The program of matrix product states on the infinite tensor product ${mathcal A}^{otimes mathbb Z}$, initiated by Fannes, Nachtergaele and Werner in their seminal paper Commun. Math. Phys. Vol. 144, 443-490 (1992), is re-assessed in a context where $
mathcal A$ is an infinite nuclear $C^ast$-algebra. While this setting presents new technical challenges, fine advances on ordered spaces by Kavruk, Paulsen, Todorov and Tomforde enabled us to push through most of the program and to demonstrate that the matrix product states accept generalizations as operator product states.
We consider the algebra $dot{mathfrak H}(mathcal L)$ of inner-limit derivations over the ${rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $mathcal L$. Under standard physical assumptions such as finite interaction range, Galil
ean invariance and continuity with respect to $mathcal L$, we demonstrate that $dot{mathfrak H}(mathcal L)$ can be completed to a groupoid-solvable pro-$C^ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^ast$-algebras for applications in the context of interacting fermions.
The bulk-boundary and a new bulk-defect correspondence principles are formulated using groupoid algebras. The new strategy relies on the observation that the groupoids of lattices with boundaries or defects display spaces of units with invariant accu
mulation manifolds, hence they can be naturally split into disjoint unions of open and closed invariant sub-sets. This leads to standard exact sequences of groupoid $C^ast$-algebras that can be used to associate a Kasparov element to a lattice defect and to formulate an extremely general bulk-defect correspondence principle. As an application, we establish a correspondence between topological defects of a 2-dimensional square lattice and Kasparovs group $KK^1 (C^ast(mathbb Z^3),mathbb C)$. Numerical examples of non-trivial bulk-defect correspondences are supplied.
Quantum Hall effect (QHE) has been theoretically predicted in 4-dimensions and higher. In hypothetical $2n$-dimensions, the topological characters of both the bulk and the boundary are manifested as quantized non-linear transport coefficients that re
late to higher Chern numbers of the bulk gap projection and winding numbers of the Weyl spectral singularities on the $(2n-1)$-dimensional boundaries. Here, we introduce the concept of phason engineering in metamaterials and use it to access QHE in arbitrary dimensions. We fabricate a re-configurable 2-dimensional aperiodic acoustic crystal displaying the 4D QHE and we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as function of the quasi-momenta. We also demonstrate novel topological wave steering enabled by this Weyl singularity.
Quasi-periodic quantum spin chains were recently found to support many topological phases in the finite magnetization sectors. They can simulate strong topological phases from class A in arbitrary dimension that are characterized by first and higher
order Chern numbers. In the present work, we use those findings to generate topological phases at finite magnetization densities that carry first Chern numbers. Given the reduced dimensionality of the spin chains, this provides a unique opportunity to investigate the bulk-boundary correspondence as well as the stability and quantization of the Chern number in the presence of interactions. The later is reformulated using a torus action on the algebra of observables and its quantization and stability is confirmed by numerical simulations. The relations between Chern values and the observed edge spectrum are also discussed.
Topological phases supported by quasi-periodic spin-chain models and their bulk-boundary principles are investigated by numerical and K-theoretic methods. We show that, for both the un-correlated and correlated phases, the operator algebras that gene
rate the Hamiltonians are non-commutative tori, hence the quasi-periodic chains display physics akin to the quantum Hall effect in two and higher dimensions. The robust topological edge modes are found to be strongly shaped by the interaction and, generically, they have hybrid edge-localized and chain-delocalized structures. Our findings lay the foundations for topological spin pumping using the phason of a quasi-periodic pattern as an adiabatic parameter, where selectively chosen quantized bits of magnetization can be transferred from one edge of the chain to the other.
It is shown that twisted $n$-layers have an intrinsic degree of freedom living on $2n$-tori, which is the phason supplied by the relative slidings of the layers and that the twist generates pseudo magnetic fields. As a result, twisted $n$-layers host
intrinsic higher dimensional topological phases and those characterized by second Chern numbers can be found in a twisted bi-layer. Indeed, our investigation of phononic lattices with interactions modulated by a second twisted lattice reveals Hofstadter-like spectral butterflies in terms of the twist angle, whose gaps carry the predicted topological invariants. Our work demonstrates how multi-layered systems are virtual laboratories for studying the physics of higher dimensional quantum Hall effect and how to generate topological edge chiral modes by simply sliding the layers relative to each other. In the context of classical metamaterials, both photonic and phononic, these findings open a path to engineering topological pumping via simple twisting and sliding.
A Thouless pump can be regarded as a dynamical version of the integer quantum Hall effect. In a finite-size configuration, such topological pump displays edge modes that emerge dynamically from one bulk-band and dive into the opposite bulk-band, an e
ffect that can be reproduced with both quantum and classical systems. Here, we report the first un-assisted dynamic energy transfer across a metamaterial, via pumping of such topological edge modes. The system is a topological aperiodic acoustic crystal, with a phason that can be fast and periodically driven in adiabatic cycles. When one edge of the metamaterial is excited in a topological forbidden range of frequencies, a microphone placed at the other edge starts to pick up a signal as soon as the pumping process is set in motion. In contrast, the microphone picks no signal when the forbidden range of frequencies is non-topological.
We use magnetic flux-tubes to stabilize zero-energy modes in a lattice realization of a 2-dimensional superconductor from class D of classification table of topological condensed matter systems. The zero modes are exchanged by slowly displacing the f
lux-tubes and an application of the adiabatic theorem demonstrates the geometric nature of the resulting unitary time-evolution operators. Furthermore, an explicit numerical evaluation reveals that the evolutions are in fact topological, hence supplying a representation of the braid group, which turns out to be non-abelian. This physical representation is further formalized using single-strand planar diagrams. Lastly, we discuss how these predictions can be implemented with and observed in classical meta-materials and how the standard Majorana representation of the braid group can be generated by measuring derived physical observables.
Non-trivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one and two-dimensional topological superconductors. Here, we generate such representations with topologically protected domain-wall m
odes in a classical analogue of the Kitaev superconducting chain, with a particle-hole like symmetry and a Z2 topological invariant. The mid-gap modes are found to exhibit distinct fusion channels and rich non-Abelian braiding properties, which are investigated using a T-junction setup. We employ the adiabatic theorem to explicitly calculate the braiding matrices for one and two pairs of these mid-gap topological defects.
