Two dimensional multiferroics inherit prominent physical properties from both low dimensional materials and magnetoelectric materials, and can go beyond their three dimensional counterparts for their unique structures. Here, based on density function
al theory calculations, a MXene derivative, i.e., i-MXene (Ta$_{2/3}$Fe$_{1/3}$)$_2$CO$_2$, is predicted to be a type-I multiferroic material. Originated from the reliable $5d^0$ rule, its ferroelectricity is robust, with a moderate polarization up to $sim12.33$ $mu$C/cm$^2$ along the a-axis, which can be easily switched and may persist above room temperature. Its magnetic ground state is layered antiferromagnetism. Although it is a type-I multiferroic material, its Neel temperature can be significantly tuned by the paraelectric-ferroelectric transition, manifesting a kind of intrinsic magnetoelectric coupling. Such magnetoelectric effect is originated from the conventional magnetostriction, but unexpectedly magnified by the exchange frustration. Our work not only reveals a nontrivial magnetoelectric mechanism, but also provides a strategy to search for more multiferroics in the two dimensional limit.
We study the symmetrized noncommutative arithmetic geometric mean inequality introduced(AGM) by Recht and R{e} $$ |frac{(n-d)!}{n!}sumlimits_{{ j_1,...,j_d mbox{ different}} }A_{j_{1}}^*A_{j_{2}}^*...A_{j_{d}}^*A_{j_{d}}...A_{j_{2}}A_{j_{1}} | leq
C(d,n) |frac{1}{n} sum_{j=1}^n A_j^*A_j|^d .$$ Complementing the results from Recht and R{e}, we find upper bounds for C(d,n) under additional assumptions. Moreover, using free probability, we show that $C(d, n) > 1$, thereby disproving the most optimistic conjecture from Recht and R{e}.We also prove a deviation result for the symmetrized-AGM inequality which shows that the symmetric inequality almost holds for many classes of random matrices. Finally we apply our results to the incremental gradient method(IGM).
