Shift Excitons: Intrinsic Quantum Displacements

• Shift excitons are excitonic states with an intrinsic, gauge-invariant shift vector that encodes the real-space displacement of bound electron–hole pairs.
• They are characterized through geometric, crystalline-topological, and response-theoretic formulations using Berry phases, Wannier centers, and optical transition operators.
• Their study provides practical insight into tuning nonlinear optical responses and bulk photovoltaic effects in noncentrosymmetric materials.

Shift excitons are excitonic states for which “shift” denotes a real-space displacement encoded in the many-body or excitonic quantum geometry, rather than merely a displaced spectral line. In the most direct recent usage, they are bound electron–hole excitations whose localization endows them with a quantum shift vector that is intrinsic and independent of light polarization (Yang et al., 9 Jul 2025). A closely related topological usage identifies exciton bands whose maximally-localised exciton Wannier states are shifted by a quantized amount relative to the electronic Wannier states, yielding interaction-induced “shift excitons” even when the underlying noninteracting bands are in a trivial atomic limit (Davenport et al., 2024). In nonlinear optics, the same structure appears as a many-body shift vector governing excitonic shift current and bulk photovoltaic response (Lai et al., 2024). A distinct strand of the literature uses “shift” for energy displacements of excitons, such as the blue-shift of bright excitons caused by electromagnetic quantum fluctuations (Combescot et al., 2022).

1. Terminology and conceptual scope

Within the cited literature, the term refers to several adjacent but non-identical constructions. One construction is geometric: an exciton carries a gauge-invariant shift vector defined from many-body wavefunctions and the optical transition operator, with a compact real-space expression once the exciton envelope is obtained from a Bethe–Salpeter equation (BSE) (Yang et al., 9 Jul 2025). A second construction is crystalline-topological: the exciton band itself has a quantized center-of-mass displacement, diagnosed by inversion data or, more generally, by exciton Berry phases and projected-position Wilson loops (Davenport et al., 2024, Davenport et al., 30 Jul 2025). A third construction is response-theoretic: the shift current of a noncentrosymmetric solid is re-expressed in a many-body excitonic basis, so that the relevant displacement is the charge-center shift between the ground state and an exciton eigenstate (Lai et al., 2024).

These usages are closely related because each elevates the exciton from a simple resonance to a many-body object with its own position-space geometry. The common ingredient is the correlated electron–hole structure of the excitation. The literature also makes clear that this should be distinguished from works where “shift” denotes only an exciton energy shift, such as Stark, polaronic, density-induced, or radiative blue shifts (Combescot et al., 2022, Cavalcante et al., 2021, Henriques et al., 2021).

2. Many-body shift vector of bound electron–hole excitations

A gauge-invariant many-body shift vector for the $n^{\text{th}}$ excitation is defined by the flux-threading expression

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$

where $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$ tracks the Berry-connection change under uniform boundary-phase $k$, and $V$ is the light–matter operator (Yang et al., 9 Jul 2025). For excitons, the excited state at total momentum $Q$ is expanded in a real-space Wannier basis as

$$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$

and the envelope $\psi_Q(r)$ satisfies the BSE in relative coordinates,

$$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$

Once the bound-state $\psi_Q(r)$ is obtained, the excitonic transition shift vector takes the real-space form

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$0

with $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$1 and $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$2 (Yang et al., 9 Jul 2025). This expression is intrinsic in the precise sense stated there: it depends only on the exciton density matrix and Wannier dipoles, not on light polarization.

The decisive distinction from free particle–hole states is localization in the relative coordinate. For a bound exciton $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$3, threading a flux through the relative coordinate produces only a pure phase, $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$4, with exponentially small boundary errors $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$5. Free particle–hole states, by contrast, are plane-wave in $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$6, cannot gauge away the flux, and retain $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$7 sensitivity to $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$8. The cited consequence is that the exciton shift vector is independent of the detailed light–matter operator in the thermodynamic limit, whereas the shift vector of non-interacting delocalized particle–hole excitations depends strongly on polarization (Yang et al., 9 Jul 2025).

3. Excitonic shift current and nonlinear optical response

In the many-body formulation of the bulk photovoltaic effect, the shift current is the light-induced shift of charge centers to many-body excited states. The DC shift-current conductivity can be written in the exciton basis as

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$9

where $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$0 is the many-body shift vector of exciton $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$1 (Lai et al., 2024). The sum-rule formulation given there shows that $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$2 is enhanced by nearly-degenerate optically-active excitons overlapping in $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$3-space. Tightly bound excitons are localized in real space and therefore spread out in $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$4-space; this increases overlap between different excitons and can enlarge the inter-exciton dipole and position matrix elements entering the shift current (Lai et al., 2024).

A symmetry consequence emphasized in the excitonic shift-vector literature is that $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$5 transforms as an ordinary vector under point-group operations. In any noncentrosymmetric but non-polar point group, there is no allowed polar axis, so $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$6 vanishes identically at $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$7. The cited implication is that the excitonic shift photocurrent is zero in non-polar crystals even though free particle–hole transitions would give a finite, polarization-dependent shift photocurrent; when a polar axis is induced, for example by uniaxial strain breaking $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$8 symmetry in $$\Delta A_k^{(0\to n)}=i\langle \Phi_n(k)|\nabla_k\Phi_n(k)\rangle-i\langle \Phi_0(k)|\nabla_k\Phi_0(k)\rangle$$9-MoS$k$0, a bulk excitonic shift current appears (Yang et al., 9 Jul 2025).

First-principles and model calculations report both enhancement and suppression, depending on the regime. In bulk BaTiO$k$1, BSE excitons suppress the shift current near threshold by roughly $k$2 relative to GW-no-BSE, whereas in monolayer SnSe the exciton-induced change in $k$3 is $k$4 because $k$5 Å $k$6 Å (Fei et al., 2018). In monolayer MoS$k$7 and GeS, $k$8-like excitons that are dark in the linear response regime yield a contribution to the photocurrent comparable to that of $k$9-like excitons; under radiation with intensity $V$0 W/cm$V$1, width $V$2m, and effective thickness $V$3 nm, the total short-circuit photocurrent is $V$4 nA for MoS$V$5 and $V$6 nA for GeS at the $V$7 resonances, while the dark $V$8 resonance in MoS$V$9 also yields $Q$0 nA (Esteve-Paredes et al., 2024). In Janus WSSe, the strongest $Q$1 exciton peak at $Q$2 eV enhances $Q$3 from $Q$4 A/V$Q$5 in the IPA to $Q$6 A/V$Q$7 with excitons, with a real-space electron–hole separation $Q$8 Å and $Q$9 Å (Mao et al., 19 Jun 2025). In BN nanotubes and a single BN sheet, the A exciton produces a giant in-gap peak whose value is more than three times larger than that of the quasiparticle shift current, and the effective exciton shift current conductivity is nearly ten times larger than the largest shift conductivity observed in ferroelectric semiconductors (Huang et al., 2023).

4. Interaction-induced crystalline topology and quantized Wannier-center shifts

A more restrictive meaning of shift excitons arises in interaction-induced crystalline topology. Starting from a centrosymmetric semiconductor with one occupied valence band and one empty conduction band, neutral excitations at total momentum $$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$0 are written as

$$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$1

and projection into this subspace yields the Bethe–Salpeter–type eigenvalue problem

$$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$2

(Davenport et al., 2024). The exciton Wannier function is obtained by Fourier transform in the total momentum,

$$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$3

and the center-of-mass shift relative to the electronic Wannier center is

$$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$4

In the presence of inversion symmetry, the exciton inversion eigenvalues $$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$5 at $$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$6 determine the center shift. If $$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$7, then

$$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$8

whereas $$|\psi_Q\rangle=\frac{1}{\sqrt N}\sum_{R_{\rm cm},r}e^{iQ\cdot R_{\rm cm}}\psi_Q(r)\, c^\dagger_{c,R_{\rm cm}+r/2}c_{v,R_{\rm cm}-r/2}|GS\rangle,$$9 gives $\psi_Q(r)$0 (Davenport et al., 2024). The corresponding $\psi_Q(r)$1 invariant is

$\psi_Q(r)$2

with $\psi_Q(r)$3. The paper distinguishes sharply between inherited topology, coming from topological electron or hole bands, and intrinsic topology encoded purely in the relative amplitudes $\psi_Q(r)$4. Shift excitons belong to the latter class.

The explicit example is the interacting spinless SSH model in its trivial electronic phase. There, purely Hubbard-type interactions cannot open a full bulk gap for the nontrivial exciton band; a pair-hopping term is required to produce a fully gapped nontrivial exciton band with $\psi_Q(r)$5, $\psi_Q(r)$6, hence $\psi_Q(r)$7 and $\psi_Q(r)$8 (Davenport et al., 2024). Under open boundary conditions with a chain terminated on a full unit cell, the nontrivial exciton band exhibits exactly one mid-gap exciton per edge. The same work shows that the local optical conductivity

$\psi_Q(r)$9

acquires an additional, sharply localised edge peak at $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$0, offering a direct spectroscopic signature of interaction-induced shift-exciton edge states.

5. Quantum geometry, exciton Berry phases, and modern polarization theory

A broader geometric framework was developed for two-dimensional exciton states in terms of exact connections on the exciton bundle. The exciton shift vector is defined as

$$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$1

where $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$2, $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$3, and $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$4 are the conduction-band, valence-band, and exciton-bundle Berry connections, and $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$5 carries the phase winding of the interaction-renormalized Bethe–Salpeter coefficient (Paiva et al., 2024). By construction, $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$6 is gauge invariant. The companion dipole vector is

$$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$7

and gives the internal polarization of the exciton. In the same framework, zeros of $$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$8 produce a singular curvature and shift the excitonic Chern number according to

$$[\varepsilon_c(\hat p+Q/2)-\varepsilon_v(\hat p-Q/2)]\psi_Q(r)+\sum_{r'}V_{\rm eff}(r,r')\psi_Q(r')=E\,\psi_Q(r).$$9

The cited implication is that interactions can introduce nontrivial topology to exciton bands beyond the topology contained in the single-particle bands (Paiva et al., 2024).

The semiclassical formulation makes the same displacement dynamical. For a wave packet in a uniform electric field,

$\psi_Q(r)$0

which yields

$\psi_Q(r)$1

Rewriting $\psi_Q(r)$2 in terms of the single-particle curvatures, the singular curvature, and $\psi_Q(r)$3 exposes a shift-velocity term exactly analogous to the single-particle shift current (Paiva et al., 2024).

A complementary “modern theory” formulation starts from an exciton projected position operator and shows that there are two unique gauge-invariant exciton Berry connections, one electron-localising and one hole-localising: $\psi_Q(r)$4

$\psi_Q(r)$5

Their difference,

$\psi_Q(r)$6

is exactly the mean electron–hole separation $\psi_Q(r)$7 (Davenport et al., 30 Jul 2025). Under inversion symmetry, $\psi_Q(r)$8 mod $\psi_Q(r)$9, with $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$00 or $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$01, and

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$02

In an inversion-symmetric trivial crystal with $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$03, a nontrivial exciton phase $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$04 implies

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$05

which is precisely the quantized shift-exciton condition. The same work shows that under $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$06, although no symmetry indicators are available, the exciton Berry phase remains quantized and still diagnoses topologically distinct exciton bands (Davenport et al., 30 Jul 2025).

6. Energetic shifts of excitons and the limits of the term

The word “shift” in the exciton literature also denotes energy shifts, and this is conceptually distinct from a shift vector or a shifted Wannier center. In the electromagnetic-fluctuation problem, the exciton Hamiltonian is

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$07

with a longitudinal interband Coulomb term and a transverse photon coupling split into resonant and nonresonant parts. Second-order perturbation theory gives the Coulomb and virtual-photon contributions

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$08

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$09

so that all $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$10-dependence cancels and

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$11

The result is an isotropic bright–dark splitting

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$12

with dark excitons unchanged because they do not couple to interband Coulomb or to photons (Combescot et al., 2022). The same work states that long-lived excitons must have a small bright–dark splitting; spatially-indirect excitons in coupled quantum wells, whose $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$13 is $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$14 that of direct excitons, should have $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$15 smaller by $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$16, in the neV range.

Other cited energy-shift problems are spectroscopically important but use “shift” differently. In atomically thin semiconductors with an in-plane electric field, neutral excitons show a purely quadratic Stark red-shift,

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$17

with no first-order contribution for the nondegenerate $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$18-type ground state (Cavalcante et al., 2021). In hBN-encapsulated monolayer WS$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$19, out-of-plane fields produce a non-monotonic Stark shift that can change sign from blue to red because the conventional red shift $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$20 competes with a binding-suppression blue shift $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$21, so that $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$22 at modest fields (Abraham et al., 2020). In monolayer WS$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$23, the temperature-dependent $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$24 exciton red-shift is described by

$$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$25

and between $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$26 K and $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$27 K the observed shift is $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$28 meV, with $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$29 meV and $$R_{0\to n}\equiv \lim_{k\to 0}\left\{\Delta A_k^{(0\to n)}+\nabla_k \mathrm{Arg}\big[\langle \Phi_0(k)|V(k)|\Phi_n(k)\rangle\big]\right\},$$30 meV (Henriques et al., 2021). These are exciton energy shifts, not shift excitons in the geometric or topological sense.

Across these lines of work, the unifying theme is that excitons possess structure beyond a single resonance energy: they carry internal geometry, symmetry data, and many-body positional information that can be read out through nonlinear response, Wilson loops, edge spectroscopy, or controlled energetic shifts. The phrase “shift excitons” is therefore most precise when reserved for excitons whose correlated wavefunction produces an intrinsic displacement—either as a many-body shift vector or as a quantized displacement of the exciton Wannier center.