Quantum Monte Carlo

Quantum Monte Carlo (QMC) is a sufficiently niche field that the best (and probably only) way to learn about it is through published journal papers. I started out in this field the summer after my sophomore year, having no idea what I was stepping into. Since then I’ve read a number of journal papers in the field and understand them to varying degrees. I did my undergraduate thesis in QMC.

Below is a collection of some of the papers I’ve read with commentary, some about the subject matter of the papers, some about how they fit into the bigger picture while others are just personal comments.

I’m not anywhere near even being a proficient student in the field but with my limited understanding, I hope to help people gain some exposure to this very interesting field, especially diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC), the two methods that I have dabbled in.

If you think this page helps you at all or just want to chat about QMC in general, feel free to shoot me an email.

• Kosztin, Faber and Schulten, Introduction to the diffusion Monte Carlo method, Am. J. Phys. 64, 633 (1996).

  This is a very accessible introduction to DMC and was among the very first papers that I read on the topic. You can understand with just intro quantum and some basic understanding of Monte Carlo integration.

  It walks you through how to transform the Schrodinger equation into the form of a diffusion equation with branching; how the path integral formulation of the imaginary-time evolution of a wave function lends itself quite well to a Monte Carlo integration and how to carry it out; why “branching” is essential for sampling efficiency; and lays out a concrete computer algorithm for a simple DMC program with some illustrative results. When you go further into the field, you’ll see some of these ideas over and over again.

• Hirsch, Discrete Hubbard-Stratonovich transformation for fermion lattice models, Phys. Rev. B 28, 4059 (1983)

  In this paper Hirsch describes what has come to be known as the Hirsch discrete spin and Hirsch discrete charge decompositions, which are discrete forms of the Hubbard-Stratonovich transformations for the Hubbard model. They are very popular in AFQMC calculations.

• Sugiyama and Koonin, Auxiliary Field Monte Carlo for Quantum Many-Body Ground States, Anns. Phys. 168, 1 (1986)

  This paper pioneered the field of ground-state AFQMC: the technique of using the Hubbard-Stratonovich transformation to introduce auxiliary fields into an interacting problem in order to break it up into many non-interacting problems.

• Hirsch, Two-dimensional Hubbard model: Numerical simulation study, Phys. Rev. B, 31, 4403 (1985)

  This is a great read. In this paper, Hirsch did a very extensive numerical study of the 2-D Hubbard model at finite temperature using Metropolis AFQMC and discussed at length the physical properties of the model.

• Santos, Introduction to Quantum Monte Carlo Simulations for Fermionic Systems, Braz. J. Phys. 33, 36 (2003)

  This is a fairly accessible article that shows how to convert the ground state projection operator with the Hubbard Hamiltonian into a huge sum which then allows you to do Monte Carlo sampling. It also touches on how to update the Green’s function efficiently; the checkerboard breakup; numerical stabilization of matrix multiplications in the algorithm; and how physical quantities can be calculated from the Green’s functions.

  Note that this is written only for finite-temperature AFQMC with Metropolis-like sampling. It took me a while to understand the subtle difference between doing AFQMC with Metropolis-like and with branching random walk.

• Fahy and Hamann, Diffusive behavior of states in the Hubbard-Stratonovich transformation, Phys. Rev. B 43, 765 (1991)

  This is a fabulous article. It essentially shows how the introduction of auxiliary fields through the Hubbard-Stratonovich transformation sets up a diffusion equation (with branching and drift) on the Grassmann manifold of normalized Slater determinants after a proper intrinsic coordinate system has been chosen on that manifold. In my opinion, this immediately links the sign problem in AFQMC with the sign problem in DMC (which is more intuitive to understand in configuration space).

  They actually explicitly wrote down that diffusion equation and give an example in the simple context of a two-site Hubbard model. There are lots of great insights peppered throughout this article.

  However, tried as I may, I couldn’t really understand the appendix which is where they did the change of variable from the auxiliary fields to the coordinates of the manifold. If you understand it, I’d love to hear from you!

  Fun trivia: you’ll notice that in the bibliography they cite one of their unpublished papers (ref 27) which for some reason was actually published in 1990, before this paper came out. In that 1990 paper (see below), they established the positive projection method based on insights developed in the current paper.

• Fahy and Hamann, Positive-Projection Monte Carlo Simulation: A New Variational Approach to Strongly Interacting Fermion Systems, Phys. Rev. Lett. 65, 3437 (1990)

  Building on the insights in the 1991 paper above, the authors propose the positive projection technique to combat the sign problem.

• Zhang, Carlson and Gubernatis, Constrained Path Quantum Monte Carlo Method for Fermion Ground States, Phys. Rev. Lett. 74, 3562 (1995)

  Zhang, Carlson and Gubernatis, Constrained path Monte Carlo method for fermion ground states, Phys. Rev. B 55, 7464 (1997)

  This pair of papers by Prof. Zhang and others pioneered the constrained path Monte Carlo method. In them, he and the co-authors combined techniques from DMC (open-ended branching random walk in imaginary time, importance sampling) and AFQMC (Hubbard-Stratonovich transformation, Slater determinants) to do ground state projection.

  However, the real deal here is the insight that by constraining the random walkers to always maintain a positive overlap with the trial wave function, they can remove the exponential signal-to-noise decay characteristic of the “fermion sign problem.” This insight was first observed by Fahy (see above) but the traditional AFQMC method makes the implementation of this constraint computationally difficult. In contrast, this constraint is very easily implemented in a branching random walk!

  I’ve read the PRB article about 30 times and each time I find new insights that I haven’t noticed before. It really showed me how much work went into developing this method. Take it slow!

• Carlson, Gubernatis and Ortiz, Issues and observations on applications of the constrained-path Monte Carlo method to many-fermion systems, Phys. Rev. B 59, 121788 (1999)

  This paper is a nice addition to the pair of papers above by Zhang et al. It very clearly illuminates the similarities and differences between first- and second-quantized QMC algorithms (in this case DMC vs CPMC). Having read papers from DMC and CPMC, this solidifies my understanding of the two methods.

  Fun trivia: Prof Zhang introduced me to Carlson when I visited W&M in Oct 2013. Of course, being an undergraduate, I didn’t have much to say to him but it was still fun to meet one of the pioneers of the method and to be able to associate a face to name on paper.

• Zhang, Finite-Temperature Monte Carlo Calculations for Systems with Fermions, Phys. Rev. Lett. 83, 2777 (1999)

  This paper by Zhang is a follow-up to the pioneering pair above (which describes ground-state CPMC). It extends the CPMC method to finite-temperature. It turns out the extension is fairly straightforward.

• Zhang and Krakauer, Quantum Monte Carlo Method using Phase-Free Random Walks with Slater Determinants, Phys. Rev. Lett. 90, 136401 (2003)

  In this paper Zhang continues developing the CPMC method by extending it to treat interactions that couple auxiliary fields to complex numbers. This extension, known as phase-free or phaseless AFQMC, allows us to deal with any kind of interactions.

• Feldbacher and Assaad, Efficient calculation of imaginary-time-displaced correlation functions in the projector auxiliary-field quantum Monte Carlo algorithm, Phys. Rev. B 63, 073105 (2001)

  Assaad, Quantum Monte Carlo Methods on Lattices: the Determinantal Approach, lecture notes

  One of the core ideas of my thesis is based on these two papers. In the first paper, based on the simple observation that the Green’s function is a projector, the authors show an elegant yet efficient and numerically stable way to calculate the imaginary-time Green’s function with AFQMC with Metropolis-like sampling.

  The second paper is a very comprehensive lecture note. It reviews the basics of AFQMC and discusses available techniques to calculate the Green’s functions.

• Motta et al, Imaginary time correlations and the phaseless auxiliary field quantum Monte Carlo, J. Chem. Phys. 140, 024107(2014)

  This paper probably comes the closest to the topic of my thesis. Here the authors used the Constrained Path Monte Carlo method (with the most general Hubbard-Stratonovich transformation) and the method by Feldbacher (above) to calculate the imaginary-time Green’s function of the jellium model. There are great discussions of the properties of Slater determinants, how to express a two-body operators as sums of square of one-body operators, the importance-sampling transformation, the back-propagation technique and so on.

• Troyer and Wiese, Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations, Phys. Rev. Lett. 94, 170201 (2005)

  This paper proves that the fermion sign problem that afflicts all QMC methods is indeed NP-hard which implies a general solution to the fermion sign problem will also prove that P = NP, a $1 million Milllennium prize problem. (If you do find one such solution, don’t forget to let me know.) The paper also provides a very clear definition of the sign problem and shows how that arises in a finite-temperature QMC method.

• Lieb and Wu, Absence of Mott Transition in an Exact Solution of the Short-range, One-band Model in One Dimension, Phys. Rev. Lett. 20, 1445 (1968)

  This is a very influential paper where the authors used the Bethe ansatz to solve the one-dimensional Hubbard model. Eq 20 is a closed-form formula for the ground state energy of the 1D Hubbard model.

• Thouless, Stability Conditions and Nuclear Rotations in the Hartree-Fock Theory, Nucl. Phys. 21, 225 (1960)

  This quite old paper from the outset doesn’t seem to have anything to do with QMC but it does in a big way. The most important result we need is the theorem in section 2 which has come to be known as Thouless theorem: the application of the exponential of a one-body operator to a Slater determinant is another Slater determinant (although you can see that it wasn’t stated that way in this paper). I provide a proof of this in an appendix of my thesis.

  This has huge implications because if we can write all operations on Slater determinants in a second-quantized QMC algorithm in the form of such operators then the number of determinants we need will not increase over time because of the operators themselves (although the number might increase due to branching but that certainly is not inherent to the algorithm).

  Fun trivia: My Reed thesis advisor (Prof. Schroeter) attended Reed with Thouless’s daughter. Thouless won the 2016 Nobel prize in physics.