Modern Statistical Mechanics Paul Fendley

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1 Modern Statistical Mechanics Paul Fendley The point of the book This book, Modern Statistical Mechanics, is an attempt to cover the gap between what is taught in a conventional statistical mechanics class (undergraduate or graduate) and between what is necessary to understand current research involving many-body physics. Put less politely, learning the virial expansion may be important, but it s not that exciting, and not that useful to much of what goes on in physics research today. The aim is to introduce some of the basics of many-body physics to a wide audience. This audience obviously includes students in condensed-matter theory, but I would like to make comprehensible the reason why experimentalists choose some systems to study: e.g. why they are working so hard to find spin liquids. Moreover, this approach has the nice feature that it makes statistical mechanics much more closely related to the kinds of research high-energy theorists do. In fact, some of the specific problems discussed (spin chains, dimers) are currently fashionable in the string-theory world. To give a simple example of the gap between traditional courses and current research reality: a typical statistical mechanics course probably does not even mention the Heisenberg model, much less explore its properties. For example, a way to illuminate the profound difference between ferromagnets and antiferromagnets is to solve say a Heisenberg chain with four sites. Using translation invariance this is simple to do, but I ve never seen it done in a text. One aim of the first part of the book is to teach such basics. In general, the first part of the book will review statistical mechanics, emphasizing the central role of the partition function for both quantum and classical cases. The idea is to remind readers that in both cases one sums over all states with some weighting, and stress how closely related quantum and classical become in this approach. Thus the only prerequisites for reading this book will be courses in statistical mechanics and quantum mechanics (and because of this review, good undergraduate-level courses should suffice). This then sets up the second part, whose aim is to introduce field theory in a very natural fashion. The emphasis is to show how a field theory arises by taking the continuum limit of a lattice model. In particular, for this to be possible, there must be a critical point nearby (a fact not understood by much of the high-energy theory community). This approach will start by following the strategy in Kogut s beautiful review article; it s amazing that no one as far as I know has incorporated the approach of this article into a textbook in the last 30 years. One nice thing about this approach is that it also allows fermions to be introduced in a very natural fashion. They provide a nice way of solving both the quantum Ising chain and classical dimer models on any planar lattice. Moreover, what s nice about introducing dimer models at an early point in the book is that the partition function is simply a counting problem, e.g. the number of ways of putting 32 dominoes, each covering two squares, onto a chessboard without any overlapping. This I think is a lot less scary-sounding than saying here we introduce the path integral approach. Another nice thing about introducing these models is that one can derive very non-trivial exact results without much trauma.

2 In both cases one can then take the continuum limit, and recover free-fermion field theories (in both Hamiltonian and Lagrangian formulations). One can then do simple computations in the field theory, but obtain non-trivial exact results, such as critical exponents. This thus leads naturally to the next topic, universality and critical points. Since there are already several good books on the renormalization group, this book won t cover that in much depth, but simple concepts like mean field theory will be introduced here. The next section of the book will treat another important topic given short shrift, the Coulomb gas approach to two-dimensional classical systems. The celebrated Kosterlitz-Thouless transition is a central part of this physics, but not the end of the story. One can obtain a variety of exact results for a number of systems; in essence, this is a simple version of the much more elaborate formalism of bosonization. Thus after discussing the Coulomb gas (and explaining how it can be used to find a variety of exact results), we can then move onto bosonization of interacting fermion field theories (the Luttinger liquid and the 1d Hubbard model). Review articles inspiring this section include an old one by Nienhuis and a more recent one by Cardy. Finally, all these topics leads to full blown conformal field theory. Since the key examples of Ising/free fermions and free bosons will have already have been discussed in depth, this will take less effort than one may expect. The famous text on conformal field theory overwhelmed the essential ingredients with technical diversions; the aim here is to take the opposite approach. Two nice (now neglected) review articles are by Ginsparg and Cardy. There are myriad applications of conformal field theory; one nice one which brings things full circle is to study non-abelian anyons and topological quantum computation. Models with nonabelian anyons have behavior closely related to spin-liquid physics, but require the full power of conformal field theory to understand. Spin-liquid physics, on the other hand, can be illustrated nicely by studying quantum dimer models, which intimately involve the classical dimer models studied earlier in the book. At this point there are many more advanced topics which can be discussed (some listed in the outline), but we ll have to see how long the book is at this point!

3 Outline I. Basic classical statistical mechanics 1. The classical partition function The Ising model 2. The transfer matrix The one and two-dimensional Ising models 3. Correlation functions II. Basic quantum statistical mechanics 1. Spin and the two-state quantum system 2. The quantum partition function The Heisenberg model 3. Quantum in d dimensions is classical in d + 1 dimensions The 2d Ising model/ising chain III. Symmetries and order parameters 1. Discrete symmetries 2. Continuous abelian symmetries and conserved currents 3. Translation symmetry and momentum 4. The XY and Heisenberg mdoels IV. Order parameters 1. Order parameters 2. Spontaneous symmetry breaking 3. Continuous non-abelian symmetries Spin V. Fermions 1. Rewriting the Ising chain in terms of fermions 2. Solving the Ising chain 3. The spectrum VI. Critical points 1. Gapped vs. gaplessness

4 2. Quantum critical points 3. Classical critical points 4. Exponential vs. algebraic decay of correlators correlation length critical exponents VII. Lattices and fields VIII. Scaling IX. Bosons 1. Dimers as a fermionic field theory 2. The Ising chain as a fermionic field theory 3. Lorentz and rotational invariance 4. The Ising model as a fermionic field theory 1. Universality 2. Mean field theory 3. Power counting/ginzburg criterion 4. Relevance and irrelevance 5. Marginality 1. Six-vertex model 2. XXZ chain/emergent U(1) symmetry 3. Dimers again 4. Free bosonic field theory X. Coulomb gas 1. The bosonic two-point function 2. Classical electrodynamics in two dimensions 3. Complex coordinates 4. Scaling dimensions in the free boson theory locality 5. Kosterlitz-Thouless transition XI. Bosonization 1. Dimers yet again monomer scaling

5 2. Ising 2 /quantum XY chain spin operator 3. Interacting fermions Tomonaga-Thirring-Luttinger-Schultz-Mattis-Lieb-Emery-Luther-Coleman-Mandelstam XII. Conformal symmetry 1. Classical conformal transformations 2. The energy-momentum tensor 3. The conformal anomaly 4. The ground-state (Casimir) energy XIII. The Ising conformal field theory 1. The spin field 2. The partition function 3. The operator product expansion XIV. The Hamiltonian formulation of conformal field theory 1. Hamiltonian formulation of free bosons and fermions 2. An infinite-dimensional symmetry algebra 3. Primary fields/states 4. Null vectors 5. Characters XV. Potts, height and minimal models 1. The six-vertex model and the Temperley-Lieb algebra 2. Potts models 3. Height/RSOS models 4. Minimal conformal field theories 5. Coulomb gas redux Feigen/Fuks/Dotsenko/Fateev XVI. The Heisenberg model and sigma models 1. Bethe s solution for the spin 1/2 chain 2. The O(3) sigma model 3. The large-spin limit

6 4. The Haldane gap 5. AKLT models 6. Higher-spin critical points XVII. Non-abelian bosonization 1. The Wess-Zumino-Witten models 2. Symmetry currents in the spin-1/2 Heisenberg model 3. Kac-Moody algebras 4. Spin-charge separation 5. Knizhnik-Zamolodchikov equations XVIII. Quantum impurity problems 1. Boundary conformal field theory 2. boundary sine-gordon 3. the Kondo problem 4. multi-channel Kondo problem XIX. Rational conformal field theories 1. Extended symmetry algebras 2. The fusion algebra 3. Verlinde formula 4. Moore-Seiberg axioms XX. Topological order 1. Quantum dimer models/toric code 2. Anyons 3. Braiding and fusing 4. Temperley-Lieb again 5. Chern-Simons field theory and the Jones polynomial Possible advanced topics (there are many other possibilities) XXI. Classical models with geometric degrees of freedom 1. Dimers

7 2. Potts completely-packed loops domain walls 3. Loops XXII. Gauge symmetries 1. Duality 2. Lattice gauge theory the 3d Ising model the toric code XXIII. Entanglement 1. What is it and why is it interesting? 2. The von Neumann entanglement entropy 3. Universal measures of entanglement CFTs Topological theories XXIV. Integrable models away from criticality XXV. Supersymmetry