—is unexpectedly well described by a damped harmonic oscillator (DHO) while approaching the zone center defining the static spin-spin correlations. A non-DHO-type, continuumlike signal is seen at higher fields, but deviations from the ideal one-dimensional TLL still remain. This change in the P/AP mode coincides with the phase transition between the longitudinal spin density wave (LSDW) and transverse AFM order. Inside the LSDW state, the DHO-type P/AP spectral weight increases and the linewidth broadens as the magnetic order parameter decreases. These results reveal the impact of three-dimensional magnetic order on the TLL spin dynamics; they call for beyond-mean-field treatment for the interchain exchange interactions.

Figure 1

Crystal structure of ${\mathrm{SrCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$. (a) Three-dimensional atomic lattice. Unit-cell boundaries are marked by dashed lines. Shaded polygons are ${\mathrm{CoO}}_6$ octahedra. (b) Fourfold screw chains of Co atoms in the $\mathit{ac}$ plane.

Figure 2

Experiment vs theory comparison for the TLL spin dynamics at (a) ${\mathbf{Q}}_{\mathrm{AFM}}$ and (b) ${\mathbf{Q}}_{\mathrm{FM}}$. The region below 0.2 meV, where the elastic peak arising from the LSDW or TAFM order prevails, has been excluded. Solid curves are theoretical line shapes for an ideal 1D TLL, which are reproduced from Ref. [10] and then scaled to match the experimental intensities. Vertical lines mark the peak positions theory curves.

Figure 3

The inelastic neutron scattering spectra measured at (a)–(d) the Néel/TAFM zone center ${\mathbf{Q}}_{\mathrm{AFM}}=(2,3,0)$, and (e)–(h) the crystal zone center ${\mathbf{Q}}_{\mathrm{FM}}=(2,2,0)$, at different magnetic fields and at a temperature of 0.05 K. The solid lines are the numerical fits described in the main text, with instrumental resolution included; the red line is the total fit and the other colors describe the four individual contributions as labeled in the legend. The black arrows mark the Bethe 2-string modes. The insets show close-up information on the elastic peaks measured at each condition, including the Gaussian fits used to characterize them.

Figure 4

The adjusted $R$-squared value [Eq. (4)] for the fits illustrated in Fig. 3 as a function of magnetic field. ${\mu }_0{H}_{\mathrm{c}1}$ and ${\mu }_0{H}_{\mathrm{c}2}$ (main text) are indicated by vertical lines. The thick shaded lines are guides for the eye.

Figure 5

The fitted parameters of the DHO used to describe the longitudinal psinon/antipsinon (P/AP) mode, as a function of magnetic field. The parameters shown are (a) the area, (b) the center, (c) the full width at half maximum (FWHM), and (d) the effective bandwidth, defined as the energy difference between the mode energies at the two zone centers. The vertical lines mark ${\mu }_0{H}_{\mathrm{c}1}$ and ${\mu }_0{H}_{\mathrm{c}2}$ (main text). The thick shaded curves are guides for the eye.

Figure 6

The FWHM of the DHO profile describing the longitudinal P/AP mode as a function of momentum transfer at 5.5 T. $K=1.0$ and 3.0 are Néel/TAFM zone centers. The thick shaded line is a guide for the eye.

Figure 7

The magnetic field dependence of the transverse continuum contributions. (a) Combined Lorentzian$+$constant term at ${\mathbf{Q}}_{\mathrm{AFM}}$. (b) Constant at ${\mathbf{Q}}_{\mathrm{FM}}$. Those at ${\mathbf{Q}}_{\mathrm{AFM}}$ inside the TAFM phase are also included for comparison. The vertical solid lines mark ${\mu }_0{H}_{\mathrm{c}1}$ and ${\mu }_0{H}_{\mathrm{c}2}$.