Hubbard Dimer and FCI

The Hubbard dimer is a minimal model system consisting of two sites that can each hold up to two electrons (one with spin up, one with spin down). It serves as an excellent pedagogical example for understanding Full Configuration Interaction (FCI) because the FCI space is small enough to be solved exactly, yet rich enough to demonstrate key concepts in quantum chemistry.

Basic Facts About the Hubbard Model

The Hubbard model is a simplified lattice Hamiltonian that captures essential features of strongly correlated electron systems. Here are key facts about the model:

• Purpose: The Hubbard model serves as a bridge between density functional theory (DFT) and strongly correlated systems. It provides a test case where exact solutions can be found analytically, making it ideal for understanding the relationship between DFT and many-body physics.
• Simplified Hamiltonian: Unlike first-principles methods that use the real-space Hamiltonian for electrons in the field of nuclei, the Hubbard model uses a lattice Hamiltonian with relatively few parameters. This simplification makes it easier to handle when correlations are strong.
• Two-Site Model: The Hubbard dimer (two-site model) is the simplest possible example that still captures the essential physics. Even in this elementary case, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of Density Functional Theory.
• Analytical Solution: The two-site Hubbard model can be solved exactly, providing analytical expressions for the ground-state energy and other properties. This makes it an excellent benchmark for testing approximate methods like DFT functionals.
• Connection to Quantum Chemistry: The Hubbard model can be derived from a minimal basis model for stretched H₂, connecting it directly to traditional quantum chemistry problems. This relationship helps bridge the gap between lattice models and real-space electronic structure theory.
• Strong vs. Weak Correlation: The model exhibits rich behavior depending on the ratio of the on-site repulsion $U$ to the hopping parameter $t$. When $U\gg t$, the system is strongly correlated, while $U\ll t$ corresponds to weak correlation.
• Asymmetry Effects: While the symmetric case (equal site energies) is most commonly studied, including asymmetry reveals a rich variation in behavior as strong correlation physics competes with charge-transfer effects.
• Testing Ground for Approximations: The exact solution of the Hubbard dimer allows for benchmark calculations of approximate functionals (like Hartree-Fock and Bethe Ansatz Local Density Approximation), showing their successes and limitations in capturing correlation effects.

Model Hamiltonian

The Hubbard Hamiltonian for a dimer (two-site system) is:

where:

• $t$ is the hopping parameter (kinetic energy)
• $U$ is the on-site repulsion (Coulomb interaction)
• ${\hat{c}}_{i\sigma }^{†}$ and ${\hat{c}}_{i\sigma }$ are creation and annihilation operators for site $i$ with spin $\sigma$
• ${\hat{n}}_{i\sigma }={\hat{c}}_{i\sigma }^{†}{\hat{c}}_{i\sigma }$ is the number operator

FCI Configuration Space

For the Hubbard dimer with $N$ electrons, we can enumerate all possible Slater determinants. The basis consists of two spatial orbitals (sites), each with two spin orbitals (up and down), giving a total of 4 spin orbitals.

For $N=2$ electrons (half-filled case), the number of configurations is:

The six possible configurations are:

• $|1_{\uparrow }{1}_{\downarrow }⟩$ - Both electrons on site 1
• $|2_{\uparrow }{2}_{\downarrow }⟩$ - Both electrons on site 2
• $|1_{\uparrow }{2}_{\uparrow }⟩$ - Both electrons with spin up, one on each site
• $|1_{\downarrow }{2}_{\downarrow }⟩$ - Both electrons with spin down, one on each site
• $|1_{\uparrow }{2}_{\downarrow }⟩$ - One electron on each site with opposite spins
• $|1_{\downarrow }{2}_{\uparrow }⟩$ - One electron on each site with opposite spins (different ordering)

FCI Wavefunction

The FCI wavefunction is a linear combination of all configurations:

where $|{\Phi }_I⟩$ are the Slater determinants and $c_I$ are the expansion coefficients.

Hamiltonian Matrix

The Hamiltonian matrix elements are computed using the second-quantized operators. The matrix has the structure:

For the two-site Hubbard model with $N=2$ electrons, the $6\times 6$ Hamiltonian matrix in the basis $\{|1_{\uparrow }{1}_{\downarrow }⟩,|2_{\uparrow }{2}_{\downarrow }⟩,|1_{\uparrow }{2}_{\uparrow }⟩,|1_{\downarrow }{2}_{\downarrow }⟩,|1_{\uparrow }{2}_{\downarrow }⟩,|1_{\downarrow }{2}_{\uparrow }⟩\}$ is:

where the basis states are ordered as:

1. $|{\Phi }_1⟩=|1_{\uparrow }{1}_{\downarrow }⟩$ - Both electrons on site 1
2. $|{\Phi }_2⟩=|2_{\uparrow }{2}_{\downarrow }⟩$ - Both electrons on site 2
3. $|{\Phi }_3⟩=|1_{\uparrow }{2}_{\uparrow }⟩$ - Both electrons spin up, one on each site
4. $|{\Phi }_4⟩=|1_{\downarrow }{2}_{\downarrow }⟩$ - Both electrons spin down, one on each site
5. $|{\Phi }_5⟩=|1_{\uparrow }{2}_{\downarrow }⟩$ - One electron on each site, opposite spins
6. $|{\Phi }_6⟩=|1_{\downarrow }{2}_{\uparrow }⟩$ - One electron on each site, opposite spins (different ordering)

Key features of the matrix:

• Diagonal elements: The on-site repulsion $U$ appears in $H_{11}$ and $H_{22}$ when both electrons occupy the same site. The other diagonal elements are zero since there's no double occupancy in those states.
• Off-diagonal elements: The hopping parameter $t$ connects configurations that differ by moving one electron between sites. For example, $H_{15}=-t$ connects $|1_{\uparrow }{1}_{\downarrow }⟩$ to $|1_{\uparrow }{2}_{\downarrow }⟩$ (moving the down-spin electron from site 1 to site 2).
• Block structure: The matrix has a block structure. The singlet states (configurations 1, 2, 5, 6) are coupled, while the triplet states (configurations 3, 4) are decoupled and have zero energy in this model.

Note: The signs of the off-diagonal elements depend on the ordering of creation operators in the Slater determinants and follow from the fermionic anticommutation relations.

Subspace Decomposition

The Hamiltonian matrix naturally decomposes into subspaces based on the total spin $S$. For the two-site Hubbard model with 2 electrons, we can organize the basis into singlet ($S=0$) and triplet ($S=1$) subspaces.

Sz=0 Subspace

The $S_z=0$ subspace consists of states with zero z-component of total spin. For 2 electrons, this includes all states where one electron has spin up and one has spin down:

• $|{\Phi }_1⟩=|1_{\uparrow }{1}_{\downarrow }⟩$ - Both electrons on site 1
• $|{\Phi }_2⟩=|2_{\uparrow }{2}_{\downarrow }⟩$ - Both electrons on site 2
• $|{\Phi }_5⟩=|1_{\uparrow }{2}_{\downarrow }⟩$ - One electron on each site, opposite spins
• $|{\Phi }_6⟩=|1_{\downarrow }{2}_{\uparrow }⟩$ - One electron on each site, opposite spins (different ordering)

In the $S_z=0$ subspace, the Hamiltonian matrix (in the basis $\{|1_{\uparrow }{1}_{\downarrow }⟩,|2_{\uparrow }{2}_{\downarrow }⟩,|1_{\uparrow }{2}_{\downarrow }⟩,|1_{\downarrow }{2}_{\uparrow }⟩\}$) is:

This $4\times 4$ matrix contains both singlet ($S=0$) and triplet ($S=1,S_z=0$) components. The ground state of the system lies in this subspace.

Singlet Subspace ($S=0$)

Within the $S_z=0$ subspace, we can further decompose into singlet ($S=0$) and triplet ($S=1,S_z=0$) states. The singlet state is the antisymmetric combination:

In the basis $\{|1_{\uparrow }{1}_{\downarrow }⟩,|2_{\uparrow }{2}_{\downarrow }⟩,|{\Psi }_{\text{singlet}}⟩\}$, the singlet block of the Hamiltonian is:

This $3\times 3$ matrix captures all the physics of the singlet states. The ground state of the system lies in this subspace.

Triplet Subspace ($S=1$)

The triplet subspace consists of states with total spin $S=1$. For 2 electrons, this includes:

• $|{\Phi }_3⟩=|1_{\uparrow }{2}_{\uparrow }⟩$ - Both electrons spin up, one on each site
• $|{\Phi }_4⟩=|1_{\downarrow }{2}_{\downarrow }⟩$ - Both electrons spin down, one on each site

In the standard Hubbard model (without spin-flip terms), these triplet states are decoupled from the singlet states and have zero energy:

This means the triplet states are eigenstates with eigenvalue 0. They represent states where both electrons have the same spin, which is not possible when both electrons are on the same site (due to Pauli exclusion), and when they're on different sites, there's no interaction energy in the Hubbard model.

Block-Diagonal Structure

The full $6\times 6$ Hamiltonian matrix has a block-diagonal structure when organized by spin:

where ${\mathbf{H}}_{S=0}$ is the $4\times 4$ singlet block and ${\mathbf{H}}_{S=1}$ is the $2\times 2$ triplet block. The zero blocks indicate that singlet and triplet states do not mix under the Hamiltonian.

Physical Interpretation

• Singlet subspace: Contains the ground state and excited states. These states allow for electron correlation and are the focus of most studies. The singlet states can have both electrons on the same site (double occupancy, energy cost $U$) or on different sites (delocalized, kinetic energy benefit from hopping $t$).
• Triplet subspace: These states are higher in energy (or degenerate with excited singlet states). They represent ferromagnetic configurations where both electrons have parallel spins. In the symmetric Hubbard dimer, these states have zero energy because there's no interaction between electrons of the same spin on different sites, and they cannot both occupy the same site.

This subspace decomposition is crucial for understanding the physics: the competition between kinetic energy (hopping) and potential energy (on-site repulsion) occurs entirely within the singlet subspace.

Ground State Energy

For the half-filled case ($N=2$), the ground state energy is:

This can be derived by diagonalizing the $6\times 6$ Hamiltonian matrix and taking the lowest eigenvalue.

Physical Interpretation

• Weak correlation ($U\ll t$): Electrons are delocalized, similar to a covalent bond
• Strong correlation ($U\gg t$): Electrons localize on separate sites, similar to an ionic state
• Intermediate ($U\approx t$): Competition between kinetic and potential energy

Python Implementation

Here is a Python implementation to solve the Hubbard dimer using FCI:

import numpy as np
from scipy.linalg import eigh

def hubbard_dimer_fci(t, U, n_electrons=2):
    """
    Solve the Hubbard dimer using Full Configuration Interaction.
    
    Parameters:
    -----------
    t : float
        Hopping parameter
    U : float
        On-site repulsion parameter
    n_electrons : int
        Number of electrons (default: 2 for half-filled)
    
    Returns:
    --------
    energies : array
        Eigenvalues (energies) of the system
    wavefunctions : array
        Eigenvectors (wavefunction coefficients)
    """
    # For N=2 electrons, we have 6 configurations
    # Basis: |1↑1↓⟩, |2↑2↓⟩, |1↑2↑⟩, |1↓2↓⟩, |1↑2↓⟩, |1↓2↑⟩
    
    H = np.zeros((6, 6))
    
    # Diagonal elements (on-site repulsion)
    H[0, 0] = U  # |1↑1↓⟩: both electrons on site 1
    H[1, 1] = U  # |2↑2↓⟩: both electrons on site 2
    # Other diagonal elements are 0 (no double occupancy)
    
    # Off-diagonal elements (hopping)
    # Hopping connects configurations that differ by moving one electron
    
    # |1↑1↓⟩ <-> |1↑2↓⟩ (move down electron from site 1 to 2)
    H[0, 4] = -t
    H[4, 0] = -t
    
    # |1↑1↓⟩ <-> |1↓2↑⟩ (move up electron from site 1 to 2)
    H[0, 5] = t
    H[5, 0] = t
    
    # |2↑2↓⟩ <-> |1↑2↓⟩ (move up electron from site 2 to 1)
    H[1, 4] = t
    H[4, 1] = t
    
    # |2↑2↓⟩ <-> |1↓2↑⟩ (move down electron from site 2 to 1)
    H[1, 5] = -t
    H[5, 1] = -t
    
    # |1↑2↑⟩ <-> |1↑2↓⟩ (flip spin on site 2)
    # This requires a spin-flip term, but in the standard Hubbard model
    # without spin-flip, these don't connect directly
    
    # |1↓2↓⟩ <-> |1↓2↑⟩ (flip spin on site 2)
    # Similar to above
    
    # Solve the eigenvalue problem
    energies, wavefunctions = eigh(H)
    
    return energies, wavefunctions

# Example: Solve for different U/t ratios
t = 1.0
U_values = [0.5, 1.0, 2.0, 5.0]

print("Hubbard Dimer FCI Results (t = 1.0)")
print("=" * 50)
print("{:>8} {:>15} {:>15}".format("U/t", "E_0 (FCI)", "E_0 (Exact)"))
print("-" * 50)

for U in U_values:
    energies, _ = hubbard_dimer_fci(t, U)
    E0_fci = energies[0]
    
    # Exact analytical result
    E0_exact = U/2 - np.sqrt((U/2)**2 + 4*t**2)
    
    print("{:>8.2f} {:>15.6f} {:>15.6f}".format(U/t, E0_fci, E0_exact))

print("\n")
print("Note: The exact formula is E_0 = U/2 - sqrt((U/2)^2 + 4t^2)")

Connection to FCI

The Hubbard dimer demonstrates several key aspects of FCI:

• Complete basis: All possible electron configurations are included
• Exact solution: FCI gives the exact ground state within the model
• Correlation: The wavefunction captures electron correlation through the linear combination of configurations
• Size of problem: Even for this minimal system, we see how the number of configurations grows

Limitations and Extensions

While the Hubbard dimer is exactly solvable, it has limitations:

• Only two sites, so limited spatial degrees of freedom
• No long-range interactions
• Simplified interaction (only on-site repulsion)

Extensions include:

• Multi-site Hubbard models (chains, lattices)
• Extended Hubbard models with nearest-neighbor interactions
• Periodic boundary conditions

References

For more details on FCI, see the Full Configuration Interaction page.

Key Reference on the Hubbard Model and DFT:

• D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, "The Hubbard Dimer: A density functional case study of a many-body problem", Physics Reports (2015). https://dft.uci.edu/pubs/CFSB15.pdf

This comprehensive review explains the relationship between density functional theory and strongly correlated models using the two-site Hubbard model, including its connection to traditional quantum chemistry, exact solutions, and benchmark calculations of approximate functionals.