Magnetic exchange interactions in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="normal">BaMn</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">As</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>: A case study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>J</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>J</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>J</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math>Heisenberg model

${\mathrm{BaMn}}_{2}{\mathrm{As}}_{2}$ is unique among Ba${T}_{2}$As${}_{2}$ compounds crystallizing in the body-centered-tetragonal ${\mathrm{ThCr}}_{2}{\mathrm{Si}}_{2}$ structure, which contain stacked square lattices of $3d$ transition metal $T$ atoms, since it has an insulating large-moment ($3.9\phantom{\rule{0.28em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}/$Mn) G-type (checkerboard) antiferromagnetic (AF) ground state. We report measurements of the anisotropic magnetic susceptibility $\ensuremath{\chi}$ versus temperature $T$ from 300 to 1000 K of single crystals of ${\mathrm{BaMn}}_{2}{\mathrm{As}}_{2}$, and magnetic inelastic neutron scattering measurements at 8 K and ${}^{75}$As nuclear magnetic resonance (NMR) measurements from 4 to 300 K of polycrystalline samples. The N\'eel temperature determined from the $\ensuremath{\chi}(T)$ measurements is ${T}_{\mathrm{N}}=618$(3) K. The measurements are analyzed using the ${J}_{1}$-${J}_{2}$-${J}_{c}$ Heisenberg model for the stacked square lattice, where ${J}_{1}$ and ${J}_{2}$ are, respectively, the nearest-neighbor (NN) and next-nearest-neighbor intraplane exchange interactions and ${J}_{c}$ is the NN interplane interaction. Linear spin wave theory for G-type AF ordering and classical and quantum Monte Carlo simulations and molecular field theory calculations of $\ensuremath{\chi}(T)$ and of the magnetic heat capacity ${C}_{\mathrm{mag}}(T)$ are presented versus ${J}_{1}$, ${J}_{2}$, and ${J}_{c}$. We also obtain band-theoretical estimates of the exchange couplings in ${\mathrm{BaMn}}_{2}{\mathrm{As}}_{2}$. From analyses of our $\ensuremath{\chi}(T)$, NMR, neutron scattering, and previously published heat capacity data for ${\mathrm{BaMn}}_{2}{\mathrm{As}}_{2}$ on the basis of the above theories for the ${J}_{1}$-${J}_{2}$-${J}_{c}$ Heisenberg model and our band-theoretical results, our best estimates of the exchange constants in ${\mathrm{BaMn}}_{2}{\mathrm{As}}_{2}$ are ${J}_{1}\ensuremath{\approx}13$ meV, ${J}_{2}/{J}_{1}\ensuremath{\approx}0.3$, and ${J}_{c}/{J}_{1}\ensuremath{\approx}0.1$, which are all antiferromagnetic. From our classical Monte Carlo simulations of the G-type AF ordering transition, these exchange parameters predict ${T}_{\mathrm{N}}\ensuremath{\approx}640$ K for spin $S=5/2$, in close agreement with experiment. Using spin wave theory, we also utilize these exchange constants to estimate the suppression of the ordered moment due to quantum fluctuations for comparison with the observed value and again obtain $S=5/2$ for the Mn spin.