Multiparticle Wannier States: Theory & Dynamics

• Multiparticle Wannier States (MWSs) are defined as orthonormal, localized basis states constructed by Fourier transforming multiparticle Bloch eigenstates in systems with co-translational symmetry.
• They enable the study of both geometric and topological properties as well as dynamical phenomena such as long-lived coherent revivals in interacting lattice models.
• MWSs fragment the many-body Hilbert space into invariant band sectors, facilitating efficient computation and deeper insights into weak ergodicity breaking in ETH-satisfying systems.

Searching arXiv for the cited papers and closely related work on multiparticle Wannier states.

Multiparticle Wannier states (MWSs) are Wannier-type states for interacting few- and many-particle lattice systems with co-translational symmetry. They are constructed as Fourier transforms of multiparticle Bloch eigenstates over the center-of-mass quasi-momentum and provide an orthonormal, localized basis for an isolated interacting band (Ke et al., 2017). In recent work on a spatially modulated Bose–Hubbard lattice, MWSs were used to identify a distinct mechanism for weak ergodicity breaking: long-lived collective revivals arising from spectral phase coherence within nearly linear multiparticle bands, even though the underlying Bloch eigenstates satisfy the eigenstate thermalization hypothesis (ETH) (Huang et al., 5 Jul 2026). The concept therefore occupies two connected roles: as a geometric and topological basis for interacting-band physics, and as a dynamical basis for analyzing coherent many-body revivals in ETH-satisfying systems.

1. Co-translational symmetry and the definition of MWSs

The defining structural assumption for MWSs is co-translational symmetry. In an interacting system of $N$ identical particles on a one-dimensional lattice with $L$ unit cells, each cell containing $q$ sites, the interaction can break single-particle translational symmetry while preserving invariance under a simultaneous shift of all particles by one cell. In the formulation of Zeng, Zhu, and Wang, the corresponding single-cell co-translation operator $T_q$ satisfies $[H,T_q]=0$, so that the total center-of-mass quasi-momentum $\kappa\in[0,2\pi/q)$ is a good quantum number (Ke et al., 2017).

Starting from the Fock basis $|n\rangle=|n_1,\dots,n_{qL}\rangle$, one forms co-translational eigenstates by summing over the orbit generated by $T_q$. The Hamiltonian then decomposes into $\kappa$-resolved blocks, and one solves

$$H(\kappa)\,|\psi_m(\kappa)\rangle = E_m(\kappa)\,|\psi_m(\kappa)\rangle,$$

with $L$0 labeling the multiparticle Bloch band (Ke et al., 2017). For a single isolated band, the multi-particle Wannier state is defined by Fourier transform over the Brillouin zone,

$L$1

where $L$2 and $L$3 labels the cell (Ke et al., 2017).

In the Bose–Hubbard superlattice construction of weak ergodicity breaking, the same logic is implemented in a finite system with $L$4 sites, period $L$5, and $L$6 unit cells. There, cotranslation momentum $L$7 is conserved, the $L$8-th multiparticle Bloch band is spanned by eigenstates $L$9, and the MWS centered at cell $q$0 is written as

$q$1

with discretized $q$2 values separated by $q$3 (Huang et al., 5 Jul 2026). This is the discrete analogue of the continuum Brillouin-zone transform used in the earlier topological formulation.

A central interpretive point is that MWSs generalize ordinary Wannier functions from single-particle band theory to interacting sectors organized by center-of-mass quasi-momentum. The 2017 work explicitly states that multi-particle Wannier states provide an orthonormal and localized basis for describing interacting Bloch bands, directly generalizing single-particle maximally localized Wannier functions (Ke et al., 2017). The 2026 work sharpens this perspective by emphasizing that the MWS localizes the center of mass of $q$4 interacting particles rather than individual particles (Huang et al., 5 Jul 2026).

2. Orthonormality, completeness, and gauge fixing

The Fourier-transform construction gives MWSs the same formal orthogonality structure as ordinary Wannier states. In the continuous Brillouin-zone setting,

$q$5

and

$q$6

where $q$7 is the projector onto band $q$8 (Ke et al., 2017). In the finite superlattice setting, orthogonality follows from the $q$9 sum,

$T_q$0

again establishing that the MWSs form an orthonormal basis within each isolated band sector (Huang et al., 5 Jul 2026).

Gauge freedom enters because the phase of each Bloch state is arbitrary:

$T_q$1

To fix this gauge, the 2017 construction introduces the spread functional

$T_q$2

with

$T_q$3

the center-of-mass position operator. Minimizing $T_q$4 yields maximally localized MPWSs (MLMPWSs), which are unique up to an overall phase (Ke et al., 2017).

This localization criterion is conceptually important because it identifies MWSs not merely as formal Fourier transforms but as band-adapted basis states optimized with respect to center-of-mass localization. A plausible implication is that the localization gauge is especially useful when the objective is geometric transport, while the bare Fourier-transform form is sufficient when the objective is spectral and dynamical analysis within a given band. That division of emphasis is reflected in the two papers: the 2017 work focuses on topology and pumping, whereas the 2026 work focuses on revival dynamics (Ke et al., 2017, Huang et al., 5 Jul 2026).

3. Band-resolved Wannier sectors and fragmentation

In the superlattice Bose–Hubbard setting, the MWS basis block-diagonalizes the Hamiltonian:

$T_q$5

Because the Bloch bands do not mix under the Hamiltonian, each band $T_q$6 defines an invariant Wannier sector of dimension $T_q$7, and any initial MWS $T_q$8 evolves entirely within its band subspace (Huang et al., 5 Jul 2026).

The paper characterizes this as band-resolved Wannier-sector fragmentation. The full Hilbert space dimension is $T_q$9, while each sector has dimension $[H,T_q]=0$0, so the decomposition is a fragmentation of the full Hilbert space by band index rather than by dynamical constraints (Huang et al., 5 Jul 2026). This formulation is central to the claim that long-lived revivals can occur without scar-like nonthermal eigenstates.

The same paper explicitly contrasts this structure with Hilbert-space fragmentation generated by dynamical constraints such as dipole conservation. In conventional fragmentation scenarios, decoupled subspaces are associated with nonthermal states. Here, by contrast, fragmentation arises from cotranslation symmetry only, and the subspace eigenstates remain thermal under ETH (Huang et al., 5 Jul 2026). This distinction addresses a likely misconception: the existence of an invariant low-dimensional subspace does not by itself imply scar physics or kinetically constrained fragmentation. The relevant invariant structure is band resolution under cotranslation symmetry.

A further implication is methodological. Since the dynamics of an initial MWS remain confined to a single $[H,T_q]=0$1-dimensional band block, one can work with a projected Hamiltonian matrix of size $[H,T_q]=0$2 per band rather than the full Hilbert space (Huang et al., 5 Jul 2026). This reduction is not a dynamical approximation in the stated setting; it follows directly from the exact band decomposition.

4. Nearly linear bands, band folding, and emergent equally spaced levels

The dynamical role of MWSs in weak ergodicity breaking depends on the spectral structure of superlattice-folded multiparticle bands. In a uniform lattice with $[H,T_q]=0$3, a multiparticle band $[H,T_q]=0$4 is generically curved, so its level spacings are irregular. Under a $[H,T_q]=0$5-site superlattice modulation, such as an onsite interaction profile

$[H,T_q]=0$6

the original Brillouin zone $[H,T_q]=0$7 is folded to $[H,T_q]=0$8, and each original band splits into $[H,T_q]=0$9 sub-bands. Near the crossing points $\kappa\in[0,2\pi/q)$0, gaps of order $\kappa\in[0,2\pi/q)$1 open. If $\kappa\in[0,2\pi/q)$2 is moderate, one of these sub-bands inherits an approximate linear segment of the original dispersion (Huang et al., 5 Jul 2026).

The linearization is expressed as

$\kappa\in[0,2\pi/q)$3

over an interval around some $\kappa\in[0,2\pi/q)$4, which becomes, in the reduced zone, a nearly linear sub-band

$\kappa\in[0,2\pi/q)$5

For discrete momenta $\kappa\in[0,2\pi/q)$6, the level spacing becomes approximately constant,

$\kappa\in[0,2\pi/q)$7

The key claim is that spatially periodic modulation folds and separates energy bands of a simple lattice into several sub-bands, among which nearly linear sub-bands inherit the linear segments of the original bands (Huang et al., 5 Jul 2026).

This equally spaced structure is emergent rather than exact in the generic case. The paper emphasizes that the nearly linear band gives rise to emergent equally spaced energy levels, and that these levels underwrite coherent revivals of MWSs despite ETH-satisfying eigenstates (Huang et al., 5 Jul 2026). A plausible implication is that the relevant spectral property is not integrability or quasiparticle protection, but a controlled miniband geometry generated by band folding and moderate modulation.

The 2026 work also states that other superlattice geometries with period $\kappa\in[0,2\pi/q)$8, as well as two- or three-dimensional lattices, produce similarly folded sub-bands, and that spatiotemporal modulation in a Floquet superlattice can engineer time $\kappa\in[0,2\pi/q)$9 space crystals or controllable miniband spacing (Huang et al., 5 Jul 2026). These statements indicate that the mechanism is framed as a band-engineering principle rather than a peculiarity of a single one-dimensional static model.

5. Revival dynamics and weak ergodicity breaking without nonthermal eigenstates

For an initial MWS in band $|n\rangle=|n_1,\dots,n_{qL}\rangle$0 and cell $|n\rangle=|n_1,\dots,n_{qL}\rangle$1, time evolution is

$|n\rangle=|n_1,\dots,n_{qL}\rangle$2

The associated auto-fidelity is

$|n\rangle=|n_1,\dots,n_{qL}\rangle$3

If the band is linear, $|n\rangle=|n_1,\dots,n_{qL}\rangle$4, every phase factor is an integer multiple of $|n\rangle=|n_1,\dots,n_{qL}\rangle$5, and at

$|n\rangle=|n_1,\dots,n_{qL}\rangle$6

one obtains exact revival with

$|n\rangle=|n_1,\dots,n_{qL}\rangle$7

Thus, an MWS in a nearly linear band exhibits long-lived collective revivals due to equal or nearly equal band spacings (Huang et al., 5 Jul 2026).

Weak curvature produces a controlled departure from exact periodicity. Writing

$|n\rangle=|n_1,\dots,n_{qL}\rangle$8

the phase differences acquire spreads $|n\rangle=|n_1,\dots,n_{qL}\rangle$9, leading to a slow envelope damping or “beat” pattern. The paper states that one finds a fidelity peak $T_q$0 at early times, and a revival envelope set by $T_q$1 only at very long times (Huang et al., 5 Jul 2026). The stated interpretation is that long-lived revivals arise from spectral phase coherence among ETH-satisfying eigenstates rather than from scar-like nonthermal eigenstates.

This point is sharpened by the ETH diagnostics reported in the same work. The multiparticle Bloch states display level statistics near Wigner–Dyson, smooth eigenstate expectation values, and volume-law entanglement; no eigenstate in the coherent band is anomalously low-entangled (Huang et al., 5 Jul 2026). The contrast with quantum many-body scars is explicit: in scar systems, revivals arise from a small set of nonthermal eigenstates embedded in an otherwise thermal spectrum, whereas here revivals stem purely from nearly equal spacings across an ETH-satisfying band (Huang et al., 5 Jul 2026).

The terminology “weak ergodicity breaking” in this context therefore refers to atypically persistent coherent revivals without invoking nonthermal eigenstates. This suggests a narrower and more spectral notion of ergodicity violation than many-body localization or scar subspaces. The paper’s abstract formulates the distinction directly: the typical mechanisms of ergodicity breaking in isolated interacting quantum systems originate from the nonthermal nature of the underlying eigenstates, whereas the MWS mechanism does not (Huang et al., 5 Jul 2026).

6. Topological transport, interacting pumping, and broader extensions

Before their dynamical use in weak ergodicity breaking, MWSs were introduced in the context of interacting topological transport. For a cyclically and adiabatically time-dependent Hamiltonian $T_q$2 satisfying

$T_q$3

the Berry connection and Berry curvature of band $T_q$4 define the multi-particle Chern number

$T_q$5

Preparing the maximally localized MWS $T_q$6 and evolving adiabatically over one cycle yields a net center-of-mass displacement

$T_q$7

so that the center of mass moves by $T_q$8 cells in one cycle (Ke et al., 2017).

The explicit example given is two interacting bosons in a Rice–Mele pump with Hamiltonian

$T_q$9

with alternating hoppings and onsite energies tracing an ellipse over one period (Ke et al., 2017). In the strong-interaction regime $\kappa$0, the two bosons form a tightly bound pair, and projection onto the bound-state subspace gives an effective single-particle Rice–Mele model for the pair. That effective model has two isolated Bloch bands with Chern numbers $\kappa$1 and $\kappa$2, and pumping an MLMPWS of the lower bound-state band shifts the pair as a whole by $\kappa$3 cell per period, while the upper band moves by $\kappa$4 cell (Ke et al., 2017).

The same work also identifies a different regime, termed topologically resonant tunneling. When the instantaneous bias $\kappa$5 passes through $\kappa$6, the bound state and a split state become nearly resonant, and one boson can tunnel across the barrier while the other remains behind. Over a full pumping cycle, the system executes four resonant Landau–Zener passages, effectively moving the two bosons sequentially one after the other by one cell, while the net Chern number of the multi-particle band remains $\kappa$7 (Ke et al., 2017). This example shows that MWSs are compatible both with bound-pair transport and with interaction-assisted sequential motion.

The two papers also outline extensions. The topological work states that the MWS concept can be extended to higher $\kappa$8 and more complex interactions, offering a route toward the study and control of interacting topological phases and fractional pumping phenomena (Ke et al., 2017). The dynamical work states that extensions to Bose–Hubbard ladders, Fermi–Hubbard models, and spin chains with cotranslation symmetry are straightforward, and that observables such as particle density $\kappa$9, two-point correlations, entanglement entropy, and out-of-time-ordered correlators can be efficiently computed within each fragment (Huang et al., 5 Jul 2026). Taken together, these statements place MWSs at the intersection of interacting-band geometry, adiabatic transport, and coherent nonequilibrium many-body dynamics.