Two-Eigenstate Model in Dirac Oscillator

• The two-eigenstate model is a framework where two eigenstates form unified analytic branches, as exemplified by the 1+1 Dirac oscillator via complex frequency continuation.
• It employs analytic continuation to connect positive and negative energy states, thereby providing a geometric basis for the particle–antiparticle relation.
• Extensions of the model are found in eigenstate thermalization, measurement bifurcation, and algorithmic filtering, highlighting its broad applicability in quantum systems.

“Two-Eigenstate Model” denotes a class of constructions in which two eigenstates are treated not as unrelated spectral points but as the two branches, outcomes, or sector representatives of a single higher-level structure. In its most explicit formulation, the term refers to the $1+1$ Dirac oscillator after analytic continuation of the oscillator frequency into the complex plane, where a positive-energy state and its negative-energy partner at fixed $n\neq 0$ become the two branches of one analytic eigenvalue function on a two-sheeted Riemann surface (Cao et al., 2019). The literature surveyed here suggests that the expression is not a single universally standardized designation. Rather, it recurs in several technically distinct senses: analytic branch pairing, pair-of-eigenstates parameterizations in ETH, bifurcation into two measurement eigenstates, effective two-state reductions inside larger Hilbert spaces, and algorithmic filters that isolate one eigenstate against a single dominant competitor.

1. Canonical formulation in the $1+1$ Dirac oscillator

The most precise and self-contained realization of a two-eigenstate model is the one-dimensional Dirac oscillator studied through analytic continuation in the frequency parameter $\omega$. The Hamiltonian is

$$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$

with $\hbar=1$ and $c=1$. Squaring the Hamiltonian gives

$$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$

or equivalently

$$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$

This reduction exhibits the problem as a nonrelativistic harmonic oscillator plus a spin term, while the later two-sheeted structure originates from the square-root relation between $\mathcal H$ and $n\neq 0$0 (Cao et al., 2019).

From the harmonic-oscillator reduction one obtains

$n\neq 0$1

and, after diagonalizing $n\neq 0$2 in the degenerate eigenspaces of $n\neq 0$3,

$n\neq 0$4

The notation is essential. The superscript sign labels the sign of the $n\neq 0$5 term under the square root, while the subscript sign labels the sign in front of the square root. For $n\neq 0$6, the conventional sector is

$n\neq 0$7

and the unconventional sector is

$n\neq 0$8

The corresponding spinors are written in terms of harmonic-oscillator functions $n\neq 0$9, with $1+1$0. In the conventional sector,

$1+1$1

while in the unconventional sector

$1+1$2

The resulting two-eigenstate interpretation is specific: the connected pair is not $1+1$3, but $1+1$4 in the conventional sector, and $1+1$5 in the unconventional sector. Each pair has the same oscillator quantum number $1+1$6.

2. Two-sheeted analytic structure and particle–antiparticle continuation

The defining move is to complexify only the oscillator frequency,

$1+1$7

while keeping $1+1$8 and $1+1$9 fixed. Because the Dirac oscillator is not symmetric under $\omega$0, the physically relevant analytic objects are not the nested square roots inherited from the ordinary harmonic oscillator, but two separate functions,

$\omega$1

Each is a genuine square-root two-sheeted structure rather than one sector of a larger physically unified four-sheeted surface (Cao et al., 2019).

For the conventional family, the branch point is determined by

$\omega$2

and the two sheets correspond exactly to the two energies

$\omega$3

Likewise, the unconventional family has branch point

$\omega$4

A branch cut may be chosen along the corresponding real axis starting at the branch point.

This makes the particle–antiparticle relation geometrically explicit. Starting from the conventional positive-energy state at real positive $\omega$5,

$\omega$6

one may vary $\omega$7 continuously along a path in the complex plane that winds around the branch point or crosses the branch cut. If $\omega$8, taking $\omega$9 moves the eigenvalue to the other branch,

$$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$0

which corresponds to $$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$1. The Hamiltonian is varied along the complex-$$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$2 path, but at the endpoint—after returning to the original positive real $$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$3—the Hamiltonian is again the original physical Hamiltonian. What changes is the sheet of the analytic eigenvalue and eigenstate. In this exact sense, the model connects a positive-energy state to its antiparticle partner without changing the final Hamiltonian (Cao et al., 2019).

The same section of the theory also identifies the standard charge-conjugation reinterpretation. After continuation to the negative-energy conventional branch, $$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$4 may be reinterpreted as an antiparticle state of positive energy in the charge-conjugated Hamiltonian

$$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$5

with charge-conjugated spinor

$$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$6

and eigenvalue

$$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$7

The two-eigenstate model is therefore simultaneously an analytic-continuation statement and a particle–antiparticle statement.

3. Exceptional sectors, branch collapse, and the PT-symmetric route

The exceptional role of $$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$8 is one of the clearest structural consequences of the model. For $$\mathcal{H}= \alpha\bigl(p-i\beta m\omega x\bigr)+\beta m, \qquad \alpha=\sigma_y,\qquad \beta=\sigma_z,$$9,

$\hbar=1$0

but the explicit spinors reduce to

$\hbar=1$1

so that

$\hbar=1$2

In the unconventional family,

$\hbar=1$3

There is therefore no nontrivial negative-energy conventional state with $\hbar=1$4. Mathematically, the conventional branch point

$\hbar=1$5

moves to $\hbar=1$6 as $\hbar=1$7, so no finite branch point remains and the two-sheeted surface collapses topologically. The absence of $\hbar=1$8 is thus given a geometric explanation rather than merely a spectroscopic one (Cao et al., 2019).

The same analysis clarifies why the formal nested expression

$\hbar=1$9

is not the correct two-state model for the original Dirac oscillator. Although it has a more elaborate branch structure, continuation across $c=1$0 does not connect eigenstates of the same Hamiltonian, because the Dirac oscillator lacks the $c=1$1 symmetry of the ordinary harmonic oscillator. Physically relevant continuation therefore decomposes into the two separate two-sheeted surfaces $c=1$2, not one unified nested construction.

Along the negative-real $c=1$3-axis, the reduced Hamiltonian in the conventional degenerate subspace is written as a $c=1$4 PT-symmetric matrix after a similarity transformation. The passage through the broken-PT region is associated with the threshold

$c=1$5

which the authors interpret as part of the analytic route connecting a state to its antiparticle branch. For $c=1$6 the threshold becomes impossible, effectively $c=1$7, again matching the absence of a finite branch structure.

4. Many-body statistical reinterpretations

Outside relativistic spectral theory, the nearest analogue of a two-eigenstate model appears in the eigenstate thermalization literature, where one asks whether matrix elements between two many-body eigenstates can be characterized solely by pair-energy data. The XXZ-chain study of eigenstate-to-eigenstate fluctuations formulates the ETH ansatz as

$c=1$8

with

$c=1$9

In that setting, a “two-eigenstate” description means that off-diagonal matrix elements are statistically controlled by the two energies of the pair, or equivalently by $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$0. For the nonintegrable XXZ chain, block submatrices at fixed $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$1 display Gaussian statistics and state-to-state equivalence, whereas in the integrable chain eigenstate-to-eigenstate fluctuations persist, so pair-energy variables alone do not suffice (Noh, 2020).

The two-dimensional XXZ study sharpens the same point by showing that any such pairwise eigenstate description must be symmetry-resolved. When the interaction is isotropic, the Hamiltonian has SU(2) symmetry, and ETH is supported only within fixed-total-spin subsectors. This suggests that a two-eigenstate parametrization is meaningful only after resolving all conserved quantum numbers; nearby energies alone do not define a valid two-state description in mixed symmetry sectors (Noh, 2022).

A more direct branch-based generalization appears near thermal first-order phase transitions in the generalized LMG-3 model. There the many-body density of states is controlled by two distinct mean-field solutions whose entropy branches exchange dominance as energy density varies. The result is a regime with coexistence of two classes of eigenstates at the same energy density, with distinct local expectation values, and another regime with Schrödinger-cat-like inter-branch superpositions, separated by an eigenstate phase transition (Serbyn et al., 13 Jan 2026). In that setting, the two-eigenstate language no longer refers to a single pair of levels, but to two competing eigenstate families.

5. Dynamical, operational, and algorithmic realizations

A distinctly operational use of the idea appears in the measurement model of a two-level system $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$2 interacting with a large apparatus $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$3. The measured observable has eigenstates $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$4 and $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$5, and the interaction is modeled as a scattering process with apparatus-dependent amplitudes $$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$6. When the apparatus is built from many independent subsystems, the multiplicative fluctuations in these amplitudes accumulate into a collective branching variable

$$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$7

The final-state distribution becomes

$$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$8

and in the large-$$\mathcal{H}^2=m^2+p^2+m^2\omega^2x^2-\sigma_z m\omega,$$9 limit

$$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$0

The measured system therefore ends in one of the two eigenstates of the measured observable, with Born weights. Here the “two-eigenstate model” is not spectral analyticity but bifurcation between two dynamically selected outcome states (Eriksson et al., 2017).

A computational use appears in eigenstate-preparation algorithms. The thesis on eigenstate preparation explicitly specializes many formulas to an initial state with support on two eigenstates,

$$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$1

In the Rodeo Algorithm, one successful cycle multiplies each eigencomponent by an energy-dependent filter factor. When the guessed energy is set to $$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$2, the target component is preserved while the undesired component is suppressed by factors

$$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$3

over successive successful cycles. The Variational Rodeo Algorithm then optimizes the input circuit precisely to increase the favorable overlap before this two-eigenstate filtering step is applied (Bonitati, 2024).

A related but different algorithmic use is Berger’s extended power method for two commuting operators $$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$4 and $$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$5. There the central object is not a dynamical two-level system but a simultaneous eigenvector labeled by an eigenvalue pair $$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$6. By pre-selecting the eigenvalue $$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$7 of $$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$8, the method iterates

$$\frac{\mathcal{H}^2-m^2}{2m} = \frac{p^2}{2m} +\frac{1}{2}m\omega^2x^2 -\frac{1}{2}\sigma_z\omega.$$9

thereby selecting a common eigenvector of $\mathcal H$0 and $\mathcal H$1 with the chosen quantum number. This is a paired-eigenvalue use of the two-eigenstate idea rather than a literal two-state dynamics (Berger, 2015).

6. Generalizations, embedded sectors, and limits of the term

Several adjacent models clarify what the expression can and cannot mean. In the Friedrichs model with one discrete state coupled to two continua, the analytic structure is no longer two-sheeted but four-sheeted. A single bare level can generate a bound state plus three virtual poles, or more than one pair of resonance poles, depending on thresholds and form factors. This extends the Dirac-oscillator logic from a two-sheeted pairing to a multi-sheet pole structure and shows that one bare state need not correspond to only one dressed eigenstate-like object (Xiao et al., 2016).

An embedded two-state sector appears in the model of two coupled two-level systems. The full Hilbert space is four-dimensional, but the one-excitation manifold spanned by $\mathcal H$2 is an exact $\mathcal H$3 problem with hybridized eigenstates

$\mathcal H$4

while $\mathcal H$5 and $\mathcal H$6 remain unchanged. The paper’s thermalization theory is therefore a four-state open system with a distinguished two-eigenstate core, not a globally two-state model (Liao et al., 2010).

The exact solution for two atoms in a $\mathcal H$7-split double well supplies an explicit limitation. It shows that a naïve two-eigenstate or two-mode truncation can be misleading: after a barrier quench, the initial odd-parity NOON state generally overlaps with the first few odd-parity eigenstates, not just two, and single-particle tunneling transfers probability between double occupancy and single occupancy. A literal two-state truncation is therefore only a restricted approximation (Liu et al., 2014).

A distinct nonlinear usage occurs in the isotropic two-electron mean-field quantum-dot model, where the two lowest $\mathcal H$8-wave nonlinear eigenstates are stationary solutions of different self-consistent equations and can have nonzero overlap,

$\mathcal H$9

That overlap is then interpreted as an intrinsic transition amplitude rather than a violation of orthogonality within one linear Hermitian eigenproblem (Reinisch, 2015).

Finally, in non-Hermitian superfluidity with two-body loss, the decisive dichotomy is not between two levels but between two eigenstate prescriptions: a right-eigenstate-based mean-field theory and its biorthogonal counterpart. The two prescriptions generate different gap equations and qualitatively different self-consistent solutions under dissipation. This suggests a further contextual usage in which a “two-eigenstate model” means a comparison between inequivalent notions of eigenstate, rather than a two-level truncation of one spectrum (Liu et al., 5 Oct 2025).

Taken together, these developments indicate that the most stable core meaning of the term remains the one supplied by the $n\neq 0$00 Dirac oscillator: two states connected as the two branches of one analytic object. Beyond that canonical case, the expression is best read contextually. In some papers it denotes a literal two-sheeted analytic pairing, in others a branch-resolved ETH structure, a measurement bifurcation, an embedded two-state manifold inside a larger spectrum, or an algorithmic two-eigenstate filter. The common theme is structural rather than terminological: two eigenstates are treated as a coupled, exchangeable, or analytically unified entity rather than as independent entries in a spectrum.