Ising order in a magnetized Heisenberg chain subject to a uniform Dzyaloshinskii-Moriya interaction

We report a combined analytical and density matrix renormalized group study of the antiferromagnetic $XXZ$ spin-$1/2$ Heisenberg chain subject to a uniform Dzyaloshinskii-Moriya (DM) interaction and a transverse magnetic field. The numerically determined phase diagram of this model, which features two ordered Ising phases and a critical Luttinger liquid, one with fully broken spin-rotational symmetry, agrees well with the predictions of Garate and Affleck [I. Garate and I. Affleck, Phys. Rev. B 81, 144419 (2010)]. We also confirm the prevalence of the $N^z$ Néel Ising order in the regime of comparable DM and magnetic field magnitudes.

Figure 1

Solution of Kosterlitz-Thouless equations (11). Different symbols and colors depict Ising phases $N^z$ (green region) and $N^y$ (yellow region) and the critical Luttinger liquid phase (purple region) according to the RG flow criteria summarized in Table 1.

Figure 2

Phase diagram for the case of relatively strong DM interaction $D/J=0.1$. Larger $D$ promotes $N^z$ state. The two phase boundaries are given by Eq. (15) (orange dot-dashed line), and Eq. (16) (red dashed line). The phase boundary between LL and $N^y$ is located at $h/D=\sqrt{2}$ and is independent of $\mathrm{\Delta }$.

Figure 3

Phase diagram for the case of a small DM interaction $D/J=0.01$ obtained via a two-stage RG process. At the first stage RG equations (8) are integrated numerically. Second-stage equations (10) are then solved analytically. The phase boundaries given by Eq. (15) (orange dot-dashed line) and Eq. (16) (red dashed line) are seen to deviate significantly from the actual ones. This shows the importance of the first-stage flow in the case of small and intermediate $D/J$.

Figure 4

Ground-state phase diagram of the system in Eq. (1) at $D/J=0.1$ as determined by DMRG (open circles) and ITEBD (open squares) calculations for system with $L=1200$ sites. For comparison, the phase diagram at $D/J=0.05$ is also provided, obtained by DMRG simulation for the system with $L=800$ sites (open triangles). The lines are guides for the eye.

Figure 5

Structure factors $M_s^y$ (blue solid line) and $M_s^z$ (green dashed line) with $L=1600$ and $M_d$ (red dot-dashed line) with $L=1200$ (a) in the $N^z$ phase at $\mathrm{\Delta }=1.1$, $D/J=0.1$, and $h/J=0.2$, (b) in the $N^y$ phase at $\mathrm{\Delta }=0.7$, $D/J=0.1$, and $h/J=0.2$, and (c) in the $LL$ phase at $\mathrm{\Delta }=0.7$, $D/J=0.1$, and $h/J=0.075$.

Figure 6

Dependence of the incommensurate peak momentum $k^{★}$ in the spin structure factor $M_s^y\left(k\right)$ (blue circles) and dimer structure factor $M_d\left(k\right)$ (green squares) as a function of the transverse magnetic field $h$ at $D/J=0.1$. The red line denotes the theoretical prediction; we used $v=\pi J/2$.

Figure 7

Critical RG ${\ell }^{*}$ for which $|y_c\left({\ell }^{*}\right)|=1$ (obtained by solving the KT equations; see Appendix pp3) as a function of the $XXZ$ anisotropy $\mathrm{\Delta }$. Here $D/J=0.1$ and $h/J=0.2$. The system is in the $N^y$ phase (red line) for $\mathrm{\Delta }<{\mathrm{\Delta }}_c\simeq 0.94$ [the phase boundary ${\mathrm{\Delta }}_c$ is determined from Eq. (15)], while at $\mathrm{\Delta }>{\mathrm{\Delta }}_c$ the system enters the $N^z$ phase (blue line). Near the transition point, ${\ell }^{*}\gg {\ell }_s=7.37$.

Figure 8

Order parameters as a function of $\mathrm{\Delta }$ for two ordered states $N^y$ and $N^z$, at $D/J=0.1$, $h/J=0.2$, and RG length scale $\ell ={\ell }_s$. Here $\mathrm{\Delta }$ is near the phase boundary ${\mathrm{\Delta }}_c\simeq 0.94$ [determined from Eq. (15)]. See the main text and Appendix pp5 for details.

Figure 9

Order parameters $N^y\left(\pi \right)$ (red circles), $N^z\left(\pi \right)$ (blue squares), and $N^y\left(k^{*}\right)$ (green diamonds) extrapolated by a second-order polynomial (17) using data from $L=600$, 800, 1000, 1200, and 1600 chains as a function of $\mathrm{\Delta }$ at (a) $h/J=0.2$ and (b) $h/J=0.05$ with $D/J=0.1$. The crossing points of the order parameters determine the phase boundary.

Figure 10

Phase diagram of the chain with $D/J=0.1$ after the finite-size extrapolation of the order parameters to $L=\infty$ using Eq. (17). The error bar are plotted at the 95% confidence interval of the order parameters.