On the spectral geometry of Liouville quantum gravity.
The talk will be given in Zoom: Link
In these talks we will discuss the spectral geometry of the Laplace-Beltrami operator associated to Liouville quantum gravity. Over the course of two lectures, our goals will be to:
- Explain how eigenvalues and eigenfunctions for LQG are defined;
- Show that the eigenvalues a.s. obey a Weyl law (joint work with Mo Dick Wong). This is closely related to the short time asymptotics of the LQG heat kernel;
- Discuss the second term in the Weyl law and its relation to the KPZ (Knizhnik-Polyakov-Zamolodchikov) scaling relation;
- Finally we will talk about some conjectures which suggest a rather beautiful connection to a phenomenon called quantum chaos.
Majoranas, parafermions, and what one can do with them
This talks is postponed till October meeting of the seminar.
In the first part we study the effect of interactions on Kitaev's toy model for Majorana wires [1]. To this end we map the model onto the axial next-nearest neighbour Ising chain and discuss the link between spinless fermions and Majoranas. We demonstrate that even though strong repulsive interaction eventually drive the system into a Mott insulating state the competition between the (trivial) band-insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions. We show that the exact ground states can be obtained analytically even in the presence of interactions when the chemical potential is tuned to a particular function of the other parameters [2]. Finally, we investigate the effect of disorder [3].
In the second part we generalise our analysis to parafermions and clock variables, with the Jordan-Wigner transformation being replaced by the so-called Fradkin-Kadanoff one. The resulting parafermion chain is shown to be equivalent to the non-chiral Z3 axial next-nearest neighbour Potts model. The phase diagram contains several gapped phases, including a topological phase where the system possesses three (nearly) degenerate ground states, and a gapless Luttinger-liquid phase [4]. We also extent Witten’s conjugation argument [5] to spin chains and use it to construct various frustration-free models [6]. If time permits, we may also briefly discuss Fock parafermions [7], which generalise spinless fermions to Z3 symmetry.
References:[1] F Hassler and D Schuricht, New J. Phys. 14, 125018 (2012)
[2] H Katsura, D Schuricht and M Takahashi, Phys. Rev. B 92, 115137 (2015)
[3] N M Gergs, L Fritz and D Schuricht, Phys. Rev. B 93, 075129 (2016)
[4] J Wouters, F Hassler, H Katsura and D Schuricht, SciPost Phys. Core 5, 008 (2022)
[5] E. Witten, Nucl. Phys. B 202, 253 (1982)
[6] J Wouters, H Katsura and D Schuricht, SciPost Phys. Core 4, 027 (2021)
[7] E Cobanera and G Ortiz, Phys. Rev. A 89, 012328 (2014)