Bosonic Bott Index in Magnonic Systems

• Bosonic Bott index is a real-space topological invariant for bosonic BdG systems that classifies positive-energy magnonic bands even when translational symmetry is broken.
• It employs metric-weighted projectors and Bott unitaries to compute indices that correspond to the sum of Chern numbers below a gap, ensuring clear bulk-boundary correspondence.
• In disordered systems, the index robustly tracks topological transitions and multi-channel edge transport, providing actionable insights into edge-state behavior and disorder effects.

The bosonic Bott index is a real-space topological invariant for bosonic band structures, especially magnons in two-dimensional magnets, that remains meaningful when translational symmetry is broken by disorder, quasicrystalline order, or finite boundaries. In current magnonic usage it denotes a $\mathbb{Z}$-valued invariant built from projected position operators and a bosonic, metric-compatible projector, thereby generalizing the Bott index of fermionic systems to bosonic Bogoliubov–de Gennes (BdG) settings with anomalous terms such as $aa$ and $a^\dagger a^\dagger$ (Huang et al., 30 Dec 2025). In clean limits it agrees with the sum of Chern numbers below a chosen gap, while in disordered systems it continues to diagnose topological phases and their bulk–boundary correspondence, including disorder-induced transitions and multi-channel edge transport (Wang et al., 2020).

1. Definition and conceptual scope

The defining feature of the bosonic Bott index is that it is a real-space invariant adapted to bosonic quasiparticles whose single-particle description is not an ordinary Hermitian band Hamiltonian but a bosonic BdG problem with a nontrivial metric. In magnonic systems this occurs when linear spin-wave Hamiltonians contain pseudodipolar or bond-anisotropy terms that generate anomalous couplings. The invariant is designed to classify positive-energy bosonic bands separated by a bulk gap, and it is computed directly in finite geometries with periodic boundary conditions so that it probes bulk topology rather than edge-state contamination (Huang et al., 30 Dec 2025).

This object differs conceptually from two nearby notions. First, it differs from the fermionic Bott index because the bosonic eigenmodes are paraunitary rather than unitary, and the projector must therefore be weighted by the bosonic metric. Second, it should not be conflated with the bosonic SPT index defined for interacting, symmetry-protected phases of lattice bosons in the thermodynamic limit. That latter construction is an $H^3(G,U(1))$-valued invariant built from local symmetry-twist associators and does not itself use the term “Bosonic Bott Index” (Sopenko, 2021).

2. Bosonic BdG structure and metric-weighted projectors

In bosonic BdG systems the diagonalization problem is formulated with an indefinite metric. For the multiband Kagome ferromagnet, the momentum-space generalized eigenvalue problem is

$$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$

with $\Sigma_z = I_3 \otimes \sigma_z$, while in real space one uses

$\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$

with $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$. The eigenvectors obey paraunitary normalization,

$\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$

rather than ordinary unitary normalization. This is the structural reason that standard projector formulas must be modified in bosonic problems (Huang et al., 30 Dec 2025).

The projector entering the bosonic Bott index is built from positive-energy bosonic modes and the metric. In real space, for an energy reference $E_*$ inside a bulk gap, the projector onto the positive-energy eigenstates below $aa$0 is

$aa$1

where the sum is restricted to positive-energy magnon bands and the eigenvectors satisfy $aa$2. In momentum space the band projector is similarly written in $aa$3-weighted form. Because bosonic BdG spectra are symmetric about zero in particle–hole-like pairs, only the positive-energy sector is counted in defining band topology and Bott indices (Huang et al., 30 Dec 2025).

A closely related formulation appears in the earlier honeycomb-magnon construction, where the bosonic commutation metric is denoted $aa$4 and the projector is

$aa$5

That work explicitly identified the essential bosonic modification as the replacement of the fermionic unit metric by the bosonic commutation metric in both diagonalization and projector construction (Wang et al., 2020).

3. Bott unitaries, branch conditions, and numerical implementation

Once the bosonic projector is fixed, the Bott index is defined from projected position operators. For an energy $aa$6 in a bulk gap,

$aa$7

Here $aa$8 and $aa$9 are the “Bott unitaries” obtained by compressing the position operators to the projected subspace. In practical calculations one uses periodic boundary conditions in both directions, with diagonal rescaled coordinate matrices $a^\dagger a^\dagger$0 and $a^\dagger a^\dagger$1, so that $a^\dagger a^\dagger$2 and $a^\dagger a^\dagger$3 implement one-unit-cell twists along $a^\dagger a^\dagger$4 and $a^\dagger a^\dagger$5 (Huang et al., 30 Dec 2025).

The logarithm is evaluated on the principal branch. The index is integer-valued and robust against small numerical perturbations when the gap is open and the relevant matrices remain nonsingular. In the underlying finite-torus Bott theory, the analogous spectral condition is that $a^\dagger a^\dagger$6 must not belong to the spectrum of

$a^\dagger a^\dagger$7

and a useful sufficient condition for unitary matrices is $a^\dagger a^\dagger$8, which ensures that the principal logarithm is well defined (Toniolo, 2017). The same branch-cut logic underlies bosonic computations.

Implementations in disordered magnon systems typically reorder eigenvalues to isolate the target band or bands, build $a^\dagger a^\dagger$9-weighted projectors, verify that $H^3(G,U(1))$0 and $H^3(G,U(1))$1 are nonsingular, and average over disorder realizations. In the multiband Kagome study, Bott-index calculations used periodic-boundary lattices of size $H^3(G,U(1))$2 and $H^3(G,U(1))$3 with disorder averages over 50 realizations, while finite samples of $H^3(G,U(1))$4 sites and stripe widths of 80 cells were used for spectral and edge analyses (Huang et al., 30 Dec 2025). In the earlier honeycomb study, Bott computations employed $H^3(G,U(1))$5 with periodic boundary conditions and averages over 100 disorder realizations (Wang et al., 2020).

4. Equality to the Chern number and bulk–boundary correspondence

The classical Bott framework on a finite torus establishes that projector-based Bott invariants and Chern numbers coincide under locality and gap hypotheses. For a gapped lattice Hamiltonian on a finite two-torus, the Bott index of projected torus-translation operators equals the projector Chern number,

$H^3(G,U(1))$6

and this equality can be proven either for quasi-unitaries or for exact unitaries such as

$H^3(G,U(1))$7

That theory is statistics-independent at the level of projector geometry, but it also emphasizes a caveat: in bosonic BdG systems one must ensure an appropriate Hermitian reduction or introduce a metric-weighted modification because the positive-frequency subspace need not be orthogonal in the standard Hilbert inner product (Toniolo, 2017).

In bosonic magnonic practice this equality is realized through the $H^3(G,U(1))$8-weighted projector formalism. The clean-limit Berry curvature of band $H^3(G,U(1))$9 is computed from the bosonic projector as

$$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$0

with

$$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$1

and in the clean limit one finds

$$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$2

For the three-band Kagome ferromagnet at $$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$3 and $$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$4, the band Chern numbers and Bott indices are

$$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$5

Stripe calculations simultaneously show chiral magnon edge states in each gap, so the number of edge branches observed in local density of states equals the magnitude of the Bott index of the bands below that gap, which is the bulk–boundary correspondence in the bosonic setting (Huang et al., 30 Dec 2025).

An operator-theoretic extension on infinite-dimensional Hilbert spaces reaches the same conclusion from a different angle. There the Bott index is defined for unitary or invertible operators under trace-class and spectral hypotheses, and in two-dimensional lattice systems it equals the noncommutative Chern number and hence the transverse Hall conductance. Because this construction depends only on locality, spectral gaps, and suitable projectors, it provides a foundation for bosonic band applications once the bosonic spectral projector is properly defined (Toniolo, 2021).

5. Multiband magnonic phases and high Bott indices

The most developed multiband realization to date is the bosonic Kagome ferromagnet with three sites per unit cell. Its spin Hamiltonian contains ferromagnetic exchange $$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$6, easy-axis anisotropy $$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$7, and a nearest-neighbor pseudodipolar interaction $$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$8,

$$\Sigma_z H_{\mathbf{k}} \Psi_{\mathbf{k}} = \Psi_{\mathbf{k}} \Sigma_z E_{\mathbf{k}},$$9

After the Holstein–Primakoff expansion, the pseudodipolar term generates $\Sigma_z = I_3 \otimes \sigma_z$0 and $\Sigma_z = I_3 \otimes \sigma_z$1 couplings, so the problem is genuinely bosonic BdG rather than a number-conserving tight-binding model (Huang et al., 30 Dec 2025).

For $\Sigma_z = I_3 \otimes \sigma_z$2 and $\Sigma_z = I_3 \otimes \sigma_z$3, the three positive-energy bands are separated by two gaps and carry invariants $\Sigma_z = I_3 \otimes \sigma_z$4. Introducing staggered anisotropy,

$\Sigma_z = I_3 \otimes \sigma_z$5

drives a sequence of topological phase transitions,

$\Sigma_z = I_3 \otimes \sigma_z$6

with edge states disappearing gap by gap. This is the simplest explicit bosonic realization in which a Bott index of magnitude greater than one is not merely formal but tied to observable multichannel edge structure (Huang et al., 30 Dec 2025).

Higher-index phases emerge when the Hamiltonian is extended by Dzyaloshinskii–Moriya interaction, next-nearest-neighbor exchange $\Sigma_z = I_3 \otimes \sigma_z$7, and hard-axis bond anisotropy $\Sigma_z = I_3 \otimes \sigma_z$8. In that setting the reported topological triples include

$\Sigma_z = I_3 \otimes \sigma_z$9

for appropriate values of $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$0 and $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$1. These phases are confirmed both by momentum-space Chern numbers and by real-space bosonic Bott indices. Stripe calculations show multiple chiral edge channels per gap, with channel number equal to the sum of Chern numbers below the gap. This establishes the possibility of “high Bott index” phases in multiband bosonic systems, rather than only the $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$2 structures common in two-band models (Huang et al., 30 Dec 2025).

6. Disorder, Anderson-like transitions, and real-space robustness

The principal advantage of the bosonic Bott index over momentum-space Chern numbers is that it remains meaningful after disorder destroys translational symmetry. In the Kagome model, disorder is introduced as onsite anisotropy randomness,

$\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$3

which preserves the bosonic BdG structure but invalidates Brillouin-zone Chern-number formulas. In that regime the Bott index still tracks topology and reveals that disorder-induced topological phase transitions can proceed in integer steps rather than in a single collapse (Huang et al., 30 Dec 2025).

For the clean phase $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$4 at $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$5 and $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$6, increasing disorder produces

$\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$7

The smaller gap loses its edge state first, and transmission at energies deep in the gaps mirrors this change: $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$8 while a chiral edge mode remains and $\Sigma_{z,r} H \Phi = \Phi \Sigma_{z,r} E,$9 once the gap becomes trivial. Near phase boundaries, disorder can also induce topological Anderson-like behavior. At $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$0, a clean $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$1 phase develops a plateau with $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$2, and at $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$3, a clean trivial phase undergoes the sequence

$\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$4

described as a nontrivial–more nontrivial–less nontrivial–trivial progression. The stated physical mechanism is that disorder in the onset energy can act as a self-energy that reduces sublattice mismatch, closing and reopening gaps (Huang et al., 30 Dec 2025).

The earlier honeycomb-magnon study already showed the same general principle in a two-band setting. There the bosonic Bott index was used to identify both robust topological phases and a magnonic analog of the topological Anderson insulator. That work also introduced energy-resolved indices $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$5 and $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$6 for disordered samples in which the gap is partially filled, thereby giving bosonic systems an effective analogue of a Fermi-energy-resolved topological diagnostic (Wang et al., 2020).

7. Transport, related invariants, and experimental context

The multiband Kagome analysis ties the bosonic Bott index directly to transport through a generalized Landauer–Büttiker formalism for magnons. For a device with open boundaries along the transport direction and damping $\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$7, the retarded Green’s function is written as

$\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$8

with transmission

$\Sigma_{z,r} = I_{N_c} \otimes \sigma_z$9

The current formulas include both lead-to-lead transmission and terms involving $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$0 that encode damping-induced exchange with the central region. In gap-pumped coherent transport, $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$1 can increase with $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$2 because linewidth broadening excites nearby bulk states, whereas $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$3 decreases because of dissipation across the sample. Under thermal bias, clean-sample currents decrease with $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$4, while in disordered samples increasing $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$5 can slightly increase currents through additional sink pathways (Huang et al., 30 Dec 2025).

These transport calculations provide an operational interpretation of the invariant. In the clean Kagome case a single right-moving edge mode in a gap gives $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$6, consistent with a Bott index of magnitude one below that gap. In high-index phases, multiple chiral branches appear in the local density of states and would yield larger $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$7 in clean ballistic devices, while disorder suppresses transport channel by channel as the Bott index decreases in a multi-step fashion. Finite-width devices can show reduced transmission relative to the ideal integer because of edge-state overlap and backscattering, but the correspondence between edge-channel count and Bott index remains the organizing principle (Huang et al., 30 Dec 2025).

A related but distinct real-space invariant for disordered bosonic BdG systems is the noncommutative-geometric index $\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$8 defined from a bosonic spectral projector and a Dirac operator,

$\Psi^\dagger \Sigma_z \Psi = \Sigma_z,$9

This invariant was applied to a disordered artificial spin ice model, where it reproduces $E_*$0 in the magnon Hall regime and $E_*$1 in a trivial localized regime, and in the clean limit it “perfectly coincides” with the Chern number. It belongs to the same family of real-space bosonic topological diagnostics, but it is formulated as an index of a pair of projections rather than through the unitary-log Bott expression (Akagi, 2020).

Experimental relevance has been identified primarily in kagome ferromagnets with significant spin–orbit coupling and Dzyaloshinskii–Moriya interaction. The reported scales $E_*$2 and Gilbert damping $E_*$3 are presented as compatible with realistic devices. Proposed probes include nonlocal magnon transport, spin Seebeck measurements, and microwave excitation. Frequency-resolved excitation inside topological gaps isolates chiral edge transport, while temperature gradients probe the interplay of bulk and edge magnons and the predicted multi-step disorder-induced transitions (Huang et al., 30 Dec 2025).