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Find the lowest excitation of a given Hamiltonian

Is there a standard way to find the lowest excitation energy of given Hamiltonian without knowing the eigenstates? In particular I have the find the lowest excitations of an 1D Ising Hamiltonian in 1D ...
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1 dimensional Ising model

How to solve the Ising model in 1D by low temperature, and high temperature expansion, and by change of variable method? Can you please give me some reference links?