Lecture 4: Microscopic Coupling and the Fate of a Crossing

Translation symmetry tells us which states may share a crystal-momentum label, but it does not determine how the internal degrees of freedom inside that sector couple. This lecture inserts microscopic dynamics into the kinematic scaffold by turning hopping, orbital content, and symmetry constraints into a Bloch-operator matrix. That distinction will return whenever the course asks whether a gap, avoided crossing, mass term, or boundary coupling is actually present—from charge localization to topological models. Symmetry organizes and forbids possibilities; a Hamiltonian supplies the amplitudes that decide among the possibilities it permits.

Lecture 3’s finite-periodic Bloch decomposition showed how translation symmetry groups a crystal-sized eigenproblem into independent momentum-labeled blocks. That is a powerful kinematic reduction, but the translation labels leave the entries inside each block undetermined.

Consider instead two schematic, course-defined tight-binding models with the same two-sublattice connectivity. In Model E, the two sites are assigned equal onsite energies. In Model I, the onsite energies are assigned different values. Translation labels and connectivity alone do not tell us how this difference changes the spectrum or the states. These are definitions inside the model, not claims about a specific material. The common connectivity organizes the states; the Hamiltonian parameters supply the unresolved dynamics.

For the crossing comparison, also assume that their common off-diagonal element ${f(k)}$ vanishes at a momentum ${k_*}$ , and that Model I has a nonzero on-site contrast. Equal on-site energies alone do not guarantee such a zero: ${f(k)=u+v\mathrm e^{-ika}}$ with unequal positive ${u,v}$ never vanishes. We will keep this extra premise visible when returning to the models.

We will construct the two-orbital Bloch matrix from localized hopping and use its entries, symmetries, and eigenvectors to decide both the fate of the crossing and the model-resolved internal character.

This comparison is a model-to-observable diagnostic, not quantitative materials modeling. The eigenvalues determine the model’s energy branches and whether its two-band spectrum contains a crossing or a gap. The eigenvectors determine the internal-state weights assigned to each branch. Any later observation model must define an observable and its coupling matrix elements before those weights can be converted into predicted signals; that additional map is outside the present Hamiltonian. Energies answer the spectral question, while eigenvectors supply the state-content information required by a separately defined observable.

The terms Bloch state and reciprocal-lattice equivalence retain their exact Lecture 3 definitions. In particular, ${k}$ and ${k+G}$ label the same translation character; this does not mean that every matrix representation built from a momentum-dependent orbital basis is an ordinary periodic function of the chosen representative.

Given localized orbitals and allowed hopping processes, which matrix element decides the fate of a crossing, and which conclusions survive a change of basis convention?

The answer is useful whenever a structural distortion, layer coupling, or symmetry-breaking field turns a visible crossing into an avoided crossing. The gap measures an allowed coupling only after the basis states and symmetry locus have been identified. The eigenvectors matter as much as the energies: their changing orbital content controls which state couples to light, strain, an impurity, or a later many-particle interaction.

The shortest path is localized hopping, Bloch matrix, coupled two-state spectrum, symmetry test, and gap diagnosis. The familiar limits, basis- covariance proof, and interface definitions then test whether that diagnosis survives a change of representation and remains usable later. They may be revisited after the main path, but they do not alter its assumptions or conclusions.

Before constructing the model, recover four operations from Lecture 3 and ordinary quantum mechanics: count the number of internal states in one fixed- ${k}$ block; Fourier transform a nearest-neighbor term; state the Hermiticity condition on a matrix element; and diagonalize ${d_z\hat\sigma_z+d_x\hat\sigma_x}$ in the limits ${d_x=0}$ and ${d_z=0}$ . If the first operation gives a crystal-sized matrix, or if the last operation predicts a crossing for nonzero ${d_x}$ , revisit the Bloch block construction before continuing.

From real-space hopping to a Bloch matrix

A Bloch matrix is not self-explanatory: its entries inherit both a chosen orbital representation and a physical downfolding. In the model below, localized orbitals and their hopping matrix elements are retained, while remote bands, interactions, and disorder are excluded. Diagonalization is exact for that matrix. The relation of fitted matrix elements to a material is phenomenological unless a controlled projection is supplied. The decisive physics is nevertheless sharp: symmetry decides whether the coupling is permitted, and an allowed matrix element converts a crossing into an avoided crossing.

Orbitals, lattice positions, and the Fourier convention

We now specialize to a finite one-dimensional chain with ${N_{\mathrm c}\geq4}$ periodic cells. Cell indices lie in ${\mathbb Z_{N_{\mathrm c}}}$ , their positions are the scalars ${R=na}$ , and a convenient representative of the one-dimensional crystal-momentum space ${\mathrm{BZ}_{1\mathrm D}=\mathbb R/(2\pi/a)\mathbb Z}$ is ${[ -\pi/a,\pi/a )}$ . Each cell contains two orthonormal localized orbitals, ${|R,A\rangle}$ and ${|R,B\rangle}$ , so the one-particle space is ${\ell^2(\mathbb Z_{N_{\mathrm c}})\otimes\mathbb C^2}$ ; the ordered internal basis is ${(|A\rangle,|B\rangle)}$ . Let ${t,g\in\mathbb R}$ have units of energy. An ${A}$ orbital hops to neighboring ${A}$ orbitals with amplitude ${t}$ , a ${B}$ orbital hops with amplitude ${-t}$ , and the two orbitals in the same cell hybridize with amplitude ${g}$ . The opposite hopping signs deliberately make one uncoupled branch rise where the other falls.

This two-orbital chain is a minimal laboratory, not a claim that all crystals have two literal atomic orbitals with opposite hoppings. The internal labels could represent atomic orbitals, sublattices, layers, or any two basis states that share a translation character. What matters is that each fixed- ${k}$ sector has internal matrix structure. The full space has dimension ${2N_{\mathrm c}}$ ; the ${N_{\mathrm c}}$ allowed crystal momenta therefore leave exactly two basis states in every Bloch block. State counting already predicts the size of the matrix we must construct.

For the finite ring, the allowed momenta form the discrete set ${K_N=\{2\pi j/(N_{\mathrm c}a):j=0,\ldots,N_{\mathrm c}-1\}}$ modulo ${2\pi/a}$ . We derive the matrix at these momenta and then regard its finite-range Fourier expression as the continuous Bloch family on the Brillouin circle. Band extrema and crossing locations below refer to that continuous family; a finite ring samples it and contains a crossing momentum only when that momentum belongs to ${K_N}$ .

Writing the real-space operator prevents the hopping diagram from becoming a cartoon detached from the algebra. With ${R+a}$ understood periodically,

$\begin{aligned} \hat H={}&\sum_R\Bigl[ t|R+a,A\rangle\langle R,A| -t|R+a,B\rangle\langle R,B|\\ &\qquad\quad +g|R,A\rangle\langle R,B|+\mathrm{h.c.}\Bigr]. \end{aligned}$

The Hermitian-conjugate terms supply the reverse neighbor hops and the reverse intracell conversion. The model contains no hidden filling assumption: it is a one-particle operator whose eigenvalues and eigenvectors can be found before deciding how many particles occupy them.

This is not yet a band structure. It is a real-space rule. Before calculating, make three predictions:

what wave-number dependence results when right- and left-neighbor hops combine?
what cell-displacement phase does an intracell hop carry?
at an uncoupled crossing, how does a nonzero ${g}$ change the two local energies, and what sets their separation?

The calculation will test these predictions.

Before reading the formulas, use Figure 4.1 to perform the representation change. Match each bond in panel (a) to a row and column in panel (b). Then decide which processes have displacement ${\Delta=0}$ and which occur in conjugate pairs at ${\Delta=\pm a}$ . The absolute cell label should disappear; only the displacement may survive in a Fourier phase.

Figure 4.1. A two-orbital chain is mapped process by process to its Bloch matrix. The same-orbital neighbor hoppings occur at displacements plus and minus one lattice constant and generate cosine diagonal entries, whereas the intracell hybridization has zero displacement and generates a phase-free off-diagonal entry.

The schematic predicts the structure before the projection: paired neighbor bonds must generate ${\mathrm e^{\mathrm i ka}+\mathrm e^{-\mathrm i ka}}$ , while the intracell bond contributes ${g}$ with no cell-displacement phase. The derivation now establishes that rule for an arbitrary hopping function, including bonds beyond the three drawn here.

The object to be constructed is a Bloch Hamiltonian acting within each translation sector. Its ${2\times2}$ array below is a representation in a declared ordered orbital basis, not the operator itself. If the columns of a unitary matrix ${U}$ express the new orbital basis in the old one, then ${H'=U^\dagger H U}$ and the coordinate column of a state changes as ${u'=U^\dagger u}$ . Thus ${Hu=Eu}$ implies ${H'u'=Eu'}$ : the energy is unchanged. Spectral projectors as operators also stay unchanged, although their coordinate matrices transform. The same symmetry selection rule holds when the symmetry and states are transformed consistently. This change of description does not switch on a physical coupling. The two-state reduction is reliable only while coupling to remote bands is small on the energy scale being resolved, a condition we will test rather than hide inside the notation.

For ${k\in K_N}$ and ${\alpha\in\{A,B\}}$ , choose the cell-phase Bloch basis. Each state superposes the same orbital over all cells with the phase pattern fixed by the chosen momentum. The two orbital choices span that momentum sector, whose internal Hamiltonian we want to calculate:

$|k,\alpha\rangle =\frac{1}{\sqrt{N_{\mathrm c}}} \sum_R \mathrm e^{\mathrm i kR}|R,\alpha\rangle, \qquad \alpha\in\{A,B\}.$

For ${k,k'\in K_N}$ , orthonormal localized orbitals give $\langle k',\alpha|k,\beta\rangle =\delta_{\alpha\beta}\frac1{N_{\mathrm c}} \sum_R e^{\mathrm i(k-k')R} =\delta_{\alpha\beta}\delta_{kk'}.$ The last equality is the finite geometric sum. It fixes the normalization before either Bloch sum is inserted into a matrix element.

Let ${\mathcal L_{N_{\mathrm c}}=a\mathbb Z_{N_{\mathrm c}}}$ be the set of cell displacements. For each pair ${\alpha,\beta\in\{A,B\}}$ , the hopping amplitude is a function ${t_{\alpha\beta}:\mathcal L_{N_{\mathrm c}}\to\mathbb C}$ . Translation invariance means that its value depends only on the displacement ${\Delta\in\mathcal L_{N_{\mathrm c}}}$ :

$t_{\alpha\beta}(\Delta) =\langle R+\Delta,\alpha|\hat H|R,\beta\rangle.$

Transforming the hopping matrix elements

Projecting ${\hat H}$ into the ordered fixed- ${k}$ basis ${\mathcal B(k)=(|k,A\rangle,|k,B\rangle)}$ gives the coordinate-matrix function ${{\bigl[\hat H(k)\bigr]}_{\mathcal B(k)}=:H(k)}$ from ${\mathrm{BZ}_{1\mathrm D}}$ to ${\operatorname{Herm}(2)}$ . Its entries are

(4.1)

$\left({\bigl[\hat H(k)\bigr]}_{\mathcal B(k)}\right)_{\alpha\beta} =H_{\alpha\beta}(k) =\langle k,\alpha|\hat H|k,\beta\rangle =\sum_{\Delta}t_{\alpha\beta}(\Delta) \mathrm e^{-\mathrm i k\Delta}.$

Let ${\mathcal B=\mathbb R^d/\Lambda^*}$ be the Brillouin torus and let ${\pi:\mathcal E\to\mathcal B}$ be the finite-rank Hermitian Bloch bundle, whose fiber ${\mathcal E_{[\mathbf k]}}$ is the retained one-particle Bloch fiber at the momentum class ${[\mathbf k]}$ . A Bloch Hamiltonian is a section $\hat H_{\mathrm B}\in \Gamma\!\left(\mathcal B,\operatorname{Herm}(\mathcal E)\right), \qquad \hat H_{\mathrm B}([\mathbf k]): \mathcal E_{[\mathbf k]}\longrightarrow\mathcal E_{[\mathbf k]},$ that assigns a self-adjoint endomorphism to every Bloch fiber. Thus the coordinate-free object is a section, not an ordinary matrix-valued function on a preidentified copy of ${\mathbb C^M}$ .

On an open set ${U\subset\mathcal B}$ , a local family of ordered orthonormal bases ${\mathcal F(\mathbf k)}$ identifies each fiber with ${\mathbb C^M}$ and produces the local Hermitian matrix function $H_{\mathcal F}:U\longrightarrow\operatorname{Herm}(M), \qquad H_{\mathcal F}(\mathbf k) :={\bigl[\hat H_{\mathrm B}([\mathbf k])\bigr]}_{\mathcal F(\mathbf k)}.$ If another such basis family satisfies ${\mathcal F'(\mathbf k)=\mathcal F(\mathbf k)U(\mathbf k)}$ , then ${H_{\mathcal F'}=U^\dagger H_{\mathcal F}U}$ on the overlap. A global periodic matrix exists only after a compatible global basis family has been chosen; a cover-defined orbital basis family may instead obey a boundary-gluing law. The operator carries a hat, while every basis-dependent coordinate matrix does not.

For the fixed- ${k}$ periodic-LCAO construction, see [SK54, pp. 1498–1501].

For the present finite ring, the ordered cell-phase states ${\mathcal F_{\mathrm{cell}}(k)=(|k,A\rangle,|k,B\rangle)}$ are bases of actual translation sectors only at the allowed discrete momenta of the ring. The stated Fourier extension defines a continuous two-orbital Bloch family, with a fixed ordered orbital coordinate basis on the half-open representative ${\mathrm{BZ}_{1\mathrm D}=[-\pi/a,\pi/a)}$ . Within this retained two-orbital model this basis family trivializes the bundle there, and the matrix derived above is precisely ${H_{\mathcal F_{\mathrm{cell}}}(k)}$ . A later change of orbital basis changes that matrix representative, not the section ${\hat H_{\mathrm B}}$ or its spectrum. This concrete example does not assert that every Bloch bundle admits a globally smooth periodic basis family.

Hermiticity and phase-convention checks

Thus ${H_{\alpha\beta}:\mathrm{BZ}_{1\mathrm D}\to\mathbb C}$ , with the two indices referring to the ordered orbital basis. The sign in the exponential follows from the phase chosen in the ket. A different but consistent Fourier convention reverses this sign. Hermiticity requires ${t_{\alpha\beta}(\Delta) =t_{\beta\alpha}^{*}(-\Delta)}$ , which in turn requires ${H_{\alpha\beta}(k)=H_{\beta\alpha}^{*}(k)}$ . This is the first strong error check on a derived Bloch matrix.

Use the rule once before the anchor specialization. Suppose a directed process takes an ${A}$ orbital in cell ${R}$ to a ${B}$ orbital in cell ${R+a}$ with amplitude ${h\in\mathbb C}$ . In the definition above this means ${t_{BA}(a)=h}$ . Supply the reverse hopping amplitude and both Bloch-matrix entries, including their phases. Test your pair at a generic momentum, not only at ${k=0}$ . Record the two entries before continuing; allow two minutes and compare them with the feedback before the following double-sum calculation.

Answer. The reverse process is $t_{AB}(-a)=h^*$ . The entries are $H_{BA}(k)=h\mathrm e^{-\mathrm i ka}$ and $H_{AB}(k)=h^*\mathrm e^{+\mathrm i ka}$ ; their conjugation holds for generic $k$ , while the phase signs follow the declared Fourier convention.

To make the displacement reduction explicit, expand one matrix element:

$\begin{aligned} \langle k,\alpha|\hat H|k,\beta\rangle &=\frac{1}{N_{\mathrm c}} \sum_{R,R'} \mathrm e^{-\mathrm i kR'} \mathrm e^{\mathrm i kR} \langle R',\alpha|\hat H|R,\beta\rangle\\ &=\frac{1}{N_{\mathrm c}} \sum_{R,\Delta} \mathrm e^{-\mathrm i k(R+\Delta)} \mathrm e^{\mathrm i kR} t_{\alpha\beta}(\Delta)\\ &=\sum_{\Delta} t_{\alpha\beta}(\Delta)\mathrm e^{-\mathrm i k\Delta}. \end{aligned}$

In the second line ${R'=R+\Delta}$ ; the sum over the absolute cell ${R}$ then supplies ${N_{\mathrm c}}$ and cancels the normalization. This is the precise place where translation invariance converts a large real-space matrix into a small matrix-valued function of ${k}$ .

Construct the Bloch matrix for the stated chain in the cell-phase convention: ${A}$ -to- ${A}$ neighbors have real amplitude ${t}$ , ${B}$ -to- ${B}$ neighbors have ${-t}$ , and the within-cell ${A}$ - ${B}$ amplitude is real ${g}$ ; onsite energies vanish. The cell spacing is ${a}$ , and the ordered orbital basis is ${(|A\rangle,|B\rangle)}$ .

Use the displacement sum entry by entry, ${H_{\alpha\beta}(k)=\sum_{\Delta}t_{\alpha\beta}(\Delta) \mathrm e^{-\mathrm i k\Delta}}$ . The cell sum in the preceding derivation cancels the ${1/N_{\mathrm c}}$ normalization, so no extra system-size factor belongs in these entries. For the ${A}$ diagonal entry, the only nonzero neighbor amplitudes are ${t_{AA}(a)=t_{AA}(-a)=t}$ . Hence

$H_{AA}(k) =t\mathrm e^{-\mathrm i ka}+t\mathrm e^{\mathrm i ka} =2t\cos ka.$

For the ${B}$ entry the amplitudes are ${-t}$ , giving ${-2t\cos ka}$ . The within-cell terms ${t_{AB}(0)=t_{BA}(0)=g}$ have ${\Delta=0}$ , so their phases are unity. No appeal to a band formula is required: every entry can be traced to one declared real-space process.

Place the two same-orbital entries on the diagonal and the intracell pair off the diagonal, in the declared orbital order:

$H(k)= \begin{pmatrix} 2t\cos ka & g\\ g & -2t\cos ka \end{pmatrix}, \qquad \hat H_{\mathrm B}(k) =2t\cos ka\,\hat\sigma_z+g\,\hat\sigma_x.$

The Pauli operators act on the ordered orbital basis ${(|A\rangle,|B\rangle)}$ , not on spin. We use hats on them because they represent operators in that two-dimensional space.

Hermiticity checks both reverse-hopping phases and the real intracell pair: ${H_{BA}=H_{AB}^*}$ and both diagonal entries are real. Periodicity gives ${H(k+2\pi/a)=H(k)}$ , consistent with the cell-phase basis. All entries have energy units, and the limiting matrices at ${t=0}$ and ${g=0}$ respectively describe isolated two-orbital cells and unmixed dispersive orbitals. The next calculation can now determine mixing from declared matrix elements rather than infer those elements from a band plot.

Pause before diagonalization. If ${t=0}$ , the cells are independent and their bonding and antibonding energies must be ${\pm|g|}$ . If ${g=0}$ , the two orbital branches must be unmixed and diagonal. Both limits are visible in the matrix. These tests probe the construction more effectively than checking a memorized eigenvalue formula after the fact.

Now hide the matrix for seven minutes and reconstruct it from the bond schematic alone: write each hopping amplitude with its phase sign and displacement index, add every Hermitian conjugate explicitly, assemble the Bloch matrix in the ordered basis ${(|A\rangle,|B\rangle)}$ , and finish by re-running the ${t=0}$ and ${g=0}$ checks on the reconstruction. A disagreement in one entry is repaired at that bond’s phase or index — not by consulting the printed matrix first.

Hybridization, eigenvectors, and the avoided-crossing gap

The constructed matrix now supplies the inputs for the two-state tools from Lecture 2. First decide which approximation and representation are justified; then the exact bands will show how those decisions control orbital mixing.

Use the unchanged anchor with ${t>0}$ , ${g>0}$ , and ${5g\leq2t}$ . Compare a resonant momentum with ${d_z=2t\cos ka=0}$ to a detuned momentum with ${d_z=5g}$ . Aim for one-percent accuracy in a nonzero band energy.

Choose a valid spectral calculation at each momentum. At the detuned point complete ${E_+\simeq d_z+\underline{\hspace{1cm}}}$ , name the dimensionless ratio, and identify the first omitted energy scale. At resonance state the two energies and the physical ${A}$ -orbital probabilities.
A classmate declares ${\hat M=\hat\sigma_z}$ a symmetry and deletes the off-diagonal term. Test that declaration before using symmetry sectors.
Under the coordinate change ${U=2^{-1/2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}}$ , with ${\boldsymbol u'=U^\dagger\boldsymbol u}$ , state the matrix that must represent the physical projector ${\hat P_A=|A\rangle\langle A|}$ . Can ${|u'_1|^2}$ automatically be called the ${A}$ -orbital probability?

Work for six minutes with the Lecture-2 notes closed, then compare with the instructor before the eigenvalue calculation. If needed, retrieve the detuned expansion, retained-subspace criterion, basis transformation, and symmetry-sector construction after recording your first choice. The cue supplies support; it is not evidence of unaided recall.

Eigenvalues and the gap

At ${g=0}$ , the two diagonal energies meet when ${\cos ka=0}$ , first at ${k_\star=\pi/(2a)}$ . At that point a nonzero hybridization leaves ${\hat H_{\mathrm B}(k_\star)=g\hat\sigma_x}$ . Its eigenvalues are ${\pm|g|}$ , so the gap must be ${2|g|}$ . To obtain the full bands without losing that prediction inside a quadratic formula, use the two basis-invariant quantities first. The matrix has zero trace and determinant ${-4t^2\cos^2(ka)-g^2}$ . Therefore

$0=\det[H(k)-E I_2] =E^2-4t^2\cos^2(ka)-g^2.$

The two roots are

(4.2)

$E_{\pm}(k) =\pm\sqrt{4t^2\cos^2(ka)+g^2}.$

For historical band-degeneracy context, see [Her37, p. 365]; the displayed eigensystem and gap criterion are derived here.

A measured two-state coupling scale

The factor of two can be checked against a measured scale without mistaking a double dot for a crystal band. In the charge-qubit experiment introduced in Lecture 2, one microwave-spectroscopy setting gives a resonant splitting ${\Delta E/(2\pi\hbar)=3.3\,\mathrm{GHz}}$ . Hence $\Delta E=(2\pi\hbar)(3.3\times10^9\,\mathrm{s}^{-1}) \simeq13.6\,\mu\mathrm{eV},\qquad |g|=\Delta E/2\simeq6.8\,\mu\mathrm{eV}.$ Here ${g}$ denotes the off-diagonal coupling of the analogous two-state Hamiltonian, not a measured Bloch-band parameter. The measurement supports the scale conversion and the distinction between coupling and splitting; it does not test the momentum dependence of Equation (4.2). [PPLG10b, fig. 1(d) and spectroscopy discussion, p. 246804-2]

Eigenvectors and orbital mixing

Write ${d_z=2t\cos ka}$ and ${r=\sqrt{d_z^2+g^2}>0}$ .

For ${g\neq0}$ , the upper eigenvector obeys $(d_z-r)u_A+g u_B=0, \qquad \frac{u_B}{u_A}=\frac{r-d_z}{g}.$ Together with ${|u_A|^2+|u_B|^2=1}$ , this gives ${|u_A|^2=(r+d_z)/(2r)}$ and ${|u_B|^2=(r-d_z)/(2r)}$ . The sign of ${g}$ fixes their relative sign. This is the two-state projector calculation with a momentum-dependent diagonal mismatch, rather than a new diagonalization method. At ${g=0}$ , choose the appropriate unmixed orbital directly; at ${r=0}$ , no unique upper-band eigenvector is selected.

On any momentum interval ${I\subset\mathrm{BZ}_{1\mathrm D}}$ on which one continuous branch may be chosen, define a real-valued mixing-angle function ${\theta:I\to\mathbb R}$ through

$\cos\theta(k) =\frac{2t\cos ka}{\sqrt{4t^2\cos^2(ka)+g^2}}, \qquad \sin\theta(k) =\frac{g}{\sqrt{4t^2\cos^2(ka)+g^2}}.$

One possible local choice is

$|u_{+}(k)\rangle =\begin{pmatrix}\cos[\theta(k)/2]\\ \sin[\theta(k)/2]\end{pmatrix}, \qquad |u_{-}(k)\rangle =\begin{pmatrix}-\sin[\theta(k)/2]\\ \cos[\theta(k)/2]\end{pmatrix}.$

Substitution gives ${H u_\pm=\pm r u_\pm}$ ; the half-angle identity gives ${\|u_\pm\|^2=1}$ and ${u_+^\dagger u_-=0}$ . At ${d_z=0}$ both orbital probabilities are ${1/2}$ , whereas ${|g/d_z|\to0}$ recovers an unmixed orbital.

Only local consistency is needed here; an overall ${k}$ -dependent phase changes neither orbital probabilities nor eigenvalues. As ${k}$ passes through ${k_\star}$ , a band that was mostly ${A}$ becomes mostly ${B}$ , and vice versa. An avoided crossing therefore means more than a nonzero energy separation: the energy eigenstates exchange basis character.

The exchange can be made quantitative without fixing an arbitrary eigenvector phase. The orbital probabilities ${P_A^{(+)},P_B^{(+)}:I\to[0,1]}$ in the upper band are

(4.3)

$P_A^{(+)}(k) ={|\langle A|u_+(k)\rangle|}^2 =\frac{1+\cos\theta(k)}{2}, \qquad P_B^{(+)}(k) =\frac{1-\cos\theta(k)}{2}.$

Orbital Bloch-sphere interpretation. In the fixed ordered orbital basis, the upper eigenstate defines the dimensionless vector $\mathbf n=\langle u_+|\hat{\boldsymbol\sigma}|u_+\rangle =\left(\frac{g}{r},0,\frac{d_z}{r}\right),\qquad r=\sqrt{d_z^2+g^2}>0.$ It has unit length and lies on the ${xz}$ great circle because the Hamiltonian has real coefficients and no ${\hat\sigma_y}$ term. Its vertical component gives ${P_A^{(+)}=(1+n_z)/2}$ and ${P_B^{(+)}=(1-n_z)/2}$ . The Bloch-vector angle is ${\theta}$ , while the real amplitude pair uses ${\theta/2}$ : an overall sign of the eigenvector leaves ${\mathbf n}$ unchanged. This sphere represents orbital state, not physical spin or a direction in real space.

Rotating ${(g,d_z)}$ at fixed ${r}$ changes orbital character but keeps the energies ${\pm r}$ fixed. Scaling ${r}$ at fixed direction changes the splitting without changing orbital probabilities. These are changes of the Hamiltonian in a fixed basis, not passive basis rotations. At ${r=0}$ , there is no distinguished upper eigenstate; the undefined direction must not be interpreted as a mixed state at the center of the sphere.

Figure 4.2. Orbital Bloch-vector geometry for the upper state. The static example has angle sixty degrees, giving A weight three quarters and B weight one quarter. In HTML, vary direction at fixed Hamiltonian length, then length at fixed direction, to separate orbital character from energy splitting. The basis stays fixed; the vector does not represent real-space motion or physical spin.

Far from the crossing, where ${|2t\cos ka|\gg|g|}$ , one probability approaches one and the other approaches zero. At the crossing they both equal ${1/2}$ . The momentum window over which strong mixing occurs is set by the competition ${|2t\cos ka|\lesssim|g|}$ . Linearizing ${2t\cos ka}$ near ${k_\star}$ gives ${-2ta(k-k_\star)}$ , so the width is of order ${|g|/(2|t|a)}$ . A smaller gap therefore produces a sharper exchange of orbital character. A separately specified observation model can distinguish this exchange when its coupling operator has unequal matrix elements for the two orbitals. The Hamiltonian supplies the orbital weights; a predicted signal also requires that coupling operator and the prepared state.

Worked reading of the eigenvector. Take ${g/t=0.4}$ with ${t>0}$ . At ${ka=\pi/3}$ , the diagonal term is ${2t\cos(ka)=t}$ , so ${\cos\theta=1/\sqrt{1+0.4^2}\simeq0.928}$ . The upper band is therefore about ${96.4\%}$ ${A}$ -like. At ${ka=\pi/2}$ , the same band is exactly a half-and-half mixture. At ${ka=2\pi/3}$ , the diagonal term has reversed sign and the upper band is about ${96.4\%}$ ${B}$ -like. The energy curve alone shows repulsion; the eigenvector shows an exchange of identity across the avoided crossing.

This exchange is often the more informative prediction. A perturbation that acts mainly on orbital ${A}$ , for example, couples strongly to the upper band on one side of the crossing and weakly on the other. This result identifies matrix elements that can vary rapidly even when the energy curve looks smooth. A predicted signal additionally requires a separately defined observation operator and state preparation.

From model parameters to observables

Return to the two-sublattice contrast. The conditional model pair is $\mathsf H_E(k)=d_0(k)\mathsf I+ \begin{pmatrix}0&f(k)\\f{(k)}^*&0\end{pmatrix},\qquad \mathsf H_I(k)=\mathsf H_E(k)+\Delta\sigma_z, \quad f(k_*)=0,\quad\Delta\ne0.$ The local splittings are ${2|f(k)|}$ and ${2\sqrt{|f(k)|^2+\Delta^2}}$ . Thus E crosses at ${k_*}$ , while I has splitting ${2|\Delta|}$ there. Equal on-site energies without the zero of ${f}$ would not establish E’s crossing. A local splitting alone does not establish a global insulating gap when ${d_0(k)}$ varies. The chain’s constant off-diagonal ${g\hat\sigma_x}$ is a different course-defined coupling mechanism. The same eigensystem calculation identifies the gap and internal-state weights in all three schematic cases.

The model-to-observation boundary now separates cleanly. The eigenvalue separation determines the local branch splitting. The calculated weights determine inputs to a separately specified sublattice-sensitive coupling operator. Neither result fixes an absolute signal because the preparation, coupling operator, and observation functional have not been defined. Longer-range hopping, spin–orbit coupling, interactions, and additional reconstruction terms are also outside this minimal comparison.

There is a second invariant check. Let ${d_0:\mathrm{BZ}_{1\mathrm D}\to\mathbb R}$ and ${\mathbf d:\mathrm{BZ}_{1\mathrm D}\to\mathbb R^3}$ . Any ${2\times2}$ Hermitian matrix-valued function can be written

$\hat H_{\mathrm B}(k) =d_0(k)\hat I+\mathbf d(k)\mathbin{\cdot}\hat{\boldsymbol\sigma}.$

Here ${\hat{\boldsymbol\sigma}}$ is the Pauli-operator tuple acting on the declared orbital space ${\mathbb C^2}$ , not an ordinary geometric vector.

For a Hermitian two-state Bloch operator $\hat H_{\mathrm B}(k)=d_0(k)\hat I+\mathbf d(k)\mathbin{\cdot} \hat{\boldsymbol\sigma}, \qquad \mathbf d(k)\in\mathbb R^3,$ the two eigenvalues are ${d_0(k)\pm|\mathbf d(k)|}$ , so their separation is ${2|\mathbf d(k)|}$ . A degeneracy at ${k_\star}$ occurs if and only if ${d_x(k_\star)=d_y(k_\star)=d_z(k_\star)=0}$ . This is a pointwise statement: a nonzero separation at the former meeting point need not remove a crossing that has shifted in momentum.

To decide the local minimum, let ${q\in\mathbb R}$ be a scalar control coordinate measured near an unperturbed crossing and linearize $\mathbf d(q) =v(q-q_0)\mathbf e_{\parallel} +m_{\parallel}\mathbf e_{\parallel} +\mathbf d_{\perp}, \qquad \mathbf e_{\parallel}\mathbin{\cdot}\mathbf d_{\perp}=0, \qquad v\ne0.$ Completing the square gives ${|\mathbf d(q)|}^2 ={\bigl[v(q-q_0)+m_{\parallel}\bigr]}^2 +{|\mathbf d_{\perp}|}^2.$ The parallel perturbation shifts the minimum to ${q=q_0-m_{\parallel}/v}$ , while the transverse component gives $\min_q\bigl(E_+(q)-E_-(q)\bigr)=2|\mathbf d_{\perp}|.$ Thus a local avoided crossing requires a nonzero component transverse to the available control direction. Beyond this linear one-control model, the unambiguous test is to solve ${\mathbf d(k)=\mathbf0}$ on the declared momentum domain. A preserved symmetry may force selected components to vanish on its invariant locus.

The eigenvalue statement follows directly from the Pauli product rule: ${{(\mathbf d\cdot\hat{\boldsymbol\sigma})}^2=|\mathbf d|^2\hat I}$ . Its traceless Hermitian part therefore has eigenvalues ${+|\mathbf d|}$ and ${-|\mathbf d|}$ ; adding ${d_0\hat I}$ shifts both together.

In the anchor model ${\mathbf d(k)=(g,0,2t\cos ka)}$ . Varying only one parameter ${k}$ cannot generically satisfy three independent equations; a crossing in a one-parameter band plot therefore normally needs either tuning or a symmetry that removes some allowed components. The matrix-element proof establishes the gap. The counting explains why avoided crossings are generic when no additional constraint is present.

The same decomposition gives the eigenspaces without choosing phases for individual eigenvectors. Wherever ${\mathbf d(k)\neq\mathbf 0}$ , define the unit-vector function ${\mathbf n(k)=\mathbf d(k)/|\mathbf d(k)|}$ . The two spectral-projector operator functions ${\hat P_{\pm}:I\to \operatorname{Herm}(\mathbb C^2)}$ are

$\hat P_{\pm}(k) =\frac{1}{2}\left[ \hat I\pm\mathbf n(k)\mathbin{\cdot}\hat{\boldsymbol\sigma} \right].$

They satisfy

$\hat P_{\pm}^2=\hat P_{\pm},\qquad \hat P_{+}\hat P_{-}=0,\qquad \hat P_{+}+\hat P_{-}=\hat I,$

and

$\hat H_{\mathrm B}(k)\hat P_{\pm}(k) =\bigl[d_0(k)\pm|\mathbf d(k)|\bigr]\hat P_{\pm}(k).$

For any normalized eigenvector spanning the corresponding one-dimensional subspace,

${\langle\hat{\boldsymbol\sigma}\rangle}_{\pm,k} =\pm\mathbf n(k).$

This is the phase-independent content of the mixing-angle construction. For the anchor model, the ${z}$ component of ${\mathbf n}$ reproduces the orbital imbalance ${P_A-P_B}$ , while the ${x}$ component records coherent hybridization. An eigenvector printed by a numerical solver may change sign or phase abruptly as ${k}$ varies; the projector, orbital probabilities, and expectation values do not. At a true degeneracy ${\mathbf d=\mathbf 0}$ , the direction ${\mathbf n}$ and the separate rank-one projectors cease to be defined because the Hamiltonian no longer selects a unique pair of one-dimensional eigenspaces. That failure is physical information, not a bad choice of phase.

One more variation separates three effects that are often conflated. Let

$\hat H_{\mathrm{gen}}(k) =\bar\varepsilon(k)\hat I +\delta(k)\hat\sigma_z +g_x(k)\hat\sigma_x +g_y(k)\hat\sigma_y,$

where all four coefficients are real-valued functions on the one-dimensional Brillouin zone. Then

$E_{\pm}(k) =\bar\varepsilon(k) \pm\sqrt{\delta^2(k)+g_x^2(k)+g_y^2(k)}.$

The common term ${\bar\varepsilon}$ moves the center of the pair without changing its separation. The detuning ${\delta}$ decides where the unmixed levels would meet. The off-diagonal coefficients ${g_x}$ and ${g_y}$ decide whether those basis states actually mix. Near a proposed crossing, this decomposition is usually more informative than the raw four matrix entries: first find where the detuning vanishes, then test which couplings symmetry allows there, and only then diagonalize.

Use Figure 4.3 as a test. Locate the uncoupled crossing in panel (a) before looking at the solid curves and, for ${g/t=0.4}$ , predict the minimum vertical separation. For the upper band in panel (b), predict ${P_A}$ , ${P_B}$ , and their sum at ${ka/\pi=0,1/2,1}$ . Finally predict the entire minimum-gap curve in panel (c) from the matrix evaluated at ${k_\star}$ .

Figure 4.3. Three computed views of the two-orbital chain. Panel a compares the uncoupled crossing with the avoided-crossing bands. Panel b shows the A- and B-orbital probabilities of the selected band (upper by default) exchanging across the crossing while summing to one. Panel c verifies the predicted minimum gap as the hybridization is varied.

At the default ${g/t=0.4}$ , the calculated separation is ${0.8t}$ . At the crossing the two upper-band probabilities are both ${1/2}$ ; on opposite sides their dominant orbital labels exchange, while ${P_A+P_B=1}$ everywhere. The sweep follows ${\Delta_{\mathrm{gap}}=2|g|}$ . Folding can put the dashed branches in the same reduced-zone plot, but it cannot generate ${g}$ . The visible gap is evidence of a nonzero microscopic coupling; whether that coupling is symmetry-allowed or merely tuned to zero requires the separate symmetry test.

Use the same ordered ${(A,B)}$ basis and ${t>0}$ . Vary ${g/t}$ , then move through ${ka/\pi=1/2}$ , comparing energies with the upper-band probabilities in Equation (4.3). For nonzero coupling the weights sum to one and exchange across the avoided crossing; reversing ${g}$ changes neither the energies nor these probabilities. At ${g=0}$ and the crossing, the degenerate eigenspace does not select a unique upper eigenvector, so its individual orbital weights are left undetermined. These are orbital probabilities, not measured intensities; setting a slider to zero is not a proof of symmetry protection.

Now reverse the visual inference: if the dashed curves appeared to cross exactly in a numerical plot, that sharpness would not prove protection. A correct diagnosis must name a symmetry operator, its invariant momentum locus, and distinct eigenvalues for the two states. “The gap is below the plot resolution” and “a matrix element is forbidden” are different claims, and the next calculation distinguishes them.

Symmetry-protected and tuned crossings

Setting ${g=0}$ produces a crossing, but it does not yet explain why the zero should survive a perturbation. Protection requires a rule that forbids the matrix element. Let ${\hat M:\mathbb C^2\to\mathbb C^2}$ be a unitary symmetry on the relevant momentum locus, and let the two crossing states satisfy

$\hat M|A\rangle=m_A|A\rangle, \qquad \hat M|B\rangle=m_B|B\rangle, \qquad m_A,m_B\in U(1).$

Let ${\hat V:\mathbb C^2\to\mathbb C^2}$ be a Hermitian perturbation within the same fixed- ${k}$ block. If it preserves the symmetry,

$\hat M^{\dagger}\hat V\hat M=\hat V,$

then the off-diagonal matrix element obeys the following selection rule.

Let ${\mathcal L\subset\mathcal B}$ be a momentum locus on which the unitary symmetry ${\hat M}$ preserves each momentum sector individually (momenta are identified modulo reciprocal-lattice vectors). At a fixed ${\mathbf k\in\mathcal L}$ , suppose ${\hat V}$ is Hermitian and symmetry preserving, ${\hat M^\dagger\hat V\hat M=\hat V}$ , and normalized states in that sector ${|A\rangle}$ and ${|B\rangle}$ satisfy ${\hat M|A\rangle=m_A|A\rangle}$ and ${\hat M|B\rangle=m_B|B\rangle}$ , with distinct unit-modulus eigenvalues ${m_A\neq m_B}$ . Then $\langle A|\hat V|B\rangle =\langle A|\hat M^{\dagger}\hat V\hat M|B\rangle =m_A^*m_B\langle A|\hat V|B\rangle=0.$ Consequently the perturbation cannot hybridize the two symmetry sectors at that momentum. The conclusion holds only while ${\hat M}$ remains a symmetry and ${\mathbf k}$ remains on ${\mathcal L}$ ; it does not by itself guarantee that the diagonal dispersions continue to meet.

For the band-symmetry context, see [Her37, p. 365]; the selection rule above is proved in full.

The anchor model specializes this statement to ${\hat M=\hat\sigma_z}$ , for which ${m_A=+1}$ and ${m_B=-1}$ . In that case the matrix element equals its own negative and must vanish.

Different symmetry labels force the coupling to vanish exactly. The term ${g\hat\sigma_x}$ fails the test because ${\hat\sigma_z\hat\sigma_x\hat\sigma_z=-\hat\sigma_x}$ . It breaks this specific symmetry and can open a gap.

In an actual crystal, ${\hat M}$ might represent a mirror, rotation, glide, or another operation whose invariant momenta can be identified independently of the band plot. In the toy chain it is an orbital-label conservation law. The algebra is the same, but the physical robustness is only as strong as the symmetry assumption: a perturbation allowed by the real crystal but excluded from the toy Hamiltonian exposes fine tuning rather than protection.

This argument has a domain. If ${\hat M}$ is a good symmetry only on a line or at a high-symmetry point in a multidimensional Brillouin zone, the matrix-element constraint holds only there. Away from the invariant locus, the two states need not retain distinct labels.

Notice the order of reasoning. We do not infer protection merely because a numerical plot seems to cross. We first declare a unitary operator ${\hat M}$ , verify that it commutes with the Hamiltonian on the stated locus, and assign different ${\hat M}$ eigenvalues to the two states. Only then does the vanishing matrix element follow. If a perturbation changes the crystal so that ${\hat M}$ is no longer a symmetry, the conclusion must be recomputed; the old labels have no power by themselves.

For ${t>0}$ , compare a diagonal, symmetry-preserving perturbation with an off-diagonal, symmetry-breaking one. Determine both whether a crossing exists and whether its states mix. Let the symmetry locus be the entire one-dimensional Brillouin circle, represented by ${-\pi/a\le k<\pi/a}$ , and take

$\hat H_0(k)=2t\cos(ka)\,\hat\sigma_z, \qquad \hat M=\hat\sigma_z.$ Compare the separate perturbations ${m\hat\sigma_z}$ and ${g\hat\sigma_x}$ , with real energy parameters ${m,g}$ . State the crossing condition in the preserving case and the minimum splitting and orbital weights in the breaking case.

First test each perturbation under ${\hat M}$ , then inspect its matrix entries and eigenvalues. The operator identity is ${\hat M^\dagger\hat V\hat M=\hat V}$ for ${m\hat\sigma_z}$ , but equals ${-\hat V}$ for ${g\hat\sigma_x}$ .

A symmetry-preserving perturbation ${m\hat\sigma_z}$ changes the crossing condition to ${2t\cos(ka)+m=0}$ but leaves the off-diagonal matrix element zero. A symmetry-breaking perturbation ${g\hat\sigma_x}$ instead gives

$E_\pm(k)=\pm\sqrt{{[2t\cos(ka)]}^2+g^2}, \qquad \min_k(E_+-E_-)=2|g|.$

This comparison separates displacement of a protected crossing from local hybridization.

For the preserving case a degeneracy exists only if ${|m|\leq2t}$ , since ${\cos(ka)=-m/(2t)}$ must have a solution. For ${|m|<2t}$ the two orbital branches cross with opposite nonzero slopes; at ${|m|=2t}$ they instead touch at a cosine extremum, where both slopes vanish. Its symmetry eigenstates may be kept as pure ${A}$ and ${B}$ orbitals even at degeneracy. If ${|m|>2t}$ , they remain unmixed but never cross. For the breaking case with ${g\ne0}$ , write ${d=2t\cos(ka)}$ , ${h=\sqrt{d^2+g^2}}$ . Use the two-level projectors ${\hat P_\pm=[\hat I\pm(d\hat\sigma_z+g\hat\sigma_x)/h]/2}$ . These are ${\hat P_\pm=|u_\pm\rangle\langle u_\pm|}$ , the operators projecting onto the two nondegenerate energy eigenstates. The formula follows from ${{(d\hat\sigma_z+g\hat\sigma_x)}^2=h^2\hat I}$ : dividing the Hamiltonian by ${h}$ gives eigenvalues plus and minus one. Thus this calculation gives the same orbital probabilities as the squared eigenvector components. The ${A}$ -orbital weight is ${w_{A,\pm}=\langle A|\hat P_\pm|A\rangle}$ ; in this orbital basis ${\langle A|\hat\sigma_z|A\rangle=1}$ and ${\langle A|\hat\sigma_x|A\rangle=0}$ . Evaluating gives $w_{A,\pm}=\frac12(1\pm d/h),\qquad w_{B,\pm}=1-w_{A,\pm}.$ They sum to one and lie in ${[0,1]}$ . At ${ka=\pi/2}$ , both are ${1/2}$ and the gap is ${2|g|}$ ; where the diagonal term is nonzero, the ${g\to0}$ limit recovers pure orbitals. The spectrum also has zero trace, checking the opposite eigenvalues. The zero-gap limit and the symmetry test answer different questions: a tuned zero coupling without a constraint is not protected. This is a local comparison; a global band gap still requires checking extrema at different momenta.

We can now separate three cases:

an allowed, nonzero coupling produces an avoided crossing;
an allowed coupling tuned to zero produces an accidental crossing, which a generic small perturbation gaps; and
a coupling forbidden by a preserved symmetry protects a crossing where the diagonal dispersions meet on the symmetry locus; it does not by itself guarantee that they meet.

Test the distinction with real energy parameters ${g_y,m\in\mathbb R}$ by adding the Hermitian operator-valued function ${\delta\hat H:\mathrm{BZ}_{1\mathrm D}\to\operatorname{Herm}(\mathbb C^2)}$ ,

$\delta\hat H(k)=g_y\hat\sigma_y+m\hat\sigma_z.$

For the ${g=0}$ anchor with this variation, the spectrum is

$E_{\pm}(k)=\pm\sqrt{ g_y^2+{\bigl[2t\cos(ka)+m\bigr]}^2}.$

This equation should be read structurally before it is minimized. The off-diagonal coefficient ${g_y}$ enters as a positive square that cannot be canceled by changing ${k}$ ; it is a mass-like gap term. The diagonal coefficient ${m}$ enters inside the expression that already varies with momentum and can therefore move the zero.

Both ${g\hat\sigma_x}$ and ${g_y\hat\sigma_y}$ are off diagonal in the ${\hat\sigma_z}$ basis and break the stated ${\hat M}$ symmetry, so either opens a gap at a would-be crossing. The term ${m\hat\sigma_z}$ preserves the symmetry and does not mix the sectors. Instead, it shifts the crossing to a solution of ${2t\cos ka+m=0}$ . If ${|m|>2|t|}$ , no such solution remains. A symmetry-preserving perturbation can therefore move a protected crossing, and can remove it by bringing two crossings together or pushing the solution out of the model’s range, without locally hybridizing the two symmetry sectors.

Two normalized states have symmetry eigenvalues ${m_A=\mathrm i}$ and ${m_B=-\mathrm i}$ . For a symmetry-preserving Hermitian perturbation, complete the multiplier in ${\langle A|\hat V|B\rangle=\underline{\hspace{1cm}}\langle A|\hat V|B\rangle}$ and infer the matrix element. Then compare two numerically identical zero-coupling matrices: one is restricted to symmetry-preserving perturbations, the other has no such restriction. Explain why their response to a small perturbation need not agree. Spend two minutes on the algebra and two on the explanation; compare with the instructor before the band-overlap question.

Local separation and global band overlap

The symmetry test decides whether states can mix at one momentum. A scalar dispersion can then move both eigenvalues together without changing that mixing. To decide whether two entire bands are separated, we must keep track of their extrema as well as their local splitting.

A band plot can display a positive vertical separation at every ${\mathbf k}$ while placing the valence maximum and conduction minimum at different momenta. We therefore need two questions: the smallest separation at a fixed crystal momentum and the smallest separation when the two momenta may differ. Let ${\mathcal V}$ and ${\mathcal C}$ be the finite index sets of nominal valence and conduction bands at a declared band partition, including every retained spin or other internal degeneracy. Assume the partition is ordered pointwise: ${E_c(\mathbf k)\geq E_v(\mathbf k)}$ for all ${c\in\mathcal C}$ , ${v\in\mathcal V}$ , and ${\mathbf k\in\mathcal B=\mathbb R^d/\Lambda^*}$ . With each continuous band function ${E_n:\mathcal B\to\mathbb R}$ measured in energy units, the three useful quantities are distinct. When the band sets overlap in energy, these labels alone do not specify occupations; Lecture 5 supplies that step.

The direct band gap of the declared one-particle band partition is the nonnegative scalar $E_{\mathrm g}^{\mathrm{dir}} =\min_{\substack{\mathbf k\in\mathcal B\\ c\in\mathcal C,\ v\in\mathcal V}} \bigl[E_c(\mathbf k)-E_v(\mathbf k)\bigr],$ which compares valence and conduction energies at the same crystal momentum. It is a same-momentum spectral threshold, not automatically the onset of a separately defined transition signal. That identification additionally requires a nonzero transition matrix element, appropriate state weights, and no lower excitation outside the declared one-particle band partition.

The signed indirect separation is the real scalar $\delta_{\mathrm{ind}} =\min_{\substack{\mathbf k\in\mathcal B\\c\in\mathcal C}} E_c(\mathbf k) -\max_{\substack{\mathbf k'\in\mathcal B\\v\in\mathcal V}} E_v(\mathbf k').$ Its sign retains information: ${\delta_{\mathrm{ind}}>0}$ means the two band sets are globally separated, ${\delta_{\mathrm{ind}}=0}$ means their extrema touch, and ${\delta_{\mathrm{ind}}<0}$ measures an energy overlap between the declared band sets. This is still a one-particle spectral diagnostic; filling and occupations enter in Lecture 5.

The indirect insulating gap derived from the signed separation is $E_{\mathrm g}^{\mathrm{ind}} =\max\bigl(0,\delta_{\mathrm{ind}}\bigr)\geq0.$ This nonnegative convention is convenient when only an insulating energy scale is wanted, but it deliberately discards the magnitude of a negative band overlap. Whenever the sign or size of that overlap is evidence, report both ${\delta_{\mathrm{ind}}}$ and ${E_{\mathrm g}^{\mathrm{ind}}}$ .

For the early optical direct/indirect momentum distinction, see [HBB54, p. 559]; the signed and clipped definitions here are course conventions.

The distinction also matters for a separately defined transition model. A same-momentum comparison can use one crystal-momentum label, whereas a different-momentum comparison requires an additional momentum-carrying degree of freedom in that model. These definitions still do not predict a signal magnitude, because the observation operator, selection conditions, and state weights must be supplied separately. Lecture 5 supplies the occupation language; later lectures add disorder and interaction effects.

Determine whether two bands with positive separation at every momentum are globally separated by calculating their direct gap and signed indirect separation. Give both orbitals the same neighbor hopping ${t_0\in\mathbb R}$ , rather than the opposite hoppings of the original anchor, and retain a positive intracell coupling ${\Delta>0}$ . In the same ordered orbital basis, $H_{\mathrm{shift}}(k)=2t_0\cos(ka)I_2 +\Delta\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad E_\pm(k)=2t_0\cos(ka)\pm\Delta.$ Here ${\Delta}$ is an energy, and ${k}$ runs over the continuous Brillouin circle. Identify the parameter ranges for separation, touching and overlap.

Compare the same-momentum splitting with extrema taken at independently chosen momenta. The common term changes both energies equally; the normalized eigenvectors remain the symmetric and antisymmetric orbital combinations. Thus their physical orbital weights remain ${1/2,1/2}$ at every momentum.

At the same momentum the common term cancels: ${E_+(k)-E_-(k)=2\Delta}$ . To compare distinct momenta, use ${-1\leq\cos(ka)\leq1}$ . The upper-band minimum and lower-band maximum are $E_{\mathrm c}^{\min}=-2|t_0|+\Delta,\qquad E_{\mathrm v}^{\max}=2|t_0|-\Delta.$ For ${t_0>0}$ these occur at ${ka=\pi}$ and ${ka=0}$ , respectively; for ${t_0<0}$ the locations exchange. Collecting the same-momentum splitting and subtracting the separate extrema gives

(4.4)

$E_{\mathrm g}^{\mathrm{dir}}=2\Delta,\qquad \delta_{\mathrm{ind}}=2\Delta-4|t_0|,\qquad E_{\mathrm g}^{\mathrm{ind}}=\max(0,2\Delta-4|t_0|).$

The band sets are separated for ${\Delta>2|t_0|}$ , just touch in energy at different momenta for ${\Delta=2|t_0|}$ , and overlap for ${0<\Delta<2|t_0|}$ . In the overlapping regime every same-momentum splitting is still positive. A local mixing gap therefore does not decide whether a chemical potential can be placed between the two whole band sets.

Two checks isolate different parts of this conclusion. Setting ${t_0=0}$ gives independent cells and makes direct and indirect gaps both ${2\Delta}$ . Changing the sign of ${t_0}$ only exchanges the extrema, leaving all three scalar diagnostics invariant. In the limit ${\Delta\to0^+}$ , the direct gap vanishes while the signed indirect separation tends to ${-4|t_0|}$ , exposing the overlap information that the clipped gap discards. Thus a filling question must use the extrema, not only the local avoided crossing. The linked band-extrema check in the notes changes the hopping sign and tests that decision.

For this family set ${t_0=-3\Delta/4}$ . Complete the two endpoint energies ${E_+(0)}$ and ${E_+(\pi/a)}$ , identify the upper minimum and lower maximum, and report the direct gap, signed indirect separation and clipped gap. Record which momentum belongs to each extremum. Take two minutes and compare with the instructor before changing microscopic descriptions.

Finish the extrema check and compare your reasoning before opening this exploration. Keep ${\Delta>0}$ as the energy unit and change ${t_0/\Delta}$ in the common-shift model of Example 4.3. The same-momentum splitting stays ${2\Delta}$ , but the upper minimum and lower maximum meet when ${|t_0|/\Delta=1/2}$ . Beyond that value the signed indirect separation is negative, while the clipped insulating-gap diagnostic is zero. Reverse the hopping sign: the extrema exchange momenta, not energies. At zero hopping the bands are flat, so their extrema occur at every momentum. The horizontal reference lines mark the analytic extrema, not fitted or measured energies.

The next comparison changes the microscopic basis rather than this global diagnostic. A weak periodic potential and localized hopping must each supply their own matrix elements and error scale, even when the retained matrix has the same two-state form.

The same two-level mechanism in familiar limits

The local-gap algebra can be reused in another microscopic basis. Its range of validity must be established again, because discarded plane waves and discarded local orbitals need not have the same separation or coupling.

Plane-wave coupling near a Bragg plane

Return briefly to the ${d}$ -dimensional continuum setting of Lecture 3. In the nearly-free electron problem, a Fourier coefficient ${V_{\mathbf G}\in\mathbb C}$ , with units of energy, of the periodic potential ${V:\Omega\to\mathbb R}$ couples plane waves ${|\mathbf k\rangle}$ and ${|\mathbf k-\mathbf G\rangle}$ . Normalize them in a periodic volume ${|\Omega|=N_{\mathrm c}|\Omega_{\mathrm{cell}}|}$ , with ${\langle\mathbf r|\mathbf p\rangle=|\Omega|^{-1/2} e^{\mathrm i\mathbf p\cdot\mathbf r}}$ , where ${\mathbf p}$ here labels a wave vector and the physical momentum is ${\hbar\mathbf p}$ . Then

$\langle\mathbf k|V|\mathbf k-\mathbf G\rangle =\frac1{|\Omega|}\int_\Omega V(\mathbf r) e^{-\mathrm i\mathbf G\cdot\mathbf r}\,\mathrm d^d r =\frac1{|\Omega_{\mathrm{cell}}|}\int_{\Omega_{\mathrm{cell}}} V(\mathbf r)e^{-\mathrm i\mathbf G\cdot\mathbf r}\,\mathrm d^d r.$

The integrand is cell periodic, so the number of cells cancels the volume normalization. The resulting matrix element is

$V_{\mathbf G} =\frac{1}{|\Omega_{\mathrm{cell}}|} \int_{\Omega_{\mathrm{cell}}} V(\mathbf r)\, \mathrm e^{-\mathrm i\mathbf G\mathbin{\cdot}\mathbf r} \,\mathrm d^d r, \qquad V_{-\mathbf G}=V_{\mathbf G}^*.$

The last relation follows because the potential is real. Periodicity prevents the potential from coupling arbitrary crystal-momentum sectors, but it permits plane-wave components inside one sector to differ by a reciprocal vector. Near their degeneracy choose the ordered basis ${(|\mathbf k\rangle,|\mathbf k-\mathbf G\rangle)}$ , write ${\varepsilon_0(\mathbf p)=\hbar^2|\mathbf p|^2/(2m)}$ , and subtract the common mean potential ${V_{\mathbf0}}$ from the energy origin. The retained matrix has the form

(4.5)

$H_{\mathrm{NFE}}(\mathbf k)= \begin{pmatrix} \varepsilon_0(\mathbf k) & V_{\mathbf G}\\ V_{\mathbf G}^{*} & \varepsilon_0(\mathbf k-\mathbf G) \end{pmatrix}.$

At exact degeneracy the splitting is ${2|V_{\mathbf G}|}$ . After rephasing one basis vector so that ${V_{\mathbf G}}$ is real and positive, the two eigenvectors are the symmetric and antisymmetric combinations of the plane waves. They form standing-wave density patterns shifted relative to one another within the unit cell, so the periodic potential can lower one energy and raise the other. This is the microscopic content of the familiar zone-boundary Bragg gap. It is not a new center of attention; it is the same two-level logic in a plane-wave basis. For the solvable periodic-potential history of allowed and forbidden energy ranges, see [KP31, pp. 500–503, Eq. (6), Figs. 2–3]; the displayed ${V_{\mathbf G}}$ matrix and normalization are derived here.

Take ${V(x)=2V_c\cos(2\pi x/a)}$ , with real ${V_c}$ , and determine the two-plane-wave splitting at the first zone edge for a particle of mass ${m}$ . Use the pair with free wave numbers ${\pm\pi/a}$ , and state when this retained matrix is a useful approximation to the full potential problem.

Compute the coupling from the potential’s Fourier coefficient rather than inferring it from the folded crossing in the free-electron folding example. The mean potential is zero. With ${G=2\pi/a}$ , the coefficient calculation gives $\begin{aligned} V_G&=\frac1a\int_0^a V(x)\mathrm e^{-\mathrm iGx}\,\mathrm dx\\ &=\frac{V_c}{a}\int_0^a \bigl(1+\mathrm e^{-2\mathrm i Gx}\bigr)\,\mathrm dx=V_c. \end{aligned}$ The oscillatory integral vanishes over the cell, leaving ${V_G=V_c}$ . The free branches labeled by ${G=0}$ and ${G=-2\pi/a}$ are degenerate at ${k=\pi/a}$ . Keeping only those two plane waves gives

$H_{\mathrm{edge}}= \begin{pmatrix} \hbar^2\pi^2/(2ma^2) & V_{2\pi/a}\\ V_{2\pi/a}^{*} & \hbar^2\pi^2/(2ma^2) \end{pmatrix}.$

The energies are the common free-particle value shifted by ${\pm|V_{2\pi/a}|}$ . If a symmetry forces this Fourier matrix element to vanish for the two states under consideration, the crossing remains. The localized-orbital coupling ${g}$ and the plane-wave coupling ${V_{2\pi/a}}$ arise from different microscopic descriptions, but they perform the same algebraic work in their respective two-state subspaces.

Here the splitting is ${2|V_c|}$ . The trace remains twice the free edge energy, while ${V_c\to0}$ restores the crossing, checking two different parts of the calculation. Changing the sign of ${V_c}$ exchanges which standing-wave combination is lower but leaves the gap unchanged. All matrix entries and the splitting have energy units. The two-state eigenvalues are exact for this retained matrix; using them for the full cosine potential requires separation from the omitted plane waves. Writing ${E_{\mathrm B}=\hbar^2\pi^2/(2ma^2)}$ , the nearest omitted wave numbers ${\pm3\pi/a}$ have free energy ${9E_{\mathrm B}}$ , so their separation is ${\Delta_{\mathrm{omit}}=8E_{\mathrm B}}$ . The controlling ratio is ${|V_c|/\Delta_{\mathrm{omit}}\ll1}$ ; individual retained energies acquire corrections of order ${V_c^2/\Delta_{\mathrm{omit}}}$ . This bound does not assert that the gap itself has a nonzero correction at that order. The microscopic input here is a potential Fourier amplitude, not a localized hopping integral.

A localized orbital and its bandwidth

For a single retained localized orbital per cell, calculate the dispersion, bandwidth and minimum from the given onsite and neighbor matrix elements. Let the onsite energy be ${\varepsilon_0}$ and the two neighbor amplitudes be ${t(\pm a)=t}$ , with ${t\in\mathbb R}$ and cell spacing ${a}$ .

Reuse the displacement-phase rule of Example 4.1 with only one orbital. With real ${t}$ , the displacement sum is $E(k)=\varepsilon_0+t\mathrm e^{-\mathrm i ka}+t\mathrm e^{\mathrm i ka}.$ Combining the conjugate phases produces

(4.6)

$E(k)=\varepsilon_0+2t\cos ka$

This scalar Bloch function contains only one state per ${k}$ and hence no intracell two-level hybridization. Since ${-1\leq\cos(ka)\leq1}$ , the energy ranges from ${\varepsilon_0-2|t|}$ to ${\varepsilon_0+2|t|}$ . Its bandwidth is ${4|t|}$ ; changing the sign of ${t}$ reverses which zone point is the band minimum but does not create a second band. The comparison reveals what the two-orbital model adds: matrix structure inside each Bloch block, and therefore a possible off-diagonal competition between basis states.

Explicitly ${E(0)=\varepsilon_0+2t}$ and ${E(\pi/a)=\varepsilon_0-2t}$ , so ${E_{\min}=\varepsilon_0-2|t|}$ , at ${k=0}$ for ${t<0}$ and at the identified zone edge for ${t>0}$ . Every ${k}$ is a minimum when ${t=0}$ . The magnitude of the endpoint-energy difference is ${|E(0)-E(\pi/a)|=4|t|}$ . The ${t=0}$ limit is a flat atomic level, and ${E(k+2\pi/a)=E(k)}$ checks reciprocal periodicity. The energy average over the zone is ${\varepsilon_0}$ , providing a further check that the two hopping phases add no constant offset. Unlike the Bragg example, this model starts with a retained orbital and its hopping amplitudes; its accuracy depends on that orbital projection, not on a weak periodic potential. This boundary case clarifies what internal matrix structure adds.

The common calculation is easiest to audit side by side:

Step	Localized-orbital Bloch block	Nearly-free plane-wave block
basis near a crossing	two orbitals in the same ${\mathbf k}$ block	${\vert{}\mathbf k\rangle,\vert{}\mathbf k-\mathbf G\rangle}$
diagonal mismatch	${\varepsilon_{\mathrm A}(\mathbf k)-\varepsilon_{\mathrm B}(\mathbf k)}$	${\varepsilon_0(\mathbf k)-\varepsilon_0(\mathbf k-\mathbf G)}$
allowed off-diagonal element	orbital hybridization ${g(\mathbf k)}$	Fourier amplitude ${V_{\mathbf G}}$
exact-degeneracy result	gap ${2\vert{}g\vert{}}$	gap ${2\vert{}V_{\mathbf G}\vert{}}$
protection check	symmetry forbids ${g}$ between distinct sectors	selection rule forces the relevant ${V_{\mathbf G}}$ matrix element to zero

The microscopic bases and selection rules differ, but the nonroutine move is identical: project to the nearly degenerate two-dimensional subspace before diagonalizing. The one-orbital cosine band is the boundary case that lacks that internal two-state subspace.

Matching the retained model to an error target

The finite matrices above are solved exactly within their declared retained spaces. Calling their relation to a larger Hamiltonian controlled requires additional scales. Reuse the Lecture-2 subspace reduction: let ${P}$ retain the near-degenerate pair and ${Q=I-P}$ denote the omitted states. In orthonormal bases define ${W=H_{PQ}}$ , with energy norm ${w=\|W\|}$ , and assume the target energy ${E}$ stays a distance ${D_{\mathrm{ex}}>0}$ from the spectrum of ${H_{QQ}}$ . Eliminating the remote coordinates gives the correction $\Sigma_P(E)=W{(EI-H_{QQ})}^{-1}W^\dagger,\qquad \|\Sigma_P(E)\|\leq\frac{w^2}{D_{\mathrm{ex}}}.$ The bound follows by multiplying operator norms and using ${\|{(EI-H_{QQ})}^{-1}\|\leq1/D_{\mathrm{ex}}}$ . The ratio ${w/D_{\mathrm{ex}}}$ controls remote-state admixture; the energy scale ${w^2/D_{\mathrm{ex}}}$ must also meet the requested spectral tolerance. These are distinct from the small splitting inside ${P}$ , where the two states are diagonalized together.

For a course-defined weak-potential case, suppose ${w=|V_{\mathbf G}|=0.05D_{\mathrm{ex}}}$ . The retained resonance gap is ${0.10D_{\mathrm{ex}}}$ , while the bound on the omitted matrix correction is ${0.0025D_{\mathrm{ex}}}$ . An energy tolerance ${0.005D_{\mathrm{ex}}}$ admits that error scale; ${0.001D_{\mathrm{ex}}}$ does not. If another plane wave approaches the pair, ${D_{\mathrm{ex}}}$ shrinks and that state must be retained. Weak potential in dimensional units alone never makes a small denominator safe.

Localized tight binding has different input data: isolated local orbitals, their separation from discarded orbitals, and the range of hopping retained. For a one-orbital chain with real longer-neighbor amplitudes ${t_n}$ , discarding neighbors ${n\geq2}$ changes the band by $\delta E(k)=2\sum_{n\geq2}t_n\cos(nka),\qquad |\delta E(k)|\leq2\sum_{n\geq2}|t_n|.$ This triangle-inequality bound tests a hopping-range truncation separately from the remote-orbital bound. For example, if the only omitted term is ${|t_2|=0.03|t_1|}$ , nearest-neighbor hopping permits an error as large as ${0.06|t_1|}$ ; it does not certify a ${0.01|t_1|}$ target. Restore ${t_2}$ or relax the target. Both corrections vanish when their omitted couplings vanish, and both have energy units. Increasing a remote separation reduces the projection correction, whereas it does not remove a declared second- neighbor hopping.

Thus the nearly-free and localized descriptions share two-state algebra but have different matching questions. The first retains resonant plane waves inside a weak periodic potential; the second retains selected local orbitals and hopping processes. Material-fitted parameters without either error audit are phenomenological. The Bloch-sector decomposition itself requires neither of these limiting approximations.

With the same normalized plane waves, the ordered pair is now ${(|k-G\rangle,|k-3G\rangle)}$ , where ${G=2\pi/a}$ , and $V(x)=u e^{2\mathrm iGx}+u^*e^{-2\mathrm iGx},\qquad u\in\mathbb C.$ Complete the phase in the integrand for the upper-right matrix element, evaluate that element, and give its reverse partner. Check the units and Hermiticity. Take three minutes and compare with the instructor before the following independent choice.

Choose and justify a retained description for each course-defined case. State what is exact within it, which discarded effect must be controlled, and the spectral conclusion. Record your choices before calculating; use seven minutes and compare with the instructor before the basis-convention section.

A continuum particle sees ${V(x)=2u\cos(Gx)}$ , with ${u>0}$ . The free states at momenta ${G/2}$ and ${-G/2}$ are degenerate. The coupling norm from their span to omitted states is ${u}$ , and the target energies remain at least ${20u}$ from the omitted spectrum. Find the leading separation and decide whether neglecting that correction meets a per-energy tolerance ${u/10}$ .
A ladder has equal neighbor hopping ${t_0>0}$ on both legs, onsite energies ${+t_0,-t_0}$ , and rung hopping ${\sqrt3\,t_0}$ . Construct its Bloch matrix, find the upper-band physical upper-leg weight, and decide whether the two entire band sets are separated. The remote-orbital coupling norm is ${t_0}$ and the target spectrum is at least ${100t_0}$ from those orbitals. Distinguish the exact answer for the declared ladder from any claim about the sign of a gap after remote-orbital corrections.

Basis covariance and invariant spectra

Why pause over a basis convention after the gap has already been found? Tight-binding codes often attach Bloch phases either to cell origins or to actual orbital positions, and later Berry-phase calculations are sensitive to mixing those choices. Return to a ${d}$ -dimensional lattice with ${M}$ orbitals per cell. For ${\alpha\in\{1,\ldots,M\}}$ , let the ordinary position vector ${\mathbf r^{\mathrm{orb}}_\alpha\in\Omega_{\mathrm{cell}}\subset\mathbb R^d}$ locate orbital ${\alpha}$ within its cell. One may use the cell-phase basis above or include that position in the phase:

${|\mathbf k,\alpha\rangle}_{\mathrm{orb}} =\frac{1}{\sqrt{N_{\mathrm c}}} \sum_{\mathbf R} \mathrm e^{\mathrm i\mathbf k\mathbin{\cdot} (\mathbf R+\mathbf r^{\mathrm{orb}}_\alpha)} |\mathbf R,\alpha\rangle.$

The two basis families are related on the reciprocal-space cover by the diagonal unitary matrix-valued function ${U:\mathbb R^d\to U(M)}$ . It is generally not a single-valued function on ${\mathbb R^d/\Lambda^*}$ . For ${\mathbf G\in\Lambda^*}$ , define ${D_{\mathbf G}=\operatorname{diag}_{\alpha} [\mathrm e^{\mathrm i\mathbf G\cdot\mathbf r^{\mathrm{orb}}_\alpha}]}$ . Then ${U(\mathbf k+\mathbf G)=D_{\mathbf G}U(\mathbf k)}$ .

The cell-phase matrix ${H_{\mathrm{cell}}:\mathbb R^d/\Lambda^*\to \operatorname{Herm}(M)}$ is periodic, whereas ${H_{\mathrm{orb}}:\mathbb R^d\to\operatorname{Herm}(M)}$ is a cover-defined representative with boundary gluing. They obey

$U_{\alpha\beta}(\mathbf k) =\delta_{\alpha\beta} \mathrm e^{\mathrm i\mathbf k\mathbin{\cdot} \mathbf r^{\mathrm{orb}}_\alpha}, \qquad H_{\mathrm{orb}}(\mathbf k) =U^{\dagger}(\mathbf k)H_{\mathrm{cell}}(\mathbf k) U(\mathbf k).$

Consequently,

$H_{\mathrm{orb}}(\mathbf k+\mathbf G) =D_{\mathbf G}^{\dagger}H_{\mathrm{orb}}(\mathbf k)D_{\mathbf G}.$

Thus ${H_{\mathrm{orb}}}$ represents the same Bloch-Hamiltonian section in a nonperiodic basis family; it is covariant across a zone boundary rather than an ordinary matrix-valued function on the quotient.

For a concrete check, take a one-dimensional cell with ${r_A^{\mathrm{orb}}=0}$ and ${r_B^{\mathrm{orb}}=a/3}$ . Then

$U(k)=\begin{pmatrix}1&0\\0&\mathrm e^{\mathrm i ka/3}\end{pmatrix}.$

An intracell entry ${H_{AB}^{\mathrm{cell}}=g}$ becomes ${H_{AB}^{\mathrm{orb}}=g\,\mathrm e^{\mathrm i ka/3}}$ , while the reverse entry acquires the conjugate phase. The trace and determinant remain unchanged, so the characteristic polynomial and both eigenvalues are identical. This is a fast implementation test: two codes using different orbital phase conventions may print different matrices, but they must agree on trace, determinant, and spectrum at each momentum class. In this example, ${D_{2\pi/a}=\operatorname{diag}(1,\mathrm e^{2\pi\mathrm i/3})}$ , so the visible matrices at ${k}$ and ${k+2\pi/a}$ need not be identical; the gluing relation, not entrywise periodicity, identifies them.

Off-diagonal entries can acquire phases, but eigenvalues and gap sizes cannot. If a rephasing changes the plotted spectrum, the two conventions have been mixed inconsistently. This covariance check is therefore useful both when comparing numerical implementations and when preparing the gauge-sensitive quantities introduced in Lectures 21–22. Derivatives of eigenvector components with respect to momentum do respond to a momentum-dependent ${U(\mathbf k)}$ , which is why a later Berry-phase calculation must carry the convention explicitly; the present lecture needs only the invariant spectrum and orbital probabilities in a declared basis.

At the anchor crossing take ${g>0}$ . The upper state has cell-phase-basis coordinates ${\boldsymbol u_+={(1,1)}^T/\sqrt2}$ . A classmate uses $U=\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}, \qquad \boldsymbol u'_+=U^\dagger\boldsymbol u_+={(1,0)}^T,$ then writes: “The upper state is now entirely orbital ${A}$ , because its first coordinate has probability one; the basis change has removed the orbital mixing.” Locate the first invalid identification, transform the physical ${A}$ -orbital projector, and repair the probability. Give a check that detects the error even though both printed state vectors are normalized. Work for four minutes and compare with the instructor before the following reconstruction.

Test the method on the local operator

$\hat H(q)=v q\,\hat\sigma_z +\eta_x\hat\sigma_x+\eta_y\hat\sigma_y +m\hat\sigma_z, \qquad q=k-k_0,$

where ${v\neq0}$ is real and has units of energy times length, while ${\eta_x,\eta_y,m}$ are real energies. Thus ${vq}$ has energy units. Before diagonalizing, answer five questions.

Which coefficients mix the two ${\hat\sigma_z}$ basis states?
What is the minimum gap if those coefficients are allowed?
What does ${m}$ do when the mixing coefficients vanish?
Which conclusion survives a momentum-dependent diagonal basis rephasing?
Why does a positive local avoided-crossing gap not, by itself, prove a positive indirect gap for the whole band structure?

The off-diagonal pair ${\eta_x,\eta_y}$ mixes the basis states and produces the gap ${2\sqrt{\eta_x^2+\eta_y^2}}$ . The diagonal term ${m}$ moves the crossing to ${q=-m/v}$ but cannot gap it by mixing. A preserved ${\hat M=\hat\sigma_z}$ forbids both off-diagonal terms on its invariant locus; without that symmetry, setting them to zero is tuning. A diagonal rephasing can rotate the visible ${\hat\sigma_x}$ and ${\hat\sigma_y}$ coefficients, but it cannot change their combined magnitude or the eigenvalue gap. The indirect gap is a separate global comparison of a conduction minimum and valence maximum that may occur at different momenta, so a local splitting cannot fix its sign. These five answers are the compact form of the method developed in Lectures 3–4.

Interfaces for later calculations

The main argument is complete. One further basis transform connects the extended states used here to the localized operators that later lectures consume.

Bloch and Wannier bases. Extended Bloch eigenstates diagonalize translations, but local interactions and lattice-particle operators are most naturally written in a localized basis. The following transform is the bridge between those descriptions. Let ${\Lambda_N\subset\Lambda}$ be the finite set of ${N_{\mathrm c}}$ cell vectors in a periodic sample and let ${K_N\subset\mathbb R^d/\Lambda^*}$ be its equally large set of allowed crystal momenta. For an isolated band label ${n}$ , the Bloch states ${|\psi_{n\mathbf k}\rangle}$ with ${\mathbf k\in K_N}$ and the Wannier states ${|\mathbf R,n\rangle}$ with ${\mathbf R\in\Lambda_N}$ are two orthonormal bases of the same band subspace. Their discrete inverse transform is

(4.7)

$|\mathbf R,n\rangle =\frac{1}{\sqrt{N_{\mathrm c}}} \sum_{\mathbf k} \mathrm e^{-\mathrm i\mathbf k\mathbin{\cdot}\mathbf R} |\psi_{n\mathbf k}\rangle,$

with inverse

$|\psi_{n\mathbf k}\rangle =\frac{1}{\sqrt{N_{\mathrm c}}} \sum_{\mathbf R} \mathrm e^{\mathrm i\mathbf k\mathbin{\cdot}\mathbf R} |\mathbf R,n\rangle.$

For the periodic-LCAO and tight-binding antecedent, see [SK54, pp. 1498–1501]; the cosine and Bloch–Wannier conventions used here are course-derived.

Lecture 7 uses this basis transform to construct lattice-particle operators and local interaction terms. Whether the resulting states are exponentially localized, whether a globally smooth gauge exists, and how topology can obstruct that choice are separate questions deferred to Lectures 21–24.

From hopping rules to a gap diagnosis

Return to the opening folded crossing. The answer is now a causal chain rather than a vocabulary list:

$\begin{aligned} \text{periodic Hamiltonian} &\Longrightarrow\text{translation characters} \Longrightarrow\mathbf k\sim\mathbf k+\mathbf G\\ &\Longrightarrow\text{independent Bloch blocks}. \end{aligned}$

That chain determines the label space and organizes the eigenproblem. It does not determine a block’s entries. Localized orbitals and hopping amplitudes produce ${H(\mathbf k)}$ ; an allowed off-diagonal entry mixes basis states and opens an avoided-crossing gap; an additional symmetry can force that entry to vanish. Thus:

$\boxed{\text{folding places branches together; a nonzero coupling makes them repel.}}$

A useful diagnostic sequence for any unfamiliar band crossing is:

identify the states and the basis in which the model is written;
identify the symmetry locus and all applicable symmetry labels;
construct or constrain the off-diagonal matrix element;
predict the gap and limiting cases before diagonalization; and
distinguish invariant eigenvalues from convention-dependent matrix entries or eigenvector phases.

The sequence separates three layers of information that a band plot often compresses into one image. Translation symmetry supplies the label and the block decomposition. A microscopic Hamiltonian supplies the entries, orbital content, and allowed couplings inside each block. A separately specified observation model supplies still more information—state preparation, coupling matrix elements, and a signal functional—not encoded by energy curves alone. Confusing these layers is the source of many incorrect claims about “a gap visible in the plot.”

The method now supports several practical diagnoses.

If two numerical bands cross, first test whether the states carry different symmetry eigenvalues on that momentum locus. A visually sharp crossing is not itself evidence of protection.
If a structural distortion breaks the symmetry that forbade mixing, an off-diagonal term can appear and the crossing may become avoided. The gap then measures that allowed coupling in the effective two-state subspace.
If two tight-binding programs use different orbital-position conventions, compare spectra, traces, and determinants before comparing individual matrix phases or eigenvector components.
If a course-defined model has an indirect rather than direct band gap, the band energies diagnose the momentum mismatch but do not calculate a transition signal; an additional momentum-carrying degree of freedom and observation matrix element must be supplied by a separate model.

The course-defined Models E and I make the diagnostic hierarchy concrete. A spectral separation is first a statement about eigenvalues. Assigning it to an onsite contrast or hybridization is a model interpretation that must agree with symmetry and eigenvector character. Predicting a signal is a further, separately specified observation problem. The lecture’s larger role is to keep those three claims—spectrum, mechanism, and observation—from being collapsed into the single phrase “the bands gap.”

The forward question is therefore not another definition of a band. Once the one-particle energies and eigenstates are known, which states are occupied, and what are the elementary low-energy excitations of the solid? Lecture 5 derives locally the statistical interpretation needed to answer that question.

For any proposed avoided crossing, write the abstract projected operator before reading its matrix entries. Then state the local ordered basis, test which coefficients symmetry permits, compare eigenvalues and projectors under a basis change, and estimate the remote-band error. Only after those checks may a gap parameter be assigned to a microscopic coupling; a predicted signal remains a separate observation-model inference.

Problems

1. A displaced intracell orbital

A chain has orbitals at ${\tau_A=0}$ and ${\tau_B=a/3}$ . In the cell-phase convention,

$H_{\mathrm{cell}}(k)= \begin{pmatrix} 2t\cos ka & g(1+\mathrm e^{-\mathrm i ka})\\ g(1+\mathrm e^{\mathrm i ka}) & \Delta \end{pmatrix}.$

Construct the cover-defined diagonal basis-change matrix ${U:\mathbb R\to U(2)}$ that includes the intracell positions in the Bloch phases, and find ${D_{2\pi/a}}$ in ${U(k+2\pi/a)=D_{2\pi/a}U(k)}$ .
Compute ${H_{\mathrm{orb}}(k)=U^{\dagger}H_{\mathrm{cell}}U}$ .
Show using trace and determinant, without rediagonalizing, that both matrices have the same eigenvalues.
Verify the boundary-gluing relation for ${H_{\mathrm{orb}}}$ . Explain which visible phases are convention dependent and which predicted gap is invariant.

2. Allowed, tuned, or protected?

Near a crossing at ${k_0}$ , consider the real-line linear model with ${q\in\mathbb R}$ , real ${v\ne0}$ of units energy times length, and real energies ${m_x,m_y,m_z}$ . The two-state operator is

$\hat H(q)=v q\,\hat\sigma_z +m_x\hat\sigma_x+m_y\hat\sigma_y+m_z\hat\sigma_z, \qquad q=k-k_0.$

Find the spectrum and the minimum gap of this real-line model. If it is used only within a finite neighborhood of ${k_0}$ , state the additional condition for the minimizing momentum to lie inside that neighborhood.
If ${\hat M=\hat\sigma_z}$ is preserved, identify the allowed values of ${m_x,m_y,m_z}$ .
For the symmetry-preserving case, determine when a crossing exists and where it moves.
Classify the crossing as avoided, tuned, or protected in each parameter regime, and state the locus on which your symmetry argument applies.

3. Transfer: diagnose a ladder

A two-leg ladder has one orbital on each leg per cell. Along each leg the nearest-neighbor hopping is ${t}$ . Rung hopping is ${g}$ , while a staggered leg energy gives ${+\Delta}$ on the upper leg and ${-\Delta}$ on the lower leg.

Choose Bloch sums and derive the ${2\times2}$ matrix.
Predict the spectrum at ${k=\pi/(2a)}$ before diagonalizing.
Represent physical exchange of the upper and lower legs by a unitary operator. Determine when it is a symmetry and whether the equal-hopping model contains any crossing for that symmetry to protect.
Now reverse the sign of the hopping on the lower leg. Distinguish physical leg exchange from conservation of the leg label. Explain which, if either, is a symmetry, and how the fate of the new crossing depends on ${g}$ .

(advanced): When Local Orbitals Overlap Between Cells

The core tight-binding construction used an orthonormal localized basis. That assumption allowed the projected Hamiltonian matrix to be diagonalized as an ordinary Hermitian matrix. Local functions used in approximate calculations need not remain orthogonal across neighboring cells. In that changed setting, the overlap matrix is part of the spectral problem rather than a small correction that may be discarded silently.

Let ${\{|\phi_{\alpha\mathbf R}\rangle\}}$ be a finite set of localized functions. Within one cell they satisfy ${\langle\phi_{\alpha\mathbf R}|\phi_{\beta\mathbf R}\rangle =\delta_{\alpha\beta}}$ , but intercell overlaps may be nonzero. Define Bloch sums using the same cell-phase convention as Lecture 4. With ${\mathbf L\in\Lambda}$ denoting a cell-displacement vector, projecting the operator ${\hat H}$ and the inner product produces two matrix-valued functions on the Brillouin zone,

$H_{\alpha\beta}(\mathbf k) =\sum_{\mathbf L} \mathrm e^{\mathrm i\mathbf k\cdot\mathbf L} \langle\phi_{\alpha\mathbf 0}|\hat H| \phi_{\beta\mathbf L}\rangle,$

$S_{\alpha\beta}(\mathbf k) =\sum_{\mathbf L} \mathrm e^{\mathrm i\mathbf k\cdot\mathbf L} \langle\phi_{\alpha\mathbf 0}| \phi_{\beta\mathbf L}\rangle.$

The first is a coefficient matrix for an operator; the second is a Gram matrix. Neither carries a hat.

Let the coefficient space be ${\mathbb C^m}$ . A generalized Hermitian eigenproblem is a matrix pencil ${(H,S)}$ with ${H=H^\dagger}$ and ${S=S^\dagger>0}$ , for which a nonzero coefficient vector ${\mathbf c_n\in\mathbb C^m}$ and a scalar ${E_n\in\mathbb R}$ satisfy $H(\mathbf k)\mathbf c_n(\mathbf k) =E_n(\mathbf k)S(\mathbf k)\mathbf c_n(\mathbf k), \qquad \mathbf c_m^\dagger S\mathbf c_n=\delta_{mn}.$ Here ${S}$ is the Gram matrix induced by the Hilbert-space inner product of the nonorthogonal basis functions. Hence ${\mathbf c_m^\dagger S\mathbf c_n}$ , not the Euclidean coefficient product, equals the overlap of the represented states.

For the fixed generalized problem and symmetric orthogonalization, see [Roo51, pp. 76–77, Eqs. (58)–(61); Low50, pp. 365–367, Eqs. (6)–(21)].

The normalization uses ${S}$ because the Euclidean dot product of coefficient vectors is not the Hilbert-space overlap of the represented states. This is the first diagnostic: imposing ${\mathbf c^\dagger\mathbf c=1}$ generally normalizes coordinates, not the state.

Indeed, if ${|\psi_{n\mathbf k}\rangle=\sum_\alpha c_{n\alpha} \vert{}\phi_{\alpha\mathbf k}\rangle}$ , then direct expansion gives

$\langle\psi_{m\mathbf k}|\psi_{n\mathbf k}\rangle =\sum_{\alpha\beta}c_{m\alpha}^* \langle\phi_{\alpha\mathbf k}|\phi_{\beta\mathbf k}\rangle c_{n\beta} =\mathbf c_m^\dagger S(\mathbf k)\mathbf c_n.$

The Gram form also proves positivity: for every nonzero coefficient vector, ${\mathbf c^\dagger S\mathbf c ={\lVert\sum_\alpha c_\alpha \lvert\phi_{\alpha\mathbf k}\rangle\rVert}^2>0}$ whenever the retained Bloch sums are linearly independent. Thus positive definiteness is not an optional numerical assumption; it is the matrix statement that the chosen basis actually spans independent states.

Assume ${S(\mathbf k)}$ is positive definite for every ${\mathbf k}$ . Its unique positive square root then exists, and the Löwdin coefficient ${\widetilde{\mathbf c}=S^{1/2}\mathbf c}$ turns the problem into

$\widetilde H(\mathbf k)\widetilde{\mathbf c}_n =E_n(\mathbf k)\widetilde{\mathbf c}_n, \qquad \widetilde H=S^{-1/2}HS^{-1/2}.$

Because ${H=H^\dagger}$ and ${S=S^\dagger>0}$ , the Löwdin matrix ${\widetilde H}$ is Hermitian. Ordinary diagonalization is recovered only after the metric has been transferred into the matrix. If an approximate overlap matrix loses positive definiteness, the problem is not a novel band singularity: the retained functions have become linearly dependent or the truncation is inconsistent.

A one-orbital chain makes every statement visible. Let the onsite Hamiltonian element be ${\varepsilon_0}$ , the nearest-neighbor hopping be ${t}$ , and the nearest-neighbor overlap be ${s\in\mathbb R}$ . Then

$H(k)=\varepsilon_0+2t\cos(ka), \qquad S(k)=1+2s\cos(ka),$

and the single band is

$E(k)=\frac{\varepsilon_0+2t\cos(ka)} {1+2s\cos(ka)}.$

The orthonormal result returns as ${s\to0}$ . Positivity for the entire Brillouin zone requires ${|s|<1/2}$ ; at the boundary a Bloch sum has zero norm for one momentum, so the assumed basis ceases to represent an independent degree of freedom. A divergence produced by crossing that boundary is a basis failure, not an infinite electronic excitation energy.

Finally, let ${B(\mathbf k)}$ be any nonsingular change of coefficient basis. The same physical state is represented by ${\mathbf c'=B^{-1}\mathbf c}$ , while

$H'=B^\dagger H B, \qquad S'=B^\dagger S B.$

Then ${H\mathbf c=ES\mathbf c}$ if and only if ${H'\mathbf c'=ES'\mathbf c'}$ . The generalized eigenvalues and ${S}$ -inner products are invariant even though individual matrix entries and coefficient components change. This is the nonorthogonal version of the core lecture’s basis-covariance lesson.

For ${\varepsilon_0=0}$ , compare ${E(k)}$ at ${s=0}$ , ${s=0.2}$ , and ${s=0.49}$ . Locate the smallest eigenvalue of ${S(k)}$ , predict where the basis becomes ill conditioned, and explain why a plot without the overlap eigenvalues could misidentify the failure as band physics.

References

[HBB54] Hall, R. N., Bardeen, J., and Blatt, F. J. (1954). “Infrared absorption spectrum of germanium.” Physical Review 95, 559–560. https://doi.org/10.1103/PhysRev.95.559.

[Her37] Herring, C. (1937). “Accidental degeneracy in the energy bands of crystals.” Physical Review 52, 365–373. https://doi.org/10.1103/PhysRev.52.365.

[KP31] Kronig, R. de L., and Penney, W. G. (1931). “Quantum mechanics of electrons in crystal lattices.” Proceedings of the Royal Society of London. Series A 130 (814), 499–513. https://doi.org/10.1098/rspa.1931.0019.

[Low50] Löwdin, P.-O. (1950). “On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals.” The Journal of Chemical Physics 18, 365–375. https://doi.org/10.1063/1.1747632.

[PPLG10b] Petersson, K. D., Petta, J. R., Lu, H., and Gossard, A. C. (2010). “Quantum coherence in a one-electron semiconductor charge qubit.” Physical Review Letters 105 (24), 246804. https://doi.org/10.1103/PhysRevLett.105.246804.

[Roo51] Roothaan, C. C. J. (1951). “New developments in molecular orbital theory.” Reviews of Modern Physics 23, 69–89. https://doi.org/10.1103/RevModPhys.23.69.

[SK54] Slater, J. C., and Koster, G. F. (1954). “Simplified LCAO method for the periodic potential problem.” Physical Review 94, 1498–1524. https://doi.org/10.1103/PhysRev.94.1498.

Lecture 4: Microscopic Coupling and the Fate of a Crossing

4.1 From real-space hopping to a Bloch matrix

4.1.1 Orbitals, lattice positions, and the Fourier convention

4.1.2 Transforming the hopping matrix elements

4.1.3 Hermiticity and phase-convention checks

4.2 Hybridization, eigenvectors, and the avoided-crossing gap

4.2.1 Eigenvalues and the gap

4.2.2 A measured two-state coupling scale

4.2.3 Eigenvectors and orbital mixing

4.2.4 From model parameters to observables

4.3 Symmetry-protected and tuned crossings

4.4 Local separation and global band overlap

4.5 The same two-level mechanism in familiar limits

4.5.1 Plane-wave coupling near a Bragg plane

4.5.2 A localized orbital and its bandwidth

4.5.3 Matching the retained model to an error target

4.6 Basis covariance and invariant spectra

4.7 Interfaces for later calculations

4.8 From hopping rules to a gap diagnosis

4.9 Problems