Extended Hubbard ring: competing spin, charge, and bond correlations

The one-dimensional extended Hubbard model adds a nearest-neighbor repulsion to the ordinary Hubbard Hamiltonian,

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

At half filling the thermodynamic model contains spin-density-wave, charge-density-wave, and bond-order-wave regimes.[1][2][3] A six-site ring cannot determine the phase boundaries, but it resolves the redistribution of finite-size correlations as \(V/t\) is increased at fixed \(U/t=4\).

Local operators for the three channels

The staggered spin and charge operators are built from

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

The bond operator is

\[ B_i=\sum_\sigma \left(c_{i\sigma}^\dagger c_{i+1,\sigma}+\mathrm{H.c.}\right). \]

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)]

bond = []
for i in range(L):
    j = (i + 1) % L
    B = 0
    for sigma in (UP, DOWN):
        hop = cd(i, sigma) * c(j, sigma)
        B += hop + hop.dag
    bond.append(B)

For any local field \(O_j\),

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

For the bond channel the disconnected contribution \(|\langle B_q\rangle|^2/L\) is subtracted.

Spin response

Staggered spin structure factor at \(U/t=4\). Increasing nearest-neighbor repulsion weakens the spin-dominated response of the small ring as the system approaches the charge-dominated side.

Charge response

Staggered charge structure factor at \(U/t=4\). The strong growth near and beyond the reference line \(V=U/2\) reflects the increasing tendency toward alternating high- and low-density sites.

Bond fluctuations

Connected staggered bond structure factor. The enhanced bond fluctuations near the spin-charge competition region are the finite-cluster signature most closely associated with the bond-order-wave regime found in larger systems.

Double occupancy

The site-averaged doublon density is

\[ D=\frac1L\sum_i\langle n_{i\uparrow}n_{i\downarrow}\rangle. \]

Double occupancy at \(U/t=4\). On the charge-dominated side, doublons become favorable because an alternating doublon-rich and low-density pattern reduces the nearest-neighbor repulsion.

Taken together, the four observables separate the three competing tendencies of the finite ring. At small \(V/t\), the staggered spin response is largest and doublon density remains suppressed. Around the spin-charge competition region, the spin response falls before the charge response has fully saturated, while connected bond fluctuations are enhanced. At larger \(V/t\), the staggered charge response and double occupancy rise together. The broad crossover and the locations of the extrema are finite-size properties, not estimates of thermodynamic phase boundaries.

Momentum profiles

The discrete momenta of the six-site ring show where the spectral weight accumulates. The three couplings below sample the spin-dominated side, the competition region, and the charge-dominated side.

Momentum-resolved spin structure factor. The \(q=\pi\) peak is strongest on the low-\(V\) side and loses weight as charge ordering becomes favorable.

Momentum-resolved charge structure factor. The \(q=\pi\) component becomes dominant on the high-\(V\) side.

Connected bond structure factor. The staggered bond channel is enhanced most strongly in the intermediate coupling region of the finite ring.

The finite-size extrema should not be interpreted as thermodynamic phase boundaries. They show instead how spin, charge, and bond fluctuations exchange spectral weight across the competition region.

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[1]

M. Nakamura, “Mechanism of CDW-SDW Transition in One Dimension,” J. Phys. Soc. Jpn. 68, 3123-3126 (1999), doi:10.1143/JPSJ.68.3123.

[2]

P. Sengupta, A. W. Sandvik, and D. K. Campbell, “Bond-order-wave phase and quantum phase transitions in the one-dimensional extended Hubbard model,” Phys. Rev. B 65, 155113 (2002), doi:10.1103/PhysRevB.65.155113.

[3]

E. Jeckelmann, “Ground-State Phase Diagram of a Half-Filled One-Dimensional Extended Hubbard Model,” Phys. Rev. Lett. 89, 236401 (2002), doi:10.1103/PhysRevLett.89.236401.