A first bite to Quantum Magnetism – Junjun's Cool Atomic Site

Magnetism is among our most familiar phenomena in daily life, however the physics behind it is so deeply rooted in the quantum world that we are still far from a complete understanding. In the following I will give you an introduction to how it is emerged and the relation to the mysterious unsolved problems in strongly correlated systems.

Our story of quantum magnetism starts from the spin of electrons. Let’s consider two electrons confined in a deep square potential well in one dimension, as shown in Fig. 1.

*Fig. 1 An illustration of two electrons confined in a one-dimensional potential well.*

The question leaves to you: what is the ground state configuration of the system?

Well, it depends on the interaction of the electrons, i.e., the strength of the Coulomb repulsion between them. To see this, we first write down the Hamiltonian of the system
\[
H=-\nabla^2_x-\nabla^2_y+V(x,y)+\frac{U}{|x-y|},
\]
where $x$ and $y$ label the coordinates of each electrons, and $V(x,y)$ accounts the trapped one-dimensional potential well. The strength of the Coulomb repulsion of these electrons are characterized by $U$.

(1) If there is no Coulomb repulsion, these two electrons will tend to occupy the same orbital ground state, in order to lower the kinetic energy (Fig. 2).

*Fig. 2 The ground state orbital wavefunction of the electrons for interaction strength $U=0$. The two electrons both occupy the lowest single-particle state to reduce the kinetic energy.*

The Fermi statistics require the spin state to be in the anti-symmetric state. However, as can be seen in the density distribution of these two electrons in Fig. 3, since they are mixed in the coordinate space, the system has no magnetic order.

*Fig. 3 The ground state density distribution of these two electrons (solid blue and dashed red respectively) for interaction strength $U=0$. The system shows no magnetism.*

(2) Now let’s give the electrons a little bit Coulomb repulsion. In this stage, the repulsion is not strong enough to break the single-particle picture, and the electrons tend to occupy higher single-particle orbitals to reduce the Coulomb repulsion, leading to an anti-symmetric orbital wavefunction as shown in Fig. 4.

*Fig. 4 The ground state orbital wavefunction of the electrons for interaction strength $U=1$. It becomes anti-symmetric to reduce the strong Coulomb repulsion, especially at positions $x=y$.*

In this case, the corresponding spin wavefunction turns to be symmetric and the system develops a ferromagnetic order (Fig. 5). This mechanism contributes to what we know as the “Hund’s rule”, and is connected to the itinerant ferromagnetism.

Fig. 5 The ground state density distribution of these two electrons (solid blue and dashed red respectively) for interaction strength $U=1$. Due to the anti-symmetric of the orbital part, the spin wavefunction now becomes symmetric, leading to a transition to a ferromagnetic ground state.

(3) We keep on increase the Coulomb repulsion. As you can imagine in Fig. 6, the repulsion is so large that the electrons have to stay separated. This leads to the strongly correlated limit, where the position of one electron is highly influenced by the other electron, and any single-particle picture in this case fails to describe the system.

It can be further shown that by including the lattice potential, the system tries to reduce its kinetic energy by forming a spin-singlet state, leading to an antiferromagnetic order in the coordinate space. This is why most strongly correlated electronic materials starts from an antiferromagnetic state.

Fig. 6 For sufficiently large Coulomb repulsion $U=500$, the ground state densities of these two electrons become separated, corresponding to a strongly correlated system. In a lattice environment, a symmetric superposition of the orbital part wins, and the system shows anti-ferromagnetism.

A further journey:

As an example of the strongly correlated systems, we show the phase diagram of the most well-known high-Tc superconductivity of copper oxides in Fig. 7. The high-Tc superconductivity grows out of the strongly correlated antiferromagnetic state of copper oxides by doping. The surprisingly pseudogap regime (a regime possibly with preformed pairing electrons without phase coherences) and quantum critical regime (a finite temperature regime where criticality behaviors of the quantum phase transition point is still felt) are still far from a complete understanding.

Fig. 7 The qualitative phase diagram of the high-Tc superconductivity of doped cooper oxides shows a tough example of the strongly correlated systems. The pseudogap and quantum critical regimes are still mysterious to us and far from a complete understanding.

Recent researches in cold atoms discovered similar behaviors in the unitary regime of Fermi gases (Fig. 8), where the system undergoes a strongly interaction regime.

Fig. 8 The unitary regime in cold atomic systems shows similar behaviors as in the high-Tc problem in copper oxides, suggesting a possible way to crack the high-Tc problem. Figure quoted from Physics Today, 61, 45 (2008).

The similarities between the unitary regime and high-Tc superconductivity in cooper oxides pave us a way to a unified description of these curious behaviors.