Crystallization of Long-Lived Frenkel Excitons

• The topic defines long-lived Frenkel excitons as tightly bound electron–hole pairs that, due to strong Coulomb binding and symmetry-forbidden recombination, become thermodynamically stable and order periodically.
• It details effective Hamiltonian models—from tight-binding to extended Bose–Hubbard—that capture the balance between minimal kinetic energy and significant inter-exciton repulsion driving crystallization.
• Experimental studies in kagome metals and moiré heterostructures reveal atomic-scale charge modulations and transport anomalies that confirm the formation of exciton crystals.

Searching arXiv for the specified papers and closely related context. arXiv search query: (Jiang et al., 2 Oct 2025) Crystallization of long-lived Frenkel excitons denotes a regime in which tightly bound electron–hole pairs become thermodynamically stable and then order periodically in real space, producing either charge order in an ionic solid or a bosonic crystalline phase in an excitonic insulator. In the correlated-ionic setting, the mechanism has been proposed as an alternative general scenario to a conventional charge-density-wave instability of the Fermi surface, while in a moiré electron–hole bilayer a thermodynamically stable exciton crystal has been reported in thermal equilibrium (Jiang et al., 2 Oct 2025, Qi et al., 27 Jan 2026). The common ingredient is the coexistence of strong Coulomb binding, suppressed recombination, and sufficiently weak or frustrated kinetic motion, so that excitons behave as long-lived interacting bosons whose spatial arrangement is controlled primarily by real-space energetics.

1. Definition of long-lived Frenkel excitons

In a correlated ionic lattice, each ion has a well-defined valence, and a Frenkel exciton is a tightly bound electron–hole pair residing on two neighboring sites. In the kagome-metal realization discussed for CsV$_3$Sb$_5$, the hole occupies a V-$d_{\parallel}$ orbital and the electron an adjacent Sb$_1$-$p_{\perp}$ orbital. The binding energy $E_b$ is large, of order $0.5$–$1\,\mathrm{eV}$, and direct recombination is symmetry-forbidden by opposite $z$-parity of the two orbitals. The resulting lifetime is therefore much longer than the phonon or electronic timescales that would otherwise re-equilibrate charge (Jiang et al., 2 Oct 2025).

The moiré excitonic-insulator realization uses a related but distinct limit: interlayer excitons in a WS$_2$/WSe$_5$0/MoSe$_5$1 heterostructure are treated as dipolar Frenkel-like bosons on a moiré superlattice. The WS$_5$2/WSe$_5$3 bilayer provides an $_5$4-period moiré potential for holes, and the monolayer MoSe$_5$5 electron sheet is separated by a $_5$6 hBN tunnel barrier. In that device, the exciton lifetime satisfies $_5$7, and in practice is limited only by an interlayer tunneling resistance of $_5$8–$_5$9, so that $d_{\parallel}$0 at low $d_{\parallel}$1, far longer than radiative lifetimes of intralayer excitons (Qi et al., 27 Jan 2026).

These two settings differ microscopically—one is an ionic solid with orbital-selective local binding, the other a gate-tunable electron–hole bilayer in a moiré potential—but both instantiate a regime in which excitons are sufficiently localized and long-lived to become the relevant low-energy particles.

2. Microscopic and effective Hamiltonians

For correlated ionic materials, the starting point is a tight-binding model with on-site $d_{\parallel}$2 and nearest-neighbor $d_{\parallel}$3:

$d_{\parallel}$4

When the inter-site repulsion $d_{\parallel}$5 between a hole on V and an electron on Sb outweighs the bare level offset $d_{\parallel}$6, it becomes favorable to form a bound particle–hole pair. The exciton creation operator is

$d_{\parallel}$7

and the reduced low-energy Hamiltonian is

$d_{\parallel}$8

Here $d_{\parallel}$9 is the bare exciton formation energy, $_1$0 arises from the internal hole-hopping around the fixed electron, and $_1$1 models hard-core repulsion when two excitons overlap (Jiang et al., 2 Oct 2025).

In the moiré platform, the effective model is an extended Bose–Hubbard Hamiltonian on the moiré superlattice:

$_1$2

with $_1$3. The nearest-neighbor hopping is strongly quenched by the moiré potential, and for exciton mass $_1$4 and $_1$5, one estimates

$_1$6

The on-site repulsion is very large, $_1$7, while the longer-range interactions are approximately dipole–dipole, with $_1$8 for nearest neighbors and $_1$9 for next-nearest neighbors (Qi et al., 27 Jan 2026).

Both formulations reduce the problem to interacting bosons with suppressed kinetic energy and sizable repulsion. This suggests a common effective description in terms of crystallization of hard-core or nearly hard-core excitons, even though the microscopic origin of the exciton differs across the two material classes.

3. Conditions for exciton crystallization

In the kagome-metal framework, exciton formation first requires

$p_{\perp}$0

equivalently $p_{\perp}$1. For CsV$p_{\perp}$2Sb$p_{\perp}$3, $p_{\perp}$4 and $p_{\perp}$5 is estimated up to $p_{\perp}$6, so bare excitons are thermodynamically stable. The exciton dispersion is written as

$p_{\perp}$7

and crystallization at wavevector $p_{\perp}$8 occurs once the band minimum touches zero:

$p_{\perp}$9

A mean-field estimate of the ordering temperature is

$E_b$0

with $E_b$1 the exciton density. Numerically, $E_b$2 implies $E_b$3, in line with the $E_b$4–$E_b$5 charge-ordering temperatures of AV$E_b$6Sb$E_b$7 (Jiang et al., 2 Oct 2025).

The ordering wavevector and supercell are selected by a competition between potential energy, which favors many small excitons and hence high density, and kinetic energy, which favors larger excitons and hence lower density. In CsV$E_b$8Sb$E_b$9, when $0.5$0 the $0.5$1 supercell minimizes

$0.5$2

while for stronger binding the $0.5$3 lattice wins (Jiang et al., 2 Oct 2025).

In the moiré excitonic insulator, the experimentally relevant crystalline filling is one exciton per three moiré sites, $0.5$4, corresponding to $0.5$5. The moiré lattice period is $0.5$6, giving site density $0.5$7, and the temperature scale for crystal melting is $0.5$8 (Qi et al., 27 Jan 2026).

Taken together, these criteria emphasize that crystallization is governed not by a weak-coupling Fermi-surface instability but by the sign of the exciton formation energy, the location of the exciton band minimum, and the balance of hopping against interaction-driven packing.

4. Relation to charge order, charge-density waves, and bosonic Wigner physics

Once excitons occupy a periodic superlattice, each occupied site carries one extra electron at Sb and one extra hole at V relative to the background. In the kagome treatment this directly produces a real-space charge modulation of atomic scale without any need for a nested Fermi surface. The contrast with a conventional charge-density wave is explicit: rather than a harmonic density modulation

$0.5$9

the exciton crystal yields an atomic density modulation in which $1\,\mathrm{eV}$0 is peaked on certain sites and nearly zero elsewhere. The same treatment states that, because the excitons carry both charge and spin-1, one often observes a co-modulated spin texture, such as spin stripes, locked to the charge pattern (Jiang et al., 2 Oct 2025).

The moiré realization is framed as the bosonic analogue of a Wigner lattice. Strong Coulomb interactions can drive electrons to crystallize into a Wigner lattice, and the reported phase is the corresponding crystal of excitons. In that setting, the crystal forms in an excitonic insulator coupled to a moiré potential, and the platform realizes an electrically tunable extended Bose–Hubbard model in thermal equilibrium (Qi et al., 27 Jan 2026).

A common misconception is to treat all periodic excitonic states as equivalent to a weakly modulated density wave. The two realizations instead point to a more discrete ordering pattern: in the kagome case an atomic-scale charge texture associated with exciton occupancy, and in the moiré case a commensurate bosonic crystal at $1\,\mathrm{eV}$1. This suggests that the relevant organizing principle is real-space localization and packing of long-lived bosons, not merely momentum-space susceptibility enhancement.

5. Kagome realization in CsV$1\,\mathrm{eV}$2Sb$1\,\mathrm{eV}$3

The kagome superconductors provide a concrete material setting for the proposed exciton-crystallization mechanism of charge order. In CsV$1\,\mathrm{eV}$4Sb$1\,\mathrm{eV}$5, the phase diagram in $1\,\mathrm{eV}$6 versus $1\,\mathrm{eV}$7 shows that excitons barely form at $1\,\mathrm{eV}$8, whereas once $1\,\mathrm{eV}$9 the $z$0 phase is stable at $z$1. The ridge $z$2 matches the measured $z$3. An order parameter may be defined as

$z$4

and in mean field $z$5 at $z$6. The analysis further reports that STM imaging sees strong $z$7 modulation of the local density of states, while scattering experiments find the same $z$8 (Jiang et al., 2 Oct 2025).

Along the $z$9 axis, weak interlayer Coulomb forces pack the two-dimensional exciton planes in ABCD or AB sequence, accounting for the observed $_2$0 and imperfect $_2$1 supercells. Within this description, the full phenomenology—$_2$2 in-plane order, stacked supercells, large atomic-scale modulation in STM, a $_2$3 transition, and weakly first-order character—follows from the Frenkel-exciton-crystal framework rather than from Fermi-surface nesting (Jiang et al., 2 Oct 2025).

The significance of this proposal lies in its scope. It is presented not as a material-specific fitting ansatz, but as a generic scenario for correlated ionic materials in which large on-site $_2$4 and sizable inter-site $_2$5 favor tightly bound, long-lived excitons whose superlattice wavevectors are fixed by real-space packing considerations.

6. Moiré excitonic-insulator realization and experimental signatures

The reported observation of an exciton crystal is based on a WS$_2$6/WSe$_2$7/MoSe$_2$8 heterostructure where the WS$_2$9/WSe$_5$00 bilayer forms an $_5$01-period moiré potential for holes and the MoSe$_5$02 monolayer supplies the electron sheet across a $_5$03 hBN barrier. The exciton binding energy is $_5$04, the effective exciton mass is $_5$05, and the moiré potential depth is $_5$06–$_5$07, which localizes the hole component into deep minima and suppresses kinetic energy, described as exactly the Frenkel exciton limit in solids (Qi et al., 27 Jan 2026).

Optical spectroscopy tracks the MoSe$_5$08 intralayer exciton resonance $_5$09 at $_5$10 at $_5$11 under net-neutral gating. As gate bias $_5$12 increases beyond the type-II gap closure, interlayer excitons populate the lattice. Exactly at $_5$13, a well-defined satellite absorption peak appears about $_5$14 above the main $_5$15 line. This is interpreted as Umklapp scattering enabled by spontaneous breaking of translational symmetry into a crystal with an enlarged $_5$16 supercell. The predicted Umklapp energy uses

$_5$17

with $_5$18, yielding $_5$19, in excellent agreement with the observed $_5$20 shift. The satellite is absent away from $_5$21 or when only electrons are doped, and it disappears above $_5$22 as the crystal melts (Qi et al., 27 Jan 2026).

Transport provides an independent signature. In Coulomb drag measurements, the excitonic-insulator regime shows perfect drag, $_5$23, signifying neutral exciton flow. The two-terminal exciton resistance $_5$24 decreases overall with density except for a pronounced peak, $_5$25, centered at $_5$26. Four-terminal hybrid optical–electrical measurements sharpen this maximum further, and the peak persists up to $_5$27. The interpretation given is that hopping vanishes, $_5$28, in the crystalline phase (Qi et al., 27 Jan 2026).

Independent graphite top- and bottom-gates control electron density $_5$29 and hole density $_5$30 through $_5$31 and $_5$32. At net neutrality the system accesses the pure excitonic crystal at $_5$33; away from neutrality it hosts mixed correlated insulating phases in which dipolar excitonic insulators form on top of a hole Mott insulator or generalized Wigner crystals with $_5$34 or $_5$35, while preserving perfect Coulomb drag when interlayer pairing is sustained (Qi et al., 27 Jan 2026).

7. Approximations, limitations, and broader significance

The kagome analysis employs Hartree–Fock $_5$36, treating excitons as non-overlapping hard-core bosons and neglecting some quantum fluctuations. The stated consequence is a likely overestimate of $_5$37 and the production of weakly first-order transitions, consistent with experiment. The treatment also absorbs short-time internal hole motion into effective $_5$38 and binding $_5$39; neglect of higher-order multiplet effects may shift critical couplings by $_5$40–$_5$41. Spin–orbit and longer-range Coulomb terms beyond nearest neighbors are omitted, and these would slightly modulate $_5$42 without changing the overall ordering pattern (Jiang et al., 2 Oct 2025).

In the moiré realization, the significance is cast in terms of platform capability. By stabilizing long-lived, strongly interacting dipolar Frenkel excitons in a moiré potential, the system is reported to realize a bosonic Wigner crystal in thermal equilibrium and its quantum melting. Because the device maps directly onto an extended Bose–Hubbard lattice with adjustable $_5$43, it is presented as a solid-state analogue to cold-atom simulators with electrical-field control, ultrafast optical readout, and tunable species mixtures of bosons and fermions (Qi et al., 27 Jan 2026).

The broader conceptual implication is that crystallization of long-lived Frenkel excitons can unify two themes that are often treated separately: charge order in correlated ionic materials and bosonic crystal formation in engineered excitonic systems. In one case, the phenomenon provides a route to understanding charge order without invoking Fermi-surface nesting; in the other, it furnishes an experimentally tunable extended Bose–Hubbard realization. This suggests that the decisive ingredients are not material family or dimensionality by themselves, but the simultaneous presence of strong binding, long lifetime, hard-core or dipolar repulsion, and sufficiently weak hopping.