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I am working in 1D here. For the ferromagnetic Ising model ##H = -\sum_k X_k X_{k+1}## (or ##H = -YY##) we know that the ground state is gapped and has a twofold degeneracy due to SSB (spontaneous symmetry breaking) of the spin flip symmetry ##P = Z_1 Z_2 Z_3 \cdots##.

I am now interested in the Hamiltonian ##H = -XX - YY##. This is known to be gapless (as can be derived using a Jordan-Wigner transformation). However, is it known whether or not this displays spontaneous symmetry breaking? Note that it has a continuous symmetry ##SO(2) = U(1)##, and I am

**not**asking whether it

*continuously*breaks in 1D (as Coleman-Mermin-Wagner implies that does not happen) but rather whether there is still the

*discrete*SSB similar to what happens for the above ##XX## Ising case. Moreover, how can one show it? (Analytically? Numerically?)