There is an exact solution of the so called Ising model. This is a model with spin 1/2 particles on a lattice with nearest neighbor interactions. The solution was found for the one dimensional chain and periodic boundary conditions by Ising in the 1920's. It can be found on http://en.wikipedia.org/wiki/Ising_model#Ising.27s_exact_solution. It is relatively easy to understand. The model was also solved in two dimensions by L. Onsager. Much more complicated. In 3D no solution has been found so far.
There are a number of similar models. You can find a list of the exactly solvable ones on wikipedia: http://en.wikipedia.org/wiki/Category:Exactly_solvable_models.
In 1D models with short-range interactions the fermion and boson statistics are identical-so called "fermion-boson democracy". The description is given as said above, the Tomonaga-Luttinger or Thirring model.
could you pl give a n exact refrence of simple treatment ofTomonaga-Luttinger or Thirring model.
Following up on Eva's point, one of the most intuitive systems with short-ranged interaction is 1D Bosons with point-like (delta-function) interactions: The most general case (arbitary particle number or interaction strength) is the Lieb-Liniger model, solved by the Bethe ansatz. A special case would be that of infinitive repulsion (Tonks-Girardeau gas), solved more simply and intuitively by the Bose-Fermi mapping.
A "pedagogical" treatment is given in Pethick/Smith: "Bose Condensation in Dilute Gases", 2nd ed. As a simple introduction, I might mention the theoretical background in my thesis, Sec 1.4 (see link); it includes original references and an illustration in terms of the simple 2-particle model.
http://archiv.ub.uni-heidelberg.de/volltextserver/8611/1/thesis2.pdf
K. Pathak: For a review, I would recommend the book by R. Rajaraman : Solitons and Instantons , North-Holland Publ.Co, 1982.
Bethe Ansatz is not the easiest thing for a first reading. A good introductory review of bosonisation is by Eggert (arXiv 0708.0003). http://lucky.physik.uni-kl.de/~eggert/papers/pedestrian.pdf
Fot Quantum Spin systems the following book contains extremely simple treatments:
"An Introduction to Quantum spin Systems"(http://www.amazon.com/Introduction-Quantum-Systems-Lecture-Physics/dp/3642132898)
contains the most basic models (i.e. XX, XY).
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