How to get the conserved quantities of a integrable quantum system?

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1. Nov 12, 2014

Billy Yang

If I have an arbitrary quantum many-body model, what is the method to calculate the the conserved quantities if the model is integrable. If it is hard to explain, can you recommend some relevant books for me? Thanks a lot!

2. Nov 12, 2014

Maybe_Memorie

This book discusses lots of different types of integrable models, both classical and quantum. As far as I'm aware this is the best intro book.

Now, it's really only spin chains I'm familiar with, but in this case the conserved quantities are given by taking the trace of the monodromy matrix on the auxiliary space. This defines the transfer matrix which generates a tower of commuting charges, one of which the is the Hamiltonian of the system. The one-dimensional spin chain with L sites and SU(2) symmetry has L degrees of freedom, whereas the transfer matrix only gives you L-1 conserved quantities. By adding a component of spin, say $S^z$, we obtain the full set of commuting charges, and the system is integrable.

3. Nov 12, 2014

Billy Yang

Dear Maybe_Memorie:
Thanks for your elaborate response and it really helps me a lot, although I can't fully understand the concepts you mentioned. But I am interested in the theory you introduced, especially its group theory parts. Thanks again!
Billy

4. Nov 18, 2014

Billy Yang

Maybe_Memorie:
Could you please tell me what the conserved quantities are in an ising model and how to express it using Pauli matrix? I am especially interested in this.

Billy