Magnetoelastic study on the frustrated quasi-one-dimensional spin-$\frac{1}{2}$ magnet ${\mathrm{LiCuVO}}_4$

We investigated the magnetoelastic properties of the quasi-one-dimensional spin-$\frac{1}{2}$ frustrated magnet ${\mathrm{LiCuVO}}_4$. Longitudinal-magnetostriction experiments were performed at 1.5 K in high magnetic fields of up to 60 T applied along the $b$ axis, i.e., the spin-chain direction. The magnetostriction data qualitatively resemble the magnetization results, and saturate at $H_{\text{sat}}\approx 54$ T, with a relative change in sample length of $\mathrm{\Delta }L/L\approx 1.8\times {10}^{-4}$. Remarkably, both the magnetostriction and the magnetization evolve gradually between $H_{\text{c3}}\approx 48$ T and $H_{\text{sat}}$, indicating that the two quantities consistently detect the spin-nematic phase just below the saturation. Numerical analyses for a weakly coupled spin-chain model reveal that the observed magnetostriction can overall be understood within an exchange-striction mechanism. Small deviations found may indicate nontrivial changes in local correlations associated with the field-induced phase transitions.

Figure 1

(a) Crystal structure of ${\mathrm{LiCuVO}}_4$ (b) Two ${\mathrm{Cu}}_2\mathrm{O}$ chains exist in the $ab$ plane with three main exchange interactions $J_1, J_2$, and $J_5$. The lattice constants $a$ and $b$ are 5.662 and 5.809 Å, respectively [7].

Figure 2

(a) Longitudinal magnetostriction $\mathrm{\Delta }L/L$ of ${\mathrm{LiCuVO}}_4$ and the normalized magnetization $M/M_{\text{sat}}$ from Ref. [28] as a function of magnetic field ($\mathbf{H}\parallel \mathrm{\Delta }L/L\parallel b$ axis). (b) Longitudinal magnetostriction $\mathrm{\Delta }L/L$ of ${\mathrm{LiCuVO}}_4$ and its derivative in the vicinity of the saturation field. The critical fields $H_{\text{c2}}, H_{\text{c3}}$, and $H_{\text{sat}}$ are estimated as 11, 48, and 54 T, respectively.

Figure 3

(a) Magnetostriction $\mathrm{\Delta }L/L$ and the spin-interaction energy $\mathrm{\Delta }{E}_{\text{int}}$ of ${\mathrm{LiCuVO}}_4$ at 1.5 K vs magnetic field. Both quantities are normalized to 1 at saturation. The spin-interaction energy is obtained from the magnetization reported in Ref. [28] through relation (3). (b) Logarithmic plot of $\mathrm{\Delta }L/L$ and $\mathrm{\Delta }{E}_{\text{int}}$ vs $M$. Dashed lines show power-law fits $\propto M^p$ with $p=1.32$ for $\mathrm{\Delta }L/L$ and $p=1.55$ for $\mathrm{\Delta }{E}_{\text{int}}$ (fit range is $0.4\leqq M/M_{\mathrm{sat}}\leqq 0.7$).

Figure 4

Comparison of the spin-interaction energy $\mathrm{\Delta }{E}_{\text{int}}$ per spin extracted from the experimental data with the DMRG results with and without the interchain coupling $J_5$.

Figure 5

Comparison of the magnetostriction results with the DMRG analyses on the local spin correlations $\overline{{\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_M}-\overline{{\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{M=0}}$ on the $J_1$ and $J_2$ bonds and the spin-interaction energy $\mathrm{\Delta }{E}_{\text{int}}$. Here, all the quantities are normalized between 0 and 1, which correspond to the zero-field case and the saturation, respectively.

Figure 6

Local spin correlations $\overline{{\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_M}$ on the $J_1, J_2$, and $J_5$ bonds calculated using DMRG and ED. Solid lines show the scaling [Eq. (5)] with $p=1.82$, 1.74, and 2 for the $J_1, J_2$, and $J_5$ bonds, respectively [39]. In the ED results, symbols with small, medium, and large sizes correspond to the cases of 20, 24, and 28 spins, respectively [screw boundary conditions [40] with $(L_x,L_y)=(5,4)$, (6,4), and (7,4) as described in Appendix pp2].

Figure 7

(a) Local spin correlations ${\langle {\mathbf{S}}_j·{\mathbf{S}}_{j+n}\rangle }_m^{\left(\mathrm{o}\right)}(n=1,2)$ for $N_{1\mathrm{D}}=168$ and $m=48/168$. Open and solid symbols respectively represent the DMRG data and fits using Eqs. (A2) and (A3). Solid lines connecting the fits are guides to the eye. (b) The DMRG data of ${\langle {\mathbf{S}}_j·{\mathbf{S}}_{j+n}\rangle }_m^{\left(\mathrm{o}\right)}$ for $m=0$. The data for $N_{1\mathrm{D}}=96$ are shown to demonstrate oscillations with a four-site period. We observed similar four-site period oscillations also for $N_{1\mathrm{D}}=120$ and 168. Dashed lines connecting the data points are guides to the eye.

Figure 8

Uniform parts of the local spin correlations, $c_1$ and $c_2$, as functions of $m$. These were obtained from the fitting of the DMRG data of ${\langle {\mathbf{S}}_j·{\mathbf{S}}_{j+n}\rangle }_m^{\left(\mathrm{o}\right)}$ for $N_{1\mathrm{D}}=96,120,$ and 168 with Eqs. (A2) and (A3) as shown in Fig. 7.

Figure 9

Comparison of the magnetization data of ${\mathrm{LiCuVO}}_4$ from Ref. [28] with the DMRG and ED results for the $J_1\text{−}{J}_2\text{−}{J}_5$ spin-chain model. Numerical results are presented for the cases with and without the interchain coupling $J_5$ to highlight its effect. In the DMRG result, $J_5$ is treated in a mean-field manner via a shift of the magnetic field by $4J_5(M/2M_{\mathrm{sat}})/g{\mu }_{\mathrm{B}}$ for each data point. For a ferromagnetic interchain coupling $J_5<0$, this treatment results in a small region near the saturation in which $M/M_{\mathrm{sat}}$ is multivalued, as seen in the figure; this should be regarded as an artifact of the mean-field treatment.