Level degeneracy of the transverse ising model

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SUMMARY

The discussion centers on the level degeneracy of the transverse Ising model, specifically defined by the Hamiltonian H = -J ∑_{l=1}^N σ_l^z σ_{l+1}^z + g σ_l^x under periodic boundary conditions. Initial expectations of a maximum degeneracy of 2 were challenged by numerical findings indicating a degeneracy of 4. The participants noted that the symmetry in the single particle excitation spectrum leads to a potentially higher degeneracy in the multiple excitation spectrum, although the underlying symmetry remains unclear.

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  • Understanding of quantum mechanics and Hamiltonians
  • Familiarity with the transverse Ising model
  • Knowledge of periodic boundary conditions in quantum systems
  • Basic concepts of excitation spectra in quantum physics
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  • Research the mathematical formulation of the transverse Ising model
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Physicists, quantum mechanics students, and researchers interested in quantum models and their degeneracies, particularly those focusing on the transverse Ising model.

wdlang
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the hamiltonian is

H=-J \sum_{l=1}^N \sigma_l^z \sigma_{l+1}^z+ g \sigma_l^x

here we assume periodic boundary condition

my problem is, what is the highest possible degeneracy of the levels?

initially i expected 2

but numerically i find that it is 4

i cannot understand it
 
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it is not even 4

it can be higher

i just noticed that since the single particle excitation spectrum is symmetric, the multiple excitation spectrum can be highly degenerate

but i cannot see the symmetry behind
 

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