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The system Ag[Li$_{1/3}$Mn$_{2/3}$]O$_{2}$ belongs to a quaternary3R-delafossite family and crystallizes in a monoclinic symmetry with spacegroup $C\,2/m$ and the magnetic Mn$^{4+}$($S=3/2$) ions form a honeycombnetwork in the $ab$-plane. An anomaly around 50 K and the presence ofantiferromagnetic (AFM) coupling (Curie-Weiss temperature $\theta_{CW}\sim-51$K) were inferred from our magnetic susceptibility data. The magnetic specificheat clearly manifests the onset of magnetic ordering in the vicinity of 48\,Kand the recovered magnetic entropy, above the ordering temperature, falls shortof the expected value, implying the presence of short-range magneticcorrelations. The (ESR) line broadening on approaching the ordering temperature$T_{{\rm N}}$ could be described in terms of a Berezinski-Kosterlitz-Thouless(BKT) scenario with $T_{{\rm KT}}=40(1)$ K. $^{7}$Li NMR line-shift probed as afunction of temperature tracks the static susceptibility (K$_{iso}$) ofmagnetically coupled Mn$^{4+}$ ions. The $^{7}$Li spin-lattice relaxation rate(1/$T$$_{1}$) exhibits a sharp decrease below about 50 K. Combining our bulkand local probe measurements, we establish the presence of an ordered groundstate for the honeycomb system Ag$_{3}$LiMn$_{2}$O$_{6}$.Our ab-initioelectronic structure calculations suggest that in the $ab$-plane, the nearestneighbor (NN) exchange interaction is strong and AFM, while the next NN and thethird NN exchange interactions are FM and AFM respectively. In the absence ofany frustration the system is expected to exhibit long-range, AFM order, inagreement with experiment.

Structural, thermodynamic, and local probe investigations of a honeycomb material Ag$_{3}$LiMn$_{2}$O$_{6}$