The discovery of magnetic and compositional effects in the low temperature properties of multi-component glasses has prompted the need to extend the standard two-level systems (2LSs) tunneling model. A possible extension cite{Jug2004} assumes that a subset of tunneling quasi-particles is moving in a three-welled potential (TWP) associated with the ubiquitous inhomogeneities of the disordered atomic structure of the glass. We show that within an alternative, cellular description of the intermediate-range atomic structure of glasses the tunneling TWP can be fully justified. We then review how the experimentally discovered magnetic effects can be explained within the approach where only localized atomistic tunneling 2LSs and quasi-particles tunneling in TWPs are allowed. We discuss the origin of the magnetic effects in the heat capacity, dielectric constant (real and imaginary parts), polarization echo and SQUID magnetization in several glassy systems. We conclude by commenting on a strategy to reveal the mentioned tunneling states (2LSs and TWPs) by means of atomistic computer simulations and discuss the microscopic nature of the tunneling states in the context of the potential energy landscape of glass-forming systems.
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