Boundary-dependent topological degeneracy in an Ising chain

Abstract: The topological degeneracy is a characteristic of quantum phase diagram in an Ising chain with transverse field. We revisit the phase diagram at nonzero temperature of an Ising chain with two types of open boundary conditions. In this work, we focus on an alternative boundary condition that not only removes the coupling between the two end sites but also eliminates the transverse field on them. We show that such a system can be exactly mapped onto two independent Kitaev chains, where spinless fermions correspond to domain-wall excitations. This results in a switch in the existence of the topological Kramers-like degeneracy in the phase diagram. The underlying mechanism is analyzed within the Majorana representation, which indicates that such a switch arises from the gauge dependence of the winding number in an SSH chain. The manifestation of bulk-boundary correspondence at nonzero temperature is demonstrated by numerical simulations on finite-size systems. This finding provides insight into the quantum spin chain.

• The paper demonstrates how distinct open boundary conditions trigger counterintuitive switches in topological degeneracy via bulk-boundary correspondence.
• It employs exact mappings to Kitaev chains and Majorana representations to reveal gauge-dependent winding number effects that reshape the zero-mode structure.
• Finite-temperature quench dynamics, illustrated by Loschmidt echo behavior, highlight practical implications for quantum state control and information processing.

Boundary-Dependent Topological Degeneracy in the Transverse Field Ising Chain

Overview

The study "Boundary-dependent topological degeneracy in an Ising chain" (2605.09510) investigates the impact of boundary conditions on the spectral degeneracy and topological features of the transverse field Ising chain, particularly at nonzero temperatures. Two distinct types of open boundary conditions are considered: one that removes only the coupling between chain ends (Type I), and another that also eliminates the transverse field at the ends (Type II). Through exact mappings to Kitaev chains and Majorana representations, the work elucidates counterintuitive boundary-driven switches in topological degeneracy, providing new insights into bulk-boundary correspondence and finite-temperature nonequilibrium dynamics in quantum spin chains.

Model, Boundary Types, and Degeneracy Analysis

Focusing on a 1D transverse field Ising model, the paper considers both periodic and open boundary conditions. Under periodic boundaries, quantum phase transitions are characterized by a ground state degeneracy change at $|g|=1$, with translational invariance allowing solutions via Jordan-Wigner and Bogoliubov transformations. The Type I boundary condition disables the interaction between chain ends, yielding persistent Kramers-like degeneracy in all eigenstates for $|g|<1$, attributed to bulk-boundary correspondence (BBC).

Type II boundaries further eliminate transverse fields at the two chain ends. Analytical and numerical analysis reveal that, although the phase diagram remains unchanged, the degeneracy characteristics under Type II boundary conditions are reversed compared to Type I, with BBC and edge spin conservation mechanisms determining the spectral structure. The degeneracy in Type II arises from edge spin z-component conservation combined with global spin-flip symmetry, and displays a robust topological nature, especially pronounced at nonzero temperature.

Domain-Wall Mapping and Kitaev Chain Correspondence

A key methodological contribution is the domain-wall excitation mapping for the Type II boundary condition. The Ising chain with this boundary can be decomposed into two independent Kitaev chains, each associated with spinless fermion states corresponding to domain-wall excitations. The spectrum of the original Ising Hamiltonian is shown to be equivalent to that of these two Kitaev chains. This mapping establishes a bijective correspondence between spin subspaces defined by fixed boundary spin configurations and complete fermionic bases, with the degeneracy rooted in edge conservation and spin-flip symmetry.

This approach demonstrates that bulk properties become sensitive to boundary modifications, and that the bulk-boundary correspondence is intricately linked to the structure of domain-wall excitations in open chains.

Majorana Representation and SSH Chain Gauge Dependence

Re-examining the problem through the Majorana fermion basis, the work maps the Ising chain onto an SSH (Su-Schrieffer-Heeger) chain with two isolated Majorana modes at the ends. This yields a (2N-2)-site SSH chain for Type II boundaries, contrasting with the 2N-site chain for Type I. Detailed analysis shows that the switching of the topological degeneracy region is governed by the gauge dependence of the SSH winding number, i.e., different boundary conditions correspond to distinct gauge choices in the SSH model, directly affecting zero-mode presence and degeneracy structure.

Closed-form expressions for Majorana zero modes are provided in the large N limit. For $|g|>1$, the degeneracy is fourfold due to the coexistence of edge and bulk zero modes; for $|g|<1$, only a twofold degeneracy survives, with the topological degeneracy determined by the robustness of Majorana (Kitaev-chain) zero modes.

Finite-Temperature Dynamics and Bulk-Boundary Manifestation

To demonstrate the bulk-boundary effects at finite temperatures, the paper explores quench dynamics using non-Hermitian perturbations. Observables related to nonlocal operators (such as $D$) constructed from edge Majorana modes reveal distinct dynamical behavior in different quantum phases. Specifically, Loschmidt echo simulations under quench show that in the topological phase ($|g|>1$) the system relaxes to a half-component state, with the echo converging to $0.5$, signifying coalescence of degenerate states. For $|g|<1$, the echo remains near unity, indicating insensitivity to perturbation.

These numerical and analytical results highlight practical implications for quantum state control, initialization, and measurement in spin chain systems, as well as the fundamental role of boundary-driven topological degeneracy in thermodynamic and dynamical behaviors.

Implications and Future Perspectives

The findings strongly indicate that boundary engineering can induce nontrivial and robust changes in the topological degeneracy landscape of quantum chains. This boundary-dependence, especially at finite temperature, extends the notion of bulk-boundary correspondence to dynamical, thermal states, and is relevant for quantum information applications where state degeneracy and stability under thermal and dynamical operations are pivotal.

From a theoretical standpoint, the demonstration of gauge-dependent winding number effects in SSH chains points toward deeper connections between symmetry, topology, and boundary constraints, with implications for topological quantum matter and finite-size engineering. Practically, these results suggest new avenues for manipulating degeneracy, encoding qubits, and probing quantum phase transitions in engineered spin systems, ultracold atom setups, and quantum simulation architectures.

Future research may generalize this approach to higher dimensions, explore other types of boundary manipulations, and investigate interactions beyond the Ising/Kitaev framework, particularly in the context of non-Hermitian dynamics and quantum information processing.

Conclusion

This paper rigorously analyzes the boundary-induced switching of topological degeneracy in the transverse field Ising chain, utilizing domain-wall mappings, Majorana representations, and quench dynamics. It establishes that boundary conditions, through gauge-dependent winding number effects in SSH models, fundamentally alter both spectral degeneracy and finite-temperature dynamical responses. The results broaden the understanding of bulk-boundary correspondence, highlighting the critical interplay between topology, symmetry, and boundaries in quantum spin chains (2605.09510).