![]() In the particular case of, the degeneracies occur at points in the parameter space, which are generically dubbed Weyl singularities. ![]() Indeed, three independent variables are needed to fine tune degeneracies in the energy spectrum, which are known to act as sources or sinks of Berry curvature. Non-trivial averaged conductances generically arise in networks with at least three independent fluxes, a dimensionality constraint mathematically rooted in the co-dimension theorem of Von Neumann and Wigner . 1(a), but rather the topology of the whole network itself : the leads connecting each vertex, the number of holes and their geometrical layout. Interestingly, these integers do not necessarily classify a property of a given physical sample, as in the quantum Hall setup of fig. The value of this topological number depends on the remaining (unaveraged) fluxes with a Φ 0-periodicity. Remarkably, when averaging them over a square slice in flux space, they become quantized in units of ,Īnd that the persistent current term averages to zero. These transport coefficients relate the current flowing around loop k to the emf around loop l. If the fluxes vary linearly in time, a constant electromotive force (emf) is generated around each loop, allowing for the definition of a conductance matrix with elements given by. The first term in eq. ( 1) can be identified as a persistent current, which may be present even in the absence of the driving. Such phase rigidity is expected to be present in superconductors over macroscopic lengths and in normal metals under more stringent conditions on length and temperature scales. As pointed out in ref. , this transport analysis strongly relies on quantum coherence of the wave function over the entire multiply connected device. The above derivation requires the adiabatic postulate to hold, which can be insured for sufficiently slow driving protocols and as long as there is a gap between the ground state of the system and the excitation spectrum. Is the Berry curvature of the corresponding eigenstate evaluated at -assumed to be non-degenerate. In ref. concrete Hamiltonian models describing such networks were analyzed, either by considering the vertices of each graph as molecular sites with a tight-binding–like description or by analyzing the dynamics of free particles moving along one-dimensional wires in the system . These carriers are constrained to move along the links of the network, and their dynamics is governed by an appropriate Schrödinger equation defined in the corresponding multiply connected domain. The synthetic parameter space defined by these fluxes can be identified with a D-dimensional torus with a fundamental period given by the flux unit, where q is the charge of the carriers in the device. 1(b)) with D loops threaded by independent, externally controllable, flux tubes. These devices can be represented as graphs (see, for example, fig. Soon afterwards, Avron and coworkers extended the study of topologically quantized transport coefficients to the realm of mesoscopic quantum networks . ![]() The angular variables defining such a torus were identified as two Aharonov-Bohm fluxes ( and ) threading the physical system and the Hall conductance as the average of the Berry curvature of the ground-state wave function in that parameter space (also known as the many-body Chern number). A couple of years later, these arguments were generalized to take into account the presence of many-body interactions and disorder by considering the Hall sample to live in a torus (two loop) geometry , as the one sketched in fig. The remarkably robust quantization of the Hall conductance of two-dimensional electron gases at high magnetic fields in units of was initially understood in terms of the topological properties of the one-particle Bloch bands in quasimomentum space . Many of the foundational concepts of the field of topological condensed matter physics have been developed during the 1980s after the experimental discovery of the quantum Hall effect .
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