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

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SUMMARY

The discussion centers on the Hamiltonian H = -XX - YY in the context of the 1D XY model and its relationship to spontaneous symmetry breaking (SSB). It is established that this model does not exhibit SSB due to its continuous symmetry SO(2) = U(1), as supported by the Coleman-Mermin-Wagner theorem. The ground state is characterized as a Fermi gas that maintains translation and time reversal invariance, thus confirming the absence of discrete symmetry breaking. The Ising model, in contrast, demonstrates a twofold degeneracy due to its discrete symmetry.

PREREQUISITES
  • Understanding of the Ising model and its Hamiltonian formulation
  • Familiarity with the XY model and its properties
  • Knowledge of spontaneous symmetry breaking (SSB) and continuous symmetries
  • Basic concepts of quantum mechanics, particularly related to fermionic systems
NEXT STEPS
  • Research the Coleman-Mermin-Wagner theorem and its implications for 1D systems
  • Explore the Jordan-Wigner transformation and its application in quantum models
  • Study the characteristics of Goldstone modes in continuous symmetry breaking
  • Investigate the differences between discrete and continuous symmetries in quantum mechanics
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Physicists, particularly those specializing in quantum mechanics, condensed matter physics, and statistical mechanics, will benefit from this discussion, especially those interested in symmetry properties of quantum models.

nonequilibrium
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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?)
 
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.
 
Physics Monkey said:
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).
 

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