it seems that there is no definite proof in experiment ?
What would you consider a definite proof? The Luttinger liquid does a good job for e.g. carbon nanotubes and edge states of fractional quantum Hall systems, but it is a linearised theory which can only deal with low energy excitations. As it is a limit, there will clearly be cases when it does not apply. There is ongoing research on the nonlinear regime.
For your and others' future reference, this review on the experimental status of the Luttinger liquids was posted on arXiv yesterday: http://arxiv.org/abs/1308.2731
The review is relatively short, but Giamarchi (the author) covers more tests than I was aware of anyway, and provides references for far more. Very readable and illuminating.
(The actual journal reference is International Journal of Modern Physics B Vol. 26, No. 22 1244004 (2012). )
Have you tried to look? And what do you define as "definite proof", is if there is such a thing in physics?
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