Anyon Gap-Closing Instabilities

• Anyon gap-closing instabilities are phase-transition mechanisms triggered when anyonic excitations or their bound states soften and condense, altering topological order.
• The phenomenon covers diverse cases, including bulk gap closures, vison-pair softening in perturbed Kitaev models, and many-body spectral transitions in non-Hermitian systems.
• These instabilities drive key changes such as confinement, symmetry breaking, and transitions between distinct topological phases without necessarily closing the single-particle gap.

Anyon gap-closing instabilities are phase-transition mechanisms in which the decisive soft mode is an anyonic excitation, an anyon-derived bound state, or a collective sector that controls topological structure more directly than an ordinary single-particle band gap. In this usage, “gap closing” encompasses several distinct phenomena: closure of the bulk energy gap of a condensing bosonic anyon, closure of a local bosonic vison-pair gap in perturbed Kitaev spin liquids, closure of a many-body gap while a single-particle gap remains finite, and, in non-Hermitian many-body systems, real–complex spectral transitions accompanied by structure in $\Im E$. Across these settings, the common problem is to identify which spectral gap becomes critical, which excitations are deconfined or confined, and how edge or surface structure responds (Long et al., 2023, Chen et al., 28 Aug 2025, Mai et al., 2024, Qin et al., 18 Mar 2026).

1. Taxonomy of gap-closing instabilities

The contemporary literature distinguishes several inequivalent notions of gap closing. In anyon condensation, the gap of a bosonic anyon vanishes and the condensed excitation becomes locally indistinguishable from vacuum; this drives confinement, identification, and sometimes splitting of other anyons. In perturbed Kitaev models, the relevant instability may instead be the closure of a single-vison gap, a fermion gap, or a bosonic vison-pair gap, with different consequences for confinement and symmetry breaking. In interacting topological insulators, the decisive transition may be tied to a many-body gap rather than a single-particle one. In non-Hermitian anyonic systems, criticality can appear through the onset of complex eigenvalues and the emergence of gaps in $\Im E$ rather than through a conventional Hermitian excitation gap (Long et al., 2023, Chen et al., 28 Aug 2025, Mai et al., 2024, Qin et al., 18 Mar 2026).

Setting	Gap whose instability is diagnostic	Consequence
Anyon condensation	Bulk energy gap of a bosonic anyon	Condensation, confinement, identification, edge symmetry breaking
Kitaev–$\Gamma$–$\Gamma'$ model	Single vison, fermion, or bosonic vison-pair gap	Conventional magnet, ordered QSL, or other transition
Interacting topological insulators	Many-body gap or spin spectrum gap	Topological change without single-particle or energy-gap closing
Non-Hermitian anyonic ladders	Real–complex transition and gaps in $\Im E$	Anyon-induced criticality and dynamical isolation of dominant states

A central misconception is that every topological transition must be accompanied by closure of a single-particle bulk energy gap. The interacting and projected-spectrum literature shows otherwise: the critical gap may be many-body, spin-resolved, or non-Hermitian in character (Mai et al., 2024, Yang et al., 2013).

2. Anyon condensation as a canonical bulk gap-closing mechanism

In anyon condensation, a bosonic anyon is tuned to proliferate and condense. The defining instability is that the gap to creating this anyon vanishes at the transition, enabling a phase change to a topologically distinct phase. Once condensed, the anyon becomes locally indistinguishable from vacuum. Anyons that braid nontrivially with the condensed anyon become confined, anyons related by fusion with it become identified, and certain anyons may split into multiple species under the new fusion rules (Long et al., 2023).

A technical obstacle is that the standard algebraic machinery, such as unitary braided fusion categories, assumes a strictly gapped bulk and therefore does not directly describe the gap-closing dynamics itself. The two-edge thought experiment introduced for condensation transitions circumvents this limitation by tuning only part of a cylindrical sample through the transition while retaining a buffer strip of the uncondensed phase between the new bulk and the physical edge. Because the strip remains gapped, edge degrees of freedom do not hybridize with bulk critical modes and can be tracked through the transition (Long et al., 2023).

This construction makes several structural results explicit. Bulk condensation induces explicit symmetry breaking of a nonlocal edge symmetry. If $c$ is the condensed anyon, then the corresponding string operator $L_c^Y(x)$ becomes local at the edge and appears as an explicit perturbation in the edge Hamiltonian. If $L_c^Y(x)$ fails to commute with the loop symmetry $L_a^X$, then condensation explicitly breaks the $L_a^X$ symmetry. The criterion for bulk confinement is the mutual braiding condition

$\Im E$0

which identifies anyons $\Im E$1 that braid nontrivially with the condensed $\Im E$2 as confined (Long et al., 2023).

The same framework shows that the number of chiral current-carrying modes at the edge cannot change through anyon condensation. Equivalently, the chiral central charge is preserved, so the low-temperature heat current

$\Im E$3

is continuous through the transition. This is a precise statement: anyon condensation changes topological data associated with bulk quasiparticles and edge operator content, but not $\Im E$4 (Long et al., 2023).

The toric code and Kitaev spin liquids furnish concrete realizations. In the toric code, condensing $\Im E$5 makes the $\Im E$6 string operator local at the edge and induces a longitudinal field in the transverse-field Ising edge Hamiltonian,

$\Im E$7

thereby explicitly breaking the global $\Im E$8 symmetry associated with $\Im E$9-anyon parity. In double-layer $\Gamma$0 Kitaev spin liquids, the boson $\Gamma$1 can condense; single-layer vortices are then confined, the edge chiral algebra is extended, and the post-condensation edge theory is a chiral boson with $\Gamma$2, matching the sum of the two layers before condensation (Long et al., 2023).

3. Microscopic anyon gap closings in the Kitaev–$\Gamma$3–$\Gamma$4 model

The Kitaev honeycomb spin liquid with $\Gamma$5 and $\Gamma$6 perturbations provides a direct microscopic setting in which phase boundaries are organized by anyon gap closings. The relevant quasiparticles are single visons, itinerant Majorana fermions, and local magnon-like bosonic vison pairs on nearest-neighbor or next-nearest-neighbor hexagons. The working procedure is explicit: compute the quasiparticle dispersions, track their minima as functions of $\Gamma$7 and $\Gamma$8, identify the critical values where the gap closes, and infer the ensuing phase from the momentum and quantum numbers of the soft mode (Chen et al., 28 Aug 2025).

The physical interpretation depends on which excitation softens first. Gap closing of a bosonic anyon, often a bosonic vison pair, signals the possibility of condensation and hence new order. Gap closing of a single vison typically signals confinement of matter fermions and favors a trivial magnetically ordered state. Gap closing of fermions indicates a distinct instability, potentially involving a change in band topology. The momentum of the soft boson determines the magnetic order: $\Gamma$9-point softening gives Néel order, $\Gamma'$0-point softening gives $\Gamma'$1 order, $\Gamma'$2-point softening gives stripe or zigzag order, and an incommensurate $\Gamma'$3 gives an incommensurate spiral (Chen et al., 28 Aug 2025).

The ferromagnetic and antiferromagnetic Kitaev limits behave very differently. For the FM Kitaev model $\Gamma'$4, $\Gamma'$5 induces strong single-vison hopping, and the vison gap closes at $\Gamma'$6 at an IC momentum and at $\Gamma'$7 at the $\Gamma'$8 and $\Gamma'$9 points. In the same regime, the fermion gap closes at $\Im E$0 and $\Im E$1, while the lowest boson gap closes at $\Im E$2 and $\Im E$3. Reported phase-boundary values include $\Im E$4, $\Im E$5, and $\Im E$6 (Chen et al., 28 Aug 2025).

For the AFM Kitaev model $\Im E$7, the hierarchy is different. Because $\Im E$8, single visons remain gapped even at larger $\Im E$9 until higher-order $c$0 processes induce a vison chemical potential. The fermion gap closes only at much larger $c$1, around $c$2 and $c$3. By contrast, the boson gap closes first at $c$4 at the $c$5 and $c$6 points, and at $c$7 at an incommensurate $c$8 between $c$9 and $L_c^Y(x)$0. Additional critical values are $L_c^Y(x)$1 and $L_c^Y(x)$2 (Chen et al., 28 Aug 2025).

The most distinctive result is the regime in which a local bosonic vison pair condenses while single visons and fermions remain gapped. In the AFM Kitaev model with a ferromagnetic $L_c^Y(x)$3 interaction, the single-vison and fermion gaps remain open while the gap of a magnon-like local boson vanishes. The resulting state therefore has coexistence of a spontaneous broken symmetry and the fractionalization pattern of the Kitaev spin liquid; depending on the sign of $L_c^Y(x)$4, the magnetic long-range order can be either a stripy antiferromagnet or an incommensurate spiral. This is a direct realization of symmetry breaking without immediate loss of deconfined fractional excitations (Chen et al., 28 Aug 2025).

4. Beyond band-gap closure: many-body and projected-spectrum instabilities

In non-interacting systems, the standard picture is that a topological phase transition without symmetry breaking requires closure of the single-particle gap. Strong correlations invalidate that identification. In the half-filled and quarter-filled Kane–Mele–Hubbard model, extensive determinantal and dynamical cluster quantum Monte Carlo simulations show that a transition between topological and trivial insulators can occur without closing the single-particle gap. The exactly solvable Kane–Mele model with extended orbital Hatsugai–Kohmoto interactions clarifies the mechanism: the many-body gap closes at the transition, while the single-particle gap can remain finite throughout. The stated conclusion is that, in interacting systems, the proper probe of topological phase transitions is the closing of the many-body rather than the single-particle gap (Mai et al., 2024).

This distinction matters directly for anyon physics. Anyon condensation and topological-order changes are governed by collective sectors, so a many-body criterion is structurally more appropriate than a purely band-theoretic one. A plausible implication is that “anyon gap-closing instability” should be understood as a collective criterion by default, even when an effective quasiparticle language exists (Mai et al., 2024).

A related generalization arises in the three-dimensional topological-insulator model with exchange field $L_c^Y(x)$5. There the Brillouin zone is sliced at fixed $L_c^Y(x)$6, and spin Chern numbers $L_c^Y(x)$7 are assigned to the resulting two-dimensional subsystems: $L_c^Y(x)$8 with

$L_c^Y(x)$9

For $L_c^Y(x)$0, the spin Chern numbers change from $L_c^Y(x)$1 to $L_c^Y(x)$2 at

$L_c^Y(x)$3

and the transition is accompanied not by energy-gap closing but by closure of the spin spectrum gap $L_c^Y(x)$4 at $L_c^Y(x)$5 and $L_c^Y(x)$6. Surface states exist only for $L_c^Y(x)$7 and vanish for $L_c^Y(x)$8. By contrast, for $L_c^Y(x)$9, the energy gap closes and the system becomes a Weyl semimetal with Weyl points and chiral Fermi-arc surface states (Yang et al., 2013).

Taken together, these results delimit a broader instability framework. The critical gap may be an anyon gap, a many-body charge gap, or a projected spin-sector gap; the phenomenology of surface or edge disappearance does not by itself identify which one is closing (Mai et al., 2024, Yang et al., 2013).

5. Anyon-induced criticality in non-Hermitian many-body systems

Non-Hermitian many-body systems introduce a different notion of instability, in which anyonic exchange statistics reshape the spectral transition itself. The model considered is a non-Hermitian interacting ladder chain of anyons with asymmetric hopping, interchain coupling $L_a^X$0, onsite Hubbard interaction $L_a^X$1, and staggered chemical potential $L_a^X$2. After a generalized Jordan–Wigner transformation, the anyonic operators map to bosonic ones with an occupation-dependent Peierls phase,

$L_a^X$3

so anyonic statistics appear as a nonlocal, occupation-dependent gauge field (Qin et al., 18 Mar 2026).

For bosons $L_a^X$4 and pseudofermions $L_a^X$5, the effective Hamiltonian respects pseudo-Hermiticity under $L_a^X$6. For anyons with $L_a^X$7, the interchain term explicitly breaks that symmetry: $L_a^X$8 This breaking is the mechanism behind the anyonic statistics-induced spectral transition. In the thermodynamic limit, complex eigenenergies appear for exponentially small $L_a^X$9, with a critical coupling that scales as

$L_a^X$0

Equivalently, $L_a^X$1 becomes finite for generic $L_a^X$2 even as $L_a^X$3, whereas it remains zero for $L_a^X$4 (Qin et al., 18 Mar 2026).

The resulting spectrum is not merely complex; it is dense along $L_a^X$5 and exhibits a finite gap separating the maximal-$L_a^X$6 state or states from the rest of the spectrum. The paper also identifies non-analyticities associated with swapping of the dominant maximal-$L_a^X$7 state, visible in the first derivative of $L_a^X$8 and in boundary correlations. By contrast, bosonic and pseudofermionic counterparts acquire complex eigenvalues only after more conventional many-body critical-skin-effect mechanisms at finite couplings (Qin et al., 18 Mar 2026).

The dynamical consequence is an anomalously stable short-time quench regime for anyons. Because the dominant eigenstates are isolated by an enhanced gap in $L_a^X$9, post-quench evolution is governed almost entirely by the initial pre-quench state for a comparatively long period. This is an instability driven purely by exchange statistics, not by local single-particle interference. In that sense, it is the non-Hermitian analogue of an anyon-specific gap-closing or gap-opening mechanism (Qin et al., 18 Mar 2026).

6. Diagnostics, misconceptions, and broader instability language

The broader instability literature emphasizes that a “gap-closing feature” is not exhausted by the statement that some spectral line reaches zero. In Majorana nanowires, the curvature of the trivial-side gap-closing feature as a function of Zeeman field constrains the microscopic spin-orbit coupling and chemical potential. Specifically, the gap-closing feature is entirely concave only for strong spin-orbit coupling relative to the chemical potential, with a boundary

$\Im E$00

The same work shows that nonlinearity of the gap-closing curve complicates the assignment of a constant effective $\Im E$01-factor; fitting the full curve can produce values differing by as much as a factor of three from naïve slope-based estimates (Pan et al., 2018).

Although this Majorana literature is not an anyon-condensation literature, it clarifies an important methodological point: the geometry of a gap-closing feature carries microscopic information, and careless identification of the relevant gap can lead to incorrect inferences. The same warning applies to anyon instabilities, where the soft mode may be a local boson, a single vison, or a many-body collective excitation rather than the fermionic quasiparticle most visible in spectroscopy (Pan et al., 2018, Chen et al., 28 Aug 2025).

A further caution comes from interaction-driven instabilities of Weyl-loop semimetals. Short-range interactions can drive a fully gapped chiral superconducting phase that spontaneously breaks time-reversal symmetry, a fully gapped insulating phase that breaks mirror symmetry, or a gapless Pomeranchuk phase that breaks rotational symmetry. The existence of both gap-opening and gapless symmetry-breaking channels shows that instability classification must specify not only that a phase transition occurs, but also whether the result is confinement, full gapping, or deformation of an existing nodal structure (Sur et al., 2016).

The cumulative lesson is that anyon gap-closing instabilities are not a single mechanism but a family of critical processes. In some cases, the bulk energy gap of a bosonic anyon closes and the condensed excitation restructures the topological order. In others, a local bosonic vison pair softens first, producing magnetic order while visons and fermions remain gapped. In still others, the decisive criticality is many-body, spin-sector, or non-Hermitian. The unifying criterion is therefore not “band gap closes,” but rather “the spectral sector that encodes the topological or fractionalized structure becomes critical” (Long et al., 2023, Chen et al., 28 Aug 2025, Mai et al., 2024, Qin et al., 18 Mar 2026).