Spinful Rice–Mele Model

• Spinful Rice–Mele model is a one-dimensional quantum chain that integrates spin degrees of freedom via explicit spin-orbit coupling and on-site interactions to explore symmetry-protected phases.
• The model exhibits a nodal loop in parameter space where degeneracies occur, establishing precise conditions for quantized charge and spin transfer through Thouless pumping.
• Incorporating interactions and edge phenomena, the model provides a robust framework for understanding boundary–bulk correspondence and is experimentally relevant in quantum simulators.

The spinful Rice–Mele model, as established in recent research, generalizes the canonical Rice–Mele chain by incorporating explicit spin degrees of freedom and, in advanced variants, spin-orbit coupling and on-site interactions. This model exhibits topologically quantized charge and spin pumping phenomena, a rich variety of symmetry-protected phases, and novel forms of boundary-bulk correspondence. Central to the theoretical framework are the emergence of degenerate nodal loops in parameter space and the precise quantization of transported spin and charge as topological invariants. The model provides a rigorous platform for exploring interaction effects, symmetry breaking, and dynamical signatures in topological one-dimensional insulators and their experimental realizations in quantum simulators.

1. Lattice Hamiltonian and Bloch Structure

The general spinful Rice–Mele (RM) Hamiltonian on a one-dimensional chain of length $2N$ is formulated to include spin index $\sigma=\uparrow,\downarrow$, dimerization ($\delta$), staggered potential ($V$), and Rashba-type spin-orbit coupling ($\lambda$):

$$H(t) = \sum_{\sigma=\uparrow,\downarrow} H_{\sigma}(t) + H_{\mathrm{so}}(t)$$

$$H_{\sigma} = \sum_{j=1}^{2N} (-1)^{\sigma} \left[\frac{1+(-1)^j\delta}{2}(c_{j,\sigma}^\dagger c_{j+1,\sigma} + \mathrm{h.c.}) - (-1)^j V\, c_{j, \sigma}^\dagger c_{j, \sigma}\right]$$

$$H_{\mathrm{so}} = \frac{\lambda}{2} \sum_{j, \sigma} \left[(-1)^\sigma c_{j, \sigma}^\dagger c_{j+1, -\sigma} + \mathrm{h.c.}\right]$$

The Bloch Hamiltonian is a $4 \times 4$ matrix operating on the spin and sublattice degrees of freedom, yielding four instantaneous bands. Key features include:

• Spin-conserving hopping and site potential modulated by dimerization and staggering.
• Spin-flipping hopping introduced by the spin-orbit term; $\lambda$ controls the strength of this mixing.
• Band structure admits nontrivial degeneracies only at $\sigma=\uparrow,\downarrow$0, as shown by explicit evaluation of the matrix elements.

The inclusion of spin-orbit coupling transforms the isolated degeneracy at $\sigma=\uparrow,\downarrow$1 in the spinless RM model into a nodal loop of degeneracies of radius $\sigma=\uparrow,\downarrow$2 in $\sigma=\uparrow,\downarrow$3 space at $\sigma=\uparrow,\downarrow$4 (Ma et al., 4 Jan 2025).

2. Topological Defects: From Point Nodes to Nodal Loops

The fundamental band-closing condition for the spinful RM model with spin-orbit coupling is

$\sigma=\uparrow,\downarrow$5

This establishes a one-parameter family of degeneracies—a nodal loop—in the $\sigma=\uparrow,\downarrow$6 parameter space. For $\sigma=\uparrow,\downarrow$7, the nodal loop collapses to a single point, recovering the topology of the spinless RM model. For $\sigma=\uparrow,\downarrow$8, cycles in parameter space encircling this loop correspond to nontrivial topological pumping. These degeneracies are robust for physically reasonable values of $\sigma=\uparrow,\downarrow$9, $\delta$0, and $\delta$1 (Ma et al., 4 Jan 2025).

3. Quantized Thouless Spin Pumping

Thouless pumping in the spinful RM model generalizes the conventional charge-pumping paradigm. The adiabatic evolution of parameters along a closed loop imparts a quantized transfer of spin across the system, calculated via the Berry curvature of the filled bands as

$\delta$2

where $\delta$3 is the pumped spin in units of $\delta$4, and $\delta$5 is the Berry curvature for the $\delta$6th band.

• For a loop in $\delta$7 not enclosing the nodal loop $\delta$8, $\delta$9.
• For a loop enclosing $V$0, $V$1.
• This dichotomy is a bulk topological invariant: the Chern number associated with the two bands corresponds to the net pumped spin across a cycle (Ma et al., 4 Jan 2025).

For the generalized interacting model with explicit Hubbard $V$2, the degeneracy splits, enabling pumping of precisely one charge per cycle when a path encloses only one singularity; this mechanism persists in models with further reduced spin symmetry (Bertok et al., 2022).

4. Boundary–Bulk Correspondence and Edge Phenomena

A key feature emergent in the spinful RM model with spin-orbit coupling is a dynamic realization of the boundary-bulk correspondence, visible under open boundary conditions:

• In the spinless RM model, quantized topological pumping is invisible at the edges (no charge transfer).
• With spin-orbit coupling, edge-localized “spin-flip currents” become quantized. Numerical results show that after two adiabatic cycles of the parameters $V$3, two spin flips are pumped from one edge to the other:

$V$4

with

$V$5

and

$V$6

This phenomenon constitutes a dynamic boundary-bulk correspondence, with quantized edge spin flips as the detectable topological invariant (Ma et al., 4 Jan 2025).

5. Symmetry, Interactions, and Topological Invariants

Extensions of the spinful RM model include on-site (Hubbard) interactions, staggered magnetic fields, and Ising-like spin couplings:

• The Hamiltonian incorporates interaction terms

$V$7

as well as staggered field and Ising coupling

$V$8

• Topological invariants constructed from generalized position operators,

$V$9

comprehensively classify symmetry-protected phases; the triplet $\lambda$0 suffices for the full phase diagram, where

$\lambda$1

• In the presence of inversion symmetry, these invariants are $\lambda$2 quantized (Aligia, 2022), giving rise to phases such as band insulator (BI), spontaneously dimerized insulator (SDI), and Mott insulator (MI), distinguished by their topological markers.

6. Experimental Realizations and Theoretical Significance

The spinful RM model is of central theoretical and experimental interest for quantum simulation platforms:

• The SU(2)-invariant case is directly accessible in optical lattices via staggered superlattice potentials and modulated hopping (Bertok et al., 2022).
• The quantized pumping of spin (or charge, for suitable interactions) is protected against disorder and finite-size effects for moderately slow cycle times.
• Measuring edge spin flips after two cycles, or tracking changes in polarization via generalized position operators or many-body Berry phases, provides robust theoretical and practical diagnostics of topological transport (Aligia, 2022).

Through the enhancement of the parameter space topology—from a point to a nodal loop via spin-orbit interactions—the spinful RM model exemplifies the deep connections between symmetry, topological invariants, and quantized transport. It provides a foundation for the study of pumping phenomena, dynamically observable topological boundary responses, and symmetry-protected topological phases in one-dimensional quantum systems (Ma et al., 4 Jan 2025, Bertok et al., 2022, Aligia, 2022).