Long-Range Topological Potential

• Long-range topological potential is the use of extended, nonlocal couplings to actively control and modify topological phases in quantum systems.
• The method employs exponentially decaying interactions to generate enriched phase diagrams with additional winding sectors and tunable edge modes in SSH and Kitaev-type models.
• This framework spans diverse platforms—from condensed matter to protein self-assembly—demonstrating practical implications in state transfer and phase transition management.

Long-range topological potential denotes the use of extended, nonlocal structure to control topology. In the generalized Su-Schrieffer-Heeger model with exponentially decaying couplings, the range of interaction is an independent lever for controlling topology: even relatively weak long-range couplings can trigger the topological transition if their range is large enough (Simonyan et al., 29 Sep 2025). In related literature, the phrase also appears in closely allied senses: as long-range hopping or pairing deformations in Kitaev-type chains (Viyuela et al., 2015), as the nonlocal influence of long-range pairing and hopping on topological critical dynamics (Defenu et al., 2019), and, in a different domain, as a weighted-total-persistence energetic bias used to guide protein self-assembly (Spirandelli et al., 21 Aug 2025). Across these usages, the common element is that topology is affected by structure extending beyond strictly local couplings.

1. Definition and conceptual scope

In the one-dimensional SSH setting studied in "Topological transitions controlled by the interaction range" (Simonyan et al., 29 Sep 2025), long-range topological potentials are identified with couplings that have a sufficiently smooth, extended spatial profile and that can destabilize or stabilize topological phases, even at very small magnitude. The central physical significance is that the range of interaction is an independent lever for controlling topology, not merely a secondary correction to coupling strength.

This usage belongs to a broader long-range-topology literature. In "Topological Massive Dirac Edge Modes and Long-Range Superconducting Hamiltonians" (Viyuela et al., 2015), long-range Hamiltonian deformations are introduced through exponentially decaying hopping amplitudes and power-law decaying superconducting pairings. In "Universal dynamical scaling of long-range topological superconductors" (Defenu et al., 2019), the long-range topological potential refers to power-law-decaying hopping and pairing in a $p$-wave superconducting quantum wire, with direct implications for quench scaling. In "Topological potentials guiding protein self-assembly" (Spirandelli et al., 21 Aug 2025), the term is used differently: the topological potential is a weighted sum of total persistence values derived from persistent homology and inserted directly into an energy function. This suggests that the phrase is not confined to a single formal definition, but is instead used for topology-sensitive nonlocal terms that act as control parameters or energetic biases.

A recurring misconception is that topology in one dimension is necessarily tied to strict locality. The available results are more differentiated. Infinite-range or slowly decaying couplings can preserve, modify, or qualitatively enrich topological structure, depending on symmetry, decay law, frustration, and boundary conditions (Patrick et al., 2016).

2. Exponentially decaying SSH formulation

The 2025 SSH-type construction is a generalized SSH model in which the canonical nearest-neighbor hoppings $J_1$ and $J_2$ are supplemented by long-range couplings between $A$ and $B$ sublattices across arbitrary distances, with exponential decay (Simonyan et al., 29 Sep 2025): $J_{(s)} = J e^{-\lambda s},$ where $J$ sets the overall scale of the long-range hopping and $\lambda$ is the inverse interaction length, so that $\lambda \rightarrow 0$ corresponds to infinite range.

The real-space Hamiltonian is

$$\begin{aligned} \hat{H} =& \sum_n \left[ J_1 \hat{a}^\dagger_n \hat{b}_n + J_2 \hat{a}^\dagger_{n+1} \hat{b}_n \right] \ & + \sum_{n} \left[ \sum_{s=1}^{\infty} J_{(s)} \hat{a}^\dagger_n \hat{b}_{n+s} + \sum_{s=2}^{\infty} J_{(s)} \hat{a}^\dagger_{n+s} \hat{b}_n \right] + \text{H.c.} \end{aligned}$$

This construction preserves chiral symmetry, which allows the use of topological invariants (Simonyan et al., 29 Sep 2025).

After Fourier transformation, the Bloch Hamiltonian retains off-diagonal structure,

$J_1$0

with

$J_1$1

Because the long-range couplings decay exponentially, the sum over long-range processes can be analytically resummed (Simonyan et al., 29 Sep 2025).

The relevant one-dimensional topological invariant is the winding number,

$J_1$2

The topological phase is determined by how $J_1$3 winds around the origin as $J_1$4 traverses the Brillouin zone, and phase transitions occur when $J_1$5 for some $J_1$6, i.e. when the bandgap closes (Simonyan et al., 29 Sep 2025).

3. Range-driven topological transition

The distinctive result of the exponentially decaying SSH model is that range and strength act as independent knobs. For fixed range $J_1$7, increasing $J_1$8 can induce a topological transition, which is consistent with previous intuition. More importantly, for fixed coupling strength, decreasing $J_1$9 can also trigger a topological phase transition. In the paper’s formulation, even very weak long-range couplings can induce a topological phase transition if they are sufficiently long-ranged (Simonyan et al., 29 Sep 2025).

The phase diagram is correspondingly enriched. Instead of only the $J_2$0 and $J_2$1 sectors of the canonical SSH model, the long-range system admits three sectors,

$J_2$2

Long-range coupling therefore enables access to a $J_2$3 phase, and a transition between $J_2$4 and $J_2$5 necessarily involves gap closing (Simonyan et al., 29 Sep 2025).

The geometry of the winding curve in the complex plane is central to this behavior. As the interaction range increases, the trajectory of $J_2$6 deforms and can cross the origin, signaling a topological transition. The bandgap can close not only at the center or edge of the Brillouin zone but also at intermediate points (Simonyan et al., 29 Sep 2025). This is a concrete mechanism by which a long-range topological potential modifies phase structure without requiring large-amplitude nonlocal couplings.

A related phenomenon appears in long-range Kitaev settings. For exponentially decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases (Viyuela et al., 2015). This parallel suggests that the range-controlled enlargement of topological regions is not restricted to the SSH geometry.

4. Boundary modes and finite-size manifestations

In the SSH-type model with exponentially decaying couplings, zero-energy edge-localized modes emerge in the topological phases with $J_2$7, and their existence is confirmed by inverse participation ratio calculations for finite systems (Simonyan et al., 29 Sep 2025). The same analysis also clarifies a finite-size subtlety: trivial sectors can have states that look localized for small system sizes, but inverse participation ratio scaling reveals their extended nature. This is an important diagnostic distinction in long-range systems, where nonlocal couplings can blur naive real-space intuition.

The broader literature shows that long-range couplings can alter edge physics in several inequivalent ways. In an infinite-range Kitaev chain, a semi-infinite geometry supports a single, exponentially localized zero energy Majorana mode, whereas a finite chain supports hybridized, non-zero energy Majorana-like edge modes whose energy is independent of chain length $J_2$8 (Patrick et al., 2016). In power-law paired Kitaev chains, the massless Majorana modes at the edges can pair together into a massive non-local Dirac fermion localized at both edges of the chain (Viyuela et al., 2015). By contrast, in an extended Kitaev chain at a topological critical point, the critical edge modes remain massless even when long-range interactions become substantially strong (Zhong et al., 2024).

These results do not point to a single universal edge response to long-range couplings. Instead, they indicate that the outcome depends on the decay profile, the distinction between gapped and critical regimes, and the boundary geometry. A plausible implication is that the phrase “long-range topological potential” captures a mechanism class rather than a unique boundary phenomenology.

5. Invariants, criticality, and bulk-boundary issues

Long-range couplings complicate the interpretation of bulk invariants. In one-dimensional insulators with long-range hopping, the introduction of next-nearest-neighbor terms can break the bipartite property and make site indexing relevant for determining bulk topological invariants. In the extended SSH chain, band inversion signals a crossover between hopping-parameter regions of influence of different chiral symmetries, and edge states become linear combinations of edge-like states with different localization lengths (Dias et al., 2021). This means that even identifying the correct unit cell can become part of the topological analysis.

Bulk-boundary correspondence can also weaken. In the long-range Kitaev chain with Aubry-André-Harper modulation, the bulk topological invariants can remain constant while dramatic changes appear in the behavior at the edge of the system, and in nonzero-energy gaps a 2D Chern invariant no longer corresponds to the number of edge mode crossings (Fraxanet et al., 2020). In non-Hermitian long-range SSH models, finite-neighbor long-range effects can produce higher winding numbers, while infinite-neighbor long-range effects can produce fractional topological invariants analyzed through pseudo-spin vectors (Kartik et al., 2022).

Critical behavior is similarly nonuniform across models. In long-range topological superconductors driven across a phase transition by a linear ramp of the chemical potential, the excitation density scales as $J_2$9, with $A$0 determined by the decay exponent of the pairing potential, and in the hopping-dominated regime the dynamical scaling can be anomalous and unrelated to equilibrium scaling (Defenu et al., 2019). Yet in the long-range critical fermionic chain studied in 2024, the critical behavior belongs to the free Majorana fermion universality class, which is stated to be entirely different from the long-range universality class in usual long-range spin models (Zhong et al., 2024). Taken together, these results show that “long-range” does not define a single universality class or a single bulk-boundary scenario.

6. Extensions across platforms and disciplines

Long-range topological potentials have been studied in band insulators, superconductors, spin chains, photonic systems, Rydberg arrays, and molecular self-assembly. The reported effects differ substantially across platforms.

System	Long-range structure	Reported effect
Generalized SSH chain (Simonyan et al., 29 Sep 2025)	Exponentially decaying $A$1-$A$2 couplings	Weak couplings can trigger a transition if range is large enough
Long-range Kitaev chain (Patrick et al., 2016)	Infinite-range and decaying long-range couplings	Zero and non-zero energy Majorana end modes depending on boundary conditions
Spin-1 chain (Gong et al., 2015)	$A$3 interactions	SPT phase survives frustrated interactions for all $A$4; unfrustrated interactions destroy it for $A$5
Cluster Ising chain (Yang et al., 2024)	Long-range antiferromagnetic interactions	Algebraic SPT phase and topological Gaussian universality
2D $A$6-wave superconductor (Viyuela et al., 2017)	Long-range hopping and pairing	Enhanced topological chiral phase or nonlocal gapped edge states
Rydberg arrays (Raupach et al., 27 Jan 2026)	Dipole-dipole long-range couplings	Enhanced gaps and robust topological quantum state transfer
Protein self-assembly (Spirandelli et al., 21 Aug 2025)	Weighted total persistence energetic term	Up to sixteen-fold improvement in assembly success rate

Several patterns recur. Long-range interactions can enlarge topological regions, generate new winding sectors, modify edge-mode energies, and enhance spectral gaps useful for protocols such as quantum state transfer (Raupach et al., 27 Jan 2026). They can also induce gapless topological behaviors, including an algebraic topological phase and a topological Gaussian universality in a cluster Ising chain with long-range antiferromagnetic interactions (Yang et al., 2024). In two-dimensional chiral superconductors, long-range couplings can greatly enlarge one chiral phase and generate nonlocal gapped edge states that remain robust and separated from the bulk (Viyuela et al., 2017).

Outside condensed-matter Hamiltonians, the phrase acquires a literal energetic meaning. In protein self-assembly, the long-range topological potential is

$A$7

where $A$8 is the total persistence in homology dimension $A$9, and the combined energy

$B$0

is used as an active bias in simulation (Spirandelli et al., 21 Aug 2025). This usage is not a topological band invariant, but it preserves the same structural idea: a topology-derived nonlocal term reshapes an otherwise rugged landscape.

The 2025 SSH result gives the concept a particularly clear form: topology can be tuned by interaction range itself (Simonyan et al., 29 Sep 2025). Higher-dimensional generalizations may allow even richer phase diagrams, especially as the number of potential long-range interacting neighbors grows (Simonyan et al., 29 Sep 2025).