Stoner Criterion from Mean-Field Hubbard Model

Hello! Welcome back to our course on condensed matter physics.

In our last lesson, we explored the Hubbard model in the atomic limit ($t=0$), where strong on-site repulsion $U$ dominates. We discovered that this interaction can completely localize electrons, leading to a Mott insulator—an insulating state driven by correlations, not band filling.

Today, we shift our focus to a regime where both kinetic energy ($t$) and interaction energy ($U$) are significant. We will investigate another profound consequence of the Hubbard $U$ term: its ability to drive a metallic system into a ferromagnetic state. This phenomenon, known as itinerant ferromagnetism, is responsible for magnetism in common metals like iron and nickel. Our goal is to apply a mean-field approximation to the Hubbard model to derive the famous Stoner criterion, which quantifies the conditions necessary for this magnetic ordering to occur.

1. The Idea Behind Itinerant Ferromagnetism

Before diving into the mathematics, let's build some physical intuition. Imagine the sea of itinerant electrons in a metal. We want to know if the system can lower its total energy by spontaneously developing a net magnetization, meaning an imbalance between the number of spin-up ($N_{\uparrow }$) and spin-down ($N_{\downarrow }$) electrons. This involves a trade-off:

• Kinetic Energy Cost: According to the Pauli exclusion principle, if we increase the number of, say, spin-up electrons, they are forced to occupy higher-energy momentum states. This increases the total kinetic energy of the system compared to the spin-balanced (paramagnetic) state where up and down electrons fill the lowest available energy levels equally.
• Potential Energy Gain: The Hubbard interaction $U$ only acts between electrons of opposite spin on the same site. By polarizing the spins (e.g., creating a majority of spin-up electrons), we reduce the probability of two electrons being on the same site, thus lowering the total interaction energy. In the extreme case of full polarization ($N_{\downarrow }=0$), the interaction energy term $U{\sum }_i{n}_{i\uparrow }{n}_{i\downarrow }$ becomes zero.

Ferromagnetism emerges if the potential energy gain from spin alignment overcomes the kinetic energy cost. The Stoner criterion provides the precise condition for this to happen.

The following video introduces the concept of band magnetism and the Stoner model, providing a good conceptual starting point for our discussion.

2. The Mean-Field Approximation to the Hubbard Model

To analyze this competition quantitatively, we must deal with the quartic interaction term $U{n}_{i\uparrow }{n}_{i\downarrow }=U{c}_{i\uparrow }^{†}{c}_{i\uparrow }{c}_{i\downarrow }^{†}{c}_{i\downarrow }$ in the Hubbard Hamiltonian. This term makes an exact solution impossible in general. We need an approximation.

The mean-field approximation is a powerful technique that simplifies the problem by replacing the instantaneous interactions between particles with an interaction between a single particle and the average field produced by all other particles.

We start by writing the number operator for spin $\sigma$ on site $i$ as its average value (the "mean field") plus a fluctuation term:
$n_{i\sigma }=⟨n_{i\sigma }⟩+\delta n_{i\sigma }\text{where}\delta n_{i\sigma }=n_{i\sigma }-⟨n_{i\sigma }⟩$

Now, we substitute this into the interaction term:
$n_{i\uparrow }{n}_{i\downarrow }=(⟨n_{i\uparrow }⟩+\delta n_{i\uparrow })(⟨n_{i\downarrow }⟩+\delta n_{i\downarrow })=⟨n_{i\uparrow }⟩⟨n_{i\downarrow }⟩+⟨n_{i\downarrow }⟩\delta n_{i\uparrow }+⟨n_{i\uparrow }⟩\delta n_{i\downarrow }+\delta n_{i\uparrow }\delta n_{i\downarrow }$

The core of the approximation is to assume the fluctuations are small and neglect the term that is quadratic in fluctuations, $\delta n_{i\uparrow }\delta n_{i\downarrow }$. After substituting $\delta n_{i\sigma }$ back, we arrive at the mean-field decomposition:
$n_{i\uparrow }{n}_{i\downarrow }\approx n_{i\uparrow }⟨n_{i\downarrow }⟩+n_{i\downarrow }⟨n_{i\uparrow }⟩-⟨n_{i\uparrow }⟩⟨n_{i\downarrow }⟩$

The physical picture is clear: a spin-up electron at site $i$ (described by $n_{i\uparrow }$) now feels an effective potential proportional to the average density of spin-down electrons $⟨n_{i\downarrow }⟩$, and vice-versa. The final term corrects for double-counting this interaction energy.

Applying this approximation to the Hubbard Hamiltonian, we get the Mean-Field Hamiltonian:
$H_{MF}={\sum }_{⟨i,j⟩,\sigma }-t(c_{i\sigma }^{†}{c}_{j\sigma }+H.c.)+U{\sum }_i(n_{i\uparrow }⟨n_{i\downarrow }⟩+n_{i\downarrow }⟨n_{i\uparrow }⟩)-U{N}_{sites}⟨n_{\uparrow }⟩⟨n_{\downarrow }⟩$
This Hamiltonian is now quadratic in the fermion operators and can be solved.

3. Deriving the Stoner Criterion

Let's assume a uniform ferromagnetic state where the average densities are site-independent: $⟨n_{i\sigma }⟩=⟨n_{\sigma }⟩$. The Hamiltonian is most easily diagonalized in momentum space:
$H_{MF}={\sum }_{\mathbf{k},\sigma }{ϵ}_{\mathbf{k}}{c}_{\mathbf{k}\sigma }^{†}{c}_{\mathbf{k}\sigma }+{\sum }_{\mathbf{k},\sigma }U⟨n_{-\sigma }⟩c_{\mathbf{k}\sigma }^{†}{c}_{\mathbf{k}\sigma }-U{N}_{sites}⟨n_{\uparrow }⟩⟨n_{\downarrow }⟩$
Here, ${ϵ}_{\mathbf{k}}$ is the dispersion of the non-interacting electrons (e.g., $-2t\cos (ka)$ in 1D). Combining terms, we see that the interaction has simply shifted the energy bands in a spin-dependent way:
$H_{MF}={\sum }_{\mathbf{k},\sigma }{E}_{\mathbf{k}\sigma }{c}_{\mathbf{k}\sigma }^{†}{c}_{\mathbf{k}\sigma }-U{N}_{sites}⟨n_{\uparrow }⟩⟨n_{\downarrow }⟩$
where the new, spin-dependent dispersion relations are:
$E_{\mathbf{k}\uparrow }={ϵ}_{\mathbf{k}}+U⟨n_{\downarrow }⟩$
$E_{\mathbf{k}\downarrow }={ϵ}_{\mathbf{k}}+U⟨n_{\uparrow }⟩$
The spin-up band is shifted by an amount proportional to the spin-down density, and vice-versa. If we have more spin-up electrons ($⟨n_{\uparrow }⟩>⟨n_{\downarrow }⟩$), the spin-down band is pushed to higher energies, while the spin-up band is lowered. This population imbalance is self-consistently maintained by the energy shifts it creates.

The Stoner criterion emerges when we find the condition for a non-magnetic (paramagnetic) state to become unstable towards a ferromagnetic state. The following reading provides two clear derivations of this. First, a simple and elegant argument based on energy competition, and second, a formal derivation using the mean-field Hamiltonian we just constructed.

Let's summarize the key argument from Section XII. We consider transferring an infinitesimal density of electrons, $\delta n$, from the spin-down to the spin-up channel, starting from a paramagnetic state ($n_{\uparrow }=n_{\downarrow }$).

1. Potential Energy Change ($\delta E_{pot}$): The change in interaction energy is $\delta E_{pot}=U(n_{\uparrow }+\delta n)(n_{\downarrow }-\delta n)-U{n}_{\uparrow }{n}_{\downarrow }\approx -U(\delta n{)}^2$. This is an energy gain.

2. Kinetic Energy Change ($\delta E_{kin}$): We must move $\delta n$ electrons from just below the Fermi energy $E_F$ to just above it. The density of states for a single spin species at the Fermi level is $N(E_F)$. The energy range we must fill to accommodate these electrons is $\delta E=\delta n/N(E_F)$. The total kinetic energy cost is the number of electrons moved multiplied by the average energy increase, $\delta E_{kin}=\delta n\times \delta E=\frac{(\delta n{)}^2}{N(E_F)}$. This is an energy cost.

3. Total Energy Change ($\delta E_{tot}$):
   $\delta E_{tot}=\delta E_{kin}+\delta E_{pot}=\left(\frac{1}{N(E_F)}-U\right)(\delta n{)}^2$

A spontaneous transition to ferromagnetism is energetically favorable if $\delta E_{tot}<0$. This leads directly to the Stoner criterion:
$\frac{1}{N(E_F)}-U<0⟹U\cdot N(E_F)>1$

This simple and beautiful result tells us that itinerant ferromagnetism is favored by:

• A large on-site interaction $U$.
• A large density of states at the Fermi level, $N(E_F)$.

This explains why elements like Fe, Co, and Ni, which have narrow and high-DOS d-bands crossing their Fermi level, are ferromagnetic, while simple s-p metals like Na or Al, with low, broad DOS, are not.

4. Alternative Derivations and a More Formal View

The simple derivation above is powerful, but it's reassuring to see the same result emerge from more formal theoretical frameworks. Given your background in QFT, you might appreciate that this mean-field approximation is equivalent to the Hartree-Fock approximation.

The following resource provides several derivations of the Stoner criterion, including the one we just saw (called the Hartree-Fock approximation there), a derivation using Feynman's variational principle, and one using Green's functions. These demonstrate the robustness of the result.

The derivation in Section 6.2.3 is particularly insightful. It shows how the condition for magnetization $m$ to be non-zero leads to an equation of the form $(1-UN(E_F))m=b{m}^3$, which is a classic Landau-Ginzburg equation for an order parameter. This firmly places Stoner ferromagnetism within the general framework of continuous phase transitions and broken symmetry.

Conclusion

Today, we've seen how the electron-electron interaction in the Hubbard model can drive a completely different phenomenon from Mott localization. By using a mean-field approximation, we tamed the complexity of the interaction term and uncovered the physics of itinerant ferromagnetism.

Key Takeaways:

• Itinerant ferromagnetism arises from a competition between the kinetic energy cost and the potential energy gain of aligning electron spins.
• The mean-field approximation linearizes the Hubbard interaction term, resulting in an effective Hamiltonian of non-interacting electrons with spin-dependent energy bands.
• The energy of the spin-$\sigma$ band is shifted by an amount proportional to the average density of the opposite spin, $⟨n_{-\sigma }⟩$.
• The system becomes unstable towards a ferromagnetic state when the Stoner criterion, $U\cdot N(E_F)>1$, is met.
• This criterion highlights that ferromagnetism is favored by strong interactions ($U$) and a high density of states at the Fermi level ($N(E_F)$).

Preview of the Next Lesson

The mean-field theory we used today is a powerful first step, but it has its limitations (e.g., it gets critical exponents wrong). We saw that ferromagnetism in the Hubbard model is possible at half-filling, though often other phases like antiferromagnetism are more stable. We have now studied the Hubbard model in the large-U limit (Mott insulator) and the weak-coupling mean-field limit (Stoner ferromagnetism). What happens at exactly half-filling? In the next lesson, we will analyze the Hubbard model at half-filling using a mean-field approach to see if we can capture the Stoner criterion for itinerant ferromagnetism there. We will then discuss the Mott metal-insulator transition in more detail and its experimental signatures, bridging the gap between the two limits we have explored.