Non-Collinear Magnetic Orderings in Mott Insulators

Non-collinear magnetic orderings of four Cu magnetic moments in Mott insulators Rd$_{2}$CuO$_{4 }$(R =Nd, Pr) of I4/mmm symmetry and associated magnetic phase transitions are of interest in studies of transformations, when correlated electron-hole carriers are introduced in R$_{2-x}$Ce$_{x}$CuO$_{4\pm \delta }$. Orderings are determined by thermodynamic potential in representation by antiferromagnetic \textbf{l}$_{1}$, \textbf{ l}$_{2 }$ and magnetic \textbf{m} vectors, with orderings of \textbf{l}$_{1 }$, \textbf{l}$_{2}$ vectors along [100] , [010] axis,$_{ }$with values \textbf{l}$_{1}^{2 }$= \textbf{l}$_{2}^{2 }$= 1/2 \textbf{l}$_{0}^{2}$, [1], which can be presented as, $\Phi \quad =$ 1/2 A( \textbf{l}$_{1}^{2 }$+ \textbf{l}$_{2}^{2})$+ 1/2 A$_{3}$\textbf{l}$_{3}^{2}$+ 1/2 B\textbf{m}$^{2}_{ }$+ 1/2 D [(\textbf{l}$_{1}$\textbf{m})$^{2}$+ (\textbf{l}$_{2}$\textbf{m})$^{2}$]+ 1/2 D$_{3}$(\textbf{l}$_{3}$\textbf{m})$^{2 }$+ 1/4 I( \textbf{l}$_{1}^{2 }$+ \textbf{l}$_{2}^{2 })^{2}$ + 1/4 I$_{3 }$\textbf{l}$_{3}^{2 }$+1/4 E ( \textbf{l}$_{1}^{2 }-$ \textbf{l}$_{2}^{2 })^{2 }$ + 1/4 a ( \textbf{l}$_{1z}^{2 }$+ \textbf{l}$_{2z}^{2 })$ + 1/4 a\textbf{l}$_{3z}^{2} \quad -$ 1/4 b$_{2 }$[ ( \textbf{l}$_{1y}^{2 }$+ \textbf{l}$_{2x}^{2 })$ - ( \textbf{l}$_{1x}^{2 }$+ \textbf{l}$_{2y}^{2 })$ ] - 1/4 b$_{4 }$[ ( \textbf{l}$_{1y}^{2 }$+ \textbf{l}$_{2x}^{2 })^{2}$ + ( \textbf{l}$_{1x}^{2 }$+ \textbf{l}$_{2y}^{2 })^{2}$ ] -\textbf{ mH} where \textbf{l}$_{3}$=0. Magnetic phase transitions, are concerned with change of \textbf{l}$_{1 }$, \textbf{l}$_{2}$ values in fields $\sim $H$_{c1}$, $\sim $H$_{c}$, where \textbf{l}$_{1}^{2}$=0, \textbf{l}$_{2}^{2}$=\textbf{l}$_{0}^{2}$, when field is oriented along [100], [110] axis respectively, and next \textbf{l}$_{2 }$rotation to orthogonal to field direction in fields $\sim $H$_{c2}$, when field is along [110] axis. Fields H$_{c1}$, H$_{c}$, H$_{c2}$ are presented as, H$_{c1}^{2}$=2BE\textbf{l}$_{0}^{4}$; H$_{c}^{2}$=H$_{c1}$H$_{c2}$, if H$_{c2}^{2}$=b$_{2}$B\textbf{l}$_{0}^{2}$; H$_{c}^{2}=\surd $2$\cdot $H$_{c1}$H$_{c2}$ if H$_{c2}^{2}$=b$_{4}$B\textbf{l}$_{0}^{4}$. Formation of charge density waves of checkerboard structure can be detected by studies of transformation of magnetic phase transitions and fields in R$_{2-x}$Ce$_{x}$CuO$_{4\pm \delta }$. [1]. A. N. Bazhan, AIP Proceedings 850 (2006) 1241