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The basic structure of quantum field theory that is used to describe the Standard Model of fundamental interactions of nature is usually formulated for zero temperature. However, the effects of temperature are extremely important for understanding a number of physical processes such as the electro-weak phase transition and quark-gluon plasma. Thermal field theory is the extension of quantum field theory to a non-zero temperature environment and is achieved by modifying the propagators in loop integrations represented by Feynman diagrams. The Python program pySecDec numerically calculates dimensionally-regularized loop integrals in quantum field theory using the sector decomposition approach. It is shown how pySecDec can be applied to thermal field theory numerical calculations using modifications within the Matsubara formalism. Using the formulated algorithm, a 2-point correlation function (such as those occurring in QCD correlation functions) at finite temperature can be numerically calculated for a variety of spacetime dimensions.

CAP Virtual Congress 2021

Numerical Loop-Integration Methods for Thermal Field Theory

finite temperature

quantum field theory

computational physics

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technical paper

It is an ongoing challenge to engineer setups in which Majorana zero modes at the ends of one-dimensional topological superconductor are well isolated which is the essence of topological protection. Recent developments have indicated that periodic deriving of a system can dynamically induce symmetries that its static counterpart does not possess [1]. We further develop the original protocol [2] where this idea [1] is applied to a system of quantum dot (QD) coupled to a Kitaev chain hosting an imperfect (overlapping) Majorana zero modes. We numerically simulate a protocol in which an electron periodically hops back and forth from the QD and the wire. We demonstrate that the current protocol reduces a non-zero hybridization energy that manifests from imperfect Majoranas by orders of magnitude. Furthermore, we examine the efficiency of the suppression and how robust it is to imperfections.

[1] K. Agarwal and I. Martin, Phys. Rev. Lett. **125**, 080602 (2020).

[2] I. Martin and K. Agarwal, PRX Quantum **1**, 020324 (2020).

Reducing Majorana Hybridization via Periodic Driving

Flat band systems are becoming popular due to special properties. For instance, the strong correlation of electrons leads to realization of unconventional superconductivity [1]. Typically, such bands are only approximately flat and are engineered by fine tuning Vanderwaal’s structures. Here we consider Kagome and Lieb lattices with perfectly flat bands. However, at some points in the Brillouin zone the bands superimpose leading to degeneracies. It has been shown that the degeneracy can be lifted when time-reversal symmetry (TRS) is broken [2]. In this presentation, we explore further means to lift the degeneracy while preserving the flat band. We show that the flatness is robust under certain changes to the lattice and that breaking TRS is not sufficient to isolate the flat band. Instead, we show that modulating the flux based on Chern-Simons field as outlined in [3] successfully gaps out the band.
[1] Cao, Y. et al. (2018). Unconventional superconductivity in magic-angle graphene superlattices. Nature, 556(7699), 43-50
[2] Green, D., Santos, L., & Chamon, C. (2010). Isolated flat bands and spin-1 conical bands in two-dimensional lattices. Physical Review B, 82(7).
[3] Maiti, S., & Sedrakyan, T. (2019). Fermionization of bosons in a flat band. Physical Review B, 99(17).

Identifying and Isolating Flat Bands in 2D Systems

Separating and analyzing DNA, RNA or proteins is often done through gel electrophoresis. This simple technique is conducted millions of times in labs every day world-wide. Analysis of the gels is typically done visually on a photo by comparing measured intensities with a well-defined reference ladder. Using digital photography and image analysis, we have developed a technique to quantitatively analyze standard gel images to determine exact sized distributions. This information can then be used to compare experimental data to theoretical models and predictions. The software consists of a smartphone-based application with an easy-to-use GUI to analyze images of any quality, dimension, or type. Within seconds, the user can sign in, choose ladder specifications, and select the image for analysis. They can then tap on the ladders of interest and send the image for analysis. The cloud-based analysis uses an exponential model to characterize the propagation of molecules within the gel and automatically measures the size and relative quantity of detectable solutes. I will provide direct evidence that the software allows measurement of RNA molecule lengths with resolution down to a single monomer. The entire analysis process takes under 10 seconds, after which the results are automatically emailed to the user. The novel way in which this data is processed and presented to the end-user in a seamlessly integrated workflow with possibilities for additional integration makes this a major step forward over existing solutions. Enabling, supporting, and promoting this research by creating easy-to-use and powerful tools has the potential to promote change in physical and biological research.

The quantitative analysis of gel electrophoresis data using a cloud-based smartphone application

We describe the **offset logarithm function** and illustrate its applications to physics. The offset log can be considered an extension of the logarithm function, and it can also be considered as a generalization of the Lambert W function.
The offset log function shows up in a variety of physical and biological models, such as the mean field model (Weiss model) of ferromagnetism, and models of carbon nanoribbons. It provides a convenient framework for describing phenomena in which the behaviour of a physical variable at one location or time, has an exponential or logarithmic relationship to the value of attribute at a displaced location or time.

The offset logarithm function of order $k$ is the multivalued function $w = L_k(z)$ of the complex variable $z$, which satisfies the equation
\begin{equation} \frac{ w-k }{w+k}e^w = z \qquad(1) \end{equation}
Each of $k$, $z$, and $w$ can be in general a complex number. However, for this, we consider $k$, $z$ and $w$ as real numbers. Approximate solutions to Equation (1) can be calculated numerically, but a study of the analytic properties of that equation can provide insight into the system being described. 

The offset log function is multi-branch. Each branch of $L_k(z)$ is an analytic function for certain regions of $ k$ and $z$ values. We present the main characteristics of the $L_k(z)$ function. For each domain of the $k$ parameter, the properties of $L_k$ in terms of the branches are given. We show that the number of solutions of Equation (1) depends upon the domains of $k$ and $z$. For these domains we give the number of solutions and we give the interval where each solution lies. We identify the cases where a solution is a multiple solution. We give a symbolic formula for the multiple solutions.

The offset logarithm function and some applications in physics: A generalization of the Lambert W function

Transitions between fractionalized and conventional quantum phases of matter in 2+1 dimensions are conceptually best understood within the framework of parton gauge theories, whereby the confinement of fractionalized excitations and spontaneous breaking of global symmetries in conventional phases is argued to result from the proliferation of gauge monopoles/instantons. To complement recent studies of the quantum numbers and scaling dimensions of monopole operators in parton gauge theories of frustrated quantum antiferromagnets, we provide an explicit semiclassical derivation of instanton contributions to the low-energy effective parton gauge theory for hardcore bosons on the honeycomb lattice using methods originally developed by 't Hooft in the solution of the U(1) problem in quantum chromodynamics. While the symmetry-breaking effect of instantons is typically associated with massless fermions and the Atiyah-Singer index theorem on compact manifolds, we show that the spontaneous breaking of the U(1) boson number conservation symmetry in the superfluid phase results here from Euclidean zero modes of massive Dirac fermions bound to instantons in noncompact spacetime.

Symmetry Breaking Effects of Instantons in Parton Gauge Theories

The creation and annihilation of particles is a fundamental feature of relativistic quantum fields. A famous example of this is provided by Schwinger’s 1951 prediction that the vacuum is unstable to particle-antiparticle production if a static electric field is applied to it. In this talk we examine the classical field limit of Schwinger pair production by mapping the Klein-Gordon equation with an electric field onto the non-relativistic Schrödinger equation with a 1/r^2 potential. This latter problem has two regimes depending on the depth of the potential: a sub-critical regime where PT symmetry is preserved and a supercritical regime where PT symmetry is broken and one finds ``fall-to-the-centre’’ where probability is absorbed at the origin. Schwinger pair production occurs in the latter regime and can be described in terms of non-Hermitian quantum mechanics.

Schwinger pair production as a non-Hermitian problem

More than 30 million individuals worldwide are living with Alzheimer’s Disease. To further the current understanding on this neurodegenerative disease, we developed a technique to create amyloid peptide clusters in synthetic, brain-like membranes, which mimic the senile plaques found in the brains of Alzheimer's patients. I compared the molecular functioning of homotaurine, a peptic anti-aggregant that binds to amyloid peptides directly, and curcumin, a non-peptic molecule that can inhibit aggregation by changing membrane properties. By using microscopy, x-ray diffraction, and UV-vis spectroscopy, we found that both curcumin and homotaurine significantly reduce the number of small, nanoscopic amyloid aggregates and the corresponding β- and cross-β sheet signals. This research shows that membrane active drugs can be as efficient as peptide targeting drugs in inhibiting amyloid aggregation in-vitro [1]. The findings can open new pathways for the developments of drugs to slow down first occurrence and progression of the disease.

[1] Xingyuan Zou, Sebastian Himbert, Janos Juhasz, Samantha Ros, Harald D. H. Stover, and Maikel C. Rheinstädter, “Curcumin and homotaurine suppress amyloid-b25-35 aggregation in synthetic brain membranes”, under review with ACS Chemical Neuroscience, Manuscript ID: cn-2021-00057r

Curcumin and homotaurine suppress amyloid-b25-35 aggregation in synthetic brain membranes

Positron Emission Tomography (PET) Imaging of the brain might become the most effective imaging technique to predict Alzheimer's disease. However, the definition of the brain in PET images is low and the lesion area is not easy to define, so the accuracy of traditional machine learning algorithms in predicting Alzheimer's disease from PET images is low. Deep learning algorithms can effectively improve the accuracy of prediction. Here, a deep learning model based on multi-attention mechanisms is constructed for the prediction of Alzheimer's disease. Firstly, in order to improve the images and reduce the loss of spatial information caused by convolution, a soft-attention mechanism was introduced to embed non-local modules and CBAM modules into the prediction model, which effectively solved the problems of the lack of detail information and the lack of connection between channels in PET images after deep convolution. Secondly, a split-attention mechanism was introduced, and the influence of different parameters of feature images with different sizes on the prediction accuracy was enhanced by using a packet network. Finally, head movement correction, image registration, craniocerebral separation, Gaussian smoothing and other image preprocessing were performed on the acquired PET images to effectively improve the image features of Alzheimer's disease focal areas in brain PET images. The experimental results showed that the prediction accuracy, sensitivity and specificity of the model in the ADNI database for Alzheimer's disease were 90.5%, 86.1% and 93.5% respectively, which could provide more accurate diagnostic results compared with the existing methods.

Prediction of Alzheimer's Disease Based on Multi-Attentional Mechanisms using Brain PET Image

The behaviour of confined polymeric systems has been the topic of much recent research. The information from these studies may help guide the development and optimization of nanofluidic devices and understand the segregation of DNA chromosomes in prokaryotic cells. Some recent experiments examined two DNA molecules confined to a box-like cavity with strong confinement in one dimension. The researchers looked at the effect of using an anisotropic cross-sectional box geometry and of having non-identical polymers. In this study, we use computer simulations to examine the effect of such anisotropies on the behaviour of the confined-polymer systems. We employ Metropolis Monte Carlo simulations where the polymers are modelled as bead spring chains. We calculate probability distributions for the centre-of-mass positions of either polymer and for the centre-of-mass displacement between the polymers. These distributions are used to examine the effect of anisotropies in polymer length for two polymers confined to a square box-like cavity and anisotropies in box dimensions for two identical polymers confined to a box-like cavity. The information obtained from this study can then be used to have a better understanding of the trends seen in the experiments.

Simulation of Two Polymers Confined to a Box-Like Cavity: The Effect of Anisotropies in Polymer Length and Confinement Dimensions

We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general dimension. These spaces are paired via an inner product called the intersection number -- an invariant which can be computed algebraically. In this language, reduction of an integrand modulo integration-by-part identities is simply its projection onto a chosen basis. The dual-forms turn out to be far simpler than their Feynman counterparts; they are uplifts of lower-dimensional forms supported on the maximal cuts of various sub-topologies (boundaries). The intersection numbers are then computed by taking residues around cuts where various subsets of propagators become on-shell, giving a systematic approach to generalized unitarity. Moreover, the dual integrals satisfy the transposed differential equation to their Feynman counterparts. Since the dual-forms are localized to cut the transposed differential equation is a much simpler to construct.

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Numerical Loop-Integration Methods for Thermal Field Theory

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