Quantum-Matter Probes, Inference, and Evidence

A measured spectrum or numerical output becomes a many-body result through an explicit relation to observables. This chapter compares that relation across experimental probes and computational methods, emphasizing resolution, uncertainty, and competing explanations.

Enter this chapter

Begin with From Measured Intensity to Many-Body Claim to map a detector signal to the correlation function it can constrain.

Helpful background. From Measured Intensity to Many-Body Claim and Transport Extraction, Inverse Problems, and Error Budgets develop the definitions used by the chapter’s opening pages.

Topics

1. From Measured Intensity to Many-Body Claim — How is a measured intensity mapped through matrix elements, resolution, backgrounds, and response functions to a bounded many-body claim?
2. Photoemission and ARPES Spectral Inference — How do photoemission matrix elements, occupied spectral weight, resolution, surfaces, and final states constrain ARPES inference?
3. Tunneling Spectroscopy and Quasiparticle Interference — How do tunneling matrix elements, surfaces, impurities, and interference map STM/STS and QPI data to local spectral claims?
4. Optical Conductivity and Spectral-Weight Diagnostics — How are reflectivity or transmission converted into optical conductivity, spectral weight, and bounded charge-dynamics claims?
5. Neutron Scattering and Dynamical Structure Factors — How do neutron scattering geometry, form factors, polarization, and resolution determine a dynamical structure-factor claim?
6. Raman, RIXS, and EELS Response — Which effective operators and resonance conditions map Raman, RIXS, and EELS intensities to charge, spin, orbital, or collective response?
7. NMR, μSR, and Local Magnetic Probes — How do hyperfine or local-field couplings convert NMR and μSR signals into static and dynamic magnetic constraints?
8. Quantum Oscillations and Thermodynamic Fermi-Surface Inference — How do oscillation frequencies, masses, phases, damping, and thermodynamics constrain extremal Fermi surfaces?
9. Cold-Atom RF, Bragg, Time-of-Flight, and Microscopy Probes — How do cold-atom RF, Bragg, time-of-flight, in-situ, and quantum-gas-microscopy measurements map to many-body observables?
10. Quantum Monte Carlo Formulations and Phase Inference — How are QMC formulations, estimators, finite-size scaling, sign constraints, and continuation errors propagated into a many-body phase claim?
11. Tensor-Network Phase Diagnostics in Quantum Matter — How do bond dimension, geometry, symmetry, contraction, and finite-entanglement scaling constrain tensor-network phase diagnostics?
12. Exact Diagonalization and Finite-Size Evidence — How do symmetry sectors, clusters, spectral statistics, observables, and drift control phase claims from exact diagonalization?
13. Model Selection and Parameter Inference — How are competing many-body models, parameters, nuisance variables, discrepancy, and identifiability inferred from heterogeneous observables?
14. Evidence Triangulation and Reproducibility — How are analytic, numerical, and experimental evidence combined without double counting shared assumptions or hiding irreproducible transformations?

How the ideas fit together

Different probes weight different operators and kinematic regions. For example, photoemission contains matrix elements and occupation factors as well as the single-particle spectral function Sobota, He, and Shen 2021, §§ II–III. A fitted feature can therefore depend on the measurement model as well as the material.

Computational comparisons add independent approximation tests only to the extent that their errors differ. Two solvers for one Hamiltonian can support its finite-system predictions without establishing that Hamiltonian’s material realization. Predictions for observables reserved from fitting provide a stronger test than repeated agreement on calibration data.

The comparison below separates the assumptions and tests needed for these different conclusions.

Probe and computation claim test matrix

Route	Forward object	Main calibration or nuisance	Covariance or prior issue	Resolution or representation limit	Held-out or negative test	Supported conclusion
Measurement contract	Detector record from a convolved normalized correlator	Background, scale, geometry, efficiency	Shared calibration and processed-bin covariance	Singular directions of the response kernel	Reconstruct a reserved standard or synthetic truth	Calibrated observable or parameter constraint
ARPES	Dipole-weighted occupied $A(\mathbf k,\omega)$	Photon energy, polarization, surface, $k_z$	Background and global energy-reference covariance	Escape depth and energy–momentum resolution	Predict another polarization, photon energy, or termination	Dispersion or effective self-energy in a declared model
STM and QPI	Tip-weighted local spectrum and impurity response	Tip orbital, set point, surface, defect vertex	Drift correction and Fourier-window covariance	Thermal, voltage, spatial, and field-of-view limits	Reproduce several defects, fields, or terminations	Local gap or scattering geometry, not unique mechanism
Optical response	Fresnel or multilayer fields mapped to $\sigma(\omega)$	Thickness, tensor geometry, extrapolation	Kramers–Kronig and common-normalization covariance	Spectral window and low-frequency access	Recover causality and the applicable sum rule	Charge-dynamics scale or stiffness after missing-area checks
Neutron scattering	Polarization-projected $S^{\alpha\beta}(\mathbf Q,E)$	Form factor, absorption, monitor, mosaic	Shared background and four-dimensional resolution	Reciprocal-space coverage and resolution ellipsoid	Test equivalent zones, polarization, temperature, and sum rules	Mode or continuum in a stated spin channel
Raman, RIXS, EELS	Probe-specific effective response operator	Resonance, polarization, self-absorption, thickness	Normalization shared across incident energies or channels	Core-hole lifetime, screening, multiple scattering	Vary incident energy, symmetry channel, momentum, or thickness	Excitation assignment within a controlled operator reduction
NMR and μSR	Hyperfine- or site-weighted local-field correlations	Site, coupling tensor, dead time, background fraction	Correlated amplitudes and relaxation rates	Larmor and detector time window	Change nucleus or site sensitivity, field, and volume fraction	Static or dynamic local-field bound in a stated window
Quantum oscillations	Extremal orbit quantization	Angle, field background, temperature, torque geometry	Close-frequency and window covariance	Finite $1/B$ span, damping, magnetic breakdown	Predict angle, harmonic, thermodynamic, or transport response	Extremal area and mass; phase only with all corrections
Cold-atom probes	Expansion, transfer, response, or site-detection kernel	Trap, pulse, loss, Wannier envelope, detection matrix	Atom-number and image-processing covariance	Finite size, entropy, PSF, pulse and observation time	Benchmark a different observable, trap, preparation, or size	Realized correlator or finite-system phase evidence
Quantum Monte Carlo	Stochastic estimator in a declared representation	Ensemble, update, discretization, sign	Autocorrelation and cross-observable covariance	Size, temperature, Trotter or projection, continuation	Predict held-out sizes and exact limits; change formulation	Controlled equilibrium observable or scaled phase result
Tensor networks	Variational state and transfer environment	Initialization, unit cell, symmetry, geometry	Correlated extrapolations and ansatz selection	Bond dimension, cylinder width, contraction dimension	Competing states, boundaries, bond dimensions, observables	Variational phase evidence after representation scaling
Exact diagonalization	Eigenpairs and dynamics of a finite sector	Symmetry, cluster, boundary, broadening	Disorder-sample and spectral-window covariance	Exponential size limit and recurrences	Hold out cluster shapes, twists, and sizes	Exact finite-system result with bounded extrapolation
Model selection	Joint generative likelihood	Nuisance parameters and discrepancy	Full covariance and prior-volume sensitivity	Candidate-set and emulator limits	Leave out a probe, regime, or defining observable	Relative predictive support, not unique truth
Triangulation	Dependence graph of evidence and claims	Versions, samples, shared Hamiltonian, preprocessing	Shared ancestors prevent naive likelihood products	Reproduction and replication scope	Independent data, representation, perturbation, and null tests	Conclusion surviving distinct failure modes

Where to go next

Continue to Quantum Matter and Emergence to compare current research questions and the observations that could discriminate their proposed explanations.

References

• Jonathan A. Sobota, Yu He, and Zhi-Xun Shen, “Angle-Resolved Photoemission Studies of Quantum Materials,” Reviews of Modern Physics 93 (2021) 025006. DOI