research35 min studyLesson 560 in Physics

Pseudogap observations and competing interpretations

Separate a partial depletion of electronic spectral weight from the hypotheses used to explain it, with explicit probe and temperature qualifications.

Before you begin

On this page

The question for today

What you’ll understand

Define a pseudogap operationally for a chosen observable.
Account for occupation, resolution and spectral-weight redistribution.
Compare pairing, ordering and correlation interpretations without presenting one as settled.

A pseudogap is an observed depletion of low-energy electronic response that is incomplete or does not satisfy the conditions of a full gap. In cuprate experiments the term can refer to features in photoemission, tunneling, spin response or thermodynamics, which weight electronic states differently. The observation must be specified before assigning a mechanism. Pairing fluctuations, competing orders and strong correlation effects make different predictions, and the relation among them remains a research question rather than a universal consequence of one spectral dip.

Define the measured depletion and its reference

For an electronic spectrum, a full gap means no spectral support over a stated energy interval in an ideal zero-resolution description. A pseudogap instead indicates reduced low-energy weight relative to a specified reference, with residual states or momentum regions still contributing. Its apparent onset can depend on the probe, subtraction procedure and chosen threshold. A reported temperature scale should therefore include what feature is being tracked rather than being treated automatically as a sharply defined thermodynamic phase boundary.

Momentum-resolved photoemission measures occupied electron-removal intensity, multiplied by matrix elements and broadened by resolution. It can distinguish strong antinodal depletion from less affected near-nodal states in particular cuprate samples, but an apparent arc can also depend on lifetime broadening and the criterion for a resolvable gap. Thermodynamic entropy integrates excitations differently. Agreement or disagreement across these probes constrains a model; it does not justify declaring every reported pseudogap scale numerically identical.

I(\mathbf k,E)\propto |M(\mathbf k,E)|^2f_F(E)A(\mathbf k,E)\quad\text{before resolution convolution and background}

I is measured intensity with instrument-dependent normalization, M a matrix element and A a canonical spectral function when properly defined. This schematic equilibrium photoemission relation is not an inversion formula without resolution, background and matrix-element information.↔ Scroll wide equations horizontally.

Preserve spectral accounting when a dip appears

The canonical spectral sum rule remains valid when low-energy weight is depleted. In an interacting model, weight can move to higher energies or between coherent and incoherent features. A finite experimental window may miss that redistribution. Normalizing a small window independently at every temperature can therefore distort the apparent depletion. Occupation changes and the temperature-dependent Fermi function must also be separated from a change in the underlying spectral function before claiming a gap has opened or closed.

A toy density of states with partial suppression illustrates why a dip is not a unique mechanism. The same integrated missing low-energy area can be produced by different momentum patterns, lifetimes or reconstruction mechanisms. Thermal convolution can fill an observed dip even if an underlying model gap parameter stays fixed, while changes in scattering can sharpen a feature without a new ordered phase. Neither filling nor closing terminology is meaningful unless the model quantity and observable are stated explicitly.

\int_{-\infty}^{\infty}A(\mathbf k,E)\,dE=1,\qquad N(E)=\frac1{N_s}\sum_{\mathbf k,\sigma}A(\mathbf k,E)

The sum rule is per canonical momentum and spin orbital; the density of states N here is spin summed per cell and has inverse-energy units. A partial energy-window integral is not generally conserved.↔ Scroll wide equations horizontally.

Compare interpretations through additional predictions

Pairing-based interpretations associate some depletion with local pair correlations that need not imply global phase coherence above the superconducting transition. Density-wave interpretations involve translational or other symmetry changes and can reconstruct electronic momenta; their connection to an observed dip requires measured ordering wavevectors, strengths and spectral consequences. Correlation-driven interpretations emphasize proximity to interaction-induced localization or fractionalized states. These descriptions can coexist in some regimes, so listing them does not establish that one must explain all compounds and dopings.

The referenced ARPES study distinguishes pairing and pseudogap scales and argues for competition in its analyzed samples; the thermodynamic study makes a different claim about whether a particular pseudogap closes at a presumed onset temperature. Those are data-informed interpretations with assumptions, not interchangeable definitions or a settled universal mechanism. A strong explanation predicts additional probe responses and survives changes of temperature, momentum and doping. The spectral lab is merely a finite sine sum and cannot generate a many-body electron spectrum or adjudicate pseudogap theories.

N_{\rm toy}(E)=\begin{cases}(1-d)N_0,&|E|<\Delta,\\N_0,&\Delta<|E|<W,\end{cases}\qquad 0<d<1

This is an illustrative partially depleted density of states over a finite model band, with depth d, energy scale Δ and background N0. It is not a normalized microscopic spectral function unless the missing weight is restored elsewhere and the full band is specified.↔ Scroll wide equations horizontally.

Work it through

A partial depletion with explicit missing weight

In the toy density of states take $N_0=2.0\,\mathrm{eV^{-1}}$ per cell, $d=0.60$ and $\Delta=0.050\,\mathrm{eV}$ . Find the remaining low-energy density and the integrated area removed from $-\Delta<E<\Delta$ relative to the flat reference.

The depleted density is $(1-d)N_0=0.40(2.0)=0.80\,\mathrm{eV^{-1}}$ per cell. It remains nonzero, so this toy model describes a partial depletion rather than a hard gap.
The missing density at each energy in the interval is $dN_0=1.20\,\mathrm{eV^{-1}}$ per cell. The interval width is $2\Delta=0.100\,\mathrm{eV}$ .
Their product is $0.120$ states per cell of missing reference area. A canonical full-band model must place that weight elsewhere; the local dip by itself does not identify pairing, a density wave or Mott correlations.

The result

The remaining density is 0.80 eV⁻¹ per cell, and the low-energy window is missing 0.120 states per cell relative to the declared flat reference.

The living notebook

Explore the model

Interactive lab

Interactive lab · spectral

Truncation changes a pattern, not an electron spectrum

Reset and set BOTH Mode count N = 2 and Coefficient decay p = 1. Predict less fine spatial structure than a fresh run: Reset and set N = 8 and p = 1. Compare time zero or equal Step counts. This exact Σ sin(nx−t)/nᵖ is a spatial sine sum, not a positive spectral function, density of states, pseudogap calculation or many-body sum-rule test.

Finite sumHighest included mode

Sum at x = π/2: 0.867
Smallest wavelength: 2π / 5
Highest-mode coefficient: 0.2

Mode count N5Coefficient decay p1

t = 0 s

A finite sum of sin(nx − t)/nᵖ. Playback changes t; increasing the cutoff includes finer spatial scales. This is not a proof about a continuum limit or renormalization. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA changed plotted pattern is not sufficient evidence for a microscopic electronic mechanism.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Photoemission intensity contains more than the underlying spectral function. A reduced finite-window signal needs probe-specific analysis before it can establish spectral redistribution or distinguish competing mechanisms.

Your turn

Try it yourself

A normalized spectral model has total area one. At one temperature its area inside a measured window is 0.70 and outside is 0.30. Later the window area falls to 0.50 while the full sum rule remains valid. Find the necessary outside area and explain whether this alone determines a pseudogap mechanism.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use conservation of the full spectral area, not a separate renormalization of the measured window.

Compare with the solution

The later outside area must be $1-0.50=0.50$ , so $0.20$ of spectral area has left the measured window. The total canonical area remains one. This bookkeeping is compatible with several possible changes in spectrum and does not specify where outside the window the weight moved, whether a hard gap developed, or what microscopic process caused the redistribution. Establishing a mechanism needs the energy and momentum pattern, probe corrections and independent predictions, rather than the missing-window area alone.

Keep the idea

Take it with you

A pseudogap must be defined for a stated observable and reference.
Low-energy depletion is compatible with a conserved full spectral sum.
Competing mechanisms must be tested through additional predictions, not selected from one dip.

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