Condensed Matter

Twisted bilayer graphene (tBLG) is a host to a variety of electronic phases, most notably superconductivity when doped away from putative correlated insulator phases. In order to understand the nature of those phases, numerical simulations such as Hartree-Fock calculation and density matrix renormalization group (DMRG) techniques are essential.

Due to the long-range Coulomb interaction and its fragile topology, however, tBLG is difficult to study with standard DMRG techniques.

In this work, we present how a recently developed MPO compression algorithm can be used to make the problem tractable, and how 1D Wannier localization can be used to circumvent the fragile topology.

As a test case, we apply this technique to the toy model of spinless/single-valley model of tBLG. We find that the ground state is essentially a k-space Slater determinant, confirming the validity of previous Hartree-Fock calculations. If time permits, I will also present our ongoing effort to apply this technique to spinful/valleyful model for tBLG.

We construct a minimal theory describing the optical activity of a thin sheet of a twisted material, the simplest example of which is twisted bilayer graphene. We introduce the notion of "twisted electrical conductivity" which parametrizes the parity-odd response of a thin film to a perpendicularly falling electromagnetic wave with wavelength larger than the thickness of the sheet, and relate the chiral response to this kinetic coefficient. We show that the low-frequency Faraday rotation angle has different behaviors in different phases: \omega^2 for insulators and \omega^0 for superconductors. In both cases the frequency dependence of the Faraday rotation angle can be obtained from a simple power counting in an effective field theory. In the metallic state, the twisted conductivity is proportional to the "magnetic helicity" (scalar product of the velocity and the magnetic moment) of the quasiparticle, averaged around the Fermi surface. Many aspects of the theory are general and applicable to strongly correlated phases.

Reference: Dung X. Nguyen and DTS, arXiv:2008.02812.

Majorana zero modes have attracted much interest in recent years because of their promising properties for topological quantum computation. A key question in this regard is how fast two Majoranas can be exchanged giving rise to a unitary gate operation. In this presentation I will first explain that the transport of Majoranas in one-dimensional topological superconductors can be formulated as a “simple” optimal control optimization problem for which we propose several different control regimes. Next I will discuss the optimization methods, Differential Programming and Natural Evolution Strategies, that were applied to the Majorana control problem and came up with a counter-intuitive transport strategy. This strategy, which we dubbed jump-move-jump, will form the focus of the last part of the presentation in which I explain the key underlying mechanisms behind the strategy by reformulating the motion of Majoranas in a moving frame. I will conclude by arguing that these results demonstrate that machine learning for quantum control can be applied efficiently to quantum many-body dynamical systems with performance levels that make it relevant to the realization of large-scale quantum technology.

The interplay between topology, strong correlations, and kinetic energy presents a new challenge for the theory of quantum matter. In this talk I will describe some recent progress on understanding a simple class of problems where these effects can all be analytically handled. I will first present results on a microscopic lowest Landau theory of the composite fermi liquid state of bosons at filling 1.  Building on work from the 1990s I will derive an effective field theory for this system that takes the form of a non-commutative field theory. I will show that an approximate mapping of this theory to a commutative field theory yield the familiar Halperin-Lee-Read action but with parameters correctly described by the interaction strength. I will describe the effect of a finite bandwidth  introduced to the Landau Level and describe the evolution between the composite Fermi liquid and a boson superfluid. Time permitting, I will describe some generalizations that will include the evolution between a Quantum Anomalous Hall state and a Landau Fermi liquid that may be experimentally accessible in moire graphene systems.