Field-induced incommensurate order for the quasi-one-dimensional $XXZ$ model in a magnetic field

We investigate phase transitions of the quasi-one-dimensional $S=1∕2$ $XXZ$ model in a magnetic field, using bosonization combined with a mean-field treatment of the interchain interaction. We then find that the field-induced incommensurate order is certainly realized in the low-field region, while transverse staggered order appears in the high-field region. On the basis of the result, we discuss the field-induced phase transition recently observed for $\mathrm{Ba}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$.

Figure 1

(Color online) Exact magnetization curve $M$, critical exponent $\eta$, and spin wave velocity $v$ for the $XXZ$ chain of $\Delta =2$.

Figure 2

(Color online) Transition temperatures for $\Delta =1.05$, 1.5, 2.0, and 3.0 with $J^{\prime }=0.01$. The intrachain coupling $J$ is set to unity. The solid and broken lines, respectively, represent $T_c^{\left(\mathit{ic}\right)}$ and $T_c^{\left(s\right)}$. The vertical dotted lines for $\Delta =2.0$ and 3.0 indicate the critical field $H_c$. $H_c$ for $\Delta =1.05$ and 1.5 is not shown here, since it is located in vicinity of $H=0$. The horizontal axis is normalized by the saturation field $H_s$.

Figure 3

(Color online) Transition temperatures for the IC and transverse staggered orders. The solid and the broken lines, respectively, represent $T_c^{\left(\mathit{ic}\right)}$ and $T_c^{\left(s\right)}$ for $\Delta =2.17$ and $J^{\prime }∕J=0.00138$. The vertical dotted line indicates $H_c=3.9\mathrm{T}$. The solid circles indicate the experimentally observed transition temperature (Ref. 4) and the solid triangles mean the phase boundary between the $z$ Néel order and the field-induced IC order.