Symmetry of Bloch States

A Bloch band acquires an exact symmetry label only when the relevant symmetry acts within the same crystal-momentum fiber. Most spatial operations instead send $\mathbf k$ to another point in its star. At an invariant point or line, reciprocal sewing brings that action back to the chosen fiber, where little-group representations constrain degeneracy, hybridization, and connectivity.

This distinction prevents several common overclaims. Orbital weight is not automatically an irrep label. Time reversal pairs generic $\mathbf k$ with $-\mathbf k$ rather than producing a same-momentum doublet everywhere. Two bands with different symmetry labels cannot mix on the invariant locus, but symmetry does not force their energies to meet. A list of high-symmetry eigenvalues can diagnose some obstructions, yet it does not by itself prove a full-zone gap, a topological invariant, or a material response.

This page owns the crystalline specialization of representation theory: the maps between Bloch fibers, the little-group labels within a fiber, and the resulting tests for degeneracy, mixing, and band connectivity. Groups and Representations owns the abstract algebra; Bloch’s Theorem owns translation sectors; Brillouin Zones owns stars and high-symmetry geometry; and the topology pages own invariants and responses.

Unconventional Superconductivity applies these one-particle sewing and irrep data to antisymmetrized zero-momentum gap matrices; this page retains Bloch-fiber symmetry actions and band-connectivity tests.

Required background. Use the translation convention and Bloch fibers from Bloch’s Theorem, the reciprocal identifications from Brillouin Zones, and irreducible representations from Groups and Representations.

Helpful background. Crystalline Symmetry, Projective Representations, Degeneracy and Multiplets, and Kramers Degeneracy supply the general theorems that are specialized below.

The Symmetry-Data Ledger

Before assigning a label to a computed band, declare:

1. the one-particle, quasiparticle, or mean-field Hamiltonian $H(\mathbf k)$ and the retained band subspace;
2. the spatial or magnetic space group of that Hamiltonian and state, not merely the parent crystal;
3. the basis embedding, Bloch-phase convention, reciprocal sewing, and gauge;
4. whether spin–orbit coupling is retained and whether single- or double-valued representations are required;
5. every antiunitary symmetry, its action on momentum, and its square in the relevant fiber;
6. the momentum point or locus, its star, and the unitary and magnetic little groups;
7. whether the target is a label, a forced degeneracy, an allowed coupling, a crossing, global connectivity, or topology; and
8. which claimed symmetries are exact, emergent, approximate, or broken by boundaries, fields, disorder, magnetism, or the numerical approximation.

The output should be equally explicit:

• an irrep or corepresentation label only where it is defined;
• the minimum enforced multiplet dimension;
• the couplings forbidden or allowed on the stated locus;
• the compatibility relations required away from that locus;
• any additional filling or global-connectivity assumption; and
• a bounded claim that stops before topology or experimental identification unless the corresponding evidence has been supplied.

Band index, orbital character, spin expectation, and valley name are useful annotations, but none is automatically an exact quantum number. At a degeneracy, energy-sorted band indices can swap under an arbitrarily small change of momentum. Projected orbital weights can exchange continuously through an avoided crossing. A physical spin component is conserved only when the Hamiltonian has the corresponding symmetry. A valley label is robust only to the accuracy with which intervalley mixing is absent.

Space-Group Actions Between Bloch Fibers

Write a unitary space-group operation as

$$S = \{p\mid\boldsymbol\tau\},$$

where $p$ is a point operation and $\boldsymbol\tau$ is a translation, possibly fractional. For $S_i=\{p_i\mid\boldsymbol\tau_i\}$, the multiplication rule is

$$\begin{aligned} S_1S_2 &= \{p_1p_2\mid\boldsymbol\tau_{12}\}, \\ \boldsymbol\tau_{12} &= \boldsymbol\tau_1+p_1\boldsymbol\tau_2. \end{aligned}$$

With the active translation convention used on this site,

$$T_{\mathbf R} |\psi_{n\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf R} |\psi_{n\mathbf k}\rangle.$$

Because

$$S T_{\mathbf R}S^{-1} = T_{p\mathbf R},$$

the transformed state belongs to the $p\mathbf k$ fiber:

$$S: \mathcal H_{\mathbf k} \longrightarrow \mathcal H_{p\mathbf k}.$$

Choose orthonormal frames for a symmetry-invariant set of bands and collect their states as columns in $\Psi_{\mathbf k}$. The sewing matrix $B_S(\mathbf k)$ is defined by

$$S\Psi_{\mathbf k} = \Psi_{p\mathbf k} B_S(\mathbf k).$$

It intertwines the Bloch Hamiltonians:

$$\begin{aligned} H(p\mathbf k)B_S(\mathbf k) = B_S(\mathbf k)H(\mathbf k). \end{aligned}$$

This equation relates different fibers in general; it is not yet a same-$\mathbf k$ commutator.

If the frame changes by a momentum-dependent unitary matrix,

$$\Psi_{\mathbf k}' = \Psi_{\mathbf k}W(\mathbf k),$$

then

$$\begin{aligned} B_S'(\mathbf k) = W^\dagger(p\mathbf k) B_S(\mathbf k) W(\mathbf k). \end{aligned}$$

Individual matrix entries and eigenvectors are gauge dependent. Eigenvalues on an invariant locus, irrep multiplicities, and compatibility data are meaningful only after the reciprocal sewing and representation convention have been fixed.

The star of a momentum

The star of $\mathbf k$ is the orbit

$$\operatorname{Star}(\mathbf k) = \left\{ p\mathbf k \bmod\Lambda^* \;\middle|\; p\in P \right\},$$

where $P$ is the crystal point group and $\Lambda^*$ is the reciprocal lattice. A symmetry makes spectra equal at star-related momenta. That is an equality between distinct fibers, not necessarily a degeneracy at one momentum.

For a square lattice with point group $C_{4v}$, a generic momentum has an eight-point star. At $\Gamma$, the full point group leaves the momentum fixed. At $X=(\pi/a,0)$, rotations by $90^\circ$ send $X$ to the distinct point $Y$, whereas a $180^\circ$ rotation and the appropriate mirrors return $X$ modulo a reciprocal vector. The star shrinks as the little group grows.

Little Groups and Exact Band Labels

The space-group little group of $\mathbf k$ is

$$G_{\mathbf k} = \left\{ S\in G \;\middle|\; p\mathbf k \sim \mathbf k \right\}.$$

Here $S=\{p\mid\boldsymbol\tau\}$ and $\sim$ means equality modulo a reciprocal vector. Translations form a normal subgroup. Quotienting them gives the little co-group. At a generic momentum it is often trivial; at high-symmetry points, lines, or planes it is larger.

When $p\mathbf k=\mathbf k+\mathbf G$, the reciprocal-boundary sewing identifies the $p\mathbf k$ fiber with the chosen representative at $\mathbf k$. The resulting matrix $D_{\mathbf k}(S)$ acts inside one fiber and satisfies

$$\left[ D_{\mathbf k}(S), H(\mathbf k) \right] = 0.$$

Only at this stage may the Bloch eigenspace be decomposed into little-group irreps.

For the unitary little group,

$$\mathcal H_{\mathbf k} = \bigoplus_\alpha \left( \mathbb C^{m_\alpha} \otimes V_\alpha \right),$$

where $V_\alpha$ has irrep dimension $d_\alpha$ and occurs $m_\alpha$ times. Schur’s lemma gives

$$\begin{aligned} H(\mathbf k) = \bigoplus_\alpha \left( h_\alpha \otimes I_{d_\alpha} \right). \end{aligned}$$

A $d_\alpha$-dimensional irrep therefore forces at least a $d_\alpha$-fold multiplet. Repeated copies of the same irrep can mix through the multiplicity-space matrix $h_\alpha$; sharing an irrep does not protect a crossing.

For nonsymmorphic groups and spinful electrons, little-co-group matrices may be projective or double valued. Fractional translations contribute momentum-dependent phases, while a $2\pi$ spin rotation contributes a minus sign for half-integer spin. A label such as $A_1$, $E$, or a glide eigenvalue is incomplete unless the space group, momentum, spin convention, and representation table are named.

Degeneracy Is a Conditional Statement

Several mechanisms produce superficially similar doublets. They should be audited separately.

Unitary multiplet degeneracy. A multidimensional unitary little-group irrep forces degeneracy at the invariant momentum or locus. One-dimensional irreps label states but do not by themselves force a doublet.

Spin-rotation degeneracy. Without spin–orbit coupling, magnetism, or a spin-dependent field, an $SU(2)$-invariant one-particle Hamiltonian may have a spin doublet at every $\mathbf k$. This is not the same mechanism as Kramers degeneracy.

Time-reversal pairing. For spinful time reversal,

$$\begin{aligned} \Theta H(\mathbf k)\Theta^{-1} &= H(-\mathbf k), \\ \Theta^2 &= -1. \end{aligned}$$

At a time-reversal-invariant momentum,

$$-\mathbf k_* = \mathbf k_*+\mathbf G,$$

reciprocal sewing returns $\Theta|\psi_{\mathbf k_*}\rangle$ to the same fiber. Kramers theorem then forces an orthogonal partner at the same energy. Away from such momenta, time reversal generally relates states in different fibers; it does not require same-$\mathbf k$ degeneracy.

Combined inversion and time reversal. Inversion sends $\mathbf k$ to $-\mathbf k$, so the antiunitary product $\mathcal P\Theta$ leaves every momentum fixed. For spinful electrons, if $\mathcal P^2=1$, $\Theta^2=-1$, and the two commute in the relevant representation, then

$$\left( \mathcal P\Theta \right)^2 = -1,$$

and every Bloch level is at least twofold degenerate. Inversion alone relates $\mathbf k$ and $-\mathbf k$ but does not generically force a same-momentum doublet.

Magnetic or nonsymmorphic antiunitary degeneracy. More general antiunitary operations can leave a momentum fiber invariant only on selected loci. Their square need not be a scalar: it may be a unitary little-group operation. In the important special case

$$\mathcal A^2 = \zeta_{\mathrm{spin}} T_{\mathbf R},$$

its restriction to the $\mathbf k$ fiber is

$$\left. \mathcal A^2 \right|_{\mathcal H_{\mathbf k}} = \zeta_{\mathrm{spin}} e^{-i\mathbf k\cdot\mathbf R} I.$$

A Kramers-like doublet follows where this scalar equals $-1$, subject to the full corepresentation algebra. The ordinary unitary-irrep argument is insufficient for magnetic little groups.

Valley degeneracy. In an ideal time-reversal-symmetric honeycomb crystal, the inequivalent corners $\mathbf K$ and $\mathbf K'$ are exchanged by time reversal, so their spectra are equal. They are distinct crystal-momentum fibers rather than a same-$\mathbf K$ doublet. Primitive translation symmetry forbids a matrix element between them unless the perturbation supplies their momentum difference modulo a reciprocal vector. An atomically sharp edge or defect can do so, and a commensurate superlattice can fold both valleys to the same reduced momentum and permit hybridization. Smooth perturbations can leave valley approximately conserved without making it an exact internal symmetry. Without time reversal or another symmetry relating the valleys, honeycomb momentum geometry alone does not enforce equal spectra. Graphene and Dirac Materials owns the full honeycomb model and its experimental valley physics; the audit here distinguishes star-related equality from same-fiber degeneracy.

Accidental degeneracy. Equal energies can occur without protection. If an allowed arbitrarily small perturbation splits or gaps the equality, it is accidental relative to the declared symmetry class. Numerical equality alone never identifies the mechanism.

Protected Crossings and Allowed Avoided Crossings

Let two nondegenerate states on a symmetry-invariant locus have unitary symmetry eigenvalues $\lambda_a$ and $\lambda_b$. For a symmetry-preserving perturbation $V$,

$$V_{ab} = \langle a|V|b\rangle.$$

Inserting the symmetry gives

$$V_{ab} = \lambda_a^* \lambda_b V_{ab}.$$

If $\lambda_a\neq\lambda_b$, then $V_{ab}=0$. The two sectors cannot hybridize while the momentum remains on that locus and the symmetry remains exact. If their dispersions meet, the crossing is protected against symmetry-preserving mixing.

The converse needs care. Equal labels permit a coupling; they do not guarantee a visible avoided crossing, because the allowed matrix element can vanish for another reason or be numerically small. Different labels forbid mixing; they do not force the two energies to become equal.

A mirror-line model

On a spinless mirror-invariant line, choose

$$M = \sigma_z.$$

Write a two-band Hamiltonian as

$$\begin{aligned} H(q) &= d_0(q)I+d_z(q)\sigma_z \\ &\quad +d_x(q)\sigma_x+d_y(q)\sigma_y. \end{aligned}$$

Mirror symmetry on the line requires

$$\left[ H(q),M \right] = 0,$$

so $d_x=d_y=0$. A zero of $d_z(q)$ is then a crossing between opposite mirror sectors. Moving away from the mirror line or breaking the mirror permits $\sigma_x$ and $\sigma_y$ terms and generally opens a gap. For spinful reflection the eigenvalues are commonly $\pm i$ rather than $\pm1$, but the selection-rule logic is the same after the double-group convention is declared.

Orbital weight through an avoided crossing

Consider instead

$$H(q) = \begin{pmatrix} \delta(q) & v \\ v^* & -\delta(q) \end{pmatrix}.$$

Its eigenvalues are

$$E_\pm(q) = \pm \sqrt{ \delta(q)^2 +|v|^2 }.$$

When $v\neq0$, the levels avoid crossing and their orbital weights exchange continuously. Calling one energy-sorted branch “orbital 1” on both sides mislabels the state. Orbital character is a projected weight; the exact label is the symmetry representation, if one exists.

Compatibility and Nonsymmorphic Connectivity

An irrep at a high-symmetry point does not remain an irrep of that entire point group as momentum leaves the point. Restrict it to the smaller little group of the outgoing line:

$$\Gamma_\alpha \downarrow G_{\mathrm{line}} = \bigoplus_\beta n_{\alpha\beta}\Lambda_\beta.$$

These subduction rules are the compatibility relations. Bands leaving the endpoint must carry the allowed line irreps, and the same line irreps must recombine into allowed irreps at the other endpoint. Compatibility constrains which branches can connect. It does not determine their energy ordering, dispersion, filling, or whether a global gap exists away from the audited loci.

Fractional translations

Suppose a glide or screw operation $S$ leaves a momentum line invariant and obeys

$$S^N = \{E\mid\mathbf R\}$$

up to a possible $2\pi$ spin rotation. Its representation satisfies

$$\begin{aligned} D_{\mathbf k}(S)^N = \zeta_{\mathrm{spin}} e^{-i\mathbf k\cdot\mathbf R}I, \end{aligned}$$

so every symmetry eigenvalue obeys

$$\lambda_S(\mathbf k)^N = \zeta_{\mathrm{spin}} e^{-i\mathbf k\cdot\mathbf R}.$$

For a spinless two-dimensional glide

$$S = \left\{ m_y \;\middle|\; \frac{a}{2}\hat{\mathbf x} \right\},$$

one has $S^2=T_{a\hat{\mathbf x}}$ on the glide-invariant lines. Hence

$$\lambda_\pm(k_x) = \pm e^{-ik_xa/2}.$$

After $k_x\to k_x+2\pi/a$, the two eigenvalue branches exchange. This enforces pairwise connectivity around the reciprocal loop; it does not mean that every momentum is degenerate. Additional symmetry and compatibility data determine where the required contact occurs. In the spinful case the mirror square changes the phase, commonly giving $\lambda_\pm=\pm i e^{-ik_xa/2}$.

The safe nonsymmorphic claim therefore has four parts:

1. name the glide or screw and its fractional translation;
2. state the invariant momentum locus;
3. include the spinful or spinless multiplication phase;
4. use compatibility around the Brillouin zone before claiming band sticking, a forced crossing, or filling-enforced gaplessness.

Band Representations and Topological Quantum Chemistry

A localized orbital at a Wyckoff position transforms under its site-symmetry group. Inducing that local representation to the full space group produces a band representation. In momentum space it decomposes into little-group irreps that obey compatibility relations across the Brillouin zone.

This construction links local chemistry and global band labels:

$$\begin{gathered} \text{site position and local irrep} \\ \downarrow \\ \text{band representation} \\ \downarrow \\ \text{little-group content}. \end{gathered}$$

It is a powerful atomic-reference test, but the interpretation must remain bounded.

• Matching a sum of atomic band representations supports a symmetry-compatible localized description, but the Wannier centers may occupy different Wyckoff positions from the chosen ions.
• A disconnected group of bands derived from an elementary band representation can carry a topological obstruction.
• Symmetry indicators compress selected representation data. A nonzero indicator can diagnose topology within its stated symmetry class, but a zero indicator does not prove triviality.
• High-symmetry eigenvalues may miss invariants requiring Berry phases, Wilson loops, generic-momentum data, or additional crystalline information.
• None of this replaces checking isolation, filling, the full-zone gap, interactions, magnetic order, surface termination, or the response operator.

Wannier Functions owns localized-basis existence and obstruction; Topological Insulators owns class-AII $\mathbb Z_2$ physics, the inversion-parity shortcut, Wilson-loop diagnostics, and boundary consequences; and Quantum Materials by Design owns material-screening workflows. This page supplies the representation and compatibility input to those analyses.

A Practical Symmetry Audit

For a computed or fitted band structure:

Band Structure Workflows owns the calculation provenance, competing states, full-zone source subspace, and numerical convergence supplied to this audit. Wannierization Workflows owns numerical symmetry constraints, sewing residuals, and localized-subspace validation. The steps below retain symmetry labels, compatibility, and representation-level conclusions.

1. Declare the actual symmetry. Include magnetic order, applied fields, strain, boundaries, and the approximation used to build $H(\mathbf k)$.

2. Fix conventions. Record basis positions, Bloch phases, spinful versus spinless representation, and reciprocal sewing.

3. Find the star and little group. Separate operations that map to another fiber from those that act within the audited fiber.

4. Construct or read the sewing matrices. Test the covariance residual

   $$\begin{aligned} \Delta_S(\mathbf k) &= H(p\mathbf k)B_S(\mathbf k) \\ &\quad - B_S(\mathbf k)H(\mathbf k). \end{aligned}$$

5. Resolve degenerate subspaces together. Diagonalizing one symmetry operator inside a degenerate space may not label the full irrep; use characters or simultaneous compatible generators.

6. Classify antiunitary structure. State which momentum is fixed, the square of the antiunitary operation, and whether a unitary partner changes the corepresentation.

7. Apply compatibility. Track how endpoint irreps subduce along lines and planes rather than joining bands by energy proximity alone.

8. Test perturbations. Add all symmetry-allowed local terms to distinguish enforced crossings from fine tuning.

9. Validate globally. Inspect the full Brillouin zone, not only a conventional high-symmetry path.

10. Stop at the licensed claim. Route invariant, response, and material-identification questions to their canonical owners.

For numerics, define the comparison scale

$$\begin{aligned} E_*(\mathbf k) &= \max \left\{ \begin{array}{c} \|H(\mathbf k)\|, \\ \|H(p\mathbf k)\|, \\ E_{\mathrm{ref}} \end{array} \right\}. \end{aligned}$$

Then report the dimensionless covariance residual

$$\eta_S(\mathbf k) = \frac{ \|\Delta_S(\mathbf k)\| }{ E_*(\mathbf k) }.$$

with a declared matrix norm and nonzero reference scale $E_{\mathrm{ref}}$. Small $\eta_S$ supports the implemented symmetry representation; it does not certify that the model is physically adequate.

Common Pitfalls

Using a basis label as an eigenstate label. Sublattice, atomic orbital, and spin components are basis coordinates. A Bloch eigenstate can mix them while carrying an exact irrep.

Treating star-related energies as a same-momentum degeneracy. A point operation can enforce $E(\mathbf k)=E(p\mathbf k)$ without producing two independent states in $\mathcal H_{\mathbf k}$.

Calling every doublet Kramers degenerate. $SU(2)$ spin degeneracy, a two-dimensional unitary irrep, $\mathcal P\Theta$ degeneracy, and an accidental equality have different hypotheses and responses to perturbations.

Using inversion alone to claim double degeneracy. Inversion relates opposite momenta. A same-$\mathbf k$ doublet needs additional structure.

Protecting a crossing off the invariant locus. Different mirror or rotation labels forbid mixing only where that operation leaves the momentum fiber invariant.

Assuming a nonsymmorphic operation forces degeneracy everywhere. Fractional translations impose momentum-dependent algebra and connectivity. The exact degeneracy locus follows from the full little-group or corepresentation relations.

Labeling individual states inside a degenerate multiplet absolutely. A unitary rotation within the multiplet changes state-by-state eigenvectors. The irrep content and symmetry-invariant subspace are robust.

Inferring topology from a high-symmetry plot. Compatibility can reveal an enforced crossing or obstruction, but a plotted path can miss a gap closure elsewhere and does not compute a response.

Applying one-particle labels to generic interacting eigenstates. The analysis applies directly to one-particle, controlled quasiparticle, or mean-field Bloch bands. Interacting many-body symmetry quantum numbers require a separately declared Hilbert space and observable.

Exercises

Exercise 1: Fiber mapping and sewing-matrix gauge covariance

First use

$$S T_{\mathbf R}S^{-1} = T_{p\mathbf R}$$

and

$$T_{\mathbf R} |\psi_{\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf R} |\psi_{\mathbf k}\rangle$$

to prove that $S|\psi_{\mathbf k}\rangle$ belongs to $\mathcal H_{p\mathbf k}$. Then, starting from

$$S\Psi_{\mathbf k} = \Psi_{p\mathbf k}B_S(\mathbf k),$$

derive the transformation of $B_S(\mathbf k)$ under $\Psi_{\mathbf k}'=\Psi_{\mathbf k}W(\mathbf k)$. Which information is gauge covariant?

Solution

The conjugation law implies

$$T_{\mathbf R}S = S T_{p^{-1}\mathbf R}.$$

Therefore

$$\begin{aligned} T_{\mathbf R} S|\psi_{\mathbf k}\rangle &= S T_{p^{-1}\mathbf R} |\psi_{\mathbf k}\rangle \\ &= e^{-i\mathbf k\cdot p^{-1}\mathbf R} S|\psi_{\mathbf k}\rangle \\ &= e^{-i(p\mathbf k)\cdot\mathbf R} S|\psi_{\mathbf k}\rangle. \end{aligned}$$

This is the translation character of the $p\mathbf k$ fiber. Apply $S$ to the new frame. Because $W(\mathbf k)$ contains frame coefficients, $S\Psi_{\mathbf k}'=S\Psi_{\mathbf k}W(\mathbf k)$. Substitute the sewing definition and $\Psi_{p\mathbf k}=\Psi_{p\mathbf k}'W^\dagger(p\mathbf k)$, then compare with the primed sewing definition. The result is

$$B_S'(\mathbf k) = W^\dagger(p\mathbf k) B_S(\mathbf k) W(\mathbf k).$$

At a little-group momentum, after reciprocal sewing, the matrices are related by a same-fiber similarity transformation. Their eigenvalue multiset and irrep multiplicities are invariant, while eigenvectors and individual matrix entries depend on the chosen frame.

Exercise 2: Star versus little group on a square lattice

For a square lattice with point group $C_{4v}$, compare a generic $\mathbf k$, $\Gamma=(0,0)$, $X=(\pi/a,0)$, and $M=(\pi/a,\pi/a)$. Which operations enlarge the little co-group, and which merely generate star partners?

Solution

A generic point not on a mirror line has no nontrivial point operation that returns it modulo a reciprocal vector, so its little co-group is trivial and its star has eight points. At $\Gamma$, every $C_{4v}$ operation fixes the point. At $M$, sign changes and quarter rotations return the point modulo reciprocal vectors, so the full $C_{4v}$ little co-group is again present. At $X$, a $180^\circ$ rotation and the horizontal and vertical mirrors return $X$ modulo reciprocal vectors, giving a $C_{2v}$-type little co-group. A $90^\circ$ rotation sends $X$ to $Y=(0,\pi/a)$, a distinct star partner. Equality of energies at $X$ and $Y$ is not a same-$X$ degeneracy.

Exercise 3: Crossing or avoided crossing?

Two bands on a mirror-invariant line have eigenvalues $+1$ and $-1$. Their energies meet at $q_0$.

1. Can a mirror-preserving perturbation gap the crossing on the line?
2. What changes if the two bands have the same mirror eigenvalue?
3. What changes away from the mirror line?

Solution

For opposite eigenvalues, the interband matrix element satisfies $V_{+-}=-V_{+-}$ and vanishes. If the energies meet, their crossing is protected on the mirror-invariant line. For equal eigenvalues, symmetry allows mixing; a generic nonzero coupling produces an avoided crossing, although symmetry does not require that coupling to be large or nonzero. Away from the invariant line, mirror maps the momentum to another fiber, so the two same-$\mathbf k$ states no longer carry those mirror eigenvalues and mixing is generally allowed.

Exercise 4: Time reversal, inversion, and same-momentum doublets

Audit the following spinful cases:

1. time reversal only at a generic momentum;
2. time reversal only at a TRIM;
3. inversion only at a generic momentum;
4. commuting inversion and time reversal with $\Theta^2=-1$.

Solution

Time reversal alone maps a generic $\mathbf k$ state to a degenerate state at $-\mathbf k$, usually a different fiber. At a TRIM, reciprocal sewing identifies the fibers and Kramers theorem gives an orthogonal same-momentum partner. Inversion alone also relates $\mathbf k$ and $-\mathbf k$ but, being unitary and squaring to $+1$ in the ordinary orbital representation, does not force a generic same-momentum doublet. With both symmetries under the stated commuting assumptions, $\mathcal P\Theta$ fixes every $\mathbf k$ and squares to $-1$, enforcing a doublet throughout the Brillouin zone.

Exercise 5: Glide connectivity

For a spinless glide with $S^2=T_{a\hat{\mathbf x}}$, derive its two eigenvalues on a glide-invariant line. What happens after one reciprocal period in $k_x$, and what is the strongest justified conclusion?

Solution

The translation convention gives

$$D_{\mathbf k}(S)^2 = e^{-ik_xa}I.$$

Thus

$$\lambda_\pm(k_x) = \pm e^{-ik_xa/2}.$$

After $k_x\to k_x+2\pi/a$, $\lambda_+$ becomes $\lambda_-$ and conversely. A single isolated band cannot return to itself with the same glide branch around the reciprocal loop. Bands must connect in at least pairs, with a contact somewhere as required by continuity and the remaining symmetries. This does not imply degeneracy at every $k_x$, nor does it locate the contact without further compatibility information.

Exercise 6: Compatibility at two endpoints

A two-dimensional irrep $\Gamma_E$ at one endpoint subduces along a line as

$$\Gamma_E \downarrow G_{\mathrm{line}} = \Lambda_+ \oplus \Lambda_-.$$

At the other endpoint, every available isolated doublet subduces as $2\Lambda_+$. Can the original pair remain isolated along the entire line?

Solution

No. One branch leaving the first endpoint carries $\Lambda_-$, but the proposed isolated endpoint pair offers no $\Lambda_-$ branch into which it can terminate. The connectivity graph is incomplete. Another band carrying the missing line irrep must join, or the assumed isolation fails through a crossing elsewhere. Compatibility determines this connection constraint, not the detailed energy ordering.

Exercise 7: Exact label or projected character?

A numerical calculation colors every band by $d_{xz}$ orbital weight. Near two close branches the colors exchange, while both branches carry the same little-group irrep. A small symmetry-preserving perturbation opens a larger gap. Diagnose the labels.

Solution

The $d_{xz}$ color is a basis-projected weight, not an exact symmetry eigenvalue. Because the two states carry the same irrep, a symmetry-preserving coupling is allowed. Hybridization produces an avoided crossing and transfers orbital character between the energy eigenstates. The robust objects are the two-band subspace, its irrep content, and the projected weights with their basis convention—not an energy-sorted assertion that one branch remains “the $d_{xz}$ band” through the avoided crossing.

Exercise 8: Representation data are not yet a topological phase

A calculation finds compatible high-symmetry irrep labels for the occupied bands and an exchange of two orbital weights at $\Gamma$. It concludes: “The material is a topological insulator.” Audit the claim and list the missing checks.

Solution

Compatibility only shows that the reported labels can connect across the audited high-symmetry loci. An apparent “band inversion” is relative to a chosen orbital basis and reference ordering; by itself it is not an invariant. A defensible insulating-topology claim must additionally:

1. declare the electron filling and the occupied projector;
2. verify positive direct occupied–unoccupied separation at every $\mathbf k$ and, for a material-insulator claim, a global indirect or chemical-potential gap with no bulk states;
3. state the exact unitary and antiunitary symmetries and use the correct single- or double-valued representations;
4. compute an appropriate invariant or a symmetry indicator whose hypotheses apply;
5. distinguish class-AII, Chern, crystalline, and interacting possibilities;
6. test robustness under allowed perturbations and numerical resolution; and
7. for a material claim, compare the predicted boundary or bulk response with probe, termination, disorder, and alternative explanations.

Compatible labels are necessary input to some of these tests. They are not a substitute for them.

References

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