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SciPost Physics Lecture Notes

Engineering of anyons on M5-probes via flux quantization

Hisham Sati, Urs Schreiber

SciPost Phys. Lect. Notes 107 (2025) · published 4 November 2025 | · pdf

These extended lecture notes survey a novel derivation of anyonic topological order (as seen in fractional quantum Hall systems) on single magnetized M5-branes probing Seifert orbi-singularities ("geometric engineering" of anyons), which we motivate from fundamental open problems in the field of quantum computing. The rigorous construction is non-Lagrangian and non-perturbative, based on previously neglected global completion of the M5-brane's tensor field by flux-quantization consistent with its non-linear self-duality and its twisting by the bulk C-field. This exists only in little-studied non-abelian generalized cohomology theories, notably in a twisted equivariant (and "twistorial") form of unstable Cohomotopy ("Hypothesis H"). As a result, topological quantum observables form Pontrjagin homology algebras of mapping spaces from the orbi-fixed worldvolume into a classifying 2-sphere. Remarkably, results from algebraic topology imply from this the quantum observables and modular functor of abelian Chern-Simons theory, as well as braid group actions on defect anyons of the kind envisioned as hardware for topologically protected quantum gates.
Lecture notes on large deviations in non-equilibrium diffusive systems

Bernard Derrida

SciPost Phys. Lect. Notes 106 (2025) · published 29 October 2025 | · pdf

These notes are a written version of lectures given in the 2024 Les Houches Summer School on Large deviations and applications. They are are based on a series of works published over the last 25 years on steady properties of non-equilibrium systems in contact with several heat baths at different temperatures or several reservoirs of particles at different densities. After recalling some classical tools to study non-equilibrium steady states, such as the use of tilted matrices, the Fluctuation theorem, the determination of transport coefficients, the Einstein relations or fluctuating hydrodynamics, they describe some of the basic ideas of the macroscopic fluctuation theory allowing to determine the large deviation functions of the density and of the current of diffusive systems.
Simplified derivations for high-dimensional convex learning problems

David G. Clark, Haim Sompolinsky

SciPost Phys. Lect. Notes 105 (2025) · published 23 October 2025 | · pdf

Statistical-physics calculations in machine learning and theoretical neuroscience often involve lengthy derivations that obscure physical interpretation. Here, we give concise, non-replica derivations of several key results and highlight their underlying similarities. In particular, using a cavity approach, we analyze three high-dimensional learning problems: perceptron classification of points, perceptron classification of manifolds, and kernel ridge regression. These problems share a common structure—a bipartite system of interacting feature and datum variables—enabling a unified analysis. Furthermore, for perceptron-capacity problems, we identify a symmetry that allows derivation of correct capacities through a naïve method.
An introduction to large deviations with applications in physics

Ivan N. Burenev, Daniël W. H. Cloete, Vansh Kharbanda, Hugo Touchette

SciPost Phys. Lect. Notes 104 (2025) · published 23 October 2025 | · pdf

These notes are based on the lectures that one of us (HT) gave at the Summer School on the "Theory of Large Deviations and Applications," held in July 2024 at Les Houches in France. They present the basic definitions and mathematical results that form the theory of large deviations, as well as many simple motivating examples of applications in statistical physics, which serve as a basis for the many other lectures given at the school that covered more specific applications in biophysics, random matrix theory, nonequilibrium systems, geophysics, and the simulation of rare events, among other topics. These notes extend the lectures, which can be accessed online, by presenting exercises and pointer references for further reading.
Stochastic resetting and large deviations

Martin R. Evans, John C. Sunil

SciPost Phys. Lect. Notes 103 (2025) · published 16 October 2025 | · pdf

Stochastic resetting has been a subject of considerable interest within statistical physics, both as means of improving completion times of complex processes such as searches and as a paradigm for generating nonequilibrium stationary states. In these lecture notes we give a self-contained introduction to the toy model of diffusion with stochastic resetting. We also discuss large deviation properties of additive functionals of the process such as the cost of resetting. Finally, we consider the generalisation from Poissonian resetting, where the resetting process occurs with a constant rate, to non-Poissonian resetting.

See all Publications in SciPost Physics Lecture Notes

Physics: Gravitation, Cosmology and Astroparticle Physics

Elements of cosmology beyond FLRW

by Pierre Fleury

Version 1 (latest version)
Submitted Oct. 31, 2025, 12:12 p.m. to SciPost Physics Lecture Notes · latest activity: Nov. 17, 2025, 10:59 a.m.
Physics: Biophysics • Statistical and Soft Matter Physics

Complexity and accessibility of random landscapes

by Sakshi Pahujani, Joachim Krug

Version 1 (latest version)
Submitted Oct. 13, 2025, 3:49 p.m. to SciPost Physics Lecture Notes · latest activity: Oct. 30, 2025, 1:32 p.m.
Physics: High-Energy Physics - Theory

Lecture Notes on Normalizing Flows for Lattice Quantum Field Theories

by Miranda C. N. Cheng, Niki Stratikopoulou

Version 1 (latest version)
Submitted Sept. 28, 2025, 6:15 a.m. to SciPost Physics Lecture Notes · latest activity: Oct. 15, 2025, 11:08 a.m.
Physics: High-Energy Physics - Theory • Mathematical Physics

Les Houches lecture notes on moduli spaces of Riemann surfaces

by Alessandro Giacchetto, Danilo Lewański

Version 1 (latest version)
Submitted Aug. 8, 2025, 2:35 p.m. to SciPost Physics Lecture Notes · latest activity: Aug. 21, 2025, 10:05 a.m.
Physics: High-Energy Physics - Theory

Les Houches lectures on two-dimensional gravity and holography

by Gustavo Joaquin Turiaci

Version 1 (latest version)
Submitted July 28, 2025, 7:05 a.m. to SciPost Physics Lecture Notes · latest activity: Sept. 10, 2025, 10:40 a.m.
Physics: Mathematical Physics

Les Houches lectures on non-perturbative topological strings

by Marcos Mariño

Version 1 (latest version)
Submitted Feb. 26, 2025, 7:36 p.m. to SciPost Physics Lecture Notes · latest activity: June 26, 2025, 2:36 p.m.

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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.

368 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.

173 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.

167 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.

125 citations
Primordial black holes as dark matter candidates

Bernard Carr, Florian Kuhnel

SciPost Phys. Lect. Notes 48 (2022) · published 2 May 2022 | · pdf

We review the formation and evaporation of primordial black holes (PBHs) and their possible contribution to dark matter. Various constraints suggest they could only provide most of it in the mass windows $10^{17}$ - $10^{23}\,$g or $10$ - $10^{2}\,M_{\odot}$, with the last possibility perhaps being suggested by the LIGO/Virgo observations. However, PBHs could have important consequences even if they have a low cosmological density. Sufficiently large ones might generate cosmic structures and provide seeds for the supermassive black holes in galactic nuclei. Planck-mass relics of PBH evaporations or stupendously large black holes bigger than $10^{12}\,M_{\odot}$ could also be an interesting dark component.

125 citations

Physics: Gravitation, Cosmology and Astroparticle Physics

Physics: Biophysics • Statistical and Soft Matter Physics

Physics: High-Energy Physics - Theory

Physics: High-Energy Physics - Theory • Mathematical Physics

Physics: High-Energy Physics - Theory

Physics: Mathematical Physics

Series contained in this Journal

368 citations

173 citations

167 citations

125 citations

125 citations