Ola Bratteli

Bio: Ola Bratteli is an academic researcher from University of Oslo. The author has contributed to research in topics: Automorphism & Cuntz algebra. The author has an hindex of 31, co-authored 108 publications receiving 7146 citations. Previous affiliations of Ola Bratteli include University of Ottawa & Research Institute for Mathematical Sciences.

Iterated Function Systems and Permutation Representations of the Cuntz Algebra

Abstract: Introduction Permutative representations of $\mathcal O_N$ Monomial representations and integers modulo $N$ Cycles of irreducible $\mathcal O_N$ representations and their atoms of UHF$_N$ representations Relations of finite type and sub-Cuntz states The shift representation The universal permutative multiplicity-free representation Some specific examples of the cycle and atom structure: The mod $N$ case Some specific examples of the cycle and atom structure: The mod $\mathbf N$ case The general mod $\mathbf N$ situation Concluding remarks Bibliography List of symbols.

Unbounded derivations of C*-algebras

Abstract: We study unbounded derivations ofC*-algebras and characterize those which generate one-parameter groups of automorphisms. We also develop a functional calculus for the domains of closed derivations and develop criteria for closeability. Some specialC*-algebras are considered\(\mathfrak{B}\mathbb{C}(\mathfrak{H}),\mathfrak{B}(\mathfrak{H})\), UHF algebras, and in this last context we prove the existence of non-closeable derivations.

Colloquium: Area laws for the entanglement entropy

Abstract: Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Finitely correlated states on quantum spin chains

Abstract: We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

Non-commutative differential geometry

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Valence Bond Ground States in Isotropic Quantum Antiferromagnets

Abstract: Haldane predicted that the isotropic quantum Heisenberg spin chain is in a “massive” phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.