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Depending on how many random numbers you need in a short amount of time, it might be worth to consider using a cryptographic PRNG. In particular AES-CTR. Now of course you might say that "but AES-CTR is soo slow". Actually it isn't. If you only re-key every GB or so of generated data you get a speed of about 12 CPU cycles needed for 16 bytes of ...

17

My question is, when I run a simulation with $N$ particles and I track the Hamiltonian per particle $(H/N)$ and the magnetization per particle $\left(\sum _i s_i /N\right)$, with $K$ values going from $0.1$ to $0.7$ in increments of $0.1$, how do I spot the region of the critical coupling constant? There has to be a signature of the critical point that is ...

14

As Anyon correctly pointed out, there is no phase transition at finite temperature in 1D. In 2D there are a number of different ways to identify the phase transition (I'm assuming you're using Monte Carlo). You could directly look at the magnetization, but a more reliable signature is the magnetic susceptibility, $\chi_m (K)$, which is strongly peaked around ...

14

It's been years since I've done Monte Carlo calculations (though it was more recent than the 90's!), so hopefully the information given below is still reasonably up-to-date. I've also had reason to look into pseudo-random number generation in the last few years, for other algorithms. My question: Are there any other modern random number generators better ...

11

You can tell if a Hamiltonian is sign-free by looking at it in the form that it is handed to you. If the Hamiltonian is real and the off-diagonals are non-positive then it is Stoquastic (which is sign-free). Moreover, every Hamiltonian is sign-free if you don't care about the computational complexity of transforming it: In the basis that it is diagonal, the ...

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The general problem of determining whether a Hamiltonian can be transformed into "stoquastic" (i.e. sign-problem-free) form by local transformations is NP-hard: https://arxiv.org/abs/1906.08800 https://arxiv.org/abs/1802.03408 On the other hand, there are lots of individual Hamiltonians for which people have figured out clever tricks to bring them into ...

10

First, some general remarks: The measurements should be made after the system has equilibrated, i.e., a large number of the first iterations should be discarded before the analysis. They should also be averaged over a number of runs, in order to reduce noise. This plot is better appreciated with a log scale in the vertical axis. Later on, it's important to ...

8

I think Anyon's and taciteloquence's answers are perfect. I just want to add an emphasis on the following fact that frequently leads to confusion for beginners. The formal definition of the magnetization $$m = \frac{\sum_i s_i}{N}$$ has a symmetry that $\mathrm{Prob}[m=+m_0]=\mathrm{Prob}[m=-m_0]$, since the energy of a particular ...

8

@stafusa's answer is great, but there is a specific phenomenon you are encountering here called critical slowing down, which is especially bad for the single-spin-flip Metropolis Algorithm. Near the critical point, the typical cluster size diverges. For the single-spin-flip algorithm, it's really hard to flip these huge clusters, so the autocorrelation ...

7

1D A famous example of a nearly ideal spin-$1/2$ isotropic Heisenberg antiferromagnetic chain (1D) system is copper pyrazine dinitrate [Cu(C$_4$H$_4$N$_2$)(NO$_3$)$_2$], which was discussed in Hammar et al. Phys. Rev. B 59, 1008 (1999) [arXiv link]. Another excellent realizations include KCuF$_3$, which has stronger (but still low) interchain coupling, and ...

6

Here I'll assume that the material is already believed to be roughly described by a model of the Heisenberg or Ising form. In that case, you just want a quantity that is easy to measure in your experiment and easy to extract from numerics (or with a well-established theoretical value). In practice, the choice of which quantity to use will depend on ...

5

As a (very) incomplete answer to this question, here is one paper discussing the Ising FM with a plaquette term. Here specifically chosen because there was not a good cluster algorithm for it (so they could try out a new algorithm: self-learning MC). Junwei Liu, Yang Qi, Zi Yang Meng & Liang Fu, Phys. Rev. B 95, 041101 (2017) Also arXiv:1610.03137

4

The equation in your question, be it Heisenberg or Ising exchange, can be calculated by Energy mapping analysis. This has to be the most popular paper that discusses this technique. Basically, you consider different spin configurations and map the exchange 'J' into the total energy values from DFT. In the simplest form, if you have two magnetic atoms in your ...

4

Actual examples of 2D magnetic systems are MXenes and metal-organic adsorption monolayers.

4

Is this the wrong assumption I am using for initializing starting_magnetization? If yes, What should be the right fact for choosing the initialization value? The initial magnetization is not so important. I will tell you how to generate the input file with a convenient method in which the initial magnetization will be considered automatically. I will assume ...

3

I think I figured this out. For most practical purposes, I think it is fine to just choose $-\pi\leq k_x\leq \pi$ and $0\leq k_y\leq \pi$ (half-BZ, with exact ranges depending on the model). I think I over-complicated the authors' work. It is unlikely that the vortices/singularities will occur near the boundary of the chosen region. So, I can just take the ...

3

No, the quantum version is not simpler There are many ways to find the ground state of an arbitrary quantum system. Quantum Monte Carlo (QMC), Density Matrix Renormalization Group (DMRG), et. al., but if you want an exact solution, you need either some very clever analytical approach (only available for special cases), or you need to do exact diagonalization....

3

NECI (N-electron configuration interaction solver) This is a free and open-source software written mainly in FORTRAN but with components in C/C++ and Python. It has been shown on a FeMoco calculation to have a parallelization efficiency of 99.7% up to at least 24800 cores. It does FCIQMC and has been used on Hubbard models in several papers, perhaps the most ...

2

ALF (Algorithms for Lattice Fermions) This is an auxiliary-field quantum Monte Carlo package. It's free, open-source, actively maintained (current version: 2.0, with 2.1 on the works — news are posted on its website) and has been cited in dozens of publications since its release in 2017. The package is very flexible, as it can simulate any Hamiltonian that ...

2

The Heisenberg formalism is often used to describe the interaction between molecules adsorbed on a surface (2D) using a cluster expansion. This has nothing to do with magnetism, but the mathematical framework is suitable for this kind of problem. Please take a look at Nielsen et al. J. Chem. Phys. 139 (2013) 224706. The application of the Heisenberg ...

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