# An Algorithmic Perspective on Strongly Correlated Systems

#### Course Logistics

• Lectures: TR 3:00-4:30 Loomis 222
• Professor: Bryan Clark
• email:
• Office: 2111 Engineering Science Building (ESB)
• Office Hours: Thursday, 4:30-5:30
• email:
• office:
• Office Hours: TBD
• Piazza Discussion Board: Find by going to piazza and search for PHYS 598 BKC. Use for questions and required for homework on literature discussions
• Text: There is no good text for this course (there isn't really even a bad text for this course which covers the material).

#### Course Information

The most interesting and difficult problems in physics are strongly correlated systems, where emergent phenomena arise that appear fundamentally different from their constituent pieces. This course will focus on how we can better understand strongly correlated phenomena from an algorithmic perspective. This includes both learning the computational methods used to simulate quantum systems as well as understanding how an algorithmic perspective, such as tensor networks, have given us a new way to think about strongly correlated physics. Algorithms that will be covered include the density matrix renormalization group, tensor networks, quantum Monte Carlo, and dynamical mean field theory. Physics examples will include area laws (we will cover the proof that entanglement is bounded in 1D gapped systems); a perspective on ADS/CFT via quantum error correcting codes and perfect tensors; understanding how the sign structure influences the physics of systems; and quantum computing.

Although there are no official prerequisites for this course, the course will be heavy on computational methods and require a willingness to program non-trivially.

This course will be challenging and is designed to push you to the edge of the research frontier and so, in many cases, will cover bleeding edge approaches, applications, and current research - you will be expected to read certain relevant papers and participate in discussing them.

#### Homework

The key to this class will be homework. This is the aspect of the course you will learn the most from. It will often involve programming and is designed to teach you important concepts in simulating quantum systems. There will be four broad classes of homework.

• Standard Problem Sets (4-6): Must be done by yourself. ~50% of the grade

• The class will develop a set of computational methods based around each lecture which will be posted on the website. You are responsible for one computational module that is based around one lecture ( ~15% of the grade). This means you can do all the work on one module,1/2 the work of two modules, 1/N the work of N modules. Multiple people can't separately get full credit for the same work. See below for coordination.

• Their is a piazza site. This site will be used for general questions, but will also be used to discuss relevant recent (or important) computational literature. You are responsible for leading the discussion on one paper and contributing to the discussion on other papers. ~10% of the grade
• Final Project ~25% of the grade

A comment about partial credit: There will not be significant partial credit for code that doesn't produce the correct answer There are a million ways a code can be incorrect and it's very hard to evaluate how close you were to the correct answer. On the other hand, you should be able to tell if your homework is correct before submitting. (this is an important skill to develop!)

Homework Submissions: Please submit through the Box.com upload folder HW_Submissions_AlgorithmicPerspective... (preferable) or through the piazza site (less preferable).

Late Homework: Please hand homework in on time. If you get behind, it will be hard to catch up. There will be a penalty for sufficiently late homework.

Solutions: The solutions will be generated by some linear combination of your homeworks and our solutions.

#### Course Outline

(currently under construction)

• Lecture 1: Introduction
• Lectures 2 and 3:
• Exact Diagonalization + Anderson Tower of States
• Vanilla Julia ED code
• Slides from Andreas Lauchli
• See page 11 of John Mcgreevy's Notes for a nice exposition.
• Lecture 4: Your First QMC Code (projector quantum Monte Carlo)
• Lecture 5: Variational Monte Carlo
• Lecture 6: Slater-Jastrow(++)
• Lecture 7: Introduction to Tensor Networks
• This was a chalkboard lecture - as such there were no slides but it closely parallels this paper of Guifre Vidal's
• Lecture 8: Resonant Valence Bonds
• Slides
• See here for a discussion of the mapping between Fermionic and RVB BCS and here for a thesis on RVB states.
• Lecture 9: Spin Liquids from RVB and ED -> MPS
• Lecture 10: Introduction to Matrix Product States
• Lecture 11: The DMRG algorithm
• Lecture 12: From $\Psi$ to $|\Psi^2|$
• Lecture 13: Reptation and the bounce algorithm
• See this paper - section 2.1 - for a description of the bounce algorithm.
• Lecture 14: Path Integral Monte Carlo I
• See this paper for a review of Path Integral Monte Carlo.
• Lecture 15: Path Integral Monte Carlo II
• Lecture 16: Continuous Time
• Lecture 17: Tensor Networks, long range entanglement
• Lecture 18: PEPS and Efficient Product State Algorithms
• Lecture 19: Linear Scaling DMRG and diffusion Monte Carlo in your head
• Lecture 20: Determinant Monte Carlo and BSS
• Lecture 21: MERA
• Lecture 22: Proof of 1D area law I
• Lecture 23: Proof of 1D area law II
• Lecture 24: Sign Problem: Origin + Marshall Sign Rule + Sign-Free Heisenberg Model
• Lecture 25: Sign Problem: 1'st vs. second quantization
• Lecture 26: Sign Problem: Anhillation and Fixed node
• Bonus Lecture I: Quantum Computing + stabilizer circuits
• see Scott's paper on improved stablizer simulations
• Bonus Lecture II: Quantum Computing + Matchgates
• Here's Valiant's original paper
• Bonus Lecture III: Simulations on quantum computers

#### Problem Sets

Problem Set 1 (Due: September 14 )
Problem Set 2 (Due: October 8 )
Problem Set 3 (ijulia notebook)(Due: November 3 )
Problem Set 4 (Due: December 2 )

#### Coordination of Lecture modules

• PQMC for Heisenberg (Wheeler)
• Reptation for Heisenberg (Jahen)
• Worldline for Heisenberg (Brian Busymeyer)
• DMRG for 1D Heisenberg (Dmitri)
• VMC for Silicon (Aneja)
• DMC for Silicon (Benjamin)
• PIMC (Paul)
• Bosonic to Fermionic Valence Bonds (Eli)
• Perfect Tensors (Garrett)
• Entanglement via VMC (Mao Lin)
• T and S matrices via VMC (Gabi)
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