The imaginary part of dielectric function related to interband transition. But how can i interpret the negative value of the real part by band theory.
You can't. Band theory is a single electron approximation. And this is a collective effect. But the quantum theory of the metal response to the electromagnetic field exists.
Yes, you, normally, can't. Also, a fast[1] outline of the Lorentz Oscillator Model (for plasmas and metals[2]) might help[1,2] you, on the negative value (slide 7[1], of the real part) of the metals' dielectric function.
1. https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/lecture-notes/MIT6_007S11_lec23.pdf
2. https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6_007S11_lorentz.pdf
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