Anyonic Tight-Binding Lattice Model

• An anyonic tight-binding model is a lattice system where fractional exchange statistics are encoded in the operator algebra, leading to occupancy-dependent hopping phases.
• This framework generates nontrivial effects such as statistics-induced bound states, asymmetric momentum distributions, and effective many-body interactions despite a quadratic Hamiltonian.
• Numerical and bosonization studies reveal unconventional correlation exponents and phase transitions, drawing parallels with fractional quantum Hall edge phenomena.

Searching arXiv for recent and foundational papers on anyonic tight-binding lattice models. An anyonic tight-binding model is a lattice model in which particles propagate by nearest-neighbor hopping on a discrete lattice while the microscopic fields obey fractional exchange algebra, so that the kinetic term is intrinsically statistical rather than a free bosonic or fermionic hopping problem. In one-dimensional realizations, this structure appears either directly in number-conserving Fock parafermion chains or, after a fractional Jordan–Wigner transformation, as occupation-dependent Peierls phases and correlated hopping in anyon-Hubbard systems. Two paradigmatic formulations are the Fock-parafermion chain, where a formally quadratic Hamiltonian is nevertheless nonintegrable and supports statistics-induced bound states, and the extended anyon-Hubbard chain at unit filling, where the statistical angle reorganizes the bosonized low-energy theory and the commensurate phase diagram (Rossini et al., 2018, Bonkhoff et al., 2024).

1. Operator algebra and the meaning of “anyonic” hopping

The defining feature of an anyonic tight-binding model is not merely lattice discreteness but the fact that exchange statistics are encoded directly in the operator algebra. In the parafermionic formulation, one starts from parafermion operators $\{\hat\gamma_j\}$ of order $p$ obeying

$$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$

together with

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$

For $p=2$ these reduce to Majorana operators. The particle-like degrees of freedom are Fock parafermions $\hat F_j,\hat F_j^\dagger$, for which each site has a local $p$-dimensional Fock space

$$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$

and onsite number operator

$$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$

The total number $\hat N=\sum_j\hat N_j$ is conserved, giving the model a genuine $p$0 symmetry. On different sites, Fock parafermions satisfy

$p$1

or equivalently

$p$2

In the anyon-Hubbard formulation, the lattice anyon operators satisfy the deformed exchange relation

$p$3

Here $p$4 gives the bosonic limit, while $p$5 gives “pseudo-fermions”: fermionic exchange for $p$6, but not ordinary fermions onsite. The exact fractional Jordan–Wigner transformation

$p$7

maps the anyons to bosons, and the kinetic term becomes

$p$8

This occupation-dependent phase is the core tight-binding manifestation of anyonic statistics: the exchange algebra is traded for a dynamical gauge phase attached to hopping rather than a removable global phase (Rossini et al., 2018, Bonkhoff et al., 2024).

2. The number-conserving Fock-parafermion chain

The simplest number-conserving anyonic tight-binding model studied for Fock parafermions is the one-dimensional nearest-neighbor chain

$p$9

usually with $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$0. Although formally quadratic in $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$1, this Hamiltonian is not a free-particle problem for $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$2. It conserves $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$3, is not inversion invariant because the FP algebra is oriented, is not time-reversal invariant, and is not particle-hole symmetric; however, it is invariant under the combined transformation $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$4. Because of this combined symmetry, densities above $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$5 can be related to lower ones.

The single-particle sector has the usual tight-binding dispersion

$$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$6

with Fourier modes

$$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$7

For $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$8, however, the momentum-space operators are not again Fock parafermions and do not satisfy a simple canonical algebra. The formal expression $$\hat\gamma_j \hat\gamma_l=\omega^{\mathrm{sgn}(j-l)}\hat\gamma_l\hat\gamma_j,\qquad \omega=e^{2\pi i/p},$$9 is therefore misleading, because

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$0

From the viewpoint of ordinary bosons or fermions, the exchange strings generate effective interactions even though the Hamiltonian is bilinear in $$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$1.

The generalized Jordan–Wigner or Fradkin–Kadanoff transformation makes this locality structure explicit. Introducing onsite Weyl hard-core boson operators $$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$2 and diagonal phase operators

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$3

one writes

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$4

so that

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$5

and hence

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$6

The mapped representation is local and suitable for DMRG, but the hopping amplitude is now occupation dependent through $$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$7: quadratic anyonic hopping becomes a correlated-hopping problem of conventional commuting onsite operators.

A further structural distinction is the relation between Fock parafermions and fractionalized fermions. For even $$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$8, the clustered operator

$$\hat\gamma_j^p=1,\qquad \hat\gamma_j^\dagger=\hat\gamma_j^{p-1}.$$9

obeys canonical fermion relations,

$p=2$0

Thus one ordinary fermion can be viewed as a bound cluster of $p=2$1 parafermions; for $p=2$2, $p=2$3. This is why the $p=2$4 chain is interpreted as a model of fractionalized fermions (Rossini et al., 2018).

3. Statistics-induced spectral structure and evidence for nonintegrability

The two-body problem already displays nontrivial consequences of anyonic statistics. With basis states

$p=2$5

the wavefunction can be written as

$p=2$6

with scattering form

$p=2$7

The energy remains additive,

$p=2$8

but periodic boundary conditions now impose statistically twisted quantization,

$p=2$9

The momenta are thus shifted by a statistical twist of order $\hat F_j,\hat F_j^\dagger$0, analogous to twisted boundary conditions in continuum anyon models.

Exact diagonalization reveals a stronger effect: branches of two-body bound states in addition to the scattering continuum. For $\hat F_j,\hat F_j^\dagger$1 they appear as two “mustaches” outside the continuum in $\hat F_j,\hat F_j^\dagger$2 versus center-of-mass momentum $\hat F_j,\hat F_j^\dagger$3. Their hallmark is that the wavefunction weight $\hat F_j,\hat F_j^\dagger$4 is exponentially localized in the relative coordinate $\hat F_j,\hat F_j^\dagger$5, strongly peaked at $\hat F_j,\hat F_j^\dagger$6, which requires complex $\hat F_j,\hat F_j^\dagger$7. The binding is strongest near $\hat F_j,\hat F_j^\dagger$8 and weakens as the branch approaches the continuum lobes. The physical origin is purely statistical: there is no explicit interaction term, but anyonic exchange phases effectively generate attraction in certain channels. As $\hat F_j,\hat F_j^\dagger$9 increases and $p$0, the bound-state branches become less visible and merge into the continuum.

The nonintegrability claim for the quadratic chain is numerical rather than a theorem. The evidence comes from level-spacing statistics at fixed particle number with open boundary conditions. Integrable systems should show Poisson statistics,

$p$1

whereas nonintegrable systems show Wigner–Dyson level repulsion,

$p$2

For $p$3, increasing system size yields clear level repulsion; for $p$4 the statistics converge rapidly toward a GOE-like form, while $p$5 also shows repulsion although finite-size convergence is slower. By contrast, $p$6 gives Poisson-like statistics, as expected for an integrable free-fermion chain. The paper also notes that a small hopping inhomogeneity can drive a GOE-to-GUE crossover, consistent with the complex nature of the Hamiltonian and symmetry sensitivity of random-matrix classes. The resulting conceptual point is precise: a quadratic anyonic model need not be integrable because the exchange algebra generates effective many-body interactions (Rossini et al., 2018).

4. Gapless regimes, anyonic correlations, and fractionalized fermions

The many-body problem of the Fock-parafermion chain was analyzed by DMRG on systems up to $p$7 with open boundary conditions. In the cases studied, the neutral gap $p$8 at fixed $p$9 scales as $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$0 except for $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$1 at unit filling $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$2, where it remains finite, indicating a gapped phase. Entanglement entropy in the gapless regimes fits

$$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$3

with $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$4, establishing a $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$5 Luttinger-liquid universality class. The gapped $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$6, $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$7 state instead obeys an area law and is interpreted as an “anyonic Mott-like” phase.

The basic one-body anyonic correlator is

$$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$8

In the local $$|m_j\rangle=\hat F_j^{\dagger m}|0\rangle,\qquad 0\le m\le p-1,$$9 representation it becomes a string correlator,

$$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$0

which is the direct lattice manifestation of anyonic statistics. In the gapless phases,

$$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$1

For $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$2, $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$3 stays around $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$4-$$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$5 over a broad density range; for $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$6, $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$7 decreases with density from about $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$8 to $$\hat N_j=\sum_{l=1}^{p-1}\hat F_j^{\dagger l}\hat F_j^l.$$9. In the gapped $\hat N=\sum_j\hat N_j$0, $\hat N=\sum_j\hat N_j$1 case, $\hat N=\sum_j\hat N_j$2 decays exponentially as $\hat N=\sum_j\hat N_j$3.

These exponents were compared with the anyonic Luttinger-liquid theory of Calabrese, Mintchev, and Bellazzini, according to which

$\hat N=\sum_j\hat N_j$4

For noninteracting anyons, $\hat N=\sum_j\hat N_j$5 and therefore $\hat N=\sum_j\hat N_j$6. The agreement improves at larger densities. The same framework predicts for

$\hat N=\sum_j\hat N_j$7

the decay

$\hat N=\sum_j\hat N_j$8

For $\hat N=\sum_j\hat N_j$9 the agreement is good; for $p$00 there are visible deviations, heuristically ascribed to truncation effects from the very small onsite Hilbert space.

The $p$01 case is distinguished by the fermionic cluster operator $p$02. Its correlator

$p$03

also decays algebraically,

$p$04

with Luttinger-liquid-inspired prediction

$p$05

so that $p$06 for $p$07. Numerically the fitted exponents approach this value at higher densities. This means that although $p$08 are genuine fermion operators, their two-point function decays roughly as $p$09 rather than as $p$10 for free fermions.

The momentum-space signatures are likewise anyonic. The anyonic momentum distribution

$p$11

is not symmetric under $p$12 and has a peak at positive momentum. The expected peak position is

$p$13

This relation is motivated from twisted momentum quantization, from the exchange-phase accumulation in $p$14, and from Luttinger-liquid theory, though significant deviations appear for $p$15. The fermionic momentum distribution

$p$16

is again asymmetric under $p$17 and peaked at nonzero momentum, unlike the standard free-fermion case.

For interpretation, the $p$18 chain was compared phenomenologically with two counter-propagating Laughlin edge modes at filling $p$19. The low-energy Hamiltonian was taken as

$p$20

with

$p$21

The correspondence is not exact: in the Laughlin-edge theory right and left movers carry opposite statistical parameter and the full theory is time-reversal invariant, whereas the lattice chain has both chiralities but only one statistical parameter and is not time-reversal invariant. Still, the scaling of the fermionic correlator for $p$22 at larger density agrees quantitatively with Wen’s hydrodynamic prediction for electron correlations at the edge of a $p$23 Laughlin state (Rossini et al., 2018).

5. Extended anyon-Hubbard chains and the interacting phase diagram

A second major realization of the anyonic tight-binding idea is the one-dimensional extended Hubbard model for lattice anyons,

$p$24

analyzed at average unit filling $p$25 and, for the main bosonization treatment, under the local constraint $p$26. After the fractional Jordan–Wigner map, the kinetic term acquires the density-dependent gauge phase

$p$27

At unit filling with maximal occupancy two, the local Hilbert space is mapped to spin-1 through

$p$28

and the kinetic term takes the occupation-dependent form

$p$29

The nontrivial amplitudes are

$p$30

This is a concrete microscopic realization of hopping in a dynamical lattice gauge field.

Bosonization yields a two-channel sine-Gordon theory in terms of $p$31 and $p$32, with low-energy Hamiltonian containing

$p$33

The angle $p$34 therefore enters not only through the cosine perturbations and Luttinger parameters $p$35, but also through a current-density coupling that produces left-right asymmetry in dynamics. A naively allowed perturbation $p$36 is absent because a modified inversion symmetry $p$37 forbids it, protecting a full $p$38 Gaussian transition line.

At unit filling, the phase diagram contains four gapped insulating phases and two gapless regimes. The insulating phases are characterized by distinct pinning patterns and order parameters:

Phase	Pinned fields	Diagnostic
Mott insulator (MI)	$p$39	$p$40
Charge-density wave (CDW)	$p$41	$p$42
Haldane insulator (HI)	$p$43	$p$44
Dimerized state	$p$45	$p$46

The corresponding diagnostics are

$p$47

$p$48

with $p$49. For small $p$50 there are also superfluid (SF) and pair-superfluid (PSF) regions; the paper identifies them numerically and does not give a full bosonization description.

The bosonization predicts two principal critical lines. A Gaussian $p$51 line occurs when $p$52, yielding the weak-coupling estimate

$p$53

An Ising $p$54 line arises from competition between $p$55 and $p$56 near $p$57, with estimate

$p$58

Numerically the transition lines are fit as

$p$59

Their opposite trends explain why the Haldane insulator shrinks and the dimerized phase grows as $p$60. For each fixed $p$61, the MI, HI, CDW, and dimer phases meet at a common crossing point, interpreted as a multicritical line in $p$62 space with bosonization suggestion $p$63.

The numerical analysis combines finite DMRG, infinite DMRG, and exact diagonalization for level spectroscopy. Phase boundaries and universality classes are identified through order parameters, fidelity susceptibility, entanglement entropy, correlation length, and level crossings. The reported critical behavior is: HI–CDW Ising with $p$64 and exponents $p$65, $p$66; dimer–MI also Ising at large $p$67; HI–MI Gaussian with $p$68; CDW–dimer continuous with $p$69 and exponents close to $p$70, $p$71; SF/PSF boundaries of BKT type; and MI–CDW first order at large $p$72 (Bonkhoff et al., 2024).

6. Scope, misconceptions, and relation to topological lattice modeling

Several distinctions are central to the subject. First, an anyonic tight-binding model is not defined by nontrivial band topology alone. The parafermion chain and the extended anyon-Hubbard model are anyonic because fractional statistics are present in the microscopic algebra or, equivalently after mapping, in the density-dependent gauge structure of the hopping. This is why a formally quadratic chain can be nontrivial, and why in the interacting anyon-Hubbard case the statistical angle reshapes the competition among commensurate instabilities. A useful concise formulation is that the characteristic lattice signature of one-dimensional anyonic tight binding is statistical transmutation encoded as local, occupation-dependent phases in nearest-neighbor hopping (Rossini et al., 2018, Bonkhoff et al., 2024).

Second, not every tight-binding model relevant to fractionalized many-body physics is itself an anyonic tight-binding model. The three-orbital moiré model for twisted transition metal dichalcogenide homobilayers is explicitly presented as a strictly local tight-binding basis with nontrivial band Chern numbers, short-range interactions, and complex symmetry-fixed phases, and it is discussed as a promising parent lattice model for fractional Chern insulators. At the same time, it explicitly states that anyons are not directly studied, and that it does not derive topological order, quasiparticle braiding statistics, emergent gauge fields, or anyonic excitations. Its relevance is therefore indirect: it provides a local topological lattice framework from which anyonic phases might emerge after interactions, rather than realizing anyonic exchange algebra at the microscopic level (Crépel et al., 2024).

Third, the evidentiary status of different claims is not uniform. In the parafermion chain, the operator algebra, Fradkin–Kadanoff map, one-particle dispersion, and existence of canonical clustered fermions for even $p$73 are exact. The two-body scattering ansatz and twisted quantization are analytical, while the existence and structure of bound states are supported by exact diagonalization and wavefunction analysis. Nonintegrability is strongly supported by level-spacing statistics rather than rigorously proven, the $p$74 low-energy description is numerically established through gap scaling and entanglement, and the comparison to Laughlin-edge physics is heuristic or phenomenological rather than a microscopic derivation. In the extended anyon-Hubbard chain, the bosonized field theory and symmetry analysis organize the phase structure, while the phase boundaries and universality classes are determined numerically.

Within condensed-matter theory, the significance of the anyonic tight-binding model is twofold. It isolates the effect of anyonic statistics without requiring explicit quartic interactions, showing that exchange and occupancy constraints alone can generate bound states, nonintegrability, asymmetric momentum distributions, and unconventional correlation exponents. It also furnishes a lattice laboratory for fractionalized fermions and for parafermion platforms related to fractional quantum Hall edges proximitized by superconductors and magnets. The general lesson is that in anyonic lattice systems, “quadratic” does not imply free behavior, and “tight binding” does not reduce the problem to ordinary single-particle band theory.