Phononic Shift Vector and Its Formulations

• Phononic shift vector is a gauge-invariant geometric quantity that extends the conventional electronic shift vector to describe phonon-assisted coordinate shifts.
• It encompasses multiple formulations, including phonon-drag in non-vertical electron transitions and displacement-space forces driving coherent phonons.
• Its construction via phase derivatives and Berry connections ensures gauge invariance, underpinning modern electron-phonon interaction theories.

Searching arXiv for papers on phononic shift vector and related shift-vector formulations. Phononic shift vector denotes a class of gauge-invariant geometric quantities that connect interband optical excitation to phonon-related coordinate shifts or forces. The recent arXiv literature does not use the term uniformly. One line of work defines a bosonic phonon-drag shift vector for non-vertical electron transitions mediated by finite-momentum phonons; another derives an electron-phonon-mediated shift vector as a specialization of a generalized many-body electronic dipole; and a more recent work introduces a phononic shift vector in displacement space, built from derivatives with respect to a Raman-active lattice coordinate and a phononic Berry connection, which governs a rectified Raman force driving coherent phonons (Wang et al., 2024, Hu et al., 22 May 2026, Pimlott et al., 23 Jul 2025).

1. Terminological scope and baseline definitions

The conventional reference point is the electronic interband shift vector for direct optical transitions,

$$R_{mn}^{a}(\mathbf{k})=A_m^{a}(\mathbf{k})-A_n^{a}(\mathbf{k})-\partial_{k_a}\arg r_{mn}^{a}(\mathbf{k}),$$

with $A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$ and $$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$. In the cited literature, phonon-related generalizations retain this gauge-covariant structure but change either the transition matrix element, the momentum-space kinematics, or the differentiation variable itself (Wang et al., 2024).

Construction	Representative formula	Physical setting
Electronic shift vector	$$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$	Direct optical transition
Phonon-drag / electron-phonon-mediated shift vector	$$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$	Non-vertical electron-phonon process
Phononic shift vector	$$\mathbb R^{mn}_{a;\ell}=\{\partial_{u_a}\mathrm{Im}\ln v_\ell^{mn}+\mathbb A_a^{mm}-\mathbb A_a^{nn}\}_{\mathbf u\to0}$$	Rectified Raman force on coherent phonons

This terminological distinction is essential. The many-body work on generalized shift vectors explicitly states that it does not introduce a standalone object called a phononic shift vector for pure phonon states; its phonon specialization is an electron-phonon-mediated shift vector associated with a phonon-assisted electronic transition, not a shift vector of a phonon wavefunction itself (Hu et al., 22 May 2026). By contrast, the coherent-phonon work defines a phononic shift vector directly in terms of lattice-displacement derivatives and a phononic or molecular Berry connection (Pimlott et al., 23 Jul 2025).

2. Electron-phonon-mediated and phonon-drag formulations

A first phonon-related formulation appears as a generalization from vertical optical excitation to scattering or drag processes between wave packets at different crystal momenta. In that setting,

$$\delta \mathbf{R}_{mn}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg V_{mn}(\mathbf{k},\mathbf{k}'), \qquad D_{\mathbf{k},\mathbf{k}'}=\partial_{\mathbf{k}}+\partial_{\mathbf{k}'} .$$

For phonons, the scattering matrix element is replaced by the electron-phonon coupling $g_{mn,\nu}(\mathbf{k},\mathbf{k}')$, yielding the bosonic phonon-drag shift vector

$$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}_b(\mathbf{k}')-\mathbf{A}_a(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}(\mathbf{k},\mathbf{k}') .$$

This object is the gauge-invariant coordinate shift associated with an electron transition mediated by a phonon mode $\nu$ between states at different momenta. It is therefore tied to non-vertical transitions and finite momentum transfer, not to ordinary vertical optical absorption (Wang et al., 2024).

A second formulation arises from a many-body dipole theory. For a correlated state

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$0

the intrinsic dipole is

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$1

and the generalized shift vector is

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$2

In the electron-hole sector with momentum transfer $A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$3, the two-band reduction is

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$4

Specializing to electron-phonon coupling replaces the correlated amplitude by the electron-phonon matrix element,

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$5

which yields

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$6

This is the electron-phonon-mediated or phonon-assisted shift vector recovered as a special case of the generalized many-body intrinsic dipole (Hu et al., 22 May 2026).

These two formulations are closely related in structure. Both replace the optical dipole matrix element phase by the phase of an electron-phonon matrix element; both describe a coordinate shift attached to a phonon-assisted electronic process; and both are explicitly finite-momentum or interaction-mediated generalizations of the direct optical shift vector. Neither, however, is presented as a shift vector of a pure phonon band eigenstate.

3. Displacement-space phononic shift vector for coherent phonons

A distinct definition appears in the theory of rectified Raman forces that drive coherent phonons. In that setting, the phononic shift vector is introduced as

$A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$7

where $A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$8 is a Raman-active lattice displacement, $A_n=i\langle n|\partial_{\mathbf{k}}|n\rangle$9 is the optical velocity matrix element, and

$$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$0

is the phononic or molecular Berry connection. The derivative is therefore with respect to lattice displacement rather than crystal momentum. The paper explicitly introduces this quantity as analogous to the electronic shift vector describing the shift current, but interprets it as measuring the change in ionic momentum from electron transitions between bands (Pimlott et al., 23 Jul 2025).

The same work expresses the rectified Raman force as

$$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$1

with low-$$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$2 expansion

$$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$3

For linear polarization, the resonant injection and shift coefficients are

$$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$4

Here the injection force is controlled by the band-diagonal EPC difference

$$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$5

whereas the shift force is controlled by the phononic shift vector itself (Pimlott et al., 23 Jul 2025).

This construction changes the ontology of the shift. The relevant quantity is no longer a real-space electronic displacement accompanying absorption, nor a coordinate shift in non-vertical electron-phonon scattering, but a geometric force quantity in phonon-coordinate space. The shift force is impulsive in time, proportional to $$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$6, and occurs in the resonant absorbing regime. The injection force is displacive, arising from the $$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$7 term and proportional to $$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$8 (Pimlott et al., 23 Jul 2025).

4. Gauge invariance, Wilson loops, and geometric structure

All three formulations are built to be gauge invariant, but the mechanism of invariance depends on the underlying parameter space. In the phonon-drag formulation, the relevant derivative operator is $$r_{mn}^{a}=i\langle m|\partial_{k_a}|n\rangle(1-\delta_{mn})$$9, and the phase derivative of $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$0 cancels the gauge dependence of the Berry connections at $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$1 and $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$2. The same paper rewrites the shift vector as the phase derivative of an interband Wilson loop and extends this to phonon drag through

$$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$3

with a Wilson object built from Bloch overlaps and electron-phonon couplings. This makes the quantity manifestly gauge invariant and suited to discretized first-principles evaluation (Wang et al., 2024).

In the many-body formulation, the phase-gradient term is not an ad hoc addition to a transition formula but the coherence contribution carried by the correlated state itself. Under the local $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$4 gauge transformation of the Bloch basis, the many-body amplitude transforms covariantly, and the induced change in $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$5 cancels the corresponding change in the diagonal Berry connections $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$6. The resulting electron-hole shift vector, including the phonon-assisted specialization, is therefore locally gauge invariant. The same work emphasizes, however, that global branch ambiguity remains, as in modern polarization theory; gauge invariance is local, while a polarization branch must be fixed by a consistent reference convention (Hu et al., 22 May 2026).

The displacement-space phononic shift vector has an analogous gauge-covariant structure. Under

$$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$7

the phase of $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$8 changes by $$R_{mn}^{a}(\mathbf{k})=A_m^{a}-A_n^{a}-\partial_{k_a}\arg r_{mn}^{a}$$9, while the diagonal phononic Berry connections transform as $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$0. The combination

$$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$1

is thus gauge invariant (Pimlott et al., 23 Jul 2025).

The geometric interpretation also differs by framework. In the phonon-drag theory, shift vectors are identified with a geodesic-curvature-like quantity, related to Berry curvature and quantum metric, and the paper derives a Gauss-Bonnet-like character

$$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$2

which is quantized to integers. The local phonon-drag shift vector itself is not claimed to be quantized; the quantized quantity is the combined curvature-plus-boundary character (Wang et al., 2024). In the coherent-phonon theory, the shift force is organized by the quantum geometric tensor $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$3, with the linear-polarization shift term taking the form $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$4 and the circular-polarization response coupling chiral phononic shift vectors to both quantum metric and Berry curvature (Pimlott et al., 23 Jul 2025).

5. Physical consequences and model realizations

In phonon-assisted electronic transport, the shift-vector difference between optical and electron-phonon-mediated processes is identified as the driving quantity of phonon-mediated shift current. Within the many-body framework, this follows from interpreting the phonon-assisted shift vector as the intrinsic dipolar content of the corresponding correlated electron-hole state. The same intrinsic dipole also appears as a real-space displacement of the joint probability density and as a linear electric-field correction in energy space, $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$5, although the paper does not develop a separate real-space probability picture for lattice displacements themselves (Hu et al., 22 May 2026).

In the phonon-drag framework, the physical claim is narrower but explicit. The bosonic phonon-drag shift vector plays a role in kinetic relaxation processes of photo-excited electrons and can generate non-vanishing shift currents in thermal equilibrium. This extension is formal and geometric rather than a complete transport theory: the paper does not derive a full phonon-drag conductivity formula with phonon occupation factors, scattering rates, and phase-space integrals (Wang et al., 2024).

In coherent-phonon dynamics, the displacement-space phononic shift vector governs a rectified Raman force rather than a current. The theory identifies injection and shift optical forces as phononic counterparts of the photogalvanic effect. For linear polarization, the shift force is allowed under inversion, forbidden by time-reversal, and also forbidden under $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$6; correspondingly, the paper states that the linear phononic shift vector is odd under $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$7 and even under $$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$8. This is why the resonant shift force requires time-reversal symmetry breaking (Pimlott et al., 23 Jul 2025).

The explicit model application is the bilayer Haldane system with interlayer shear phonons. There the phononic shift vector is proportional to the Haldane gap parameter at small gap, vanishes at zero Haldane gap, grows as time-reversal breaking is turned on, and decreases at larger gap. The shift susceptibility is odd in the Haldane gap, whereas the injection susceptibility is even. The paper also reports strong tunability in both magnitude and direction of the rectified force by varying driving frequency and magnetic flux, with additional higher-energy steps associated with multiband transition channels (Pimlott et al., 23 Jul 2025).

A related but distinct development appears in the theory of shift heat current. There, the geometric object remains the usual electronic shift vector,

$$\mathbf{R}_{mn,\nu}(\mathbf{k},\mathbf{k}')=\mathbf{A}(\mathbf{k}')-\mathbf{A}(\mathbf{k})-D_{\mathbf{k},\mathbf{k}'}\arg g_{mn,\nu}$$9

and the phonon-driven mechanism acts through electron-phonon coupling and virtual electronic excitation. The paper is explicit that even if only the phonons are excited by an external field, the amplitude of the shift heat current is determined by the energy scale of electrons, not of phonons. It therefore extends shift-vector physics to phonon-driven heat transport without defining a phononic shift vector intrinsic to phonon bands (Onishi et al., 2022).

6. Limits, misconceptions, and unresolved distinctions

A common misconception is that any phonon-assisted shift-current formula already defines a phononic shift vector in the strict sense of a geometric quantity intrinsic to phonon bands. The cited works do not support that identification. The generalized many-body theory treats correlated electronic states expanded in a Bloch-product basis; phonons enter through perturbation matrix elements, not as explicit dynamical phonon coordinates in the many-body state, and the paper does not provide a standalone derivation for a pure phonon many-body state or a shift vector built from phonon Berry connections (Hu et al., 22 May 2026). The shift-heat-current theory likewise contains no phonon-band Berry connection, no phonon interband dipole, and no geometric invariant intrinsic to phonons; its geometry remains electronic (Onishi et al., 2022).

A second misconception is that the local phononic or phonon-drag shift vector is itself quantized. The geometric quantization result applies instead to a combined character involving Berry-curvature flux and a boundary integral of the shift vector, analogous to Gauss-Bonnet or Euler characteristic. The local field is not asserted to take integer values (Wang et al., 2024).

The coherent-phonon formulation is the only cited work that explicitly introduces the phononic shift vector as such. Even there, the object is not a property of a phonon normal mode in isolation. It is constructed from electronic Bloch eigenstates, optical velocity matrix elements, and electron-phonon couplings differentiated with respect to lattice displacement. This suggests a specifically electron-phonon geometric quantity in displacement space rather than a standalone phonon-band analog of the conventional electronic shift vector.

Several technical limits remain explicit in the literature. The many-body gauge proof is carried out for nondegenerate bands and Abelian $$\mathbb R^{mn}_{a;\ell}=\{\partial_{u_a}\mathrm{Im}\ln v_\ell^{mn}+\mathbb A_a^{mm}-\mathbb A_a^{nn}\}_{\mathbf u\to0}$$0 gauges; non-Abelian degeneracies are deferred to future work (Hu et al., 22 May 2026). The phonon-drag construction is given a Wilson-loop representation and a quantization formula but no explicit material implementation (Wang et al., 2024). The coherent-phonon theory warns of singular behavior when the optical matrix element $$\mathbb R^{mn}_{a;\ell}=\{\partial_{u_a}\mathrm{Im}\ln v_\ell^{mn}+\mathbb A_a^{mm}-\mathbb A_a^{nn}\}_{\mathbf u\to0}$$1 vanishes, and its practical formula contains intermediate-band denominators $$\mathbb R^{mn}_{a;\ell}=\{\partial_{u_a}\mathrm{Im}\ln v_\ell^{mn}+\mathbb A_a^{mm}-\mathbb A_a^{nn}\}_{\mathbf u\to0}$$2 and $$\mathbb R^{mn}_{a;\ell}=\{\partial_{u_a}\mathrm{Im}\ln v_\ell^{mn}+\mathbb A_a^{mm}-\mathbb A_a^{nn}\}_{\mathbf u\to0}$$3, so band degeneracies require care (Pimlott et al., 23 Jul 2025).

Taken together, these works establish that phononic shift vector is not a single settled term but a family resemblance across three neighboring geometric constructions: the coordinate shift of a phonon-drag scattering event, the intrinsic dipole of a phonon-assisted correlated electronic state, and the displacement-space geometric factor governing resonant rectified Raman forces on coherent phonons. The last of these is the most literal realization of a phononic shift vector, while the former two are best understood as electron-phonon-mediated extensions of shift-vector physics rather than shift vectors of phonon states themselves.