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In a many-body system, we can know the spectrum function at a particular temperature from Green function. It means density of states. A peak of spectrum represents one mode. My question is that in the same model why the peak of the spectrum gets broader with lower dimension?

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Do you have an example of the spectral peak broadening in lower dimension? The broadening is directly related to large imaginary part of the self energy, and I can not see how is that related to the dimensionality. – Everett You Feb 7 at 17:16

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