Spin-Resolved Tight-Binding Excitonic Model

• The spin-resolved tight-binding excitonic model defines a lattice Hamiltonian with explicit spin and orbital degrees of freedom coupled to an excitonic sector representing electron–hole interactions.
• It employs diverse formulations—including Bethe–Salpeter equations, configuration interaction, and tensor-network methods—to capture symmetry breaking, spin-triplet order, and fine excitonic structure.
• The model addresses practical challenges in accurately modeling exchange interactions and excitonic fine structure in materials like TMDCs, semiconductor nanostructures, and bulk systems.

Searching arXiv for the cited works to ground the article in the referenced literature. arXiv search query: (Antão et al., 1 Jul 2026) A spin-resolved tight-binding excitonic model is a lattice many-body framework in which the single-particle sector is described by a spinful tight-binding Hamiltonian, while the neutral excitation sector is built from interacting electron–hole configurations, excitonic order parameters, or explicit two-particle eigenstates. Across the literature, this designation covers several closely related constructions: two-band lattice models with spin-triplet excitonic order and cross-hopping-driven symmetry breaking; momentum-space Bethe–Salpeter formulations for spin- and valley-resolved excitons in monolayer transition-metal dichalcogenides; atomistic empirical tight-binding plus configuration interaction schemes for nanostructures; direct-space triangular-lattice exciton models for monolayer TMDs; six-orbital $5p$-Wannier tight-binding plus screened-Coulomb models for single-layer antimony; and tensor-network formulations in which spin, layer, valley, and electron–hole degrees of freedom are appended to a quantics tensor representation of ultra-large tight-binding problems (Nishida et al., 2018, Chaves et al., 2017, Różański et al., 2016, Gunlycke et al., 2015, Rudenko et al., 2017, Antão et al., 1 Jul 2026).

1. Definition and model classes

The common structural element is the separation between a spinful lattice Hamiltonian and an excitonic sector. In the simplest case, the lattice sector consists of conduction and valence orbitals, or $c$ and $f$ orbitals in a two-band Hubbard model, with explicit spin labels $\sigma=\uparrow,\downarrow$. The excitonic sector is then constructed either as a two-particle electron–hole Hamiltonian, as a Bethe–Salpeter eigenproblem for the interband polarization, or as a mean-field hybridization channel $\langle c^\dagger f\rangle$ or $\langle v^\dagger c\rangle$. In more elaborate realizations, the single-particle basis may be a Dirac-like valley Hamiltonian, a multi-orbital three-band model around $\Gamma$, an atomistic $sp^3d^5s^\*$ or $sp^3s^\*$ empirical tight-binding basis, or a six-orbital Wannier basis for $5p$ states in a buckled honeycomb lattice (Chaves et al., 2017, Różański et al., 2016, Rudenko et al., 2017).

A concise taxonomy of representative realizations is useful because the phrase “spin-resolved tight-binding excitonic model” is not tied to a single formalism.

Realization	Single-particle basis	Excitonic treatment
Two-band square/triangular lattice	$c$0 orbitals with nearest-neighbor intra-orbital and cross-hopping	Spin-triplet excitonic mean field and spin-current analysis
Monolayer TMDCs	Dirac-like $c$1 model and three-band $c$2-point TB	EOM-derived momentum-space BSE with exchange-renormalized optical bands
Atomistic semiconductor nanostructures	Empirical TB with explicit spin and SOC	Configuration interaction with direct and exchange Coulomb integrals
Triangular-lattice TMD model	Nearest-neighbor triangular-lattice TB	Direct-space exciton Hamiltonian with screened Coulomb and on-site exchange
Ultra-large moiré and related systems	Quantics tensor-network TB with auxiliary spin site	Interleaved two-particle MPO for $c$3

These model classes span weakly bound and strongly localized regimes. The triangular-lattice TMD model is stated to be appropriate “across the spectrum from Wannier to Frenkel excitons,” while the atomistic ETB+CI framework targets nanostructures whose fine structure depends on atomic-scale symmetry and exchange. The tensor-network formulation adds a distinct dimension: it is not a different exciton Hamiltonian, but a different representation and solver architecture for the same class of spinful and excitonic operators (Gunlycke et al., 2015, Różański et al., 2016, Antão et al., 1 Jul 2026).

2. Spinful tight-binding sector

In the generic spinful construction, the electronic Hamiltonian contains hopping, on-site energies, and spin-dependent terms such as spin–orbit coupling, Zeeman coupling, and interlayer tunneling. In the tensor-network formulation, sites $c$4 are encoded by $c$5 pseudospins, while physical spin $c$6 is carried by an auxiliary site of dimension $c$7. This yields a block-structured Hamiltonian in which position is encoded in the quantics chain and spin acts through the auxiliary index (Antão et al., 1 Jul 2026).

A representative lattice-level form is

$c$8

with $c$9. In this setting, $f$0 may include long-range terms and spatial modulation, $f$1 may be spin dependent, Rashba-like and on-site $f$2 couplings can be represented through auxiliary spin blocks, and twisted-layer couplings $f$3 may be exponentially decaying and spatially modulated (Antão et al., 1 Jul 2026).

In the two-band Hubbard realization without SOC, the noninteracting momentum-space Hamiltonian is

$f$4

with

$f$5

and

$f$6

The symmetry of $f$7 encodes the cross-hopping type: $f$8- and $f$9-type are even in $\sigma=\uparrow,\downarrow$0, whereas $\sigma=\uparrow,\downarrow$1- and $\sigma=\uparrow,\downarrow$2-type are odd in $\sigma=\uparrow,\downarrow$3. This even/odd distinction is central because it determines whether the excitonic phase breaks time-reversal or inversion symmetry once the order parameter acquires a nontrivial phase (Nishida et al., 2018).

In monolayer TMDCs, the low-energy spin–valley structure near $\sigma=\uparrow,\downarrow$4 is encoded by

$\sigma=\uparrow,\downarrow$5

with

$\sigma=\uparrow,\downarrow$6

Here the spin resolution is inseparable from valley resolution because time-reversal symmetry couples $\sigma=\uparrow,\downarrow$7 and $\sigma=\uparrow,\downarrow$8. Around $\sigma=\uparrow,\downarrow$9, the same work instead uses a three-band tight-binding model based on metal $\langle c^\dagger f\rangle$0 orbitals, with SOC neglected there because it is small at $\langle c^\dagger f\rangle$1 (Chaves et al., 2017).

Other spinful lattice sectors follow the same template but with different microscopic bases. In atomistic ETB, explicit spin-up and spin-down components are propagated in $\langle c^\dagger f\rangle$2 or $\langle c^\dagger f\rangle$3 bases and on-site SOC is included following Chadi’s formulation. In single-layer Sb, the relevant Wannier basis comprises three $\langle c^\dagger f\rangle$4-like orbitals per atom, six per cell, with a local SOC term

$\langle c^\dagger f\rangle$5

where $\langle c^\dagger f\rangle$6 mixes local $\langle c^\dagger f\rangle$7 orbitals through Pauli matrices with $\langle c^\dagger f\rangle$8 eV. Because the lattice is centrosymmetric, the bands remain spin-degenerate even though SOC strongly reshapes the valence edge (Różański et al., 2016, Rudenko et al., 2017).

3. Electron–hole sector and excitonic formulations

The excitonic sector can be introduced at three distinct levels: as an interacting electron–hole Hamiltonian, as a Bethe–Salpeter problem for interband polarization, or as an excitonic mean field.

The tensor-network two-particle construction uses two copies of the position chain, one for the electron and one for the hole, plus spin auxiliary sites. The central operator is

$\langle c^\dagger f\rangle$9

where $\langle v^\dagger c\rangle$0 and $\langle v^\dagger c\rangle$1 are conduction and valence kinetic MPOs, and $\langle v^\dagger c\rangle$2 is the electron–hole interaction MPO. The corresponding density–density interaction can be written as

$\langle v^\dagger c\rangle$3

Screened Coulomb kernels, moiré-modulated interactions, and exchange-like terms $\langle v^\dagger c\rangle$4 are described as admissible extensions (Antão et al., 1 Jul 2026).

At the level of exciton operators, the same formalism introduces

$\langle v^\dagger c\rangle$5

The label $\langle v^\dagger c\rangle$6 may encode singlet, triplet, or spin–valley texture, and $\langle v^\dagger c\rangle$7 partition the center-of-mass momentum between electron and hole sectors (Antão et al., 1 Jul 2026).

The momentum-space BSE formulation is most explicit in the TMDC literature. There, the homogeneous equation for the interband polarization yields

$\langle v^\dagger c\rangle$8

A distinctive feature is that exchange is treated predominantly as a self-energy correction to the optical bands, not primarily as an explicit short-range kernel. The exchange self-energy at $\langle v^\dagger c\rangle$9,

$\Gamma$0

renormalizes the spin-resolved optical gap and is stated to be essential for obtaining the correct position of the $\Gamma$1-exciton peak around $\Gamma$2 (Chaves et al., 2017).

In mean-field excitonic-insulator treatments, the excitonic degree of freedom appears as an interband hybridization. In the two-band Hubbard model, the spin-triplet order parameter is

$\Gamma$3

and the off-diagonal mean-field element becomes

$\Gamma$4

In the tensor-network mean-field excitonic-insulator formulation, the local electron–hole pairing field is instead

$\Gamma$5

which is explicitly stated to be solved self-consistently by iterating the density matrix and the excitonic self-energy, analogously to BdG pairing but across $\Gamma$6 sectors (Nishida et al., 2018, Antão et al., 1 Jul 2026).

In atomistic ETB+CI, the excitonic Hamiltonian is built in a configuration basis $\Gamma$7, with

$\Gamma$8

This explicit decomposition into direct and exchange terms makes the spin dependence of bright–dark and anisotropic splittings especially transparent (Różański et al., 2016).

4. Symmetry breaking, spin textures, and excitonic fine structure

Spin resolution in these models is not reducible to the presence or absence of SOC. One of the clearest counterexamples is the two-band Hubbard model without SOC, where spin textures and spin currents arise purely from spin-triplet excitonic symmetry breaking. In that setting, even-parity $\Gamma$9 cross-hopping preserves inversion but breaks time-reversal symmetry in the excitonic phase, so

$sp^3d^5s^\*$0

whereas odd-parity $sp^3d^5s^\*$1 cross-hopping breaks inversion for $sp^3d^5s^\*$2, so

$sp^3d^5s^\*$3

This yields $sp^3d^5s^\*$4-asymmetric spin-split Fermi surfaces in the absence of SOC, a point that is often misconstrued in discussions of Rashba-like textures (Nishida et al., 2018).

The same work distinguishes local from global spin transport. For $sp^3d^5s^\*$5-type cross-hopping and $sp^3d^5s^\*$6, both diagonal and off-diagonal local spin currents are finite, but the total global spin current vanishes identically: $sp^3d^5s^\*$7 The vanishing follows from the Bloch theorem under periodic boundary conditions and does not contradict the existence of finite local bond currents. For triangular $sp^3d^5s^\*$8-type cross-hopping, threefold symmetry cancels the local spin currents as well (Nishida et al., 2018).

Excitonic fine structure in semiconductors is controlled by exchange, orbital character, and lattice symmetry. In the TMDC BSE framework, spin–valley coupling splits the $sp^3d^5s^\*$9 and $sp^3s^\*$0 exciton series through the spin-resolved gaps $sp^3s^\*$1, while Berry curvature produces a fine structure such as the splitting of $sp^3s^\*$2-like $sp^3s^\*$3 states. Only $sp^3s^\*$4 and $sp^3s^\*$5 excitons contribute to the optical response in the Dirac formulation, with the $sp^3s^\*$6 contribution typically negligible for realistic parameters (Chaves et al., 2017).

The triangular-lattice TMD model sharpens a second misconception: the two-dimensional hydrogen model is inadequate for the lowest-energy exciton bands. Explicitly including exchange and lattice symmetry shows lack of subshell degeneracy and places the $sp^3s^\*$7-like states below the $sp^3s^\*$8-like state. In that model, the $sp^3s^\*$9–$5p$0 exciton split depends not only on spin–orbit interaction but also on the electrostatic environment (Gunlycke et al., 2015).

In bulk GaAs, a microscopic TB+STO plus BSE treatment resolves the short-range and long-range exchange structure of the fundamental exciton. Reported values are a binding energy $5p$1 meV, a short-range bright–dark exchange splitting $5p$2, a dark-state anisotropy splitting $5p$3, and a longitudinal–transverse splitting $5p$4. These quantities exemplify the level of fine structure that a spin-resolved tight-binding excitonic model can encode when the microscopic wavefunctions are retained (Benchamekh et al., 2013).

5. Representations, algorithms, and scaling

The computational realization of these models varies widely because the excitonic Hilbert space grows much faster than the single-particle basis. Three algorithmic strategies recur: direct-space sparse diagonalization, momentum-space integral-equation methods, and tensor-network compression.

In the direct-space triangular-lattice exciton model, the relative-coordinate Hamiltonian is sparse because only six nearest-neighbor kinetic links appear explicitly, while the screened interaction enters diagonally: $5p$5 This makes iterative sparse eigensolvers natural for the lowest bound states (Gunlycke et al., 2015).

In the ETB+CI approach for million-atom nanostructures, the numerically dominant step is the evaluation of Coulomb and exchange integrals. The reported solution is wave-function reconstruction on a uniform real-space grid, followed by FFT-based convolution of quasidensities. This replaces the $5p$6 or $5p$7 summations typical of TB-LCAO Coulomb evaluations with an overall near-linear strategy for the two-particle integrals. The method was benchmarked on self-assembled InAs/GaAs dots with $5p$8 million atoms and crystal-phase InP quantum dots embedded in nanowires with up to $5p$9 million atoms (Różański et al., 2016).

In the TMDC EOM/BSE approach, the major numerical issue is convergence of the $c$00-space integral equation rather than matrix storage. The work states that Gauss–Legendre/Laguerre quadrature is much more efficient than uniform meshes, and that the homogeneous BSE near $c$01 converges within minutes on a standard laptop. Around $c$02, the angular hierarchy is truncated at $c$03, which is described as numerically stable and fast (Chaves et al., 2017).

The tensor-network formulation generalizes these strategies to ultra-large lattices. A system of $c$04 sites is mapped to $c$05 pseudospin sites, operators are encoded as MPOs, and observables are evaluated entirely with tensor-network algebra, without explicit matrix storage or diagonalization. For compressible real-space structure, the MPO bond dimension is described as typically of order a few tens, with $c$06 and memory scaling with $c$07. The two-particle electron–hole Hamiltonian is placed on an interleaved chain $c$08 so that electron–hole interactions remain local in the tensor topology (Antão et al., 1 Jul 2026).

This representation supports a broad algorithmic stack. Spectral functions are computed by KPM in MPO/MPS form through Chebyshev recursion; momentum-resolved spectra are obtained through a tensor-network quantum Fourier transform; real-time dynamics are evolved with TDVP for pure states or RK4 on the von Neumann equation for density matrices; projectors for real-space topological invariants are built by purification or KPM; and non-Hermitian phenomena are treated by hermitization,

$c$09

The reported demonstrations include exciton LDOS and DOS for $c$10, corresponding to $c$11 electron–hole pairs, and billion-site-scale calculations for spectral functions, TN-QFT momentum maps, and self-consistent-field magnetism in compressible models (Antão et al., 1 Jul 2026).

6. Representative material and platform realizations

The literature shows that spin-resolved tight-binding excitonic models are not confined to one material family.

In monolayer TMDCs, the Dirac-like $c$12 model uses material-specific parameters $c$13 and the Keldysh interaction

$c$14

Example parameter sets are given for MoS$c$15, MoSe$c$16, WS$c$17, and WSe$c$18. For MoS$c$19 in the neutral case with vacuum screening, reported $c$20-series binding energies are $c$21 eV, $c$22 eV, $c$23 eV, and $c$24 eV. For WS$c$25 on SiO$c$26, $c$27 and $c$28 Å are stated to give deviations below $c$29 meV for the low-lying $c$30-series states. The same framework attributes the placement of $c$31-exciton resonances to the exchange correction around $c$32 (Chaves et al., 2017).

In the triangular-lattice TMD model, the lattice constant is the transition-metal sublattice spacing and the nearest-neighbor hopping is parameterized as

$c$33

Representative values reported in the paper are $c$34 eV for MoS$c$35, $c$36 eV for MoSe$c$37, $c$38 eV for WS$c$39, and $c$40 eV for WSe$c$41, together with the corresponding valence-band SOC splittings $c$42. This model is explicitly intended to interpolate between Wannier and Frenkel regimes and to expose the nonhydrogenic ordering of the low-lying exciton manifold (Gunlycke et al., 2015).

In atomistic nanostructures, the ETB+CI approach was demonstrated for InAs/GaAs and InP systems. For the InAs/GaAs lens-shaped quantum dot of diameter $c$43 nm, height $c$44 nm, wetting layer $c$45 nm, and $c$46 atoms, reported direct Coulomb integrals are $c$47 meV, $c$48 meV, and $c$49 meV for Herman–Skillman and TB-LCAO bases, while short-range exchange is shown to be strongly basis sensitive. For InP crystal-phase quantum dots, the exciton binding energy decreases from $c$50 meV to $c$51 meV as diameter increases to $c$52 nm, and the excitonic FSS vanishes because of $c$53 symmetry (Różański et al., 2016).

In bulk GaAs, the microscopic empirical-TB plus Slater-orbital wavefunction construction shows that one can fit optical momentum matrix elements by matching real-space STO matrix elements to the $c$54 matrix elements of the TB Hamiltonian. This route is then used to compute electron–hole exchange and excitonic fine structure directly from TB wavefunctions (Benchamekh et al., 2013).

In single-layer Sb, the spin-resolved excitonic model is built on a six-orbital $c$55-Wannier tight-binding Hamiltonian with a local SOC constant $c$56 eV and RPA-screened Coulomb interactions. The fully screened on-site interaction is reported as $c$57 eV for $c$58, with $c$59NN, $c$60NN, $c$61NN, and $c$62NN values $c$63, $c$64, $c$65, and $c$66 eV, respectively. The quoted ratio $c$67 is presented as evidence for strongly correlated $c$68 electrons (Rudenko et al., 2017).

In moiré and super-moiré settings, the tensor-network implementation supports spinful bilayers with Rashba SOC, Zeeman coupling, twist-modulated interlayer tunneling, and electron–hole attraction, all represented in MPO form and implemented in TensorBinding.jl. The stated motivation is the study of quantum matter at meso and macroscopic scales where explicit matrix representations become prohibitively costly (Antão et al., 1 Jul 2026).

7. Approximations, limitations, and recurrent misconceptions

The main approximations are model dependent but structurally similar. In the TMDC EOM/BSE formulation, the kernel is built within the ladder approximation, screening is static, $c$69 is treated as constant even though a density dependence is physically expected, nonlinear and density terms in the EOM are neglected, and electron–phonon interactions are omitted. The work states explicitly that exchange is treated as a static self-energy rather than as a fully explicit kernel and that a self-consistent treatment would further refine band positions and splittings (Chaves et al., 2017).

In atomistic ETB+CI, the accuracy depends on basis locality, screening assumptions, and CI truncation. Long orbital tails in unmodified Slater-type orbitals are shown to strongly overestimate short-range exchange and hence fine-structure splittings, whereas Herman–Skillman or optimized Slater orbitals avoid this artifact. The CI basis is necessarily truncated, and the integral count scales as $c$70, which is the principal reason the linear-scaling Coulomb strategy is needed (Różański et al., 2016).

In the tensor-network setting, the critical assumption is real-space compressibility. The reported bond dimensions remain modest only for compressible real-space structure; strong disorder, many competing ranges, and non-smooth kernels can increase $c$71 and cost. Non-Hermitian calculations are stated to require careful damping and moment control (Antão et al., 1 Jul 2026).

Several recurrent misconceptions are directly addressed by the literature. One is that spin textures in lattice excitonic phases necessarily require SOC; the two-band Hubbard analysis shows otherwise, because spin textures and local spin currents can emerge solely from spin-triplet excitonic symmetry breaking (Nishida et al., 2018). Another is that any finite local spin current contradicts the Bloch theorem; the same work shows that the theorem constrains only the global equilibrium current, not the existence of compensating local currents. A third is that low-lying excitons in monolayer TMDs are well captured by a two-dimensional hydrogenic model; the triangular-lattice model is explicitly presented as evidence of the inadequacy of that approximation for the lowest exciton bands (Gunlycke et al., 2015).

Taken together, these formulations define a broad methodological family rather than a single canonical Hamiltonian. Their unifying idea is that spin, orbital, valley, layer, and electron–hole structure are all retained at the lattice level, so that exchange, symmetry breaking, optical selection rules, and large-scale spatial modulation can be treated within one microscopic framework. This suggests that the decisive choice is rarely whether to use a spin-resolved tight-binding excitonic model, but rather which microscopic basis, interaction kernel, and numerical representation best match the target material and length scale (Rudenko et al., 2017, Antão et al., 1 Jul 2026).