Singlet-Triplet Splitting and Antiferromagnetic Coupling

The Full Low-Energy Spectrum

So far, we have focused on the ground state, which we found by diagonalizing the $2\times 2$ Hamiltonian matrix within the even-parity ($P=+1$) sector of the Hilbert space. The basis states were $|1+⟩$ and $|2+⟩$. To find the excited states, we must consider the other sectors.

Let's first examine the odd-parity ($P=-1$) sector for two electrons with total spin projection $S_z=0$. The basis states are:

The Hamiltonian matrix elements in this basis are:

• $⟨1-|H|1-⟩=0$
• $⟨2-|H|2-⟩=U$
• $⟨1-|H|2-⟩=⟨2-|H|1-⟩=0$ (You can verify this as an exercise).

Since the off-diagonal element is zero, the Hamiltonian is already diagonal in this basis. The eigenstates are simply $|1-⟩$ with energy $E=0$ and $|2-⟩$ with energy $E=U$.

Finally, we must consider the states with total spin projection $S_z=\pm 1$. These are the states with parallel spins:

• $|\uparrow \uparrow ⟩=c_{1\uparrow }^{†}{c}_{2\uparrow }^{†}|0⟩$
• $|\downarrow \downarrow ⟩=c_{1\downarrow }^{†}{c}_{2\downarrow }^{†}|0⟩$

What are their energies? The interaction term $U{\sum }_i{n}_{i\uparrow }{n}_{i\downarrow }$ gives zero since there are no doubly occupied sites. The hopping term $-t{\sum }_{\sigma }(c_{1\sigma }^{†}{c}_{2\sigma }+c_{2\sigma }^{†}{c}_{1\sigma })$ also gives zero when acting on these states. For instance, if an electron hops from site 2 to site 1 in the $|\uparrow \uparrow ⟩$ state, it would create $c_{1\uparrow }^{†}{c}_{1\uparrow }^{†}{c}_{2\uparrow }^{†}{c}_{1\uparrow }|0⟩$, which is zero by the Pauli exclusion principle ($(c_{1\uparrow }^{†}{)}^2=0$). Thus, these two states are also eigenstates with energy $E=0$.

Identifying Singlet and Triplet States

We have found the complete spectrum of the two-electron system. Now let's organize the states by their total spin. Recall that for two spin-1/2 particles, the total spin can be $S=0$ (a singlet) or $S=1$ (a triplet).

• Singlet States ($S=0$):

  • The ground state, $|{\Psi }_0⟩=u|1+⟩+v|2+⟩$, is a spin singlet. Its energy is $E_S=\frac{U}{2}-\sqrt{(\frac{U}{2}{)}^2+4t^2}$.
  • There is another singlet excited state with energy $E_{S^{\prime }}=\frac{U}{2}+\sqrt{(\frac{U}{2}{)}^2+4t^2}$.

• Triplet States ($S=1$):

  • The state $|1-⟩$ is the $S_z=0$ component of the spin triplet. Its energy is $E_T=0$.
  • The states $|\uparrow \uparrow ⟩$ ($S_z=1$) and $|\downarrow \downarrow ⟩$ ($S_z=-1$) are the other two members of the triplet. Their energy is also $E_T=0$.

At low energies, the relevant states are the singlet ground state $|{\Psi }_0⟩$ and the three degenerate triplet states. The energy splitting between them is:

This is the singlet-triplet splitting. The ground state is always a singlet, meaning the system energetically favors the state where the two electron spins are antiparallel.

The Strong-Coupling Limit: Emergence of Superexchange

The physics becomes particularly illuminating in the strong-coupling limit, $U\gg t$, which is the regime where the Hubbard model describes Mott insulators. Let's find the approximate form of the singlet-triplet splitting in this limit.

We can expand the square root using the Taylor series $\sqrt{1+x}\approx 1+x/2$ for small $x$:

So, in the strong-coupling limit, the singlet state is lower in energy than the triplet states by approximately $4t^2/U$.

This energy splitting is the hallmark of an effective magnetic interaction. We can describe this low-energy physics with an effective Hamiltonian that acts only on the spin degrees of freedom. This is the Heisenberg Hamiltonian:

where ${\vec{S}}_1$ and ${\vec{S}}_2$ are the spin operators on the two sites. The eigenvalues of ${\vec{S}}_1\cdot {\vec{S}}_2$ are $-3/4$ for a singlet and $+1/4$ for a triplet. The energy splitting is therefore $E_T-E_S=J(1/4-(-3/4))=J$.

By comparing this with our result from the Hubbard model, we can identify the effective exchange coupling constant $J$:

Since $J>0$, the interaction is antiferromagnetic: it favors antiparallel alignment of spins.

The following resources provide a detailed derivation and interpretation of this crucial result.

The physical mechanism behind this antiferromagnetic coupling is called superexchange. In the $U\gg t$ limit, the electrons are localized on their respective sites to avoid the large energy cost $U$. However, quantum mechanics allows for "virtual" processes. An electron can hop to the adjacent, occupied site (costing energy $U$) and then immediately hop back. This process is only possible if the two electrons have opposite spins (due to the Pauli principle). According to second-order perturbation theory, such virtual fluctuations always lower the energy of the ground state. The state with antiparallel spins (the singlet) can take advantage of these virtual hopping processes to lower its energy, while the state with parallel spins (the triplet) cannot. The energy gained by the singlet state through this virtual process is precisely $J=4t^2/U$.

This is a profound result: a model containing only kinetic energy ($t$) and charge repulsion ($U$) gives rise to an effective magnetic interaction between spins in the strong-correlation limit.

To see a lecture-style explanation of this derivation via degenerate perturbation theory, you can watch the following video.

Conclusion

In this lesson, we completed our analysis of the two-site Hubbard model at half-filling by examining its excited states and their spin character.

Key Takeaways:

• The low-energy spectrum of the two-site Hubbard model consists of a singlet ground state and a three-fold degenerate triplet excited state.
• The ground state is always a singlet, with energy $E_S=\frac{U}{2}-\sqrt{(\frac{U}{2}{)}^2+4t^2}$. The triplet states have energy $E_T=0$.
• In the strong-coupling limit ($U\gg t$), the singlet-triplet energy splitting is $\Delta E\approx 4t^2/U$.
• This energy splitting can be described by an effective Heisenberg Hamiltonian $H_{\text{eff}}=J{\vec{S}}_1\cdot {\vec{S}}_2$, with an antiferromagnetic exchange coupling $J=4t^2/U$.
• The physical origin of this interaction is superexchange: virtual hopping processes, only allowed for antiparallel spins, lower the energy of the singlet state.

The two-site model, despite its simplicity, has revealed the two cornerstone phenomena of the Hubbard model: the U-driven localization leading to a Mott insulator (seen via double occupancy) and the emergence of antiferromagnetism at strong coupling.

In the final lesson of this module, we will briefly step away from the half-filled case to see how the model behaves for other electron numbers, completing our exact analysis of this toy system before moving on to larger lattices.