Tessellations of the hyperbolic spaces by regular polygons support discrete quantum and classical models with unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our Letter delivers a practical and rigorous solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and construct all but one topological models that deliver the topological gaps predicted by the K theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.
Tessellations of the hyperbolic spaces by regular polygons support discrete quantum and classical models with unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our Letter delivers a practical and rigorous solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and construct all but one topological models that deliver the topological gaps predicted by the K theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.
We construct an anomalous Floquet topological phase on a negatively curved hyperbolic lattice. The model is a tight-binding Hamiltonian with a periodically repeated four-color edge-hopping sequence and a sublattice-staggered onsite potential step. The topological regime is reached near the limit in which a single hopping step transfers amplitude completely across an active edge, while the trivial regime is reached near the point where two full hops occur along an active edge during a single hopping step, returning the amplitude to its starting site. In finite open patches, the topological regime is characterized by bulk quasienergy gaps at 0 and π that are populated by in-gap states, in contrast to a trivial regime where these gaps remain empty. Using compact periodic lattices, we map the bulk 0 and π quasienergy gaps and identify the gapped regions connected to the trivial and anomalous open-boundary spectra. We diagnose the in-gap states as chiral boundary modes by their real-space dynamics. Finally, we introduce a small-boundary spectral-flow diagnostic based on punctured periodic hyperbolic lattices, which avoids the ambiguity associated with the extensive outer boundary of finite hyperbolic patches. This puncture-based diagnostic should be useful for studying other topological hyperbolic systems.
Symmetry-protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap.Here we introduce a notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions.More specifically, given a symmetric boundary termination we say two bulk Hamiltonians belong to distinct boundary obstructed topological phases (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but cannot be deformed to each other in the open system without closing the gap at at least one high-symmetry surface.BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus, but, similar to SPTs, they are associated with boundary signatures in open systems such as surface states or fractional corner charges.In contrast to SPTs, these boundary signatures are not anomalous and can be removed by symmetrically adding lower-dimensional SPTs on the boundary, but they are stable as long as the spectral gap at high-symmetry edges/surfaces remains open.We show that the double-mirror quadrupole model of [W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017)] is a prototypical example of such phases, and present a detailed analysis of several aspects of boundary obstructions in this model.In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges.Furthermore, we provide a complete characterization of boundary obstructed phases in terms of symmetry representations.Namely, two distinct BOTP phases correspond to equivalent band representations in the periodic system which become inequivalent upon restricting the symmetry group to that of the open system.This is used to shown that for a given open boundary, there is only one class of BOTPs which corresponds to a local representation of the symmetry of the open system and thus can be designated as the trivial phase.All other BOTP classes do not correspond to local representation of the open system and as a result necessarily exhibit a filling anomaly or gapless surface states.
We report our systematic construction of the lattice Hamiltonian model of topological orders on open surfaces, with explicit boundary terms. We do this mainly for the Levin-Wen string-net model. The full Hamiltonian in our approach yields a topologically protected, gapped energy spectrum, with the corresponding wave functions robust under topology-preserving transformations of the lattice of the system. We explicitly present the wavefunctions of the ground states and boundary elementary excitations. The creation and hopping operators of boundary quasi-particles are constructed. It is found that given a bulk topological order, the gapped boundary conditions are classified by Frobenius algebras in its input data. Emergent topological properties of the ground states and boundary excitations are characterized by (bi-) modules over Frobenius algebras.
Abstract Whether two boundary conditions of a two-dimensional topological order can be continuously connected without a phase transition in between remains a challenging question. We tackle this challenge by constructing an effective Hamiltonian of anyon interaction that realizes such a continuous deformation. At any point along the deformation, the model remains a fixed point model describing a gapped topological order with gapped boundaries. That the deformation retains the gap is due to the anomaly cancelation between the boundary and bulk. Such anomaly inflow is quantitatively studied using our effective Hamiltonian. We apply our method of effective Hamiltonian to the extended twisted quantum double model with boundaries (constructed by two of us in ref. [1]). We show that for a given gauge group G and a three-cocycle in H 3[G, U(1)] in the bulk, any two gapped boundaries for a fixed subgroup K ⊆ G on the boundary can be continuously connected via an effective Hamiltonian. Our results can be straightforwardly generalized to the extended Levin-Wen model with boundaries (constructed by three of us in ref. [2]).
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.
The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.