Matryoshka Sine-Cosine Chains

• Matryoshka-type sine–cosine chains are hierarchical models defined by recursive squaring that embeds lower-level sine–cosine structures.
• They exhibit robust chiral symmetry and protect edge states, with energy spectra governed by nested square-root formulas.
• These chains bridge topological condensed-matter physics and quantum computing through layered Hamiltonian inheritance and functional equation analogues.

Searching arXiv for the cited papers to ground the article in current arXiv records. arXiv search query: (Dias et al., 2021) Matryoshka approach to Sine-Cosine topological models Matryoshka-type Sine–Cosine chains are hierarchical constructions in which sine–cosine structures reappear under an iterated reduction or “unsquaring” procedure, so that an apparently extended model contains lower-level sine–cosine models as embedded components. In the condensed-matter setting, the term refers most specifically to the Sine–Cosine topological chains introduced as particular extended Su–Schrieffer–Heeger models with recursively organized hopping amplitudes and a block structure preserved under repeated squaring (Dias et al., 2021). In a broader mathematical sense, the same “Matryoshka” idea appears whenever cosine–sine objects are built layer by layer from simpler inner data, as in semigroup functional equations, nested trigonometric formulas, and phase-shifted sine–Gordon reductions (Ajebbar et al., 2023). Across these settings, the defining feature is not merely the presence of sine and cosine terms, but a nested dependence in which each outer level is generated from the previous one by a fixed transform.

1. Topological Sine–Cosine chains as the canonical Matryoshka realization

The most direct use of the term arises in the “Matryoshka approach to Sine-Cosine topological models” (Dias et al., 2021). There, Sine–Cosine models SC$(n)$ are defined as extended SSH chains with $2n$ sites in the unit cell and hoppings constrained by

$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$

A unit cell therefore carries the ordered hopping pattern

$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$

Because the Hamiltonian is purely off-diagonal in the sublattice basis, chiral symmetry is automatic (Dias et al., 2021).

The Matryoshka subclass is denoted SSC$(n)$. These are the SC$(2^{n-1})$ chains that are “$n$-times squarable,” meaning that after squaring the Bloch Hamiltonian, applying an energy shift, and renormalizing the energy unit, one of the resulting blocks is again a Sine–Cosine model of the previous level, SSC$(n-1)$ (Dias et al., 2021). Repeating this operation $n$ times reduces the chain to SSC$(0)$, which is a uniform tight-binding chain with one site per unit cell. This recursive embedding is the central reason for the Matryoshka designation.

The construction is parameterized by a sequence of positive hopping factors

$2n$0

Once this sequence is fixed, the authors state that the full SSC$2n$1 band structure and the angles $2n$2 are determined by it, up to a discrete ambiguity associated with the choice of sublattice block at each squaring step (Dias et al., 2021). This produces a family of spectrally equivalent realizations sharing the same nested hierarchy.

A closely related development appears in “Scalable topological quantum computing based on Sine-Cosine chain models” (Lykholat et al., 26 Mar 2026), where Matryoshka-type Sine–Cosine chains are described as a specific family of 1D “$2n$3-root” topological insulators obtained by recursively taking square roots of the SSH chain. There the order is denoted $2n$4, the unit cell grows as $2n$5, and the Hamiltonian is written in terms of sine–cosine modulated hoppings

$2n$6

This formulation makes the same nesting explicit, but emphasizes square-root inheritance rather than repeated squaring (Lykholat et al., 26 Mar 2026).

2. Recursive squaring, spectral renormalization, and nested gaps

For any bipartite Hamiltonian written as

$2n$7

squaring yields

$2n$8

In SC$2n$9 chains, this operation produces sublattice blocks with uniform on-site potentials

$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$0

and effective nearest-neighbor hoppings

$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$1

along the chosen sublattice block (Dias et al., 2021). After subtracting the identity, one obtains a pure hopping model again. SSC$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$2 is the special case in which this block can be identified, after rescaling, with SSC$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$3.

The recursive matching conditions are given by

$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$4

$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$5

for $$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$6, together with

$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$7

These equations are the recursive constraints that enforce Matryoshka self-similarity across levels (Dias et al., 2021).

The spectral recursion is equally explicit. If $$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$8 is an energy of SSC$$t_{2j-1} = \sin\theta_j,\qquad t_{2j} = \cos\theta_j,\qquad j=1,\dots,n.$$9, then at the next level

$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$0

Iterating from the uniform SSC$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$1 chain yields the full nested band-edge formula

$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$2

This is the characteristic spectral signature of the Matryoshka chain (Dias et al., 2021).

The same nested-square-root pattern is reported in the quantum-computing formulation, where the general dispersion obeys

$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$3

and the protected edge-state energies lie in the set

$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$4

continuing through the full nested hierarchy (Lykholat et al., 26 Mar 2026). This parallel strongly indicates that the SSC$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$5 construction and the $$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$6-root chain construction are two descriptions of the same recursive spectral architecture.

A useful summary is the following.

Level	Structural operation	Result
SC$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$7 / SSC$$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$8	Square $$\{\sin\theta_1,\cos\theta_1,\sin\theta_2,\cos\theta_2,\dots,\sin\theta_n,\cos\theta_n\}.$$9, shift, rescale	One block becomes SSC$(n)$0
SSC$(n)$1	Uniform tight-binding chain	Terminal element of the hierarchy
$(n)$2-root chain	Recursive square root of SSH	Parent recovered after squaring

This suggests that “Matryoshka-type” is best understood as a recursive closure property under squaring or square-rooting, rather than as a mere hopping parametrization.

3. Chiral symmetry, band topology, and edge-state inheritance

All Sine–Cosine chains in the topological construction possess chiral symmetry because they contain only inter-sublattice hoppings. With

$(n)$3

one has

$(n)$4

This enforces spectral symmetry $(n)$5 and protects zero-energy edge states in the central gap of each chiral level (Dias et al., 2021).

For SSC$(n)$6, which is the ordinary SSH chain, the Bloch Hamiltonian is

$(n)$7

and the corresponding winding number is

$(n)$8

The topological characterization for higher SSC$(n)$9 is discussed in terms of Zak phases of the positive bands; the authors state that adding all Zak phases of the positive bands of SSC$(2^{n-1})$0 in the non-trivial phase gives $(2^{n-1})$1, signaling protected edge states (Dias et al., 2021).

The distinctive Matryoshka phenomenon is edge-state inheritance across levels. Zero-energy edge states in a lower-level chain do not disappear when embedded in a higher-level one; rather, they are mapped to finite-energy edge states in non-central gaps of the outer chain (Dias et al., 2021). In the $(2^{n-1})$2-root formulation, the same principle is stated as follows: each zero level of the parent SSH model “unfolds” into two edge levels in the next Matryoshka order, and each of the $(2^{n-1})$3 gaps can host a pair of edge states under open boundaries (Lykholat et al., 26 Mar 2026).

For open chains with lengths of the form

$(2^{n-1})$4

the nested blocks produced by squaring retain uniform local potentials, allowing the Matryoshka sequence to persist in the OBC setting (Dias et al., 2021). This is crucial for interpreting the OBC spectrum as a superposition of levels generated from a single zero mode by successive energy shifts, renormalizations, and square roots, plus additional folding-energy levels associated with the complementary sublattice block (Dias et al., 2021).

A common misconception is that all edge modes in these models are central-gap zero modes. The recursive construction shows otherwise: only the innermost SSH-like level is tied to zero energy, while higher-level descendants generally occupy non-central gaps at finite energies fixed by the nested-root formulas (Dias et al., 2021).

4. Quantum-information interpretation: qudits, braiding, and memory

The quantum-computing proposal based on Matryoshka-type Sine–Cosine chains uses the same nested edge-state hierarchy as an information-bearing Hilbert space (Lykholat et al., 26 Mar 2026). Because the number of gaps grows as

$(2^{n-1})$5

and each gap can host two edge states, the maximum number of edge states is

$(2^{n-1})$6

under open boundary conditions (Lykholat et al., 26 Mar 2026). These localized edge modes are then used for high-dimensional qudit encoding and multi-qubit memories.

A concrete example is given for a $(2^{n-1})$7 chain with 80 sites and angles

$(2^{n-1})$8

which has a 7-site defect on the left and 7-site defect on the right, producing 14 localized edge states in total and hence a defect Hilbert space of dimension $(2^{n-1})$9 (Lykholat et al., 26 Mar 2026). The authors state that this can be used as a 14-level qudit or as a storage space for several qubits.

For $n$0, each defect carries three levels $n$1. In a Y-junction braiding setup, the left/right defect states are organized into a 6-dimensional basis,

$n$2

and the resulting braiding operator acts blockwise on the $n$3, $n$4, and $n$5 sectors (Lykholat et al., 26 Mar 2026). The explicit operator is presented as

$n$6

with Pauli-like action on the respective two-level sectors (Lykholat et al., 26 Mar 2026). The stated implication is that a single braid can realize a composed gate acting simultaneously on several logical subspaces.

The same paper extends the SSH memory protocol to Matryoshka chains. At $n$7, a qubit may be coupled to an $n$8, $n$9, or $(n-1)$0 edge mode, already increasing the storage options relative to a single SSH chain (Lykholat et al., 26 Mar 2026). For higher $(n-1)$1, the memory dimension grows exponentially with the order. The authors summarize this directly: “This qudit (or multiple qubit) memory architecture exponentially scales with the order $(n-1)$2 of the Matryoshka chain” (Lykholat et al., 26 Mar 2026).

The topological protection in this context is explicitly qualified as partial. The parent SSH zero modes are strictly topologically protected under the standard chiral-symmetry conditions, but the finite-energy descendants in the $(n-1)$3-root hierarchy inherit a weaker form of protection that becomes diluted as $(n-1)$4 increases (Lykholat et al., 26 Mar 2026). This limits any interpretation of these chains as fully topological analogues of Majorana or non-Abelian platforms.

5. Broader mathematical meanings of “Matryoshka” in sine–cosine systems

Beyond topological chains, the same nested logic appears in functional-equation theory. In “Cosine-sine functional equation on semigroups” (Ajebbar et al., 2023), the main equation

$(n-1)$5

is solved hierarchically in terms of multiplicative functions $(n-1)$6, sine-law solutions

$(n-1)$7

and cosine-sine solutions

$(n-1)$8

The authors explicitly describe this as a layered structure: $(n-1)$9 This is a rigorous functional-equation realization of a Matryoshka-type sine–cosine chain (Ajebbar et al., 2023).

A related generalization on semigroups with involution,

$n$0

likewise reduces solutions to multiplicative functions, $n$1-additive sine-type functions, and twisted cosine–sine equations (Ajebbar et al., 2023). Here the Matryoshka principle is the successive reduction of the outer Levi–Civita-type equation to inner sine and cosine–sine equations.

An analytically distinct but structurally similar example is provided by the combined sine–cosine–Gordon equation

$n$2

which can be rewritten as

$n$3

The paper shows that traveling-wave solutions can be recast in the canonical form

$n$4

so the apparently two-term sine–cosine potential is revealed as a phase-shifted sine–Gordon kink with nested dependence on $n$5 and $n$6 (Kuo et al., 2012). The same source explicitly interprets

$n$7

as a one-step “matryoshka,” with the original coefficients hidden inside amplitude and phase parameters (Kuo et al., 2012).

These examples indicate that “Matryoshka-type” has at least two precise meanings in current research usage: recursive Hamiltonian inheritance under squaring in lattice topological models, and layered solution-building in sine–cosine functional or nonlinear equations.

6. Related transform, spectral, and geometric chain constructions

The language of sine–cosine chains also appears in transform theory and harmonic geometry, though usually without the topological square-root meaning. In “Supercharacters and the discrete Fourier, cosine, and sine transforms” (Garcia et al., 2017), the DCT and DST are derived from supercharacter theory on $n$8, and the matrices they diagonalize are described by structured algebras with Toeplitz+Hankel-like forms. The paper itself does not construct Matryoshka chains, but it explicitly states that this is the kind of structure one wants to build “nested chains of cosine/sine operators” (Garcia et al., 2017). A plausible implication is that operator-level Matryoshka constructions can be organized spectrally via supercharacter algebras.

For continuous transforms, “Eigenfunctions of the Cosine and Sine Transforms” (Katsnelson, 2012) decomposes $n$9 into the eigensubspaces

$(0)$0

and constructs continuous orthogonal chains of generalized eigenfunctions $(0)$1. These are built from Mellin waves $(0)$2 with phase factors $(0)$3 and $(0)$4, so that the cosine and sine eigenspaces become nested inside a common Mellin framework (Katsnelson, 2012). This suggests a spectral notion of Matryoshka structure in which common carrier functions are wrapped by different cosine/sine symmetry shells.

A different form of nesting appears in “Nested formulas for cosine and inverse cosine functions based on Viète's formula for $(0)$5” (Kawalec, 2020). There the degree-2 Chebyshev map

$(0)$6

generates a forward cosine chain

$(0)$7

while the inverse cosine is represented by nested radicals using

$(0)$8

This produces a literal nested-radical Matryoshka chain for $(0)$9, together with Gray-code sign patterns that enumerate branches (Kawalec, 2020). Although this is unrelated to topological lattices, it is a mathematically exact instance of repeated sine–cosine shelling.

Finally, in “Periodic functions: self-intersections, local singular points, and folds” (Sakhnovich, 2023), an $2n$00-member chain is defined as the finite Fourier-type curve

$2n$01

Here the “chain” is a superposition of harmonics, not a recursive square-root structure. The paper nevertheless interprets higher harmonics as algebraic layers over the base oscillation and extends them to periodic helices and S-torus knots (Sakhnovich, 2023). This is a different use of chain terminology and should not be conflated with the SSC$2n$02 Matryoshka construction.

7. Conceptual synthesis and scope

In the most specific and technically established sense, Matryoshka-type Sine–Cosine chains are chiral bipartite lattice models whose sine–cosine hopping pattern is closed under repeated squaring, so that SSC$2n$03 contains SSC$2n$04, SSC$2n$05, and ultimately a uniform chain as nested spectral ancestors (Dias et al., 2021). Their energy landscape is controlled by nested square roots, their edge-state content by inherited chiral structures, and their open-boundary phenomenology by the conversion of inner zero modes into outer finite-energy edge states (Dias et al., 2021).

In the quantum-information reinterpretation, the same nested edge-state hierarchy becomes a resource for high-dimensional qudit encoding, Y-junction braiding, and extended memories, with edge-state counts scaling as $2n$06 and with robustness that is partial rather than absolute (Lykholat et al., 26 Mar 2026). In the broader mathematical literature, “Matryoshka-type” extends naturally to functional-equation hierarchies, phase-shifted nonlinear waves, nested transform eigenstructures, and iterative trigonometric formulas (Ajebbar et al., 2023).

A final distinction is essential. Not every sine–cosine chain is Matryoshka-type. Finite harmonic sums, DCT/DST operator algebras, and periodic helices are chain constructions in their own right, but only some of them satisfy the stronger nested criterion that an outer sine–cosine object contains a lower-level sine–cosine object as an exact structural or spectral core. The SSC$2n$07 and $2n$08-root topological chains satisfy this criterion explicitly (Dias et al., 2021). The functional-equation hierarchies satisfy it algebraically (Ajebbar et al., 2023). The Viète–Chebyshev formulas satisfy it analytically (Kawalec, 2020). Taken together, these works establish Matryoshka-type Sine–Cosine chains as a cross-disciplinary pattern of recursive sine–cosine organization rather than a single isolated model class.