Rice–Mele Hubbard Model Analysis

• The Rice–Mele model with Hubbard interaction is a one-dimensional lattice Hamiltonian that combines alternating tunneling and staggered potentials with short-range interactions to enable topological charge pumping.
• It interpolates between bosonic Mott insulators and fermionic regimes, linking the interacting SSH and ionic Hubbard limits while revealing complex edge state and polarization behavior.
• Advanced analytical and numerical methods such as iDMRG, tensor-network simulations, and functional RG are employed to precisely capture gap renormalization, bulk-boundary correspondence, and interaction-driven phase transitions.

The Rice–Mele model with Hubbard interaction denotes a family of one-dimensional dimerized lattice Hamiltonians in which alternating hoppings and staggered on-site energies are supplemented by short-range interactions. In bosonic form it provides an interacting realization of a Thouless pump in a Mott insulator; in fermionic form it interpolates between the interacting Su–Schrieffer–Heeger and ionic Hubbard limits, supports correlation-driven restructuring of pump singularities, and exhibits nontrivial edge and boundary-charge phenomena (Hayward et al., 2018, Bertok et al., 2022). Closely related spinless formulations replace onsite Hubbard repulsion by a nearest-neighbor density-density term and are used to analyze bulk and boundary observables in a controlled renormalization-group setting (Lin et al., 2020).

1. Hamiltonians and model variants

At the lattice level, the common Rice–Mele structure consists of alternating nearest-neighbor tunneling amplitudes and a staggered on-site energy. The bosonic interacting Rice–Mele Hamiltonian studied for topological pumping is

$$H = \sum_i \biggl[ -J\,\bigl(1+(-1)^i\delta\bigr)\,\hat a_i^\dagger \hat a_{i+1}^{\phantom\dagger} + \text{h.c.} + (-1)^{i}\frac{\Delta}{2} \hat n_i + \frac{U}{2} \hat n_i(\hat n_i-1) \biggr],$$

where $J$ is the average hopping, $\delta$ the dimerization, $\Delta$ the staggered potential, and $U$ the on-site Hubbard interaction for bosons. At half filling, defined there as one boson per unit cell or average density $\bar n=1/2$ per site, sufficiently large $U/J$ produces a Mott insulator with a many-body gap (Hayward et al., 2018).

For spinful fermions, several equivalent conventions appear. One open-chain half-filled Rice–Mele–Hubbard Hamiltonian is

$\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$

with $t$ the hopping scale, $\delta$ the dimerization, $J$0 the staggered potential, and $J$1 the on-site repulsion (Bisht et al., 28 Aug 2025). A closely related formulation used for phase-diagram and pumping analysis is

$J$2

which explicitly contains the interacting SSH model at $J$3 and the ionic Hubbard model at $J$4 (Aligia, 2022).

A third, spinless interacting variant employs a nearest-neighbor density-density term,

$J$5

rather than onsite $J$6. In the cited work this is treated as a Hubbard-type interaction and used to analyze the interacting Rice–Mele model’s bulk and boundary observables (Lin et al., 2020).

Variant	Interaction term	Principal focus
Bosonic Rice–Mele	$J$7	Mott insulator, topological pumping
Spinful Rice–Mele–Hubbard	$J$8 or particle–hole-symmetric equivalent	BI/SDI/MI structure, charge and spin pumping, edge states
Spinless interacting Rice–Mele	$J$9	Gap renormalization, boundary charge, effective edge states

In the noninteracting limit, all formulations reduce to the standard Rice–Mele problem: a two-band one-dimensional insulator with dimerized hopping and staggered potential, gapless only at the point $\delta$0 or $\delta$1, depending on notation (Hayward et al., 2018, Yahyavi et al., 2017).

2. Many-body topology, polarization, and charge pumping

For interacting pumps, the central topological quantity is the change of many-body polarization over a pump cycle. In the bosonic model, the polarization is defined through Resta’s formula

$\delta$2

with $\delta$3, and the transported charge satisfies

$\delta$4

For an adiabatic cycle of period $\delta$5, the net pumped charge is the winding of $\delta$6,

$\delta$7

which is quantized because $\delta$8 is only defined modulo $\delta$9. At half filling and for pump paths encircling the degeneracy while remaining gapped, the bosonic calculation gives $\Delta$0, hence one particle pumped per cycle (Hayward et al., 2018).

This many-body description is required because the single-particle picture can fail at finite interaction. In the hardcore-boson limit $\Delta$1, the Jordan–Wigner mapping to free spinless fermions restores the conventional Chern-number description. By contrast, at finite interaction strength the physical bosons do not occupy a single band uniformly in momentum space, and weighting the noninteracting Berry curvature by the actual bosonic momentum distribution does not, in general, reproduce the quantized transport. For the cited “path II” at $\Delta$2, the weighted lower-band integral gives $\Delta$3, while the true pumped charge extracted from the many-body polarization remains exactly quantized (Hayward et al., 2018).

For spinful interacting Rice–Mele models, the many-body polarization is complemented by generalized position-operator invariants

$\Delta$4

The cases $\Delta$5, $\Delta$6, and $\Delta$7 provide charge, spin, and spin-resolved topological diagnostics. The paper shows that $\Delta$8 should be integers, and in some cases of magnitude larger than $\Delta$9 to lead to well defined expectation values; in inversion-symmetric regions these invariants are complementary and are necessary and sufficient to construct the phase diagrams (Aligia, 2022).

In the two-component fermionic pump, the Hubbard interaction has an additional topological effect: it splits the single noninteracting critical point at the origin into two separate critical points. As a result, an adiabatic loop can enclose none, one, or both singularities, yielding quantized charge transport of $U$0, $U$1, or $U$2 in a system that at $U$3 permits only $U$4 or $U$5 charges per cycle (Bertok et al., 2022).

3. Interaction-driven phases and singularity structure

The phase structure depends strongly on statistics and spin content, but in all cases the fate of the many-body gap controls topology. In the bosonic Rice–Mele model at $U$6, finite $U$7 produces a Mott-insulating phase at large $U$8 and a superfluid phase at smaller $U$9. The iDMRG phase diagram contains a central superfluid “bubble” around $\bar n=1/2$0 that shrinks as $\bar n=1/2$1 increases. Quantized pumping survives exactly as long as the pump cycle lies fully in the Mott-insulating region and avoids the superfluid phase; if the path crosses the superfluid, the many-body polarization becomes ill-defined and the pumped charge is no longer a universal topological invariant (Hayward et al., 2018).

For spinful fermions, the $\bar n=1/2$2 ionic Hubbard limit organizes the half-filled system into a band insulator, a spontaneously dimerized insulator, and a Mott insulator. In the generalized-position-operator analysis, the BI$\bar n=1/2$3SDI transition is detected by a jump in $\bar n=1/2$4, while the SDI$\bar n=1/2$5MI transition is detected by a jump in $\bar n=1/2$6. In the $\bar n=1/2$7 interacting SSH limit with additional Ising exchange, the relevant phases are a Néel antiferromagnet and two dimerized bond-order-wave phases; there $\bar n=1/2$8 detects the Néel$\bar n=1/2$9BOW transition, whereas $U/J$0 distinguishes the two SSH-like dimerized sectors (Aligia, 2022).

In the SU(2)-symmetric spinful pump, the interaction-induced splitting of the noninteracting singularity is tied to ionic Hubbard physics. The line $U/J$1 corresponds to the ionic Hubbard model, and the original degeneracy at $U/J$2 becomes two critical points at $U/J$3 and $U/J$4. A loop centered at one of the split singularities can transport one charge per cycle, while an origin-centered loop can still transport two charges if it encloses both points, or zero if increasing $U/J$5 moves the singularities outside the loop (Bertok et al., 2022).

This suggests a unifying rule across bosonic and fermionic variants: the interacting Rice–Mele pump remains topological whenever the adiabatic path stays in a gapped many-body phase, but the structure of the relevant singularities and the allowed quantized transports depend on whether the interaction primarily stabilizes a Mott insulator, splits a critical point, or produces gapless intermediate regions.

4. Edge states, quasiparticles, and boundary charge

The interacting Rice–Mele model exhibits two distinct boundary notions: localized spectral features in open chains, and boundary charge extracted from the density profile. These do not respond to interactions in the same way. In the spinless interacting Rice–Mele model, the main bulk effect of the interaction is a power-law renormalization of the gap with an interaction-dependent exponent, while the important characteristics of the boundary charge are unaltered and can be understood from the renormalized bulk properties. The same work emphasizes that the interaction spoils the relation between the bulk properties and the number of edge states, so edge-state counting ceases to be a reliable bulk indicator once the interaction is finite (Lin et al., 2020).

For open spinful Rice–Mele–Hubbard chains at half filling, the quasiparticle edge spectrum itself evolves nontrivially with $U/J$6. At $U/J$7, the bulk charge gap is

$U/J$8

and the edge-state energy is

$U/J$9

As $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$0 increases, $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$1 decreases, reaches zero at a critical interaction $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$2, then rises again until it merges with the bulk states at $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$3. For stronger interaction, the low-energy edge mode disappears, and for $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$4 edge states reappear in a high-energy gap between Hubbard-like bands. The paper terms this sequence a transmigration of edge states from the physical charge gap to a high-energy gap (Bisht et al., 28 Aug 2025).

That analysis is supported by an effective charge-only Hamiltonian derived in Kumar representation, where the charge sector experiences an effective potential

$\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$5

The special point $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$6, equivalently $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$7, explains the zero-energy edge state as a repulsion-driven cancellation of the effective staggered potential in the charge sector (Bisht et al., 28 Aug 2025).

A broader boundary perspective emerges from generalized Aubry–André–Harper models with related unit-cell structures: interaction-induced effective edge states can appear in the local single-particle spectral function close to a boundary, while the characteristics of the boundary charge are not modified by the interaction. The cited work explicitly states that its results for the Rice–Mele and Su–Schrieffer–Heeger model are generic and can be found in lattice models with more complex unit cells as well (Lin et al., 2021).

5. Analytical and numerical methods

The interacting Rice–Mele literature is methodologically heterogeneous because distinct questions require distinct observables. For bosonic pumping, the principal tool is infinite-size matrix-product-state simulation. The cited study uses iDMRG with bond dimension up to $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$8, preserving U(1) particle-number symmetry, and directly accesses the ground-state energy, correlation length, one-body density matrix, momentum distribution, Schmidt values, entanglement spectrum, and many-body polarization in the thermodynamic limit (Hayward et al., 2018).

For charge and spin pumping in two-component fermions, the calculations combine real-time simulations in finite and infinite systems with gap and polarization diagnostics. The reported implementations use Lanczos exact diagonalization for systems up to $\begin{split} H &= -t\sum_{l=1}^{L-1} \sum_{s=\uparrow,\downarrow} \left[1 + (-)^{l}\delta \right] \left(c^\dagger_{l,s}c^{}_{l+1,s} + {\rm H.c.}\right) \ &\quad + V\sum_{l=1}^{L}\sum_{s=\uparrow,\downarrow} (-)^l n_{l,s} + U\sum_{l=1}^L\left(\hat{n}^{ }_{l,\uparrow} - \frac{1}{2}\right)\left(\hat{n}^{ }_{l,\downarrow} - \frac{1}{2}\right), \end{split}$9 with pumping times $t$0, and infinite-system tensor-network methods based on VUMPS plus iTEBD with bond dimension $t$1 and $t$2, together with charge and spin Berry phases and many-body polarizations $t$3 and $t$4 (Bertok et al., 2022).

For bulk and boundary properties in the spinless interacting Rice–Mele model, the central technique is truncated functional RG with a static self-energy. This approach resums the series of leading logarithms for small gaps, is controlled for small interactions, can be applied directly to the microscopic lattice model, and is benchmarked against DMRG. In that framework the self-energy renormalizes both the low-energy gap and the high-energy bandwidth, which is essential for reconstructing the interacting boundary charge from renormalized bulk parameters (Lin et al., 2020).

Open-chain edge-state studies rely on finite-system DMRG, exact diagonalization, and effective quasiparticle reductions. The edge-state paper uses DMRG on $t$5 chains for $t$6 and $t$7, DMRG on $t$8 chains for full excitation spectra, and exact diagonalization on $t$9 chains. Its effective description is formulated in Kumar representation, which separates charge and spin degrees of freedom and yields a Bogoliubov quasiparticle picture for the edge states (Bisht et al., 28 Aug 2025).

An additional diagnostic unique to the bosonic pump is the entanglement spectrum. Because each Schmidt state carries a particle imbalance $\delta$0 across the cut, the polarization can be written as

$\delta$1

Over one pump cycle in the Mott phase, the entanglement levels exhibit a spectral flow with $\delta$2, implying $\delta$3. In that sense, the pumped charge is encoded directly in the winding of particle-imbalance sectors in the entanglement spectrum (Hayward et al., 2018).

6. Experimental relevance, terminology, and conceptual issues

The experimental motivation is strongest in ultracold-atom platforms. For bosons, the Rice–Mele lattice is implemented with a bichromatic potential

$\delta$4

where varying $\delta$5 and the lattice depths controls $\delta$6 and $\delta$7. The pumped charge is measured by center-of-mass motion, and the momentum distribution $\delta$8 by band mapping and time-of-flight. The paper proposes comparing the center-of-mass-based quantized charge to the noninteger weighted-Berry-curvature estimate to demonstrate the breakdown of a naive single-particle invariant in an interacting bosonic Mott insulator (Hayward et al., 2018).

For two-component fermions, the most accessible experimental scenario identified in the literature is the SU(2)-invariant interacting Rice–Mele pump in ultracold gases, where only hopping modulation, a staggered potential, and onsite interactions are needed. The cited analysis interprets interaction-induced changes in pumped charge as motion of the split singularities relative to the pump loop, rather than as a generic destruction of topology, and discusses the relevance of this picture for a related quantum-gas experiment by Walter et al. (Bertok et al., 2022).

Open-chain edge phenomena are also experimentally addressable. The repulsion-induced zero-energy edge state at $\delta$9 and its subsequent migration to a high-energy gap were proposed as targets for local spectroscopy at the edges, site-resolved detection of density and excitations in ultracold-atom chains, and dynamical probes that inject particles at the boundaries (Bisht et al., 28 Aug 2025).

A recurrent terminological issue concerns the phrase “on-site interaction.” In the noninteracting Rice–Mele study of polarization distributions, that phrase refers to the staggered on-site potential $J$00, not to a many-body Hubbard term; the paper explicitly states that there is no term of the form $J$01 or similar (Yahyavi et al., 2017). That distinction matters because the genuine Hubbard extensions discussed above replace single-particle Zak-phase diagnostics by many-body polarization, Berry-phase, entanglement-spectrum, and boundary-charge constructions.

Taken together, the published record supports a precise picture. In interacting Rice–Mele systems, topology is controlled by gapped many-body evolution rather than by a naive band picture; Hubbard interactions can stabilize Mott-insulating pumps, split critical points, reshape or even transmigrate edge states, and renormalize gaps and bandwidths without destroying the boundary-charge manifestation of bulk–boundary correspondence (Lin et al., 2020).