Algebraic States in Continuum (AICs)

• AICs are defined through operator-algebraic frameworks on infinite-dimensional Hilbert spaces, with quasi-free states characterized by covariance operators and distinct GNS representations.
• In non-Hermitian contexts, AICs manifest as impurity-induced eigenstates displaying algebraic 1/r decay, while bilayer systems use dispersion polynomial factorization to yield defect bound states.
• AICs also emerge in continuum particle systems via current algebra representations, unifying stochastic processes and quantum observables under common algebraic principles.

Algebraic States in Continuum (AICs) denotes several distinct but structurally related notions in contemporary mathematical physics. In operator algebra and algebraic quantum theory, it refers to states on infinite-dimensional CAR or CCR $C^*$-algebras, especially quasi-free states whose GNS representations obey a Kakutani-type dichotomy (Matsui et al., 2012). In recent non-Hermitian band theory, the same acronym denotes impurity-induced eigenstates whose energies lie inside a bulk continuum while their real-space envelopes decay algebraically, typically as $1/r$ in two dimensions (Yang et al., 9 Jul 2025). Closely related work on bilayer materials identifies defect bound states in the continuum whose existence is controlled not by symmetry protection but by algebraic reducibility of the dispersion polynomial (Massatt et al., 2023). A further algebraic-probabilistic usage arises for continuum particle systems, where $\mathfrak{su}(1,1)$ current algebras, Pascal point processes, and self-intertwining unitaries furnish a concrete continuum representation of algebraic states and their symmetries (Floreani et al., 2023). The shared motif is that algebraic structure governs continuum behavior, but the underlying objects—operator-algebraic states, embedded eigenstates, and stochastic states—are not interchangeable.

1. Operator-algebraic AICs on CAR and CCR algebras

In the algebraic approach to continuum quantum systems, AICs are modeled as states on CAR or CCR $C^*$-algebras built over infinite-dimensional one-particle spaces. For a real Hilbert space $V$, the CAR algebra $\mathcal{C}(V)$ is generated by $V$ with relations

$x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$

For a real vector space $V$ with alternating bilinear form $\sigma$, the CCR algebra $1/r$0 is generated in Weyl form by $1/r$1 satisfying

$1/r$2

In both cases, the continuum aspect is the infinite-dimensionality of $1/r$3, typically separable, as in one-particle spaces such as $1/r$4 (Matsui et al., 2012).

The privileged class is that of quasi-free states, the algebraic analogue of Gaussian measures. In the CAR case, a quasi-free state $1/r$5 is determined by a bounded self-adjoint covariance operator $1/r$6 on $1/r$7 with

$1/r$8

and its higher even moments are fixed by the fermionic Wick rule while the odd part vanishes. In the CCR case, a quasi-free state $1/r$9 is determined by a positive sesquilinear covariance form $\mathfrak{su}(1,1)$0 via

$\mathfrak{su}(1,1)$1

subject to

$\mathfrak{su}(1,1)$2

These states model free Fermi and Bose fields, ground states, and thermal equilibrium states of quadratic Hamiltonians.

The central representational objects are the GNS triples. A state $\mathfrak{su}(1,1)$3 yields $\mathfrak{su}(1,1)$4 with

$\mathfrak{su}(1,1)$5

In the standard $\mathfrak{su}(1,1)$6 realization used in the paper, the cyclic subspace

$\mathfrak{su}(1,1)$7

encodes the GNS representation class. Two states are quasi-equivalent when these cyclic subspaces coincide, and disjoint when they are orthogonal. For continuum algebraic quantum theory, this distinction separates states lying in the same representation class from states belonging to different superselection sectors.

2. Kakutani-type dichotomy for quasi-free continuum states

The main structural statement for quasi-free operator-algebraic AICs is a sharp dichotomy: two such states generate GNS representations that are either quasi-equivalent or disjoint, with no intermediate possibility. In the CAR case, quasi-equivalence of quasi-free states $\mathfrak{su}(1,1)$8 and $\mathfrak{su}(1,1)$9 is characterized by the Hilbert–Schmidt condition

$C^*$0

and otherwise the representations are disjoint. In the CCR case, the analogous criterion is formulated in terms of the symmetric covariances and an associated Hilbert–Schmidt equivalence condition; the paper also proves that quasi-equivalence is equivalent to positivity of the transition amplitude, whereas failure of that condition forces disjointness (Matsui et al., 2012).

A useful quantity is the algebraic transition amplitude

$C^*$1

which plays the role of a continuum Hellinger integral. For standard states, vanishing of this quantity is equivalent to disjointness. The paper derives determinant formulas for quasi-free states: in the CAR setting the amplitude is expressed through

$C^*$2

with

$C^*$3

while in the CCR setting an analogous determinant formula involves the geometric mean of positive forms. These formulas are the operator-algebraic analogues of infinite products of Hellinger overlaps.

The analogy with Kakutani’s theorem for infinite product measures is exact at the level of representation theory. Infinite products of scalar overlaps are replaced by operator determinants, equivalence of measures by quasi-equivalence of GNS representations, and mutual singularity by disjointness. A common misconception is that transition amplitude alone always decides the issue. The paper explicitly notes that this fails in the CAR case: there can be quasi-equivalent CAR quasi-free states with zero amplitude. The binary dichotomy remains valid, but the CCR situation is stronger because positivity of the amplitude is then equivalent to quasi-equivalence.

3. Non-Hermitian impurity AICs in $C^*$4

A different usage defines AICs as impurity-induced eigenstates embedded in the continuum spectrum of non-Hermitian systems in dimensions $C^*$5. The setting is a translationally invariant non-Hermitian Hamiltonian $C^*$6 perturbed by a single on-site impurity,

$C^*$7

An AIC is an eigenstate $C^*$8 of $C^*$9 whose eigenvalue lies inside the bulk continuum spectrum of $V$0 under periodic boundary conditions, while its real-space profile is localized around the impurity in an algebraic rather than exponential manner. In two dimensions the asymptotic envelope is

$V$1

and more generally the leading tail is $V$2 with direction-dependent prefactor (Yang et al., 9 Jul 2025).

The mechanism is formulated through the unperturbed Green’s function

$V$3

Any impurity eigenstate satisfies

$V$4

so the asymptotics of the eigenstate are those of the Green’s function. The eigenvalue condition is the impurity equation

$V$5

When the complex band $V$6 fills a finite area in the complex-energy plane, the equation $V$7 has a discrete set of momentum solutions $V$8. Expanding around such a point yields a first-order “Dirac-like” PDE with complex gradient ratio

$V$9

and for $\mathcal{C}(V)$0 the Fourier integral gives

$\mathcal{C}(V)$1

The threshold for producing these states is encoded in the image of $\mathcal{C}(V)$2 as $\mathcal{C}(V)$3 ranges over the continuum. If some continuum energy $\mathcal{C}(V)$4 satisfies $\mathcal{C}(V)$5, equivalently $\mathcal{C}(V)$6, arbitrarily small impurities can generate an AIC. This divergence is tied to a Bloch saddle point $\mathcal{C}(V)$7 with $\mathcal{C}(V)$8. Without such points, a finite threshold

$\mathcal{C}(V)$9

is required. The paper further states that these AICs are forbidden in Hermitian systems and in one-dimensional non-Hermitian systems. In the Hermitian case, constant-energy manifolds are continuous contours rather than discrete points, the relevant Fourier integral diverges for $V$0, and no algebraically localized impurity eigenstate of this type exists. In one dimension, the two-derivative structure responsible for the $V$1 tail is absent.

A further subtlety is normalization. In an infinite two-dimensional system, a $V$2 profile is not square-integrable because

$V$3

Accordingly, the paper treats the state either as a finite-size normalizable eigenstate on $V$4 lattices with periodic boundary conditions or as a generalized eigenstate in the thermodynamic limit. “Localized” therefore refers to algebraic concentration near the impurity, not to strict $V$5 normalizability in infinite volume.

4. Algebraic BICs in bilayer materials

Related work on bilayer electronic materials studies defect bound states in the continuum that are explicitly described as arising from the algebraic structure of the tight-binding Hamiltonian rather than symmetry protection. In AA-stacked bilayer graphene, the Hamiltonian can be block-diagonalized by symmetry, so BICs are symmetry-protected. In AB-stacked bilayer graphene, by contrast, the full $V$6 Hamiltonian is not block-diagonalized by lattice symmetry, but its dispersion polynomial is reducible for fixed energy when trigonal warping is neglected. This algebraic reducibility is the core mechanism for the embedded bound state (Massatt et al., 2023).

For monolayer graphene, the dispersion polynomial is

$V$7

with the composite momentum variable

$V$8

In AB stacking with $V$9, the bilayer dispersion function becomes quadratic in $x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$0,

$x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$1

and factorizes as

$x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$2

where

$x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$3

The Green’s-function construction then uses a localized source or defect supported on a single $x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$4–$x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$5 pair. Through polynomial matrix factorization, one isolates a central $x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$6 block whose eigenvectors are momentum-independent. Choosing the source proportional to one such eigenvector cancels one factor, say $x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$7, from the denominator of the momentum-space wavefunction, leaving only $x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$8. If the chosen energy lies in a band of $x^* = x,\qquad xy + yx = (x,y)\,\mathbf{1}.$9 but in a gap of $V$0, the real-space state is exponentially localized even though the same energy belongs to the continuum of the other channel. That is the algebraic bound state in the continuum.

This mechanism is distinct from the non-Hermitian $V$1-tail AICs. Here the state is exponentially localized in the idealized model, and the algebraic ingredient is the factorization of the dispersion polynomial, not an algebraic spatial envelope. The paper further shows that $V$2 preserves reducibility, while trigonal warping $V$3 destroys exact factorization and broadens the exact bound state into a long-lived resonance. A notable misconception addressed by this analysis is that “without symmetry protection” means “without structure.” In AB-stacked graphene the relevant protection is algebraic, coming from reducibility in the composite variable $V$4, not from a symmetry block decomposition.

5. Continuum particle systems and algebraic-probabilistic states

A further continuum meaning arises in interacting particle systems, where an algebraic approach to intertwinings is developed through a $V$5 current algebra acting on continuum configurations. The underlying space is a Borel space $V$6, the configuration space is the space $V$7 of finite counting measures on $V$8, and the Hilbert space is

$V$9

where $\sigma$0 is the Pascal point process with parameter $\sigma$1 and finite measure $\sigma$2 (Floreani et al., 2023).

For bounded measurable test functions $\sigma$3, the paper defines raising, lowering, and neutral current operators on a dense domain $\sigma$4: $\sigma$5

$\sigma$6

$\sigma$7

These satisfy the $\sigma$8 current-algebra commutation relations

$\sigma$9

$1/r$00

Adjointness of $1/r$01 and $1/r$02 is proved using the Papangelou identity for the Pascal process.

For the constant test function $1/r$03, the global currents generate unitary operators

$1/r$04

where $1/r$05 is essentially self-adjoint on $1/r$06. If $1/r$07 is a Markov semigroup that is consistent and reversible with respect to $1/r$08, then

$1/r$09

These are self-intertwiners of the continuum dynamics. For a special choice

$1/r$10

the corresponding unitary maps $1/r$11-particle-supported observables into generalized Meixner chaos components: $1/r$12 and $1/r$13. This gives a continuum Meixner-chaos analogue of the discrete results of Carinci, Franceschini, Giardinà, Groenevelt, and Redig cited by the paper.

In this usage, “algebraic states in continuum” does not denote embedded impurity eigenstates. It refers instead to a continuum representation of an algebra of observables, a distinguished probabilistic state given by the Pascal process, and unitary morphisms intertwining the dynamics. A plausible implication is that the term AIC here is best understood in the operator-algebraic and representation-theoretic sense of a state on an algebra realized over a continuum configuration space.

6. Comparative structure and recurrent misunderstandings

The literature therefore supports several non-equivalent meanings of AIC. In the CAR/CCR setting, an AIC is a state on a $1/r$14-algebra, classified by covariance operators or forms and by the quasi-equivalence class of its GNS representation. In non-Hermitian impurity physics, an AIC is an embedded eigenstate with algebraic spatial decay, typically $1/r$15 in two dimensions. In bilayer materials, the closest analogue is a defect BIC produced by algebraic reducibility of the dispersion polynomial, with exponential localization in the idealized model. In continuum particle systems, the relevant object is a probabilistic or vector state on a current-algebra representation space, together with self-intertwining unitaries.

Several points are easily conflated. First, “continuum” has different meanings across the papers: infinite-dimensional one-particle Hilbert space in algebraic quantum theory, continuum spectrum in band theory, continuum configuration space in stochastic particle systems, and continuum limit or bulk Brillouin-zone description in lattice models. Second, “algebraic” can refer to Wick-Gaussian algebraic structure, to polynomial reducibility of a dispersion relation, to current-algebra commutation relations, or to algebraic spatial decay. Third, localization properties differ sharply: quasi-free CAR/CCR states are not localized wavefunctions; the non-Hermitian AICs are algebraically localized and non-$1/r$16 in infinite 2D; the bilayer defect states are exponentially localized when exact; and the continuum particle-system states are states on configuration space rather than single-particle spatial profiles.

The cited works also delimit several misconceptions. In quasi-free CAR theory, zero transition amplitude does not by itself imply disjointness; the dichotomy is still quasi-equivalent versus disjoint, but amplitude is not a complete discriminator. In the non-Hermitian impurity setting, “localized” does not mean square-integrable in the infinite system. In bilayer graphene, lack of symmetry protection does not mean accidental fine-tuning; the relevant mechanism is algebraic factorization. In the current-algebra framework, self-intertwining unitaries are symmetries of a semigroup and generators of Meixner-chaos transforms, not spectral embedded states.

Taken together, these strands show that AIC is presently a polysemous research term rather than a single established concept. What unifies the usages is not a common ontology but a recurring principle: algebraic structure—covariance data, dispersion-factor reducibility, current-algebra relations, or Green’s-function singularity structure—organizes continuum phenomena into sharply classifiable sectors.