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October 8, 2025Open Access

Duality symmetry, zero energy modes and boundary spectrum of the sine-Gordon/massive Thirring model

Key Points

The system exhibits duality symmetry, connecting models with opposite mass parameters and boundary conditions.
For negative mass parameters and open boundary conditions, the system shows a unique trivial ground state.
Zero energy modes localized at boundaries indicate a topological phase for certain conditions.
Differences in boundary excitation structures arise between attractive and repulsive regimes.

Abstract

We solve the one-dimensional massive Thirring model, which is equivalent to the one-dimensional sine-Gordon model, with two types of Dirchlet boundary conditions: open boundary conditions (OBC) and twisted open boundary conditions (OBC). The system exhibits a duality symmetry which relates models with opposite bare mass parameters and boundary conditions, i. e: m₀ - m₀, OBC. For m₀0 and OBC, the trivial phase occurs, whereas the topological phase occurs for m₀>0 and OBC. In addition, we analyze the structure of the boundary excitations, finding significant differences between the attractive (g>0) and the repulsive (g<0) regimes.

Duality symmetry, zero energy modes and boundary spectrum of the sine-Gordon/massive Thirring model

Key Points

Abstract

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