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Dynamics of Chiral Fermions



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Onder, Kaan 


This thesis studies the dynamics of a variety of two and four dimensional quantum field theories containing Weyl fermions respecting some chiral symmetry. There are severe challenges in lattice regularising such theories and chiral gauge theories, where a non-anomolous chiral symmetry is gauged, can display interesting strong coupling dynamics such as confinement without chiral symmetry breaking. We explore such gauge dynamics in two and four dimensions using a variety of different techniques. Furthermore, we use tensor network methods to study chiral fermions on the lattice in two dimensions.

We start by studying the dynamics of chiral SU(N) gauge theories in four dimensions. These contain Weyl fermions in the symmetric or anti-symmetric representation of the gauge group, together with further fermions in the fundamental and anti-fundamental. We revisit an old proposal of Bars and Yankielowicz who match the ‘t Hooft anomalies of this theory to free fermions. We show that there are novel and, in some cases, quite powerful constraints on the dynamics in the large N limit. In addition, we study these SU(N) theories with an extra Weyl fermion transforming in the adjoint representation. Here we show that all 21 ‘t Hooft anomalies for global symmetries are matched with those of a Spin(8) gauge theory. This suggests a non-supersymmetric extension of the duality of Pouliot and Strassler. We then discuss some non-supersymmetric dualities with vector-like matter content for SO(N) and Sp(N) gauge theories and the constraints imposed by Weingarten inequalities.

We then move on to study the dynamics of analogous chiral gauge theories in two dimensions which also contain Weyl fermions in the symmetric, antisymmetric, and fundamental representations. A consistent infrared limit of these theories consists of certain coset conformal field theories. There is also a free-fermion phase which shares the same central charge and ’t Hooft anomalies but does not coincide with the coset models. We show that these two theories sit on a conformal manifold of infrared theories and are related by a current-current deformation. We further consider extensions of these theories by adding Dirac fermions and comment on possible renormalization group flows.

Finally, we present matrix product state simulations of the 3450 lattice chiral fermion model in two dimensions. We consider a lattice setup introduced by Wang and Wen which realises two left and two right-moving fermions on one edge of a thin Chern insulator. The partner mirror fermions are localised on the opposing edge. We turn on symmetry preserving six-fermion gapping interactions on one edge of the Chern insulator to gap the doublers whilst preserving an anomaly free U(1) chiral symmetry and leaving the opposing edge gapless. This provides a candidate lattice regularisation of chiral fermions. We present numerical results for the entanglement entropy scaling to extract the central charge and study excited states by using a quasi-particle ansatz. We observe a BKT transition at six-fermion coupling strength g=7 and a central charge c=2 scaling regime at g=25. We conclude by discussing subtleties of working with symmetric MPS at a fixed charge density.





Tong, David


Chiral Fermion, Quantum Field Theory


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge
Cambridge Trust