Does H = XX+YY spontaneously break symmetry in 1D?

[nonequilibrium]

Hello,

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?)

 

[Physics Monkey]

Hey nonequilibrium,

This is the XY model. It is exactly solvable using free fermions and there is no symmetry breaking in the model. The fermion ground state is a Fermi gas which is translation invariant, time reversal invariant, and preserves spin rotation symmetry.

Note that, if I understood what you are suggesting correctly, it is not possible to "discretely" break a continuous symmetry. If you break a continuous symmetry you will get a Goldstone mode. You could imagine breaking some discrete symmetry, like time reversal or lattice translation, but that does not happen in this case.

Hope this helps.

 

[DrDu]

Note that, if I understood what you are suggesting correctly, it is not possible to "discretely" break a continuous symmetry. If you break a continuous symmetry you will get a Goldstone mode. You could imagine breaking some discrete symmetry, like time reversal or lattice translation, but that does not happen in this case.

I think what is meant is that the Ising model has only a discrete symmetry as sz can only point either up or down.
In the XY model, the spin can lie anywhere in the xy plane, so the symmetry is continuous (which can't be broken in 1d).