# Tag Info

## Hot answers tagged model-hamiltonians

30

In an era of ab initio methods and many-body methods like the $GW$, there is not too much room for methods like the extended Hückel model to be the main method in any particular field of materials modeling. However, the method is still very much appreciated by the solid-state community particularly for it's accessibility. It is especially popular with those ...

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One area where the extended-Huckel method continues to see use is to form the initial guess for an SCF calculation or even just a more accurate semi-empirical method. While most electronic structures packages use the Superposition of Atomic Densities (SAD) guess as the default, the option is available in almost all of them and Psi4 uses it as the default for ...

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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 ...

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There's three scenarios that come to my mind, for when ab initio methods get abandoned: The cost becomes prohibitive (e.g. too many electrons) The insight is lost It is simply not required for what we want to do Prohibitive cost: If solving the Schrödinger equation (for example) is no longer possible, we may very well still wish we could solve the ...

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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 ...

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I know of at least one place, where it is relatively common to use the extended Hückel theory in practise: in generating initial guess orbitals for further electronic structure calculations. The most popular example I can think of is Turbomole, see its manual (pdf, chapter 4.3, p. 75). They claim that the starting vectors are better than a core Hamiltonian ...

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Introduction Your question reminds me of a quote by Paul Dirac, The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes ...

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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 ...

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A locally interacting system displaying a continuous phase transition belongs to a universality class that is determined solely by the system symmetries and dimensionality. Drawing from Wikipedia's list (itself mostly based on Ódor's paper) and this answer from Physics SE, here's a partial list of universality classes and critical exponents: \begin{array}{|...

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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 ...

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One way of determining this is using the projected density of states (P-DOS) This resolves the DOS into specific orbitals thereby allowing you to discretize each orbitals weight for a specific energy. $\mathrm{PDOS}_\nu(E) = \sum_i \psi^*_{i,\nu} [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i)$ Note here that $|\psi_i\rangle$ is the $i$th eigenvector and ...

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Indeed, due to the uncertainty relation, the electrons in a molecule do not have a definite potential energy, nor do they have a definite kinetic energy (yet they have well-defined expectation values of potential energy and kinetic energy). But, assuming that the molecule is in an energy eigenstate, the two uncertainties exactly cancel out, and gives an ...

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Realizing this Hamiltonian in a natural material I cannot imagine a material in which all nearest-neighbor spin-spin interactions can be adjusted arbitrarily at the same time. Spin-spin couplings that are stronger when the spins are closer together, and weaker when the spins are farther apart, can be adjusted by moving the spins relative to each other; so ...

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QuSpin QuSpin is an open-source Python code that can do exact diagonalization of spin, fermion, and boson systems. It has a wide support for use of symmetries, constrained Hilbert spaces, various models, and time evolution. The combination of fairly simple Python syntax and a large number of tutorials make it a great choice for beginners, for small-scale ...

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You're correct, but there's also an easier way. I tend to rephrase such problems in terms of an overall energy scale and a ratio, e.g. $$H=J\left[ \sum \limits_{\langle i,j \rangle} \vec S_i \cdot \vec S_j -\frac{g ~\mu_B}{J} \sum \limits_{i} \vec S_i \cdot \vec B \right] = J\left[ \sum \limits_{\langle i,j \rangle} \vec S_i \cdot \vec S_j -\alpha \sum \... 9 Following your arguments, we would see also a 'violation' of the Heisenberg uncertainty principle (HUP) in single-particle quantum mechanics, e.g. in a Hamiltonian of a particle in an external potential:$$ H = \hat{p}^2/2m + V(\hat x) \quad .\tag{1}$$But writing down such a Hamiltonian does not mean that we know the position or momentum (or both) of this ... 8 I think Anyon's and taciteloquence's answers are perfect. I just want to add an emphasis on the following fact that frequently leads to confusion for beginners. The formal definition of the magnetization $$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 ... 8 Yes! It is definitely possible and it is useful for calculating other things like electronic transport and first-principles values of Hubbard U (i.e. the ACBN0 method, which I have used a bit). Some good papers to read are from Prof. Marco Buongiorno Nardelli's group at UNT: L. A. Agapito, A. Ferretti, A. Calzolari, S. Curtarolo, and M. Buongiorno Nardelli, ... 8 For this type of calculation, I find MATLAB/Octave to be easier, at least for demonstrating what to do. You can add the np. everywhere afterwards if you want to use Python. Assuming you have the following matrices already defined in your workspace: \eta_z = eta \tau_x = taux \tau_y = tauy \tau_z = tauz m_z = tauz \sigma_z = sigmaz, and the ... 7 1D A famous example of a nearly ideal spin-1/2 isotropic Heisenberg antiferromagnetic chain (1D) system is copper pyrazine dinitrate [Cu(C_4H_4N_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 ... 7 How do you get the Hamiltonian formula for transition metal dichalcogenides? How to derivate the expression above (equation (1))? You can use the  k \cdot p method to derive this effective Hamiltonian. Please take a look at this paper for the details. k \cdot p theory for two-dimensional transition metal dichalcogenide semiconductors. What does the ... 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 ... 6 A common textbook statement of the uncertainty principle is:$$\tag{1} \Delta x \Delta p \ge \frac{\hbar}{2},  where $\Delta x$ and $\Delta p$ are the uncertainties in measurements of the position $x$ and momentum $p$ respectively. What mathematically is an "uncertainty"? In this case it's the standard deviation: \begin{align} \tag{2} \Delta x &...

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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

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In the attached paper they have used the broken symmetry method to calculate the exchange. Assuming that you're dealing with a 1D chain of spins. you can do this using quantum ESPRESSO via the following steps: Create a 2x1x1 super cell from your CIF file using VESTA (assuming that your 1d chain is along the first basis vector) For the SCF calculation using ...

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The Bernevig-Hughes-Zhang (BHZ) model for topological insulators comes to mind, see e.g. this pedagogical introduction or the review by Qi and Zhang (arXiv link here). It's often justified to set the block-off-diagonal terms zero, as was done in the paper introducing the model (arXiv version here), which simplifies analytical calculations considerably (it ...

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As @Jack suggested, try the $\vec{k} \cdot \vec{p}$ method. The hopping integral $t_{ij}$ is "borrowed" from the Hubbard model. It is the kinetic term in the Hamiltonian that explains electrons being able to "hop" from one atomic site to another. In real space, it is an integral of the orbital overlap between the two neighboring atomic ...

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I felt that it's important to note that not all universality classes are determined solely by spacial dimension and symmetry, even when the interactions are local. The easiest model that exemplifies this is probably the Ashkin-Teller model in 2D. It has a continuously varying (thus infinite variety of) critical exponents (universality classes) depending on ...

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