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In literature ladder operators of different nature exist. The most famous arethose obeying canonical (anti-) commutation relations, but they are not theonly ones. In our knowledge, all ladder operators have a common feature: thelowering operators annihilate a non zero vector, the {\em vacuum}. This isconnected to the fact that operators of these kind are often used infactorizing some positive operators, or some operators which are { bounded frombelow}. This is the case, of course, of the harmonic oscillator, but not only.In this paper we discuss what happens when considering lowering operators withno vacua. In particular, after a general analysis of this situation, we proposea possible construction of coherent states, and we apply our construction tographene.

Ladder operators with no vacuum, their coherent states, and an application to graphene