EFT, RG, and Computation

Autumn, 2009

Welcome to the
Physics 880.05 Many-Body Physics: EFT, RG, and Computation home page!

URL: http://www.physics.ohio-state.edu/physics/ntg/880_2009/manybody.php

The course information is available here plus lots of supplementary
info. Please check this page regularly.

- 11-Dec-2009 --- Some hints for PS#4 problem 2(b) are available.
- 07-Dec-2009 --- Some notes relevant for the final problem set on computing observables on the lattice are available.
- 02-Dec-2009 --- The nineteenth set of lecture notes are available.
- 01-Dec-2009 --- Problem Set #4 is linked below.
- 24-Nov-2009 --- *** Typo alert! *** Two zeta's were missing in equation (2) of PS#3 (thanks to Ken Patton for spotting this). You need them to get the matrix correct! A corrected version (with the other typo's fixed) is linked below.
- 18-Oct-2009 --- Some notes from the 780 Computational Physics course on Monte Carlo methods are linked under Supplementary Reading.
- 13-Oct-2009 --- Solving the many-particle Schrodinger equation: Take a look at the beginning of Walter Kohn's Nobel Lecture for what he says about the "exponential wall".
- 08-Sep-2009 --- The website from a past version (from 2003) of Physics 880.05 is still available (ask about the username/password to look at the old lecture notes). The present course will be only one quarter and is not a repeat of the first quarter of the 2003 version.

Reading | Topic | Reading Due | |
---|---|---|---|

excerpt from Suzuki/Varga chapter 3 | "Introduction to Variational Methods" (for SVM discussion) | 09/28/09 | |

overview of SVM | "Stochastic variational methods in few-body systems" by Suzuki, Varga, Usukura | 09/28/09 | |

excerpt from
Morse, Thermal Physics chapter 8 |
Thermodynamic potentials and Legendre transformations between them, including for "extra" pairs of variables. | 09/28/09 | |

excerpts from F&W chapter two | Statistical mechanics review and application to non-interacting Fermi and Bose gases in second quantization | 09/28/09 | |

excerpt from Zinn-Justin chapter 1 | Summary of formulas for gaussian integration, including Grassmann variables. | 09/30/09 | |

excerpts from Negele/Orland text | Functional integral formulation; basics of perturbation theory with path integrals | 10/08/09 | |

excerpt from Atland/Simons text | Functional integral formulation for quantum mechanics. | 10/08/09 | |

Notes from 780.20 on Monte Carlo methods | Excerpts from the class notes for Computational Physics on random numbers, Monte Carlo methods, the Metropolis algorithm, and more. | 10/18/09 | |

excerpts from Negele/Orland text | Coherent states and Gaussian integrals | 10/21/09 | |

excerpts from Negele/Orland text | Hugenholtz diagrams and Feynman rules | 10/21/09 | |

excerpts from Negele/Orland text | Irreducible diagrams and integral equations | 11/04/09 |

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Click on the problem set number to get a pdf copy of the assignment.

Due Date(s) | Assignment | Comments | Solutions |
---|---|---|---|

10/12/09 and 10/16/09 | #1 | Hints, suggestions, etc. | solutions |

10/27/09 and 10/30/09 | #2 | Programs: svm_test.zip, MCIntEval.zip, expm_demos.zip. Hints, suggestions, etc. | solutions |

11/24/09 | #3 | Hints, suggestions, etc. | solutions |

12/11/09 | #4 | Programs: one_d_fermions.zip. Notes on computing observables. Hints, suggestions, etc. | solutions |

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Date Out | Handout | Comments |
---|---|---|

23-Sep-2009 | overview 1 | Graphs of phenomenological central nucleon-nucleon potential and potential between He-3 atoms plus lattice QCD calculations. |

23-Sep-2009 | overview 2 | Nuclear three-body forces. |

23-Sep-2009 | overview 3 | Nuclear dof's and Weinberg RG quote |

23-Sep-2009 | overview 4 | Digital potential and resolution analogy |

23-Sep-2009 | overview 5 | Full Configuration Interaction (FCI) scaling with basis size (from James Vary talk, 2008) |

28-Sep-2009 | Sample SVM results | Some figures and tables from the Suzuki/Varga SVM book. |

28-Sep-2009 | Ill-conditioning example | A simple MATLAB example of an ill-conditioned matrix because of near linear dependence (as encountered with the SVM). |

30-Sep-2009 | Approximations to model Z | Error plots for approximations to a simple integral that serves as a model partition function. Ordinary perturbation theory is compared to expanding about saddlepoints (one in this case). |

02-Oct-2009 | Using MATLAB for Linear Algebra | Summary guide to using MATLAB matrix functions. |

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This is a list of excerpts from texts or review articles handed out in class (extras are available from Prof. Furnstahl). In most cases, they can't be scanned because of copyright issues, but there will be some that have links to downloadable versions (e.g., from the arXiv).

Date Out | Handout | Comments |
---|---|---|

23-Sep-2009 | First pages from review article | "Effective Field Theory and Finite-Density Systems" |

23-Sep-2009 | First pages from Blume/Daily article | "Illustration of universal relations for trapped four-fermion system with arbitrary s-wave scattering length" |

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The notes are in PDF format only.

Class Date | Notes | Comments |
---|---|---|

23-Sep-2009 | lecture 1 | Course logistics and references, physics overview. |

28-Sep-2009 | lecture 2 | Solving the (many-body) Schrodinger equation, variational/basis wave function methods, Stochastic Variation Method (SVM). |

30-Sep-2009 | lecture 3 | Review of thermodynamics/statistical mechanics relevant for many-body path integrals; introduction of model partition function. |

05-Oct-2009 | lecture 4 | Symmetry factors for the model partition function. |

07-Oct-2009 | lecture 5 | Loose ends on symmetry factors and partial resummations. Saddlepoint/steepest descent and auxiliary field manipulations for the model partition function. |

12-Oct-2009 | lecture 6 | Introduction to the functional integral formalism, using a single-particle quantum mechanics example. Also an introduction to the stochastic evaluation of path integrals. |

14-Oct-2009 | lecture 7 | Recap and further discussion of stochastic evaluation of path integrals and follow-ups to the one-particle path integral, leading to its generalization. |

19-Oct-2009 | lecture 8 | Perturbation theory for the one-particle path integral, functional derivatives, intro to 2nd quantization and coherent states. |

21-Oct-2009 | lecture 9 | Functional derivative and Feynman rule practice. 2nd quantization and field operators. Coherent states and classical harmonic oscillator. |

26-Oct-2009 | lecture 10 | Coherent state path integral, Grassmann numbers and coherent states, dilute Fermi gas with delta function interaction, Feynman rules. |

28-Oct-2009 | lecture 11 | Molecular dynamics and preliminary inversion strategies. |

02-Nov-2009 | lecture 12 | Comments on PS#2; background on non-interaction fermion and boson noninteracting gases. |

04-Nov-2009 | lecture 13 | Feynman rules recap. Results for non-interacting system. Spin-sum simplification with Mathematica. Spin-dependent force. Beachball diagram. Dyson's equation. |

09-Nov-2009 | lecture 14 | Momentum space Feynman rules example. Beachball lead-in to effective field theory (EFT). |

11-Nov-2009 | lecture 15 | Further discussion of hybrid Monte Carlo. |

16-Nov-2009 | lecture 16 | Renormalization in the EFT for short-range interactions, with application to the dilute Fermi gas. Cutoff and dimensional regularization. |

18-Nov-2009 | lecture 17 | Large nu limit of dilute Fermi gas. Start of effective action discussion. |

23-Nov-2009 | lecture 18 | Application of effective action to large N (large nu) example. |

30-Nov-2009 | lecture 19 | Effective action for pairing. |

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On some machines clicking the link will start Mathematica directly. If not, "right click" to save the file and then read it into Mathematica separately.

Date Out | Notebook | Comments |
---|---|---|

10/15/09 | Rolling Dice | Mathematica notebook that simulates the rolling of dice, where the outcome is a random integer from 1 to 6. Histograms are generated for different numbers of "trials" (i.e., throws of the dice), providing visualization of how the fluctuations scale with the number of trials. |

11/04/09 | DeltaSimplify | Mathematica package defining DeltaSimplify to simplify spin sums. |

11/04/09 | Spin Sums I | Mathematica notebook with examples of evaluating spin sums using the simplification package (deltasimplify.m). |

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The call number is linked to the OSCAR entry. In some cases, this entry includes the table of contents of the book.

Author(s) | Title | Call no. | Comments |
---|---|---|---|

A. Atland and B. Simons | Condensed Matter Field Theory | QC173.454 .A48 2006 | Recent book very much in the spirit of the RG/EFT approach to many-body physics with emphasis on low-energy field theories, path integrals, universality. More a condensed matter than particle theory outlook. Expensive! |

A.L. Fetter and J.D. Walecka | Quantum Theory of Many-Particle Systems | QC174.5.F43 1971 | Classic text, but pre-path integrals. Now available in an inexpensive (about $20) Dover reprint. Get it! |

J.W. Negele and H. Orland | Quantum Many-Particle Systems | QC174.17.P7 N44 1988 | Detailed and careful use of path integrals. Full of good physics but most of the examples are in the problems, so it can be difficult to learn from. |

J. Zinn-Justin | Quantum field theory and critical phenomena | QC174.45 .Z56 2002 | Encyclopedic, rigorous reference. Excellent on path integrals and renormalization. |

N. Nagaosa | Quantum Field Theory in Condensed Matter Physics | QC174.45.N27 1999 | Covers path integral methods and symmetry breaking. |

M. Stone | The Physics of Quantum Fields | QC174.45.S79 2000 | A combined introduction to quantum field theory as applied to particle physics problems and to nonrelativistic many-body problems. Some very nice explanations. |

R.D. Mattuck | A Guide to Feynman Diagrams in the Many-Body Problems | QC174.5.M34 1967 | This is a nice, intuitive guide to the meaning and use of Feynman diagrams. The second edition is from 1976, but the library may only have the older edition. A paperback version is available from Amazon (and probably elsewere) for under $11. |

W.H. Dickhoff and D. van Neck | Many-body theory exposed! : propagator description of quantum mechanics in many-body systems | QC174.17.P7 D53 2008 | Up-to-date propagator-based description of many-body physics. No path integrals, so complementary to our treatment in many respects. |

N. Goldenfeld | Lectures on Phase Transitions and the Renormalization Group | QC175.16.P5 G65 1992 | The discussion of scaling, dimensional analysis, and phase transitions is wonderful. |

G.D. Mahan | Many-Particle Physics | QC176.M24 2000 | Standard, encyclopedic reference. |

K. Huang | Statistical Mechanics | QC174.8.H83 1987 | Excellent choice for general treatment of statistical mechanics, with good sections on many-body physics. |

J. Kohanoff | Electronic structure calculations for solids and molecules : theory and computational methods | QD462.6.D45 K64 2006 | Good, up-to-date introductory guide to quantum chemistry methods. |

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The call number is linked to the OSCAR entry. In some cases, this entry includes the table of contents of the book.

Author(s) | Title | Call no. | Comments |
---|---|---|---|

P. Nozieres | Theory of Interacting Fermi Systems | QC174.5 .N651 1964 | Outdated in many ways, but great physics explanations. (Three copies are available.) |

J.F. Donoghue, E. Golowich, B.R. Holstein | Dynamics of the Standard Model | QC794.6.S75 D66 1991 | Good introduction to low-energy, standard model physics. |

P. Ring and P. Schuck | The Nuclear Many-Body Problem | QC174.17.P7 R56 1980 | Somewhat out of date, but still a good, encyclopedic guide to the nuclear many-body problem. Doesn't discuss Green's function methods much and no path integrals. |

P.J. Siemens and A.S. Jensen | Elements of Nuclei: Many-Body Physics with the Strong Interaction | QC793.3.S8 S54 1987 | Good survey of nuclear phenomenology and basic methods. |

Abrikosov, Gorkov and Dzyalozinskii. | Methods of Quantum Field Theory in Statistical Physics | QC174.4 .A21 1975 | Old but classic. Referred to as "AGD". |

P.W. Anderson | Basic notions of condensed matter physics | QC173.4.C65 A53 1984 | Another classic. |

A.M. Tsvelik | Quantum Field Theory in Condensed Matter Physics | QC174.45.T79 1995 | Lots on one-dimensional systems. |

E. Fradkin | Field theories of condensed matter systems | QC611.98.H54 F73 1991 | Good reference for fractional statistics and quantum hall effect. |

Y. Suzuki and K. Varga | Stochastic variational approach to quantum-mechanical few-body problems | QC174.17.P7 S89 1998 | The bible on the SVM method. Very accessible. |

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- P. Lepage, "How to Renormalize the Schrodinger Equation" (1997).
- R. Shankar, "Renormalization-group approach to interacting
fermions",
Rev. Mod. Phys.
**66**, 129 (1994). - R.J. Furnstahl, G. Rupak, and T. Schafer,
"Effective
Field Theory and Finite-Density Systems",
Annual Review of Nuclear and Particle Science
**58**, 1 (2008). - J. Drut, R.J. Furnstahl, and L. Platter, "Toward ab initio density functional theory for nuclei", to appear in Progress in Particle and Nuclear Physics (2009).
- S. Aoki, T. Hatsuda, N. Ishii, "Nuclear Force from Monte Carlo Simulations of Lattice Quantum Chromodynamics" (2008).
- S. Weinberg, "Why the Renormalization Group is a Good Thing". Great essay!
- Path Integral Monte Carlo references
- Online notes on the path integral Monte Carlo method.
- More Online notes on the path integral Monte Carlo method.
- Open source path integral simulation program developed by the Shumway research group.
- D. M. Ceperley, "Path integrals in the theory of condensed helium", Rev. Mod. Phys. 67 279.

- W. Kohn, "Nobel Lecture: Electronic structure of matter --- wave functions and density functionals", Rev. Mod. Phys. 71 (1999) 1253.
- N.J. Higham, "The Scaling and Squaring Method for the Matrix Exponential Revisited", SIAM. J. Matrix Anal. & Appl. 26 (2005) 1179.

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Your comments and suggestions are appreciated.

[OSU Physics] [Math and Physical Sciences] [Ohio State University]

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furnstahl.1@osu.edu