Checkerboard Quantum Spin Hall Hubbard Model

• The paper demonstrates that tuning the spin-dependent interaction parameter λ produces two distinct fractional quantum spin Hall phases at ν=1/3 with ninefold and threefold ground-state degeneracies.
• It employs exact diagonalization on a nearly flat-band checkerboard lattice, using spectral flow and Berry phase analysis to probe the topological order.
• Both phases share an identical many-body spin Chern number and quantized spin Hall response, highlighting an interaction-driven reorganization of the fractionalized state.

The checkerboard quantum-spin-Hall-Hubbard model denotes an interacting-electron problem on a time-reversal-invariant checkerboard lattice with a nearly flat-band structure, formulated to investigate fractional quantum spin Hall (FQSH) physics in the presence of short-range repulsion. In the exact-diagonalization study of Sheng, Guo, Sun, and Sheng, the model supports two distinct fractionalized phases at filling factor $\nu=\frac13$: a ninefold-degenerate phase at small nearest-neighbor spin-dependent interaction $\lambda$, and a threefold-degenerate phase at large $\lambda$. These phases are separated by an interaction-driven quantum phase transition near $\lambda_c \approx 0.17$, yet share the same many-body spin Chern number and quantized spin Hall response, so the transition is not a topological-to-trivial one but a reorganization of the FQSH ground-state manifold (Li et al., 2014).

1. Hamiltonian and lattice formulation

The many-body Hamiltonian studied on the checkerboard lattice is

$$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$

Here $\hat H_0$ is the single-particle hopping Hamiltonian of the checkerboard model, $U$ is the onsite Hubbard repulsion between opposite spins on the same site, $V$ is the nearest-neighbor density interaction, and $\lambda$ is a dimensionless parameter tuning the spin-dependent part of the nearest-neighbor interaction (Li et al., 2014).

The exact-diagonalization analysis focuses on the regime in which $U$ is neglected. The stated reason is that a dominant $\lambda$0 tends to favor a symmetry-breaking spin-polarized state, whereas the main objective is to study the evolution induced by $\lambda$1. In this formulation, $\lambda$2 corresponds to a nearest-neighbor interaction acting only within the same spin sector, while $\lambda$3 turns on opposite-spin nearest-neighbor repulsion.

This structure is central to the model’s interpretation. The onsite Hubbard term makes the model “Hubbard” in the usual lattice sense, but the many-body phases emphasized in the flat-band study are controlled primarily by the projected nearest-neighbor interaction and, in particular, by its spin anisotropy through $\lambda$4. This suggests that the operative distinction between phases is not simply interaction strength, but interaction structure.

2. Single-particle checkerboard flat-band background

The single-particle sector is written as a two-copy, spinful version of the $\lambda$5-flux checkerboard flat-band model: $\lambda$6 with sublattice spinor

$\lambda$7

and $\lambda$8 acting on the $\lambda$9 sublattice space.

The $\lambda$0 components are

$\lambda$1

$\lambda$2

$\lambda$3

In this parametrization, $\lambda$4, $\lambda$5, and $\lambda$6 are respectively the nearest-neighbor, next-nearest-neighbor, and third-neighbor hoppings. The resulting noninteracting model has a large bulk gap and supports time-reversal-protected edge states, which is why it is treated as a candidate platform for fractional quantum spin Hall physics after projection to the flat bands (Li et al., 2014).

The single-particle construction matters because the many-body phases are not built from a generic dispersive band, but from nearly flat topological bands with time-reversal symmetry. A plausible implication is that the fractionalized phases inherit both the reduced kinetic-energy scale characteristic of flat-band problems and the two-copy structure associated with opposite chiralities for opposite spins.

3. Fractional phases and interaction-driven transition

At filling

$\lambda$7

the projected model exhibits two distinct incompressible phases as $\lambda$8 is tuned (Li et al., 2014). The total number of lattice sites is

$\lambda$9

At small $\lambda_c \approx 0.17$0, the ground-state manifold is ninefold degenerate. This phase is interpreted as two decoupled fractional quantum Hall-like states, one for each spin species, and the paper identifies it as an FQSH state with a strong “two-copy” character. The ninefold structure is also noted to match earlier work at $\lambda_c \approx 0.17$1 filling.

At large $\lambda_c \approx 0.17$2, the ground state becomes threefold degenerate. This phase is also identified as an FQSH state. No spontaneous symmetry breaking is found, and the description given is that the two spin species become more strongly correlated, with the system favoring local opposite-spin occupancy but without developing long-range order.

The two phases are separated by a quantum critical point near

$\lambda_c \approx 0.17$3

The transition is identified by splitting and rearrangement of low-lying eigenstates, closing of the quasispin excitation gap, and reorganization of the finite-size spectrum. Crucially, the transition is not marked by a change in the topological spin Chern number. The central conclusion is therefore that the ninefold and threefold states are distinct fractional quantum spin Hall phases rather than a topological phase and a trivial phase.

Regime of $\lambda_c \approx 0.17$4	Ground-state degeneracy	Characterization
Small $\lambda_c \approx 0.17$5	9	Two decoupled fractional quantum Hall-like spin sectors
Large $\lambda_c \approx 0.17$6	3	Correlated FQSH phase without spontaneous symmetry breaking

Both phases remain separated from excited states by a finite gap and show robust finite-size evidence for fractionalization. The difference lies in the structure of the ground-state manifold and in the excitation counting, not in the loss of topological response.

4. Exact diagonalization and twisted-boundary diagnostics

The phase identification is based on finite-size exact diagonalization after projection onto the lowest flat bands. The calculations are carried out on finite tori of size $\lambda_c \approx 0.17$7, and the analysis includes low-lying many-body energy spectra, momentum-sector structure, spectral flow under twisted boundary conditions, spin and charge correlations, and quasispin excitation spectra (Li et al., 2014).

Both spin-independent twists and spin-dependent twists are introduced through boundary phases $\lambda_c \approx 0.17$8. These twisted boundary conditions probe topological response by tracking the evolution of the low-energy manifold under flux insertion. The spectral flow shows that the ground states evolve into each other under flux insertion and return to the original configuration after three flux quanta. This is the expected behavior of a $\lambda_c \approx 0.17$9-type fractional topological state.

The low-energy spectra display a clear gap separating the ground-state manifold from higher excitations, and the finite-size splitting within the ground-state manifold decreases with size. The momentum-sector relations among degenerate states are described as consistent with Abelian fractional quantum Hall-type topological order.

Berry phases enter as a complementary diagnostic. Although the analysis is not presented as an explicit derivation-heavy Berry-curvature construction, the paper states that the evolution of low-lying spectra and Berry phases under twisted boundary conditions demonstrates that the two phases share the same topological spin Chern number. In this sense, the spectral flow and Berry-phase data jointly establish that both the ninefold and the threefold states are topological FQSH phases.

5. Spin Chern number and quantized response

The topological invariant emphasized in the study is the many-body spin Chern number. It is defined through Berry curvature under twisted boundary conditions and is associated with spin Hall response in the many-body setting (Li et al., 2014).

For each of the three lowest states in the ground-state manifold, the reported value is

$$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$0

From this, the quantized spin Hall conductance is given as

$$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$1

The significance of this result is explicit. The change from ninefold to threefold degeneracy does not alter the topological response: both phases have the same spin Chern number and the same quantized spin Hall conductance. Accordingly, the transition driven by $$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$2 is interpreted as a change in the internal organization of the fractionalized state rather than a destruction of topological order.

This point addresses a common potential misconception. A degeneracy change in a finite-size many-body spectrum need not imply a transition from a topological phase to a trivial one. In the checkerboard quantum-spin-Hall-Hubbard setting studied here, the invariant and the transport response remain unchanged across the transition, while the degeneracy structure and excitation content reorganize.

6. Quasispin excitation spectra and counting rules

A distinctive part of the analysis is the use of quasispin excitation spectra as fingerprints of the underlying fractionalized order. These spectra are obtained by changing particle number while keeping the system size fixed, thereby probing a quasispin excitation sector (Li et al., 2014).

Near the critical region, the quasispin gap closes, which is one of the numerical signatures used to locate the transition. Away from the transition, the counting of low-lying states below the gap differs sharply between the two regimes.

At $$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$3, the low-lying quasispin spectrum has $$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$4 states below the gap for the finite system shown. The counting is explained in terms of two independent fractional quantum Hall states, one for each spin component, obeying a generalized Pauli principle or $$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$5-admissible rule. The formula given is

$$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$6

which yields exactly

$$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$7

for the system size considered.

At $$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$8, only $$\hat H = \hat H_0 + U \sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} + V \sum_{\langle i,j\rangle} \Big[ \hat n_{i\uparrow}\hat n_{j\uparrow} +\hat n_{i\downarrow}\hat n_{j\downarrow} +\lambda\big(\hat n_{i\uparrow}\hat n_{j\downarrow} +\hat n_{i\downarrow}\hat n_{j\uparrow}\big) \Big].$$9 states lie below the gap. The counting argument given is that there are $\hat H_0$0 ways to remove a spin-up particle in the chosen finite system, the spin-down sector has threefold degeneracy, and opposite spins prefer to occupy the same site in this phase, leading to

$\hat H_0$1

states.

These counting rules are presented as fingerprints of the two FQSH phases. In the weak-$\hat H_0$2 regime, the larger counting reflects the effectively decoupled two-copy structure. In the strong-$\hat H_0$3 regime, the reduced counting reflects a correlated threefold FQSH state with reorganized spin structure. This suggests that excitation counting, together with spectral flow and spin Chern data, is sufficient to distinguish phases that share the same quantized topological response but differ in their internal fractionalization pattern.