Magnetic-field control of electric polarization in coupled spin chains with three-site interactions

The linear perturbation renormalization group (LPRG) is used to study coupled $XY$ chains with Dzyaloshinskii-Moriya (DM) and three-spin interactions in a magnetic field. Starting with a minimal model exhibiting the magnetoelectric effect, a spin-$\frac{1}{2} XY$ chain with nearest, next-nearest $\left(J_2^x\right)$, and DM $\left(D_1^y\right)$ interactions in a magnetic field, the recursion relations for all effective interactions generated by the LPRG transformation are found. The evaluation of these relations allows us to analyze, among others, the influence of $J_2^x,D_1^y$, three-spin ($S_i^x{S}_{i+1}^y{S}_{i+2}^z-S_i^y{S}_{i+1}^x{S}_{i+2}^z$), and interchain interactions on the thermodynamic properties. The field and temperature dependences of the polarization, specific heat, and correlation functions are found. It is shown that an interchain coupling triggers a phase transition indicated by the divergence of the renormalized coupling parameters.

Figure 1

Field dependences of polarization for $k_{xyz}=0.5,k_{xzy}=0$ (thick lines) and $k_{xyz}=0,k_{xzy}=0.5$ (thin lines) at reduced temperature $t=0.5$ (solid lines) and $t=0.2$ (dashed lines).

Figure 2

Field dependences of polarization for $k_{xyz}=0.5,k_{xzy}=0$ (a) for several values of temperature, $t=2$ (thick dashed line), 1 (thick solid line), 0.5 (thin dashed line), and 0.2 (thin solid line), and (b) for several values of the electric field, $d_1^y=0$ (thin solid line), 0.3 (thick dashed line), 0.5 (dotted line), and 0.8 (thick solid line), at $t=0.5$.

Figure 3

Field dependences of polarization for three models: (i) $k_{xyz}=0.5,k_2^x=0,d_1^0=0$ (thin solid line), (ii) $k_{xyz}=0.5,k_2^x=-0.5,d_1^y=0$ (dashed line), and (iii) $k_{xyz}=0.5,k_2^x=0,d_1^y=0.3$ (thick solid line) at $t=0.2$.

Figure 4

Field dependences of (a) polarization $P_y$ and (b) nearest-neighbor correlation function $G_{xx}$ for (i) $k_{xyz}=0.5,k_2^x=0.0,d_1^y=0.0$ (thin solid line), (ii) $k_{xyz}=0.5,k_2^x=-0.5,d_1^y=0.0$ (thick solid line), (iii) $k_{xyz}=0.5,k_2^x=0.0,d_1^y=0.3$ (dashed line), and (iv) $k_{xyz}=0.5,k_2^x=-0.5,d_1^y=0.3$ (dotted line) at $t=2$.

Figure 5

Temperature dependences of polarization $P_y$ and specific heat $C_h$ for $k_{xyz}=0.5,k_2^x=0.0,d_1^y=0.0$, and several values of magnetic field.

Figure 6

Temperature dependences of polarization $P_y$ and specific heat $C_h$ for several values of $k_{xyz}$ and magnetic field.

Figure 7

Critical temperature as a function of magnetic field for $k_{xyz}=0$ (solid lines) and $k_{xyz}=0.5$ (dashed lines), $j_1^x=0.5$ or 0.9, and $d_1^y=0$ (bottom and top curves) or $d_1^y=0.3$ (middle curve).

Figure 8

Field dependences of polarization for 2D ($k_{xyz}=0.5$) and $j_1^x=0.5,k_2^x=0$ (dashed lines), $j_1^x=0.1,k_2^x=0$ (thin solid line), and $j_1^x=0.1,k_2^x=-0.5$ (thick solid line) at temperature $t=0.5$ and 0.2.

Figure 9

Temperature dependences of polarization $P_y$ and specific heat $C_h$ for $k_{xyz}=0.5$ and $j_1^x=0$ (solid lines) and $j_1^x=0.5$ (dashed lines) at several values of field.

Figure 10

Comparison of the polarization found in this paper (solid lines) with the exact results [6] (dashed lines) for the model (2)–(3) with $J_1=1,J_2=0,D_1^y=0,K=0$, and $E=0.5$ at reduced temperatures $t=0.5$ (thin lines) and 2 (thick lines).