Effective $S=\frac{1}{2}$ description of the $S=1$ chain with strong easy-plane anisotropy

We present a study of the one-dimensional $S=1$ antiferromagnetic spin chain with large easy-plane anisotropy, with special emphasis on field-induced quantum phase transitions. Temperature and magnetic field dependence of magnetization, specific heat, and thermal conductivity is presented using a combination of numerical methods. In addition, the original $S=1$ model is mapped into the low-energy effective $S=1/2XXZ$ Heisenberg chain, a model which is exactly solvable using the Bethe ansatz technique. The effectiveness of the mapping is explored, and we show that all considered quantities are in qualitative, and in some cases quantitative, agreement. The thermal conductivity of the considered $S=1$ model is found to be strongly influenced by the underlying effective description. Furthermore, we elucidate the low-lying electron spin resonance spectrum based on a semianalytical Bethe ansatz calculation of the effective $S=1/2$ model.

Figure 1

The magnetic field dependence of magnetization $M$ at fixed temperature (a) $T/J=0.02$ and (b) $T/J=0.2$. The solid line corresponds to TMRG results obtained for the $S=1$ large-$D$ chain and the dashed line corresponds to TBA results obtained for the $S=1/2XXZ$ chain. Vertical lines indicate the location of critical fields $H_1/J=2.28$ and $H_2/J=8$. Satisfactory agreement between the two models is achieved, particularly close to $H_2$ where the two curves are indistinguishable.

Figure 2

The temperature dependence of magnetization for (a) the $S=1$ large-$D$ model and (b) the $S=1/2XXZ$ model, for various fields. Dots indicate the position of extrema that correspond to the Luttinger liquid crossover. $T_c$ decreases toward $T=0$ as $H$ approaches $H_1$ or $H_2$.

Figure 3

Magnetic phase diagram of the $S=1$ chain with a strong easy-plane anisotropy (full points) and of the $S=1/2XXZ$ chain (open points). Symbols indicate the crossover into a finite-temperature LL regime present for both models.

Figure 4

The magnetic field dependence of specific heat ${\mathcal{C}}_v$ at fixed temperature $T/J=0.1$. The solid line corresponds to TMRG results for the $S=1$ large-$D$ model and the dashed line corresponds to TBA results of the $S=1/2XXZ$ model.

Figure 5

The temperature dependence of specific heat for various fields, calculated for the $S=1$ model using TMRG.

Figure 6

The magnetic field dependence of specific heat ${\mathcal{C}}_v$ at fixed temperature $T/J=0.5$ as calculated with TMRG (solid line) and FTLM (points) for the $S=1$ model. Deviations are due to finite-size effects of FTLM data.

Figure 7

Integrated conductivity $I_{QQ}\left(\omega \right)$ for (a) $T/J=1$ and (b) $T/J=10$ as calculated for $L=16$ sites and different fields $H$. Dashed vertical line represents ${\omega }_0/J=2\pi /L\sim 0.4$.

Figure 8

Frequency dependence of ${\kappa }_{QQ}\left(\omega \right)$ at $H=2$ and $T/J=1$. Labels ${\omega }_{\alpha ,\beta }$ indicate the boundaries of the band with nonvanishing weight at low $T$.

Figure 9

System size scaling of Drude weight $D_{QQ}$ at (a) $T/J=10$ and (b) $T/J=1$, obtained for systems with $L=6,\cdots ,16$ sites with various magnetic fields $H/J=2,4,8,10$.

Figure 10

Comparison of $S=1$ integrated conductivity $I_{QQ}\left({\omega }_0\right)$ at ${\omega }_0=2\pi /L$ for $L=16$ with exact $S=1/2$ Drude weight ${\tilde{D}}_{QQ}$ calculated in the thermodynamic limit for $T=0.5,1$, and 2 as a function of the magnetic field $H$.

Figure 11

Magnetic field dependence of ${\tilde{D}}_{QQ},{\tilde{K}}_{\text{th}}$, and MTC term at fixed temperature (a) $T/J=0.5$ and (b) $T/J=1$. Vertical lines indicate the critical fields.

Figure 12

Field dependence of $T=0$ low-lying ESR lines calculated from the effective $S=1/2$ model diagonalized through the Bethe ansatz. Lines $B$ and $C$ are the straight lines ${\omega }_B$ and ${\omega }_C$ given in Eq. (19) for fields outside the intermediate phase but bend downwards in a nontrivial manner upon entering the intermediate phase to meet at the center and thus form a $V$-like structure. The inset depicts the field dependence of the matrix element $|\langle m^{*}|{\tilde{S}}_{\text{tot}}^{-}|\tilde{\Omega }{\rangle |}^2$, which is directly relevant for the calculation of the intensity of ESR modes. Vertical dotted lines indicate the location of the critical fields $H_1$ and $H_2$ calculated from Eq. (18).