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| 2. |
- Hannukainen, Julia D., et al.
(författare)
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Local Topological Markers in Odd Spatial Dimensions and Their Applicationto Amorphous Topological Matter br
- 2022
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Ingår i: Physical Review Letters. - : American Physical Society (APS). - 0031-9007 .- 1079-7114. ; 129:27
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Tidskriftsartikel (refereegranskat)abstract
- Local topological markers, topological invariants evaluated by local expectation values, are valuable forcharacterizing topological phases in materials lacking translation invariance. The Chern marker-the Chernnumber expressed in terms of the Fourier transformed Chern character-is an easily applicable local markerin even dimensions, but there are no analogous expressions for odd dimensions. We provide general analyticexpressions for local markers for free-fermion topological states in odd dimensions protected by localsymmetries: aChiral marker, a localZmarker which in case of translation invariance is equivalent to thechiral winding number, and aChern-Simons marker, a localZ2marker characterizing all nonchiral phases inodd dimensions. We achieve this by introducing a one-parameter familyP theta of single-particle densitymatrices interpolating between a trivial state and the state of interest. By interpreting the parameter theta as anadditional dimension, we calculate the Chern marker for the familyP theta. We demonstrate the practical use ofthese markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions:a topological superconductor (Zclassification) and a topological insulator (Z2classification).
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| 3. |
- Hansson, Thors Hans, et al.
(författare)
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Effective field theories for topological states of matter
- 2020
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Ingår i: Springer Proceedings in Physics. - Cham : Springer. - 0930-8989 .- 1867-4941. ; 239, s. 1-68
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Tidskriftsartikel (refereegranskat)abstract
- Since the discovery of the quantum Hall effect in the 1980s it has been clear that there exists states of matter characterized by subtle quantum mechanical effects that renders certain properties surprisingly stable against dirt and noise. The theoretical understanding of these topological quantum phases have continued to develop during the last few decades and it has really surged after the discovery of the time-reversal invariant topological insulators. There are many examples of topological phases that have been important for the theoretical understanding of topological states of matter as well as being of great physical relevance. In this chapter we will focus on some examples that we find particularly enlightening and relevant, but we will not make a complete classification. Some of the most important tools for the understanding of topological quantum matter are based on effective field theory methods. We shall employ two different types of effective field theories. The first, which is valid at intermediate length and time-scales, will not capture the physics at microscopic scales. Such theories are the analogs, for topological phases, of the Ginzburg–Landau theories used to describe the usual symmetry breaking non-topological phases. The second type of theories describe the physics on scales where non-topological gapped states would be very boring, namely at distances and times much larger than the correlation length and the time set by the inverse gap. On these scales everything is independent of any distance and the theories will be topological field theories, which do not describe any dynamics in the bulk, but do carry information about topological properties of the excitations, and also about excitations at the boundaries of the system. Finally, we will also study effective response actions. In a strict sense these are not effective theories, since they do not have any dynamical content, but encode the response of the system to external perturbations, typically an electromagnetic field. As we shall see, however, the effective response action for topological states can be used to extract parts of the dynamic theory through a method called functional bosonization.
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- Klein Kvorning, Thomas, et al.
(författare)
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Nonlocal order parameters for states with topological electromagnetic response
- 2020
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Ingår i: Physical Review B. - 2469-9950 .- 2469-9969. ; 101:20
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Tidskriftsartikel (refereegranskat)abstract
- Chern insulators are states of matter characterized by their quantized Hall conductance but also by their singular response to monopole configurations of an external electromagnetic field. In this paper, we exploit this response to provide a classification for these states. We demonstrate that for each Chern-insulator state, including the trivial state, the response defines an associated operator that can be interpreted as the insertion of local charges and monopoles at two points separated in space. This operator decays algebraically in the monopole separation distance only for its associated state but exponentially in all other states. Crucially, the operators do not depend on any microscopic properties, and therefore constitute a general set of nonlocal order parameters. We support this claim with numerical evaluation of the order parameters in a simple lattice model, and find excellent agreement. Our construction is well suited for generalization to other states with topological electromagnetic response, and we use the states with a quantized magnetoelectric effect in three dimensions as an example. Aside from providing insights into topological states of matter, our construction can also be exploited to efficiently diagnose such states numerically.
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| 5. |
- Klein Kvorning, Thomas, et al.
(författare)
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Time-evolution of local information : Thermalization dynamics of local observables
- 2022
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Ingår i: SciPost Physics. - : Stichting SciPost. - 2542-4653. ; 13:4
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Tidskriftsartikel (refereegranskat)abstract
- Quantum many-body dynamics generically result in increasing entanglement that eventually leads to thermalization of local observables. This makes the exact description of the dynamics complex despite the apparent simplicity of (high-temperature) thermal states. For accurate but approximate simulations one needs a way to keep track of essential (quantum) information while discarding inessential one. To this end, we first introduce the concept of the information lattice, which supplements the physical spatial lattice with an additional dimension and where a local Hamiltonian gives rise to well-defined locally conserved von Neumann information current. This provides a convenient and insightful way of capturing the flow, through time and space, of information during quantum time-evolution, and gives a distinct signature of when local degrees of freedom decouple from long-range entanglement. As an example, we describe such de-coupling of local degrees of freedom for the mixed-field transverse Ising model. Building on this, we secondly construct algorithms to time-evolve sets of local density matrices without any reference to a global state. With the notion of information currents, we motivate algorithms based on the intuition that information for statistical reasons flows from small to large scales. Using this guiding principle, we construct an algorithm that, at worst, shows two-digit convergence in time-evolutions up to very late times for diffusion process governed by the mixed-field transverse Ising Hamiltonian. While we focus on dynamics in 1D with nearest-neighbor Hamiltonians, the algorithms do not essentially rely on these assumptions and can in principle be generalized to higher dimensions and more complicated Hamiltonians.
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