Topology under the Assault of Interactions: Polarization in the SSH Model ✛ ✎
Context
Topological insulators are usually described by noninteracting electrons, but in one dimension interactions are crucially important: fermions cannot avoid each other, and a non-Fermi-liquid state emerges. This work fits into the study of interacting topological phases, where numerical methods often show that topology survives up to an interaction threshold. Unlike related works that use lattice models and numerical calculations, here an analytical field theory is applied in the continuum limit.
Methods
The authors formulate the polarization in terms of the Green's function and compute it in the low-energy limit of the SSH model with nearest-neighbor interactions. Using bosonization, they map the model onto the sine-Gordon theory and apply exact results from integrable field theory: scaling dimensions and form factors. The polarization is expressed through the anomalous dimension of the fermionic field, which can be found exactly.
Results
The polarization in the continuum theory equals half the sign of the mass multiplied by the factor y = (1/(1-g/π) + 1 - g/π)/2, where g is the interaction constant. At g=0, the noninteracting result 1/2 is recovered. As g increases, the polarization changes smoothly, remaining nonzero up to the threshold g=π/2, where the gap opens and the theory becomes massless. The change in polarization is interpreted as a renormalization of the quasiparticle charge: contributions from quasiparticle smearing and the soliton's topological charge.
Implications
This work shows that in one-dimensional systems, interaction does not destroy the topological phase but merely redefines the charge of excitations. This differs from higher dimensions, where interactions can induce new phases. The analytical result provides a benchmark for testing numerical methods and extends our understanding of topology beyond the Fermi liquid.
Future development
A natural extension is to generalize the approach to other types of interactions, such as long-range or symmetry-breaking ones. One could also investigate how the anomalous dimension affects edge states and shot noise in mesoscopic systems.
Impact
The result is important for the physics of one-dimensional conductors, cold atoms in optical lattices, and quantum simulators, where interactions are controlled.
Next steps
Verify the prediction of fractional charge in numerical calculations of the interacting SSH lattice model and in experiments with cold atoms.
Key open problems
This work connects topological invariants with the concept of a non-Fermi liquid in one dimension, where a quasiparticle description is absent. It is a step toward understanding topology in strongly correlated systems, such as ladder models or Hubbard models with nontrivial band structure.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The paper computes the polarization in the SSH model with nearest-neighbor interactions. The authors use bosonization and exact results from integrable field theory to obtain an analytic expression for the polarization in terms of the anomalous dimension of the fermionic field. It is shown that the polarization remains quantized, but its amplitude depends on the interaction strength.
Where its strength is
A strong point is the analytic approach, which yields an exact result in the continuum limit, providing a benchmark for numerical methods and clarifying the role of interactions in one-dimensional topological phases.
Where it could go next
- Nearby on the arXiv there is a paper on topology determined by spin dynamics in quantum systems. It might be interesting to see how the approach of measuring spin correlations could be adapted to detect fractional charges in the interacting SSH model.based on: Topology by Aroma: How One Spin Gives Away the Quantum Phase arXiv:2607.10656
- The arXiv also features an analysis of ultrathin films that retain memory through electrical properties. One could consider how similar ideas of controlling polarization with external fields might be applied to one-dimensional interacting chains to test the stability of the topological phase.based on: Electric Dance at the Atomic Edge: How Ultrathin Films Retain Memory arXiv:2606.23508
Worth trying
- Check whether the quantization of polarization persists when disorder or a periodic potential is added.
- Compare the analytic result with numerical calculations in the interacting SSH lattice model.
- Investigate how polarization behaves near the interaction threshold where the gap opens.
Why this may matter
The result is important as an exact analytic statement about the survival of topology under interactions in one dimension, which can serve as a guide for more complex systems and numerical methods.
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The archive is sparse around this work: the closest review is only 0.51, against a usual 0.64. It looks like few have drilled nearby.
The SSH model was proposed to describe polyacetylene, a polymer where solitons behave as particles with fractional charge.
Key numbers
- interaction threshold: g = π/2
- noninteracting polarization: 1/2
- polarization change at transition: 1 (in units of e)