In the pursuit of designing complex materials with desired properties, un- derstanding their design parameter space is crucial. However, this space’s convolution often hinders comprehension of...Show moreIn the pursuit of designing complex materials with desired properties, un- derstanding their design parameter space is crucial. However, this space’s convolution often hinders comprehension of complex materials’ responses as a function of their design parameters. Machine Learning has recently emerged as a promising tool for capturing patterns in complex design spaces, although this performance often comes at the cost of interpretabil- ity. This thesis aims to explore the design parameter space of interact- ing hysterons using interpretable Machine Learning, specifically Decision Tree inspired methods. Despite the complexity of the design parameter space of even small systems of interacting hysterons, interpretable Ma- chine Learning can classify coarse-grained properties of the system effec- tively. Introducing the Support Vector Classifier (SVC) inspired Decision Tree, we achieve almost perfect isolation of these properties. This model preserves interpretability while effectively probing the statistical structure of design parameter space of systems of interacting hysterons.Show less
Since the 80s, strange metals, metals where the electrons are so densely entangled that the conventional condensed matter paradigm of Short Ranged Entanglement fails, have eluded any form of study...Show moreSince the 80s, strange metals, metals where the electrons are so densely entangled that the conventional condensed matter paradigm of Short Ranged Entanglement fails, have eluded any form of study due to the sign problem, which renders numerical calculations impossible. However, holography, a duality between strongly coupled quantum field theory problems and classical general relativity problems of one spatial dimension higher, grants us a way to circumvent the sign problem. In this thesis, we will run a modified version of code that was once used to simulate binary black holes on a supercomputer to calculate the properties of two $2+1$-dimensional holographic models for strange metals, the Reissner-Nordstr\"om metal and the Gubser-Rocha metal, subject to an ionic lattice potential: the code needed to simulate the Gubser-Rocha metal was only finished last year. We then investigate whether the DC electrical conductivity $\sigma$, thermopower $\alpha$ and thermal conductivity $\bar{\kappa}$ obey four different Drude models: one basic relativistic model and three models with different extra incoherent terms, models A, B and C. We find that model A, the most conventional model, fails, while the conductivities obey model C ($\kappa$-dominated transport) for low lattice strength $A$ and model B ($\sigma_{Q=0}$-dominated transport) for high $A$. We suspect this surprising result is caused by a pole collision causing a crossover between two regimes, but more research needs to be done to verify this.Show less
Neural networks have been an active field of research for years, but relatively little is understood of how they work. Specific types of Neural Networks have a layer structure with decreasing width...Show moreNeural networks have been an active field of research for years, but relatively little is understood of how they work. Specific types of Neural Networks have a layer structure with decreasing width which acts like coarse-graining, reminiscent of the renormalization group (RG). We examine the Restricted Boltzmann Machine (RBM) and discuss it’s possible relation to RG. The RBM is trained on the 1D and 2D Ising model, as well as the MNIST dataset. In particular for the 2D Ising model showing a flow towards the critical point Tc ≈ 2.27, opposite to the RG-flow. Examining the behaviour of the RBM on the MNIST dataset shows that sparse datasets can allow multiple fixed points which can be removed by artificially creating new samples. We conclude that this RBM-flow exists due to the multiple relevant length scales at the critical point and we briefly discuss why.Show less
We compute the quantum back reaction of Schwinger pair production in the context of spinor QED. Using path integral and Schwinger’s proper time methods, we employ a WKB approximation to find the...Show moreWe compute the quantum back reaction of Schwinger pair production in the context of spinor QED. Using path integral and Schwinger’s proper time methods, we employ a WKB approximation to find the current-current correlation function and relate it to the current expectation value in a manner analogous to fluctuation-dissipation theory. The same expression for the current expectation value in terms of the electric field is derived using point-splitting to cure UV divergences. We use this current value to compute the back reaction on the electric field via Maxwell’s equation, obtaining a non-linear complex differential equation. A stable numerical solution is found, whose real part is the electric field. The field, initially just below the Schwinger limit, undergoes a sharp drop followed by exponential-like decay. Further analysis on the behavior of the current, production rate, and the energy density is performed using numerical integration.Show less
Monstrous moonshine describes the unexpected relation between the modular J-function and the largest sporadic simple group known as the Monster. Both objects arise naturally in a conformal vertex...Show moreMonstrous moonshine describes the unexpected relation between the modular J-function and the largest sporadic simple group known as the Monster. Both objects arise naturally in a conformal vertex algebra V♮; the J-function as the character and the Monster as the automorphism group. This vertex algebra can be interpreted as the quantum theory of a bosonic string living on a Z2-orbifold of spacetime compactified by the Leech lattice. We will discuss basic properties of vertex algebras and construct vertex algebras associated to even lattices, which form the main building block of V♮. We will calculate the characters of these vertex algebras and discuss their modular properties. Subsequently, we will develop the quantum theory of free bosonic open and closed strings, including the toroidal and orbifold compactificaton of the latter. We will conclude with calculating the one-loop partition function of the closed string on the Z2-orbifold compactified by the Leech lattice, and show it to be equal to the J-function.Show less
We first introduce the concept of partner potentials in non-relativistic quantum mechanics, i.e. a pair of potentials with the same spectrum, possibly except for a zero-energy ground state. We use...Show moreWe first introduce the concept of partner potentials in non-relativistic quantum mechanics, i.e. a pair of potentials with the same spectrum, possibly except for a zero-energy ground state. We use this to define a family of partner potentials, giving us a technique to calculate the entire spectrum of a potential. The mechanism of partner potentials is then used for a quantum mechanical model of supersymmetry. It turns out that a special class of potentials exists where the spectrum can be determined very quickly using the techniques developed. We explore some of these potentials, called shape invariant potentials or SIPs and discover some simple properties of them. Finally, we take a quick look at a Hamiltonian with a p 4 term in it, discovering that for a small class of potentials, we can make a supersymmetric quantum mechanical model out of it.Show less
In this thesis we present Dijkgraaf-Witten theory. We start by considering a two-dimensional topological quantum field theory that can be used to prove Mednykh’s formula along the way. Subsequently...Show moreIn this thesis we present Dijkgraaf-Witten theory. We start by considering a two-dimensional topological quantum field theory that can be used to prove Mednykh’s formula along the way. Subsequently, we define the Dijkgraaf-Witten invariant as a partition function where we assign a specified weight to the 3-simplices of a compact, oriented and triangulated 3-manifold with boundary. The partition function depends on how we assign elements of a finite, discrete group G to all the oriented edges of the manifold. We prove that, whenever the triangulation of the boundary is fixed, the invariant does not depend on the triangulation of the manifold. Finally, we define a similar invariant where we model the weight of the 3-simplices to mimic the action of Chern-Simons theory. We demonstrate that by demanding invariance, we obtain the Dijkgraaf-Witten invariant.Show less
