Quasi-particle Interference (QPI)

• QPI is the quantum interference pattern from electron scattering off surface defects, mapping electronic band structure and many-body effects.
• STM/STS techniques capture QPI by measuring the local density of states and employing Fourier transforms to convert real-space modulations into momentum-space insights.
• The analysis of QPI is highly sensitive to defect types, enabling the study of superconducting coherence, inelastic tunneling, and topological signatures in various quantum materials.

Quasi-particle interference (QPI) is the quantum-mechanical standing-wave pattern produced when electronic excitations at a crystal surface scatter from defects and interfere with themselves. In scanning tunneling microscopy and spectroscopy, these modulations appear in the local density of states (LDOS) and, after Fourier transformation, provide access to momentum-space information on band dispersions, constant-energy contours, superconducting gaps, symmetry breaking, spin and orbital selection rules, and defect scattering. QPI has been used in superconducting, topological, Rashba, Dirac, Weyl, moiré, and other quantum materials, but its interpretation depends jointly on the electronic structure, the impurity potential, and the surface projection of the measured states (Butler et al., 2017).

1. Core definition and experimental observable

In the standard surface-scattering picture, quasiparticles with well-defined wave vectors $\mathbf{k}_i$ and $\mathbf{k}_f$ are connected by the momentum transfer

$\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$

Because the incident and scattered waves remain coherent, the LDOS acquires a spatially modulated contribution,

$$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$

and the Fourier-transformed modulation

$$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$

defines the QPI pattern in $\mathbf{q}$-space (Butler et al., 2017).

Scanning tunneling microscopy accesses this quantity through the differential conductance. In the formulations used across the literature surveyed here, $dI/dV(\mathbf{r},V)$ is proportional to the LDOS at energy $E=eV$, and the LDOS itself is obtained from the full Green’s function as

$$N(E,\mathbf{r}) = -\frac{1}{\pi}\,\mathrm{Im}\,\mathrm{Tr}\left[G(\mathbf{r},\mathbf{r},\omega_n)\right]_{i\omega_n\rightarrow E+i0^+}.$$

A conductance map at fixed bias is therefore a real-space image of $\delta\rho(\mathbf{r},E)$; its two-dimensional Fourier transform is the experimentally observed FT-STM or FT-STS QPI map (Knolle et al., 2010).

The physical interpretation is organized by constant-energy contours. Allowed scattering processes connect pairs of states on the same contour $\mathbf{k}_f$0, so dispersing QPI peaks $\mathbf{k}_f$1 encode the underlying quasiparticle band structure. In superconductors, the same logic applies to Bogoliubov quasiparticles and therefore carries information on both the normal-state dispersion and the gap structure (Dutt et al., 2017).

2. Green-function, $\mathbf{k}_f$2-matrix, and reduced-response formulations

The standard microscopic description treats disorder through a $\mathbf{k}_f$3-matrix. In momentum space, the impurity-corrected Green’s function takes the form

$\mathbf{k}_f$4

with the $\mathbf{k}_f$5-matrix determined by a Dyson equation. The QPI response then has the generic structure

$\mathbf{k}_f$6

so that both the clean propagator and the impurity matrix elements enter on equal footing (Knolle et al., 2010).

A widely used simplification replaces the full $\mathbf{k}_f$7-matrix response by a joint-density-of-states or nesting-function construction, schematically

$\mathbf{k}_f$8

This captures the basic kinematic content of scattering between regions of large spectral weight, but it suppresses channel-dependent matrix elements and can miss or misrepresent singular structures (Butler et al., 2017). A systematic refinement is the reduced response function approach, in which the generalized joint density of states appears as the imaginary part of a reduced response function,

$\mathbf{k}_f$9

Within that framework, singular QPI features are tied to solutions of the simultaneous constant-energy conditions together with

$\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$0

which identifies the loci where the two group velocities become parallel or antiparallel (Zhang et al., 2019).

QPI can also be pushed beyond band mapping. In a cuprate-like setting with an interaction-generated self-energy $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$1, the dispersion and width of QPI peaks can, in principle, be used to re-extract the real and imaginary parts of the self-energy along certain nested momentum directions. The same analysis also shows the main limitation: the procedure requires sharp, well-isolated QPI peaks, reliable knowledge of the bare band structure, and restricted momentum dependence of the self-energy over the probed region (Dahm et al., 2013).

For surface states and realistic defects, ab-initio Green-function formulations extend the same logic to first-principles electronic structure. In Bi$\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$2Te$\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$3, such a formalism reproduces the Fourier-transformed QPI images around magnetic and non-magnetic defects and connects the measured $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$4-space structure directly to the scattering properties of topologically protected surface electrons (Rüßmann et al., 2020).

3. Selection rules, coherence factors, and defect specificity

QPI intensity is often interpreted as if defects were generic, isotropic scatterers that illuminate all allowed $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$5-vectors, with missing features then attributed to intrinsic selection rules such as spin-momentum locking or superconducting coherence factors. The accumulated literature shows that this picture is only partially correct.

In multiband nodeless $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$6-wave superconductors, the QPI response near an interband vector $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$7 contains a coherence factor

$\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$8

and the energy dependence near $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i.$9 shows three singularities. Two are associated with the gap scales, and a third depends on both the gaps and band-mismatch parameters. Within that framework, only the singularity near the smallest gap is universal: its sign is positive for $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$0 and negative for $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$1, irrespective of variations in ellipticity, chemical-potential shift, momentum offset, temperature, and coupling strength (Dutt et al., 2017).

At the same time, a different analysis of Fe-based superconductors shows that the “extinction rule” familiar from the cuprate octet model does not carry over to a fully gapped $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$2-wave state. Under experimentally relevant conditions, the resonance condition $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$3 is not satisfied, and the QPI peak around $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$4 can be explained in both $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$5 and $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$6 states; its field-induced suppression can likewise be accounted for in both cases (Yamakawa et al., 2015). A missing or weakened $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$7-vector therefore does not, by itself, establish a sign-changing order parameter.

Odd-frequency superconducting pairing introduces a different and more direct selection rule. In a conventional $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$8-wave superconductor under applied magnetic field, the full QPI response contains even-even, odd-odd, and mixed even-odd anomalous contributions. The defining feature is a bias asymmetry in the QPI peak positions between positive and negative energies, present generically in materials with odd-frequency pairing irrespective of its origin (Chakraborty et al., 2022).

Defect chemistry adds a separate layer of selectivity. On the vacuum-cleaved (001) surface of ZrSiS, impurities on the Zr and S lattice sites generate very different QPI patterns. Zr-site defects predominantly scatter the $$\rho(\mathbf{r},E) = \rho_0(E) + \delta\rho(\mathbf{r},E),$$9 subset of the predominantly Zr $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$0-derived band structure, while S-site defects predominantly scatter the $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$1 subset. The paper explicitly concludes that the “usual assumption of generic scattering centers allowing observations of quasiparticle interference to shed light indiscriminately and isotropically upon the \textit{q}-space of scattering events does not hold,” and that interpretation can be strongly contingent on material defect chemistry (Butler et al., 2017).

4. Superconductivity, density waves, and nematic order

QPI has been especially productive in superconductors and electronically anisotropic phases because it ties energy-resolved scattering vectors to reconstructed band structures and to the phase structure of the order parameter.

In the spin-density-wave phase of iron-based superconductors, a four-band itinerant model with stripe order at $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$2 yields a reconstructed Fermi surface with pronounced $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$3 symmetry. The calculated QPI exhibits quasi-one-dimensional structure because only one electron pocket participates in the SDW while the other remains essentially unreconstructed, and because the participating pocket is elliptic. The result is a pronounced one-dimensional pattern at low energy, a crossover to two-dimensionality on an energy scale set by the SDW gap, estimated to be around $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$4, and a predicted $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$5 rotation of the QPI pattern at sufficiently negative energies around $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$6 (Knolle et al., 2010).

A complementary five-orbital study of iron pnictide nematicity distinguishes magnetic and orbital scenarios through their QPI dimers. In the magnetic scenario, the SDW state produces dimers oriented along the ferromagnetic direction of the SDW order, and these dimer structures persist as the SDW correlation length decreases, although near the Fermi level they tend to merge. In the orbital scenario, the QPI exhibits a dimer structure over a wide energy region, but the dimer undergoes a $$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$7 rotation with increasing energy because the two Dirac nodes generated by orbital order occur at inequivalent energies (Zhang et al., 2016).

In the unconventional metamagnetic compound Sr$$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$8Ru$$\delta\rho(\mathbf{q},E) = \int d^2r\, e^{-i\mathbf{q}\cdot\mathbf{r}}\,\delta\rho(\mathbf{r},E)$$9O$\mathbf{q}$0, a $\mathbf{q}$1-orbital tight-binding model including spin-orbit coupling, bilayer splitting, and the staggered rotation of the RuO octahedra yields a zero-field QPI spectrum with a hollow square-like feature. That structure arises from nesting of the quasi-1D $\mathbf{q}$2 and $\mathbf{q}$3 orbital bands and agrees with the corresponding STM measurements. Rotational symmetry breaking in the nematic metamagnetic state is predicted to appear directly in the QPI spectra as well (Lee et al., 2010).

Within superconductivity proper, QPI can become explicitly phase sensitive once in-gap defect bound states are used as the source of the interference. In LiFeAs, defect bound state QPI shows that in-gap bound states induced by defect scattering are formed from Bogoliubov quasiparticles with significant spatial extent. The phase-referenced Fourier-transform analysis of the bound-state QPI distinguishes the phase of the order parameter and the nature of the defect. The experimental comparison favors an $\mathbf{q}$4 order parameter and indicates that the analyzed native defects are better described as nonmagnetic scatterers (Chi et al., 2017).

5. Topological, moiré, fractionalized, and three-dimensional QPI

QPI is not confined to conventional metals and superconductors. It has been extended to topological surface states, moiré bands, quantum spin liquids, and bulk materials with strong $\mathbf{q}$5 dispersion.

For the topological insulator Bi$\mathbf{q}$6Te$\mathbf{q}$7, the Fourier-transformed QPI around non-magnetic and magnetic defects captures the central topological signature: in the non-magnetic case backscattering is strongly suppressed, whereas magnetic impurities partially reopen that channel. The calculated maps also show that forward and $\mathbf{q}$8-type scattering remain stronger than the backscattering signal even when the impurity is magnetic (Rüßmann et al., 2020). More abstractly, the reduced-response-function analysis shows that global QPI patterns can serve as indicators of topological numbers in gapless systems: odd-even backscattering suppression, channel-dependent hot-arc counting, and rotation of global patterns encode winding information that is robust against complicated local band-structure features (Zhang et al., 2019).

In moiré systems, realistic QPI calculations for twisted bilayer structures show that the moiré supercell greatly reduces the Brillouin-zone size, yet the dominant QPI scattering vectors retain characteristics of the individual monolayers. In twisted bilayer graphene, the principal $\mathbf{q}$9-vectors remain tied to the original Dirac-cone structure, while distinct perturbations from the twisted geometry and the van Hove singularities can be linked back to the electronic structure through unfolded spectral functions (Rhodes et al., 2024).

For a Kitaev quantum spin liquid, the relevant tunneling electron fractionalizes into a bosonic chargon and a fermionic spinon, and the local tunneling conductance around a spin vacancy or localized vison exhibits features associated with fractionalized Majorana fermions, chargons, and visons. In certain parameter regimes, the derivative of the electron LDOS directly tracks the single-spinon LDOS, and the momentum-space QPI allows extraction of the single-spinon density of states and dispersion from a charged tunneling probe (Jahin et al., 2024).

Three-dimensionality introduces a distinct interpretive problem because STM is surface sensitive whereas the electronic structure disperses strongly along $dI/dV(\mathbf{r},V)$0. A continuum surface Green-function method for three-dimensional systems shows that defects at different depths from the surface produce unique sets of scattering vectors, and that these can still be related to the three-dimensional electronic structure of the bulk (Rhodes et al., 2022). A related tomographic analysis of galena demonstrates that the dominant QPI signal at a given bias is controlled by bias-dependent cuts through the 3D electronic structure, enabling a form of quasiparticle tomography of the hidden dimension and, with orbital decomposition, of the orbital character of the contributing bands (Marques et al., 2021).

6. Interpretation, misconceptions, and current directions

Several recurrent misconceptions now have explicit counterexamples. First, missing QPI vectors do not automatically imply symmetry-forbidden scattering. They can also arise because the available defects do not couple efficiently to the relevant bands, as in ZrSiS (Butler et al., 2017). Second, in multiband superconductors a field dependence or a $dI/dV(\mathbf{r},V)$1-centered peak does not by itself resolve $dI/dV(\mathbf{r},V)$2 versus $dI/dV(\mathbf{r},V)$3, as shown in Fe-based systems (Yamakawa et al., 2015). Third, the elastic-tunneling picture is not always sufficient.

LiFeAs provides a direct example of the last point. Experimental QPI maps and the extracted dispersions show replica features that track the elastic QPI dispersions but are shifted in energy. Their comparison with theory identifies an inelastic mode with a resonance between $dI/dV(\mathbf{r},V)$4 and $dI/dV(\mathbf{r},V)$5, and comparison with inelastic neutron scattering attributes the replica features to inelastic tunneling through spin fluctuations (Chi et al., 15 Aug 2025). This establishes that QPI can probe not only static band structure and order-parameter phase but also electron-boson coupling.

A plausible implication is that future QPI analysis must treat the measured signal as the product of several filters: the surface projection of the states, the real-space form and depth of the defects, the orbital and spin matrix elements of the impurity potential, and, where relevant, inelastic tunneling channels. The literature already points toward three practical consequences. Defect identification becomes an essential part of QPI interpretation (Butler et al., 2017). In three-dimensional materials, depth-dependent and $dI/dV(\mathbf{r},V)$6-resolved modeling is not optional (Rhodes et al., 2022). And along carefully nested directions, QPI remains a possible probe of interaction effects such as self-energy renormalization, albeit with significant restrictions on linewidth, band knowledge, and peak isolation (Dahm et al., 2013).

QPI therefore occupies a hybrid position among spectroscopies. It is a real-space interference phenomenon, but its content is intrinsically momentum-space and increasingly many-body. Its most mature formulations no longer treat impurities as generic or the surface as a passive window. Instead, QPI is a defect-conditioned, channel-dependent, and sometimes inelastic probe whose strength lies precisely in that sensitivity.