Adiabatic Continuity Conjecture

• Adiabatic continuity is a principle stating that different regimes of a theory remain in the same phase if they connect continuously without phase transitions, gap closing, or symmetry changes.
• It is applied in diverse contexts such as compactified gauge theories, topological matter, and variational many-body theory, with semiclassical mechanisms and lattice simulations supporting its claims.
• Evidence from gauge theory, topological band theory, and computational models shows smooth transitions under controlled conditions, while counterexamples highlight limitations like symmetry breaking and lattice artifacts.

Searching arXiv for papers on adiabatic continuity and related uses of the term. The adiabatic continuity conjecture is a family of closely related claims asserting that two regimes of a theory are in the same phase when they can be connected by a continuous deformation of parameters without a phase transition, a bulk gap closing, or a change in the relevant symmetry or topological data. A prominent use concerns asymptotically free quantum field theories on $\mathbb{R}^3\times S^1$ or $\mathbb{R}\times S^1$, where one asks whether the weakly coupled small-circle regime with center-symmetric or twisted boundary conditions is continuously connected to the strongly coupled large-circle regime (Fujimori et al., 2020). Other uses appear in topological band theory, fractional quantum Hall physics, variational many-body theory, adiabatic quantum computation, and functional-analytic treatments of Floer-type moduli problems. This suggests that the phrase denotes not a single universal theorem but a common structural principle instantiated in several technically distinct settings.

1. Terminological range and common structure

Context	Interpolating parameter	Typical invariant or diagnostic
Compactified gauge theory	Circle size $L$ or $L_\tau$	Polyakov loop, center symmetry, chiral condensate
Topological matter	Hamiltonian interpolation	Gap, Chern number, overlap field, entanglement spectrum
Variational many-body theory	Correlation parameter on a reference state	Analyticity, Drude weight, symmetry realization
Adiabatic computation and Fredholm theory	Hamiltonian path or adiabatic parameter $\varepsilon$	Complexity, finite-dimensional reduction, continuity of charts

Across these settings, the basic question is whether a controlled regime and a difficult regime belong to the same phase. In the gauge-theory literature this usually means that small-$S^1$ semiclassics can be extrapolated to large $S^1$ because center symmetry remains unbroken and no thermodynamic singularity intervenes (Sheu et al., 2022). In topological phases it means that two gapped states lie in the same phase if a symmetry-preserving interpolation exists without gap closing, and one operational sufficient condition is nonvanishing wavefunction overlap throughout the Brillouin zone (Gu et al., 2016). In variational many-electron theory it means that a family of correlated trial states remains in the same qualitative phase as its reference state unless the reference state itself is changed (Baeriswyl, 2011).

A persistent distinction in the literature is between adiabatic continuity and the adiabatic theorem. The former is usually a statement about phase structure or equivalence classes of Hamiltonians; the latter concerns slow real-time evolution. This distinction becomes explicit in work that tests whether an overlap-based remnant of the adiabatic theorem survives even for sudden quenches (Damerow et al., 9 May 2025).

2. Compactified asymptotically free field theories

In its contemporary field-theoretic form, the conjecture is tied to compactification on a circle with boundary conditions chosen to stabilize a center-symmetric or center-like vacuum. For the two-dimensional ${\mathbb C}P^{N-1}$ model on $S_s^1\times S_\tau^1$, the relevant twisted boundary condition is

$$\phi(x,\tau+L_\tau)=\Omega\,\phi(x,\tau),\qquad \Omega=\mathrm{diag}\!\left[1,e^{2\pi i/N},\dots,e^{2(N-1)\pi i/N}\right],$$

which produces an exact intertwined $\mathbb{R}\times S^1$0 symmetry and $\mathbb{R}\times S^1$1 classical vacua (Fujimori et al., 2020). In this setting, adiabatic continuity means that the $\mathbb{R}\times S^1$2-symmetric vacuum at small $\mathbb{R}\times S^1$3, where fractional instantons and bions dominate semiclassically, is continuously connected to the vacuum at large $\mathbb{R}\times S^1$4 without a phase transition.

For four-dimensional gauge theories the same logic is implemented on $\mathbb{R}\times S^1$5. In the $\mathbb{R}\times S^1$6 chiral Yang–Mills model, center-symmetric vacua are stabilized at small $\mathbb{R}\times S^1$7 by a double-trace deformation; weak coupling then permits a semiclassical description in terms of monopole-instantons and bions, and adiabatic continuity is invoked to infer that the weak-coupling condensate $\mathbb{R}\times S^1$8 persists on $\mathbb{R}\times S^1$9 (Sheu et al., 2022). In $L$0 supersymmetric Yang–Mills with periodic adjoint fermions, the small-circle regime is expected to remain confining and chirally broken, so the conjecture becomes the absence of a deconfinement or chiral-restoration transition between semiclassical and strongly coupled confinement (Bergner et al., 2018).

The common semiclassical mechanism involves topological molecules. In $L$1, fractional instantons of charge $L$2 interpolate between adjacent vacua, while neutral bions stabilize the $L$3 symmetry and contribute to the mass gap (Fujimori et al., 2020). In adjoint gauge theories on $L$4, monopole-instantons and magnetic or neutral bions generate the holonomy potential and confinement at small circle size (Bergner et al., 2018). In this sense, the conjecture is not merely about smooth parameter dependence; it is also a claim that semiclassical objects at small $L$5 encode the same vacuum structure that appears nonperturbatively at large volume.

3. Lattice and reduced-model evidence

The most direct nonperturbative tests come from lattice compactifications and large-$L$6 reduced models. In the lattice $L$7 model with $L$8-twisted boundary condition, the Polyakov loop

$L$9

serves as the order parameter for the intertwined $L_\tau$0 symmetry, and the pseudo-entropy density

$L_\tau$1

probes possible nonanalytic behavior of the vacuum as $L_\tau$2 varies (Fujimori et al., 2020). Simulations for $L_\tau$3 found $L_\tau$4 across the accessible range of $L_\tau$5, with the Polyakov-loop distribution evolving from an isotropic cloud near the origin to a regular $L_\tau$6-polygon on the complex plane. The spatial profile of the local Polyakov loop $L_\tau$7 exhibits kink, anti-kink, and kink–antikink structures interpreted as fractional instantons and bions, while the pseudo-entropy varies smoothly and shows no sign of a phase transition (Fujimori et al., 2020).

For $L_\tau$8 supersymmetric Yang–Mills on the lattice, periodic and antiperiodic fermion boundary conditions lead to sharply different behavior. With antiperiodic boundary conditions, the Polyakov loop, susceptibility, and Binder cumulant display an ordinary thermal deconfinement transition. With periodic boundary conditions, the Polyakov loop remains small, the susceptibility peak does not show Ising critical behavior, and the per-site Polyakov-loop eigenvalue distribution interpolates from Haar-like non-Abelian confinement at large $L_\tau$9 to localization near the center-symmetric holonomy at small $\varepsilon$0, consistent with Abelian confinement (Bergner et al., 2018). The same study also emphasizes that Wilson-fermion lattice artifacts become important at very small $\varepsilon$1, where the lattice one-loop potential behaves qualitatively like continuum QCD(adj) with $\varepsilon$2, so continuity claims must be separated from cutoff effects (Bergner et al., 2018).

Large-$\varepsilon$3 twisted Eguchi–Kawai reductions provide a complementary test. A partially reduced model on $\varepsilon$4 with one adjoint Dirac fermion found that, for periodic boundary conditions, the Polyakov loop around $\varepsilon$5 remains near zero in the light-fermion regime as the circle shrinks, while antiperiodic boundary conditions show a clear deconfinement transition (Hamada et al., 20 Apr 2026). With two adjoint fermions, simulations of both the fully reduced and partially reduced TEK models likewise found that heavy adjoint fermions stabilize the $\varepsilon$6 center-symmetric vacuum in the reduced theory and that, in the partially reduced geometry mimicking $\varepsilon$7, periodic adjoint fermions support a center-symmetric confined phase as the circle is made small (Hamada et al., 27 Jul 2025). These results are explicitly framed as support for the adiabatic continuity conjecture at large $\varepsilon$8.

4. Topological matter and quantum Hall formulations

In topological band theory, adiabatic continuity is often formulated in terms of a momentum-space overlap criterion. For two band insulators with occupied-band overlap matrix $\varepsilon$9, the scalar field

$S^1$0

is defined over the Brillouin zone. If $S^1$1 for all $S^1$2, an explicit gapped interpolation can be constructed, and the path preserves all symmetries shared by the endpoints; consequently the two insulators are in the same topological phase (Gu et al., 2016). Zeros of $S^1$3 signal topological obstruction and are associated with gap closing at a phase transition.

A many-body realization appears in the relation between fractional Chern insulators and fractional quantum Hall states. Using a gauge-fixed Wannier representation of a $S^1$4 kagome Chern band, one can interpolate between projected lattice interactions and continuum Landau-level model Hamiltonians by

$S^1$5

For fermionic Laughlin $S^1$6, fermionic Moore–Read $S^1$7, and their bosonic analogues, the many-body gap remains open and the low-lying orbital entanglement spectrum remains qualitatively unchanged along the interpolation, while squared overlaps between FCI and FQH ground states reach $S^1$8 for the 8-electron Laughlin state and $S^1$9 for the 10-electron Moore–Read state (Liu et al., 2012). By contrast, for fermions at $S^1$0 the interpolation undergoes a phase transition, showing that adiabatic continuity between FCI and FQH states is not universal (Liu et al., 2012).

An allied construction connects Hofstadter bands to Chern-insulator bands. Explicit interpolations show that the lowest $S^1$1 band of checkerboard or kagome Chern insulators is adiabatically connected to a Landau-like Hofstadter band, and that $S^1$2 Laughlin and $S^1$3 Moore–Read states remain gapped along this path, whereas $S^1$4 is far more fragile (Wu et al., 2012). In that setting, nonuniformity of the Berry curvature weakens FQH/FCI gaps and can destroy delicate composite-fermion fractions (Wu et al., 2012).

Spinful quantum Hall states furnish a related flux-transmutation picture. In an extended Hubbard model of spinful anyons on a torus, adiabatic deformation by transmutation of statistical fluxes shows that the $S^1$5 family remains spin-polarized, while singlet QH states at $S^1$6 and $S^1$7 are adiabatically continuous. The many-body Chern number of the ground-state multiplet acts as an adiabatic invariant and explains the dramatic variation of topological degeneracy along the path (Kudo et al., 2022).

5. Variational, computational, and functional-analytic extensions

In many-electron variational theory, adiabatic continuity refers to the persistence of a phase within a variational family built on a fixed reference state. A canonical example is the Gutzwiller ansatz

$S^1$8

where $S^1$9 is typically the filled Fermi sea (Baeriswyl, 2011). Within this framework, a single adiabatically linked variational state cannot generate long-range order absent from ${\mathbb C}P^{N-1}$0, cannot destroy long-range order already present, and cannot convert a metallic reference into an insulator for finite correlation parameter. Quantum phase transitions are therefore described not by one adiabatically continuous family but by competing families built from distinct reference states, such as weak-coupling Gutzwiller and strong-coupling Baeriswyl-type ansätze (Baeriswyl, 2011).

A computational analogue appears in adiabatic quantum computation. For Simon’s problem, an explicitly constructed adiabatic algorithm has the same complexity as the circuit-based algorithm and is exponentially faster than the classical counterpart, supporting the conjecture that suitably constructed adiabatic algorithms can match the complexity of their circuit analogues in a stronger sense than generic polynomial equivalence (Hen, 2013). The same paper presents this as evidence for an adiabatic version of period finding, with implications for the possibility of an adiabatic analogue of Shor’s factoring algorithm (Hen, 2013).

A different extension treats sudden quenches rather than slow evolution. In the transverse-field Ising model and the ANNNI model, the conjectured extension is

${\mathbb C}P^{N-1}$1

for quenches within the same phase, where ${\mathbb C}P^{N-1}$2 is the initial ground state and ${\mathbb C}P^{N-1}$3 are eigenstates of the post-quench Hamiltonian (Damerow et al., 9 May 2025). This was confirmed analytically and numerically in the TFIM and analytically in a special ANNNI case, with numerical support beyond that special case (Damerow et al., 9 May 2025). The proposal reframes adiabatic continuity as an overlap dominance property even for maximally non-adiabatic changes.

At a more abstract level, adiabatic Fredholm theory develops a functional-analytic framework in which an adiabatic parameter ${\mathbb C}P^{N-1}$4 enters a family of Fredholm problems ${\mathbb C}P^{N-1}$5. The theory constructs adiabatic Fredholm families, proves uniform finite-dimensional reductions, and shows that these inherit continuity and differentiability from the family; the paper explicitly indicates applications to strip-shrinking in quilted Floer theory and to adiabatic limits relevant to the nondegenerate Atiyah–Floer conjecture (Bottman et al., 2024). In this setting, adiabatic continuity is recast as continuity of local moduli-space charts and obstruction models across a degeneration parameter.

6. Limitations, counterexamples, and interpretive cautions

The literature also delineates sharp limits to the conjecture. In gauge theory, continuity requires the relevant center symmetry to remain unbroken; antiperiodic boundary conditions generically produce thermal deconfinement rather than adiabatic continuity, and even with periodic adjoint fermions lattice artifacts can distort the very small-radius regime (Bergner et al., 2018). In the lattice ${\mathbb C}P^{N-1}$6 study, the authors explicitly note that very large ${\mathbb C}P^{N-1}$7 is plagued by long autocorrelation times, so apparent ${\mathbb C}P^{N-1}$8 breaking can arise from vacuum freezing and insufficient statistics rather than a genuine phase transition (Fujimori et al., 2020).

In topological matter, adiabatic continuity is not automatic even when the endpoint phases are superficially related. The overlap-field criterion is powerful for noninteracting band insulators, but its many-body generalization is obstructed in generic thermodynamic systems by the orthogonality catastrophe (Gu et al., 2016). FCI/FQH interpolations give both positive and negative results: robust Laughlin and Moore–Read phases show smooth continuity, whereas ${\mathbb C}P^{N-1}$9 provides a clear counterexample with a phase transition (Liu et al., 2012). In variational many-electron theory, a single Gutzwiller-type family enforces adiabatic continuity by construction and therefore misses transitions that in the exact system occur at infinitesimal coupling or involve symmetry breaking not built into the reference state (Baeriswyl, 2011).

These limitations clarify the status of the conjecture. It is most credible when supported simultaneously by a stable gap, unbroken controlling symmetry, preserved topological or order-parameter data, and, where possible, an explicit semiclassical or overlap-based mechanism. Where any of these fail, the phrase “adiabatic continuity” ceases to be a theorem-like property and becomes a hypothesis to be checked model by model.