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In this paper, we show that the nonlocal discrete focusing nonlinearSchr\"odinger (NLS) and nonlocal discrete defocusing NLS equation are gaugeequivalent to the discrete coupled Heisenberg ferromagnet (HF) equation and thediscrete modified coupled HF equation, respectively. Under the continuouslimit, the discrete coupled HF equation and the modified discrete coupled HFequation lead to the corresponding coupled HF equation and modified coupled HFequation. This means that the nonlocal focusing and defocusing NLS equationsare gauge equivalent to the coupled HF and coupled modified HF equations. Thesolution of the modified coupled HF equation is obtained by using the solutionof nonlocal defocusing NLS equation.

From discrete nonlocal nonlinear Schr\"odinger equation to coupled discrete Heisenberg ferromagnet equation