Extended Rice-Mele Model

• Extended Rice-Mele model is a family of one-dimensional dimerized lattices that generalizes the original framework by incorporating varied control parameters including counter-diabatic driving and non-Hermitian terms.
• The model enables finite-time Thouless pumping and alters Berry-phase and Chern-number formulations through extensions like disorder, interaction effects, and spatial modulation.
• These extensions create new regimes in topological physics, revealing altered bulk-edge correspondence, many-body polarization, and innovative platform-specific realizations.

Searching arXiv for papers on extended Rice-Mele model and closely related generalizations. arXiv search query: "Rice-Mele model extension counter-diabatic non-Hermitian disorder interaction acoustic" The extended Rice-Mele model denotes a family of non-equivalent generalizations of the Rice-Mele chain, a one-dimensional dimerized two-sublattice lattice with alternating hoppings and a staggered onsite potential. In its standard form, the model is a minimal setting for Thouless pumping, Zak-phase polarization, and boundary-state physics; in extended forms it acquires additional structures such as counter-diabatic driving, non-Hermiticity, unequal bond disorder, Hubbard and bosonic interactions, long-period spatial modulation, spin-orbit coupling, intrinsic nonlinearity, and platform-specific implementations. Across these variants, the recurrent organizing principles are the two-band Bloch description, Berry-phase and Chern-number formulations, and the persistence, deformation, or reformulation of bulk-edge correspondence under altered symmetry, dynamics, or many-body structure (Chiel et al., 2024, Lin et al., 2015, Lin et al., 2020).

1. Canonical framework and the meaning of “extended”

The standard time-dependent Rice-Mele Hamiltonian is

$$\hat{H}_0(t) = -\sum_j \left( J_1(t)\hat{a}_j^{\dagger}\hat{b}_j + J_2(t)\hat{a}_{j+1}^{\dagger}\hat{b}_j + h.c.\right) + \Delta(t)\sum_j\left(\hat{a}_j^{\dagger}\hat{a}_j - \hat{b}_j^{\dagger}\hat{b}_j\right),$$

where $J_1(t)$ and $J_2(t)$ are intra-cell and inter-cell hoppings and $\Delta(t)$ is the staggered onsite potential. In momentum space it is a two-band Bloch Hamiltonian,

$$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$

As long as the gap stays open, adiabatic cyclic evolution pumps an integer amount of charge per cycle (Chiel et al., 2024).

In the recent literature, “extended Rice-Mele model” is therefore not a single canonical Hamiltonian but a class of constructions that preserve the dimerized two-site backbone while changing the control parameters, the symmetry class, the microscopic statistics, or the interpretation of the auxiliary parameter space. Some extensions modify the drive so that the pump is finite-time rather than quasi-adiabatic; some introduce non-Hermitian onsite or hopping terms; others add interactions, disorder, long-period spatial modulation, or spin structure; still others realize the model in emergent or classical-wave lattices (Asaga et al., 2021, Ma et al., 4 Jan 2025, Xia et al., 3 Mar 2026).

Extension class	Added structure	Representative consequence
Counter-diabatic pump	$H_{CD}(t)$	Arbitrary-speed ground-state following
Non-Hermitian RM	Imaginary potential or asymmetric hopping	Complex Berry phase or biorthogonal topology
Interacting RM	Hubbard, bosonic, or nearest-neighbor interaction	Many-body polarization replaces single-particle invariant
Disordered RM	Unequal bond disorder	Disorder-driven topological transition
Spatially modulated RM	Rational phase $p/q$	Diophantine equation and spatial pump
Spinful/SOC RM	Spin-orbit coupling	Nodal loop and quantized spin pumping
Nonlinear RM	Onsite cubic term	Auxiliary-eigenvalue-based anomalous BEC

This plurality is not merely terminological. It indicates that the Rice-Mele structure functions as a topological kernel onto which distinct physical mechanisms are grafted: adiabatic transport, Floquet engineering, gain/loss, strong correlations, synthetic dimensions, and boundary-sensitive observables all reuse the same dimerized backbone but alter the meaning of topology and transport in model-specific ways (Arceci et al., 2019, Hattori et al., 2024, Bai et al., 5 Jan 2025).

2. Finite-time pumping and non-adiabatic control

A major dynamical extension adds a counter-diabatic term to the Rice-Mele pump so that the system exactly follows the instantaneous ground state of the bare Hamiltonian at arbitrary driving speed. For the two-band Bloch problem,

$$\hat{H}(k,t)=\hat{H}_0(k,t)+\hat{H}_{CD}(k,t) =\left(\mathbf{R}+\frac{1}{R^2}\mathbf{R}\times \partial_t\mathbf{R}\right)\cdot \boldsymbol{\sigma},$$

with

$$\hat{H}_{CD}(k,t)=\frac{1}{R^2}\left(\mathbf{R}\times \partial_t\mathbf{R}\right)\cdot \boldsymbol{\sigma}.$$

The pumped charge over a full cycle remains

$Q_{\text{pump}}(T)=C,$

where $J_1(t)$0 is the Chern number of the map $J_1(t)$1. The extension is therefore a finite-time Thouless pump, not a different transport mechanism. Its central practical difficulty is that the exact counter-diabatic term generically produces long-range hopping in real space. Two mitigation routes were given: a special “control-freak” protocol with exact nearest-neighbor realization, and a numerical inverse-design strategy constraining the effective $J_1(t)$2 to nearest-neighbor form (Chiel et al., 2024).

A different finite-speed analysis treats weak non-adiabaticity rather than eliminating it. In the Landau-Zener description of the periodically driven Rice-Mele model, two avoided crossings per cycle generate transition amplitudes, a Stokes phase, and interference between non-adiabatic paths. The lower-band population after one cycle contains both direct $J_1(t)$3 loss and an interference cosine term involving the dynamical phase and $J_1(t)$4. In this regime, a non-adiabatic extension of the Zak phase can be defined through the lower-band Wannier-center shift, but the pumped charge ceases to be integer-valued once lower-band occupation is not preserved (Kuno, 2018).

An open-system extension reaches a different conclusion: weak low-temperature dissipation can improve pumping quantization at finite frequency. In the Floquet picture, dissipation increases the population of the lowest-energy Floquet band, whose quasi-energy winding carries the topological transport, and the asymptotic pumped charge is controlled by Floquet-band populations. The work formulates each momentum sector as a driven dissipative two-level system and finds good agreement between Bloch-Redfield dynamics and exact MPS numerics in the weak-coupling regime (Arceci et al., 2019).

3. Non-Hermitian Rice-Mele families

One non-Hermitian extension adds an imaginary staggered onsite term and a time-dependent flux to a ring geometry,

$J_1(t)$5

In momentum space, each $J_1(t)$6-sector is a non-Hermitian two-level problem with a fully real spectrum in two regimes: $J_1(t)$7, or $J_1(t)$8. Under a slow flux sweep $J_1(t)$9, the geometric phase becomes complex,

$J_2(t)$0

and its imaginary part controls amplitude amplification or attenuation. In the adiabatic limit,

$J_2(t)$1

This establishes a flux-controlled amplitude mechanism based on complex Berry phases rather than on transport quantization (Lin et al., 2015).

A second non-Hermitian extension keeps time-reversal symmetry but replaces reciprocal hoppings by imbalanced hopping amplitudes. The Bloch Hamiltonian

$J_2(t)$2

has real eigenvalues

$J_2(t)$3

independent of the asymmetry parameter $J_2(t)$4. In the biorthonormal framework, the Berry connection may be complex, but the Chern number remains quantized. Exact open-chain mid-gap edge modes can be constructed analytically, and the pumped charge, defined from a biorthonormal local current, equals the Chern number under cyclic adiabatic evolution. This model shows that geometric concepts familiar from Hermitian topological insulators extend to a non-Hermitian regime once left and right eigenstates are used consistently (Wang et al., 2018).

A more elaborate construction couples $J_2(t)$5 non-Hermitian Rice-Mele chains with staggered gain and loss into an effectively two-dimensional system. In the thermodynamic limit, exceptional points are zeros of a real auxiliary vector field built from Bloch-state expectation values and carry half-integer topological charge $J_2(t)$6 via a winding-number calculation. The same topological characterization survives finite $J_2(t)$7 and even the single-chain limit, where the two-dimensional defects project to quasi-one-dimensional kinks (Li et al., 2020).

4. Interactions, disorder, and nonlinear generalizations

For interacting bosons, the Rice-Mele model becomes

$J_2(t)$8

At filling $J_2(t)$9, the appropriate topological invariant is the many-body polarization,

$\Delta(t)$0

whose winding gives the pumped charge. Quantization survives as long as the pump path avoids the superfluid phase and remains in the gapped Mott-insulating regime. In the hardcore-boson limit, the pump can again be interpreted by single-particle Berry curvature and Chern number, but that interpretation breaks down at finite $\Delta(t)$1. The entanglement spectrum supplies a complementary diagnostic: its spectral flow shifts the particle-imbalance labels by one over a pump cycle, encoding one transported charge (Hayward et al., 2018).

For spinless fermions with nearest-neighbor interaction,

$\Delta(t)$2

the bulk gap is power-law renormalized, while the fractional part of the boundary charge remains determined by renormalized bulk quantities. Functional RG and DMRG show that the bulk-boundary correspondence survives for boundary charge but not for edge-state counting: interactions and boundary self-energy modulations can create or remove in-gap peaks in the local spectral function without a corresponding bulk topological change (Lin et al., 2020).

In the two-component fermionic Rice-Mele Hubbard model, repulsive onsite interaction splits the original topological singularity into two critical points. Pump cycles can then enclose none, one, or both singularities, yielding $\Delta(t)$3, $\Delta(t)$4, or $\Delta(t)$5 transported charges. Adding a staggered magnetic field produces quantized pumping of one charge and one unit of spin, whereas an Ising-type spin coupling yields a pure single-charge pump. In the SU(2)-symmetric case, the cycle passes through the ionic Hubbard model and encounters a gapless spin sector in the thermodynamic limit, although finite-size and finite-time simulations still show robust first-cycle transport (Bertok et al., 2022).

Unequal bond disorder provides a nonchiral extension. With random intracell and intercell hoppings,

$\Delta(t)$6

and staggered mass $\Delta(t)$7, the model remains classifiable by a real-space invariant $\Delta(t)$8. For small $\Delta(t)$9, the transition is set by

$$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$0

equivalently by equality of geometric means, and is accompanied by anomalous localization with envelope $$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$1. For sufficiently large $$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$2, the phase boundary instead follows

$$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$3

so arithmetic means replace geometric means as the controlling quantities (Hattori et al., 2024).

A further extension is intrinsically nonlinear. In the nonlinear Rice-Mele model with onsite cubic term $$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$4, the eigenproblem becomes self-consistent. To formulate bulk-edge correspondence, an auxiliary eigenvalue problem

$$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$5

is introduced, with physical states corresponding to $$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$6. The resulting correspondence between the auxiliary bulk invariant and nonlinear edge or soliton states is termed an anomalous bulk-edge correspondence. The construction distinguishes intrinsic nonlinear eigenvalues from auxiliary-eigenvalue-induced nonlinearity and identifies regimes where pumping persists, becomes non-quantized, or is suppressed by self-trapping (Bai et al., 5 Jan 2025).

5. Spatial modulation, spin structure, and emergent lattices

A generalized Rice-Mele model introduces a rational spatial phase $$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$7 into both bond alternation and staggered potential,

$$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$8

This produces a long-period modulation periodic in both time and position and yields a Diophantine equation for the $$\hat{H}_0(k,t)=\mathbf{R}(k,t)\cdot \boldsymbol{\sigma}, \qquad \mathbf{R}(k,t)=\left\{-J_1(t)-J_2(t)\cos(ka),\,-J_2(t)\sin(ka),\,\Delta(t)\right\}.$$9-th gap,

$H_{CD}(t)$0

The corresponding one-dimensional Středa formula is

$H_{CD}(t)$1

Adiabatically changing the spatial period therefore pumps charge in space rather than in time: the rightmost occupied charge is displaced relative to the leftmost by the Chern number, and the chain length changes by an integer number of lattice spacings (Asaga et al., 2021).

A spinful extension with spin-orbit coupling changes the parameter-space topology itself. When

$H_{CD}(t)$2

the single degeneracy point of the spinless Rice-Mele model expands into a nodal loop,

$H_{CD}(t)$3

Under periodic boundary conditions, loops that enclose the nodal circle pump spin $H_{CD}(t)$4 in units of $H_{CD}(t)$5, while loops inside or outside it pump spin $H_{CD}(t)$6; the total pumped charge remains zero because the two spin sectors contribute oppositely. Under open boundary conditions, the topological response becomes an edge-pumping-spin flip, quantified only after a double period $H_{CD}(t)$7, since the edge states return to themselves only after two cycles (Ma et al., 4 Jan 2025).

A different route to extension is emergent rather than explicit. In periodically shaken optical lattices with attractively interacting, population-imbalanced fermions, composites self-organize into a density-wave background, and the excess fermions experience an effective Rice-Mele Hamiltonian with alternating inter-orbital tunnelings $H_{CD}(t)$8 and staggered onsite term $H_{CD}(t)$9. The resulting defect structure hosts topological solitons with fractional particle number

$p/q$0

Exact diagonalization confirms the emergent density-wave regime near the Bessel zero $p/q$1, a clustered phase far away, and a defect-rich intermediate phase that persists beyond the simplest mean-field picture [(Przysiezna et al., 2014); (Biedroń et al., 2015)].

6. Realizations, response functions, and boundary-sensitive diagnostics

A direct classical-wave realization was achieved in an acoustic cavity-tube metamaterial. Each site is an acoustic resonator supporting a localized $p/q$2-mode; a square hole drilled in a low-field region shifts the effective onsite potential linearly,

$p/q$3

while the coupling-tube cross section controls the hopping linearly,

$p/q$4

Using

$p/q$5

the synthesized $p/q$6 band structure has Chern number $p/q$7, and the acoustic field distribution shifts adiabatically from the left edge through the bulk to the right edge, in agreement with the tight-binding Rice-Mele prediction (Xia et al., 3 Mar 2026).

Real-space and thermodynamic diagnostics provide a different extension of the model’s interpretation. In a continuum Dirac formulation,

$p/q$8

the spectral consequences depend sharply on the domain and boundary condition. For a semi-infinite geometry, an edge state at $p/q$9 exists when $$\hat{H}(k,t)=\hat{H}_0(k,t)+\hat{H}_{CD}(k,t) =\left(\mathbf{R}+\frac{1}{R^2}\mathbf{R}\times \partial_t\mathbf{R}\right)\cdot \boldsymbol{\sigma},$$0. For a finite domain with symmetric boundary conditions, the critical point shifts to

$$\hat{H}(k,t)=\hat{H}_0(k,t)+\hat{H}_{CD}(k,t) =\left(\mathbf{R}+\frac{1}{R^2}\mathbf{R}\times \partial_t\mathbf{R}\right)\cdot \boldsymbol{\sigma},$$1

whereas it is zero for the semi-infinite domain. Particle-number fluctuations, energy fluctuations, and entropy acquire in-gap peaks reflecting the emergence of edge states, with asymmetric or symmetric peak structure depending on the boundary condition (You et al., 2018).

Polarization diagnostics can be refined beyond the first Berry phase. Gauge-invariant cumulants associated with the Zak phase can be converted into moments and used in a maximum-entropy reconstruction of the underlying polarization distribution. When Wannier functions are localized within one unit cell, the reconstructed distribution corresponds to the Wannier probability density. In the fully dimerized limit all moments have the same magnitude, while cycles around the topological singularity shift the reconstructed distribution into the next unit cell with substantial distortion of its shape (Yahyavi et al., 2017).

The Rice-Mele framework also supports strong-field response diagnostics. In high-harmonic generation, the current operator in the band basis contains not only the usual band-velocity term and interband polarization term but an additional intraband contribution induced by interband transitions,

$$\hat{H}(k,t)=\hat{H}_0(k,t)+\hat{H}_{CD}(k,t) =\left(\mathbf{R}+\frac{1}{R^2}\mathbf{R}\times \partial_t\mathbf{R}\right)\cdot \boldsymbol{\sigma},$$2

This term becomes crucial when the gap is smaller than or comparable to the excitation frequency and the system is close to half filling, indicating that response theory for extended Rice-Mele systems must retain the full gauge-consistent current decomposition (Nagai et al., 2022).

Taken together, these developments show that the extended Rice-Mele model is best understood as a research program rather than a single modification. The unifying object is the dimerized two-site lattice with staggered onsite structure; what changes across the literature is the rule by which topology is encoded, probed, or deformed—through finite-time control, non-Hermiticity, many-body polarization, disorder statistics, synthetic spatial phases, spin transport, nonlinear spectra, or platform-specific observables.