Thermodynamics of multiferroic spin chains

The minimal model to describe many spin-chain materials with ferroelectric properties is the Heisenberg model with ferromagnetic nearest-neighbor coupling $J_1$ and antiferromagnetic next-nearest-neighbor coupling $J_2$. Here we study the thermodynamics of this model using a density-matrix algorithm applied to transfer matrices. We find that the incommensurate spin-spin correlations—crucial for the ferroelectric properties and the analog of the classical spiral pitch angle—depend not only on the ratio $J_2/\left|J_1\right|$ but also strongly on temperature. We study small easy-plane anisotropies which can stabilize a vector chiral order as well as the finite-temperature signatures of multipolar phases, stable at finite magnetic field. Furthermore, we fit the susceptibilities of ${\text{LiCuVO}}_4$, ${\text{LiCu}}_2{\text{O}}_2$, and ${\text{Li}}_2{\text{ZrCuO}}_4$. Contrary to the literature, we find that for ${\text{LiCuVO}}_4$ the best fit is obtained with $J_2\sim 90\text{K}$ and $J_2/\left|J_1\right|\sim 0.5$ and show that these values are consistent with the observed spin incommensurability. Finally, we discuss our findings concerning the incommensurate spin-spin correlations and multipolar orders in relation to future experiments on these compounds.

Figure 1

(Color online) (a) Susceptibilities near the quantum critical point $\alpha =1/4$. (b) Susceptibilities for $\alpha =2.0,1.6,1.2,1.0,0.8,0.6$ (solid lines from bottom to top). For comparison, the result for the nearest-neighbor Heisenberg chain obtained by Bethe ansatz is shown (dashed line).

Figure 2

(Color online) Specific heat for $\alpha =0.4$,0.6, 1.0, 1.4, and 2.0 (from bottom to top).

Figure 3

(Color online) The wave vector $k$ of the spin-spin correlation function depends not only on $\alpha$ but also on temperature.

Figure 4

(Color online) (a) $k$ as a function of temperature for $\alpha =0.6,0.8,\dots ,2.0$ (from bottom to top). (b) Extrapolated $k$ (pitch angle) for zero temperature (circles) and $k$ at $T/J_2=1.0$ (squares). The dashed lines are a guide to the eyes. For comparison, the classical pitch angle $\varphi =\text{arccos}\left(1/4\alpha \right)$ (solid line) is shown.

Figure 5

(Color online) Various correlation functions for $\alpha =2$ and $T/J_2=0.05$. The circles denote the spin-spin $\left(O_r={\mathit{S}}_r\right)$, the squares the dimer $\left(O_r={\mathit{S}}_r{\mathit{S}}_{r+1}\right)$, and the diamonds the chiral correlation function $\left(O_r={\mathit{S}}_r\times {\mathit{S}}_{r+1}\right)$. The lines are a guide to the eyes.

Figure 6

(Color online) The static spin-structure factor $S\left(q\right)$ for temperatures $T/\left|J_1\right|=0.5,0.2,0.1,0.05,0.035,0.02$ (along the arrow direction) with (a) $\alpha =1/4$ (quantum critical point), (b) $\alpha =0.3$, (c) $\alpha =0.6$, and (d) $\alpha =1.0$. The inset of (a) shows the deviation from the sum rule $S\equiv {\sum }_{q}S\left(q\right)=3/4$.

Figure 7

(Color online) The longitudinal spin (solid line), the dimer (dashed line), and the chiral correlation length (dot-dashed line) for $\alpha =0.4$ and $\Delta =0.8$. At low temperatures the chiral correlation length diverges stronger than $1/T$ indicating long-range order at zero temperature. (a) The longitudinal spin-spin and the chiral correlation functions at $T/J_2=0.125$ showing that chiral correlations dominate. (b) The wave vector $k$ of the longitudinal spin-spin correlation function.

Figure 8

(Color online) The longitudinal spin (solid line), the dimer (dashed line), and the chiral (dot-dashed line) correlation lengths for $\Delta =0.8$ with (a) $\alpha =0.6$ and (b) $\alpha =2.0$.

Figure 9

(Color online) The longitudinal spin (solid line), the dimer (dashed line), and the chiral (dot-dashed line) correlation lengths for $\alpha =0.4$ with $\Delta =1$ and $h=0.25J_2$. (a) The wave vector $k$ of the spin-spin correlation. (b) The magnetization $m$ as a function of temperature.

Figure 10

(Color online) $S^{zz}\left(q\right)$ for $\alpha =0.32$, $h=0.02J_1$ and temperatures $T/\left|J_1\right|=2.0,1.0,0.5,0.2,0.1,0.07,0.05$ (from bottom to top). The dashed line marks $q/\left[\pi \left(1–2m\right)\right]=1/3$. The inset shows the magnetization $m$ as a function of temperature.

Figure 11

(Color online) The experimentally measured susceptibility of ${\text{LiCuVO}}_4$ for $H\parallel c$ (solid line) compared to fits (dots) following Eq. (6) with a gyromagnetic ratio $g=2.313$ (Refs. 6, 32). (a) The best fit is obtained with ${\chi }_0=6\times {10}^{-5}\text{emu}/\text{mol}$, $J_2=91\text{K}$, and $\alpha =0.5$. (b) An alternative fit with ${\chi }_0=1.4\times {10}^{-4}\text{emu}/\text{mol}$, $J_2=72\text{K}$, and $\alpha =0.6$. (c) An exchange coupling $J_2=52\text{K}$ similar to the one in Ref. 5 can be used with ${\chi }_0=2.7\times {10}^{-4}\text{emu}/\text{mol}$, however, the frustration parameter $\alpha =1.0$ chosen to obtain the best fit is still much smaller than the one in Ref. 5 and the fit is much worse than the ones shown in (a) and (b).

Figure 12

(Color online) The measured susceptibility (solid line) for ${\text{Li}}_2{\text{ZrCuO}}_4$ taken from Ref. 4 compared to fits using the $J_1\text{−}{J}_2$ model. An excellent fit (circles) is obtained using $\alpha =0.3$, $g=2$, and $J_1=-273\text{K}$ $\left(J_2=81.9\text{K}\right)$ with ${\chi }_0=0$ confirming the analysis in Ref. 4 based on exact diagonalization data. A reasonable fit (squares) is also possible with $\alpha =0.4$, $g=2.2$, ${\chi }_0=0$, and $J_1=-75\text{K}$. Inset: Blow up of the low-temperature region.

Figure 13

(Color online) The measured susceptibility (solid line) for ${\text{LiCu}}_2{\text{O}}_2$ with $H\parallel c$ taken from Ref. 1 compared to fits using the $J_1\text{−}{J}_2$ model. With $g=2.2$ the best fit is obtained with $\alpha =0.6$, $J_2=120\text{K}$, and ${\chi }_0=0$ (circles) while for $g=2.3$ a fit with $\alpha =1.6$, $J_2=83\text{K}$, and ${\chi }_0=0$ works best (squares).