---
title: Matryoshka Sine-Cosine Chains
url: https://www.emergentmind.com/topics/matryoshka-type-sine-cosine-chains
type: topic
---

# Matryoshka Sine-Cosine Chains

Searching arXiv for the cited papers to ground the article in current arXiv records.
arXiv search query: 2102.00887 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 [2102.00887]. 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 [2306.04666]. 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” [2102.00887]. 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 [2102.00887].

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\((n-1)\) [2102.00887]. 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
\[
t^{(0)}, t^{(1)}, \dots, t^{(n-1)}.
\]
Once this sequence is fixed, the authors state that the full SSC\((n)\) band structure and the angles \(\{\theta_j^{(m)}\}\) are determined by it, up to a discrete ambiguity associated with the choice of sublattice block at each squaring step [2102.00887]. 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” [2603.25952], where Matryoshka-type Sine–Cosine chains are described as a specific family of 1D “\(2^n\)-root” topological insulators obtained by recursively taking square roots of the SSH chain. There the order is denoted \(P\), the unit cell grows as \(2^P\), and the Hamiltonian is written in terms of sine–cosine modulated hoppings
\[
\begin{aligned}
H =\; t^{(P)} \sum_{m=1}^N \Bigg[
& \sum_{j=1}^{2^P} \sin\theta_j \, \ket{m,B_j}\bra{m,A_j} \\
& + \sum_{j=1}^{2^P-1} \cos\theta_j \, \ket{m,A_{j+1}}\bra{m,B_j} \\
& + \cos\theta_{2^P} \ket{m+1,A_1}\bra{m,B_{2^P}}
\Bigg] + \text{H.c.}
\end{aligned}
\]
This formulation makes the same nesting explicit, but emphasizes square-root inheritance rather than repeated squaring [2603.25952].

## 2. Recursive squaring, spectral renormalization, and nested gaps

For any bipartite Hamiltonian written as
\[
H=\begin{pmatrix}0&T\\T^\dagger&0\end{pmatrix},
\]
squaring yields
\[
H^2=\begin{pmatrix}TT^\dagger&0\\0&T^\dagger T\end{pmatrix}.
\]
In SC\((n)\) chains, this operation produces sublattice blocks with uniform on-site potentials
\[
\varepsilon_j=\sin^2\theta_j+\cos^2\theta_j=1,
\]
and effective nearest-neighbor hoppings
\[
t_j=\cos\theta_j\sin\theta_{j+1}
\]
along the chosen sublattice block [2102.00887]. After subtracting the identity, one obtains a pure hopping model again. SSC\((n)\) is the special case in which this block can be identified, after rescaling, with SSC\((n-1)\).

The recursive matching conditions are given by
\[
t^{(n-1)} \sin\theta_{j}^{(n-1)}
  = \cos\theta_{2j-1}^{(n)} \,\sin\theta_{2j}^{(n)},
\]
\[
t^{(n-1)} \cos\theta_{j}^{(n-1)}
  = \cos\theta_{2j}^{(n)}   \,\sin\theta_{2j+1}^{(n)},
\]
for \(j=1,\dots,2^{n-2}\), together with
\[
t^{(n-1)} = \sqrt{\left(\cos\theta_{2j-1}^{(n)}\sin\theta_{2j}^{(n)}\right)^2
      +\left(\cos\theta_{2j}^{(n)}\sin\theta_{2j+1}^{(n)}\right)^2}.
\]
These equations are the recursive constraints that enforce Matryoshka self-similarity across levels [2102.00887].

The spectral recursion is equally explicit. If \(\varepsilon^{(j-1)}\) is an energy of SSC\((j-1)\), then at the next level
\[
\varepsilon^{(j)}=\pm\sqrt{\,1+t^{(j-1)}\varepsilon^{(j-1)}\,}.
\]
Iterating from the uniform SSC\((0)\) chain yields the full nested band-edge formula
\[
\varepsilon_{\pm\pm\cdots\pm}
= \pm \sqrt{1 \pm t^{(n-1)}\sqrt{1 \pm t^{(n-2)}\sqrt{\cdots\sqrt{1 \pm t^{(1)}\sqrt{1 \pm \sqrt{2}\,t^{(0)}} }.
\]
This is the characteristic spectral signature of the Matryoshka chain [2102.00887].

The same nested-square-root pattern is reported in the quantum-computing formulation, where the general dispersion obeys
\[
E_{\pm\pm\ldots\pm}^{(P)}(k)
 = \pm\sqrt{\,1 + t^{(P-1)} E_{\pm\pm\ldots\pm}^{(P-1)}(k)\,},
\]
and the protected edge-state energies lie in the set
\[
\begin{aligned}
\varepsilon \in
\Big\{ & \pm 1,\quad \pm \sqrt{1\pm t^{(P-1)}}, \\
& \pm\sqrt{1\pm t^{(P-1)} \sqrt{1\pm t^{(P-2)}}}, \\
& \ldots \Big\},
\end{aligned}
\]
continuing through the full nested hierarchy [2603.25952]. This parallel strongly indicates that the SSC\((n)\) construction and the \(2^n\)-root chain construction are two descriptions of the same recursive spectral architecture.

A useful summary is the following.

| Level | Structural operation | Result |
|---|---|---|
| SC\((n)\) / SSC\((n)\) | Square \(H\), shift, rescale | One block becomes SSC\((n-1)\) |
| SSC\((0)\) | Uniform tight-binding chain | Terminal element of the hierarchy |
| \(2^n\)-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
\[
\Gamma = \sum_{j\in A} |j\rangle\langle j| - \sum_{j\in B} |j\rangle\langle j|,
\]
one has
\[
\{H,\Gamma\}=0.
\]
This enforces spectral symmetry \(E\leftrightarrow -E\) and protects zero-energy edge states in the central gap of each chiral level [2102.00887].

For SSC\((1)\), which is the ordinary SSH chain, the Bloch Hamiltonian is
\[
H(k)=\begin{pmatrix}0&q(k)\\q(k)^*&0\end{pmatrix},\qquad
q(k)=t^{(1)}\bigl(\sin\theta+\cos\theta\,e^{-ik}\bigr),
\]
and the corresponding winding number is
\[
\nu=\frac{1}{2\pi i}\int_{-\pi}^{\pi} dk\, \partial_k\log q(k).
\]
The topological characterization for higher SSC\((n)\) 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\((j)\) in the non-trivial phase gives \(\pi\), signaling protected edge states [2102.00887].

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 [2102.00887]. In the \(2^n\)-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 \(2^{P+1}-1\) gaps can host a pair of edge states under open boundaries [2603.25952].

For open chains with lengths of the form
\[
N=2^n p-1,\qquad p>1,
\]
the nested blocks produced by squaring retain uniform local potentials, allowing the Matryoshka sequence to persist in the OBC setting [2102.00887]. 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 [2102.00887].

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 [2102.00887].

## 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 [2603.25952]. Because the number of gaps grows as
\[
2^{P+1}-1
\]
and each gap can host two edge states, the maximum number of edge states is
\[
2\times(2^{P+1}-1)=2^{P+2}-2
\]
under open boundary conditions [2603.25952]. These localized edge modes are then used for high-dimensional qudit encoding and multi-qubit memories.

A concrete example is given for a \(P=2\) chain with 80 sites and angles
\[
\theta_1=\pi/6,\qquad \theta_2=\pi/4,\qquad \theta_3=\pi/6,\qquad \theta_4=0,
\]
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 \(d=14\) [2603.25952]. The authors state that this can be used as a 14-level qudit or as a storage space for several qubits.

For \(P=1\), each defect carries three levels \((\varepsilon=1,-1,0)\). In a Y-junction braiding setup, the left/right defect states are organized into a 6-dimensional basis,
\[
\ket{1},\dots,\ket{6},
\]
and the resulting braiding operator acts blockwise on the \(\varepsilon=1\), \(\varepsilon=-1\), and \(\varepsilon=0\) sectors [2603.25952]. The explicit operator is presented as
\[
\mathbf{OP}
=
\begin{bmatrix}
e^{-i t/\hbar} Y_{\varepsilon=1} & \mathbf{0}_{2\times2} & \mathbf{0}_{2\times2} \\
\mathbf{0}_{2\times2} & e^{i t/\hbar} X_{\varepsilon=-1} & \mathbf{0}_{2\times2} \\
\mathbf{0}_{2\times2} & \mathbf{0}_{2\times2} & Y_{\varepsilon=0}
\end{bmatrix},
\]
with Pauli-like action on the respective two-level sectors [2603.25952]. 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 \(P=1\), a qubit may be coupled to an \(\varepsilon=1\), \(\varepsilon=-1\), or \(\varepsilon=0\) edge mode, already increasing the storage options relative to a single SSH chain [2603.25952]. For higher \(P\), 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 \(P\) of the Matryoshka chain” [2603.25952].

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 \(2^n\)-root hierarchy inherit a weaker form of protection that becomes diluted as \(P\) increases [2603.25952]. 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” [2306.04666], the main equation
\[
f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y)
\]
is solved hierarchically in terms of multiplicative functions \(\chi\), sine-law solutions
\[
\varphi(xy)=\varphi(x)\chi(y)+\chi(x)\varphi(y),
\]
and cosine-sine solutions
\[
\psi(xy)=\psi(x)\chi(y)+\chi(x)\psi(y)+\varphi(x)\varphi(y).
\]
The authors explicitly describe this as a layered structure:
\[
\chi \Longrightarrow (\varphi,\chi) \Longrightarrow (\psi,\varphi,\chi) \Longrightarrow (f,g,h).
\]
This is a rigorous functional-equation realization of a Matryoshka-type sine–cosine chain [2306.04666].

A related generalization on semigroups with involution,
\[
f(x\sigma(y))=f(x)g(y)+g(x)f(y)+h(x)h(y),
\]
likewise reduces solutions to multiplicative functions, \(\chi\)-additive sine-type functions, and twisted cosine–sine equations [2312.05285]. 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
\[
u_{tt}-k\,u_{xx}+a\sin u+\beta\cos u=0,
\]
which can be rewritten as
\[
u_{tt}-k u_{xx}+R\sin(u+\varphi)=0,
\qquad
R=\sqrt{a^2+\beta^2},\qquad
\varphi=\arctan\frac{\beta}{a}.
\]
The paper shows that traveling-wave solutions can be recast in the canonical form
\[
u(x,t)=-\varphi+4\arctan\left(e^{(x-ct-\xi_0')/2}\right),
\]
so the apparently two-term sine–cosine potential is revealed as a phase-shifted sine–Gordon kink with nested dependence on \(R\) and \(\varphi\) [1209.6113]. The same source explicitly interprets
\[
a\sin u+\beta\cos u=R\sin(u+\varphi)
\]
as a one-step “matryoshka,” with the original coefficients hidden inside amplitude and phase parameters [1209.6113].

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” [1702.02689], the DCT and DST are derived from supercharacter theory on \(\mathbb{Z}/n\mathbb{Z}\), 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” [1702.02689]. 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” [1203.2427] decomposes \(L^2(\mathbb{R}_+)\) into the eigensubspaces
\[
L^2(\mathbb{R}_+)=C_{+1}\oplus C_{-1}=S_{+1}\oplus S_{-1},
\]
and constructs continuous orthogonal chains of generalized eigenfunctions \(e_\bullet^\pm(t,T)\). These are built from Mellin waves \(t^{-1/2\pm iT}\) with phase factors \(c(T)\) and \(s(T)\), so that the cosine and sine eigenspaces become nested inside a common Mellin framework [1203.2427]. 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 \(\pi\)” [2007.08639]. There the degree-2 Chebyshev map
\[
T(x)=-1+2x^2
\]
generates a forward cosine chain
\[
\cos(x)=\lim_{n\to\infty}T^{\circ n}\Big(1-\frac{x^2}{2^{2n+1}}\Big),
\]
while the inverse cosine is represented by nested radicals using
\[
T^{-1}(y)=\sqrt{\frac{1+y}{2}}.
\]
This produces a literal nested-radical Matryoshka chain for \(\arccos\), together with Gray-code sign patterns that enumerate branches [2007.08639]. 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” [2304.09940], an \(n\)-member chain is defined as the finite Fourier-type curve
\[
x(t)=\sum_{k=1}^{n}c_k\cos(m_k t),\qquad
y(t)=\sum_{k=1}^{n}d_k\sin(m_k t).
\]
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 [2304.09940]. This is a different use of chain terminology and should not be conflated with the SSC\((n)\) 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\((n)\) contains SSC\((n-1)\), SSC\((n-2)\), and ultimately a uniform chain as nested spectral ancestors [2102.00887]. 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 [2102.00887].

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 \(2^{P+2}-2\) and with robustness that is partial rather than absolute [2603.25952]. In the broader mathematical literature, “Matryoshka-type” extends naturally to functional-equation hierarchies, phase-shifted nonlinear waves, nested transform eigenstructures, and iterative trigonometric formulas [2306.04666].

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\((n)\) and \(2^n\)-root topological chains satisfy this criterion explicitly [2102.00887]. The functional-equation hierarchies satisfy it algebraically [2312.05285]. The Viète–Chebyshev formulas satisfy it analytically [2007.08639]. 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.

Source: https://www.emergentmind.com/topics/matryoshka-type-sine-cosine-chains