Nature of the bosonic edge phase in a double quantum spin Hall insulator

The gist The interaction-driven transition from fermionic edge modes to a bosonic edge phase in a double quantum spin Hall insulator is numerically verified, showing that single-fermion excitations are gapped while fermion pairs remain gapless, matching field-theoretic predictions <ref:2610.30251#pg2>.

Theoretical Background of the Bosonic Edge Phase

Field-theoretic analysis predicted that the edge of a double quantum spin Hall insulator (DQSHI) can undergo an interaction-driven transition from a phase with fermionic edge modes into one with only bosonic modes <ref:2610.30251#pg3>. All single-fermion edge excitations are gapped in this bosonic edge phase, but fermion pairs remain gapless, leading to distinct experimental signatures <ref:2610.30251#pg3>. The DQSHI is a topological phase protected by charge conservation and S z-spin conservation, with spin Chern number Cs = 2 <ref:2610.30251#pg3>. Experimentally, the fermionic and bosonic edge phases can be distinguished in shot-noise measurements using a quantum point contact geometry <ref:2610.30251#pg3>. In the fermionic edge phase, charge is backscattered in units of e, leading to a Fano factor of F = 1 <ref:2610.30251#pg3>. In the bosonic edge phase, however, the single-fermion gap only allows for backscattering in units of 2e, producing a doubled Fano factor of F = 2 <ref:2610.30251#pg3>.

Numerical Simulation Setup using Segment DMRG

To realize the bosonic edge phase in an explicit microscopic model, segment DMRG was employed to verify the existence of this phase <ref:2610.30251#pg4>. The calculation requires an explicit bulk embedding of the DQSHI edge, due to its mixed anomaly between charge and S z conservation <ref:2610.30251#pg3>. The simulation uses segment DMRG on an infinite two-dimensional cylinder, where the DQSHI bulk is modeled in terms of two copies of the spin-up lowest Landau level, combined with their time-reversed counterparts <ref:2610.30251#pg3>. The setup partitions the cylinder into a semi-infinite bulk segment, a semi-infinite vacuum segment, and a finite edge segment <ref:2610.30251#pg4>. DMRG is then performed on the finite edge segment, keeping the bulk and vacuum segments as fixed boundary conditions <ref:2610.30251#pg4>. This method circumvents obstacles in simulating the DQSHI edge in a strictly 1D geometry by using segment DMRG which allows for a generic 2D edge embedding <ref:2610.30251#pg4>.

Results of the Numerical Verification

In parameter regimes where bosonization predicts the bosonic edge phase, the simulation finds a single-fermion gap that extrapolates to a finite value in the thermodynamic limit <ref:2610.30251#pg5>. Moreover, this single-fermion gap has an exponential dependence on interaction strength that is predicted by bosonization <ref:2610.30251#pg5>. The fermion-pair gaps, in contrast, remain consistent with zero for all interaction strengths studied <ref:2610.30251#pg5>. These results show that the bosonic edge phase can arise in a microscopic DQSHI model, supporting moiré TMDs as candidate platforms for its realization <ref:2610.30251#pg5>. The extrapolated edge gaps are shown in Fig. 2 for 0 ≤ U0/αl ≤ 1 <ref:2610.30251#pg5>. As can be seen in Fig. 2, Egap(1, 1/2) increases with increasing U0/αl, while the fermion-pair gaps, Egap(2, 0), Egap(0, 1), and Egap(2, 1), all remain close to zero <ref:2610.30251#pg5>. This is exactly the behavior predicted for the bosonic edge phase of the DQSHI <ref:2610.30251#pg5>.

Scaling and Convergence Analysis

The scaling of the single-fermion gap with interaction strength is consistent with field-theoretic calculations <ref:2610.30251#pg5>. Renormalization group analysis of the bosonized theory predicts a single-fermion gap ∝ exp(−παl/U0) for small U0/αl <ref:2610.30251#pg5>. Fitting Egap(1, 1/2) to ∆0 exp(−κ αl/U0) over 0 < U0/αl ≤ 0.75, we indeed find ∆0/αl = 11.72 ± 0.48 and κ = 3.167 ± 0.028 <ref:2610.30251#pg5>. The segment DMRG calculations probe the physical edge, which forms a ring of circumference Ly, so we extrapolate the gaps in 1/Ly <ref:2610.30251#pg9>. The selected fits do not bend artificially as 1/Ly → 0 <ref:2610.30251#pg9>.

Implications and Future Directions

The numerical confirmation motivates the possibility that the bosonic edge phase can be realized in real DQSHIs, specifically in twisted MoTe2 and WSe2 <ref:2610.30251#pg5>. The explicit DQSHI model assumed that both sets of Chern bands were degenerate, unlike in the moiré TMDs <ref:2610.30251#pg8>. However, our results still hold if the bands are shifted in energy relative to each other <ref:2610.30251#pg8>. Determining which of these effects turns out to be dominant in the twisted TMD-based DQSHIs is an important future work task <ref:2610.30251#pg8>. If the interactions in these systems are suitable for stabilizing the bosonic edge phase, shot-noise measurements should yield a Fano factor F = 2 <ref:2610.30251#pg8>. Because the doubling is an interaction effect, it is also tunable: weakening the interactions shrinks the single-fermion gap, and once the gap falls below the bias voltage or temperature, the Fano factor should cross over to F = 1 <ref:2610.30251#pg8>. The segment construction is not specific to the DQSHI <ref:2610.30251#pg4>. For any topological phase whose bulk ground state is available as an infinite matrix product state (MPS), obtained from a microscopic Hamiltonian using iDMRG or VUMPS, the filled bulk segment can be replaced by this MPS, which then serves as the boundary condition for the edge segment <ref:2610.30251#pg8>. The extracted Luttinger parameters show strong repulsive interactions between the edge modes as a consequence of large bulk gaps and hence small edge localization lengths imposing dimensional quenching of Coulomb interactions <ref:2610.30251#pg8>. Such repulsive interactions can be beneficial to the formation of the bosonic edge phase: Repulsion between opposite spins in opposite bands stabilizes the bosonic edge phase, while repulsion between opposite spins in the same band instead can destabilize it <ref:2610.30251#pg8>. The paper concludes by noting that observing a doubled Fano factor that reverts to F = 1 under increased screening, or at elevated bias or temperature, would provide strong evidence for the bosonic edge phase <ref:2610.30251#pg8>.

Computational Details

The MPS bond dimension χ = 4000 underestimates the cost of our calculations, which is dominated by the size of the MPO <ref:2610.30251#pg10>. The LLL-projected contact interaction couples all four flavors and orbitals over a range that grows with Ly <ref:2610.30251#pg10>. Constructing the MPO exactly as a finite-state machine with merged equivalent paths yields, for the largest calculation (Ne = 144, U0/αl = 1), a graphcompressed bond dimension DW = 4039 <ref:2610.30251#pg10>. The cost of the edge-segment calculation is set by the MPS bond dimension χ at the segment boundary, which is inherited from the bulk and vacuum segments, and by the entanglement generated within the edge segment <ref:2610.30251#pg4>. A single data point (U0, Ly) therefore costs about 1400 core-hours <ref:2610.30251#pg10>. The absolute difference between the single-fermion gaps obtained with ED and segment DMRG as a function of the kinetic energy cutoff of the ED decays exponentially <ref:2610.30251#pg10>. The calculations were performed with code based on ITensors.jl as well as TensorKit.jl and MPSKit.jl <ref:2610.30251#pg10>. The bond dimension DW is reduced to 592 when using SVDbased compression of Refs. 86 and 87 with cutoff 10−8 <ref:2610.30251#pg10>.

Benchmark Comparison

The absolute difference between the single-fermion gaps obtained with ED and segment DMRG as a function of the kinetic energy cutoff of the ED decays exponentially <ref:2610.30251#pg5>. The ED gap thus converges rapidly to the DMRG result, until it saturates at the DMRG truncation error <ref:2610.30251#pg5>. The edge wave function extrapolates to that of the bulk near x = -Lx <ref:2610.30251#pg4>. Similarly, the densities all quickly approach 0 as x → Lx, as expected for approaching the vacuum boundary <ref:2610.30251#pg4>. The density approaches the vacuum value of ν = 0 as x approaches Lx = 12l <ref:2610.30251#pg5>. This justifies keeping the bulk and vacuum segments fixed even for the largest value of U0 considered here <ref:2610.30251#pg5>. A single data point (U0, Ly) therefore costs about 1400 core-hours <ref:2610.30251#pg10>. The bond dimension DW is reduced to 592 when using SVDbased compression of Refs. 86 and 87 with cutoff 10−8 <ref:2610.30251#pg10>. The calculations were performed with code based on ITensors.jl as well as TensorKit.jl and MPSKit.jl <ref:2610.30251#pg10>. The bond dimension DW is reduced to 592 when using SVDbased compression of Refs.

1. Segment DMRG implementation for topological phases allows for a generic 2D edge embedding, enabling numerical simulation of anomalous edges like those in DQSHI without breaking relevant symmetries or resorting to volume-law entanglement entropy. This capability enables AI systems to explore and verify microscopic models for complex topological states that are difficult to treat with standard 1D methods.

2. The ability to calculate the single-fermion gap and fermion-pair gaps as a function of interaction strength U0/alpha l allows AI systems to predict the transition between fermionic and bosonic edge phases based on calculated scaling, matching predictions from bosonization theory. This helps in designing materials or simulating interactions that drive these phase transitions.

3. The result that all single-fermion excitations are gapped in the bosonic edge phase allows AI systems to accurately characterize the energy spectrum of these edges under interaction, distinguishing them from fermionic phases where all Egap's should vanish in the thermodynamic limit. This provides a robust signature for experimental verification via shot-noise measurements (Fano factor F = 2).

4. The method of using MPS as a boundary condition for the finite edge segment allows AI systems to generalize DMRG techniques to study more general topological phases, including those with fractionalized degrees of freedom and enriched symmetries. This expands the scope of topological phase characterization beyond canonical examples like DQSHI.