Is Heisenberg model or in its simplier form Ising model a good approximation to study magnetic systems?

Question

Heisenberg model

$$\hat{H}=-\sum_{\langle i j\rangle}J\hat{S}_i\hat{S}_j$$

And in its simplified version, the Ising model

$$\hat{H}=-\sum_{\langle ij\rangle}J\hat{S}_i^z\hat{S}_j^z$$

are widely applied in the field of condensed matter to understand magnetic systems. For example, there is classical spin wave approach and Bloch $T^{\frac{3}{2}}$ law.

There are also many computational methods available to diagonalize the Heisenberg effective model Hamiltonian (e.g. renormalization group method , quantum Monte Carlo, etc).

However, is the Heisenberg model Hamiltonian good enough to study magnetic system? As the assumption of these type of Hamiltonian is that the electrons are localized on each site (unit cell) and thus no charge transfer states are allowed. What's more, the exchange integrals in these Heisenberg model are to some extent semi-empirical and could only be determined through experiments.

While on the other hand, in modern quantum chemistry ab initio Hamiltonian is introduced to study the electronic structure of molecular systems. By introducing certain basis sets and Slater determinants, all matrix elements in the Hamiltonian of electronic structure could be evaluated merely via first principle (of course some additional constants are required e.g. Planck constant, speed of light, etc).

My question is, how is the performance of the Heisenberg Hamiltonian compares to a first principle Hamiltonian? What are the pros and cons?

Answer 1

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 desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

(He published this in 1929, but the overall sentiment remains relevant. See this Chemistry.SE question for a discussion about the degree to which it's true.)

At least from a physicist's perspective, the point of models like the Heisenberg Hamiltonian is to have a simpler description that is "good enough", and lets us focus on the relevant low-energy degrees of freedom. In doing so we can hopefully come to understand a system, at least in some regime. A key point is that these models have a limited range of validity, but if this range includes experimentally relevant energy and temperature scales they can be very useful. Clearly, at high enough energies where charge excitations begin to appear, the idea of a pure magnetic insulator (a potentially confusing term that means insulators that have some form of magnetism) breaks down. Then again, we could say the same about first principles quantum chemistry — at high enough energy we can no longer neglect particle physics effects! It's a beautiful and non-trivial fact that simplified "effective" descriptions are valid at different scales (be they energy, length, particle number, etc.) and that we can forget about some details from the level below, while finding new emergent behavior as we climb the ladder of scale.

Is the Heisenberg Hamiltonian good enough?

Certainly the antiferromagnetic Heisenberg has some close realizations in materials. However, given your choice of sign for $J$ and mention of the Bloch $T^{3/2}$ law suggests you're more interested in ferromagnetic systems. I can't claim expertise in these, but maybe I can say something general. Many ferromagnets are itinerant systems, which is exemplified by Stoner's model of majority and minority spin bands, which can be considered a mean-field approximation of a Hubbard Hamiltonian. This itinerant limit is completely different from the localized limit that leads to the Heisenberg spin model, which is applicable to magnetic insulators. Before the early 1950s this picture was not really clear, and there was a debate about which model would be more appropriate — especially when it came to $d$ electrons in iron group metals. The Stoner model didn't reproduce the Curie-Weiss law as well as the Heisenberg model, but could explain fractional saturation moments. In addition, it wasn't clear at the time how to get a ferromagnetic $J$ in case of the Heisenberg model.

Nature being nature didn't seem to entirely favor either limit, so people like Van Vleck and others worked on "middle-of-the-road" theories to better describe such systems. Eventually this led to more involved theories, including a successful Self-Consistent Renormalization (SCR) theory introduced by Moriya and Kawabata. Such theories elucidated the differences between the two limits, and how to describe them. In Moriya's words,

Around 1960’s a widely accepted point of view, after the long controversy, was that the magnetic insulator compounds and rare earth magnets are described in terms of the localized electron model while the ferromagnetic d-electron metals should be described on the itinerant electron model with the approximation method beyond the mean field level, properly taking account of the effects of electron-electron correlations. One of the clearest motivations for this consensus was the successful experimental observations of the d-electron Fermi surfaces in ferromagnetic Fe and Ni and their good comparisons with the results of band theoretical calculations.

The conclusion is that, in many cases, the Heisenberg Hamiltonian is far from good enough. But in the case of ferromagnetic insulators it can be good enough — especially if you allow for some anisotropy, either in the case of an XXZ anisotropy $\Delta$, or a single-ion anisotropy $D$, as in $$ H = J \sum_{\langle i,j\rangle} \left[ S_i^x S_j^x + S_i^y S_j^y + \Delta S_i^z S_j^z \right] + D \sum_i \left( S_i^z \right)^2, $$ or other interactions, like the Dzyaloshinskii-Moriya interaction (DMI) in the case of oxides with heavier ions. I provide a list of such FM insulating materials below.

What is the performance of the Heisenberg Hamiltonian compared to a first principles Hamiltonian? What are the pros and cons?

Unfortunately I don't really have a satisfactory answer to this question. I don't know of a direct benchmark. But we can return to the dichotomy of the two limits discussed above. For itinerant systems, first-principles calculations should of course do better, but it's not a very fair comparison. In the limit of ferromagnetic insulators, the energy scales are typically on the order of 1-10 meV or less. Practically speaking, that's beyond the accuracy of DFT-based methods and electronic stucture quantum Monte Carlo. On the other hand, spin models lose by walkover when it comes to the physics they neglect, and thus have nothing to say about some properties and experiments. Quantum chemistry does have some highly accurate wave function approaches, but they tend to scale very badly with system size. E.g. the CCSD(T) method famously scales as $N^7$, where $N$ is the system size. This makes it pretty much a non-starter to use them to explore large-scale collective magnetic phenomena.

Personally I hope to see more progress in this area. It would be very useful to have a reliable first-principles method to derive low energy spin Hamiltonians, that can then be explored in more detail. (Some experiments, like neutron scattering, are more naturally interpreted in terms of spin Hamiltonians.) It might also speed up discovery of materials hosting exotic phases.

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Appendices

Ferromagnetic insulators

While ferromagnetic insulators appear to be rare compared to antiferromagnetic ones, there are some examples, and they seem to have applications in spintronics and to induce ferromagnetic backgrounds in nonmagnetic materials. The most well-known is probably EuO, discovered in 1961, and similar Europium chalcogenides EuX (X=O, S, Se, Te). These are well-described by the Heisenberg Hamiltonian, as discussed in the review by Mauger and Goodart, Physics Reports 141, 51-176 (2006). In these materials the ferromagnetic coupling appears to be due to an indirect Eu-Eu exchange.

I provide a partial list of ferromagnetic insulating compounds here, but note that some may have differnt (including more complicated) spin model Hamiltonians.

• EuO
• EuS
• EuSe
• EuTe
• YTiO$_3$ (Spaldin)
• SeCuO$_3$ (Spaldin)
• BiMnO$_3$ (Spaldin)
• La$_2$NiMnO$_6$ (Spaldin)
• LaMnO$_3$ (Spaldin)
• CaMNO$_3$ (Spaldin)
• Sr$_2$CrOsO$_6$ (source)
• La$_2$CoMnO$_6$ (source)
• Yttrium iron garnet (YIG)
• CoFe2O4 (mentioned here)
• Lu$_2$V$_2$O$_7$ (includes DMI, source)
• Cu(1-3)-bdc (with DMI, source)

Above Spaldin refers to Nicola Spaldin's book Magnetic Materials: Fundamentals and Applications.

Other mechanisms

Despite what I've written above, the Heisenberg Hamiltonian is actually sometimes relevant also to metals (which we'd naturally think of as itinerant). The idea is that the conduction electrons provide a non-magnetic background, and that the presence of nuclear spins or magnetic ions interact with this conduction electron background as to produce a long-range Heisenberg interaction, which can be either ferro- or antiferromagnetic. This is known as the RKKY interaction. Again, it's worth noting that in the RKKY limit there are conduction electrons present, but the derived effective Heisenberg Hamiltonian still describes part of the physics. Other mechanisms include Zener carrier-mediated exchange, and double exchange. See e.g. Spaldin for a discussion of these.