# Undergraduate Thesis

My undergraduate thesis, entitled *Quantum Monte Carlo calculation of the imaginary-time Green’s function in the Hubbard model*, is in the area of quantum Monte Carlo, which uses stochastic algorithms to solve problems in many-body quantum physics.
If you are interested in learning what QMC is, here is my curated collection of papers on the subject.
I’m also the lead author of a journal paper in this field.

My thesis was selected by the American Physical Society (APS) as one of six finalists for the 2014 LeRoy Apker Award, recognizing the best undergraduate physics research works in the US annually. It is also the highest honor for a physics major in the country. Every college and university in the country is eligible to nominate one student. The selection committee will select six finalists and, out of those, two winners. Click here to see the announcement from the Reed physics department (under the heading of July 2014). Here is the presentation I gave to the selection committee that summarizes the theory and shows the algorithm’s accuracy against exact benchmark calculations.

This work was advised by Prof. Darrell Schroeter from Reed and was done in close collaboration with Prof. Shiwei Zhang from the College of William & Mary. It was supported by grants totaling $1,500 from Reed College in the form of an A.V. Davis Foundation Initiative Grant and from the Reed physics department. Most of the computations were performed on the SciClone high-performance computing cluster at the College of William & Mary.

Two years after the completion of my thesis, Prof. Zhang’s group published a research paper built upon the work done in my thesis.

Here is the link to my thesis in PDF format. If you have any questions, comments, suggestions or error corrections, please email me.

Fun facts: Since the establishment of the award in 1978, there have only been 5 Apker finalists from Reed (including one winner in 1992). Interestingly, the two most recent finalists are my adviser (Darrell Schroeter–1995) and one of his advisees (Eliot Kapit–2005).

This is a picture of me receiving the Apker finalist certificate from Prof. Robert Byer, the then-President of the American Physical Society and chair of the selection committee. (The content of the certificate is not visible because of the direct on-camera flash.)

## Description

In my thesis, I formulated and implemented an algorithm to compute imaginary-time correlation functions in the ground state of the one-band, one-dimensional repulsive Hubbard model using the constrained path Monte Carlo (CPMC) method. The algorithm and implementation that I developed are general and has been extended to the two-dimensional Hubbard model to obtain imaginary-time correlation functions for doped systems, which are believed to be a starting point for the description of high-temperature superconductivity. I will briefly introduce the Hubbard model and quantum Monte Carlo before detailing my contribution.

Strongly-correlated systems are quantum mechanical systems in which the interactions between particles are strong enough that their behavior cannot be described in terms of non-interacting entities. These systems represent one of the most exciting frontiers in condensed matter physics today because of their novel properties, such as high-temperature superconductivity, heavy-fermion metals and colossal magnetoresistance.

One of the most important models in this area is the Hubbard model, named after John Hubbard who invented it in 1963. The model describes a lattice on which electrons can hop between lattice sites and interact through the Coulomb repulsion. Whereas the kinetic energy favors electrons being as mobile as possible, the repulsive potential energy encourages electrons to stay apart from each other. This competition leads to fascinating properties, such as ferromagnetism, antiferromagnetism and metal-insulator transition.

Despite the promise of the Hubbard model as the key to understanding high-temperature superconductivity and intense research into its properties, there are many obstacles standing in the way of its solution. Because the computational cost of exact calculations increases exponentially with the number of particles, exact numerical calculations are limited to small systems, leaving us with only approximate numerical methods to tackle large, realistic systems. Furthermore, the strong interactions render traditional electronic structure methods (such as mean-field methods) ineffective because they treat particles individually instead of collectively and because they only treat the interaction in an average way. These challenges are also encountered in other strongly-correlated models. Thus, there exists a pressing need for new computational methods with more favorable computational cost that can effectively tackle strongly-correlated systems. One very promising candidate is quantum Monte Carlo (QMC).

QMC is a class of stochastic algorithms that use the Monte Carlo technique to compute properties of quantum systems. Instead of giving definite numerical results, QMC calculations give results that have associated statistical uncertainties that can be reduced algebraically with more computer run time. Its computational cost increases polynomially (typically to the third or fourth power of system size) instead of exponentially.

Auxiliary-field quantum Monte Carlo (AFQMC) refers to a group of QMC algorithms whose common feature is the introduction of auxiliary fields (and thereby stochasticity) through the Hubbard-Stratonovich (HS) transformation in order to convert an interacting system into many non-interacting systems (which are easier to solve). The auxiliary fields are sampled stochastically by Monte Carlo. AFQMC was pioneered in the mid-1980s by Sugar, Scalapino, Sugiyama, Koonin and others. These early methods used the Metropolis algorithm, invented in 1953 by Metropolis et al., to sample the auxiliary fields. Within the framework of Metropolis AFQMC, Feldbacher and Assaad invented an elegant and efficient way of calculating imaginary-time correlation functions in 2001.

Nevertheless, Metropolis AFQMC methods had to contend with the “fermion sign problem,” which arises because of the antisymmetry of fermionic wave functions and causes large statistical errors at low temperatures, large imaginary times or strong correlations, precisely the regimes that yield interesting physics. Progress was made on this front in 1990 when Fahy and Hamann invented the “positive projection” technique that greatly alleviates or even eliminates the sign problem, even though the technique was very computationally expensive to implement in Metropolis AFQMC.

A breakthrough came in 1997 when Zhang *et al.* invented the constrained path Monte Carlo method, which combines features from Metropolis AFQMC methods (such as the HS transformation) with the open-ended branching random walk technique from earlier QMC methods, such as Diffusion Monte Carlo.
Using a branching random walk, instead of the Metropolis algorithm, to sample the auxiliary fields has several advantages.
It makes the implementation of the aforementioned positive projection technique (now known as the “constrained path” approximation) very simple and it improves the efficiency of the Monte Carlo sampling.
In the same paper, Zhang also invented the “back propagation” technique in CPMC to calculate expectation values of observables that do not commute with the Hamiltonian, such as correlation functions.

This is where my thesis comes in. The algorithm that I formulated combines Feldbacher’s technique with Zhang’s back-propagation technique to calculate the imaginary-time correlation functions using CPMC. The key to integrating these two components is the estimator used in the back-propagation technique, which is similar in form to the estimator used in Metropolis AFQMC. The result is an algorithm that behaves like a branching random walk for parts of the calculation and like the Metropolis algorithm for other parts. With this algorithm, it is now possible to efficiently compute correlation functions at long imaginary times, which so far has been very difficult due to the fermion sign problem. These correlation functions, in turn, allow the computation of many dynamical quantities of the Hubbard model, such as spin and charge dynamical structure factors, optical conductivity and so on.

The starting point of my implementation of this algorithm is CPMC-Lab, a MATLAB package that performs CPMC calculations. Extensive testing of this algorithm was conducted in the one-dimensional Hubbard model. In addition to comparison with many analytical results, which are available in one dimension, I developed an exact diagonalization (ED) program to benchmark the CPMC results in finite supercells.

I wrote my thesis with the intention of making it accessible to advanced undergraduates who possess some familiarity with the second quantization formalism (which I also review at the beginning of chapter 1). In this notation, in chapter 1, I derive analytical expressions for the imaginary-time correlation functions for electrons in a one-dimensional lattice with nearest-neighbor hopping, subject to an external potential. In chapter 2, I develop the ED program to calculate the imaginary-time correlation functions of the full interacting Hubbard model exactly. In chapter 3, I describe the algorithm that I developed and test it against the ED program in chapter 2. The algorithm I developed gives excellent agreement with exact results. Finally, in chapter 4, I use imaginary-time correlation functions to examine the magnetic ordering of a large system that is beyond the reach of exact calculations.