Optimal bounds from current-density calculations constrain the energy density versus number density in the massive Thirring/sine-Gordon model by a factor of two at high densities for any coupling, with the lower bound becoming exact at low densities.
Rajaraman,Solitons and instantons
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The massive Thirring / sine-Gordon model with non-zero current density
Optimal bounds from current-density calculations constrain the energy density versus number density in the massive Thirring/sine-Gordon model by a factor of two at high densities for any coupling, with the lower bound becoming exact at low densities.