Universität zu Köln

Arenz, Julian (2026). Field theory for Anderson transitions in high dimensions. PhD thesis, Universität zu Köln.

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Abstract

Disordered electron systems can undergo a quantum phase transition, known as the Anderson transition, from a metallic phase for weak disorder to an insulating phase at strong disorder. In low dimensions (d = 2 + ε), an efficient tool for studying the transition is an effective field theory called the nonlinear σ-model. The renormalization group treatment of this effective model supports a one-parameter scaling hypothesis for the Anderson transition, where the conductance is the only relevant scaling variable. In the present work, we investigate the Anderson transition in the opposite case of high lattice dimensions. Here the critical point moves into the strong coupling regime, and a field-theoretical description, similar to the nonlinear σ−model in low dimensions, is lacking at present. The plan here is to identify a field-theoretical mechanism that describes the critical regime in high dimensions. The main finding is a distinct mechanism of spontaneous symmetry breaking. In the metallic phase, the hyperbolic symmetry of the field theory breaks down to a maximal compact subgroup, whereas at criticality it breaks down to a nilpotent subgroup. This distinct symmetry-breaking mechanism naturally leads to a singular continuous spectrum of the underlying random Hamiltonian–a spectral type that has not yet been widely appreciated in the physics literature. The explicit microscopic model we analyze is a variant of the standard Anderson model, namely the Wegner N = 1-orbital model, which belongs to symmetry class A in the Tenfold Way. The model is analyzed using an approximation that goes back to Abou-Chacra, Anderson, Thouless (AAT). In this approximation, one obtains self-consistent equations for advanced and retarded Green’s functions. It is exact on tree-like lattices and is expected to capture the essential physics of Anderson transitions in high lattice dimensions.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Arenz, Julian julian.arenz@web.de UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-806300
Date:	2026
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Physics > Institute for Theoretical Physics
Subjects:	Physics
Uncontrolled Keywords:	Keywords Language Field theory for Anderson transitions in high dimensions UNSPECIFIED Spontaneous symmetry breaking in non-compact field theories UNSPECIFIED Singular continuous spectrum for random Hamiltonians UNSPECIFIED AAT approximation UNSPECIFIED
Date of oral exam:	2 June 2026
Referee:	Name Academic Title Zirnbauer, Martin Prof. Dr. Diehl, Sebastian Prof. Dr. Disertori, Margherita Prof. Dr.
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/80630