What gets measured, what gets computed in condensed matter

Condensed matter has more techniques on both sides than almost any other field — dozens of experimental probes and as many computational methods, partly because the field spans non-interacting band theory through strongly-correlated Mott insulators and superconductors. The question, as on the other quantities pages, is which numbers a condensed-matter physicist actually uses, and how those numbers correspond to what a diffractometer, an ARPES beamline, a tunneling microscope, or a SQUID magnetometer measures.

What an experiment measures

Four broad categories, each with its dominant instrumental technique(s):

1. Structural / crystallographic

X-ray diffraction (XRD) and neutron diffraction give lattice constants, crystal symmetry, atomic positions in the unit cell, and through diffraction-pattern temperature dependence, phase transitions. Low-energy electron diffraction (LEED) does the same for surfaces. X-ray absorption fine structure (EXAFS) gives local atomic coordination — useful when crystallinity is poor. The output is a 3D atomic structure with bond lengths typically known to $\sim 0.001$ Å.

2. Electronic structure (occupied and unoccupied)

Angle-resolved photoemission (ARPES) is the flagship technique: shine photons at a sample, measure the energy and momentum of emitted electrons, reconstruct the occupied band structure $E_n(\mathbf{k})$ directly. Modern ARPES can resolve to ~1 meV and ~0.01 Å⁻¹. Inverse photoemission (IPES) does the same for unoccupied bands above the Fermi level. Scanning tunneling microscopy (STM) gives the local DOS at the sample surface; scanning tunneling spectroscopy (STS) gives $dI/dV(V)\propto \rho (E_F+eV)$, revealing energy gaps, impurity states, quasiparticle interference. X-ray photoelectron spectroscopy (XPS) probes core levels and chemical state.

3. Transport and dynamical response

Resistivity $\rho (T)$, Hall coefficient $R_H(T)$, and thermopower (Seebeck coefficient $S(T)$) probe charge and heat transport. Optical reflectivity / ellipsometry gives the complex dielectric function $ϵ(\omega )$, from which the optical conductivity $\sigma (\omega )$ follows. Inelastic neutron scattering (INS) measures the dynamic structure factor $S(\mathbf{q},\omega )$ — phonon dispersion, magnon dispersion, paramagnon spectra. Inelastic X-ray scattering (IXS) and resonant inelastic X-ray scattering (RIXS) probe charge and orbital excitations. Specific heat $C(T)$ integrates the full excitation density of states.

4. Magnetic and superconducting

Magnetization $M(H,T)$ and magnetic susceptibility $\chi (T)$ from SQUID magnetometry. Neutron magnetic diffraction determines magnetic structure (FM, AFM, helical, …); polarized inelastic neutron scattering gives spin-wave dispersion $\omega (\mathbf{q})$. Muon spin rotation (μSR) measures local magnetic fields. NMR Knight shift probes the local spin susceptibility. Critical temperature $T_c$, superconducting gap $\Delta$, and critical field $H_{c2}$ characterize a superconductor; tunneling spectroscopy reads $\Delta$ directly as the DOS gap.

What a calculation produces

The condensed-matter method zoo is larger than the quantum-chemistry one because of strong correlation. Here are the headline computable quantities, grouped by method family:

Equilibrium structure and total energy

From DFT in a plane-wave or muffin-tin basis: total energy $E(\mathbf{R},\mathbf{a})$ as a function of atomic positions and lattice parameters. Minimize to get the equilibrium lattice constant, bulk modulus, elastic constants, and energetic stability of competing crystal structures.

Band structure and density of states

Solve the Kohn-Sham equations at each $\mathbf{k}$ in the Brillouin zone: ${\hat{H}}_{\mathrm{KS}}{\psi }_{n\mathbf{k}}={\epsilon }_{n\mathbf{k}}{\psi }_{n\mathbf{k}}$. The set of ${\epsilon }_n(\mathbf{k})$ across the BZ is the band structure; $\rho (E)={\sum }_{n\mathbf{k}}\delta (E-{\epsilon }_{n\mathbf{k}})$ is the density of states. DFT systematically underestimates band gaps; GW corrects this.

Optical response

Dielectric function $ϵ(\mathbf{q},\omega )$ from linear-response TDDFT (Casida-style for clusters; matrix-free Liouville-Lanczos for solids). For materials with strong excitonic effects (insulators, semiconductors), the Bethe-Salpeter equation (BSE) on top of GW captures bound electron-hole pairs that TDDFT-LDA misses.

Phonons and electron-phonon coupling

Density-functional perturbation theory (DFPT) computes the second derivative of the total energy with respect to atomic displacements at each $\mathbf{q}$: the dynamical matrix, whose eigenvalues are phonon frequencies ${\omega }_{\nu }(\mathbf{q})$. Mixed second derivatives give the electron-phonon coupling $g_{mn\nu }(\mathbf{k},\mathbf{q})$, which feeds Eliashberg theory for superconductivity.

Quasiparticle and many-body spectra

The GW approximation computes self-energy $\Sigma (\omega )$ and corrects DFT band energies to proper quasiparticle energies. DMFT (dynamical mean-field theory) maps a strongly-correlated lattice problem onto a self-consistent Anderson-impurity problem, solved by ED, CT-QMC, or NRG; the output is the local spectral function $A(\omega )$ and the lattice self-energy.

Magnetism and superconductivity

Magnetization and magnetic structure from spin-polarized DFT or DFT+U. Exchange constants $J_{ij}$ mapped onto a Heisenberg model from total-energy comparisons, then solved by QMC / DMRG / linear spin-wave theory. Superconducting $T_c$ from BCS / Eliashberg with electron-phonon coupling from DFPT, or from solving a Migdal-Eliashberg integral equation.

The bridge

Row by row, experimental observable to computed quantity:

Which method for which quantity

Condensed matter's method landscape is bigger than chemistry's because strong correlation, finite temperature, and lattice symmetry add axes. Pick the method to match the physics:

One-sentence summary

A condensed-matter calculation almost always answers "what does an ARPES, neutron-scattering, transport, optical, or NMR experiment measure?" — and despite the field's method zoo being bigger than chemistry's or nuclear's, the menu of observables is small enough to fit on this page. Pick the experimental quantity, find its row in the bridge table, choose a method that matches both the row and the correlation regime of your material, and run.

Related on this site

• Crystals — the structural basis for everything else.
• Zone folding — how supercell band structures relate to primitive-cell ones.
• APW and LAPW — the basis functions that underlie all-electron band-structure codes.
• Liouville-Lanczos TDDFT — the matrix-free Krylov method for optical response in solids; the right tool when Casida's matrix is too big.
• Casida equation — the molecular analog, with the same Padé-via-Lanczos correspondence.
• Hubbard dimer FCI — the two-site exactly-solvable model that's the toy version of strongly-correlated condensed-matter physics.
• Ising model — the prototype of magnetic and ordered-phase calculations.
• Parallel pages: quantum chemistry, nuclear physics, computational neuroscience.