#### IMSc Webinar

#### The shift invariant and a new Hofstadter butterfly

#### Naren Manjunath,

##### Department of Physics, Condensed Matter Theory Center, and Joint Quantum Institute, The University of Maryland, College Park, Maryland

*The 'shift' is a universal property of certain gapped topological*

phases of matter, such as the quantum Hall states. It encodes how certain

ground state properties depend on the curvature of the underlying spatial

manifold (for example, a sphere versus a torus). The notion of shift can be

defined for continuum as well as lattice systems in (2+1) dimensions. In

this ongoing work, we numerically obtain the shift in the Hofstadter model

of free fermions on a square lattice with a uniform magnetic field. We do

this in two ways, by computing (i) the charge bound to disclination defects

on the square lattice; (ii) the rotation eigenvalue of the ground state

with an additional uniformly inserted magnetic flux. We find that the shift

forms a 'Hofstadter butterfly' with seemingly fractal properties. Our work

provides an additional characterization of the square lattice Hofstadter

model apart from the usual Hall conductivity, and also suggests a route

towards a complete characterization.

Collaborators: Yuxuan Zhang, Gautam Nambiar, and Maissam Barkeshli (UMCP)

Here is the link to the Google meet:

meet.google.com/ujh-dswt-bso

Done