# Half-filled Hubbard ring: Mott correlations on a finite cluster

The one-dimensional repulsive Hubbard model is a standard setting for the interplay between itinerancy, local-moment formation, and antiferromagnetic correlations. At half filling the thermodynamic chain is insulating for every $U>0$ by the Lieb-Wu solution.[^liebwu] A finite ring does not exhibit a thermodynamic transition, but it resolves how charge fluctuations are suppressed and how spin correlations become increasingly dominant as $U/t$ grows.

For a periodic ring of $L=6$ sites,

$$
H=-t\sum_{i,\sigma}
\left(c_{i\sigma}^\dagger c_{i+1,\sigma}+\mathrm{H.c.}\right)
+U\sum_i n_{i\uparrow}n_{i\downarrow},
$$

with $c_{L\sigma}=c_{0\sigma}$ and $N=L=6$. The fixed-$N$ Hilbert space contains

$$
\binom{12}{6}=924
$$

states.

## Ground state and low-energy spectrum

```python
import numpy as np
from edinpy import fermion as edf

L = 6
UP, DOWN = 0, 1

site = edf.DoF(L, name="site")
spin = edf.DoF(2, name="spin")
modes = edf.FermionModes(site, spin)
sector = edf.NParticleSector(modes, N=L).build()

c = edf.set_notation(edf.Annihilation, modes)
cd = edf.set_notation(edf.Creation, modes)
n = edf.set_notation(edf.Number, modes)

H = 0
for i in range(L):
    j = (i + 1) % L
    for sigma in (UP, DOWN):
        hop = cd(i, sigma) * c(j, sigma)
        H += -(hop + hop.dag)
    H += 4.0 * n(i, UP) * n(i, DOWN)

hamiltonian = edf.Hamiltonian(H, sector)
energies, _ = hamiltonian.eigsolve(k=8, which="SA")
psi0 = hamiltonian.eigenstate(0)
```

```{figure} ../../_static/figures/example_hubbard_chain_spectrum.svg
:width: 78%
:alt: Lowest excitation energies of a six-site Hubbard ring versus U over t

Lowest excitation energies of the half-filled six-site ring. Interaction reorganizes the low-energy manifold continuously on the finite cluster; the decreasing spin scale at large $U/t$ is consistent with the emergence of an exchange scale proportional to $t^2/U$.
```

## Local charge fluctuations and moment formation

The site-averaged double occupancy is

$$
D=\frac1L\sum_i
\langle n_{i\uparrow}n_{i\downarrow}\rangle,
$$

while the local moment is

$$
\mu^2=\frac1L\sum_i
\left\langle(n_{i\uparrow}-n_{i\downarrow})^2\right\rangle.
$$

```python
def double_occupancy(psi):
    return sum(
        np.real(psi.dag * n(i, UP) * n(i, DOWN) * psi)
        for i in range(L)
    ) / L


def local_moment(psi):
    return sum(
        np.real(
            psi.dag
            * (n(i, UP) - n(i, DOWN))
            * (n(i, UP) - n(i, DOWN))
            * psi
        )
        for i in range(L)
    ) / L
```

```{figure} ../../_static/figures/example_hubbard_chain_double_occupancy.svg
:width: 76%
:alt: Double occupancy of the six-site Hubbard ring versus U over t

Site-averaged double occupancy. Increasing repulsion suppresses local charge fluctuations and transfers weight toward singly occupied configurations.
```

```{figure} ../../_static/figures/example_hubbard_chain_local_moment.svg
:width: 76%
:alt: Local moment of the six-site Hubbard ring versus U over t

Site-averaged local moment. The growth of $\mu^2$ accompanies the loss of doublon weight and marks the formation of well-defined local spins on the finite ring.
```

## Real-space correlations

Define

$$
S_i^z=\frac12(n_{i\uparrow}-n_{i\downarrow}),
\qquad
\delta n_i=n_i-1,
$$

and translationally averaged equal-time correlations

$$
C_s(r)=\frac1L\sum_i\langle S_i^zS_{i+r}^z\rangle,
\qquad
C_c(r)=\frac1L\sum_i\langle\delta n_i\delta n_{i+r}\rangle.
$$

With the local spin and charge operators defined explicitly, the correlation function is

```python
spin_z = [edf.SpinZ((i, UP), (i, DOWN), modes) for i in range(L)]
delta_n = [n(i, UP) + n(i, DOWN) - 1 for i in range(L)]


def correlation(psi, operators, r):
    return sum(
        np.real(psi.dag * operators[i] * operators[(i + r) % L] * psi)
        for i in range(L)
    ) / L
```

```{figure} ../../_static/figures/example_hubbard_chain_spin_correlations.svg
:width: 76%
:alt: Real-space spin correlations of the six-site Hubbard ring

Spin correlations on the half-filled ring. Repulsion strengthens the alternating sign pattern expected from antiferromagnetic correlations at half filling.
```

```{figure} ../../_static/figures/example_hubbard_chain_charge_correlations.svg
:width: 76%
:alt: Real-space connected charge correlations of the six-site Hubbard ring

Connected charge correlations on the same ring. Their amplitude decreases as double occupancy is suppressed and charge motion becomes increasingly costly.
```

## Momentum-space structure factors

For local operators $O_j$, define

$$
O_q=\sum_j e^{-iqj}O_j,
\qquad
S_O(q)=\frac1L\langle O_q^\dagger O_q\rangle.
$$

In code,

```python
def structure_factor(psi, operators, q):
    Oq = sum(
        (np.exp(-1j * q * j) * op for j, op in enumerate(operators)),
        start=0,
    )
    return np.real(psi.dag * Oq.dag * Oq * psi) / L
```

```{figure} ../../_static/figures/example_hubbard_chain_spin_structure.svg
:width: 76%
:alt: Spin structure factor of the six-site Hubbard ring

Spin structure factor. The $q=\pi$ response grows with repulsion, reflecting the increasingly staggered spin correlations of the half-filled ring.
```

```{figure} ../../_static/figures/example_hubbard_chain_charge_structure.svg
:width: 76%
:alt: Charge structure factor of the six-site Hubbard ring

Charge structure factor. Repulsion suppresses the staggered charge response relative to the noninteracting ring.
```

For onsite singlet pairs,

$$
\Delta_i=c_{i\downarrow}c_{i\uparrow},
$$

and

$$
P(q)=\frac1L\langle\Delta_q^\dagger\Delta_q\rangle.
$$

```{figure} ../../_static/figures/example_hubbard_chain_pair_structure_q.svg
:width: 76%
:alt: Onsite pair structure factor of the six-site repulsive Hubbard ring

Onsite pair structure factor. Repulsive $U$ suppresses onsite pairing correlations throughout momentum space.
```

The real-space and momentum-space observables give a consistent finite-cluster picture. Increasing $U/t$ suppresses doublons and connected charge fluctuations while strengthening the staggered spin response. The onsite pair channel is suppressed at the same time. These trends diagnose the redistribution of low-energy weight on the finite ring; they do not by themselves establish a thermodynamic order parameter.

## Finite-size sequence

Finite-size dependence can be examined without changing the model. For even open chains at half filling, the ground state lies in the $S_z=0$ sector, so fixing

$$
N_\uparrow=N_\downarrow=L/2
$$

reduces the basis before diagonalization. A compact size sweep is

```python
def open_chain_energy_density(L, U=4.0, t=1.0):
    site = edf.DoF(L, name="site")
    spin = edf.DoF(2, name="spin", labels=("up", "down"))
    modes = edf.FermionModes(site, spin)
    sector = (
        edf.NParticleSector(modes, N=L)
        .project_particles("spin", up=L // 2, down=L // 2)
        .build()
    )

    c = edf.set_notation(edf.Annihilation, modes)
    cd = edf.set_notation(edf.Creation, modes)
    n = edf.set_notation(edf.Number, modes)

    H = 0
    for i in range(L - 1):
        for sigma in (UP, DOWN):
            hop = cd(i, sigma) * c(i + 1, sigma)
            H += -t * (hop + hop.dag)
    for i in range(L):
        H += U * n(i, UP) * n(i, DOWN)

    ham = edf.Hamiltonian(H, sector)
    E, _ = ham.eigsolve(k=1, which="SA", tol=1e-10)
    return E[0] / (L * t)


sizes = (4, 6, 8, 10)
energy_density = [open_chain_energy_density(L) for L in sizes]
```

At $U/t=4$, the Lieb-Wu thermodynamic result can be written

$$
\frac{E_0}{Lt}\bigg|_{L\to\infty}
=-4\int_0^\infty
\frac{J_0(\omega)J_1(\omega)}
{\omega\left(1+e^{2\omega}\right)}\,d\omega
\simeq -0.573729.
$$

```{figure} ../../_static/figures/example_hubbard_chain_finite_size_energy.svg
:width: 76%
:alt: Ground-state energy density of finite half-filled open Hubbard chains versus inverse length

Ground-state energy density at $U/t=4$ for even open chains. The finite-system values move toward the Lieb-Wu thermodynamic result as $1/L$ decreases.
```

The energy density is already smooth over this short sequence, but that behavior cannot be transferred automatically to gaps or correlation functions. Different observables can carry different finite-size corrections, and periodic clusters can show additional shell effects.

[^liebwu]: E. H. Lieb and F. Y. Wu, "Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension," Phys. Rev. Lett. 20, 1445 (1968), [doi:10.1103/PhysRevLett.20.1445](https://doi.org/10.1103/PhysRevLett.20.1445).