Physics Resources

Differential Forms in Algebraic Topology

Differential Geometry & Topology

3 items

▣ PDF / Notes Graduate

Bott & Tu

The best account of de Rham theory, spectral sequences, and characteristic classes for physicists who want honest mathematics.

◈ Notes Advanced UG

Lecture Notes on Differential Geometry

Compiled notes covering smooth manifolds, tangent bundles, Lie derivatives, and connections — with examples drawn from classical mechanics.

◈ Notes Advanced UG

Lecture Notes on Differential Geometry

Compiled notes covering smooth manifolds, tangent bundles, Lie derivatives, and connections — with examples drawn from classical mechanics.

Topology from the Differentiable Viewpoint

John Milnor

A masterclass in economy of thought. Milnor covers Sard's theorem, degree theory, and the Hopf invariant in under 70 pages of pure elegance.

Group Theory & Symmetry

3 items

Systematic treatment of finite groups, Lie groups, and their representations — directly aimed at physicists. Excellent for learning how representations constrain physical Hamiltonians.

Group Theory in Physics

Wu-Ki Tung

▣ PDF / Notes Advanced UG

Notes on Lie Algebras and Lie Groups

Pavel Etingof

Covering su(2), su(3), root systems, Dynkin diagrams, and the classification of semisimple Lie algebras with physical applications throughout.

A narrative account of the classification of finite simple groups, remarkable for making an abstract mathematical achievement feel genuinely dramatic.

Symmetry and the Monster

Mark Ronan

Functional Analysis & Operator Theory

3 items

Mathematical Methods of Classical Mechanics

V. I. Arnold

The gold standard for geometric mechanics — symplectic manifolds, Hamiltonian flows, and integrable systems treated with uncompromising mathematical precision.

◈ Notes Advanced UG

Spectral Theory for Quantum Mechanics

Self-adjoint operators, spectral theorem, Stone's theorem, and the rigorous foundations of quantum mechanics beyond Dirac notation.

▣ PDF / Notes Research

Analysis on Manifolds

James Munkres

Rigorous multivariable analysis culminating in Stokes' theorem on manifolds. The preparation every theoretical physicist should have before differential geometry.

Superconductivity

Cooper Pairs & Broken Symmetry

From BCS theory and Ginzburg-Landau phenomenology to unconventional pairing, Andreev physics, and topological superconductivity. Resources spanning experiment, theory, and computation.

9Resources

Foundational

de Gennes — Superconductivity of Metals and Alloys

P. G. de Gennes · Addison-Wesley · Graduate Level

Arguably the most elegant pedagogical treatment of superconductivity ever written. de Gennes moves from pairing to London theory to the microscopic picture with a clarity that few textbooks match.

BCS Theory & Conventional Superconductivity

3 items

Introduction to Superconductivity

Michael Tinkham

The standard reference for conventional superconductivity. Comprehensive treatment from Meissner effect and flux quantization through BCS and Josephson junctions.

Notes on BCS Theory and Gap Equation

Derivation of the BCS ground state via variational approach, gap equation, density of states, quasiparticles, and the pairing interaction in momentum space.

▣ Lecture Notes Advanced UG

Ginzburg-Landau Theory

Various

Phenomenological GL theory, coherence length, penetration depth, type-I vs type-II classification, Abrikosov vortex lattice, and the free energy landscape.

Unconventional Pairing & Bogoliubov–de Gennes

3 items

◉ Textbook Research

Bogoliubov-de Gennes Method and Its Applications

Jian-Xin Zhu

Self-contained treatment of the BdG formalism, quasiparticle eigenstates, and its applications to inhomogeneous superconductors, interfaces, and vortices.

Notes on p-wave Superconductivity

Triplet pairing, spin structure of the order parameter, Kitaev chain, and the emergence of Majorana zero modes at the ends of a 1D p-wave wire.

▣ Review Article Research

Andreev Reflection and Crossed Andreev Reflection

Blonder, Tinkham & Klapwijk + extensions

The BTK formalism, subgap conductance, and the physics of non-local Andreev processes at SC–normal and SC–SC–normal junctions.

Majorana Modes & Topological SC

3 items

▣ Review Article Research

Alicea — New Directions in the Pursuit of Majorana Fermions

Jason Alicea · Rep. Prog. Phys. 2012

Excellent review of the Kitaev chain, semiconductor nanowires, and topological superconducting phases. Clearly written with experiment in mind.

Non-Abelian Braiding Statistics

Fusion rules, braid group representations, and why Majorana zero modes implement non-Abelian anyons relevant for fault-tolerant quantum computation.

Topological Quantum Computing — Lecture Notes

Nayak, Simon, Stern, Freedman, Das Sarma

The canonical review. From anyons and topological quantum field theory to physical platforms and error correction. Dense but rewarding.

Topological Matter

Topology Meets Condensed Matter

Chern insulators, topological magnons, Berry phases, symmetry-protected topological phases, and the rich interplay between band geometry and physical observables. My primary research domain.

10Resources

Core Reference

Bernevig & Hughes — Topological Insulators and Superconductors

B. A. Bernevig & T. L. Hughes · Princeton University Press · Graduate

The textbook that defined a generation's approach to topological band theory. Model Hamiltonians, topological invariants, surface states, and the periodic table of topological phases, all developed from first principles.

Berry Phase & Topological Invariants

3 items

Berry Phase, Berry Curvature, and Chern Numbers

Adiabatic theorem, Berry's geometric phase, the Berry connection and curvature as a gauge field, Chern number as an integral invariant, and TKNN formula.

▣ Review Advanced UG

Xiao, Chang & Niu — Berry Phase Effects on Electronic Properties

Rev. Mod. Phys. 82, 1959 (2010)

The go-to review on Berry phase effects — anomalous velocity, orbital magnetization, Hall effects — written with admirable clarity. Essential reading before research.

Geometry, Topology and Physics — Nakahara

Mikio Nakahara

Fibre bundle perspective on Berry phase — the mathematical structure underlying gauge fields in quantum mechanics and band theory.

Topological Magnons & Magnonic Phases

3 items

▣ Research Paper Research

Topological Magnon Insulator in Insulating Ferromagnet

L. Zhang, J. Ren, J.-S. Wang, B. Li

Pioneering paper establishing topological characterization of magnon bands in honeycomb ferromagnets — the bosonic analogue of electronic Chern insulators.

Linear Spin-Wave Theory and Magnon Bands

Holstein-Primakoff transformation, spin-wave Hamiltonian, Bogoliubov transformation for bosons, and Chern number computation for magnon bands.

Dirac Nodal Line Magnons in Honeycomb Antiferromagnets

Ganguly (2024) — Preprint Notes

Personal research notes on PT-symmetric magnonic phases in layered honeycomb AFMs, interlayer spin canting, and topological phase transitions.

Symmetry-Protected Topological Phases

4 items

▣ Review Graduate

Hasan & Kane — Colloquium: Topological Insulators

Rev. Mod. Phys. 82, 3045 (2010)

The review that introduced the field to much of the physics community. Time-reversal symmetry, Z₂ invariant, surface states, and experimental discoveries.

Chiu et al. — Classification of Topological Quantum Matter with Symmetries

Rev. Mod. Phys. 88, 035005 (2016)

The Altland-Zirnbauer classification, topological K-theory, and the periodic table of topological insulators and superconductors — systematic and comprehensive.

Bulk-Boundary Correspondence

The SSH model, winding numbers, edge states, and the precise statement of the bulk-boundary correspondence in 1D and 2D topological systems.

Topological Band Theory — Vanderbilt Group Notes

David Vanderbilt

Wannier functions, modern theory of polarization, electric polarization as a Zak phase, and the connection to topological classification.

Quantum Field Theory

Fields, Path Integrals & Renormalisation

QFT from canonical quantisation and path integrals through renormalisation group, anomalies, and topological field theories. Both the particle physics and condensed matter perspectives.

9Resources

Essential

Altland & Simons — Condensed Matter Field Theory

Alexander Altland & Ben Simons · Cambridge University Press · Graduate

The indispensable field theory textbook for condensed matter physicists. Path integrals, coherent states, RG, broken symmetry, and topological field theories all developed through physically motivated examples.

Path Integrals & Canonical Quantisation

3 items

Peskin & Schroeder — An Introduction to Quantum Field Theory

M. E. Peskin & D. V. Schroeder

The standard first graduate QFT course. Canonical quantization, Feynman diagrams, QED, renormalization, and non-Abelian gauge theories.

Path Integral Formulation of QM and QFT

Feynman's sum over paths, coherent state path integrals for bosons and fermions, generating functionals, and the Hubbard-Stratonovich transformation.

◉ Textbook Advanced UG

Zee — Quantum Field Theory in a Nutshell

Anthony Zee

A conceptual tour of QFT with exceptional physical intuition. Better for building understanding than for calculation — the ideal companion to Peskin-Schroeder.

Renormalisation Group & Critical Phenomena

3 items

Shankar — Renormalisation Group for Interacting Fermions

R. Shankar · Rev. Mod. Phys. (1994)

The clearest pedagogical account of Wilsonian RG applied to Fermi liquids. Shows how BCS superconductivity emerges as a marginal instability. Beautifully written.

RG Flow, Fixed Points, and Universality

Wilson's RG in real and momentum space, beta functions, relevant vs irrelevant operators, universality classes, and the epsilon expansion near d=4.

Kardar — Statistical Physics of Fields

Mehran Kardar

Fluctuations, scaling, and the RG developed through the lens of statistical mechanics. Kardar writes with unusual clarity on a technically demanding subject.

Topological Field Theories & Anomalies

3 items

Chern-Simons Theory and Topological Order

E. Witten & others

Chern-Simons gauge theories, their relation to knot invariants, fractional quantum Hall effect, and anyon statistics emerging from topological field theory.

Anomalies in Quantum Field Theory

Chiral anomaly, 't Hooft anomaly matching, anomaly inflow, and the deep connection between bulk anomalies and boundary physics in topological phases.

▣ Lecture Notes Research

Wen — Quantum Field Theory of Many-Body Systems

Xiao-Gang Wen

Topological order, string-net condensation, long-range entanglement, and the QFT of quantum Hall states. Wen's unique perspective on what makes a phase of matter truly new.

Computational & Data-Driven Physics

Algorithms, Numerics & ML for Physics

Exact diagonalization, tensor networks, Monte Carlo, and machine learning approaches to quantum many-body problems. Resources for both the mathematical foundations and practical implementation.

9Resources

Practical

Sandvik — Computational Studies of Quantum Spin Systems

Anders W. Sandvik · AIP Conference Proceedings · Graduate

Comprehensive lecture notes on exact diagonalization, quantum Monte Carlo, and stochastic series expansion for quantum spin systems. An invaluable practical guide to numerical many-body physics.

Exact Diagonalization & Tight-Binding

3 items

Exact Diagonalization of Quantum Many-Body Systems

Personal Notes + Weisse & Fehske

Hilbert space construction, symmetry sectors, sparse matrix methods, Lanczos algorithm, and spectral functions via continued fractions.

▣ Tutorial Advanced UG

PythTB and Tight-Binding Models

Vanderbilt Group

Practical tutorial for constructing and solving tight-binding models using PythTB — computing band structures, Berry curvature, Chern numbers, and edge spectra.

◈ Code Notes Graduate

Computing Topological Invariants Numerically

Discretized Berry curvature, link variables for Chern number on a lattice, Z₂ invariant by parity eigenvalues, and Wannier charge centres.

Tensor Networks & DMRG

3 items

Orús — A Practical Introduction to Tensor Networks

Román Orús · Ann. Phys. (2014)

MPS, PEPS, MERA, and the variational manifold perspective on tensor network methods. Accessible entry point to a technically deep subject.

Schollwöck — DMRG in the Age of Matrix Product States

Ulrich Schollwöck · Ann. Phys. (2011)

The definitive modern account of DMRG. Unifies the original White algorithm with MPS formalism, time evolution, and finite-temperature methods.

Entanglement and Area Laws

Entanglement entropy in many-body systems, area laws for ground states, volume laws for excited states, and their implications for classical simulability.

Machine Learning for Quantum Matter

3 items

Carrasquilla & Melko — Machine Learning Phases of Matter

Nature Physics 13, 431 (2017)

Seminal paper demonstrating that neural networks can learn to identify quantum phases of matter from Monte Carlo configurations — opening a rich research direction.

▣ Review Graduate

A Review of ML Approaches to Phase Diagrams

Mehta et al. · Physics Reports (2019)

High-dimensional survey connecting statistical mechanics to machine learning — renormalization group as a deep learning analogy, restricted Boltzmann machines, variational autoencoders.

Decoherence & Open Quantum Systems

Environment, Noise & Quantum Dissipation

Study of quantum systems coupled to environments — decoherence, dissipation, Lindblad dynamics, quantum trajectories, and non-equilibrium quantum statistical mechanics. Essential for quantum computing and realistic condensed matter systems.

6Resources

Core Reference

Breuer & Petruccione — The Theory of Open Quantum Systems

Oxford University Press · Graduate

The definitive reference on open quantum systems: density matrices, master equations, Markovian approximations, and quantum noise. A must-read for decoherence and quantum information.

Decoherence & Density Matrix Formalism

2 items

Density Matrix and Reduced Dynamics

Partial trace, mixed states, von Neumann entropy, and how tracing out environment leads to decoherence and classicality.

▣ Review Graduate

Decoherence and the Quantum-to-Classical Transition

W. H. Zurek

Classic review explaining how environment-induced decoherence selects preferred basis and explains classical emergence.

Master Equations & Lindblad Dynamics

2 items

Lindblad Master Equation

GKSL form, completely positive trace-preserving maps, dissipators, and physical interpretation of jump operators.

Open Quantum Systems — Lecture Notes

Various

Born-Markov approximation, Redfield equation, and derivation of Lindblad form from system-bath coupling.

Quantum Trajectories & Measurement

2 items

Quantum Jump Method

Stochastic evolution, quantum trajectories, and Monte Carlo wavefunction methods.

Continuous Quantum Measurement

Wiseman & Milburn

Measurement backaction, quantum trajectories, and feedback control in open quantum systems.

Pop Physics

Physics for the Curious

Books, articles, and talks that convey the wonder of physics to a broad audience — without sacrificing honesty. A selection of what I consider genuinely excellent popular science writing.

8Resources

Remarkable

Feynman — The Character of Physical Law

Richard P. Feynman · MIT Press · General Audience

Seven Messenger Lectures from 1964 in which Feynman distils what is most strange and beautiful about the laws of nature. On gravity, symmetry, the quantum, and what it means to understand something at all.