Tight binding

A tight-binding model expands a particle's state in localized orbitals and represents tunnelling between sites by off-diagonal Hamiltonian matrix elements.

A nearest-neighbour tight-binding model on a bipartite lattice, with identical on-site energy ${ϵ}_0$, has paired single-particle energies ${ϵ}_0\pm x$. At half filling, $f({ϵ}_0+x)+f({ϵ}_0-x)=1$ for the Fermi-Dirac distribution with $\mu ={ϵ}_0$. Thus its fixed-density chemical potential equals ${ϵ}_0$ at all positive temperatures within the isolated-band model. An anisotropic rectangular nearest-neighbour band retains this property; additional same-sublattice hopping or unequal on-site energies can remove it.

A tight binding chain of one species of fermions with nearest-neighbour hopping and density interaction. At half filling and $V/|t|\to +\infty$, minimizing the interaction requires alternating occupied and empty sites. An even periodic chain has two such configurations at $t=0$; finite hopping admixes defects through virtual hopping. A finite ring can select superpositions rather than one ordered product state. Two uncoupled copies have four limiting ordered choices when each is half filled, without a general claim of exact finite-hopping fourfold state degeneracy.

For a chain along $x$, translated $p_z$ orbitals form a side-on bonding combination at $k=0$, whereas translated $p_x$ orbitals form an end-on bonding combination at $k=\pi /a$. With $E=ϵ+2t\cos ka$ and identically translated orbitals, these correspond to $t_{\pi }<0$ and $t_{\sigma }>0$. Changing the sign convention of the hopping Hamiltonian changes the parameter signs, not the physical band extrema.

Opposite parity of same-centre s orbitals and axial p orbitals makes their left and right nearest-neighbour hopping integrals opposite in sign. Their Bloch wave sum is consequently odd in wavevector and vanishes at the zone centre and boundary. Away from those points, orbital hybridization repels coupled levels and can produce an avoided crossing.

In an orthogonal one-orbital nearest-neighbor tight-binding model, the hopping integral is $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$ and the dispersion is the displayed cosine. Using the same orbital-phase convention at every site, equal s-orbital coefficients usually make a bonding state at the zone center, so $t<0$ and $k=0$ is a band minimum. For $p_x$ orbitals along the chain, with every positive lobe pointing right, equal site coefficients bring opposite-sign lobes together between atoms. This is antibonding; alternating coefficients are bonding, giving the usual $t_{\sigma }>0$ and a band maximum at zero. Changing conventions requires transforming the Bloch basis consistently. The physical statement concerns the curvature and orbital character of a specified Bloch state, not an isolated hopping sign divorced from its basis.

For a dispersion relation $E(k)=E_0+t\cos (ka)$, $t<0$, occupied states at zero temperature form the interval $|k|<k_F$ in the first Brillouin zone. Its occupied fraction is $f=2k_F/(2\pi /a)=k_{F}a/\pi$. A spin-degenerate band therefore holds $2f$ electrons per site. If the band is independent of the other two wavevector coordinates, its Fermi surface consists of two planes. For $t>0$ the occupied region is instead near the zone edges, bounded by $|k|=\pi (1-f)/a$.

In an orthonormal basis of localized atomic orbitals, a ring of three equivalent sites with real hopping $t$ has Hamiltonian $H=(E_0-t)I+tJ$, where $J$ is the three-by-three all-ones matrix. The normalized symmetric state $(|1⟩+|2⟩+|3⟩)/\sqrt{3}$ has energy $E_0+2t$. Its two-dimensional orthogonal complement has energy $E_0-t$. Thus the symmetric state is the unique spatial ground state for $t<0$, with an energy difference $3|t|$ to the excited eigenstates. Equivalently, the three discrete Bloch states have energies $E_0+2t\cos (2\pi l/3)$, $l=0,\pm 1$. Nonorthogonal atomic orbitals instead require a generalized eigenvalue problem.

For one localized orbital per site in a chain, take on-site matrix elements $H_{jj}=E_0$, nearest-neighbour matrix elements $H_{j,j\pm 1}=h$, overlaps $S_{jj}=1$ and $S_{j,j\pm 1}=s$, with other terms neglected and $h,s$ real. A Bloch state reduces the generalized eigenvalue problem $Hc=ESc$ toThe overlap metric must be positive for the retained model; $|s|<1/2$ is sufficient for all $k$. Dropping it yields the cosine band of the orthogonal tight-binding approximation.

In the orthogonal tight-binding approximation, localized basis states satisfy $⟨i|j⟩={\delta }_{ij}$ while the Hamiltonian retains nonzero hopping integrals. These are independent matrix elements: orthogonal states need not have vanishing Hamiltonian coupling. One can construct localized orthonormal orbitals from overlapping atomic orbitals; truncating the resulting Hamiltonian to nearest neighbours is a further approximation. An overlap matrix must instead be retained for raw nonorthogonal orbitals.

For one orthonormal localized orbital per simple-cubic site and real equal nearest-neighbour hopping integral $t$, the six neighbours give the displayed energy band. Each cosine ranges from minus one to one, so the positive bandwidth is $12|t|$, independently of the sign convention for $t$. Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$ and the band effective mass is $-{\hslash }^2/(2t{a}^2)$. It is positive there for $t<0$; for $t>0$ the zone centre is a maximum instead.

A hopping integral is an off-diagonal Hamiltonian matrix element between localized orbitals on different lattice sites. It has units of energy and controls the dispersion in a tight-binding model. Its sign and phase depend on the orbital phase convention; observable band energies do not. The sign of a real nearest-neighbour $t$ in $E(k)=E_0+2t\cos (ka)$ determines the location of the minimum in that convention.

A hopping process through an intermediate configuration of different unperturbed energy, contributing to a low-energy effective description through perturbation theory. If an intermediate state costs $\Delta >0$, second-order nondegenerate perturbation theory gives a contribution $-|⟨a|T|0⟩{|}^2/\Delta$. Its amplitude is of order $t/\Delta$ and its energy correction of order $-t^2/\Delta$; a degenerate low-energy sector requires the corresponding degenerate perturbation theory before identifying the ground state.

If a tight-binding Hamiltonian is written with coefficient $-t_{ij}$ multiplying $c_i^{†}{c}_j$, projecting the one-electron Hamiltonian gives the displayed relation. An alternative convention defines the coefficient itself as the hopping integral and has no leading minus. Orthogonality removes constant scalar potentials from off-diagonal matrix elements, but does not remove a spatially varying potential. Orbital rephasing changes both matrix elements together and cannot repair an algebraic mismatch between these two conventions in a fixed basis.

A two-state degree of freedom represented mathematically like spin one-half, without necessarily being physical spin. The sublattice pseudospin in graphene labels amplitudes on two sites per primitive cell.

For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta \tau /\hslash {)}^{r+s}/(r!s!)$ and applying the Fourier representation of a Kronecker delta gives the displayed modified Bessel function kernel in a normalized localized-site basis. Its Fourier exponent gives the tight-binding model energy $E(\theta )=E_{\mathrm{well}}-2\Delta \cos \theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.

Near a nondegenerate minimum $x_0$, a smooth potential satisfiesThe localized levels are therefore approximately those of a harmonic oscillator of frequency $\sqrt{V^{\prime \prime }(x_0)/m}$; tunnelling broadens each level into a narrow band.

For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficientsThus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi /a$.

For hopping amplitude $-\lambda$ from each lattice site to its neighbours at $\pm a_1$ and $\pm a_2$, a Bloch state has energyThe band ranges from $E_0-4|\lambda |$ to $E_0+4|\lambda |$ and therefore has width $8|\lambda |$.