Tight-Binding Model

The tight-binding model describes coherent motion among a retained set of localized orbitals. Instead of resolving the continuous wavefunction between neighboring orbital centers, it records that motion through Hamiltonian matrix elements called hopping amplitudes. Translation-invariant hopping turns localized site states into extended crystal-momentum eigenstates and, on a periodic chain, produces the characteristic cosine dispersion.

The basic model uses unconstrained canonical bosonic or fermionic modes and is quadratic: it contains onsite one-body energies and hopping, but no particle–particle interaction. It can therefore be solved by diagonalizing a one-particle Hermitian matrix. The same matrix may describe one particle, many noninteracting fermions, or many noninteracting bosons; statistics and filling determine how its modes are occupied.

Helpful background. Lattice models explains local spaces and constraints; momentum representation fixes the Fourier normalization used below.

Coherent hopping between localized orbitals

Choose orthonormal localized states

$$\lvert i,\alpha\rangle,$$

where $i$ labels a site or unit cell and $\alpha$ labels an orbital, sublattice, spin, or another internal mode. A particle initially localized near one site can evolve into neighboring orbitals because the Hamiltonian has off-diagonal matrix elements.

For one orbital per site, a useful schematic form is

$$h_{\mathrm{tb}} = \sum_i \epsilon_i \lvert i\rangle\langle i\rvert - \sum_{i\ne j} t_{ij} \lvert i\rangle\langle j\rvert.$$

The onsite energy $\epsilon_i$ is diagonal in the localized basis. The hopping $t_{ij}$ couples distinct orbitals. Hopping is coherent unitary coupling: a localized preparation develops amplitudes on other sites. The basic dimer calculation is in two-site hopping; material orbital choices and projection are developed in tight-binding methods for quantum matter.

Localized Orbitals from a Continuum Problem

Suppose a one-particle continuum Hamiltonian $\widehat h$ is projected onto orthonormal localized orbitals $w_{i\alpha}(\mathbf r)$. Its retained matrix elements are

$$h_{i\alpha,j\beta} = \int d^dr\, w_{i\alpha}^*(\mathbf r) \widehat h w_{j\beta}(\mathbf r).$$

The projected Hamiltonian is

$$h_{\mathrm{proj}} = \sum_{i,\alpha} \sum_{j,\beta} h_{i\alpha,j\beta} \lvert i,\alpha\rangle \langle j,\beta\rvert.$$

A common sign convention defines off-diagonal hopping by

$$t_{i\alpha,j\beta} = -h_{i\alpha,j\beta}, \qquad (i,\alpha)\ne(j,\beta).$$

The minus sign is conventional. A trustworthy calculation states whether $t_{ij}$ denotes the matrix element itself or its negative.

Projection and Truncation

Tight binding is not defined solely by drawing a lattice. It is an approximation or effective model only after specifying:

• which localized orbitals are retained;
• how well separated they are from omitted orbitals;
• which hopping ranges are kept;
• whether overlap between basis functions has been orthogonalized;
• whether interactions, spin–orbit terms, or external fields are omitted or added;
• which observables and energy window the model is intended to reproduce.

Exponential localization often makes distant hopping amplitudes small, but it does not make every nearest-neighbor truncation exact. Symmetry can also force a nominally short-range matrix element to vanish while a longer-range one remains important.

Nonorthogonal Basis Caveat

Atomic-orbital trial functions are not always orthogonal. If

$$S_{ab} = \langle a\vert b\rangle \ne \delta_{ab},$$

then the coefficients obey a generalized eigenvalue problem,

$$\sum_b h_{ab}u_b = E \sum_b S_{ab}u_b.$$

Replacing $S$ by the identity without an orthogonalization or a controlled small-overlap argument changes the model. For linearly independent orbitals $S$ is positive definite. Symmetric orthogonalization gives the ordinary Hermitian matrix $S^{-1/2}hS^{-1/2}$ and normalized coefficients $v=S^{1/2}u$. If $S$ is singular, redundant orbitals must first be removed. The formulas below assume an orthonormal retained basis.

Hermiticity and Bond Orientation

A directed hopping amplitude has an orientation:

$$t_{ab} \quad \text{multiplies} \quad \lvert a\rangle\langle b\rvert.$$

Hermiticity fixes the reverse process to have amplitude $t_{ab}^*$. For real hopping, both directions carry the same real coefficient. For complex hopping, the phase changes sign under reversal.

A frequent factor-of-two error comes from summing both ordered pairs $(a,b)$ and $(b,a)$ and also appending a Hermitian conjugate. Use either an ordered-pair sum or an unordered-bond sum, not both.

Graph Interpretation

For one orbital on each vertex of a graph, uniform real nearest-neighbor hopping gives

$$h_{\mathrm{tb}} = \epsilon_0 I-tA,$$

where $A$ is the graph adjacency matrix:

$$A_{ij} = \begin{cases} 1,&i\text{ and }j\text{ are connected},\\ 0,&\text{otherwise}. \end{cases}$$

The graph spectrum of $A$ therefore determines the one-particle tight-binding spectrum. Geometry enters through connectivity, loops, boundaries, coordination, and any spatial dependence assigned to the amplitudes.

Adjacency Matrix Versus Discrete Laplacian

The graph Laplacian is

$$L = D-A,$$

where $D_{ii}=z_i$ is the degree of vertex $i$. On a regular graph with degree $z$,

$$tL = tzI-tA.$$

Thus a Laplacian kinetic term and a nearest-neighbor tight-binding term differ only by the constant onsite shift $tz$ on a regular graph. On an irregular graph or at an open boundary, $D$ is not generally proportional to the identity, so replacing $-tA$ by $tL$ changes boundary onsite terms and can change the spectrum.

The continuum-discretization meaning of this distinction is developed in Real-Space Representation.

Second-Quantized Hamiltonian

Introduce a creation operator $c_a^\dagger$ for each retained orbital. The Fock-space lift of the one-particle matrix is

$$H_{\mathrm{tb}} = \sum_{a,b} h_{ab} c_a^\dagger c_b.$$

For onsite energies and unordered hopping bonds,

$$H_{\mathrm{tb}} ={} \sum_a \epsilon_a n_a - \sum_{\langle a,b\rangle} \left( t_{ab}c_a^\dagger c_b + t_{ab}^*c_b^\dagger c_a \right),$$

where

$$n_a=c_a^\dagger c_a.$$

The operators are canonical bosonic or fermionic modes. Every term preserves total number. Hard-core or other constrained operators require a separate treatment: a bilinear in those operators need not be a free-particle Hamiltonian. Quadratic pairing terms also need a Bogoliubov transformation rather than the number-conserving diagonalization used here.

Exact Diagonalization on an Arbitrary Graph

Solve the one-particle eigenproblem

$$\sum_b h_{ab}u_{b\nu} = \varepsilon_\nu u_{a\nu}.$$

Choose the eigenvectors orthonormally:

$$\sum_a u_{a\nu}^*u_{a\mu} = \delta_{\nu\mu}.$$

Define normal-mode operators

$$d_\nu = \sum_a u_{a\nu}^*c_a, \qquad c_a = \sum_\nu u_{a\nu}d_\nu.$$

The transformation preserves the canonical commutation or anticommutation relations. Substitution gives

$$H_{\mathrm{tb}} = \sum_\nu \varepsilon_\nu d_\nu^\dagger d_\nu.$$

This is the complete solution of the number-conserving quadratic model on a finite graph.

Many-Body Spectrum

In the normal-mode occupation basis,

$$\lvert\{n_\nu\}\rangle,$$

the energy is

$$E_{\{n_\nu\}} = \sum_\nu \varepsilon_\nu n_\nu.$$

For spinless fermions,

$$n_\nu\in\{0,1\},$$

while for ordinary bosons,

$$n_\nu\in\{0,1,2,\ldots\}.$$

Adding a nearest-neighbor density interaction to a one-dimensional spinless hopping band gives the Spinless Fermion Chains $t$–$V$ model. That interaction changes the problem from a one-body band-filling exercise into an interacting Luttinger-liquid and charge-order problem.

If spin or another conserved flavor is present, each flavor has its own mode occupation. The one-body eigenvalues are shared when the hopping is flavor independent.

The Nearest-Neighbor Chain

Take $N\ge3$ sites at

$$x_j=ja, \qquad j=0,\ldots,N-1,$$

with lattice spacing $a$, uniform onsite energy $\epsilon_0$, and real nearest-neighbor hopping $t$. The standard Hamiltonian is

$$H ={} \epsilon_0 \sum_j n_j - t \sum_j \left( c_{j+1}^\dagger c_j + c_j^\dagger c_{j+1} \right).$$

The meaning of the final bond depends on the boundary condition. This small piece of model data changes the finite-size eigenstates.

Periodic Boundary Conditions

For a ring,

$$c_{j+N}=c_j.$$

There are $N$ bonds, including the bond from $N-1$ back to $0$. Translation by one site is an exact symmetry.

The allowed wave numbers satisfy

$$e^{ikNa}=1,$$

so

$$k_m = \frac{2\pi m}{Na}.$$

Choose any $N$ inequivalent representatives modulo $2\pi/a$. A common first Brillouin zone is

$$-\frac{\pi}{a} \le k < \frac{\pi}{a}.$$

The endpoints differ by a reciprocal-lattice vector and are not distinct.

Lattice Fourier Modes

Use the unitary transform

$$c_j = \frac1{\sqrt N} \sum_k e^{ikx_j}c_k,$$

with inverse

$$c_k = \frac1{\sqrt N} \sum_j e^{-ikx_j}c_j.$$

Discrete orthogonality gives

$$\frac1N \sum_{j=0}^{N-1} e^{i(k-q)x_j} = \delta_{kq}.$$

Consequently,

$$\sum_j n_j = \sum_k n_k.$$

The transform changes basis; it does not change the number of one-particle modes.

Fourier Diagonalization

For the forward hopping sum,

$$\sum_j c_{j+1}^\dagger c_j = \sum_k e^{-ika} c_k^\dagger c_k.$$

The reverse hopping gives the complex conjugate phase:

$$\sum_j c_j^\dagger c_{j+1} = \sum_k e^{ika} c_k^\dagger c_k.$$

Therefore

$$H = \sum_k \varepsilon(k) c_k^\dagger c_k,$$

with

$$\varepsilon(k) = \epsilon_0 - t \left( e^{ika}+e^{-ika} \right).$$

Using $e^{ix}+e^{-ix}=2\cos x$,

$$\varepsilon(k) = \epsilon_0-2t\cos(ka).$$

The localized site basis and the energy basis are different whenever $t\ne0$.

A periodic nearest-neighbor chain is diagonalized by the lattice Fourier transform. The six-site ring samples the cosine curve at six allowed momenta, marked by dots. For $t>0$ the continuous band has edges $\epsilon_0\mp2t$; finite rings need not sample both edges.

Dispersion Relation

The function $\varepsilon(k)$ is periodic:

$$\varepsilon \left( k+\frac{2\pi}{a} \right) = \varepsilon(k).$$

This periodicity reflects crystal-momentum equivalence, not repeated physically distinct states. A finite ring samples the dispersion at $N$ allowed $k$ values; the continuous curve is the large-$N$ interpolation.

For real hopping, the dispersion is even:

$$\varepsilon(-k) = \varepsilon(k).$$

That equality follows from the combination of real hopping and inversion-symmetric nearest-neighbor geometry.

Band Edges and Bandwidth

For $t>0$,

$$\varepsilon_{\min} = \epsilon_0-2t \quad \text{at} \quad k=0,$$

and

$$\varepsilon_{\max} = \epsilon_0+2t \quad \text{at} \quad k=\frac{\pi}{a}.$$

The bandwidth of the continuous band is

$$W = \varepsilon_{\max} - \varepsilon_{\min} = 4t.$$

For arbitrary real $t$,

$$W=4\lvert t\rvert.$$

The onsite energy shifts the whole band but does not change its width. A finite ring only samples this band. At zero twist, an even-$N$ ring contains both $k=0$ and $k=\pi/a$, while an odd-$N$ ring misses the upper edge for $t>0$. Its actual spectral width is $2t[1+\cos(\pi/N)]$, smaller than $4t$. A nonzero twist can also move the sampled minimum.

Group Velocity

For a wave packet narrow in crystal momentum, the semiclassical group velocity is

$$v(k) = \frac1{\hbar} \frac{d\varepsilon}{dk} = \frac{2ta}{\hbar} \sin(ka).$$

Thus

$$\lvert v(k)\rvert \le \frac{2\lvert t\rvert a}{\hbar}.$$

At a band extremum, the group velocity vanishes even though the eigenstate is spatially extended. Localization and zero group velocity are different statements.

Effective Mass Near a Band Edge

For $t>0$ and $\lvert ka\rvert\ll1$,

$$\cos(ka) = 1-\frac{(ka)^2}{2} + O((ka)^4).$$

The band bottom is therefore

$$\varepsilon(k) = \epsilon_0-2t + ta^2k^2 + O(tk^4a^4).$$

Comparing with

$$\varepsilon(k) = \varepsilon_{\min} + \frac{\hbar^2k^2}{2m^*} +\cdots$$

gives

$$m^* = \frac{\hbar^2}{2ta^2}.$$

At the top of the same band, the curvature is negative and the electron effective mass defined by curvature is negative. Hole variables reorganize nearly filled-band dynamics with a positive hole mass. The compact curvature formula is collected in Effective Mass.

Density of states in the infinite chain

For one orbital per site and $t\ne0$, the density of states per site is

$$D(E)=\frac{1}{\pi\sqrt{4t^2-(E-\epsilon_0)^2}}, \qquad |E-\epsilon_0|<2|t|,$$

and zero outside the band. To derive it, replace $N^{-1}\sum_k$ by $(a/2\pi)\int_{-\pi/a}^{\pi/a}dk$ in $N^{-1}\sum_k\delta(E-\epsilon(k))$. Each interior energy has two roots, and the delta-function Jacobian is $|d\epsilon/dk|^{-1}$. Integrating $D(E)$ over the band gives one state per site. The square-root divergences at the edges are integrable. A finite ring instead has a sum of discrete delta peaks; broadening those peaks is a separate plotting or experimental-resolution choice.

Open Boundary Conditions

For an open chain, sites are labeled

$$j=1,\ldots,N,$$

and the hopping sum contains only

$$j=1,\ldots,N-1.$$

There is no bond between sites $N$ and $1$. Translation by one site is not an exact symmetry, so periodic plane waves are not exact finite-chain eigenstates.

The one-particle difference equation in the interior is

$$E\psi_j = \epsilon_0\psi_j - t \left( \psi_{j-1}+\psi_{j+1} \right).$$

Open ends are encoded by

$$\psi_0=0, \qquad \psi_{N+1}=0.$$

These are endpoint conditions for the discrete recurrence, not extra physical sites.

Standing-Wave Eigenstates

The normalized open-chain eigenvectors are

$$\psi_m(j) = \sqrt{\frac{2}{N+1}} \sin(q_mj),$$

where

$$q_m = \frac{\pi m}{N+1}, \qquad m=1,\ldots,N.$$

Their energies are

$$E_m = \epsilon_0-2t\cos q_m.$$

The variable $q_m$ is dimensionless. A wave number may be defined as $k_m=q_m/a$, but it is not an eigenvalue of an exact finite-chain translation symmetry.

The open chain has $N$ standing-wave modes, exactly matching its $N$ sites.

Twisted Boundary Conditions

A twist $\theta$ imposes

$$c_{j+N} = e^{i\theta}c_j.$$

Allowed plane waves satisfy

$$e^{ikNa} = e^{i\theta},$$

so

$$k_m(\theta) = \frac{2\pi m+\theta}{Na}.$$

The energies are

$$\varepsilon_m(\theta) = \epsilon_0 - 2t \cos \left( \frac{2\pi m+\theta}{N} \right).$$

A twist shifts the finite momentum grid through the same bulk dispersion. Because $\theta$ and $\theta+2\pi$ describe the same boundary condition,

$$\operatorname{Spec}H(\theta+2\pi) = \operatorname{Spec}H(\theta)$$

as an unordered set, although individual level labels can permute.

Boundary Phase and Flux

The twist can be placed on the boundary bond,

$$-t \left( e^{-i\theta}c_0^\dagger c_{N-1} + e^{i\theta}c_{N-1}^\dagger c_0 \right),$$

or distributed uniformly over all bonds by a site-dependent phase redefinition. The spectrum depends on the total loop phase, not on where a particular gauge places it.

For a charged particle on a ring, this total phase can represent an Aharonov–Bohm flux. The gauge principle and its limits are developed in Peierls Phase Preview.

Boundary Comparison

Boundary	Final bond	Exact one-site translation	Natural modes
open	absent	no	standing waves
periodic	real hopping	yes	discrete plane waves
twisted	phase $e^{i\theta}$	gauge-dependent representation of a twisted translation	shifted plane waves

All three require exactly $N$ one-particle eigenstates for $N$ single-orbital sites.

Sign of the Hopping

On an open nearest-neighbor chain, the transformation

$$c_j \longmapsto (-1)^j c_j$$

changes

$$t\longmapsto -t.$$

For a periodic or infinite bipartite chain it shifts the momentum label by $\pi/a$; an open chain has no exact crystal-momentum label. The unitary transformation preserves the energy spectrum, even though the apparent location of the band minimum changes. On a bipartite graph with hopping only between the two sublattices, the same sign change can be made with opposite phases on the two sublattices.

This does not mean that every hopping sign is unphysical. On an odd ring, a uniform sign flip changes the phase accumulated around the loop. With several hopping ranges or frustrated loops, no single site rephasing need remove all relative signs.

Local Basis Phases

Under a rephasing

$$c_i' = e^{i\chi_i}c_i,$$

the hopping coefficients in the primed basis are

$$t_{ij}' = e^{i(\chi_i-\chi_j)} t_{ij}.$$

Individual link phases therefore depend on the localized-orbital convention. Around a closed oriented loop $C$, the phase

$$\Phi_C = \arg \prod_{(ij)\in C} t_{ij}$$

is invariant under site rephasings. Spectra can depend on such loop phases.

Real Hopping and Time Reversal

For a spinless model written in a basis where every hopping and onsite energy is real, complex conjugation $K$ satisfies

$$KhK^{-1}=h.$$

This is the standard spinless time-reversal symmetry. A loop phase not congruent to $0$ or $\pi$ modulo $2\pi$ generally obstructs a globally real representation and can break this symmetry. A $\pi$ loop phase can still be represented by real signed hoppings.

For spinful particles, time reversal also acts on spin and squares to $-1$ for a single spin-$1/2$ particle. Spin-dependent hopping then requires a matrix treatment beyond the scalar real-hopping criterion.

Hypercubic Lattices

On a $d$-dimensional hypercubic lattice with primitive spacings $a_\mu$ and real axis-dependent hoppings $t_\mu$,

$$\varepsilon(\mathbf k) = \epsilon_0 - 2 \sum_{\mu=1}^d t_\mu \cos(k_\mu a_\mu).$$

For isotropic $t_\mu=t>0$,

$$\varepsilon_{\min} = \epsilon_0-2dt,$$

and

$$\varepsilon_{\max} = \epsilon_0+2dt.$$

The bandwidth is

$$W=4d\lvert t\rvert.$$

Coordination and bandwidth are related for this simple lattice, but the exact bandwidth of a general graph is not determined by coordination alone.

General Translation-Invariant Hopping

Let $\boldsymbol\delta$ run over directed displacement vectors and let

$$t(-\boldsymbol\delta) = t(\boldsymbol\delta)^*.$$

Then

$$H = - \sum_{j,\boldsymbol\delta} t(\boldsymbol\delta) c_{j+\boldsymbol\delta}^\dagger c_j$$

has dispersion

$$\varepsilon(\mathbf k) = - \sum_{\boldsymbol\delta} t(\boldsymbol\delta) e^{-i\mathbf k\cdot\boldsymbol\delta},$$

up to any separately displayed onsite energy. Hermiticity makes $\varepsilon(\mathbf k)$ real.

This compact formula is useful, but its bond-counting convention must be stated: the displacement set here includes both directions.

Longer-Range Hopping

For a one-dimensional inversion-symmetric chain with real hopping $t_r$ across $r$ lattice spacings,

$$\varepsilon(k) = \epsilon_0 - 2 \sum_{r\ge1} t_r\cos(rka).$$

Nearest-neighbor tight binding retains only $t_1$. A next-nearest-neighbor term adds

$$-2t_2\cos(2ka).$$

It can shift band extrema, change curvature, break the simple bipartite spectral symmetry, and alter the density of states. A fit that needs substantial $t_2$ is still tight binding, but it is not a nearest-neighbor model.

Several Orbitals per Unit Cell

Let $\alpha=1,\ldots,M$ label orbitals or sublattices within each cell. Translation symmetry block-diagonalizes the Hamiltonian as

$$H = \sum_{\mathbf k} \mathbf c_{\mathbf k}^\dagger h(\mathbf k) \mathbf c_{\mathbf k},$$

where

$$\mathbf c_{\mathbf k} = \begin{pmatrix} c_{\mathbf k1}\\ \vdots\\ c_{\mathbf kM} \end{pmatrix},$$

and $h(\mathbf k)$ is an $M\times M$ Hermitian matrix.

Fourier transformation resolves translation sectors. A second, generally $\mathbf k$-dependent unitary transformation diagonalizes $h(\mathbf k)$ inside each sector.

Two-Orbital Bloch Matrix

A generic two-orbital block is

$$h(k) = \begin{pmatrix} \epsilon_A(k) & f(k)\\ f(k)^* & \epsilon_B(k) \end{pmatrix}.$$

Its two band energies are

$$E_\pm(k) ={} \frac{ \epsilon_A(k)+\epsilon_B(k) }{2} \pm \sqrt{ \left[ \frac{ \epsilon_A(k)-\epsilon_B(k) }{2} \right]^2 + \lvert f(k)\rvert^2 }.$$

Hybridization $f(k)$ generally splits levels that would cross in its absence. Symmetry may force $f(k)$ to vanish at selected momenta, in which case crossings can remain.

The SSH Model and Graphene Dirac Model are compact examples of multi-sublattice hopping, with further discussion of topology and orbital interpretation.

Wannier and Bloch descriptions

Within a retained band subspace, a discrete Fourier transform connects a chosen Bloch frame to Wannier orbitals. The transformation is exact; localization is a separate question. An isolated set with a nonzero total Chern number cannot have an exponentially localized orthonormal Wannier basis spanning that set with ordinary lattice translations. Extra symmetry requirements can introduce further obstructions (Marzari et al., 2012).

The Wannier-function construction explains frame choice, topology and disentanglement. Starting from declared lattice orbitals avoids assuming that every isolated band is independently Wannierizable. Selecting a subspace and truncating the resulting hopping range are additional approximations.

Tight Binding Versus Nearly Free Particles

Tight binding begins from localized orbitals and treats intersite hybridization through hopping. A nearly-free-particle expansion begins from plane waves and treats a periodic potential as coupling reciprocal-space modes.

Both can describe the same periodic Hamiltonian in different regimes. Neither slogan alone guarantees quantitative accuracy. The useful basis is the one in which the retained subspace is small and omitted couplings are controlled.

Filling and Statistics

The dispersion does not determine a many-body state until statistics and filling are specified. For noninteracting fermions at zero temperature, the ground state fills the lowest available one-particle modes up to the particle number. For noninteracting bosons, repeated occupation of the lowest mode is allowed.

In a grand-canonical description,

$$K = H-\mu\widehat N = \sum_\nu \left( \varepsilon_\nu-\mu \right) d_\nu^\dagger d_\nu.$$

The chemical potential changes equilibrium occupations; it does not alter the eigenvectors. At positive temperature, the bosonic grand-canonical trace over these finitely many modes requires $\mu<\min_\nu\varepsilon_\nu$. A negative one-particle energy is harmless at fixed particle number, but without a number constraint the corresponding bosonic Hamiltonian is unbounded below. A zero lowest eigenvalue also makes the unrestricted zero-chemical-potential thermal trace diverge.

Calling a tight-binding model metallic, insulating, magnetic, or superconducting requires more than displaying $\varepsilon(k)$. One must specify filling, degeneracies, gaps to other bands, interactions, disorder, dimension, and the observable criterion.

Spin and Flavor

If hopping is independent of a flavor $\sigma$,

$$H = \sum_{\mathbf k,\sigma} \varepsilon(\mathbf k) c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma}.$$

Each one-particle energy is flavor degenerate. Spin-independent hopping alone does not select a unique polarized equilibrium phase. A polarized preparation or a partially filled degenerate shell may carry a moment, but stable magnetic order needs a specified state-selection mechanism and thermodynamic analysis.

Disorder and Inhomogeneity

An onsite potential

$$H_V = \sum_j V_j n_j$$

or position-dependent hopping $t_{ij}$ breaks translation symmetry unless its pattern is periodic. The model remains quadratic and exactly reducible to a one-particle matrix, but ordinary Fourier modes no longer diagonalize it.

Real-space eigenvectors, inverse participation ratios, local densities, and boundary sensitivity then become useful. Disorder-induced localization is a deeper subject; a localized finite-size eigenvector by itself does not establish a thermodynamic localization transition.

Sublattice Symmetry

Consider a bipartite graph with only hopping between sublattices $A$ and $B$ and no onsite term after subtracting a common $\epsilon_0$. Define

$$\Gamma = \begin{cases} +1,&A,\\ -1,&B. \end{cases}$$

As a one-particle operator,

$$\{\Gamma,h-\epsilon_0 I\} = 0.$$

If $(h-\epsilon_0 I)\lvert u\rangle=E\lvert u\rangle$, then

$$(h-\epsilon_0 I) \Gamma\lvert u\rangle = -E \Gamma\lvert u\rangle.$$

The spectrum is symmetric about $\epsilon_0$. Same-sublattice hopping or nonuniform onsite energies generally break this symmetry.

Continuum Limit Near a Band Minimum

For the nearest-neighbor chain,

$$\varepsilon(k)-\varepsilon_{\min} = 2t \left[ 1-\cos(ka) \right].$$

Using

$$1-\cos(ka) = \frac{(ka)^2}{2} + O((ka)^4),$$

one obtains

$$\varepsilon(k)-\varepsilon_{\min} = ta^2k^2 + O(tk^4a^4).$$

To approximate a continuum particle of mass $m$, choose

$$t = \frac{\hbar^2}{2ma^2},$$

and subtract the divergent energy offset associated with the band minimum as $a\to0$. Merely sending $a\to0$ at fixed $t$ does not preserve a fixed continuum mass.

Physical Lattice Versus Numerical Grid

For a physical lattice, $a$ is an atomic, molecular, optical, or synthetic spacing and need not be taken to zero. The band is a physical low-energy structure.

For a numerical grid, the lattice is a regulator. Then $a\to0$ is a refinement limit, hopping must scale with $a$, and boundary terms must reproduce the intended continuum operator. The Real-Space Representation explains that continuum–lattice dictionary.

Exact Limits and Checks

Several limits expose mistakes quickly.

Atomic limit

At $t=0$,

$$H = \epsilon_0 \sum_j n_j.$$

Every site orbital has the same one-particle energy $\epsilon_0$ in the uniform model.

Dimer limit

For two sites with one bond,

$$h = \begin{pmatrix} \epsilon_0&-t\\ -t&\epsilon_0 \end{pmatrix},$$

with energies

$$\epsilon_0-t, \qquad \epsilon_0+t.$$

This is the Tight-Binding Dimer. A two-site periodic-chain sum needs special bond-counting care because the left and right neighbor coincide.

Trace check

For any finite matrix,

$$\sum_\nu\varepsilon_\nu = \operatorname{Tr}h.$$

If all onsite energies equal $\epsilon_0$ and hopping has no diagonal part,

$$\sum_\nu\varepsilon_\nu = N\epsilon_0.$$

The hopping redistributes one-particle energies around their mean but does not change the trace.

Mode-count check

One orbital on each of $N$ sites gives exactly $N$ one-particle eigenvalues, including multiplicities. A Fourier transform, boundary twist, or basis rotation cannot change this count.

Exercises

Open-chain standing waves

Verify that

$$\psi_m(j) = \sqrt{\frac{2}{N+1}} \sin \left( \frac{\pi mj}{N+1} \right)$$

solves the open-chain eigenproblem, and find its energy.

Solution

Set

$$q_m = \frac{\pi m}{N+1}.$$

The endpoint conditions hold because

$$\sin(0)=0, \qquad \sin(q_m(N+1))=\sin(\pi m)=0.$$

Using

$$\sin(q(j-1)) + \sin(q(j+1)) = 2\cos q\sin(qj),$$

the difference equation gives

$$E_m\psi_m(j) ={} \epsilon_0\psi_m(j) - 2t\cos(q_m)\psi_m(j).$$

Thus

$$E_m = \epsilon_0-2t\cos q_m.$$

The displayed normalization follows from the discrete sine orthogonality relation.

Flipping the hopping sign

Show that $c_j\mapsto(-1)^jc_j$ flips the sign of nearest-neighbor hopping on an open chain. Why does the same argument fail on an odd periodic ring?

Solution

Each nearest-neighbor bilinear transforms as

$$c_{j+1}^\dagger c_j \longmapsto (-1)^{j+1} (-1)^j c_{j+1}^\dagger c_j = - c_{j+1}^\dagger c_j.$$

Thus every open-chain bond changes sign.

On a periodic ring, the boundary bond connects $N-1$ to $0$. Its sign factor is

$$(-1)^{N-1}(-1)^0.$$

For odd $N$, this factor is $+1$, whereas every ordinary nearest-neighbor bond has factor $-1$. One cannot flip all bonds consistently. Equivalently, an odd cycle is not bipartite, and the loop phase is invariant under site rephasings.

Next-nearest-neighbor extrema

Consider

$$\varepsilon(k) = \epsilon_0 - 2t_1\cos(ka) - 2t_2\cos(2ka).$$

Find the stationary-point condition and identify when extrema away from $ka=0,\pi$ are possible.

Solution

Let $x=ka$. Differentiation gives

$$\frac{d\varepsilon}{dx} = 2t_1\sin x + 4t_2\sin(2x).$$

Using $\sin(2x)=2\sin x\cos x$,

$$\frac{d\varepsilon}{dx} = 2\sin x \left( t_1+4t_2\cos x \right).$$

The usual stationary points satisfy $\sin x=0$, so $x=0$ or $\pi$ modulo $2\pi$. Additional stationary points satisfy

$$\cos x = - \frac{t_1}{4t_2}.$$

For $t_2\ne0$, real solutions exist when

$$\left| \frac{t_1}{4t_2} \right| \le 1.$$

Equality places the solution at an existing center or edge point. Additional stationary points away from these require the strict inequality $|t_1|<4|t_2|$. Their curvature is $-8t_2[1-(t_1/4t_2)^2]$, so they are minima for $t_2<0$ and maxima for $t_2>0$. When both hoppings vanish, the band is flat and every point is stationary.

Two-orbital bands

Diagonalize

$$h(k) = \begin{pmatrix} \Delta & f(k)\\ f(k)^*&-\Delta \end{pmatrix}.$$

When can the two bands touch?

Solution

The characteristic equation is

$$E^2 - \Delta^2 - \lvert f(k)\rvert^2 = 0.$$

Therefore

$$E_\pm(k) = \pm \sqrt{ \Delta^2+\lvert f(k)\rvert^2 }.$$

The bands touch only if the square root vanishes, requiring both

$$\Delta=0$$

and

$$f(k)=0$$

at the same momentum. A nonzero sublattice offset $\Delta$ opens a gap in this two-band model.

References

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