---
title: 'Shift Excitons: Intrinsic Quantum Displacements'
url: https://www.emergentmind.com/topics/shift-excitons
type: topic
---

# Shift Excitons: Intrinsic Quantum Displacements

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 [2507.07182]. 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 [2405.19394]. In nonlinear optics, the same structure appears as a many-body shift vector governing excitonic shift current and bulk photovoltaic response [2402.02002]. 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 [2202.10652].

## 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) [2507.07182]. 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 [2405.19394; 2507.22983]. 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 [2402.02002].

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 [2202.10652; 2103.16485; 2102.07368].

## 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 [2507.07182]. 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
\[
R_{0\to ex}=\sum_{r,r'}\rho(r,r')\,[r\,\delta_{r,r'}+D_{cv}(r'-r)],
\]
with \(\rho(r,r')=\psi^*(r)\psi(r')\) and \(D_{cv}(R)=e^{+iQ\cdot R/2}d_c(R)-e^{-iQ\cdot R/2}d_v(R)\) [2507.07182]. 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 \(\psi_Q(r)\sim e^{-|r|/\xi}\), threading a flux through the relative coordinate produces only a pure phase, \(\psi_Q^k(r)=e^{-ik\cdot r}\psi_Q(r)\), with exponentially small boundary errors \(\sim e^{-L/(2\xi_M)}\). Free particle–hole states, by contrast, are plane-wave in \(r\), cannot gauge away the flux, and retain \(O(1)\) sensitivity to \(k\). 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 [2507.07182].

## 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
\[
\sigma^{(\alpha,{\rm shift})}_{\beta\beta}(\omega)
=
\mathrm{Re}\left\{
C\,\omega^{-2}\sum_S
R^{(\alpha)}_{S0}\,|p^{(\beta)}_{0S}|^2\,
\delta(\Omega_S-\Omega_0-\omega)
+
(\omega\leftrightarrow -\omega)
\right\},
\]
where \(R^{(\alpha)}_{S0}=\langle S|\hat r^{(\alpha)}|S\rangle-\langle 0|\hat r^{(\alpha)}|0\rangle\) is the many-body shift vector of exciton \(S\) [2402.02002]. The sum-rule formulation given there shows that \(R_{S0}\) is enhanced by nearly-degenerate optically-active excitons overlapping in \(k\)-space. Tightly bound excitons are localized in real space and therefore spread out in \(k\)-space; this increases overlap between different excitons and can enlarge the inter-exciton dipole and position matrix elements entering the shift current [2402.02002].

A symmetry consequence emphasized in the excitonic shift-vector literature is that \(R_{0\to ex}\) transforms as an ordinary vector under point-group operations. In any noncentrosymmetric but non-polar point group, there is no allowed polar axis, so \(R_{0\to ex}\) vanishes identically at \(Q=0\). 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 \(C_3\) symmetry in \(3R\)-MoS\(_2\), a bulk excitonic shift current appears [2507.07182].

First-principles and model calculations report both enhancement and suppression, depending on the regime. In bulk BaTiO\(_3\), BSE excitons suppress the shift current near threshold by roughly \(50\%\) relative to GW-no-BSE, whereas in monolayer SnSe the exciton-induced change in \(J\) is \(\lesssim 10\%\) because \(d\approx 5.5\) Å \(\ll \alpha^{-1}\approx 2000\) Å [1811.05287]. In monolayer MoS\(_2\) and GeS, \(2p\)-like excitons that are dark in the linear response regime yield a contribution to the photocurrent comparable to that of \(1s\)-like excitons; under radiation with intensity \(I\simeq 1.6\times 10^4\) W/cm\(^2\), width \(w=2\ \mu\)m, and effective thickness \(d\simeq 0.7\) nm, the total short-circuit photocurrent is \(J_x\simeq 0.5\) nA for MoS\(_2\) and \(\simeq 5\) nA for GeS at the \(1s\) resonances, while the dark \(2p\) resonance in MoS\(_2\) also yields \(\sim 0.5\) nA [2406.14215]. In Janus WSSe, the strongest \(C_1\) exciton peak at \(3.02\) eV enhances \(\sigma^{yyy}\) from \(\approx 0.8\times 10^{-8}\) A/V\(^2\) in the IPA to \(\approx 3.5\times 10^{-8}\) A/V\(^2\) with excitons, with a real-space electron–hole separation \(\sim 1.5\) Å and \(R^y_\lambda\approx 0.5\) Å [2506.16067]. 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 [2305.12439].

## 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 \(p\) are written as
\[
|\phi^p\rangle=\sum_k \phi^p_k\,c^\dagger_{p+k,{\rm c}}c_{k,{\rm v}}|GS\rangle,
\]
and projection into this subspace yields the Bethe–Salpeter–type eigenvalue problem
\[
\sum_{k'}\Big[(E_{\rm c}(p+k)-E_{\rm v}(k))\delta_{k,k'}-V_p(k,k')\Big]\phi^p_{k'}
=E_{\rm exc}(p)\phi^p_k
\]
[2405.19394]. The exciton Wannier function is obtained by Fourier transform in the total momentum,
\[
W_n(R,\Delta)=\frac{1}{\sqrt L}\sum_p e^{-ipR}\phi^p_\Delta,
\]
and the center-of-mass shift relative to the electronic Wannier center is
\[
s_{\rm exc}
=
\sum_{R,\Delta}|W_n(R,\Delta)|^2R
=
\langle W_n^0|\hat x_{\rm e}|W_n^0\rangle-x_{\rm e}.
\]

In the presence of inversion symmetry, the exciton inversion eigenvalues \(\lambda_I^{\rm exc}(\tilde p)=\pm 1\) at \(\tilde p=0,\pi\) determine the center shift. If \(\lambda_I^{\rm exc}(0)\neq \lambda_I^{\rm exc}(\pi)\), then
\[
s_{\rm exc}=\frac12 \quad (\bmod\,1),
\]
whereas \(\lambda_I^{\rm exc}(0)=\lambda_I^{\rm exc}(\pi)\) gives \(s_{\rm exc}=0\) [2405.19394]. The corresponding \(\mathbb Z_2\) invariant is
\[
\nu_{\rm exc}
=
\frac12\big[1-\lambda_I^{\rm exc}(0)\lambda_I^{\rm exc}(\pi)\big]\in\{0,1\},
\]
with \(\nu_{\rm exc}=1\iff s_{\rm exc}=1/2\). The paper distinguishes sharply between inherited topology, coming from topological electron or hole bands, and intrinsic topology encoded purely in the relative amplitudes \(\phi_k^p\). 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 \(\lambda_I^{\rm exc}(0)=+\), \(\lambda_I^{\rm exc}(\pi)=-\), hence \(\nu_{\rm exc}=1\) and \(s_{\rm exc}=1/2\) [2405.19394]. 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
\[
\sigma(\omega,r)
=
-\frac1L
\sum_n
\frac{\langle GS|\hat J|\phi^n\rangle\langle \phi^n|\hat j(r)|GS\rangle}
{\hbar\omega-(E_n-E_0)+i\eta}
\]
acquires an additional, sharply localised edge peak at \(\omega=E_{\rm edge}-E_0\), 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
\[
S_{(Q)}
\equiv
A^{f}(Q)-A^{f}_{\rm sing}(Q)
=
[A^{\rm c}(Q/2)-A^{\rm v}(-Q/2)]-A^{\rm exc}(Q)-i\,d_Q\log F(Q),
\]
where \(A^{\rm c}\), \(A^{\rm v}\), and \(A^{\rm exc}\) are the conduction-band, valence-band, and exciton-bundle Berry connections, and \(F(Q)\) carries the phase winding of the interaction-renormalized Bethe–Salpeter coefficient [2408.10300]. By construction, \(S_{(Q)}\) is gauge invariant. The companion dipole vector is
\[
P_p(Q)
=
iF^*\nabla_pF
+
|F|^2\,[A^{\rm c}(p+Q/2)-A^{\rm v}(p-Q/2)],
\]
and gives the internal polarization of the exciton. In the same framework, zeros of \(F(Q)\) produce a singular curvature and shift the excitonic Chern number according to
\[
C^{\rm exc}=C^{\rm c}-C^{\rm v}-\bar N_V.
\]
The cited implication is that interactions can introduce nontrivial topology to exciton bands beyond the topology contained in the single-particle bands [2408.10300].

The semiclassical formulation makes the same displacement dynamical. For a wave packet in a uniform electric field,
\[
L
=
Q\cdot \dot R
+
\dot Q\cdot A^{\rm exc}(Q)
-
eE\cdot r_\omega(Q)
-
\varepsilon^{\rm exc}(Q),
\]
which yields
\[
\dot R
=
\nabla_Q\varepsilon^{\rm exc}(Q)
-
\dot Q\times \Omega^{\rm exc}(Q)
-
e\nabla_Q\sum_p[E\cdot P_p(Q)].
\]
Rewriting \(\Omega^{\rm exc}\) in terms of the single-particle curvatures, the singular curvature, and \(\nabla_Q\times S_{(Q)}\) exposes a shift-velocity term exactly analogous to the single-particle shift current [2408.10300].

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:
\[
A_{{\rm exc},e}(Q)
=
\sum_k \bar\phi^Q_k\,i\partial_Q\phi^Q_k
+
\sum_k |\phi^Q_k|^2 A_{{\rm elec},c}(Q+k),
\]
\[
A_{{\rm exc},h}(Q)
=
\sum_k \bar\phi^Q_k\,i\partial_Q\phi^Q_k
+
\sum_k |\phi^Q_k|^2 A_{{\rm elec},v}(k).
\]
Their difference,
\[
{\cal F}(Q)=A_{{\rm exc},e}(Q)-A_{{\rm exc},h}(Q),
\]
is exactly the mean electron–hole separation \(\langle r_e-r_h\rangle_Q\) [2507.22983]. Under inversion symmetry, \(\gamma_e=\gamma_h\equiv \gamma_{\rm exc}\) mod \(2\pi\), with \(\gamma_{\rm exc}=0\) or \(\pi\), and
\[
W_{{\rm exc},e}=W_{{\rm exc},h}
=
\prod_{Q=0,\pi}\lambda_I^{\rm exc}(Q).
\]
In an inversion-symmetric trivial crystal with \(x_c=x_v=0\), a nontrivial exciton phase \(\gamma_{\rm exc}=\pi\) implies
\[
s_e=s_h=\frac{\gamma_{\rm exc}}{2\pi}=\frac12,
\]
which is precisely the quantized shift-exciton condition. The same work shows that under \(C_2{\cal T}\), although no symmetry indicators are available, the exciton Berry phase remains quantized and still diagnoses topologically distinct exciton bands [2507.22983].

## 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
\[
H=H_0+V_C^{(\rm inter)}+W_{ph}^{(\rm inter)},
\]
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
\[
\Delta E_{\rm Coul}^{(\mu'\mu)}(\mathbf K)
=
\frac{2|\Lambda_{\nu_0}|^2}{E_{\rm gap}^2}
\left(\frac{\mathbf K}{K}\cdot e_{\mu'}\right)
\left(\frac{\mathbf K}{K}\cdot e_\mu\right),
\]
\[
\Delta E_{ph}^{(\mu'\mu)}(\mathbf K)
=
\frac{2|\Lambda_{\nu_0}|^2}{E_{\rm gap}^2}
\sum_{\lambda=X,Y}
(e_{\lambda,\mathbf K}\cdot e_{\mu'})
(e_{\lambda,\mathbf K}\cdot e_\mu),
\]
so that all \(\mathbf K\)-dependence cancels and
\[
\Delta E^{(\mu'\mu)}(\mathbf K)=\delta_{\mu'\mu}\,\frac{2|\Lambda_{\nu_0}|^2}{E_{\rm gap}^2}.
\]
The result is an isotropic bright–dark splitting
\[
\Delta E_{BD}\equiv \Delta E_{\rm bright}-\Delta E_{\rm dark}
\simeq
\frac{2|\Lambda_{\nu_0}|^2}{E_{\rm gap}^2}
\propto
\frac{|\Omega_{ph-X}|^2}{E_{\rm gap}},
\]
with dark excitons unchanged because they do not couple to interband Coulomb or to photons [2202.10652]. The same work states that long-lived excitons must have a small bright–dark splitting; spatially-indirect excitons in coupled quantum wells, whose \(\tau_{\rm rad}\) is \(\gtrsim 10^3\times\) that of direct excitons, should have \(\Delta E_{BD}\) smaller by \(\gtrsim 10^6\), 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,
\[
\Delta E^{(2)}=-\frac12\alpha F^2,
\]
with no first-order contribution for the nondegenerate \(s\)-type ground state [2103.16485]. In hBN-encapsulated monolayer WS\(_2\), out-of-plane fields produce a non-monotonic Stark shift that can change sign from blue to red because the conventional red shift \(\Delta E_1(F)\approx -\gamma''F^2\) competes with a binding-suppression blue shift \(\Delta E_2(F)\simeq +\mu F\), so that \(\Delta E(F)\simeq \mu F-\gamma''F^2\) at modest fields [2011.00221]. In monolayer WS\(_2\), the temperature-dependent \(1s\) exciton red-shift is described by
\[
\Delta E_{\rm total}(T)=\Delta E_{\rm gap}(T)+\Delta E_{\rm exc}(T),
\]
and between \(4\) K and \(300\) K the observed shift is \(\sim 80\) meV, with \(\Delta E_{\rm gap}\simeq -65\) meV and \(\Delta E_{\rm exc}\simeq -15\) meV [2102.07368]. 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.

Source: https://www.emergentmind.com/topics/shift-excitons