Paired Wigner Crystal

• Paired Wigner crystals are charge-ordered phases where the localized objects are electron pairs, exhibiting intrinsic spin-singlet or spin-triplet bonds.
• Studies employ frustrated lattice models, variational Monte Carlo, and DMRG to reveal cooperative charge disproportionation, bond modulation, and enhanced singlet-triplet gaps.
• Investigations in moiré potentials and quantum-geometric electron gases showcase practical examples where paired crystallization provides novel insights into correlated electronic systems.

Searching arXiv for recent and foundational papers on paired Wigner crystals and closely related concepts. First, I’ll look for papers explicitly using the term “paired Wigner crystal” and related formulations such as “paired electron crystal.” Searching arXiv for "paired Wigner crystal" and "paired electron crystal". A paired Wigner crystal is a charge-ordered insulating state in which the objects that crystallize are pairs rather than single carriers. Across the literature, the term does not denote a single universal phase: in frustrated quarter-filled lattice models it closely corresponds to the paired-electron crystal, a commensurate crystal of nearest-neighbor Heitler-London spin singlets; in moiré systems it can denote a molecular Wigner crystal of delocalized opposite-spin two-electron singlets; and in quantum-geometric electron gases it can denote a crystal of local two-electron bound states, including a spin-triplet state with relative orbital angular momentum $m=-1$. Other works establish important negative or limiting cases, including models in which Wigner-crystal charge order suppresses pairing rather than promoting it, and settings where only defect pairing, rather than electron pairing, occurs (Dayal et al., 2011, Smith et al., 18 Feb 2026, Zverevich et al., 8 Jan 2026, Clay et al., 2023, Zhuang et al., 2024).

1. Conceptual definition and scope

In the strict sense, a paired Wigner crystal is a crystalline electronic phase whose elementary localized object is a two-electron composite rather than an individual electron. The phase therefore differs from an ordinary Wigner crystal, where single carriers localize one-by-one at crystal sites. It also differs from a generic charge-density wave pinned to an external lattice, because the paired object has an internal structure—typically a spin singlet or, in one proposal, a spin-triplet state with nontrivial relative angular momentum.

The literature represented here uses several closely related but nonidentical formulations. The paired-electron crystal on the anisotropic triangular lattice is explicitly described as a “WC of Heitler-London spin-singlets,” making it the clearest quarter-filled lattice realization of a paired Wigner crystal in the strong-correlation literature (Dayal et al., 2011). In a honeycomb moiré potential, the paired Wigner crystal is a molecular crystal of opposite-spin electrons delocalized over a six-minimum hexagon, with the molecular centers forming a triangular lattice (Smith et al., 18 Feb 2026). In a quantum-geometric two-dimensional electron gas, the paired Wigner crystal is a low-density crystal with two electrons per unit cell, and can occur in both singlet $m=0$ and spin-triplet $m=-1$ versions (Zverevich et al., 8 Jan 2026).

A central interpretive issue is that not every Wigner crystal displaying clustering or pair-like language is a paired Wigner crystal in this sense. Some papers study only ordinary Wigner crystallization of individual particles, using density pair correlations but no pair operators; others study defect pairing in a pre-existing Wigner crystal rather than electron pairing. These distinctions are essential for correct usage of the term (Jaworowski et al., 2017, Zhuang et al., 2024).

2. Frustration-stabilized paired-electron crystal at $\rho=\frac{1}{2}$

The canonical lattice realization is the paired-electron crystal on the anisotropic triangular lattice at carrier density $\rho=\frac{1}{2}$, described in the quarter-filled-band language customary for charge-transfer solids. The Hamiltonian includes anisotropic hopping, on-site and intersite Coulomb repulsion, and self-consistent intersite and intrasite electron-phonon couplings,

$$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$

with

$$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$

Its defining real-space pattern is a charge disproportionation of charge-rich and charge-poor sites, with

$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$

arranged as

$\cdots 1100 \cdots$

along two directions and

$\cdots 1010 \cdots$

along the third. The nearest-neighbor charge-rich sites are connected by the strongest bond and form a local spin singlet. The phase is therefore not merely charge ordered: it has coexisting charge disproportionation, bond modulation, local singlet formation, and a spin gap. The singlet-triplet gap is

$m=0$0

This state competes with two more familiar quarter-filled broken-symmetry states: a dimerized antiferromagnet at weak frustration and a checkerboard Wigner crystal at sufficiently large nearest-neighbor Coulomb repulsion. The paired-electron crystal is stabilized by moderate to strong frustration, introduced by the diagonal hopping $m=0$1, and can occur even for $m=0$2. That feature is central: nearest-neighbor Coulomb repulsion is not essential to the phase, although moderate $m=0$3 can strengthen it and reduce the critical frustration $m=0$4 for its onset (Dayal et al., 2011).

The paper’s phase-diagram trends sharpen the distinction between ordinary and paired Wigner crystallization. For $m=0$5, $m=0$6, and $m=0$7, increasing $m=0$8 drives a transition from the paired-electron crystal to the checkerboard Wigner crystal near $m=0$9, with a WC-SG crossover or subphase boundary near $m=-1$0. At $m=-1$1 and $m=-1$2, the checkerboard Wigner-crystal threshold is estimated as $m=-1$3. When $m=-1$4, checkerboard Wigner order is destabilized, while the AFM–PEC transition remains over $m=-1$5. This establishes the PEC as a frustration-stabilized paired analogue of Wigner crystallization rather than a simple deformation of checkerboard charge order (Dayal et al., 2011).

The paper also advances a broader physical interpretation: at $m=-1$6, charge disproportionation and singlet bonding are cooperative rather than competing. A plausible implication is that the paired Wigner crystal is the quarter-filled analogue of a valence-bond solid, but with charge degrees of freedom intrinsically active.

3. Competition with ordinary Wigner-crystal order on the square lattice

A distinct question is whether proximity to an ordinary quarter-filled Wigner crystal enhances superconducting pairing, or even produces a paired Wigner crystal, in the two-dimensional square-lattice extended Hubbard model. The model studied is

$m=-1$7

at density

$m=-1$8

with the main calculations at $m=-1$9. In this convention, the quarter-filled Wigner crystal is the checkerboard charge order with wavevector

$\rho=\frac{1}{2}$0

the two-dimensional analogue of the one-dimensional $\rho=\frac{1}{2}$1 pattern (Clay et al., 2023).

The numerical study combines exact diagonalization on a 20-site periodic cluster with DMRG on width-4 rectangular and diamond cylinders. Charge correlations are diagnosed through

$\rho=\frac{1}{2}$2

spin correlations through

$\rho=\frac{1}{2}$3

and superconducting correlations through the singlet pair operator

$\rho=\frac{1}{2}$4

with pair-pair correlator

$\rho=\frac{1}{2}$5

The outcome is explicitly negative for a paired-Wigner-crystal interpretation. The estimated checkerboard charge-ordering transition lies around $\rho=\frac{1}{2}$6 for $\rho=\frac{1}{2}$7, with $\rho=\frac{1}{2}$8 peaking near $\rho=\frac{1}{2}$9 on the periodic cluster. However, the dominant $\rho=\frac{1}{2}$0 pairing channel weakens monotonically with increasing $\rho=\frac{1}{2}$1 at long distance, and $\rho=\frac{1}{2}$2 shows only a negative peak near $\rho=\frac{1}{2}$3, rather than the positive-then-negative structure expected from a metal $\rho=\frac{1}{2}$4 superconductor $\rho=\frac{1}{2}$5 charge-order sequence. On the rectangular cylinder, long-range $\rho=\frac{1}{2}$6 correlations are already significantly weaker than the free-fermion case for $\rho=\frac{1}{2}$7; on the diamond cylinder, both nearest-neighbor and next-nearest-neighbor singlet correlations decay faster than $\rho=\frac{1}{2}$8 for all studied $\rho=\frac{1}{2}$9, and their magnitudes are smaller for all interacting cases than for $$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$0 (Clay et al., 2023).

This establishes a major caution in the paired-Wigner-crystal literature: ordinary Wigner-crystal charge order and pairing need not be proximate in a constructive way. In the square-lattice extended Hubbard model at $$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$1, the checkerboard Wigner crystal competes with superconducting pairing rather than promoting it. The paper therefore does not support coexistence of Wigner-crystal order and strong pairing, an intervening paired phase, or a pretransitional rise of long-distance pair correlations as $$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$2 (Clay et al., 2023).

4. Moiré and quantum-geometric paired Wigner crystals

Two recent directions realize the paired-Wigner-crystal concept in qualitatively different ways.

In a honeycomb moiré potential at filling factor

$$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$3

the paired Wigner crystal is an interaction-driven insulating phase of a two-dimensional electron gas in which opposite-spin electrons form singlet-like valence-bond pairs delocalized over a hexagonal ring of six moiré minima. The continuum Hamiltonian is

$$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$4

with

$$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$5

The molecular centers form a triangular lattice, one quarter of the moiré minima are mostly depleted, the local spin density remains $$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$6, and the internal two-electron structure favors relative angle $$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$7, indicating that the two electrons prefer opposite sides of the same ring. The phase is diagnosed by the complex polarization

$$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$8

with $$H= -\sum_{\nu,\langle ij\rangle_\nu} t_\nu(1+\alpha_\nu \Delta_{ij}) B_{ij} +\frac{1}{2}\sum_{\nu,\langle ij\rangle_\nu} K_\alpha^\nu \Delta_{ij}^2 +\beta \sum_i v_i n_i +\frac{1}{2}K_\beta \sum_i v_i^2 +U\sum_i n_{i\uparrow}n_{i\downarrow} +\frac{1}{2}\sum_{\langle ij\rangle} V_{ij} n_i n_j,$$9 for a metal and $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$0 for an insulator, and by a molecular localization measure

$$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$1

with $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$2 for uniform charge and $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$3 for perfect ring localization. The reported evolution is two-stage: $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$4 begins to grow around $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$5, while $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$6 rises strongly only for $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$7, and at $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$8, $$B_{ij}=\sum_\sigma \left(c^\dagger_{i\sigma}c_{j\sigma}+ \text{H.c.}\right).$$9. This suggests local molecule formation before full insulating crystallization. The state was discovered using a neural-network-based variational Monte Carlo ansatz, and fixed-node DMC with neural-network-derived orbitals yields lower energy than DMC with LDA orbitals at $$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$0, supporting the ring-like paired molecular crystal as a genuine variational improvement (Smith et al., 18 Feb 2026).

A different realization emerges in a low-density two-dimensional electron gas with nontrivial band geometry. There the conventional Wigner crystal has one electron per unit cell, whereas the paired Wigner crystal has two electrons per unit cell localized as an intracell bound state. The one-band Hamiltonian includes band form factors

$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$1

and the key control variables are the density parameter

$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$2

and the geometric scale

$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$3

The paired crystal admits both a singlet $$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$4 state and a spin-triplet $$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$5 state. The internal relative-motion problem obeys

$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$6

and the central analytical mechanism is the geometric correction

$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$7

The second term is a metric contribution that raises all pair energies, while the first lowers negative-$$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$8 states. At $$\rho_i = 0.5+\delta \quad \text{or} \quad 0.5-\delta,$$9, the variational phase diagram gives a transition from the ordinary Wigner crystal to a singlet paired crystal near $\cdots 1100 \cdots$0. Increasing Berry curvature extends the paired regime and can stabilize the $\cdots 1100 \cdots$1 triplet paired crystal; for $\cdots 1100 \cdots$2, the paper finds a direct transition from the monatomic crystal to the $\cdots 1100 \cdots$3 paired crystal, whereas for $\cdots 1100 \cdots$4 the $\cdots 1100 \cdots$5 phase is absent (Zverevich et al., 8 Jan 2026).

These two realizations share a core idea—crystallization of paired objects—but the microscopic pairing mechanism differs. In artificial graphene the pair is a singlet-like valence-bond molecule delocalized over six minima; in the quantum-geometric electron gas the pair is an intracell bound state whose internal angular momentum is selected by Berry curvature.

5. Related but distinct phenomena

Several adjacent literatures are relevant precisely because they are not, strictly speaking, paired Wigner crystals.

The first is ordinary Wigner crystallization in partially filled topological flat bands. In that setting the many-body problem is solved by exact diagonalization in a single-band projected, flat-band approximation for spinless particles on kagome, honeycomb/Haldane, and checkerboard lattices. The diagnostic is the pair correlation density

$\cdots 1100 \cdots$6

together with its Cartesian and angular Fourier transforms,

$\cdots 1100 \cdots$7

Crystallization strength increases as filling decreases, with a rapid increase beginning around $\cdots 1100 \cdots$8, and strong Wigner crystals most common below roughly $\cdots 1100 \cdots$9. But there is no pairing-channel analysis, no pair binding energy, no pair-pair correlator, and no charge-$\cdots 1010 \cdots$0 density operator. The paper therefore provides a diagnostic baseline for ordinary Wigner crystals, not evidence for a pair crystal (Jaworowski et al., 2017).

A second neighboring concept is the Wigner molecular crystal observed in twisted bilayer WS$\cdots 1010 \cdots$1 moiré superlattices. There the experimentally imaged objects are multi-electron or multi-hole intracell clusters hosted by moiré artificial atoms. The relevant parameter is

$\cdots 1010 \cdots$2

with quoted hole-side scales $\cdots 1010 \cdots$3, $\cdots 1010 \cdots$4, and $\cdots 1010 \cdots$5. The observed intracell structures include a two-hole ring-like or, under strain, dimer-like state, a three-particle trimer, and a four-hole molecular pattern. This is a cluster crystal of charges pinned to moiré sites rather than a demonstrated crystal of bound charge-$\cdots 1010 \cdots$6 pairs. The two-hole state is highly relevant as an analogue, but the mechanism is confinement plus Coulomb repulsion inside one moiré well, not a many-body paired crystal in the narrower sense (Li et al., 2023).

A third related phenomenon is defect pairing in a weakly density-imbalanced bilayer Wigner crystal. The underlying bilayer electron gas has Hamiltonian

$\cdots 1010 \cdots$7

with imbalance accommodated by vacancies, interstitials, or interstitial-vacancy bound states. The paired object here is not an electron pair but an interlayer interstitial-vacancy bound state with binding energy

$\cdots 1010 \cdots$8

The paired-defect liquid is favored when

$\cdots 1010 \cdots$9

In the classical limit, paired defects are favored at small $m=0$00 and again in a narrow phase-V window $m=0$01. This is an important extension of pairing phenomena in Wigner crystals, but the pairing occurs at the defect level rather than at the level of the electrons comprising the crystal (Zhuang et al., 2024).

6. Diagnostics, misconceptions, and interpretive boundaries

Because “paired Wigner crystal” spans several non-equivalent constructions, diagnosis depends on the object whose crystallization is being asserted. For the paired-electron crystal, the essential evidence is the cooperative appearance of charge disproportionation, enhanced bond order on a charge-rich nearest-neighbor bond, strongly negative spin correlation on that same bond, and an increased singlet-triplet gap, together with the specific $m=0$02 directional pattern (Dayal et al., 2011). For the artificial-graphene molecular phase, the decisive combination is long-range charge order of molecular centers, restored local $m=0$03 symmetry within each occupied hexagon, short-range singlet-like spin correlations, and insulating behavior inferred from the complex polarization $m=0$04 (Smith et al., 18 Feb 2026). For the quantum-geometric pair crystal, the diagnosis is variational: the minimized energy of a crystal with two electrons per unit cell must beat that of the monatomic Wigner crystal, and the internal two-body problem must select the relevant angular-momentum channel (Zverevich et al., 8 Jan 2026).

A recurring misconception is to equate any use of “pair correlation” or any clustered charge texture with a paired Wigner crystal. The topological-flat-band study demonstrates why this is incorrect: its pair correlation density is an ordinary conditional density of constituent particles, not a pair operator, and the observed crystals are interpreted throughout as crystals of individual particles (Jaworowski et al., 2017). Another misconception is that proximity to Wigner-crystal charge order generically enhances electronic pairing. The square-lattice extended Hubbard results directly contradict that inference in the simplest checkerboard-WC setting at $m=0$05, where increasing $m=0$06 suppresses long-distance pairing (Clay et al., 2023).

The following comparison summarizes the principal variants discussed in the literature represented here.

Variant	Localized object	Status
Paired-electron crystal	Nearest-neighbor Heitler-London spin-singlet pair	Direct paired-WC realization
Artificial-graphene PWC	Opposite-spin singlet-like hexagonal molecule	Direct paired-WC realization
Quantum-geometric PWC	Intracell two-electron bound state, $m=0$07 or $m=0$08	Direct paired-WC realization
Square-lattice checkerboard WC near $m=0$09	Single electrons in $m=0$10 charge order	Negative result for paired-WC formation
Bilayer defect liquid	Interlayer interstitial-vacancy bound defect	Defect pairing, not electron pairing
Wigner molecular crystal in moiré artificial atoms	Intracell few-body cluster	Related cluster crystal, not a narrow PWC

Taken together, these works imply that the phrase “paired Wigner crystal” is best reserved for phases where pair formation is intrinsic to the crystallizing degree of freedom. Within that restricted usage, three distinct mechanisms are presently represented: frustration-stabilized singlet pairing at quarter filling, emergent molecular singlets in a honeycomb moiré potential, and Berry-curvature-stabilized intracell pairing—including spin-triplet pairing—in a low-density electron crystal. The negative and neighboring cases are equally important, because they delimit what the term does not mean (Dayal et al., 2011, Smith et al., 18 Feb 2026, Zverevich et al., 8 Jan 2026, Clay et al., 2023, Jaworowski et al., 2017, Zhuang et al., 2024, Li et al., 2023).