Magnetic transitions, exchange constants, spin-flop transition, and spin-phonon coupling in Ising-like antiferromagnet <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Mn</mml:mi> <mml:mn>4</mml:mn> </mml:msub>

${\mathrm{Mn}}_{4}{\mathrm{Ta}}_{2}{\mathrm{O}}_{9}$ (MTO) is a magnetoelectric material with Ising-like antiferromagnetism (AFM) in which an applied magnetic field ($H$) induces electric polarization and an applied electric field produces magnetization. Here, we report results from our detailed investigations of the temperature ($T$) and $H$ dependence of magnetization ($M$) and heat capacity of MTO. These results show AFM ordering in MTO below ${T}_{N} \ensuremath{\sim}$ 102 K followed by additional transitions near ${T}_{S1} \ensuremath{\sim}$ 50 K and ${T}_{S2} \ensuremath{\sim}$ 20 K. The $M$ vs $H$ variations at 3 K show the presence of weak ferromagnetism (FM) below ${T}_{S1}$ superposed on AFM, with coercivity becoming zero for $T\ensuremath{\ge} {T}_{S1}$. This weak FM for $T\ensuremath{\le} {T}_{S1}$ is likely due to two inequivalent ${\mathrm{Mn}}^{2+}$ ions present in MTO. The magnetic susceptibility $\ensuremath{\chi}$ vs $T$ data for $T&gt; {T}_{N}$ is used to determine two dominant AFM exchange constants ${J}_{2}/{k}_{B}$ = --11.3 K and ${J}_{3}/{k}_{B}$ = --22.9 K, which are in good agreement with those determined from the first-principles density-functional theory ($\mathrm{DFT}+U$) calculations. It is argued that the $H$-induced transition observed at 3 K at the critical field ${H}_{c} \ensuremath{\sim}$ 20 kOe is a spin flop transition whose surprising shift to higher $H$ values with increasing $T$ is explained in terms of $T$ dependence of anisotropic $\ensuremath{\chi}$, magnetic anisotropy, and exchange fields. The temperature dependence of magnetic specific heat and entropy shows the presence of short-range spin correlations ordering up to 140 K, a $\ensuremath{\lambda}$-type anomaly at ${T}_{N}$, and a broad anomalous peak near 40 K covering the region of ${T}_{S1}$ and ${T}_{S2}$. To further explore the nature of these transitions at ${T}_{N}, {T}_{S1}$, and ${T}_{S2}$, the $T$ dependence of various Raman modes of MTO was measured and compared with the lattice dynamics calculations from $\mathrm{DFT}+U$. For $T&gt; {T}_{N}, T$ dependencies of the frequency and linewidth of the Raman modes are described by the three-phonon anharmonic approximation with considerable departures in these quantities observed at ${T}_{N}, {T}_{S1}$, and ${T}_{S2}$, which are then related to the strong spin-phonon coupling evident in MTO.