Exciton Projected Position Operator

• Exciton projected position operator is defined by projecting the standard position operator onto the excitonic subspace, revealing distinct electron and hole centers.
• It employs Wilson loops and Berry-phase techniques to quantify exciton polarization and infer topological invariants in semiconductor systems.
• Real-space Wannier analyses and ab initio methods integrate with this framework to evaluate exciton localization and shift, bridging theoretical models with computational practice.

Searching arXiv for the cited exciton projected position operator papers and closely related work. The exciton projected position operator is a position operator defined after projection onto an excitonic subspace, so that the geometric and topological structure of bound electron–hole states is encoded directly at the level of exciton bands. In the recent formulation of exciton Berryology, translationally invariant semiconductors with exciton bound states admit an infinite number of possible exciton Berry connections because a many-body exciton state at fixed total momentum can be decomposed into free electron and hole Bloch states in multiple ways. The projected position construction isolates two unique choices—one associated with the electron and one with the hole—whose spectra determine corresponding exciton Berry phases, Wannier-state centers, and the internal polarization of the exciton (Davenport et al., 30 Jul 2025). Related work places this construction in a broader framework of interaction-renormalized exciton quantum geometry, where projected center-of-mass operators generate non-Abelian and Abelian exciton Berry connections, curvatures, shift vectors, dipole vectors, and topological invariants (Paiva et al., 2024). A complementary real-space formulation based on Wannier-function decomposition expresses projected excitonic position observables as weighted sums over electron–hole separations, thereby connecting Berry-geometric quantities to ab initio Bethe–Salpeter analyses of exciton localization (Tao et al., 8 Oct 2025).

1. Formal definition

In the one-dimensional formulation of "Exciton Berryology," unit cells are labeled by $R\in\{0,\dots,L-1\}$, orbitals or sites by $i$, and the lattice constant is set to $a=1$. The conduction-band and valence-band projected number operators are defined as

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$

Periodic position operators are then formed by inserting the phase $e^{i\Delta R}$, with $\Delta=2\pi/L$,

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$

Projecting these operators into the exciton band with

$$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$

gives the exciton projected position operators

$$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$

This construction yields two operators rather than one: $\hat{\mathcal Z}_c$ and $i$0. Their distinction is central. The electron-projected operator tracks the position structure associated with the conduction-band component of the exciton, while the hole-projected operator tracks that of the valence-band component. The paper emphasizes that these are the two unique projected operators whose associated Wilson loops define physically meaningful exciton Berry phases (Davenport et al., 30 Jul 2025).

A broader projected-operator viewpoint appears in the two-dimensional framework of "Shift and Polarization of Excitons from Quantum Geometry," where the projected center-of-mass operator is introduced as

$i$1

with $i$2 the projector onto the low-energy excitonic subspace spanned by Bethe–Salpeter eigenstates $i$3. Its matrix elements in the Bloch-like exciton basis are

$i$4

so the projected operator directly generates the non-Abelian exciton Berry connection $i$5 (Paiva et al., 2024). This suggests that the one-dimensional electron and hole projected operators may be viewed as component-resolved analogues of a more general excitonic projected-position formalism.

2. Spectral structure and Wannier-state centers

The spectral problem for the projected operators is formulated as

$i$6

In the thermodynamic limit $i$7, the eigenvalues take the form

$i$8

so that

$i$9

is the real-space center of the exciton Wannier state $a=1$0 (Davenport et al., 30 Jul 2025). The quantities $a=1$1 and $a=1$2 are therefore shifts, in units of the lattice constant, of the electron and hole within the exciton.

This spectral interpretation directly parallels the modern theory of polarization. The projected operator does not merely encode a basis-dependent phase; its eigenvalues determine physically meaningful Wannier centers. In the excitonic setting, however, there are two such center assignments because the exciton is a composite object. The electron and hole centers generally do not coincide, and their displacement is a measure of internal exciton polarization.

A complementary real-space representation emerges in the Wannier-function decomposition of excitons. In the WFDX framework, the exciton wavefunction is expanded in products of maximally localized Wannier functions for electron and hole sectors,

$a=1$3

Within this basis, the expectation value of the relative position operator is

$a=1$4

where $a=1$5 (Tao et al., 8 Oct 2025). This real-space result is consistent with the projected-operator spectral interpretation: the position content of the exciton can be reduced to weighted Wannier-center separations.

3. Berry connections and Wilson-loop formulation

A central result of "Exciton Berryology" is that the two projected operators produce two Wilson loops,

$a=1$6

whose phases define the two unique exciton Berry phases $a=1$7 and $a=1$8 (Davenport et al., 30 Jul 2025). The corresponding discrete overlaps are

$a=1$9

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$0

These expressions exhibit how the exciton envelope $$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$1 and the underlying Bloch overlaps jointly determine the geometric phase.

In the continuous smooth-gauge limit, one may define the associated Berry connections as

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$2

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$3

with

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$4

The significance of these formulas is that exciton Berry geometry is not inherited trivially from single-particle Berry connections. Each connection includes both an envelope contribution and a single-particle geometric contribution, weighted by the exciton state.

In two dimensions, the projected center-of-mass operator yields the non-Abelian exciton Berry connection

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$5

whose diagonal part

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$6

defines the Abelian exciton Berry connection, and whose curl defines the exciton Berry curvature

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$7

(Paiva et al., 2024). Together, these results place the projected position operator at the origin of both the one-dimensional Wilson-loop description and the higher-dimensional non-Abelian geometric structure.

4. Gauge invariance and exciton polarization

An important practical and conceptual feature of the exciton projected position operator is the availability of a discrete Wilson-loop formulation that does not require a smooth gauge. "Exciton Berryology" states that in practice one only needs the single-$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$8 wavefunction and raw Bloch overlaps. The relative exciton polarization

$$\hat n^c_{R,i}\;=\;\sum_{k,k'}\bar\psi^{k,c}_{R,i}\,\psi^{k',c}_{R,i}\; c^\dagger_{k,c}\,c_{k',c}, \qquad \hat n^v_{R,i}\;=\;\sum_{k,k'}\psi^{k,v}_{R,i}\,\bar\psi^{k',v}_{R,i}\; c^{\phantom\dagger}_{k,v}\,c^\dagger_{k',v}.$$9

can be computed from

$e^{i\Delta R}$0

Equivalently, each Wilson loop may be evaluated directly as a product of discrete overlaps, for example

$e^{i\Delta R}$1

and similarly for $e^{i\Delta R}$2 (Davenport et al., 30 Jul 2025). No smooth gauge over $e^{i\Delta R}$3 is required.

The same work proves that

$e^{i\Delta R}$4

where $e^{i\Delta R}$5 are the single-particle Wannier-center shifts and $e^{i\Delta R}$6 is the unit-cell separation operator. Thus $e^{i\Delta R}$7 is precisely the average electron–hole separation in the exciton at total momentum $e^{i\Delta R}$8. It is real and gauge invariant under both electronic and exciton-state gauge changes, and it controls the difference of the two exciton Berry phases through

$e^{i\Delta R}$9

This identification resolves a possible misconception that exciton Berry phases are purely formal constructs unrelated to real-space structure. In the formulation of (Davenport et al., 30 Jul 2025), the difference between the two Berry phases directly tracks the internal polarization of the exciton. A related gauge-invariant expression appears in the two-dimensional quantum-geometric framework, where the exciton dipole vector is

$\Delta=2\pi/L$0

which is again gauge invariant and describes the polarization of the exciton (Paiva et al., 2024). The parallel between $\Delta=2\pi/L$1 and $\Delta=2\pi/L$2 suggests a common physical content: projected excitonic position operators furnish gauge-invariant measures of internal electron–hole separation.

5. Symmetry quantization and topological diagnosis

The projected-operator framework leads to symmetry-protected quantization results. Under crystalline inversion symmetry $\Delta=2\pi/L$3, if the many-body exciton state $\Delta=2\pi/L$4 at $\Delta=2\pi/L$5 carries inversion eigenvalues $\Delta=2\pi/L$6 and $\Delta=2\pi/L$7, then in the thermodynamic limit

$\Delta=2\pi/L$8

Hence the two Berry phases coincide and are quantized to $\Delta=2\pi/L$9 or $$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$0 (Davenport et al., 30 Jul 2025). In this case, the projected electron and hole centers are forced to the same quantized value modulo the lattice.

For spinless $$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$1, which is anti-unitary, the same paper shows that

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$2

and therefore

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$3

No symmetry eigenvalues exist for $$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$4, but the Berry phase remains quantized and still diagnoses topologically distinct exciton bands (Davenport et al., 30 Jul 2025). This is significant because quantization persists beyond the symmetry-indicator setting available for inversion.

The paper further states that the theory generalizes the notion of shift excitons, whose exciton Wannier states are displaced from those of the non-interacting bands by a quantized amount, beyond symmetry indicators. In this sense, the projected position operator supplies a diagnostic of exciton-band topology even when conventional symmetry-eigenvalue formulas are unavailable.

A related but broader topological statement appears in (Paiva et al., 2024), where the first Chern number of an exciton band is defined as

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$5

In the weak-coupling limit $$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$6, one finds

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$7

When the Bethe–Salpeter amplitude develops zeros, each zero carries an integer vorticity and contributes to the exciton topology through

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$8

This establishes that projected excitonic position operators can encode topology not reducible to the topology of the parent single-particle bands (Paiva et al., 2024).

6. Real-space formulations and ab initio evaluation

The Wannier-function decomposition of excitons provides a real-space language for evaluating projected position observables within the ab initio Bethe–Salpeter framework. Starting from the Bloch exciton expansion

$$\hat Z_{c/v}\;=\;\sum_{R,i}e^{i\Delta R}\,\hat n^{c/v}_{R,i}.$$9

the amplitudes are rotated into a maximally localized Wannier basis using the unitary matrices $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$0, producing real-space coefficients

$$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$1

In this basis, the relative exciton-position expectation value is

$$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$2

and all matrix elements of $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$3 and $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$4 are diagonal in the exciton index $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$5 and expressible in terms of $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$6 and Wannier-center coordinates (Tao et al., 8 Oct 2025).

The implementation workflow given in (Tao et al., 8 Oct 2025) consists of DFT $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$7 GW, followed by the BSE in the Tamm–Dancoff approximation to obtain exciton energies $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$8 and amplitudes $$\hat P_{\rm exc}=\sum_p|p_{\rm exc}\rangle\langle p_{\rm exc}|$$9, Wannierization with Wannier90, discrete Fourier transformation to real-space exciton amplitudes, and then evaluation of $$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$0, $$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$1, or their difference via the real-space weighted sums. The same framework also supports interpolation back to a fine $$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$2-grid by exploiting the rapid decay of $$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$3.

The real-space approach is particularly useful for localization diagnostics. The quantity

$$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$4

directly measures Frenkel versus charge-transfer character, and the second moment

$$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$5

gives a size measure which, in the Wannier–Mott limit, recovers the Bohr radius (Tao et al., 8 Oct 2025). This demonstrates that projected exciton position operators are not only geometric tools but also practical observables in first-principles studies.

7. Conceptual scope and related constructs

The exciton projected position operator sits at the intersection of several lines of inquiry: modern polarization theory, Wilson-loop band topology, Bethe–Salpeter exciton physics, and real-space Wannier analysis. In the one-dimensional Berryology framework, its most distinctive feature is the existence of two unique projected operators, one for the electron and one for the hole, whose Wilson-loop phases have clear physical content: one measures the shift of the electron center, the other the shift of the hole center, and their difference tracks the exciton’s internal polarization (Davenport et al., 30 Jul 2025).

This construction also clarifies the status of the many possible exciton Berry connections. Since a fixed-total-momentum exciton can be decomposed into entangled electron and hole Bloch states in multiple ways, Berry connections defined from arbitrary decompositions are not unique. The projected position operators pick out two privileged choices by tying them to spectral data of projected observables. A plausible implication is that these two Berry phases play for composite excitons a role analogous to Wannier-center phases in single-particle band theory, but with a resolved internal degree of freedom corresponding to the electron–hole partition.

The relation to the exciton shift vector and dipole vector developed in (Paiva et al., 2024) further expands the conceptual scope. There, the gauge-invariant shift vector

$$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$6

and the dipole vector $$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$7 arise from the interaction-renormalized exciton bundle. These quantities do not appear in a single-particle theory; they result from projecting the full two-particle position operators onto the bound-state subspace (Paiva et al., 2024). That observation guards against another common misunderstanding: exciton geometry is not merely inherited from the constituent bands but is genuinely interaction-renormalized.

Finally, the WFDX framework shows that projected exciton position physics can be rendered fully in real space, where nonsymmorphic and other crystal symmetries act directly on the Wannier indices $$\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$$8, enforcing corresponding symmetry relations on $\hat{\mathcal Z}_{c/v} \;=\; \hat P_{\rm exc}\, \hat Z_{c/v}\, \hat P_{\rm exc}.$9 (Tao et al., 8 Oct 2025). This suggests a unifying perspective in which projected position operators, Wilson loops, and Wannier decompositions are different representations of the same excitonic positional structure: spectral in the first case, geometric in the second, and spatially resolved in the third.