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SciPost Physics Lecture Notes

Galilei particles revisited

José Miguel Figueroa-O'Farrill, Simon Pekar, Alfredo Pérez, Stefan Prohazka

SciPost Phys. Lect. Notes 93 (2025) · published 13 March 2025 | · pdf

We revisit the classifications of classical and quantum galilean particles: that is, we fully classify homogeneous symplectic manifolds and unitary irreducible projective representations of the Galilei group. Equivalently, these are coadjoint orbits and unitary irreducible representations of the Bargmann group, the universal central extension of the Galilei group. We provide an action principle in each case, discuss the nonrelativistic limit, as well as exhibit, whenever possible, the unitary irreducible representations in terms of fields on Galilei spacetime. Motivated by a forthcoming study of planons we pay close attention to the mobility of the less familiar massless Galilei particles.
Scattering from an external field in quantum chromodynamics at high energies: From foundations to interdisciplinary connections

Athanasia-Konstantina Angelopoulou, Anh Dung Le, Stéphane Munier

SciPost Phys. Lect. Notes 92 (2025) · published 12 March 2025 | · pdf

We review the factorization of the $S$-matrix elements in the context of particle scattering off an external field, which can serve as a model for the field of a large nucleus. The factorization takes the form of a convolution of light cone wave functions describing the physical incoming and outgoing states in terms of bare partons, and products of Wilson lines. The latter represent the interaction between the bare partons and the external field. Specializing to elastic scattering amplitudes of onia at very high energies, we introduce the color dipole model, which formulates the calculation of the modulus-squared of the wave functions in quantum chromodynamics with the help of a branching random walk, and the scattering amplitudes as observables on this classical stochastic process. Methods developed for general branching processes produce analytical formulas for the asymptotics of such observables, and thus enable one to derive exact large-rapidity expressions for onium-nucleus cross sections, from which electron-nucleus cross sections may be inferred.
Two lectures on Yang-Lee edge singularity and analytic structure of QCD equation of state

Vladimir V. Skokov

SciPost Phys. Lect. Notes 91 (2025) · published 7 March 2025 | · pdf

These lecture notes, prepared for the 2024 XQCD PhD, provide an introduction to the analytic structure of an equation of state near a second-order phase transition and its most prominent landmark: the Yang-Lee edge singularity. In addition to discussing general properties, the notes review recent theoretical progress in locating the QCD critical point by tracking the trajectory of the Yang-Lee edge singularity.
Introduction to string theory

Carlo Maccaferri, Fabio Marino, Beniamino Valsesia

SciPost Phys. Lect. Notes 90 (2025) · published 29 January 2025 | · pdf

These are lecture notes of the introductory String Theory course held by one of the authors for the master program of Theoretical Physics at Turin University. The world-sheet approach to String Theory is pedagogically introduced in the framework of the bosonic string and of the superstring.
Population genetics: An introduction for physicists

Andrea Iglesias-Ramas, Samuele Pio Lipani, Rosalind J. Allen

SciPost Phys. Lect. Notes 89 (2024) · published 10 December 2024 | · pdf

Population genetics lies at the heart of evolutionary theory. This topic forms part of many biological science curricula but is rarely taught to physics students. Since physicists are becoming increasingly interested in biological evolution, we aim to provide a brief introduction to population genetics, written for physicists. We start with two background chapters: chapter 1 provides a brief historical introduction to the topic, while chapter 2 provides some essential biological background. We begin our main content with chapter 3 which discusses the key concepts behind Darwinian natural selection and Mendelian inheritance. Chapter 4 covers the basics of how variation is maintained in populations, while chapter 5 discusses mutation and selection. In chapter 6 we discuss stochastic effects in population genetics using the Wright-Fisher model as our example, and finally we offer concluding thoughts and references to textbooks in chapter 7.

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Physics: Atomic, Molecular and Optical Physics - Theory • Condensed Matter Physics - Theory • Condensed Matter Physics - Computational • Quantum Physics

Lecture Notes: Many-body quantum dynamics with MCTDH-X

by Paolo Molignini, Sunayana Dutta, Elke Fasshauer

Version 1 (latest version)
Submitted 2025-01-29 22:40 to SciPost Physics Lecture Notes · latest activity: 2025-02-10 11:23

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Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

Johannes Hauschild, Frank Pollmann

SciPost Phys. Lect. Notes 5 (2018) · published 8 October 2018 | · pdf

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [https://github.com/tenpy/tenpy]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

318 citations
Lecture notes on Generalised Hydrodynamics

Benjamin Doyon

SciPost Phys. Lect. Notes 18 (2020) · published 28 August 2020 | · pdf

These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.

150 citations
Tangent-space methods for uniform matrix product states

Laurens Vanderstraeten, Jutho Haegeman, Frank Verstraete

SciPost Phys. Lect. Notes 7 (2019) · published 15 January 2019 | · pdf

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

142 citations
Phase transitions in the early universe

Mark Hindmarsh, Marvin Lüben, Johannes Lumma, Martin Pauly

SciPost Phys. Lect. Notes 24 (2021) · published 15 February 2021 | · pdf

These lecture notes are based on a course given by Mark Hindmarsh at the 24th Saalburg Summer School 2018 and written up by Marvin L\"uben, Johannes Lumma and Martin Pauly. The aim is to provide the necessary basics to understand first-order phase transitions in the early universe, to outline how they leave imprints in gravitational waves, and advertise how those gravitational waves could be detected in the future. A first-order phase transition at the electroweak scale is a prediction of many theories beyond the Standard Model, and is also motivated as an ingredient of some theories attempting to provide an explanation for the matter-antimatter asymmetry in our Universe. Starting from bosonic and fermionic statistics, we derive Boltzmann's equation and generalise to a fluid of particles with field dependent mass. We introduce the thermal effective potential for the field in its lowest order approximation, discuss the transition to the Higgs phase in the Standard Model and beyond, and compute the probability for the field to cross a potential barrier. After these preliminaries, we provide a hydrodynamical description of first-order phase transitions as it is appropriate for describing the early Universe. We thereby discuss the key quantities characterising a phase transition, and how they are imprinted in the gravitational wave power spectrum that might be detectable by the space-based gravitational wave detector LISA in the 2030s.

97 citations
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Jünemann, Daniel Jaschke, Matteo Rizzi, Simone Montangero

SciPost Phys. Lect. Notes 8 (2019) · published 18 March 2019 | · pdf

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

92 citations

Physics: Atomic, Molecular and Optical Physics - Theory • Condensed Matter Physics - Theory • Condensed Matter Physics - Computational • Quantum Physics

Series contained in this Journal

318 citations

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142 citations

97 citations

92 citations