Master's projects

Exploring the Half-Links Schwinger Model with Tensor Networks

There are many different ways of putting gauge symmetry on the lattice in the Hamiltonian formulation. Quantum Link Models provide one possible route where gauge links are replaced by spins. Choosing different total spins per link gives rise to different models. This offers a wide variety of models which can potentially be addressed by quantum simulation experiments.

A variation of these models, dubbed Half-Links Schwinger (HLS), has been studied extensivelly in the group. In this model, every other gauge link is removed. We have uncovered the phase diagram, i.e. the ground state for different parameters. In this project we will extend explore the real time dynamics using tensor networks. We will explore thermalization or its absence through the presence of Quantum Many-Body Scars. This project will establish a connection between these phenomena in HLS and standard formulation o gauge theories.

Scars and Time evolution in 2 + 1D Gauge Ladders

In our group, we have identified Quantum Many-Body Scars in 2+1D gauge theories. These are special eigenstates that can hinder thermalization in many-body systems under specific initial conditions. It is not yet full understood to what degree of precision one has to prepare the state in order to observe scarring phenomena. In this project, we will employ Tensor Networks, Exact Diagonalization, and analytical methods to investigate the thermalization properties of a diverse range of initial states in ladder systems. We aim at full characterization of the dynamics which can aid experiments and improve our understanding of Quantum Many-Body Scars in gauge theories.

Abelian-Higgs Model at Finite Lattice Spacing

While taking the lattice spacing to zero is, in the context of quantum field theory, the natural thing to do, this does not need to be the case if we are able to implement the lattice theory in a quantum simulator. In that case, the lattice spacing can be seen as a new parameter that experimentalists might be able to control.
In recent works, the phase diagram at finite lattice spacing has been investigated for this model. The results indicate a new critical point where a second order phase transitions occurs. At this critical point, a new field theory emerges and a continuum limit, different from the standard one, can be taken. Specifically, at the critical point, a conformal field theory emerges. This conformal theory has been investigated in these works and some puzzling phenomena has been found. In this project we want to investigate this model in the Lagrangian formulation, testing and extending the results present in the literature though a different formulation.

Complex Langevin and diffusion models

Essential: QFT I, and familiarity with Machine Learning. Proficiency in Python and familiarity with Pytorch.

Langevin dynamics describes the evolution of a system involving fast-varying fluctuating forces. The governing equation, which is called the Langevin equation, is a stochastic differential equation containing both deterministic and random forces. The random force gives a stochastic nature to the dynamics, and the deterministic force shapes the resulting steady-state distribution. Taking advantage of these features, one can use Langevin-based Monte Carlo simulations to draw samples from distributions of interest. In the context of Machine Learning, the diffusion models, which are based on the Langevin method,  are used for image generation. In a similar manner, one can use the diffusion models to generate samples/configurations for lattice QFTs. As an interesting extension, the complex Langevin method has been introduced as an approach to circumvent the numerical sign problem arising in computations involving complex weights. Although it looks very powerful, there have been difficulties in applying the complex Langevin method. In this project, you study the complex Langevin method for a quantum system involving a complex potential. In addition to using conventional complex-Langevin-based Monte Carlo simulations, you investigate constructing a diffusion model to tackle the problem using the power of ML.

This project requires both numerical methods and analytical techniques.

Auto-regressive models for generating lattice QFT configurations

Essential: QFT I, and familiarity with Machine Learning. Proficiency in Python and familiarity with Pytorch.

Many problems in quantum field theories can be solved using Monte Carlo simulations by generating appropriate configurations. Although very powerful, there are limitations in these simulations. Machine Learning based approaches have been proposed to circumvent some of these limitations. In this project, you will explore ML-based models for image and text generations, as well as generation of lattice QFT configurations. In particular, you will investigate auto-regressive models. The goal is to incorporate QFT symmetries, such as translational invariance, in auto-regressive models and apply them to scalar $\phi^4$ and gauge field theories.

This project mainly requires numerical simulations, provides a context for applying your ML & QFT knowledge, and improves your understanding of various theoretical aspects of QFTs.

Non-perturbative renormalization in lattice QCD with $N_\mathrm{f}=4$ Wilson fermions

Essential: QFT I/II, recommended: QED. Proficiency with a programming language like C/python.

The goal here is to define and investigate renormalization schemes for composite operators such as the electromagnetic current using finite-temperature simulations of lattice QCD. Schemes which are both precise and free of systematic uncertainties are essential to compute precision input for the Standard Model of Particle Physics such as the muon anomalous magnetic moment. This project will require both analytic and numerical work.

Multi-level integration for Yang-Mills theory at finite temperature

Essential: QFT I, Proficiency with a programming language like C/python.

Importance sampling algorithms for two-point correlation functions suffer from exponential signal-to-noise ratio problems as a function of the separation between operators. The solution to this is known via multi-level integration algorithms. In this thesis, you will investigate numerically and theoretically the variance-reduction when these algorithms are applied to study Yang-Mills theory at finite temperature.

A declarative language for Wick-contractions in lattice field theory

Writing evaluations of Wick-contractions is a challenging task, specially if they involve communication with neighbouring ranks as well as sparse intermediate objects. They may become increasingly complicated with multiple propagators plugged in (all-to-all, point-to-all, ...). A general language to describe contractions that is independent on the underlying data format would speed up the developing and testing phase of our workflow immensely.
We can borrow paradigms from data science, such as data independence and come up with a SQL-like language to combine and aggregate data in contractions. An execution engine parses the written contraction and comes up with an optimised execution plan that is finally executed in parallel on a modern supercomputing facility.
High-stakes lattice determinations of quantities like the muon $g-2$ are examples of observables which require many contractions, when including iso-spin breaking effects. We should be able to write these contractions in a declarative language telling an underlying engine what to do, not how to do it. Such expressions would add substantial value to a result, its correctness and its reproducability – problems that we have not solved yet as a scientific community in lattice field theory. A standardised language could be a way to approach that.

Multiple RHS solvers

It can be shown that linear algebra operations are usually bound by memory bandwidth. That means, if we manage to reduce memory bandwidth of a certain operation, we directly observe speedups of the running application. The most important computational kernel in lattice field theory is the Dirac operator, a large sparse linear operator. We repeatedly have to solve linear systems of equations involving this operator using iterative Krylov solver algorithms. These solvers usually apply the Dirac operator in every iteration.
One can calculate the amount of memory traffic produced by one application of the Dirac matrix to multiple source spinor fields at the same time (multiple right-hand sides, RHS). Since in this scenario the data for the operator has to be read only once, but applied to multiple input spinors we observe a substantial decrease of memory traffic and by this an asymptotic speedup of 2.5x for the Wilson-Dirac operator and 4x for the Wilson-Clover Dirac operator. Solver algorithms operating on the multiple RHS variant of the original operator experience a similar speedup.

Exploring the Phase Diagram of the Half-Links Schwinger Model

There are many different ways of putting gauge symmetry on the lattice in the Hamiltonian formulation. Quantum Link Models provide one possible route where gauge links are replaced by spins. Choosing different total spins per link gives rise to different models. This offers a wide variety of models which can potentially be addressed by quantum simulation experiments.

A variation of these models, dubbed Half-Links Schwinger (HLS), has been studied more ex-tensivelly in the group. In this model, every other gauge link is removed. This reduces the number of gauge symmetries. It is an open question what is the effect of such modification. In particular it would be itneresting to udnerstand if the theories would share similar continuum limits and phase diagrams.

Abelian-Higgs Model at Finite Lattice Spacing

While taking the lattice spacing to zero is, in the context of quantum field theory, the natural thing to do, this does not need to be the case if we are able to implement the lattice theory in a quantum simulator. In that case, the lattice spacing can be seen as a new parameter that experimentalists might be able to control. 

In recent works, the phase diagram at finite lattice spacing has been investigated for this model. The results indicate a new critical point where a second order phase transitions occurs. At this critical point, a new field theory emerges and a continuum limit, different from the standard one, can be taken. Specifically, at the critical point, a conformal field theory emerges. This conformal theory has been investigated in these works and some puzzling phenomena has been found. In this project we want to investigate this model in the Lagrangian formulation, testing and extending the results present in the literature though a different formulation.

Analytical techniques for enhanced (g-2)-related lattice QCD observables

This project aims to implement and extend the analytical continuation techniques presented by Greynat and de Rafael [arXiv:2311.11597] for calculating the hadronic vacuum polarization (HVP) contribution to g-2. The core idea of the paper is to use an artificially heavy muon mass in the calculation, which shifts the focus to a region where lattice data is more precise, thereby improving overall accuracy. An analytical model is then used to extrapolate these results back to the physical muon mass.

Main goals: 

• Thoroughly understand and apply this method to our existing lattice data.
• Assess its limitations, applicability, and potential for statistical error reduction.
• Adapt the methodology to related observables such as moments and window functions.
• Explore leveraging the dimensionless nature of a_μ to maximize information from existing lattice ensembles.

This project will provide hands-on experience with g-2 calculations, analytical continuation, and lattice QCD techniques.

Quantum Link Models in Large Winding Sectors

Offloading the Hybrid Monte Carlo algorithm to GPUs

Correlation Functions in the Lattice Schwinger Model at Finite Temperature

Accelerating lattice field theory using GPUs

Towards Measuring Hadronic Vacuum Polarization in the Schwinger Model 

The screening spectrum of Yang-Mills theory at high temperatures

Cluster Algorithms for Quantum Link Models

Generalised Simulated Tempering Approach to Topological Freezing in Lattice Gauge Theories

Lattice Gauge Theory of the Half-Links Schwinger Model