# Tag Cloud

## Lie algebra

20 Mar 2019 Lie Algebra as Generator   Paused

In this post generator of Lie groups as well as its subgroup are considered. This short post is preparation for the application of Mathematical theory into QM and CM. This post is the third of a series of posts that start from Lie group and Lie algebra.

11 Mar 2019 Lie Group as Differential Manifold   Completed

In this post Lie groups are regarded as a differential manifold, and one-parameter subgroups are introduced. This post is the second of a series of posts that start from Lie group and Lie algebra.

12 Jan 2019 Lie Groups and "Actions"   Completed

In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.

06 Jun 2018 李群和李群的李代数   Archived

My understanding of Lie-group and its Lie-algebra. The statement “Lie-algebra is approximation of Lie-group” is inaccurate since they essentially live in different spaces. This post is not finished. Check posts under Lie Group and/or Lie Algebra where I rewrote and added more aspects of Lie group and Lie algebra.

## Lie group

20 Mar 2019 Lie Algebra as Generator   Paused

In this post generator of Lie groups as well as its subgroup are considered. This short post is preparation for the application of Mathematical theory into QM and CM. This post is the third of a series of posts that start from Lie group and Lie algebra.

11 Mar 2019 Lie Group as Differential Manifold   Completed

In this post Lie groups are regarded as a differential manifold, and one-parameter subgroups are introduced. This post is the second of a series of posts that start from Lie group and Lie algebra.

12 Jan 2019 Lie Groups and "Actions"   Completed

In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.

06 Jun 2018 李群和李群的李代数   Archived

My understanding of Lie-group and its Lie-algebra. The statement “Lie-algebra is approximation of Lie-group” is inaccurate since they essentially live in different spaces. This post is not finished. Check posts under Lie Group and/or Lie Algebra where I rewrote and added more aspects of Lie group and Lie algebra.

## QFT

07 Jun 2018 量子场论   Archived

My Study notes on QFT lesson for the final. Each step of deduction is present in this post. This post covers K-G equations and Dirac equations.

## study notes

07 Jun 2018 量子场论   Archived

My Study notes on QFT lesson for the final. Each step of deduction is present in this post. This post covers K-G equations and Dirac equations.

## Statistical mechanics

08 Jun 2018 第一性原理热统   Archived

Class project. My attempt at building Statistical Mechanics from First Principles. I think Statistical Mechanics can be applied to distinguishable particles such as millions of footballs. This post tries to develop such Statistical Mechanics, especially on the entropy. This post is not finished.

## First Principle

08 Jun 2018 第一性原理热统   Archived

Class project. My attempt at building Statistical Mechanics from First Principles. I think Statistical Mechanics can be applied to distinguishable particles such as millions of footballs. This post tries to develop such Statistical Mechanics, especially on the entropy. This post is not finished.

## One form

21 Aug 2018 N-Forms and Tensors on Manifold   Completed

We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.

20 Aug 2018 Vectors and One-Forms on Manifold   Completed

Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.

## Topology

16 Jan 2019 Fiber Bundles   Paused

Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.

23 Nov 2018 Introduction to de Rham Cohomology   Completed

Cohomology is viewed as a natural dual space of homology in this post. The bilinear map (i.e., the inner product) between these two spaces is just integration. At the end of this post, the cohomology group as an indicator of “holes” in space is discussed.

01 Nov 2018 Introduction to Homology   Completed

Euler characteristics is a topological invariant, and can be interpreted as a “hole”-indicator. Homology is just a natural way of defining Euler characteristics in topological spaces. With triangulation of a manifold, we can define cycles and boundaries and combine them to homology groups. We see that the group is trivial for trivial spaces, and can distinguish manifolds in terms “holes” in them.

21 Aug 2018 N-Forms and Tensors on Manifold   Completed

We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.

20 Aug 2018 Vectors and One-Forms on Manifold   Completed

Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.

## Vector

20 Aug 2018 Vectors and One-Forms on Manifold   Completed

Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.

## Tangent space

20 Aug 2018 Vectors and One-Forms on Manifold   Completed

Since in general there is no way to define a “straight arrow” connecting two points, Vectors can only be “tangent vectors” on manifolds. In this post, tangent vectors are introduced heuristically, with emphasis on how and why should we define vectors as operators. Co-vectors, also called one-forms, are introduced as the dual. The reason for defining one-forms as differentials are introduced heuristically. This post also addresses the problem of inconsistency when the basis of vector act on that of one form.

## Differential Forms

21 Aug 2018 N-Forms and Tensors on Manifold   Completed

We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.

## Tensor

21 Aug 2018 N-Forms and Tensors on Manifold   Completed

We are going to generalize the concept of vectors and one-forms to tensors and differential forms. In the meantime, the wedge product and exterior derivative were introduced. Exterior derivative lies the foundation for cohomology.

## Fortran

30 Dec 2019 Computer Code For Toric Code   Completed

The post is my initial set up for Kitaev’s toric code. The work is published on arXiv:1912.12964.

18 Apr 2019 The Fortran Gram-Schmidt Process Pitfall   Completed

I tried using Gram-Schmidt Process the other day and just couldn’t get the correct result. It took me a few days trying to figure out why. Turns out that it’s just a simple precision error problem.

06 Oct 2018 You Should Try Fortran   Completed

I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.

01 Sep 2018 Fortran with VS2017   Completed

Instructions on using Fortran with VS2017

## Coding

18 Apr 2019 The Fortran Gram-Schmidt Process Pitfall   Completed

I tried using Gram-Schmidt Process the other day and just couldn’t get the correct result. It took me a few days trying to figure out why. Turns out that it’s just a simple precision error problem.

06 Oct 2018 You Should Try Fortran   Completed

I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.

01 Sep 2018 Fortran with VS2017   Completed

Instructions on using Fortran with VS2017

## Thoughts

06 Oct 2018 You Should Try Fortran   Completed

I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.

## Python

06 Oct 2018 You Should Try Fortran   Completed

I found Fortran incredibly fast compared to pure Python. Fortran’s reputation for hard to maintain is true only if you read the code without an understanding of the Physics. It’s still an active language. You should give it a try.

## Homology

01 Nov 2018 Introduction to Homology   Completed

Euler characteristics is a topological invariant, and can be interpreted as a “hole”-indicator. Homology is just a natural way of defining Euler characteristics in topological spaces. With triangulation of a manifold, we can define cycles and boundaries and combine them to homology groups. We see that the group is trivial for trivial spaces, and can distinguish manifolds in terms “holes” in them.

## Euler Characteristic

01 Nov 2018 Introduction to Homology   Completed

Euler characteristics is a topological invariant, and can be interpreted as a “hole”-indicator. Homology is just a natural way of defining Euler characteristics in topological spaces. With triangulation of a manifold, we can define cycles and boundaries and combine them to homology groups. We see that the group is trivial for trivial spaces, and can distinguish manifolds in terms “holes” in them.

## de Rham Cohomology

23 Nov 2018 Introduction to de Rham Cohomology   Completed

Cohomology is viewed as a natural dual space of homology in this post. The bilinear map (i.e., the inner product) between these two spaces is just integration. At the end of this post, the cohomology group as an indicator of “holes” in space is discussed.

## Stokes' Theorem

23 Nov 2018 Introduction to de Rham Cohomology   Completed

Cohomology is viewed as a natural dual space of homology in this post. The bilinear map (i.e., the inner product) between these two spaces is just integration. At the end of this post, the cohomology group as an indicator of “holes” in space is discussed.

## word cloud

23 Nov 2018 Thank You Word Cloud with Mathematica   Completed

Code for generating thank-you word cloud with Mathematica. It’s perfect for thank-you page of your slide-shows.

## Mathematica

23 Nov 2018 Thank You Word Cloud with Mathematica   Completed

Code for generating thank-you word cloud with Mathematica. It’s perfect for thank-you page of your slide-shows.

## Jekyll

01 Dec 2018 Build Your Blog with GitHub Pages   Paused

Build Your Blog with GitHub Pages. This is a sketch of how to use my theme PointingToTheMoon to write your blog. This theme is great for academic use, for it features simple post page with mathjax support and a side bar with toc. The main page on the other hand is somewhat fancy.

## VPN

09 Dec 2018 Shadowsocks + Digital Ocean = VPN of 1TB/month   Completed

Code for using digital ocean service for VPN. It is cheap for .edu-e-mail owners (free 50\$ on GitHub Student Pack).

## Pullback

12 Jan 2019 Lie Groups and "Actions"   Completed

In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.

## Pushforward

12 Jan 2019 Lie Groups and "Actions"   Completed

In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.

## representation

12 Jan 2019 Lie Groups and "Actions"   Completed

In this post Lie groups and it’s actions are introduced. This is the first of a series posts start from Lie group and Lie algebra, where I try to understand “infinitesimal operators” and “generators” used by physicists from a mathematical standpoint. Hopefully, this series ends with a good explanation of what “generators” are in Classical Mechanics as well as Quantum Mechanics.

## Fiber

16 Jan 2019 Fiber Bundles   Paused

Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.

## Fiber bundles

16 Jan 2019 Fiber Bundles   Paused

Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.

## Berry's Phase

16 Jan 2019 Fiber Bundles   Paused

Fiber bundle is introduced with intuitive examples of pasta and pancakes, Berry phase and Calabi-Yau space. Structure group is introduced as a natural consequence of transition functions.

## Exponential map

11 Mar 2019 Lie Group as Differential Manifold   Completed

In this post Lie groups are regarded as a differential manifold, and one-parameter subgroups are introduced. This post is the second of a series of posts that start from Lie group and Lie algebra.

## Generator

20 Mar 2019 Lie Algebra as Generator   Paused

In this post generator of Lie groups as well as its subgroup are considered. This short post is preparation for the application of Mathematical theory into QM and CM. This post is the third of a series of posts that start from Lie group and Lie algebra.

## Quantum computation

11 Apr 2019 Introduction to Topological Quantum Computation: Basics of Quantum Computation   Completed

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post establishes the barebone basics of quantum computations.

## Topological quantum computation

23 May 2019 Introduction to Topological Quantum Computation: Crash Course on Knots Theory   Completed

This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.

14 May 2019 Introduction to Topological Quantum Computation: Ising Anyons Case Study   Completed

This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.

01 May 2019 Introduction to Topological Quantum Computation: Side Notes on Simulations   Writing

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post started as introducing a pitfall of using Stern-Gerlach experiment as quantum computers, and end with a discussion on simulations of QC and TQC using classical computers.

13 Apr 2019 Introduction to Topological Quantum Computation: Anyons Model   Completed

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.

11 Apr 2019 Introduction to Topological Quantum Computation: Basics of Quantum Computation   Completed

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post establishes the barebone basics of quantum computations.

## Anyons

13 Apr 2019 Introduction to Topological Quantum Computation: Anyons Model   Completed

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.

## Braiding

14 May 2019 Introduction to Topological Quantum Computation: Ising Anyons Case Study   Completed

This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.

13 Apr 2019 Introduction to Topological Quantum Computation: Anyons Model   Completed

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.

## Pentagon and Hexagon equation

14 May 2019 Introduction to Topological Quantum Computation: Ising Anyons Case Study   Completed

This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.

13 Apr 2019 Introduction to Topological Quantum Computation: Anyons Model   Completed

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post discusses anyon model in general. Fusion diagram and hexagon and pentagon identities are introduced.

## Gram Schmidt

18 Apr 2019 The Fortran Gram-Schmidt Process Pitfall   Completed

I tried using Gram-Schmidt Process the other day and just couldn’t get the correct result. It took me a few days trying to figure out why. Turns out that it’s just a simple precision error problem.

## pitfall

01 May 2019 Introduction to Topological Quantum Computation: Side Notes on Simulations   Writing

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post started as introducing a pitfall of using Stern-Gerlach experiment as quantum computers, and end with a discussion on simulations of QC and TQC using classical computers.

## Classical simulation of Quantum computers

01 May 2019 Introduction to Topological Quantum Computation: Side Notes on Simulations   Writing

This is a series of posts on topological quantum computations. The aim of this series is to work my way to understanding the diagrams of “strands” widely used in the field. This post started as introducing a pitfall of using Stern-Gerlach experiment as quantum computers, and end with a discussion on simulations of QC and TQC using classical computers.

## Wild thoughts

07 May 2019 Computation and Currency   Completed

This is my wild thoughts on computation and currency. The story begins when a spaceship appeared near a lake one day…

## Complexity

07 May 2019 Computation and Currency   Completed

This is my wild thoughts on computation and currency. The story begins when a spaceship appeared near a lake one day…

## Ising anyons

14 May 2019 Introduction to Topological Quantum Computation: Ising Anyons Case Study   Completed

This is a series of posts on topological quantum computations. In this post, the most promising candidate for TQC, Ising anyons, are discussed. A theoretical topological quantum computer is realized via Ising anyons’ initialization, braiding, and fusion. F and R matrices are calculated from the consistency requirement, i.e. Hexagon and Pentagon equations. Braiding matrices are introduced heuristically. A set of Clifford gates is implemented as the result of braiding. This post features lots of diagrams.

## Knot theory

23 May 2019 Introduction to Topological Quantum Computation: Crash Course on Knots Theory   Completed

This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.

## Jones polynomial

23 May 2019 Introduction to Topological Quantum Computation: Crash Course on Knots Theory   Completed

This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.

## Alexander polynomial

23 May 2019 Introduction to Topological Quantum Computation: Crash Course on Knots Theory   Completed

This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.

## Kauffman bracket

23 May 2019 Introduction to Topological Quantum Computation: Crash Course on Knots Theory   Completed

This is a series of posts on topological quantum computations. To address the reason why we introduce such “strange-looking” equations to calculate Jones polynomials, we have to know the history of knot theory, and understand how the pioneers came up with their ideas. This post provide a somewhat natural way to define Alexander Polynomials and skein relations from the coloring of knots, and ended on the note that the author is currently incapable of giving an equivalently convincing reason behind the definition of the Jones or the Kauffman polynomial.

## PIE

06 Aug 2019 This Summer and the Site   Completed

After a long absence I decided to recap what happened during the last few months and start updating this blog again.

## Meta

06 Aug 2019 This Summer and the Site   Completed

After a long absence I decided to recap what happened during the last few months and start updating this blog again.

## topology

02 Sep 2019 Introduction To Topological Insulator   Writing

This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.

## SSH model

02 Sep 2019 Introduction To Topological Insulator   Writing

This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.

## Tight binding

02 Sep 2019 Introduction To Topological Insulator   Writing

This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.

## Chern number

02 Sep 2019 Introduction To Topological Insulator   Writing

This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.

## Winding number

02 Sep 2019 Introduction To Topological Insulator   Writing

This post is to familiarize myself with the common concepts arise in the field of Topological insulator. It covers winding number, Chern number, symmetries and bulk-boundary correspondence.

## Fundamental Group

29 Oct 2019 Introduction To Fundamental Group   Writing

This post is to prepare myself for future study. As the fundamental group plays an important role in many parts of topological insulators.

## Homotopy

29 Oct 2019 Introduction To Fundamental Group   Writing

This post is to prepare myself for future study. As the fundamental group plays an important role in many parts of topological insulators.

## Genus

29 Oct 2019 Introduction To Fundamental Group   Writing

This post is to prepare myself for future study. As the fundamental group plays an important role in many parts of topological insulators.

## Triangulation

30 Dec 2019 Computer Code For Toric Code   Completed

The post is my initial set up for Kitaev’s toric code. The work is published on arXiv:1912.12964.

## Kitaev's Toric Code

30 Dec 2019 Computer Code For Toric Code   Completed

The post is my initial set up for Kitaev’s toric code. The work is published on arXiv:1912.12964.

## Algebra

05 Mar 2020 The Amazing World Of Diagrams   Completed

This post is the first chapter of a long note that I have been writing and organizing aiming at telling the stories in the languages of mathematical concepts mostly used in condensed matter Physics.

## Diagrams

05 Mar 2020 The Amazing World Of Diagrams   Completed

This post is the first chapter of a long note that I have been writing and organizing aiming at telling the stories in the languages of mathematical concepts mostly used in condensed matter Physics.