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Quasiparticle interference as a tool to study quantum materials

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Pith's one-line read Quasiparticle interference has matured into a quantitative probe of quantum materials — able to map both occupied and unoccupied electronic states, at sub-100 mK temperatures and in magnetic fields, and to read the symmetry of superconducti

desk verdict A comprehensive, honest review of QPI that deserves to become a standard reference; the central argument holds, and the main weakness (inelastic tunneling) is openly acknowledged. read the letter →

arxiv 2607.22487 v1 pith:J6SRMECZ submitted 2026-07-24 cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-scicond-mat.supr-con

Luke C. Rhodes , Yuhki Kohsaka , Tetsuo Hanaguri , Carolina de Almeida Marques , Peter Wahl This is my paper · ORCID
classification cond-mat.str-elcond-mat.mes-hallcond-mat.mtrl-scicond-mat.supr-con
keywords quasiparticleinterferencescanningtunnelingmicroscopyelectronicstructuresuperconductingorderparameterselectionrulesGreen'sfunctionT-matrixheavyfermions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review argues that quasiparticle interference (QPI) imaging in a scanning tunneling microscope has matured into a quantitative electronic-structure probe, complementary to ARPES and quantum oscillations. QPI uses defects as scattering centers: quasiparticles scattered off impurities create standing-wave modulations of the local density of states, and the Fourier transform of those modulations reveals the band structure, spin texture, and superconducting gap symmetry. The paper establishes a unified theoretical framework — Green's function and T-matrix scattering theory — and shows how selection rules (spin, time-reversal, spin-orbit, coherence factors, and velocity) determine which scattering vectors appear. It surveys applications from noble-metal surfaces to cuprates, iron-based superconductors, heavy-fermion compounds, and putative triplet superconductors. The central claim is that, with continuum modelling and low-temperature instrumentation, QPI can deliver momentum-resolved information, including phase sensitivity of the superconducting order parameter, that is otherwise difficult to obtain.

What carries the argument

The central machinery is the T-matrix (or Green's function) description of a point impurity in a periodic potential, combined with the low-temperature relation dI/dV ∝ ρ_s(eV) for elastic tunneling. The T-matrix formalism yields the QPI response as a product of two Green's functions, so scattering vectors q connecting states with substantial spectral weight at the same energy dominate; the velocity selection rule further favors states with antiparallel group velocities. Selection rules are encoded in matrix elements: spin selection suppresses scattering between opposite spin states; time-reversal-odd impurity potentials cancel in the Born approximation; spin-orbit scattering introduces a σ·(

What would settle it

A direct test would be to measure QPI in a strongly correlated material at base temperature and compare the energy dependence of sharp QPI features with a model that includes only elastic single-particle scattering. If inelastic spin-fluctuation tunneling is significant, the QPI contrast near the Fermi energy should deviate from the predicted dI/dV = C ρ_s(eV) form and show replica features or anomalous broadening; alternatively, verifying the predicted setpoint constraint (Eq. 12) by recording maps at different setpoint voltages would confirm or refute the setpoint-effect interpretation.

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Extended reading notes

Core claim

The paper's central claim is that QPI is now a quantitative, complementary technique to ARPES and quantum oscillations, with three unique advantages: it probes occupied and unoccupied states, works below 100 mK and in high magnetic fields, and is one of only two techniques able to provide momentum-resolved information about the symmetry of a superconducting order parameter. The argument rests on the T-matrix description of impurity scattering, which shows that the Fourier-transformed differential conductance is governed by scattering vectors connecting states on constant-energy contours, modulated by internal selection rules. In superconductors, coherence factors make QPI phase-sensitive: ti

Load-bearing premise

The analysis relies on the relation dI/dV ∝ ρ_s(eV), which holds only if the tip density of states and tunneling matrix element are energy independent and elastic tunneling dominates; the paper itself notes that inelastic spin-fluctuation tunneling can be large in correlated materials and may limit sharp QPI features near the Fermi energy.

Editorial extensions

If this is right

  • If QPI is quantitative, it can be used as a routine band-structure tool at energy scales and temperatures inaccessible to ARPES, including unoccupied states and sub-100 mK regimes.
  • The selection rules imply that QPI can map spin-orbital texture in Rashba systems, topological insulators, and Weyl semimetals, and can distinguish sign-changing from sign-preserving superconducting order parameters via coherence-factor contrast.
  • The setpoint effect produces non-dispersive artifacts; the paper's Eq. (12) gives a testable constraint — any finite-q modulation at one energy must be compensated within the setpoint window.
  • Phase-referenced QPI, comparing modulations at ±V, provides a field-free method to extract the sign structure of the superconducting gap, which has been demonstrated in cuprates and Fe(Se,Te).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If inelastic tunneling from spin fluctuations is as large as the paper suggests in correlated materials, then sharp QPI features near the Fermi energy may be systematically broadened; this would mean the 'disentangling' the conclusion calls for must include a bosonic background, not just single-particle bands.
  • The velocity selection rule could be tested more directly: in a system with known Fermi surface, one could predict which scattering vectors are suppressed (parallel group velocities) and verify their absence, as was done in bilayer graphene.
  • A natural extension is to use the setpoint-effect constraint (Eq. 12) as a diagnostic for mirage features: any QPI feature that disappears under the conductance-ratio normalization is likely setpoint-induced, which could help re-evaluate reports of checkerboard electronic crystals.
  • Combining spin-polarized or superconducting tips with QPI could yield separate maps of spin and orbital character, turning the technique into a spin-resolved band-structure probe beyond current ARPES capabilities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. This review article surveys quasiparticle interference (QPI) imaging with scanning tunneling microscopy as a probe of the low-energy electronic structure of quantum materials. It covers the experimental basis of QPI, the elastic-tunneling approximation underlying Eq. (3), the setpoint effect and its correction, the T-matrix scattering formalism, selection rules from spin, orbital, spin-orbit, and superconducting coherence factors, numerical modeling approaches from joint density of states to continuum Wannier-based simulations, and a broad range of applications from noble-metal surfaces to topological insulators, graphene, cuprates, iron-based superconductors, heavy-fermion systems, and putative triplet superconductors. The review argues that QPI has matured into a quantitative technique that complements ARPES and quantum oscillations and can provide momentum-resolved information about superconducting order-parameter symmetry. I checked the key internal derivation of Eq. (12) from Eq. (11); it is correct within the stated constant-current setpoint model, and the ratio and Feenstra-function normalizations in Eqs. (13)–(15) are also internally consistent.

Significance. If taken at face value, this review provides a valuable, up-to-date synthesis of a technique that has grown from a surface-state curiosity into a quantitative probe of correlated electron systems. Its strengths are the clear derivation of the T-matrix QPI response, the systematic catalog of selection rules, the emphasis on the continuum Wannier transformation for realistic STM modeling, and the candid discussion of limitations. The authors explicitly acknowledge the most serious caveat — inelastic spin-fluctuation tunneling, which can reach 30–50% conductance changes and may produce replica features (Secs. III B and VII C). Because this limitation is stated rather than hidden, it does not overturn the review's central claim, though it tempers the word 'routine' in the abstract. The paper also benefits from references to publicly available simulation code (Refs. 68–69) and from cross-checks against ARPES and quantum-oscillation data in several materials.

minor comments (5)
  1. [Sec. V A] The sentence 'mathematically it is not possible to derive the JDOS expression Eq. (36) under any algebraic approximation' refers to the wrong equation: Eq. (36) is the T-matrix QPI response, not the JDOS expression. The intended reference should be Eq. (84) or (85). This is a typo, but it will confuse readers who try to verify the claim.
  2. [Abstract vs. Secs. V D, VII C, VII E] The abstract states that 'recent theoretical progress now enables routine modelling of QPI,' but the text itself notes that the impurity potential is the least controlled parameter (Sec. V D), that inelastic tunneling may limit sharp QPI features near the Fermi energy (Sec. VII C), and that calculated QPI often lacks quantitative agreement because of impurity distributions and self-energy effects (Sec. VII E). I suggest softening 'routine' to 'increasingly routine' or adding a qualifier such as 'for a range of quantum materials,' to match the body of the review.
  3. [Sec. VII C] The discussion of inelastic tunneling is honest but stops short of giving the reader a practical criterion for when Eq. (3) can be trusted. A short quantitative statement — for example, an order-of-magnitude threshold for the inelastic-to-elastic conductance ratio, or an energy window in which the elastic approximation is expected to hold — would make the caveat more actionable without requiring new theory.
  4. [Sec. III E, Eq. (14)] The conductance ratio Z(r,V) is stated to 'largely cancel' the setpoint normalization. Under the model assumptions used to derive Eq. (11), the cancellation is exact, because the setpoint denominator is independent of the sign of V. The qualifier 'largely' is unnecessarily weak and could be replaced by 'exactly within the stated model assumptions.'
  5. [Sec. VI E 2] The magnetic-field BQPI analysis is described as phenomenological and lacking microscopic support (Refs. 51–52 issue). This is a fair self-critique, but the following paragraph already points to the two-band theory of Ref. 51 as a more robust route. Consider making that connection explicit in the earlier discussion, since it directly addresses the stated limitation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the review's claims rest on standard tunneling theory and external experimental results, with explicit caveats.

full rationale

The paper is a review and does not derive a new prediction from fitted inputs. Its central mapping, Eq. (3) g(V) ∝ ρ_s(eV), is presented as a standard consequence of the Bardeen/Tersoff–Hamann tunneling expression (Eq. (1)) under stated assumptions (energy-independent tip DOS and matrix element, elastic tunneling), and the paper explicitly identifies these assumptions and later flags (Sec. VII C) that inelastic spin-fluctuation tunneling can be large and complicate QPI interpretation. No parameter is fitted to data and then renamed a prediction; the octet-model gap extraction uses measured scattering-vector dispersions and is cross-checked against ARPES, not derived from itself. The many self-citations (e.g., Refs. 32, 47, 65, 89, 98, 135, 171) are supporting published experimental and modelling results, and the theoretical framework (Green's functions, T-matrix, continuum Wannier transformation) is set out in the text with equations rather than imported as an unverified premise. There is no uniqueness theorem or ansatz smuggled in via self-citation. The acknowledged inelastic-tunneling caveat weakens the breadth of the central claim but does not make the argument circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The review makes no new postulates. Its central claim rests on standard condensed-matter assumptions (weak scattering, elastic tunneling, point defects). The free parameters listed are those used in the modeling approaches the review describes, not parameters fit to a new result.

free parameters (3)
  • Wannier orbital radius = not specified
    Sec. V C: 'This leaves the radius of these wavefunctions as a free parameter, however suitable values can be chosen based on the distribution of hopping parameters.' The choice affects cLDOS simulations.
  • Impurity potential V(r) = not specified
    Sec. V C: 'The impurity potential is typically the least controlled parameter.' It is estimated from DFT or chosen ad hoc in many QPI models, and the QPI pattern depends on its momentum dependence (Eq. 38).
  • Phenomenological broadening η = not specified
    Eq. (40) introduces η as a phenomenological lifetime broadening, often chosen by hand to match experimental linewidths.
assumptions (4)
  • domain assumption Elastic tunneling dominates; tip DOS and tunneling matrix element are energy independent (Eq. 3)
    Used throughout to identify dI/dV with ρ_s(eV); the authors themselves note inelastic contributions can be significant (Sec. VII C).
  • domain assumption Point-like impurity and Born or T-matrix approximation for scattering
    The entire QPI response is derived from a single point defect with a weak potential (Sec. IV A-B); the Born approximation (Eq. 31) is used for selection rules.
  • domain assumption Host time-reversal symmetry for the cancellation rule
    Sec. IV D 2 assumes a time-reversal-symmetric host to derive cancellation of time-reversal-odd scattering.
  • domain assumption Single-band spin-singlet superconductor for Table I
    Sec. IV D 4 derives selection rules for a reduced two-component Nambu basis and states the single-band, singlet, real-gap simplification for Table I.

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Cite this review

Pith. "Pith review of Quasiparticle interference as a tool to study quantum materials." pith.science (2026). https://pith.science/paper/J6SRMECZ

@misc{pith2026260722487,
  author       = {Pith},
  title        = {Pith review of: Quasiparticle interference as a tool to study quantum materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6SRMECZ}},
  note         = {Machine review of arXiv:2607.22487}
}
abstract

To understand the properties of quantum materials a detailed knowledge of the material's low energy electronic structure is key. Details of the electronic structure drive the ground state through electronic instabilities, electronic correlation effects, new electronic orders or just the absence of electronic states near the Fermi energy - making a realistic and detailed understanding crucial to be able to control and design properties of quantum materials. The past 25 years have seen a significant improvement in experimental techniques to observe the true electronic structure, in particular in techniques such as Angle resolved photoemission spectroscopy (ARPES) where energy resolutions of 2meV are routinely achievable now, which however is limited to zero magnetic field and only provides information about the occupied states. Scanning tunneling microscopy (STM) achieves a significantly better energy resolution <100${\mu}$eV and can operate at temperatures well below 50mK and in magnetic fields. While per se a real-space technique, by imaging quasiparticle interference (QPI) STM can also provide information about the electronic structure. This technique has been used over the past decades to study a wide range of quantum materials to understand correlated electron behaviour. Recent theoretical progress now enables routine modelling of QPI, a key requirement to interpret the complex data. Here, we review the principles of QPI, its origin, experimental detection, and the physical insight gained from the study of QPI and possible future directions for this technique.

Figures

Figures reproduced from arXiv: 2607.22487 by the authors.

Figure 1
Figure 1. FIG. 1. First experiments on QPI in noble metal (111) surface states. (a) Topographic [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the working principle of STM. (a) Sketch of the measurement geometry [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Measurement of Quasi-particle interference. (a) Topographic STM image [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Signal from noise. (a) synthetic conductance map [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Linear transformation and Lawler-Fujita algorithm. (a) Constant-current topography, [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Unfolding procedure. (a) FT of a map layer of a differential conductance map [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a, b) Simulated real space image of the LDOS (a) and differential conductance (b) for [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of the JDOS, FT of the LDOS using a dLDOS and FT of the LDOS using [PITH_FULL_IMAGE:figures/full_fig_p044_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. QPI imaging for a simple nearest-neighbour tight-binding model. (a) Real space image [PITH_FULL_IMAGE:figures/full_fig_p047_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Quasi-particle interference imaging of the surface state of Ag(111). (a) shows [PITH_FULL_IMAGE:figures/full_fig_p049_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Spectroscopic signatures in Rashba systems. (a) A d [PITH_FULL_IMAGE:figures/full_fig_p055_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Quasi-particle interference imaging of a topological insulator. (a) JDOS, (b) ex [PITH_FULL_IMAGE:figures/full_fig_p058_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Quasi-particle interference imaging of a Weyl semimetal. (a) Constant energy contour [PITH_FULL_IMAGE:figures/full_fig_p060_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Quasi-particle interference imaging of graphene. (a) Topographic STM image of a [PITH_FULL_IMAGE:figures/full_fig_p062_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Schematic illustration of the Bogoliubov quasiparticle dispersion near one of [PITH_FULL_IMAGE:figures/full_fig_p065_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. (a) Experimentally observed BQPI pattern in Bi [PITH_FULL_IMAGE:figures/full_fig_p066_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Superconducting-gap dispersions in multi-band iron-based superconductors deduced [PITH_FULL_IMAGE:figures/full_fig_p067_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Phase-sensitive BQPI experiments. (a) The sign structure (gray and white [PITH_FULL_IMAGE:figures/full_fig_p070_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. QPI in the heavy fermion material URu [PITH_FULL_IMAGE:figures/full_fig_p074_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Quantum confinement at the surface of URu [PITH_FULL_IMAGE:figures/full_fig_p075_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. QPI at the surface of PdCoO [PITH_FULL_IMAGE:figures/full_fig_p077_21.png]

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