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Personal profile

Research interests

Overview

In our everyday lives the behaviour of observed objects does not depend on whether they are identical or not. However, the microscopic world is governed by laws of quantum physics that open up new possibilities for the collective behaviour of identical quantum particles. Research exploring this phenomenon has led to inventing lasers, superconductors and explaining stability of matter around us. Current studies of identical quantum particles are leading to new exciting developments in quantum information and condensed matter physics.

In my work, I aim to provide computationally efficient tools for approximately constructing the so-called generalised Pauli principles. The insights of this work will be subsequently applied to develop next-generation numerical methods of computing the electronic structure of chemical molecules.

The second branch of my work aims to improve our understanding of anyonic quasi-particles that play key roles in fault-tolerant quantum computing. I am focusing on recently discovered simple models of anyons for particles constrained to move on complex networks. The main goal is to provide new robust architectures for quantum computers and investigate long-standing problems concerning the behaviour of anyonic particles in complex quantum materials.

 

PhD Projects 

(in joint supervision with Professor Jonathan Robbins)

Topic

 Effective hamiltonians for anyons on graphs via self-adjoint extensions of the Landau operator

Short description 

This is a project in mathematical physics that mainly involves functional analysis and quantum mechnaics. More specifically, it relies on applications of the theory of self-adjoint extensions to many-body quantum mechanics with a limited use of topology.

Anyons are (quasi)particles that obey so-called fractional quantum statistics – their statis- tical properties are neither bosonic nor fermionic. Such particles are known to exist in two- dimensional systems (2D lattices or thin metallic strips) and one-dimensional systems (quan- tum wires). Much less is known about the behaviour of anyons on networks formed from quantum wires, i.e. on quantum graphs. Formally, we consider a graph Γ as a one-dimensional CW-complex. We form its configuration space, Cn(Γ) by considering the space of all un- ordered tuples of length n that consist of distinct points from Γ, i.e.

Cn(Γ) = (Γn − n)/Sn,

where n {(x1,...,xn) : xi ∈ Γ and xi xj for some ≠ jand Sn is the permutation group. Graph braid group on n strands is defined as the fundamental group of Cn(Γ), Brn(Γ) := π1(Cn(Γ)).

Similarly, one can consider a configuration space of a topological space, XCn(X). Some important special cases are

  • X = R3 – in three dimensional space there are only bosons or fermions (sometimes in disguise). The braid group Brn(R3) is simply the permutation group, Sn.
  • X = R2 - this setting leads to exotic statistics. The nontrivial topology of the configuration space of R2 supports anyons whose fractional statistics is realised in physical models as unitary representations of the planar braid group, Brn(R2). They appear in solid state physics in certain models of superconductors, in fault-tolerant quantum computing and in Chern-Simons theories.

Initial reading:

  • Leinaas, J., M., Myrheim, J. (1977) On the theory of identical particles, Nuovo Cim. 37B, 1-23
  • David Tong, The Quantum Hall Effect, TIFR Infosys Lectures, http://www.damtp. cam.ac.uk/user/tong/qhe/qhe.pdf

Leinaas and Myrheim (1977) show that the dynamics of anyons can be studied by inserting magnetic fluxes in the "holes" of the configuration space. The corresponding hamiltonian is then found via the minimal coupling principle which says that the momentum of kth particle is given by pk = i∂k + Ak, where Ak is the local magnetic potential. The corresponding hamiltonian, H = Σpk2, is called the Landau operator. To solve the time-independent Schrödinger equation, HΨ = EΨ, we first need to find the correct gluing conditions for Ψ corresponding to situations where i) a particle is on a junction of the graph and ii) two particles come close to each other. The mathematical theory that tells us how to find such gluing conditions is the theory of self-adjoint extensions of symmetric operators. The aim is to look at specific families of graphs, starting with the simplest T-junction and then proceeding to general star graphs, the lasso graph, wheel graphs, etc. The project is open-ended.

Education/Academic qualification

Center for Theoretical Physics of the Polish Academy of Sciences

30 Sep 201530 Nov 2018

Award Date: 30 Nov 2018

External positions

Research Assistant, Center for Theoretical Physics of the Polish Academy of Sciences

24 Aug 201430 Jul 2020

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