Abstract
This dissertation is based almost entirely on numerical methods. Chief among these is the stochastic series expansion quantum Monte Carlo method, although I have also used Lanczos exact diagonalization for some cases. I will describe exact diagonalization methods briefly; the rest of the chapter is devoted to developing the quantum Monte Carlo methods that I have used in this dissertation. I have attempted to make this chapter a pedagogically useful guide for the reader interested in replicating or building upon this work. I begin by describing the foundations of classical Monte Carlo. I then derive the stochastic series expansion formulation of quantum Monte Carlo, and show applications of this method to the Heisenberg model, the J-Q model, and the Heisenberg model in an external field (including directed loop updates). I then synthesize the previous sections to build the QMC method used here for the J-Q model in an external field. I describe the supplementary techniques quantum replica exchange and β-doubling and finish with a brief discussion of random number generators.
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Notes
- 1.
A side effect of the exponential growth is that only a factor of two separates the size of a system that can be solved in an hour by a simple code running on a laptop and the largest that can be solved using a state-of-the-art code on a supercomputer.
- 2.
Usage and installation procedures for QuSpin can be found in [4]. QuSpin can be installed from the package manager Anaconda or from Github: https://github.com/weinbe58/QuSpin.
- 3.
In the case of evaluating π, this example is especially contrived because there are far more precise specialized procedures for calculating π.
- 4.
The discretization error is a kind of systematic error (as opposed to statistical or random error) which is not Gaussian and does not have well-defined error bars.
- 5.
The uniformly weighted average is the most “efficient” estimator for the mean (i.e., it has the lowest variance) [6, p. 135].
- 6.
Often abbreviated MCMC.
- 7.
Said by Lode Pollet during his lecture on QMC at the Arnold Sommerfeld Center at LMU Munich as part of the Arnold Sommerfeld School on 13 September 2017.
- 8.
In fact, there are infinitely many solutions to the detailed balance condition.
- 9.
This algorithm is usually referred to simply as the Metropolis Algorithm although perhaps it should be called the Rosenbluth or Rosenbluth–Teller Algorithm. Metropolis was first author on the original paper [8], but according to Marshall Rosenbluth [9, 10] Metropolis was merely the head of the computer lab and made no scientific contribution to the paper. In Metropolis’ memoirs, he makes no claim to have invented the algorithm either [11]. See Sect. 1.2.2 for a more complete discussion.
- 10.
For example, in an implementation of the Metropolis Algorithm for the Ising model, one selects a spin at random and proposes to flip that spin; the probability of proposing this change is therefore 1∕N and the probability of proposing the reverse change is also 1∕N [3, Sec. 3.2].
- 11.
For example, in the Heisenberg model, after some simple transformations there are only four nonzero local matrix elements which all have the same value, Eq. (5.35).
- 12.
A brief history of the development of stochastic series expansion is available in [14].
- 13.
The exact fraction used here is not important; any number greater than unity will work.
- 14.
- 15.
As a rough definition, you can think of the weight as the unnormalized probability.
- 16.
There are, after all, only about 1080 atoms in the universe.
- 17.
Here we use “timeslice” to refer to the individual time-propagated states αi. In the literature this term sometimes refers instead to well-defined intervals of imaginary time composed of many time-propagated states.
- 18.
Removing an off-diagonal operator (and replacing it with the identity) would result in a zero-valued matrix element and therefore a zero-weighted, “invalid,” configuration.
- 19.
A full FORTRAN implementation of the SSE method for the \(S=\frac {1}{2}\) Heisenberg model can be found here: http://physics.bu.edu/~sandvik/vietri/index.html.
- 20.
The Heisenberg antiferromagnet will be frustrated and suffer from the sign problem on non-bipartite lattices such as the triangular lattice.
- 21.
One might ask why we would use \(u=\frac {1}{2}\) and not some other fraction. It is easy to convince oneself that this is optimal. Multiplying by 0 or 1 would clearly generate bad updates and it seems logical that there should be symmetry between u and 1 − u, thus the optimal choice would be where u = 1 − u therefore \(u=\frac {1}{2}\).
- 22.
In some of my simulations I have also stored the spin configuration of the operator legs in an array legs[4*cutoff]. This imposes a cost in memory use and is not strictly necessary, but is nonetheless useful for debugging when the operator types become more complicated.
- 23.
This step is not strictly necessary, but it helps.
- 24.
There is also a variant that uses three singlet projection operators called the J-Q3 model.
- 25.
Another word for sign-problem-free is Marshall positive.
- 26.
It might seem strange or inefficient to decide this by chance without using any information about the state (like if a Q-type operator even can be inserted), but this method of deciding is simple, unbiased, and (most importantly) makes it easy to calculate the proposal probability g.
- 27.
In the sublattice-rotated version of Pi,j, the Sz operators have the opposite sign from the ladder operators.
- 28.
In 1D the number of bonds nb is just N, in 2D it is 2N, etc.
- 29.
The heat bath solution to detailed balance is a good example of a solution to the detailed balance condition that is not the Metropolis Algorithm.
- 30.
This first set corresponds to the upper left quadrant of Fig. 8 of [2].
- 31.
This second set corresponds to the lower left quadrant of Fig. 8 of [2].
- 32.
This method can also be done just a single replica, sampling the temperature stochastically without swapping, but then typically a bias in the temperature acceptance rates must be imposed in order to ensure that the desired temperature regime is sampled.
- 33.
In principle, we could allow swaps between any two replicas, but in practice, the acceptance rates of swaps involving large changes in field is nearly zero. Considering swaps only between neighbors results in a higher acceptance rate without violating the detailed balance condition.
- 34.
Strictly speaking, simulation time is not the same as physical time, but the effect is often similar.
- 35.
C++ implementation of the Mersenne Twister: http://www.bedaux.net/mtrand/.
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Iaizzi, A. (2018). Methods. In: Magnetic Field Effects in Low-Dimensional Quantum Magnets. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-01803-0_5
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