© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2511.11261

Topological states and flat bands in exactly solvable decorated Cayley trees

Wanda P. Duss, Askar Iliasov, Tomáš Bzdušek

cond-mat.mes-hallcond-mat.stat-mechquant-phSubmitted Nov 14, 2025• Announced Nov 17, 2025• Revised Jul 16, 2026 (v2)

v1 · Nov 14, 2025 v2 · Jul 16, 2026

Abstract

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices; namely of the Lieb, the double Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.

Journal reference: Phys. Rev. Res. 8, 033036 (2026)
DOI: 10.1103/t8gg-xqls
Comments: 35 pages, 24 figures

Related papers

Topological bound states in interacting Su–Schrieffer–Heeger rings

A M Marques, R G Dias

We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.

arXiv:1710.09148•Journal of Physics Condensed Matter•Jun 19, 2018•38 citations

Glued tree lattices with only compact localized states

Andrew Osborne, Ciro Salcedo, Andrew Houck

Flat band physics is a central theme in modern condensed matter physics. By constructing a tight--binding single particle system that has vanishing momentum dispersion in one or more bands, and subsequently including more particles and interactions, it is possible to study physics in strongly interacting regimes. Inspired by the glued trees that first arose in one of the few known examples of quantum supremacy, we define and analyze two infinite families of tight binding single particle Bose--Hubbard models that have only flat bands, and only compact localized states despite having any nonnegative number of translation symmetries. The first class of model that we introduce is constructed by replacing a sufficiently large fraction of the edges in a generic countable graph with glued trees modified to have complex hoppings. The second class arises from thinking of complex weighted glued trees as rhombi that can then be used to tile two dimensional space, giving rise to the familiar dice lattice and infinitely many generalizations thereof, of which some are Euclidian while others are hyperbolic.

arXiv:2503.22843•Mar 28, 2025•0 citations

Lattice Realizations of Flat Gauging and T-duality Defects at Any Radius

Riccardo Argurio, Giovanni Galati, Nathan Godechal

We analyze non-invertible topological interfaces and defects in the two-dimensional compact boson, focusing on the more exotic ones obtained by gauging continuous symmetries with flat connections on a half-space. These include interfaces between mutually irrational radii and T-duality symmetries at arbitrary boson radius. Using the modified Villain discretization on both a Euclidean two-dimensional square lattice and a quantum one-dimensional chain, we show that all these topological interfaces survive discretization and give rise to non-compact edge modes localized at the defect sites. Such non-compact edge modes imply a continuous defect spectrum and an infinite quantum dimension. In the special case of rational radii, we show how the defect action or Hamiltonian can be modified in order to compactify the edge modes and produce more standard defects with finite quantum dimension.

arXiv:2604.09126•Apr 10, 2026•0 citations

Topological Phases on Quantum Trees

Alex Westström, Wenbu Duan, Kangpei Yao +3 more

In this work, we present a theory for topological phases for quantum systems on tree graphs. Conventionally, topological phases of matter have been studied in regular lattices, but also in quasicrystals and amorphous settings. We consider specific generalizations of regular tree graphs, and explore their topological properties. Unlike conventional systems, infinite quantum trees are not finite-dimensional, allowing for novel phenomena. We find a proliferation of topological zero modes present throughout the entire system, indicating that the bulk also acts as a boundary. We then go on to show that only three symmetry classes host stable topological phases in contrast to the usual five symmetry classes per dimension. Finally, we introduce what we call the Su-Schrieffer-Heeger tree which is topologically non-trivial even in the absence of inner degrees of freedom and does not possess any gapped trivial phases. We realize this system in an electronic circuit and show that our theory matches with experiments.

arXiv:2302.03166•Feb 6, 2023•0 citations

Topological quantum phase transitions and topological flat bands on the star lattice

Wen-Chao Chen, Ru Liu, Yi-Fei Wang +1 more

We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the $1/2$ fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.

Physical Review B•Aug 16, 2012•52 citations

Engineering the Bogoliubov modes through geometry and interaction: From collective edge modes to flat-band excitations

Maryam Darvishi, Fatemeh Pouresmaeeli, Saeed H. Abedinpour

We propose a procedure to engineer solid-state lattice models with superlattices of interaction-coupled Bose-Einstein condensates. We show that the dynamical equation for the excitations of Bose-Einstein condensates at zero temperature can be expressed in an eigenvalue form that resembles the time-independent Schr{ö}dinger equation. The eigenvalues and eigenvectors of this equation correspond to the dispersions of the collective modes and the amplitudes of the density oscillations. This alikeness opens the way for the simulation of different tight-binding models with arrays of condensates. We demonstrate, in particular, how we can model a one-dimensional Su-Schrieffer-Heeger lattice supporting topological edge modes and a two-dimensional Lieb lattice with flat-band excitations with superlattices of Bose-Einstein condensates.

arXiv:2408.06125•Physical Review A•Jun 5, 2025•1 citation

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices; namely of the Lieb, the double Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.

We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.

Flat band physics is a central theme in modern condensed matter physics. By constructing a tight--binding single particle system that has vanishing momentum dispersion in one or more bands, and subsequently including more particles and interactions, it is possible to study physics in strongly interacting regimes. Inspired by the glued trees that first arose in one of the few known examples of quantum supremacy, we define and analyze two infinite families of tight binding single particle Bose--Hubbard models that have only flat bands, and only compact localized states despite having any nonnegative number of translation symmetries. The first class of model that we introduce is constructed by replacing a sufficiently large fraction of the edges in a generic countable graph with glued trees modified to have complex hoppings. The second class arises from thinking of complex weighted glued trees as rhombi that can then be used to tile two dimensional space, giving rise to the familiar dice lattice and infinitely many generalizations thereof, of which some are Euclidian while others are hyperbolic.

We analyze non-invertible topological interfaces and defects in the two-dimensional compact boson, focusing on the more exotic ones obtained by gauging continuous symmetries with flat connections on a half-space. These include interfaces between mutually irrational radii and T-duality symmetries at arbitrary boson radius. Using the modified Villain discretization on both a Euclidean two-dimensional square lattice and a quantum one-dimensional chain, we show that all these topological interfaces survive discretization and give rise to non-compact edge modes localized at the defect sites. Such non-compact edge modes imply a continuous defect spectrum and an infinite quantum dimension. In the special case of rational radii, we show how the defect action or Hamiltonian can be modified in order to compactify the edge modes and produce more standard defects with finite quantum dimension.

In this work, we present a theory for topological phases for quantum systems on tree graphs. Conventionally, topological phases of matter have been studied in regular lattices, but also in quasicrystals and amorphous settings. We consider specific generalizations of regular tree graphs, and explore their topological properties. Unlike conventional systems, infinite quantum trees are not finite-dimensional, allowing for novel phenomena. We find a proliferation of topological zero modes present throughout the entire system, indicating that the bulk also acts as a boundary. We then go on to show that only three symmetry classes host stable topological phases in contrast to the usual five symmetry classes per dimension. Finally, we introduce what we call the Su-Schrieffer-Heeger tree which is topologically non-trivial even in the absence of inner degrees of freedom and does not possess any gapped trivial phases. We realize this system in an electronic circuit and show that our theory matches with experiments.

We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the

1/2

fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.

We propose a procedure to engineer solid-state lattice models with superlattices of interaction-coupled Bose-Einstein condensates. We show that the dynamical equation for the excitations of Bose-Einstein condensates at zero temperature can be expressed in an eigenvalue form that resembles the time-independent Schr{ö}dinger equation. The eigenvalues and eigenvectors of this equation correspond to the dispersions of the collective modes and the amplitudes of the density oscillations. This alikeness opens the way for the simulation of different tight-binding models with arrays of condensates. We demonstrate, in particular, how we can model a one-dimensional Su-Schrieffer-Heeger lattice supporting topological edge modes and a two-dimensional Lieb lattice with flat-band excitations with superlattices of Bose-Einstein condensates.