Siddharth Vadnerkar

Life and education:

I was born in the city of Pune, India (A city that was colloquially known as Oxford of the East, is famous for its constantly pleasant weather and strong Marathi culture)
My undergrad was in Engineering Physics at IIT Delhi, in the gorgeous city of Delhi (Which probably has the best street food in the entire world)
Then I flew to New York to attend Stony Brook University for my Masters in physics (It was quite a culture shock to go from a big Indian City to the village of Stony Brook)
And finally, I've landed at UC Davis for my PhD in physics (and it's quite pleasant to be close to happening places like Tahoe or San Francisco but still be in a smaller town)

Research Interests: 

I am interested in the physical applications of topology in a variety of discrete and continuous systems: From lattice-based implementations of topology in the form of Toric Code and Levin-Wen models, to realizations of topology on continuous systems like Chern Simons theory. To that effect, I am very interested in fields (pun intended) that result from it: quantum computation using anyons, fractional quantum hall effect, SPT and SET phases and so on and so forth. Most of my work is looking at these systems from a mathematically rigorous point of view, which is where my thesis advisor, Bruno Nachtergaele comes in.
Currently, I'm getting interested in finding the lattice interpretation of some incredibly interesting results obtained by Category Theorists. Some basic questions that motivate me: Lattice interpretation of anyons and their higher counterparts, fusion and braiding structures, bulk-boundary correspondence in lattice systems, finding signatures of condensation/confinement etc.

Ongoing projects: 

Papers:

Classification of the anyon sectors of Kitaev's quantum double model (A. Bols, S. Vadnerkar)
Abstract: We give a complete classification of the anyon sectors of Kitaev's quantum double model on the infinite triangular lattice and for finite gauge group $G$, including the non-abelian case. As conjectured, the anyon sectors of the model correspond precisely to the irreducible representations of the quantum double algebra of $G$. Our proof consists of two main parts. In the first part, we construct for each irreducible representation of the quantum double algebra a pure state and show that the GNS representations of these pure states are pairwise disjoint anyon sectors. In the second part we show that any anyon sector is unitarily equivalent to one of the anyon sectors constructed in the first part. Purity of the states constructed in the first part is shown by characterising these states as the unique states that satisfy appropriate local constraints. These constraints are of two types, namely flux constraints and gauge constraints. The flux constraints single out certain string-net states, while the gauge constraints fix the way in which these string-nets condense. At the core of the proof is the fact that certain groups of local gauge transformations act freely and transitively on collections of local string-nets. The proof that the GNS representations of these states are anyon sectors relies on showing that they are unitarily equivalent to amplimorphism representations which are much easier to compare to the ground state representation. For the second part, we show that any anyon sector contains a pure state that satisfies all but a finite number of the constraints characterising the pure states of the first part. Using known techniques we can then construct a pure state in the anyon sector that satisfies all but one of these constraints. Finally, we show that any such state must be a vector state in one of the anyon sectors constructed in the first part.
https://arxiv.org/abs/2310.19661

Superselection sectors in the 3d Toric Code (S. Vadnerkar)
Abstract: We rigorously define superselection sectors in the 3d (spatial dimensions) Toric Code Model on the infinite lattice $\mathbb{Z}^3$. We begin by constructing automorphisms that correspond to infinite flux strings, a phenomenon that's only possible in open manifolds. We then classify all ground state superselection sectors containing infinite flux strings, and find a rich structure that depends on the geometry and number of strings in the configuration. In particular, for a single infinite flux string configuration to be a ground state, it must be monotonic. For configurations containing multiple infinite flux strings, we define "infinity directions" and use that to establish a necessary and sufficient condition for a state to be in a ground state superselection sector. Notably, we also find that if a state contains more than 3 infinite flux strings, then it is not in a ground state superselection sector.
https://arxiv.org/abs/2308.06883

Dynamical abelian anyons with bound states and scattering states  (S. Bachmann, B. Nachtergaele, S. Vadnerkar)
Abstract: We introduce a family of quantum spin Hamiltonians on \mathbb{Z}^2 that can be regarded as perturbations of Kitaev's abelian quantum double models that preserve the gauge and duality symmetries of these models. We analyze in detail the sector with one electric charge and one magnetic flux and show that the spectrum in this sector consists of both bound states and scattering states of abelian anyons. Concretely, we have defined a family of lattice models in which abelian anyons arise naturally as finite-size quasi-particles with non-trivial dynamics that consist of a charge-flux pair. In particular, the anyons exhibit a non-trivial holonomy with a quantized phase, consistent with the gauge and duality symmetries of the Hamiltonian.
https://arxiv.org/abs/2303.07379, journal link: https://doi.org/10.1063/5.0151232

Current news about me:

Nov '23: My third paper! I really should stop counting... arxiv link: https://arxiv.org/abs/2310.19661
Aug '23: Attended the workshop Paths to QFT 2023, Durham and met some great researchers.
Aug '23: Gave a seminar at Cambridge University for the TCM group on the Dynamic Toric Code Model.
Aug '23: Published my second paper! Here is the arxiv link: https://arxiv.org/abs/2308.06883
July '23: Visited PSSCMP 2023 and presented a poster based on the "Dynamic Toric Code Model".
March '23: Published my first paper! Here is the arxiv link: https://arxiv.org/abs/2303.07379
January '23: Gave an informal talk on "Superselection sectors in the 3d Toric Code". Here is the recording.
January '23:  Invited to speak on "Topology in physics" at the physics department colloquium at UC Davis. Here are the slides.
September '22: Attended the "Generalised Global Symmetries" summer conference at the Simons Center. Had an excellent time learning about the applications of UMTCs in HEP/Condensed Matter.
September '22: Gave a talk at QMATH15 on "Dynamic Toric Code Model". Here are the slides.
June '22: Gave a talk at GLaMP '22 on "Dynamic Toric Code model". Here are the slides.
September '21: Attended the "Mathematics of Topological Phases of Matter" summer school, and met some great researchers.
May '21: Passed my qualifying exam! Now I can look forward to publishing with the blessings of the university.
February '20: Gave a talk in the "Quantum Lattice Systems" seminar series on my current research, the "Dynamic Toric Code Model".
September '20: Gave a talk in a seminar series "Quantum Lattice Systems" on "Chern Simons theory and topology in higher dimensions" hosted by Professor Martin Fraas.