Publications

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• Introduction to the theory of open quantum systems

  Piotr Szańkowski

  SciPost Phys. Lect. Notes 68 (2023) · published 18 April 2023 | Toggle abstract · pdf

  This manuscript is an edited and refined version of the lecture script for a one-semester graduate course given originally at the PhD school in the Institute of Physics of Polish Academy of Sciences in the Spring/Summer semester of 2022. The course expects from the student only a basic knowledge on graduate-level quantum mechanics. The script itself is largely self-contained and could be used as a textbook on the topic of open quantum systems. The program of this course is based on a novel approach to the description of the open system dynamics: It is showed how the environmental degrees of freedom coupled to the system can be represented by a multi-component quasi-stochastic process. Using this representation one constructs the super-quasi-cumulant (or super-qumulant) expansion for the system's dynamical map-a parametrization that naturally lends itself for the development of a robust and practical perturbation theory. Thus, even an experienced researcher might find this manuscript of some interest.

  5 citations

• Topological insulators and geometry of vector bundles

  Alexander S. Sergeev

  SciPost Phys. Lect. Notes 67 (2023) · published 12 April 2023 | Toggle abstract · pdf

  For a long time, band theory of solids has focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.

• Parametric couplings in engineered quantum systems

  Anja Metelmann

  SciPost Phys. Lect. Notes 66 (2023) · published 13 March 2023 | Toggle abstract · pdf

  Parametric couplings in engineered quantum systems are a powerful tool to control, manipulate and enhance interactions in a variety of platforms. It allows us to bring systems of different energy scales into communication with each other. This short chapter introduces the basic principles and discusses a few examples of how one can engineer parametric amplifiers with improved characteristics over conventional setups. Clearly, the selected examples are author-biased, and other interesting proposals and implementations can be found in the literature. The focus of this chapter is on parametric effects between linearly coupled harmonic oscillators, however, parametric modulation is also applicable with nonlinear couplings and anharmonic systems.

  1 citation

• Seven Études on dynamical Keldysh model

  Dmitri V. Efremov, Mikhail N. Kiselev

  SciPost Phys. Lect. Notes 65 (2022) · published 6 December 2022 | Toggle abstract · pdf

  We present a comprehensive pedagogical discussion of a family of models describing the propagation of a single particle in a multicomponent non-Markovian Gaussian random field. We report some exact results for single-particle Green's functions, self-energy, vertex part and T-matrix. These results are based on a closed form solution of the Dyson equation combined with the Ward identity. Analytical properties of the solution are discussed. Further we describe the combinatorics of the Feynman diagrams for the Green's function and the skeleton diagrams for the self-energy and vertex, using recurrence relations between the Taylor expansion coefficients of the self-energy. Asymptotically exact equations for the number of skeleton diagrams in the limit of large $N$ are derived. Finally, we consider possible realizations of a multicomponent Gaussian random potential in quantum transport via complex quantum dot experiments.

  1 citation

• The hitchhiker's guide to 4d $\mathcal{N}=2$ superconformal field theories

  Mohammad Akhond, Guillermo Arias-Tamargo, Alessandro Mininno, Hao-Yu Sun, Zhengdi Sun, Yifan Wang, Fengjun Xu

  SciPost Phys. Lect. Notes 64 (2022) · published 26 October 2022 | Toggle abstract · pdf

  Superconformal field theory with $\mathcal{N}=2$ supersymmetry in four dimensional spacetime provides a prime playground to study strongly coupled phenomena in quantum field theory. Its rigid structure ensures valuable analytic control over non-perturbative effects, yet the theory is still flexible enough to incorporate a large landscape of quantum systems. Here we aim to offer a guidebook to fundamental features of the 4d $\mathcal{N}=2$ superconformal field theories and basic tools to construct them in string/M-/F-theory. The content is based on a series of lectures at the Quantum Field Theories and Geometry School (https://sites.google.com/view/qftandgeometrysummerschool/home) in July 2020.

  5 citations

• Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function

  Hiroshi Shinaoka, Naoya Chikano, Emanuel Gull, Jia Li, Takuya Nomoto, Junya Otsuki, Markus Wallerberger, Tianchun Wang, Kazuyoshi Yoshimi

  SciPost Phys. Lect. Notes 63 (2022) · published 22 September 2022 | Toggle abstract · pdf

  This lecture note reviews recently proposed sparse-modeling approaches for efficient ab initio many-body calculations based on the data compression of Green's functions. The sparse-modeling techniques are based on a compact orthogonal basis, an intermediate representation (IR) basis, for imaginary-time and Matsubara Green's functions. A sparse sampling method based on the IR basis enables solving diagrammatic equations efficiently. We describe the basic properties of the IR basis, the sparse sampling method and its applications to ab initio calculations based on the GW approximation and the Migdal-Eliashberg theory. We also describe a numerical library for the IR basis and the sparse sampling method, sparse-ir, and provide its sample codes. This lecture note follows the Japanese review article with major revisions [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].

  12 citations

• Quantum Field Theory Anomalies in Condensed Matter Physics

  R. Arouca, Andrea Cappelli, T. H. Hansson

  SciPost Phys. Lect. Notes 62 (2022) · published 8 September 2022 | Toggle abstract · pdf

  We give a pedagogical introduction to quantum anomalies, how they are calculated using various methods, and why they are important in condensed matter theory. We discuss axial, chiral, and gravitational anomalies as well as global anomalies. We illustrate the theory with examples such as quantum Hall liquids, Fermi liquids, Weyl semi-metals, topological insulators and topological superconductors. The required background is basic knowledge of quantum field theory, including fermions and gauge fields, and some familiarity with path integral and functional methods. Some knowledge of topological phases of matter is helpful, but not necessary.

  10 citations

• Quantum Neural Network Classifiers: A Tutorial

  Weikang Li, Zhide Lu, Dong-Ling Deng

  SciPost Phys. Lect. Notes 61 (2022) · published 17 August 2022 | Toggle abstract · pdf

  Machine learning has achieved dramatic success over the past decade, with applications ranging from face recognition to natural language processing. Meanwhile, rapid progress has been made in the field of quantum computation including developing both powerful quantum algorithms and advanced quantum devices. The interplay between machine learning and quantum physics holds the intriguing potential for bringing practical applications to the modern society. Here, we focus on quantum neural networks in the form of parameterized quantum circuits. We will mainly discuss different structures and encoding strategies of quantum neural networks for supervised learning tasks, and benchmark their performance utilizing Yao.jl, a quantum simulation package written in Julia Language. The codes are efficient, aiming to provide convenience for beginners in scientific works such as developing powerful variational quantum learning models and assisting the corresponding experimental demonstrations.

  23 citations

• Phase transitions above the upper critical dimension

  Bertrand Berche, Tim Ellis, Yurij Holovatch, Ralph Kenna

  SciPost Phys. Lect. Notes 60 (2022) · published 12 August 2022 | Toggle abstract · pdf

  These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension $d_{\rm uc}$. After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above $d_{\rm uc}$. We recount how Fisher's dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above $d_{\rm uc}$. We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.

  11 citations

• SIMP Dark Matter

  Yonit Hochberg

  SciPost Phys. Lect. Notes 59 (2022) · published 29 July 2022 | Toggle abstract · pdf

  These notes summarize the zoom-course on Strongly Interacting Massive Particle (SIMP) dark matter given at the Les Houches Summer School in the summer of 2021. An alternative title for this course would be `SIMPS, Cannibals and ELDERs', where our focus is on high-point interactions amongst dark matter particles. The spirit of the course is to give students a taste of these exciting developments in dark matter, while primarily teaching tricks and methods that are tough to captivate when reading textbooks or papers. These notes are written in similar spirit: I will put emphasis on how quantities scale and how to perform back of the envelope estimates, which should serve to help you when you develop the next great dark matter idea.

  2 citations