--- # System prepended metadata title: Reading Materials and Discussion --- Here, I am providing a list of materials and some thoughts on how they might best be used. ## Mathematics Provided one is already familiar with single-variable calculus (differential and integral), I believe these resources to be a good starting point for the mathematical foundations of the work we do in condensed matter (and physics generally). If you are already solid on the material below and desire a more formal overview of vector spaces and linear algebra, I suggest Shankar's Principles of Quantum Mechanics, Chapter 1 and Appendix A1. Shankar is more "mathy" than one necesarily needs to become productive in research during undergrad, but can be useful for a deeper understanding. More generally, for the study of mathematical physics, I suggest Arfken and Weber, as I know it to be a standard text thatI used in both undergraduate and graduate mathematical physics classes. I also know there is a text by Boas, which I think is a nice place to start learning Fourier transforms (see below). #### Vectors/Multivariable Calculus : Griffith's Introduction to Electromagnetism (E&M) The first chapter of Griffiths E&M is a nice introduction to vectors and multivariable calculus, along with Appendix A as a technical supplement. The 'flyleaf' of Griffith's E&M is also very useful when one needs to recall how to change units of integration, such as from rectilinear to spherical or different vector product rules. This (the flyleaf) is usually found at the start or end of pdf versions of the text. Furthermore, for a deeper dive on vector calculus, there is this set of [lecture notes](https://www.damtp.cam.ac.uk/user/tong/vc/vc.pdf) by David Tong, a physicist at Cambridge. #### Linear Algebra : Griffith's Introduction to Quantum Mechanics (QM) Griffith's E&M introduces what vectors are and how to do calculus with them, but there is an entire other use for vectors, namely linear algebra. An introduction to this and its relavance to QM is discussed in Appendix A of Griffiths QM, although the visuals there are lacking. Furthermore, there is a wonderful Youtube series, [The Essence of Linear Algebra](https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&si=kDiIkIlo3ja_d7VU) by Grant Sanderson (3Blue1Brown), that was extremely helpful for me in my undergraduate studies, and is the first place where I felt like I understood what a matrix determinant represents (and why it is zero for non-invertible matrices). He also has quite wonderful sets of videos on [calculus](https://youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr&si=hfn2_eKMeLmycWdD) (single variable), [differential equations](https://youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6&si=1IVymKSklYf9Dufh), and [neural networks](https://youtube.com/playlist?list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi&si=Y3ARdodTTPSb1iby), as well as many other great standalone videos discussing math, physics, computer science, etc. Note that the videos made by Grant Sanderson should not replace regular learning of material. Rather, I find they work best as a supplement before, during, and/or after learning a subject (in a more traditional/pedagogical setting with exercises) to gain intuition/insight. #### Fourier Series/Transform Fourier transforms are fundamental in physics. In QM/condensed matter, a Fourier transform of basis vectors can exremely simplify some problems, or at the very least offer a different viewpoint. The basic idea is that the Fourier transform is a specific decomposition of some function $f(x)$. Recall that, in calculus, one has the [Taylor/Maclaurin Series](https://en.wikipedia.org/wiki/Taylor_series) $$ f(x) = f(0) + \sum_{i=1}^\infty \frac{x^n}{n!}\left(\frac{d}{dx}\right)^n f(x) $$ which decomposes a function into polynomial components. In a similar way, one can decompose a function into waves, i.e., sine and cosine components. This decomposition is the referred to as a Fourier transform. The references here are many, as there are many approaches to teaching mathematical physics. I offer 2 different levels of difficulty in approach. Boas, Mathematical Methods in the Physical Sciences, Chapter 7 : I am less familiar with this text, but it seems to be a nice and intuitive introduction to both the idea and machinery of Fourier transforms. Arfken and Weber, Mathematical Methods for Physicists, chapter 14 : A standard text, although it is not very introductory and assumes both complex analysis and series (as in summations) knowledge by that point in the text. ## Quantum Mechanics : Now, with a grip on vectors, calculus, and linear algebra, we can turn to the foundations of condensed matter, namely quantum mechanics (QM). While not at a introductory level, I feel obligated to mention that chapter one of Sakurai's QM also contains good foundational knowledge for quantum mechanics. Particularly, the answer to problem 1.11 on the eigenvectors of ${\mathbf{S}}\cdot\hat{\mathbf{n}}$ (i.e., eigenvectors of an arbitrary sum of Pauli matrices) is useful to know. #### Griffith's Introduction to Quantum Mechanics (QM) Griffith's E&M introduces what vectors are and how to do calculus with them, but there is an entire other use for vectors, namely linear algebra. For this, we look to Griffith's once again. Chapters 1-4 provide a good foundation, and chapter 5 gets into solids, but may not be the most direct approach for what we are doing. ## Statistical Mechanics : While not required (at the start), statistical mechanics will be useful when looking at standard textbooks on condensed matter/solid state physics as they often start with thermodynamical and statistical quantities like specific heat, free energies and partition functions (e.g., phonons of a chain) before doing the kind of condensed matter we do in this lab. For a undergrad textbook, I enjoyed Thermal Physics by Schroeder, but I'm sure there are others. ## Condensed Matter : Now that we have our math and quantum mechanics, we can start looking to condensed matter. The introductory pathway here is less clear, as many standard undergraduate textbooks assume familiarity with QM, statsical mechanics, and classical mechanics (for a Classical mechanics book, I enjoyed John R Taylor's text). So, in addition to textbooks, we offer some note/paper based resources on the subject that move faster into the kind of work we do in this lab. #### Lecture Notes : [These](https://arxiv.org/abs/1509.02295) notes by Asboth et al, which cover exactly the kind of foundational material for what we do in this lab. The first four chapters are a good introduction for 1D systems (and the rest is also good). However, this may be a better resource once one is comfortable with things like Bloch Hamiltonians/States, tight-binding descriptions, and the pathway between position and reciprocal spaces (see Simon chapters 12 and 13 below) [Here](https://www.damtp.cam.ac.uk/user/tong/aqm/solidstate.pdf) is a set of lecture notes on introductory condensed matter by David Tong that I believe to be good. For getting started on tight-binding calculations, one can likely skip the first chapter, although the content there is important. Note that David Tong has a whole [website](https://www.damtp.cam.ac.uk/user/tong/teaching.html) filled with lecture notes on topics ranging from Newtonian mechanics to supersymmetry #### Textbooks : For textbooks, a standard one is The Oxford Solid State Basics by Steve Simon. While this book won't be a "shortest path" option to producing results in this lab, it is a pretty good book and has [companion lectures](https://youtu.be/XQk25fSJkL8?si=XAuNUCEv9x2cwK01) from Steve Simon himself, which I found useful when going through the text. Below I briefly discuss some of the relevant sections of the book. Section I (Chapters 2-4) : This is the canonical solid state physics, e.g., the story of Dulong-Petit $\rightarrow$ Boltzmann $\rightarrow$ Einstein $\rightarrow$ Debye pathway through our understanding of the specific heat of solids and the Drude/Sommerfield models of electrons. Good stuff. Section II (chapters 5-7) : These chapters are on a chemical viewpoint of matter. Things such as different types of bonding and electronic screening of nuclear charge are discussed. In my undergraduate course, these chapters were skipped as they aren't really required for the later chapters. However, Section 6.2.2 has a discussion related to the tight-binding method, and would be good to look at for an understanding of why we can use it. In chemistry, tight-binding is often referred to as "linear combination of atomic orbitals" (LCAO). Section III (chapters 8-11) : In this section, we are now looking at 1D systems and single chains of atoms. Phononcsonatomic and diatomic chains are discussed. Chapter 9 on the monatomic chain introduces the notion of "crystal momentum," which is fundamental to condensed matter physics when studying crystalline systems under periodic boundary conditions (these are sometimes called Born-Von Karman boundary conditions). The discussions in Chapters 9 and 10 can be viewed of as vibrational analogies of the tight-binding metal and SSH chains for electronic systems. Chapter 11 introduces the tight binding method as we use it today and energy bands/band structures. Section IV (chapters 12 and 13) : Here, the formalism for describing crystals is discussed. Concepts such as the direct lattice vectors and the Wigner-Seitz unit cell, as well as the reciprocal counterparts, the reciprocal lattice and (first) Brillouin zone. These chapters can often feel dull upon learning, but are fundamental for discussing crystalline systems. Section V (chapter 14) : Not necessary, but can be interesting. Section VI (chapters 15-18) : Chapters 15 and 16 revisit band structures in multiple spatial dimensions and discuss Bloch's theorem, which is a lot of the reason for why working in crystal momentum space can be so advantageous. It is worth noting that, while Bloch's theorem is often just stated for the reader to accept, the idea is simple once one is familiar with the fact that any two commuting operators (matrices) share a common basis of eigenvectors. Given that, we then note that, if our system (Hamiltonian, $H$) is a pristine (defectless) crystal under periodic boundary conditions, then $H$ will commute with the discrete translation operator $T_{\mathbf{R}}$ that translates by a direct lattice vector $\mathbf{R}$ (definition 13.1 in the text is the first mention of the word ***direct***). If these operators commute ($[H,T_{\mathbf{R}}]=HT_{\mathbf{R}}-T_{\mathbf{R}}H=0$), then their simultaneous eigenvectors will be the Bloch states $\psi_\mathbf{k}(\mathbf{R} + \mathbf{r})$ ($\mathbf{k}$ is crystal momentum and $\mathbf{r}$ is a vector the denotes position within the unit cell) with a unit-cell periodic component $u_\mathbf{k}(\mathbf{r}) = u_\mathbf{k}(\mathbf{r} + \mathbf{R})$ such that $\psi_\mathbf{k}(\mathbf{R} + \mathbf{r})=e^{\mathrm{i}\mathbf{k}\cdot(\mathbf{R}+ \mathbf{r})} u_\mathbf{k}(\mathbf{r})$. In chapter 16, Simon refers to the periodic component of a Bloch state, $u_\mathbf{k}$, as the Bloch function and the eigenstate $\psi_\mathbf{k}$ as a *modified plane wave*, but they are commonly referred to as the Bloch state $\psi_\mathbf{k}$ and its periodic component $u_\mathbf{k}$. Remaining Sections : The rest of this book can be viewed as special topics. The later chapter, discussing the Hubbard model, could be good for one interested in doing interacting/many-body research, although there are likely better places to learn about the Hubbard model (and second quantization).