hamop

One tight-binding Hamiltonian, every observable, strictly consistent. Build a Hamiltonian once, as real-space blocks in an orthogonal or nonorthogonal basis, and compute its band structure, density of states, Kubo-Greenwood optical conductivity and Landauer (NEGF) transmission from the same matrices.

The point of the package is the consistency, not any single solver. When the optics of a model and its spectrum are computed by different codes with different conventions, they drift: a different gauge for the velocity operator, a different treatment of the overlap matrix, a different broadening, and suddenly the absorption edge no longer sits at the band gap. Here every observable diagonalizes the same Bloch matrices through the same canonically orthogonalized solver, and the Kubo velocity operator is built from the exact k-derivative of the same assembly, so spectral, optical and transport statements about one model cannot disagree with each other.

What it does

• TightBindingModel: sites with any number of orbitals, directed hopping blocks with automatic Hermitian completion, optional overlap blocks (LCAO-style nonorthogonal bases), periodic in any dimension or finite. Assembles H(k), S(k) and their exact k-derivatives in the atomic gauge.
• Spectrum (bands, dos, fermi_level, band_edges, k_path): band structures along arbitrary k-lists or interpolated high-symmetry paths, Gaussian-broadened densities of states, chemical potential at a given filling by bisection, band edges and gap about a chemical potential.
• Optics (sigma_optical, sigma_tensor, drude_weight): Kubo-Greenwood real sheet conductivity in units of e²/(4ℏ) with Gaussian or Lorentzian broadening, the full complex interband conductivity tensor σ_ab(ω) — including the finite-frequency Hall component σ_xy(ω), whose ω → 0 limit reproduces the TKNN quantization σ_xy = C e²/h against the package's own Chern number, sign included — plus the intraband (Drude) weight. All built on the nonorthogonal velocity correction v = dH/dk − (eₙ+eₘ)/2 dS/dk that makes them exactly invariant under a shift of the energy zero. Intra-atomic dipole blocks (model.set_dipole) add the on-site velocity contribution i(Eₙ−Eₘ)X_nm, restoring transitions the site-diagonal position approximation leaves dark — validated against a hand-derived atomic s→p line, in orthogonal and nonorthogonal bases (the eigenstate identity ⟨n|[H,x]|m⟩ = (Eₙ−Eₘ)⟨n|x|m⟩ makes the same expression exact with overlap too).
• Topology (berry_phase, berry_curvature, chern_number, chern_marker): Wilson-loop Berry phases and the gauge-invariant lattice field strength of Fukui, Hatsugai and Suzuki (J. Phys. Soc. Jpn. 74, 1674 (2005)), whose Brillouin-zone sum is an exact integer — the Chern number. Nonorthogonal bases are handled in two conventions: the smooth Löwdin frame d = S(k)^½ c (a bundle isomorphism under which the Chern number is invariant), and the atomic frame, whose links carry the midpoint overlap metric implied by the same site-diagonal position operator the velocity uses. Berry-phase values are convention-dependent; the integers must agree between frames and are asserted to. Large cells with few occupied bands can use solver="sparse". For finite systems the Bianco-Resta real-space Chern marker (Phys. Rev. B 84, 241106(R) (2011)) gives the local topological density, its bulk average reproducing the periodic Chern number and its whole-system total vanishing identically.
• Magnetic fields (with_peierls, magnetic_supercell): uniform out-of-plane fields on finite models by Peierls substitution (Peierls, Z. Phys. 80, 763 (1933)) — the midpoint line integral is exact for linear gauges, so gauge invariance, ring flux spectra, plaquette fluxes and flux-quantum periodicity all hold to machine precision — and, on periodic 2D models, Hofstadter magnetic supercells at rational flux p/q with a self-validating gauge check, anchored on exact zero-flux band folding, the π-flux square-lattice closed form, and the TKNN consistency of the lowest Hofstadter band's Chern number with σ_xy on the magnetic cell.
• Spin (with_spin, PAULI, kane_mele): spin doubling as a stated convention (spin innermost, blocks tensored with Pauli matrices), so Zeeman and intrinsic spin-orbit terms are ordinary hopping blocks; the Kane-Mele model ships as the canonical spin-orbit anchor.
• Transport (sancho_rubio, transmission, transmission_direct, transmission_sparse, principal_layers, buttiker_transmission, scba_transmission): two-probe Landauer transmission with Sancho-Rubio lead surface Green functions and a recursive Green function sweep, nonorthogonal bases included, plus dense and sparse-LU reference implementations of the same quantity — and automatic partitioning of a finite model into principal layers, which verifies that no coupling skips a layer instead of silently truncating it. User-supplied retarded interaction self-energies Σ(E) can be attached per layer; a current-conserving Büttiker dephasing probe (Phys. Rev. B 33, 3020 (1986)) is built in; and scba_transmission iterates the elastic self-consistent Born self-energy for uncorrelated on-site disorder, Σᵢ = W² diag(Gᵢᵢ), to a verified fixed point — cross-checked against the independent bulk scalar SCBA equation of the chain — and multiprobe_transmission puts current-conserving dephasing probes on many layers (D'Amato and Pastawski, Phys. Rev. B 41, 7411 (1990)), solved by exact linear-response current conservation, which reproduces Ohmic (linear-in-length) resistance scaling.
• Sparse / large systems (bloch_sparse, bloch_derivative_sparse, lowest_bands, kpm_dos, kpm_sigma): CSR assembly of the identical Bloch matrices and their k-derivatives, Lanczos diagonalization of just the low-energy window (generalized eigenproblem included), the kernel polynomial method for the density of states — nonorthogonal bases included, via a sparse LU of S — and the KPM Kubo-Greenwood optical conductivity (dipole term included) from the double Chebyshev expansion of the velocity-velocity spectral density (Weisse et al., Rev. Mod. Phys. 78, 275 (2006)), deterministic or stochastic trace.
• k-mesh reduction (monkhorst_pack(..., time_reversal=True), symmetry_fold, find_point_group): the time-reversal fold pairs k with −k for k-even observables (offered only when every real-space block is real); symmetry_fold folds by a point group, verifying before folding both that each operation maps the reciprocal lattice to itself and that it leaves the spectrum invariant at random test k-points; and find_point_group detects that group automatically by exact lattice-automorphism enumeration filtered through the same spectral check — nothing asserted about a Hamiltonian that is not verified on it. Valid for spectral observables, stated plainly.
• Band interpolation (fourier_interpolation): the exact- arithmetic Fourier backbone of Wannier interpolation — sample H(k), S(k) on a grid, transform to H(R) on the torus, evaluate anywhere; machine-precision exact when the hopping range fits the sampling window, with a residual check that detects undersampling.
• gen_eigh: generalized eigensolver with canonical orthogonalization (Szabo and Ostlund, Modern Quantum Chemistry, sec. 3.4.5), so mildly overcomplete overlaps cannot blow up the spectrum — the standard remedy used inside electronic-structure codes.

Dependencies: NumPy and SciPy. Nothing else.

Validation against closed forms

Every physical claim in the package is pinned by a test against an exact result, not a stored number:

• the single-orbital chain reproduces E(k) = e₀ + 2t cos ka to machine precision, and its nonorthogonal variant reproduces E(k) = 2t cos ka / (1 + 2s cos ka);
• the chain density of states matches 1/(π√(4t² − E²)) and integrates to the orbital count;
• graphene's nearest-neighbour model gives Dirac-point closure at K exactly, ±3|t| at Γ exactly, and the universal optical sheet conductivity e²/(4ℏ) on the interband plateau (Kuzmenko et al., Phys. Rev. Lett. 100, 117401 (2008)) — which is also the absolute anchor for the package's conductivity unit;
• the two-site molecule absorbs at exactly 2|t| with the hand-derived velocity matrix element |M| = |a t|;
• σ(ω) is invariant to 10⁻¹⁰ under H → H + cS with μ → μ + c, which pins the nonorthogonal velocity term;
• the chain's lead surface Green function matches its closed form (E − i√(4t² − E²))/(2t²); a pristine chain transmits exactly one channel inside the band and nothing outside; two decoupled chains transmit two; an on-site impurity ε reproduces T = (4t² − E²)/((4t² − E²) + ε²);
• the recursive Green function sweep agrees with dense direct inversion to machine precision, disorder and overlap included;
• the Haldane model returns its known phase diagram (Haldane, Phys. Rev. Lett. 61, 2015 (1988)) with the Chern number an exact integer to 10⁻¹²: ±1 inside the topological phase, 0 outside, sign reversal with the flux direction, and zero total over all bands;
• the SSH chain's Zak phase is quantized to 0 or π and the two dimerizations differ by exactly π — the convention-free statement;
• the Drude weight of the half-filled chain reproduces its closed form 8·spin·|t|·a and is exactly invariant under a shift of the energy zero in a nonorthogonal basis;
• the automatic principal-layer partition reproduces hand-built blocks exactly, reproduces the single-impurity closed form end to end, and refuses a layer width smaller than the interaction range;
• σ_xy(0) of the gapped Haldane model equals its Chern number times e²/h (TKNN; Phys. Rev. Lett. 49, 405 (1982)) to 10⁻⁶, sign included, computed by two independent routes through the package (Kubo tensor vs. lattice field strength); it vanishes in the trivial phase, and the tensor is antisymmetric to machine precision;
• the Chern number survives a nonorthogonal deformation of the basis unchanged (the Löwdin frame is a bundle isomorphism), and the overlap-SSH chain keeps its quantized Zak phases with the exact π difference;
• the Kane-Mele model equals two Haldane copies to machine precision, its spin-orbit gap at K is exactly 6√3 λ_so, its total Chern number vanishes and its spin sectors carry ±1 (Kane and Mele, Phys. Rev. Lett. 95, 226801 (2005)); spin doubling is an exact double degeneracy, and a Zeeman term splits it by exactly 2B;
• a constant self-energy on one layer reproduces the impurity closed form; the recursive sweep with complex Σ(E) agrees with direct inversion to machine precision; the Büttiker probe at γ = 0 is the coherent result exactly, matches the hand-written scalar closed form on a single-site device, and suppresses the double-barrier resonance;
• the time-reversal-folded k-mesh reproduces full-grid DOS, σ(ω) and Drude weight to 10⁻¹² with roughly half the points, and refuses complex-block models;
• the sparse assembly equals the dense assembly element for element; the Lanczos window reproduces the open chain's closed form 2t cos(πj/(N+1)) (nonorthogonal variant included); the KPM density of states matches the chain's closed form at the band center and integrates to the orbital count — in the nonorthogonal chain too, where the band-center DOS is again 1/(2π|t|) and the band edges sit at 2t/(1±2s), outside of which the KPM DOS vanishes identically;
• the atomic-frame links return the same exact Chern integers as the Löwdin frame (orthogonal and overlap Haldane, both phases), the atomic-frame Zak phases of the orthogonal SSH chain are ∓π/2 with the exact π difference, and inversion antisymmetry of the Zak pair survives the overlap; the sparse Berry solver returns the same integers as the dense one (C = 2 on stacked Haldane copies);
• a flux-threaded ring reproduces 2t cos((2πj + Θ)/N) to machine precision, the Landau and symmetric gauges give identical spectra to 10⁻¹², the plaquette-flux product is exactly e^{2πiφ}, the spectrum is exactly periodic in the flux quantum, and the lowest Landau level of the square lattice sits at −4|t| + ħω_c/2 with ħω_c = 4π|t|φ to 3%, macroscopically degenerate;
• the dark on-site s→p transition acquires exactly the hand-derived peak spin·4π(Δd)²/(η√(2π)Δ) once the dipole block is set, and a dipole that commutes with H changes nothing identically;
• SCBA: W² = 0 is the coherent result exactly; the converged Σ satisfies its own equation below 10⁻¹⁰ with Im Σ ≤ 0; the central layer of a long chain reproduces the independent bulk scalar SCBA fixed point to 10⁻³;
• the C6-folded graphene grid (≈ ×6 fewer points) reproduces full-grid DOS and chemical potentials to 10⁻¹²; the same fold works on the time-reversal-broken Haldane model (C6 is still a spectral symmetry, verified not assumed); a 90° rotation on the hexagonal lattice and a C6 request on a bond-stretched model are both refused;
• the KPM conductivity's integrated molecular line weight matches the kernel-independent closed form spin·4π(at)²/(2|t|) to 3%, and the KPM route agrees with the dense eigenpair Kubo route on a dimerized chain to 2%; transmission_sparse equals dense direct inversion to machine precision, overlap and complex Σ(E) included;
• the zero-flux magnetic supercell reproduces exact band folding; the π-flux square lattice reproduces E = ±2|t|√(cos²kₓ + cos²k_y) to machine precision; the lowest 1/3-flux Hofstadter band's Chern number matches σ_xy on the magnetic cell (TKNN, sign included), and an incompatible flux gauge is refused with the exact equivalent-flux remedy;
• find_point_group returns group orders 12 (hexagonal), 8 (square) and 2 (chain), a strict subset on a bond-stretched crystal, and its detected group folds the DOS exactly;
• the multi-probe network conserves current to machine precision, reduces to the single Büttiker probe exactly, and gives Ohmic linear-in-length resistance (R² > 0.9999);
• the Bianco-Resta Chern marker's whole-system total vanishes to 10⁻⁸, its bulk average equals the periodic Chern number (sign included) and vanishes in the trivial phase; the impossibility of a finite-system DC Hall conductivity, Im Tr[PxQy] = 0, is itself a test;
• intra-atomic dipoles work identically in a nonorthogonal basis (two uncoupled atoms double exactly; energy-zero gauge invariance holds with overlap and dipole to 10⁻¹⁰), and the KPM conductivity picks up the same dipole line through the exact operator i(HX − XH);
• Fourier interpolation reproduces the direct bands to 10⁻¹² for the chain, nonorthogonal chain, graphene, SSH and Haldane models, and flags an undersampled grid through its residual.

Run them yourself: pip install -e .[test] then pytest — 119 tests across Python 3.9 through 3.13.

Install and use

pip install hamop

import numpy as np
from hamop import graphene, bands, dos, sigma_optical

g = graphene(t=-2.7, a=2.46)          # eV, Angstrom
omega = np.linspace(0.5, 2.0, 60)
sigma = sigma_optical(g, omega, mu=0.0, mesh=120, eta=0.12)
# sigma is ~1.0 on the plateau: the universal e^2/(4 hbar)

Building your own model:

from hamop import TightBindingModel, band_edges

m = TightBindingModel(positions=[[0.0], [0.7]], norb=1, cell=[[2.0]])
m.add_hop(0, 1, (0,), [[-1.0]])       # intra-cell bond
m.add_hop(1, 0, (1,), [[-0.6]])       # inter-cell bond
print(band_edges(m, mu=0.0, mesh=2001))   # the SSH gap, 2|t1 - t2|

Conventions, stated once: energies in eV, positions in Angstrom, k in 1/Angstrom, Cartesian. Each directed hopping block is added once and its Hermitian partner is implied. Optical conductivity is the real sheet conductivity in units of e²/(4ℏ) with spin degeneracy as an explicit factor (default 2). The velocity operator uses the standard atomistic position gauge (position operator diagonal at the sites) by default — the common approximation in tight-binding optics — and the intra-atomic dipole contribution is added when on-site position blocks are supplied through model.set_dipole.

Relation to existing tools

Excellent tools cover parts of this space: PythTB and pybinding build tight-binding models and their spectra, and Kwant is the standard for quantum transport. hamop does not replace any of them, and for their core use cases they are more capable. Its niche is the combination they leave open: nonorthogonal (LCAO-style) overlap matrices as first-class citizens across all observables, optics and transport computed from the same Bloch assembly as the spectrum so the three can never disagree, and a deliberately small NumPy/SciPy-only core validated line by line against closed forms -- the shape of engine an LCAO electronic-structure pipeline exports its Hamiltonians into.

Status

v0.5.0 (alpha). Implemented and tested (119 closed-form-anchored tests, Python 3.9–3.13): the model container with exact k-derivatives and intra-atomic dipole blocks, canonical-orthogonalization eigensolver, band structures and k-paths, densities of states, filling-resolved chemical potentials, band edges; Kubo-Greenwood optical conductivity (Gaussian or Lorentzian broadening, dipole term included in orthogonal and nonorthogonal bases), the complex interband conductivity tensor σ_ab(ω) including the finite-frequency Hall component, and the intraband Drude weight; Wilson-loop Berry phases, lattice Berry curvature and Chern numbers in orthogonal and nonorthogonal bases, two frame conventions (Löwdin and atomic), dense or sparse solver, plus the real-space Chern marker for finite systems; spin doubling, Pauli-block spin-orbit terms and the Kane-Mele builder; uniform magnetic fields on finite models by Peierls substitution and Hofstadter magnetic supercells for periodic ones; Sancho-Rubio surface Green functions, recursive, direct-inversion and sparse-LU Landauer transmission, verified automatic principal-layer partitioning, per-layer interaction self-energies, the Büttiker probe, the multi-probe dephasing network and the elastic SCBA disorder self-energy; time-reversal folding, verified point-group folding and automatic point-group detection; sparse Bloch and velocity assembly, Lanczos low-energy bands, KPM densities of states (nonorthogonal included) and KPM optical conductivity; and exact Fourier band interpolation.

Deliberate scope, stated plainly — designed-out, not overlooked: inelastic (Keldysh) electron-phonon SCBA (the SCBA here is elastic, disorder-type, and the disorder-averaged conductance carries no vertex corrections); a finite-system KPM Hall conductivity (impossible in the site-diagonal position formulation — Im Tr[PxQy] = 0 for any bounded system, which is itself a test — so the real-space Chern marker is the honest finite-system observable); maximally localized Wannier functions (fourier_interpolation is the exact Fourier step those methods build on, not the localization procedure); and electron-electron interactions beyond a supplied static self-energy (no mean-field or GW self-consistency). Everything the package's original design roadmap named is now implemented; these remaining items are research-frontier machinery with genuine methodological choices, kept out precisely so a user can tell a designed boundary from a gap.

Where it comes from

Methodological basis:

"Learning the quantum Hamiltonian of defective monolayer MoS2 reveals collective vacancy brightness decoupled from defect count"; code for the paper: https://github.com/Tanvir-Mahmud-Mahim/mos2-vacancy-optics

That study computes the optics, the electronic structure and the transport of vacancy-disordered MoS2 supercells from one density-functional Hamiltonian, so that a defect configuration's optical and electronic signatures are strictly consistent — and its conclusions depend on that consistency. This package is the general-purpose engine distilled from that pipeline: the same observables for any Hamiltonian a user supplies, with the material-specific machinery (DFT extraction, machine-learned Hamiltonians, MoS2 structures) left in the paper repository.

Support and governance

The package is written and maintained by Tanvir Mahmud Mahim (Department of Electrical and Electronic Engineering, BRAC University), who reviews every change and takes the final decision on scope and releases. There is no separate governance body; design questions are discussed in the open in issues and pull requests, and the standing rule of CONTRIBUTING.md binds the maintainer exactly as it binds contributors: a change that touches physics arrives with a test, and a constant arrives with its source.

Support runs through the issue tracker at https://github.com/TaN-MM-Org/hamop/issues. Usage questions are welcome there alongside bug reports; a docstring that left a unit or a sign convention unclear is treated as a documentation bug, not as user error. The maintainer aims to respond within a week.

While the version is below 1.0 the API may still move between minor versions; such changes are called out in the release notes. The limitations named under Status are deliberate scope, recorded there precisely so that a user can tell a designed-out feature from an oversight.

License

Apache-2.0 (see LICENSE). Citation metadata is in CITATION.cff; every release is archived on Zenodo under the concept DOI 10.5281/zenodo.22311381, which always resolves to the latest version.