The product ab = (2k + 1)(2l + 1) = 2(k + l) + 1

which seems par for the course.

]]>Hi Jiahai, the order parameter q has two possible values, as you say, since it is the solution to a quadratic equation. The positive root is the physical one. Theoretically this makes sense from the replica symmetry ansatz Q_{ab} = 1/D Delta_a * Delta_b = q delta_{ab} + q0. The diagonals of this matrix Q_{aa} = q + q0 must be larger than the off-diagonals Q_{ab} = q0 for a =/= b since off-replica vectors Delta_{a} and Delta_{b} for a =/= b cannot have higher dot product than within replica overlap Delta_{a} * Delta_{b}.

For this problem, one can also see that the q order parameter has an interpretation in terms of the trace of a positive semidefinite matrix. See equation 87 here https://arxiv.org/pdf/2006.13198.pdf (note that kappa = q + lambda ).

Hope this helps!

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