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Staggered loop currents should be directly visible in scanning tunneling spectroscopy: a 10–20 meV partial gap in the density of states and, in the presence of spin-orbit coupling, a 15–20% spin polarization detectable by spin-polarized STM

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2026-08-01 10:59 UTC pith:J3QJCCND

load-bearing objection A solid, practice-oriented paper that turns loop-current order into concrete STM/SP-STM predictions; the main caveat is the linear scaling of signal size with the uncertain moment-to-current conversion. the 2 major comments →

arxiv 2607.20030 v1 pith:J3QJCCND submitted 2026-07-22 cond-mat.str-el

How to measure loop currents in scanning tunneling microscopy

Victoire Morisseau , Luke C. Rhodes , Peter Wahl , Carolina A. Marques This is my paper
classification cond-mat.str-el
keywords loop current orderscanning tunneling microscopyquasiparticle interferencespin-polarized STMcuprate pseudogapSr2RuO4 surfacevan Hove singularityspin-orbit coupling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that the loop-current ordered states proposed for the cuprate pseudogap and the surface of Sr2RuO4 leave specific, measurable traces in STM. In a square-lattice tight-binding model with staggered loop currents at ordering vector q=(1,1), the complex hopping term folds the Brillouin zone and splits the Van Hove singularity at the zone corner, opening a partial gap of roughly 10–20 meV for the ordered moments (0.01–0.1 μB) reported from neutron and muon experiments. In a multiband model of the Sr2RuO4 surface that includes spin-orbit coupling, the loop currents split the edges of the SOC-induced hybridization gap by about 2 meV and generate a staggered spin polarization of order 15–20%, which spin-polarized STM should detect. The paper thus converts a debated ordering scenario into concrete, parameter-dependent predictions—including the minimum loop-current strength needed for a signal—so that STM experiments can confirm or rule out this specific staggered loop-current geometry.

Core claim

The central claim is that a staggered loop-current order with ordering vector q=(1,1) is not spectroscopically invisible. Adding an imaginary hopping i tLC to bonds makes the nearest-neighbor hopping complex, doubles the unit cell, and hybridizes bands at the Brillouin-zone corner, splitting the Van Hove singularity. For reported local moments m ≈ 0.01–0.1 μB, the implied tLC = 2.6–26 meV yields a partial density-of-states gap of order 10–20 meV in both a one-band cuprate model and a multiband model of the Sr2RuO4 surface. With atomic spin-orbit coupling, the same orbital currents couple to spin and split the edges of the SOC-induced hybridization gap by about 2 meV, producing a spin polariz

What carries the argument

The central object is the imaginary nearest-neighbor hopping term H_LC = i tLC Σ (−1)^{n+m}(c†_{0,0} c_{n,m} + h.c.), which turns the hopping amplitude t0 into t0 e^{i φLC} with tLC = t0 sin φLC. It carries the loop-current order: it breaks time-reversal and mirror symmetry, folds the Brillouin zone at q=(1,1), splits the Van Hove singularity, and—through spin-orbit coupling—converts the orbital angular momentum of the loop currents into a spin-splitting and a staggered spin polarization. The paper anchors all signal sizes by converting reported magnetic moments m into tLC through the classical dipole relation m μB = I·A, with I = (e/ℏ) tLC and A = 14.4 Ų.

Load-bearing premise

The load-bearing premise is that the reported magnetic moments m map onto the loop-current hopping through the classical dipole formula m μB = I·A with a fixed loop area of 14.4 Ų; if the moments are not purely orbital or the effective loop area differs, all predicted gap sizes and spin polarizations shrink in proportion and could fall below detectability, and the model also assumes one specific staggered full-unit-cell geometry while neglecting self-energy corrections.

What would settle it

Measure the tunneling conductance of an underdoped cuprate or the Sr2RuO4 surface at the energy of the dxy van Hove singularity with ~1 meV energy resolution while cooling through the proposed loop-current transition: if no partial gap of order 10–20 meV develops, or if spin-polarized STM finds no staggered spin polarization above a few percent at the SOC gap edges, then the specific q=(1,1) full-unit-cell staggered loop-current order considered here is ruled out for the reported moment range.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • In a one-band cuprate model, staggered loop currents with m ≈ 0.01 μB split the Van Hove singularity and open a partial density-of-states gap of order 20 meV, clearly resolvable in STS at 4.2 K.
  • In the Sr2RuO4 surface model, the same order splits the dxy Van Hove singularity and shifts the strongest quasiparticle-interference features at q=(0,0) and (0.5,0.5) to energies offset from the no-loop-current case.
  • With spin-orbit coupling, loop currents split the edges of the SOC-induced hybridization gap by about 2 meV and produce a spin polarization of order 15–20% in a staggered spatial pattern, detectable by spin-polarized STM.
  • The spin polarization stays below ~15% yet remains detectable for ordered moments as small as about half of the reported values, setting a concrete lower bound for observability.
  • The partial gap itself is not a unique fingerprint of loop-current order, so the paper emphasizes the energy-shifted QPI features and the spin-polarization pattern as the more specific signatures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mapping from measured moment m to loop-current hopping tLC assumes the moment is purely orbital and that the current encircles the full 14.4 Ų Cu/Ru plaquette; if either is wrong, the predicted gap and polarization scale linearly with tLC and could fall below detection, so the numerical thresholds should be read as order-of-magnitude estimates.
  • The predicted checkerboard spin pattern shares its periodicity with the checkerboard charge order already observed at the Sr2RuO4 surface; a spin-resolved STM experiment could determine whether that charge order is spin-polarized or a separate phenomenon.
  • Because the paper neglects self-energy corrections, correlation effects could renormalize the 10–20 meV gap and the 15–20% polarization; comparing predicted and measured spectra near the van Hove energy would quantify this renormalization.
  • For loop-current orders that break inversion symmetry instead of mirror symmetry, the same mechanism would produce a staggered Rashba-type spin splitting, giving a distinct signature testable with the same experimental approach.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper provides concrete STM predictions for staggered loop-current order in square-lattice models inspired by cuprates and the Sr2RuO4 surface. Loop currents are introduced as an imaginary hopping tLC, calibrated to reported ordered moments m via a classical dipole formula (Eq. 4). Using continuum Green's-function cLDOS/QPI simulations, the authors show a Van Hove singularity splitting at the zone corner leading to a partial gap of ~10–20 meV for m = 0.01–0.1, additional QPI scattering, and, when spin-orbit coupling is included, a spin-splitting of the SOC-induced hybridization gap edges yielding a spin polarization of ~15–20% detectable by SP-STM. Results are presented for a one-band cuprate model and a multiband Sr2RuO4 surface model, with all computational parameters stated.

Significance. If the predicted signatures are correct, the paper fills a clear gap by giving experimentalists testable criteria for a controversial order. The central band-folding mechanism is symmetry-based and robust, and the calculations are transparent: model parameters, impurity potential, broadening, and k-grids are all specified, and the simulation code (CalcQPI) is openly available. The paper also makes a falsifiable comparison to existing Sr2RuO4 QPI data. The main quantitative bridge from measured moments to tLC, however, is not yet accompanied by an uncertainty analysis, and the spin-polarization number is quoted inconsistently.

major comments (2)
  1. [Eq. (4) and surrounding paragraph] The conversion m μB = I·A with I=(e/ℏ)tLC and A=14.4 Ų is the quantitative bridge for all predictions. It assumes a purely orbital moment, a single full unit-cell loop area, and the point-dipole approximation. Because ΔE and P_z scale linearly with tLC, a factor of 5–10 error (e.g., a 20% orbital fraction) would reduce the predicted gap to a few meV and P_z to a few percent, weakening the claim that the signatures are 'clear'. Please add an explicit sensitivity analysis and state how non-orbital contributions or a different effective loop area would change the predicted observables.
  2. [Abstract/Conclusion and Fig. 4] The spin-polarization magnitude is reported as ~15% in the main text and Fig. 4, but as 'on the order of 20%' in the abstract and conclusion. The maximum P_z in Fig. 4(f) appears to saturate near 15%. Since this number is central to the SP-STM detectability claim, the authors should reconcile the inconsistency and quote the actual computed maximum.
minor comments (5)
  1. [General] Define 'Van Hove singularity' (VHs) at first use; the abbreviation is used without definition in Fig. 2 and elsewhere.
  2. [Main text / S1] Define 'cLDOS' in the main text; it is introduced in S1 but used in Fig. 3 before definition.
  3. [Eqs. (3)–(4)] Clarify the notation: in Eq. (3) m is a number such that m μB is the moment, while Eq. (4) writes m directly; specify that m is in units of μB.
  4. [Fig. 4(f)] The sentence 'The spin-polarization stays below ~15%' should specify that this is the maximum over energy for each m; otherwise it reads as a statement about the functional form.
  5. [Conclusions, QPI discussion] The phrase 'extra scattering vectors at q=(0,0) and (1,0)' could confuse readers because q=(0,0) is the trivial scattering channel; clarify that the authors mean dispersive features near q=0, as shown in the energy cuts.

Circularity Check

0 steps flagged

No significant circularity: the paper forward-models STM signatures from externally reported magnetic moments without fitting to the target STM data.

full rationale

The paper's derivation chain is self-contained: experimentally reported ordered moments m (from neutron diffraction and muSR, refs. [11,25,26]) are converted to a loop-current hopping amplitude tLC via the classical dipole relation of Eq. (4); this tLC is then used as a parameter in a tight-binding Hamiltonian to compute band structures, density of states, QPI maps, and spin polarization. The predicted partial gap (10–20 meV) and spin polarization (~15–20%) are computed outputs of the model, not fitted to the target STM observables, and no STM data are used to define or refine m or tLC. The self-citations (CalcQPI code, prior Sr2RuO4 tight-binding models, prior experimental QPI/checkerboard data) are methodological or comparative inputs, not load-bearing justifications for the central prediction. The main uncertainty—whether Eq. (4) correctly converts measured moments into orbital currents—is a modeling assumption, not circularity, because the predicted quantities are not equivalent by construction to the input moments. There is no imported uniqueness theorem, no ansatz hidden in a self-citation, and no renaming of a known result as a new derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The predictions rest on translating experimental magnetic moments into a hopping parameter via a classical dipole formula and on a specific staggered loop-current geometry borrowed from prior literature. The model also assumes loop currents can be represented as imaginary hoppings and neglects self-energy effects. These are not invented entities, but they are load-bearing modeling choices.

free parameters (3)
  • tLC (loop-current hopping) = 2.6–26 meV (corresponding to m = 0.01–0.1 μB)
    Central control parameter. Determined from reported experimental ordered moments via Eq. (4), then varied across this range to compute spectra.
  • A (loop area) = 14.4 Ų
    Approximate area enclosed by a Cu/Ru square with bond length ~3.8 Å. Used in the dipole formula m μB = I·A; a geometric input.
  • λ (spin-orbit coupling) = 0.15 eV
    Atomic SOC at the Ru site, a standard value from prior DFT/literature; needed for the spin-splitting prediction.
axioms (5)
  • domain assumption The loop current order is a staggered full-unit-cell pattern (q=(1,1)) with opposite current directions in adjacent cells, as in Chakravarty et al. (ref. 4).
    This specific geometry determines where the VHs splitting appears and the symmetry of the resulting spin polarization. Other intra-unit-cell geometries would give different signatures, as the authors note.
  • domain assumption The magnetic moment of a loop current is given by the classical dipole relation m μB = I·A with I = (e/ℏ) tLC.
    Used to translate reported moments into the hopping parameter tLC. This is a simplified estimate; the paper does not derive it from a microscopic model.
  • domain assumption Loop currents can be modeled by adding an imaginary nearest-neighbor hopping while preserving |t| (HLC in Eq. 1).
    Standard representation of orbital currents in tight-binding models; assumed without derivation.
  • domain assumption Self-energy effects accompanying the loop current order are neglected.
    Explicitly stated in the text: 'We here neglect possible self energy effects...' This could modify the spectral functions and the quantitative size of the predicted gap and spin polarization.
  • domain assumption Reported bulk or surface magnetic moments in cuprates and Sr2RuO4 correspond to the surface layer modeled here.
    The model for Sr2RuO4 is a free-standing layer; the experimental moments come from μSR/neutron measurements that may sample different regions.

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Pith. "Pith review of How to measure loop currents in scanning tunneling microscopy." pith.science (2026). https://pith.science/paper/J3QJCCND

@misc{pith2026260720030,
  author       = {Pith},
  title        = {Pith review of: How to measure loop currents in scanning tunneling microscopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3QJCCND}},
  note         = {Machine review of arXiv:2607.20030}
}
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read the original abstract

The emergence of loop current phases, where spontaneous loops of orbital currents give rise to a weak local magnetic moments, has been proposed to exist in a number of quantum materials based on measurements that pick up weak signatures of time reversal symmetry breaking or small magnetic moment order. The most prominent example is as an explanation of the pseudogap phase on the underdoped side of the phase diagram of the high-temperature cuprate superconductors, but more recently, it has been proposed to occur in Kagome materials and at the surface layer of Sr$_2$RuO$_4$. Experimental results have, however, been inconclusive so far, some detecting signatures that can be understood as emerging due to loop current phases, whilst others have not detected any significant proof. One of the techniques that should be able to pick up local signatures of loop current orders is low temperature scanning tunneling microscopy and spectroscopy (STM/STS), however firm predictions of how to detect them are missing. Here, we provide specific predictions for how loop current orders in a square lattice can be seen in spectroscopic maps, using models of the cuprate high-temperature superconductors and of the surface layer of Sr$_2$RuO$_4$. We find that, besides lifting degeneracies at the specific ordering vector of the loop current order, a finite spin polarisation emerges when spin-orbit coupling is present, signatures of which can be detected in spin-polarised STM.

Figures

Figures reproduced from arXiv: 2607.20030 by Carolina A. Marques, Luke C. Rhodes, Peter Wahl, Victoire Morisseau.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) shows the resulting electronic structure. The impact of the loop currents on the electronic structure is most clearly seen at the zone corner, at the X-point, where the dxy Van Hove singularity splits proportionally to the strength of the loop currents. As in the cuprate model, this splitting results in a partial energy gap in the density of states, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a). The introduction of the loop current order re￾sults in a small splitting of the edges of the SOC-induced hybridization gap, inset in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 1, 2026.