Exact ground-state status of the central projected state at finite perturbation

Determine whether the state \(\left|\psi_q^N\right\rangle\), selected as the lowest-energy state within first-order degenerate perturbation theory for \(\Delta<\Delta_q=\cosh q\), remains an exact ground-state eigenstate of the perturbed XXZ Heisenberg chain for finite perturbation strength \(\lambda=\Delta-\cosh q\).

Background

Along the exact phase boundary Δq=cosh⁡q\Delta_q=\cosh q, the staggered-field XXZ chain has a highly degenerate ground-state manifold spanned by the states {∣ψqn⟩}\left\{\left|\psi_q^n\right\rangle\right\}. The authors perturb the Hamiltonian by an Ising interaction with strength λ=Δ−cosh⁡q\lambda=\Delta-\cosh q. For λ<0\lambda<0, first-order degenerate perturbation theory identifies the central-sector state ∣ψqN⟩\left|\psi_q^N\right\rangle as the lowest-energy state in the projected degenerate manifold and predicts a finite-size gap that vanishes in the thermodynamic limit. However, unlike the ferromagnetic-side ground states, the exact ground-state character of this central state for finite perturbation strength is not analytically established; the authors therefore use numerical calculations to investigate the issue.

References

In contrast to the case of \Delta >\Delta _{q}, where the ground states can be established exactly for finite \lambda , we cannot analytically establish that \left\vert \psi _{q}{N}\right\rangle remains an exact ground-state eigenstate for finite \lambda in the present regime.

— Exact phase diagram of the XXZ Heisenberg chain in a staggered magnetic field  (2609.29728 - Zhang et al., 24 Sep 2026) in Section "Quantum phase transition", case (ii), immediately before Section "DMRG results"