Two-Orbital Hubbard–Kanamori Chain

• Two-Orbital Hubbard–Kanamori Chain is a 1D lattice model with two orbitals per site and Kanamori-type Coulomb interactions, crucial for studying Mott insulation and Hund-driven physics.
• Model variants—including degenerate, crystal-field-split, and orbital-directional formulations—highlight distinct regimes such as spin–charge separation, orbital ordering, and excitonic dynamics.
• Advanced simulations (DMRG, DMFT) reveal in-gap holon–doublon bands, fractionalized excitations, and effective superconducting pairing, underpinning practical insights into correlated transition-metal systems.

The two-orbital Hubbard–Kanamori chain is a one-dimensional multiorbital lattice model with two orbitals per site, orbital-resolved hopping, and local Coulomb interactions of Kanamori type. In the 1D literature, it appears in a degenerate form without inter-orbital hybridization, in crystal-field-split forms where one orbital is half-filled and the other empty, and in anisotropic $e_g$ or $t_{2g}$ realizations tied to specific orbital geometries. Across these variants, it is used to study Mott insulating behavior, Hund-driven local moments, holon–doublon excitations, spin–charge–orbit fractionalization, orbital ordering, and superconducting tendencies in correlated transition-metal systems (Boidi et al., 2021).

1. Hamiltonian structure and model variants

A widely used 1D realization is the degenerate two-orbital Kanamori–Hubbard chain with nearest-neighbor, orbital-diagonal hopping and no inter-orbital hybridization. In that formulation,

$$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$

with $t_1=t_2=0.5$, $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$, and

$$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$

Here $U$ is the intra-orbital Hubbard repulsion, $V$ the inter-orbital Coulomb repulsion, and $J>0$ the ferromagnetic Hund exchange, with explicit spin-flip and pair-hopping terms. The site energy is chosen as $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$, so that $t_{2g}$0 corresponds to half-filling. The rotationally invariant choices highlighted in this formulation are $t_{2g}$1 at $t_{2g}$2 and $t_{2g}$3 at finite $t_{2g}$4 (Boidi et al., 2021).

A second important 1D variant introduces a crystal-field splitting and uses the rotationally invariant Kanamori relation $t_{2g}$5. Its kinetic term is

$t_{2g}$6

with $t_{2g}$7, $t_{2g}$8, and $t_{2g}$9, so orbital $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$0 is lower in energy. The interaction contains the standard density–density, spin-exchange, and pair-hopping Kanamori terms, and the chosen crystal field makes orbital $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$1 effectively empty while orbital $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$2 is half-filled in the non-interacting ground state (Pandey et al., 2021).

A third line of work uses orbital-directional hopping. In the $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$3 chain along the $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$4 direction, only the $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$5 orbital has finite nearest-neighbor hopping,

$$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$6

while the $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$7 orbital is localized. The on-site interaction is again of Kanamori form, with $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$8, $$H=\sum_{\langle ij\rangle \alpha\sigma} t_{\alpha}\, c_{i\alpha\sigma}^{\dagger}c_{j\alpha\sigma} - (\mu -\epsilon )\sum_{i}n_{i} + \sum_{i}{H}_{i},$$9, exchange $t_1=t_2=0.5$0, and pair hopping $t_1=t_2=0.5$1, constrained by $t_1=t_2=0.5$2 and $t_1=t_2=0.5$3 (Onishi, 2011).

These model choices define a family rather than a single Hamiltonian. What remains common is the coexistence of local multiorbital Coulomb physics and one-dimensional kinematics.

2. Filling conventions, local states, and strong-coupling backgrounds

In the degenerate chain, half-filling means two electrons per site on average, and $t_1=t_2=0.5$4 realizes that condition (Boidi et al., 2021). In the crystal-field chain with $t_1=t_2=0.5$5 sites, the studied filling is $t_1=t_2=0.5$6, described as quarter-filling of the two-orbital system, with orbital $t_1=t_2=0.5$7 a half-filled Mott insulator and orbital $t_1=t_2=0.5$8 empty in the chosen parameter regime (Pandey et al., 2021). In the $t_1=t_2=0.5$9 chain, quarter-filling likewise means one electron per site on average, and the ground state effectively places that electron in the itinerant $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$0 orbital (Onishi, 2011).

At strong coupling, Hund’s exchange organizes the local Hilbert space into high-spin and low-spin sectors. A particularly clear example is the zigzag $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$1 chain relevant to CaV$$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$2O$$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$3: with two electrons per site and active $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$4 orbitals, a local spin $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$5 state is formed by two electrons in different orbitals, and the system is described as a Haldane system with active $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$6-orbital degrees of freedom (Onishi, 2013). In that setting, orbital-state transitions induced by crystal-field splittings $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$7 and $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$8 reorganize the effective spin problem. The orbital-ordered backgrounds yield, respectively, a spin $$n_i=\sum_{\alpha\sigma} c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$$9 antiferromagnetic chain, a spin $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$0 zigzag chain with ferromagnetic $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$1 and antiferromagnetic $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$2, and a spin $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$3 antiferromagnetic ladder (Onishi, 2013).

A complementary strong-coupling formulation is the Hund-projected Kanamori model. There the local high-spin manifold favored by Hund’s first rule is retained while low-spin local states are projected out. For $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$4 orbitals at half-filling, the maximal local spin is $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$5, the Hund gap is $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$6, and the undoped limit reduces to a spin-$$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$7 Heisenberg system; upon doping, carrier motion couples strongly to the spin background through spin-dependent effective hopping amplitudes (Carlström, 25 Nov 2025). This suggests a useful low-energy language for many two-orbital chains close to a Mott insulating regime.

3. Mott state, Hubbard bands, and the in-gap holon–doublon band

The best-resolved single-particle spectroscopy in a 1D two-orbital Kanamori chain shows a clear Mott-insulating state at half-filling. For the degenerate chain, the zero-temperature local density of states exhibits a lower Hubbard band and an upper Hubbard band separated by a gap at $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$8, with no spectral weight at the Fermi level (Boidi et al., 2021).

Upon hole doping, the spectral structure changes qualitatively if the inter-orbital Coulomb interaction $$\begin{aligned} H_{i} =&\; U\sum_{\alpha} n_{i\alpha\uparrow}n_{i\alpha\downarrow} + \sum_{\sigma\sigma'}\left(V-J\delta_{\sigma\sigma'}\right) n_{i1\sigma}n_{i2\sigma'} \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}c_{i2\downarrow}^{\dagger}c_{i2\uparrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}c_{i1\downarrow}^{\dagger}c_{i1\uparrow}\right) \ &- J\left(c_{i1\uparrow}^{\dagger}c_{i1\downarrow}^{\dagger}c_{i2\uparrow}c_{i2\downarrow} + c_{i2\uparrow}^{\dagger}c_{i2\downarrow}^{\dagger}c_{i1\uparrow}c_{i1\downarrow}\right). \end{aligned}$$9 is finite. A new in-gap band appears between the lower and upper Hubbard bands and is pulled down from the upper Hubbard band as $U$0 increases. Projecting the density of states onto local configurations shows that, at sufficiently large dopings, this band is dominated by inter-orbital holon–doublon excitations: one orbital is holon-like and the other doublon-like on the same site (Boidi et al., 2021). For $U$1, the corresponding atomic-limit excitation energy is

$U$2

and the in-gap band in the full 1D chain approximately follows this energy as $U$3 is varied (Boidi et al., 2021).

The interaction dependence is sharply resolved. At $U$4, the model is just two decoupled single-orbital Hubbard chains and no isolated in-gap holon–doublon band appears. At $U$5, spectral weight is transferred from the upper Hubbard band into a distinct in-gap band, and at fixed doping the band moves to lower energy as $U$6 increases. Hund’s coupling $U$7 does not primarily shift the band position; instead, it broadens the in-gap band through the spin-flip and pair-hopping terms, which enhance two-particle fluctuations (Boidi et al., 2021).

Momentum-resolved spectra confirm that this is a genuine dispersive band rather than a purely local excitation. For $U$8, holon–doublon-related weight is concentrated near the zone boundaries and remains merged with the Hubbard bands. For $U$9, a separated dispersive holon–doublon band appears at lower energies, and finite $V$0 broadens it further in both energy and momentum (Boidi et al., 2021).

Related DMFT work on the degenerate two-orbital Kanamori–Hubbard model on the Bethe lattice identifies the same local multiplet mechanism in higher dimension: hole doping and finite inter-orbital Coulomb interaction produce a holon–doublon in-gap subband, and finite Hund’s coupling splits that subband by $V$1. The lower Hubbard band also resolves into several subbands associated with different local configurations (Hallberg et al., 2020). In one dimension, the chain study shows that hole doping is required for the in-gap holon–doublon band, because the half-filled system is already a Mott insulator (Boidi et al., 2021).

4. Real-time dynamics, fractionalization, and nonequilibrium carriers

Time-dependent DMRG studies reveal that the two-orbital chain supports richer real-time fractionalization than a single-band Hubbard chain. In the crystal-field-split chain, an exciton is created by promoting an electron from the lower-energy half-filled orbital $V$2 into the empty higher-energy orbital $V$3, using a Gaussian wave packet. Tracking local charge, spin, and orbital observables shows charge–spin separation in orbital $V$4: the charge wave packet in $V$5 moves predominantly to the left, while the spin wave packet in $V$6 moves predominantly to the right. In contrast, the charge and spin wave packets in the initially empty orbital $V$7 move together, so no spin–charge separation occurs there (Pandey et al., 2021).

The same simulation exhibits spin–orbit separation. The orbital pseudospin

$V$8

defines an orbiton wave packet that moves to the left, while the spinon associated with the antiferromagnetic background in orbital $V$9 moves to the right. From peak tracking, the representative velocities at $J>0$0 and $J>0$1 are $J>0$2 for the orbiton and $J>0$3 for the spinon (Pandey et al., 2021). Increasing $J>0$4 decreases $J>0$5, weakens the exciton binding, and markedly increases the orbiton velocity, whereas the spinon velocity remains essentially unchanged over the same range (Pandey et al., 2021).

The spin-flip exciton channel shows an additional Hund-driven effect. For $J>0$6, the spin wave packet in orbital $J>0$7 splits into two distinct packets moving in opposite directions, interpreted as two fractionalized spinons; for $J>0$8, that splitting is absent (Pandey et al., 2021).

A different nonequilibrium protocol starts from a localized holon–doublon pair in the $J>0$9 chain. There, the holon motion is governed by nearest-neighbor hopping in the itinerant orbital, while the doublon can also transfer between orbitals through pair hopping. When $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$0, holon and doublon propagate with the same speed. When $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$1, the pair-hopping term mixes $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$2 and $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$3 doublon configurations into local eigenstates with energies $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$4, and the effective doublon hopping amplitude becomes $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$5. Numerically, the doublon then propagates at about half the holon velocity (Onishi, 2011). This establishes pair hopping as a direct dynamical source of orbital differentiation in charge transport.

5. Magnetic correlations, orbital order, and pairing tendencies

Near half-filling, the two-orbital Hubbard–Kanamori chain supports a well-defined pairing regime. DMRG calculations for a degenerate two-orbital chain with $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$6 define the two-hole binding energy as

$\epsilon = -\frac{U}{2} - V + \frac{J}{2}$7

Negative $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$8 signals bound holes. For two holes, $\epsilon = -\frac{U}{2} - V + \frac{J}{2}$9 becomes negative only in an intermediate-coupling window, crossing zero around $t_{2g}$00, reaching a minimum near $t_{2g}$01, and remaining negative up to $t_{2g}$02; finite-size extrapolation at $t_{2g}$03 gives $t_{2g}$04 (Patel et al., 2017).

The dominant superconducting channel is a nearest-neighbor inter-orbital spin singlet,

$t_{2g}$05

Pair-pair correlations in this channel are dominant near half-filling, or at least of similar strength as charge and spin correlations, whereas on-site pair operators and spin-triplet channels are subleading (Patel et al., 2017). The physical picture is explicitly Hund-driven: robust local moments and antiferromagnetic correlations are necessary for hole binding, and without sizable $t_{2g}$06 the binding tendency disappears (Patel et al., 2017).

The half-filled background itself is magnetic in a specifically multiorbital way. In the same chain, the local moments approach $t_{2g}$07, the spin structure factor develops a clear peak at $t_{2g}$08, and the low-energy picture is consistent with an effective spin-1 Haldane chain (Patel et al., 2017). The zigzag $t_{2g}$09 chain shows how orbital order reorganizes that magnetic background: depending on the orbital configuration selected by $t_{2g}$10 and $t_{2g}$11, the effective spin model becomes a uniform S=1 antiferromagnetic chain, a zigzag chain with ferromagnetic and antiferromagnetic exchanges, or an S=1 ladder (Onishi, 2013).

Orbital order can also be boundary-sensitive. In the zigzag chain with open boundaries, DMRG finds that edge geometry biases the orbital occupancy and generates a kink in the antiferro-orbital $t_{2g}$12 pattern at the chain center (Onishi, 2013). This shows that in one dimension, orbital order is not merely an internal quantum number but can control the effective exchange topology of the chain.

6. Orbital selectivity, symmetry limits, and broader formulations

One recurring issue is whether a two-orbital chain can undergo an orbital-selective Mott transition. Within the Composite Operator Method, a minimal two-orbital Hubbard model with density–density $t_{2g}$13 and $t_{2g}$14 and different bandwidths shows a clear signature of an orbital selective Mott transition: the critical coupling $t_{2g}$15 for the wide band is essentially unchanged as the bandwidth ratio $t_{2g}$16 is varied, whereas $t_{2g}$17 for the narrow band is highly sensitive to $t_{2g}$18 and appears roughly linear in it (0903.1544).

A later DMFT+NRG/DMRG study of the SU(4)-symmetric limit $t_{2g}$19, $t_{2g}$20, and $t_{2g}$21 reaches a different conclusion. In that setting, even with very different bandwidths, there is no orbital-selective Mott transition; instead, both bands undergo a simultaneous Mott transition. The narrow band develops a pseudo-gap-like feature with a very narrow central peak whose width depends strongly on the hopping ratio, but the density of states at $t_{2g}$22 remains finite in the metallic phase (Gusmão et al., 2 Apr 2025). This suggests that orbital selectivity in a two-orbital chain is highly sensitive to symmetry, bandwidth structure, and approximation scheme.

The same SU(4)-symmetric study also interprets the Mott transition through Green’s-function topology: in the insulating phase the self-energies of both bands diverge at $t_{2g}$23, implying zeros of the interacting Green’s functions and a change in the winding number of

$t_{2g}$24

That formulation treats the Mott transition as a topological transition in the Green’s-function sense (Gusmão et al., 2 Apr 2025).

A broader strong-coupling route is provided by the Hund-projected Kanamori model. Starting from the multiorbital Hubbard–Kanamori Hamiltonian and projecting onto the high-spin manifold, one obtains for $t_{2g}$25 at half-filling a spin-1 Heisenberg system; upon doping, the effective carriers are spinless fermions whose hopping amplitudes depend explicitly on the local spin configuration, and the model develops Hund-enhanced kinetic ferromagnetism (Carlström, 25 Nov 2025). For the one-dimensional two-orbital chain, this formulation supplies a compact low-energy description of the regime where local S=1 moments, carrier motion, and Hund’s coupling must be treated on equal footing.

Taken together, these results define the two-orbital Hubbard–Kanamori chain not as a single canonical phase diagram but as a multiorbital 1D framework with several distinct regimes. At half-filling it can realize Mott insulating and spin-1 backgrounds; with doping it supports in-gap holon–doublon bands, heavy composite carriers, and inter-orbital pairing; and under changes in bandwidth asymmetry, crystal field, or Hund’s coupling it can move between orbital-selective, excitonic, magnetic, and superconducting sectors.