Autocorrelated ARPES in Cuprate Superconductors

• Autocorrelated ARPES is a technique that convolves ARPES intensity over momentum space to reveal preferred momentum transfers and a unique octet scattering pattern.
• It identifies sharp peaks corresponding to hot spots on the reconstructed Fermi surface, with clear energy and doping dependencies in cuprate superconductors.
• This method complements conventional ARPES by encapsulating many-body renormalization effects, thereby offering actionable insights into quasiparticle scattering and electron correlation phenomena.

Searching arXiv for recent and foundational papers on ARPES autocorrelation and related ARPES methodology. Autocorrelated angle-resolved photoemission spectroscopy refers to the analysis of ARPES intensity through a momentum-space self-convolution at fixed energy, producing an autocorrelation function whose peaks identify preferred momentum transfers between regions of large spectral weight. In the cuprate literature, ARPES autocorrelation is formulated as a direct fingerprint of strong-correlation-driven electron Fermi surface reconstruction: when low-energy weight is concentrated near hot spots, the autocorrelation develops sharp peaks at the wave vectors connecting those hot spots, yielding an octet scattering pattern with a well-defined energy and doping dependence (Tan et al., 2020).

1. Formal definition within the ARPES framework

ARPES is a direct probe of electronic structure with energy- and momentum-resolved information. In the standard spectral-function description, the measured intensity is written as

$$I(\mathbf{k},\omega)=I_0(\mathbf{k},\nu,\mathbf{A})\,A(\mathbf{k},\omega)\,f(\omega),$$

or, in the cuprate autocorrelation formulation,

$$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$

with the matrix element treated as essentially constant in the latter case. The single-particle spectral function is therefore the central object, encoding both renormalized dispersion and finite lifetime broadening through the self-energy [(Zhang et al., 2022); (Park et al., 2010)].

The ARPES autocorrelation is then defined as

$$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$

with $\mathbf k$ restricted to the first Brillouin zone. This is the momentum-space self-convolution of the ARPES intensity at fixed energy. Physically, it measures the joint density of states for pairs of states separated by $\mathbf q$, so it becomes large when many intense ARPES states are connected by the same momentum transfer (Tan et al., 2020).

Because the underlying ARPES map is itself a product of the occupied spectral function and matrix-element weighting, autocorrelation does not replace spectral analysis; it reorganizes the same information into momentum-transfer space. In that sense, it is complementary to direct inspection of energy distribution curves and momentum distribution curves, which remain the conventional routes for extracting dispersions and linewidths from ARPES spectra (Zhang et al., 2022).

2. Microscopic formulation in electron-doped cuprates

The most explicit development of ARPES autocorrelation in the supplied literature is for electron-doped cuprate superconductors. There the calculation is based on the square-lattice $t$-$J$ model, with $t<0$ and $t'<0$ for electron doping and the on-site constraint

$\sum_\sigma C^\dagger_{l\sigma}C_{l\sigma}\ge 1.$

A particle-hole transformation,

$$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$0

together with the charge-spin separation fermion-spin representation,

$$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$1

rewrites the problem in terms of constrained hole operators and spins, with

$$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$2

Superconductivity is treated within the kinetic-energy-driven mechanism, in which charge carriers exchange spin excitations, generating a $$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$3-wave charge-carrier pairing state and, after charge-spin recombination, a $$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$4-wave electron superconducting state (Tan et al., 2020).

Within that framework, the electron spectral function is written as

$$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$5

Here $$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$6 is the renormalized quasiparticle dispersion and $$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$7 is the quasiparticle scattering rate. The autocorrelation analysis is therefore not independent of many-body renormalization; it is built from the same $$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$8 and $$I(\mathbf k,\omega)=|M(\mathbf k,\omega)|^2 n_F(\omega) A(\mathbf k,\omega),$$9 that determine the measured ARPES intensity (Tan et al., 2020).

This formulation gives ARPES autocorrelation a precise status: it is a derived observable constructed from a many-body spectral function, rather than a purely geometric transform of noninteracting bands. That distinction is essential in strongly correlated systems, where the momentum distribution of spectral weight is governed by the self-energy.

3. Reconstructed electron Fermi surface, hot spots, and octet scattering

The central physical mechanism is the reconstruction of the electron Fermi surface by strong electron correlation. In the superconducting state, the low-energy spectral intensity is not distributed uniformly around a large Fermi surface. Instead, the antinodal spectral weight is strongly suppressed, while the remaining low-energy weight concentrates near the nodal-side arc tips, which coincide with hot spots on the electron Fermi surface (Tan et al., 2020).

This reconstruction is attributed to the momentum-dependent scattering rate $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$0. On the constant-energy contour defined by

$$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$1

the spectral weight is effectively controlled by $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$2. Since $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$3 is smallest at the hot spots and much larger near the antinodes, the quasiparticles pile up at the hot spots. The reconstructed electron Fermi surface is thus broken into Fermi-arc-like pieces whose ends dominate low-energy physics (Tan et al., 2020).

There are eight hot spots on the reconstructed electron Fermi surface. Because the strongest spectral weight is concentrated at these eight points, scattering between them produces a discrete set of preferred momentum transfers, the $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$4 vectors of an octet scattering model. The literature explicitly discusses $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$5 and $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$6 along the Brillouin-zone parallel direction, and $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$7 and $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$8 along the diagonal direction. The sharp ARPES autocorrelation peaks are directly correlated with these scattering wave vectors, and are weakly dispersive in momentum space (Tan et al., 2020).

A recurrent misconception is that the autocorrelation peaks are merely a numerical byproduct of convolving a structured intensity map. The cuprate analysis argues the opposite: once the low-energy spectral weight becomes concentrated near a discrete set of hot spots, the peaks are a kinematic consequence of the reconstructed electron Fermi surface and its associated quasiparticle scattering structure (Tan et al., 2020).

4. Energy evolution and doping dependence

As the binding energy changes, the hot spots on the constant-energy contour move slightly, so the associated $$\bar C(\mathbf q,\omega)=\frac{1}{N}\sum_{\mathbf k} I(\mathbf k+\mathbf q,\omega)\,I(\mathbf k,\omega),$$9 also move. The peak positions therefore show a smooth energy dependence, but the peaks remain identifiable and track the octet vectors over the relevant energy range. The reported behavior is weak dispersion with energy rather than the disappearance of the octet pattern (Tan et al., 2020).

The doping evolution is a defining result. In electron-doped cuprates, the positions of the autocorrelation peaks associated with

$\mathbf k$0

move in one direction in momentum space as electron doping increases. In hole-doped cuprates, the corresponding peak positions move in the opposite direction as hole doping increases. The same contrast is stated for the charge-order wave vector: in the electron-doped case it increases with electron doping, whereas in hole-doped materials it decreases with hole doping (Tan et al., 2020).

The significance of this asymmetry is twofold. First, it preserves the underlying universality of the hot-spot and octet-scattering mechanism across electron- and hole-doped cuprates. Second, it identifies a clear electron-hole asymmetry in how reconstructed Fermi-surface geometry evolves with carrier concentration. The autocorrelation thus functions not only as a structural probe of hot-spot connectivity, but also as a sensitive indicator of doping-dependent electronic reconstruction (Tan et al., 2020).

5. Relation to quasiparticle scattering interference

ARPES autocorrelation has an intrinsic connection to quasiparticle scattering interference, but the two observables are not identical. What they share is the same reconstructed electron Fermi surface and the same octet scattering geometry: hot spots dominate the spectral weight, the same $\mathbf k$1 connect the dominant states, and both quantities therefore show peaks at related momentum transfers (Tan et al., 2020).

What differs is the underlying observable. ARPES autocorrelation is the self-convolution of ARPES intensity in momentum space. QSI, by contrast, is the impurity-induced Fourier-transformed density-of-states modulation. In the cited cuprate theory, the Fourier-transformed local density-of-states modulation $\mathbf k$2 is calculated from a single point-like impurity potential,

$\mathbf k$3

using the same electron Green’s functions, and its momentum-space pattern is shown to be qualitatively consistent with the ARPES autocorrelation pattern (Tan et al., 2020).

The distinction matters methodologically. ARPES autocorrelation does not require an impurity potential and does not measure impurity scattering directly; QSI does. Their similarity arises because both are controlled by the same hot-spot quasiparticles and the same octet scattering vectors. The relation is therefore one of common origin rather than formal identity (Tan et al., 2020).

6. Broader ARPES context, interpretive limits, and related correlation-sensitive uses

Any interpretation of ARPES autocorrelation must remain anchored to the fact that ARPES itself measures a matrix-element-weighted spectral function rather than bare bands. In first-principles work on graphene, the ARPES spectrum is written as

$\mathbf k$4

and the spectral function is explicitly controlled by the self-energy, with $\mathbf k$5 shifting the quasiparticle energies and $\mathbf k$6 producing finite linewidths and finite quasiparticle lifetimes. This establishes the broader principle that ARPES intensity maps encode many-body renormalization and scattering, not only geometric band contours (Park et al., 2010).

The same caution is emphasized in more general ARPES theory. The one-step model treats photoemission as a single coherent quantum process including excitation, transport through the crystal, transmission through the surface barrier, and escape into vacuum, and stresses that the experimentally measured intensity is the full photocurrent with matrix-element effects, multiple scattering, surface-barrier effects, lifetime broadening, and, where needed, disorder, phonons, and correlations. Likewise, the Huygens formulation recasts ARPES as the interference of dipole-allowed Coulomb wavelets emitted from orbital components of the initial state, retaining final-state phase shifts and kinetic-energy-dependent interference effects that a plane-wave approximation misses [(Minár et al., 2010); (Moser, 2022)].

A plausible implication is that autocorrelation inherits the interpretive strengths and limitations of the underlying ARPES map. Peaks in $\mathbf k$7 can reveal preferred momentum transfers, but the intensity landscape being convolved is already shaped by self-energy effects, matrix elements, and final-state interference. For that reason, autocorrelation is best understood as an analysis layer built on top of conventional ARPES theory rather than as an autonomous replacement for it.

Related but distinct are correlation-sensitive ARPES applications that do not use an autocorrelation formalism. In Sr$\mathbf k$8Cu$\mathbf k$9Zn$\mathbf q$0O$\mathbf q$1Cl$\mathbf q$2, the spectral weight beyond the antiferromagnetic zone boundary, quantified by the ratio $\mathbf q$3, was shown to track the spin correlation length $\mathbf q$4 estimated from $\mathbf q$5Cl NMR. Even for $\mathbf q$6, substantial back-bended spectral weight remains visible, demonstrating that ARPES is sensitive to short-range antiferromagnetic correlations as well as to long-range order (Wallauer et al., 2015). This does not constitute ARPES autocorrelation in the narrow sense, but it clarifies the broader landscape in which ARPES-derived observables are used to interrogate correlation physics.