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Topological order, mixed states and open systems

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 Added by Paolo Facchi
 Publication date 2019
  fields Physics
and research's language is English




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The role of mixed states in topological quantum matter is less known than that of pure quantum states. Generalisations of topological phases appearing in pure states had received only quite recently attention in the literature. In particular, it is still unclear whether the generalisation of the Aharonov-Anandan phase for mixed states due to Uhlmann plays any physical role in the behaviour of the quantum systems. We analyse from a general viewpoint topological phases of mixed states and the robustness of their invariance. In particular, we analyse the role of these phases in the behaviour of systems with a periodic symmetry and their evolution under the influence of an environment preserving its crystalline symmetries.



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227 - Umberto Lucia 2011
The principle of maximum irreversible is proved to be a consequence of a stochastic order of the paths inside the phase space; indeed, the system evolves on the greatest path in the stochastic order. The result obtained is that, at the stability, the entropy generation is maximum and, this maximum value is consequence of the stochastic order of the paths in the phase space, while, conversely, the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation at the stability.
Bilayer quantum Hall (BLQH) systems, which underlie a $U(4)$ symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian $mathbb G_2^4=U(4)/U(2)^2$, orthonormal basis, $U(4)$ generators and their matrix elements in the reproducing kernel Hilbert space $mathcal H_lambda(mathbb G_2^4)$ of analytic square-integrable holomorphic functions on $mathbb G_2^4$, which carries a unitary irreducible representation of $U(4)$ with index $lambdainmathbb N$. A many-body representation of the previous construction is introduced through an oscillator realization of the $U(4)$ Lie algebra generators in terms of eight boson operators. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index $lambda$ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor $ u=2$. The simpler, and better known, case of spin-$s$ CS on the Riemann-Bloch sphere $mathbb{S}^2=U(2)/U(1)^2$ is also treated in parallel, of which Grassmannian $mathbb G_2^4$-CS can be regarded as a generalized (matrix) version.
A topological measure characterizing symmetry-protected topological phases in one-dimensional open fermionic systems is proposed. It is built upon the kinematic approach to the geometric phase of mixed states and facilitates the extension of the notion of topological phases from zero-temperature to nonzero-temperature cases. In contrast to a previous finding that topological properties may not survive above a certain critical temperature, we find that topological properties of open systems, in the sense of the measure suggested here, can persist at any finite temperature and disappear only in the mathematical limit of infinite temperature. Our result is illustrated with two paradigmatic models of topological matter. The bulk topology at nonzero temperatures manifested as robust mixed edge state populations is examined via two figures of merit.
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427 - Umberto Lucia 2011
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