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arXiv:2410.23532 · Physical Review A

Topologically protected Bell-cat states in a simple spin model

B. Lajci, Duncan H J O’Dell, J. Mumford

Published Dec 23, 2025• 2 citations
arXivHTMLPDF

Abstract

We consider the topological properties of the so-called central spin model that consists of $N$ identical spins coupled to a single distinguishable central spin which arises in physical systems such as circuit-QED and bosonic Josephson junctions coupled to an impurity atom. The model closely corresponds to the Su-Schrieffer-Heeger (SSH) model except that the chain of sites in the SSH model is replaced by a chain of states in Fock space specifying the magnetization. We find that the model accommodates topologically protected eigenstates that are `Bell-cat' states consisting of a Schrödinger cat state of the $N$ spins that is maximally entangled with the central spin, and show how this state can be adiabatically created and moved along the chain by driving the central spin. The Bell-cat states are visualized by plotting their Wigner function and we explore their robustness against random noise by solving the master equation for the density matrix. We also explain the essential topological difference between identical spins and the excitations of a bosonic mode.

DOI
10.1103/y4g4-bksr

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We consider the topological properties of the so-called central spin model that consists of NNN identical spins coupled to a single distinguishable central spin which arises in physical systems such as circuit-QED and bosonic Josephson junctions coupled to an impurity atom. The model closely corresponds to the Su-Schrieffer-Heeger (SSH) model except that the chain of sites in the SSH model is replaced by a chain of states in Fock space specifying the magnetization. We find that the model accommodates topologically protected eigenstates that are `Bell-cat' states consisting of a Schrödinger cat state of the NNN spins that is maximally entangled with the central spin, and show how this state can be adiabatically created and moved along the chain by driving the central spin. The Bell-cat states are visualized by plotting their Wigner function and we explore their robustness against random noise by solving the master equation for the density matrix. We also explain the essential topological difference between identical spins and the excitations of a bosonic mode.
Topological properties of quantum systems could provide protection of information against environmental noise, and thereby drastically advance their potential in quantum information processing. Most proposals for topologically protected quantum gates are based on many-body systems, e.g., fractional quantum Hall states, exotic superconductors, or ensembles of interacting spins, bearing an inherent conceptual complexity. Here, we propose and study a topologically protected quantum gate, based on a one-dimensional single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain. The proposed YYY gate acts in the two-dimensional zero-energy subspace of a Y junction assembled from three chains, and is based on the spatial exchange of the defects supporting the zero-energy modes. With numerical simulations, we demonstrate that the gate is robust against hopping disorder but is corrupted by disorder in the on-site energy. Then we show that this robustness is topological protection, and that it arises as a joint consequence of chiral symmetry, time-reversal symmetry, and the spatial separation of the zero-energy modes bound to the defects. This setup will most likely not lead to a practical quantum computer; nevertheless it does provide valuable insight to aspects of topological quantum computing as an elementary minimal model. Since this model is noninteracting and nonsuperconducting, its dynamics can be studied experimentally, e.g., using coupled optical waveguides.
We investigate interacting Su-Schrieffer-Heeger (SSH) chains with two- and three-site unit cells using density matrix renormalization group (DMRG) simulations. By selecting appropriate filling fractions and sweeping across interaction strength Jz{J}_{z}Jz​ and dimerization $\ensuremath{\delta}$, we map out their phase diagrams and identify transition lines via entanglement entropy and magnetization measurements. In the two-site model, we observe the emergence of an interaction-induced antiferromagnetic intermediate phase between the topologically trivial and nontrivial regimes, as well as a critical region at negative Jz{J}_{z}Jz​ with suppressed magnetization and finite-size scaling of entanglement entropy. In contrast, the three-site model lacks an intermediate phase and exhibits asymmetric edge localization and antiferromagnetic ordering in both positive and negative Jz{J}_{z}Jz​ regimes. We further examine the response of edge states to Ising perturbations. In the two-site model, zero-energy edge modes are topologically protected and remain robust up to a finite interaction strength. However, in the three-site model, where the edge states reside at finite energy, this protection breaks down. Despite this, the edge-localized nature of these states survives in the form of polarized modes whose spatial profiles reflect the noninteracting limit.
We investigate the topological properties of one-dimensional weakly interacting topological insulators using bosonization. To do that we study the topological edge states that emerge at the edges of a model realized by a strong impurity or at the boundary between topologically distinct phases. In the bosonic model, the edge states are manifested as degenerate bosonic kinks at the boundaries. We first illustrate this idea on the example of the interacting Su-Schrieffer-Heeger (SSH) chain. We compute the localization length of the edge states as the width of an edge soliton that occurs in the SSH model in the presence of a strong impurity. Next, we examine models of two capacitively coupled SSH chains that can be either identical or in distinct topological phases. We find that weak Hubbard interaction reduces the ground-state degeneracy in the topological phase of identical chains. We then prove that, similarly to the noninteracting model, the degeneracy of the edge states in the interacting case is protected by chiral symmetry. We then study topological insulators built from two SSH chains with interchain hopping that represent models of different chiral symmetric universality classes. We demonstrate in bosonic language that the topological index of a weakly coupled model is determined by the type of interchain coupling, invariant under one of two possible chiral symmetry operators. Finally, we show that a general one-dimensional model in a phase with topological index $\ensuremath{\nu}$ is equivalent at low energies to a theory of at least $\ensuremath{\nu}$ SSH chains. We illustrate this idea on the example of an SSH model with longer-range hopping.
We show that topological edge states can be realized in a simple superconducting circuit composed of an inductively shunted flux qubit. The Josephson potential of that qubit forms a honeycomb lattice similar to the case of graphene. The additional shunting element produces a quartic potential along one flux direction, which effectively creates a boundary in flux space. We map analytically the low-energy dynamics of the circuit to an extended Su--Schrieffer--Heeger (SSH) chain, in which a quartic potential is added to the usual SSH Hamiltonian. This mapping is confirmed by numerical simulations of the full superconducting circuit. We argue that chiral symmetry still holds for the edge states of the extended model around a chiral point in parameter space. Close to this point, we numerically find that edge states in our superconducting circuit are topologically protected against perturbations that obey chiral symmetry, as is the case for charge noise. Our results show that topological effects previously thought to occur in multi-qubit systems, such as protection against decoherence, can also be found in much simpler single superconducting qubits.
We consider spin chain families inspired by the Su, Schrieffer and Hegger (SSH) model. We demonstrate explicitly the topologically induced spatial localisation of quantum states in our systems. We present detailed investigations of the effects of random noise, showing that these topologically protected states are very robust against this type of perturbation. Systems with such topological robustness are clearly good candidates for quantum information tasks and we discuss some potential applications. Thus, we present interesting spin chain models which show promising applications for quantum devices.
We study the interplay of spontaneous symmetry breaking and topological properties in interacting one-dimensional models. We solve these models using bozonization and identify topologically non-trivial phases by counting the additional degeneracy (affiliated with the edge modes) of a finite-size system relative to the infinite one. We find even if the mean-field solution is topological, this may not be true when it arises from spontaneous symmetry breaking, including in the Su-Schrieffer-Heeger (SSH) model. This implies that the original SSH model, as presented by Su, Schrieffer, and Heeger, is topologically trivial, as opposed to its mean-field version. A spinful version, on the other hand, does exhibit a topologically non-trivial phase. In that state, both mean-field solutions are topologically non-trivial and correspond to non-interacting SSH chains in the opposite phases with the winding number ν=1ν=1ν=1. We show that this phase is protected by a chiral symmetry, similar to the non-interacting phases.