Experimental Observables in Electronic Systems

“Our narrative is by no means a recommendation of how research should be done, it simply reflects what we thought, how we acted and what we felt. However, it would certainly be gratifying if it encouraged a more relaxed attitude towards doing science.”

Gerd Binnig and Heinrich Rohrer, Nobel Lecture, December 8, 1986 [1]

Throughout, calligraphic symbols denote propagators, while boldface symbols denote variational mean-field tensors.

Green’s Functions and Physical Observables

This chapter collects the Green’s functions associated with the Bogoliubov–de Gennes mean-field Hamiltonian and shows how physical observables are obtained from them. No reference to the variational derivation or self-consistency equations is required.

Nambu—Gor’kov Green’s function

The terminology “Green’s function” traces back to George Green’s original 1828 essay. [2]

Let the Bogoliubov–de Gennes Hamiltonian satisfy

\hat{H}_{BdG}^{MF} Ψ_{n} = E_{n} Ψ_{n}, (BdG eigenproblem)

with particle–hole–symmetric spectrum (\pm\mathcal E_n).

The Nambu–Gor’kov Green’s function is defined as the resolvent

G (z) \equiv (z - \hat{H}_{BdG}^{MF})^{- 1}, z \in C . (NG resolvent)

In the eigenbasis,

G (z) = n \sum \frac{Ψ _{n} Ψ _{n}^{†}}{z - E _{n}} + \frac{Ψ ˉ _{n} Ψ ˉ _{n}^{†}}{z + E _{n}} . (Spectral representation)

Matrix structure

In superconductivity, the anomalous (pair) propagator was introduced by Gor’kov in his Green’s-function formulation of BCS theory. [3]

In Nambu space the Green’s function decomposes as

G (z) = (G (z) \overset{ˉ}{F} (z) F (z) \overset{ˉ}{G} (z)) . (Nambu structure)

It is convenient to bundle spin and orbital indices into a single multi-index (\alpha\equiv(\sigma,m)). Then the normal and anomalous blocks admit the BdG spectral representations

G_{α β} (z; i, i^{'}) = n \sum [\frac{u _{i α, n} u _{i^{'} β, n}^{*}}{z - E _{n}} + \frac{v _{i α, n}^{*} v _{i^{'} β, n}}{z + E _{n}}], (Normal Green)

F_{α β} (z; i, i^{'}) = n \sum [\frac{u _{i α, n} v _{i^{'} β, n}^{*}}{z - E _{n}} + \frac{v _{i α, n}^{*} u _{i^{'} β, n}}{z + E _{n}}] . (Anomalous Green)

The particle–hole–conjugate blocks (\bar{\mathcal G}) and (\bar{\mathcal F}) are fixed by the symmetry relations below, so we do not restate them as separate sums.

Symmetry relations

\overset{ˉ}{G} (z) = - G^{T} (- z), \overset{ˉ}{F} (z) = F^{†} (z^{*}) . (Block symmetries)

Hence the full Nambu–Gor’kov Green’s function may be written as

G (z) = (G (z) F^{†} (z^{*}) F (z) - G^{T} (- z)) . (Final Nambu form)

From Green’s functions to experimental observables

Experimentally accessible quantities fall into two broad classes. The first comprises single-particle spectra, which are obtained from the retarded Nambu–Gor’kov Green’s function

G^{R} (ω) = G (ω + i 0^{+}), A (ω) \equiv - \frac{1}{π} ℑ G^{R} (ω) . (8.9)

Physical spectra measured by probes such as ARPES and STM are extracted from the electron block (\mathcal G^R), traced over spin and orbital indices.

The second class comprises linear-response functions, obtained from retarded correlators of densities, currents, and spins,

χ_{A B}^{R} (t) \equiv - i Θ (t) ⟨[\hat{A} (t), \hat{B} (0)]⟩, χ_{A B}^{R} (ω) = \int_{- \infty}^{\infty} d t e^{iω t} χ_{A B}^{R} (t), (8.10)

where (\hat A) and (\hat B) may denote charge-density, current, or spin operators. Microscopically these quantities are built from products of one-particle Green’s functions, supplemented by vertex corrections when required by symmetry or conservation laws.

Single-particle observables (ARPES/STM)

In a translationally invariant system,

A (k, ω) \equiv - \frac{1}{π} ℑ Tr_{σ, m} G^{R} (k, ω), (8.11)

and, up to matrix-element effects, ARPES measures (I(\mathbf k,\omega)\propto f(\omega),A(\mathbf k,\omega)). This is the momentum-resolved single-particle spectral function.

The real-space counterpart is the local density of states measured by STM/STS,

N (r, ω) \equiv - \frac{1}{π} ℑ σ, m \sum G_{σσ}^{R m, m} (ω; r, r), \frac{d I}{d V} (r, V) \propto N (r, ω = e V), (8.12)

for a featureless tip density of states and weak energy dependence of the tunnelling matrix element.

In the presence of impurities, one considers the modulation of the local density of states,

δ N (r, ω) \equiv N (r, ω) - N_{0} (ω), g (q, ω) \equiv FT [δ N (r, ω)], FT - STS (q, ω) = ∣ g (q, ω) ∣. (8.13)

The Fourier amplitude (g(\mathbf q,\omega)) is the basic quasiparticle-interference observable. In a (T)-matrix treatment, (\delta N) is controlled by (\mathcal{\mathbf G},\mathcal{\mathbf T},\mathcal{\mathbf G}) in Nambu space.

Linear-response observables (Kubo dictionary)

Let (\Pi^R_{ij}(\omega)) be the retarded current–current correlator (\Pi^R_{ij}(\omega)\equiv \chi^R_{j_i j_j}(\omega)). Then the conductivity follows from

σ_{ij} (ω) = \frac{1}{i ( ω + i 0 ^{+} )} [Π_{ij}^{R} (ω) - Π_{ij}^{R} (0)], (8.14)

where the subtraction enforces gauge invariance together with the diamagnetic term. This quantity governs both the optical conductivity and the microwave response.

The London kernel (K_{ij}) is the static long-wavelength current response,

K_{ij} \equiv q \to 0 lim (Π_{ij} (q, ω = 0) + K_{ij}^{dia}), λ^{- 2} \propto K_{T}, (8.15)

where (K_T) is the transverse part. The temperature dependence of (\lambda(T)) is correspondingly a sensitive probe of nodal structure through the superfluid stiffness.

At optical frequencies Kerr rotation is controlled by the antisymmetric part of the optical response, commonly expressed through (\sigma_{xy}(\omega)) (from (\Pi_{xy})) together with the sample’s electrodynamics (dielectric function / refractive index). Microscopically, (\sigma_{xy}) is again a current–current Kubo response in the TRSB state.

For spin operators (\hat S^\alpha),

χ^{α β} (q, ω) \equiv χ_{S^{α} S^{β}}^{R} (q, ω) . (8.16)

The Knight shift is proportional to the uniform static susceptibility, (K(T)\propto \chi^{\alpha\alpha}(\mathbf q{=}0,\omega{=}0)). The NMR relaxation rate probes the low-frequency spin response,

\frac{1}{T _{1} T} \propto q \sum ω \to 0 lim \frac{ℑ χ ^{+-} ( q , ω )}{ω} . (8.17)

Neutron scattering, in turn, measures

S (q, ω) \propto [1 - e^{- β ω}]^{- 1} ℑ χ (q, ω) . (8.18)

From the grand potential (\Omega) (computable from BdG eigenvalues, equivalently (\ln\det \mathcal{\mathbf G}^{-1})), one obtains the specific heat and condensation energy via

C (T) = - T \frac{\partial ^{2} Ω}{\partial T ^{2}}, Δ F = Ω_{SC} - Ω_{N} . (8.19)

Thermal conductivity (κ_{ij}) follows from heat-current correlators (Kubo), and (κ(T)) again strongly constrains nodal structure.

Field and current probes (real-space equilibrium observables)

Experiments such as ZF-(\mu)SR, scanning SQUID, or Hall magnetometry probe internal fields (\mathbf B(\mathbf r)) produced by equilibrium supercurrents (\mathbf j(\mathbf r)) and/or magnetization (\mathbf M(\mathbf r)),

\nabla \times B (r) = μ_{0} (j (r) + \nabla \times M (r)) . (8.20)

In BdG/quasiclassical formalisms, (\mathbf j(\mathbf r)) can be expressed directly in terms of Green’s functions (paramagnetic contribution plus diamagnetic term), so that (\mathcal{\mathbf G}\Rightarrow \mathbf j \Rightarrow \mathbf B\Rightarrow P(B)) (for (\mu)SR).

Observable dictionary (experiment (\leftrightarrow) correlator)

Experimental method	Measured quantity	Theory object	Green’s-function content
ARPES	(I(\mathbf k,\omega))	(A(\mathbf k,\omega))	(-\frac{1}{\pi}\Im,\mathrm{Tr}_{\sigma,m},\mathcal G^R(\mathbf k,\omega))
STM/STS	(dI/dV(\mathbf r,V))	(N(\mathbf r,\omega)) (LDOS)	(-\frac{1}{\pi}\Im \sum_{\sigma,m}\mathcal G^{R,m,m}_{\sigma\sigma}(\omega;\mathbf r,\mathbf r))
QPI / FT-STS	(\lvert g(\mathbf q,\omega)\rvert)	(\delta N(\mathbf q,\omega))	(\mathcal{\mathbf G}\mathcal{\mathbf T}\mathcal{\mathbf G}) (impurity (T)-matrix)
Optical/microwave	(\sigma_{ij}(\omega))	conductivity	current–current (\Pi_{ij}=\chi_{j_i j_j})
Penetration depth	(\lambda(T))	superfluid stiffness	static transverse current response (K_T)
Kerr rotation	(\theta_K(\omega))	TRSB optical response	antisymmetric (\sigma_{xy}(\omega)) from (\Pi_{xy})
Knight shift	(K(T))	(\chi_s(0,0))	spin–spin (\chi_{SS})
NMR (1/T_1)	relaxation rate	low-(\omega) spin fluctuations	(\sum_{\mathbf q}\Im\chi^{+-}(\mathbf q,\omega)/\omega)
Neutrons	(S(\mathbf q,\omega))	dynamical susceptibility	(\Im,\chi_{SS}(\mathbf q,\omega))
ZF-(\mu)SR / scanning SQUID	(P(B)), (\mathbf B(\mathbf r))	fields/currents	(\mathcal{\mathbf G}\Rightarrow \mathbf j,\mathbf M \Rightarrow \mathbf B)
Specific heat	(C(T))	(\Omega(T))	BdG spectrum or (\ln\det \mathcal{\mathbf G}^{-1})

Retarded and advanced Green’s functions

G^{R / A} (ω) = G (ω \pm i 0^{+}), (Retarded/advanced)

ℑ G^{R} (ω) = - ℑ G^{A} (ω) . (Spectral antisymmetry)

Density of states

Local density of states (LDOS)

N (r, ω) \equiv - \frac{1}{π} ℑ σ, m \sum G_{σσ}^{R m, m} (ω; r, r) . (LDOS)

(Global) density of states

DOS (ω) = \frac{1}{V} \int d r N (r, ω) . (DOS)

Anomalous density of states

ADOS_{σ τ}^{mn} (ω, i) = - \frac{1}{π} ℑ, F_{σ τ}^{R, mn} (ω; i, i) . (ADOS)

Equal-time observables (real-frequency representation)

Normal density matrix

ρ_{σ σ^{'}}^{m m^{'}} (i, i^{'}) = \int d ω, [- \frac{1}{π} ℑ G_{σ σ^{'}}^{R} (ω; i, i^{'})] f (ω) . (Normal density)

At (T=0),

ρ_{σ σ^{'}}^{m m^{'}} (i, i^{'}) = - \frac{1}{π} \int_{- \infty}^{ω_{F}} d ω, ℑ G_{σ σ^{'}}^{R} (ω; i, i^{'}) . T = 0 (Normal density,)

Anomalous (pair) density

χ_{σ σ^{'}}^{m m^{'}} (i, i^{'}) = - \frac{1}{2 π} \int d ω, ℑ [F_{σ σ^{'}}^{R} (ω; i, i^{'}) + F_{σ σ^{'}}^{A, T} (- ω; i, i^{'})] (1 - 2 f (ω)) . (Pair density)

At (T=0),

χ_{σ σ^{'}}^{m m^{'}} (i, i^{'}) = - \frac{1}{2 π} \int_{- \infty}^{ω_{F}} d ω, ℑ [F_{σ σ^{'}}^{R} (ω; i, i^{'}) + F_{σ σ^{'}}^{A, T} (- ω; i, i^{'})] . T = 0 (Pair density,)

Equal-time observables (Matsubara representation)

ω_{n} = (2 n + 1) π T . (Fermionic Matsubara frequencies)

G (i ω_{n}) = (i ω_{n} - \hat{H}_{BdG}^{MF})^{- 1} . (Matsubara Green)

Normal density matrix

ρ_{σ σ^{'}}^{m m^{'}} (i, i^{'}) = T n \sum G_{σ σ^{'}}^{m m^{'}} (i ω_{n}; i, i^{'}) . (Normal density (Matsubara))

Anomalous (pair) density

χ_{σ σ^{'}}^{m m^{'}} (i, i^{'}) = T n \sum F_{σ σ^{'}}^{m m^{'}} (i ω_{n}; i, i^{'}) . (Pair density (Matsubara))

Interpretation

(\mathcal G) encodes single-particle propagation together with charge, spin, and orbital densities, while (\mathcal F) encodes pairing correlations. Their spectral weights determine the density of states and anomalous density of states, whereas frequency integrals or Matsubara sums yield the corresponding equal-time observables.

No reference to the interaction or mean-field self-consistency is required at this stage.

Scanning tunnelling microscopy

History

The scanning tunnelling microscope is the finest resolution microscope ever developed, making manifest the complete departure of quantum physics from classical. The device itself uses the quantum wave-particle duality of matter –“all things possess a portion of every thing”, believed Anaxagoras[4], and indeed, the wavefunction describing any piece of confined matter extends beyond limits of its barriers¹. The wavefunction of the electrons from the tip of the probing stylus overlaps with that of those in the sample. Quantum mechanics dictates that an overlap implies a finite probability amplitude for the electrons to tunnel. Nevertheless, the idea was bounced around between theorists for twenty years, but not taken seriously enough to attempt realising until the work of Gerd Binnig and Heinrich Rohrer in 1981[1].

Theory

The scanning tunnelling microscope is an instrument consisting of a sharp conducting tip which scans the surface of a flat conducting sample. When a voltage bias $V$ is applied between the tip and sample, a tunnelling current $I$ flows.

In order to calculate the probability amplitude of a current, we use time-dependent perturbation theory, which Fermi famously referred to as the ‘golden rule’ for transition rates[5]. The elastic tunnelling current at bias $V$ from the sample to the tip is

I_{sample \to tip} (V) = σ \sum e \frac{2 π}{ℏ} ∣ M ∣^{2} filled e^{-} states in the sample ρ_{s} (ϵ) f (ϵ) hole states in the tip ρ_{t} (ϵ - e V) [1 - f (ϵ - e V)] (8.34)

where we have summed over spin degrees of freedom $σ$ , (e>0) is the elementary charge; $\frac{2 π}{ℏ}$ comes from time-dependent perturbation theory; $∣ M ∣^{2}$ is the matrix element, $ρ_{s (t)} (ϵ)$ is the density of states of the sample (tip), and $f (ϵ)$ is the Fermi distribution. There will also be a smaller tunnelling amplitude from the tip to the sample

I_{tip \to sample} (V) = σ \sum e \frac{2 π}{ℏ} ∣ M ∣^{2} filled e^{-} state in the tip ρ_{t} (ϵ - e V) f (ϵ - e V) hole states in the sample ρ_{s} (ϵ) [1 - f (ϵ)] (8.35)

The total current from the sample to the tip is the sum of the two individual currents, integrated over all energies. Up to a sign convention for the current direction,

I (V) \propto \int d ω ρ_{t} (ω - e V) ρ_{s} (ω) [f (ω - e V) - f (ω)], (8.36)

where (e>0) is the elementary charge and energies are measured relative to the sample chemical potential. For a featureless tip DOS and weak energy dependence of the tunnelling matrix element, this reduces (at low (T)) to the familiar proportionality

\frac{d I}{d V} (V) \propto ρ_{s} (ω = e V), (8.37)

and in real space (\rho_s(\omega)) is replaced by the local density of states (N(\mathbf r,\omega)) of the sample.

In practice one therefore chooses a tip material with a relatively flat density of states within the probing energy range. For these reasons, the tip material is usually chosen to be tungsten², sharpened in situ by field emission onto a gold surface. Conveniently, gold also has a flat density of states, so scanning its surface provides another check of the flatness of the tungsten density.

The first theoretical calculation of the tunnelling current was described by Bardeen in 1961[6]. He assumed the tip and the sample density of states are independent of one-another; decay exponentially through the tunnelling barrier; and the wavefunctions of the sample and tip insignificantly influence one-another. In the case of these three minimal assumptions, the tunnelling matrix elements are independent of the energy difference between the two systems. Further, this implies the matrix will remain unchanged even if either of the systems enters the superconducting state. A more Hamiltonian reformulation was published in 1961 by Cohen et al[7]. Under these minimal assumptions one often takes (M) energy independent and approximates the tip DOS as constant in the relevant window, leading to (dI/dV\propto \rho_s(eV)) at low (T). According to basic quantum mechanics[8], the tunnelling probability through a square barrier is given by the WKB approximation as

γ = = \int_{0}^{s} \frac{2 m ϕ}{ℏ ^{2}} d x \frac{s}{ℏ} 2 m ϕ,

where (s) is the barrier width and (\phi) is the effective barrier height.

Realisation

In practice, STM requires an atomically flat and chemically clean surface so that the measured tunnelling matrix elements are controlled by the electronic structure of interest rather than by uncontrolled surface disorder or contamination.

Experimental observables: topographic maps, spectral maps and quasiparticle interference

Within the present framework of multiorbital two-dimensional superconductivity with impurity scattering, STM provides a direct route from the theoretical Green’s functions to experimentally accessible observables. The role of impurities is particularly important, since they generate the interference patterns that make it possible to visualise quasiparticle structure at atomic scales. This real-space-to-momentum-space STM program was worked out in detail in the Davis-group cuprate literature, where spectroscopic imaging, FT-STS, and quasiparticle-interference analysis were explicitly tied together. [9, 10, 11]

2026-04-01T14:33:32.167882 image/svg+xml Matplotlib v3.10.8, https://matplotlib.org/ *{stroke-linejoin: round; stroke-linecap: butt}

Native QTT geometry view of the Friedel benchmark model used as the running STM example in this chapter: a circular open square lattice with a single central impurity site highlighted and a stylized STM tip positioned above it. The schematic is generated directly from the same canonical lattice builder used by the chapter-local native driver.

Density of states

Even without impurities, the local spectrum is directly accessible through the differential tunnelling conductance measured at a given point. Electrons tunnel between sample and tip under an applied bias voltage (V), and at low temperature one has

\frac{d I}{d V} (r, V) \propto N (r, ω = e V),

where the local density of states is

N (r, ω) \equiv - \frac{1}{π} ℑ σ, m \sum G_{σσ}^{R m, m} (ω; r, r) .

The spatial average gives the global density of states,

DOS (ω) = \frac{1}{V} \int d r N (r, ω) .

Spectroscopic-imaging STM also made this local spectral structure directly visible in the cuprates, including the spatial evolution of pairing and pseudogap phenomenology. [12]

The same observable can be evaluated on a projected orthorhombic material-basis diagnostic. As a concrete control example, the figure below compares the local spectrum of the compact four-Wannier LaNiGa $_{2}$ singlet-control calculation in the clean system, at the impurity site, and a few unit cells away from it. In the present methodology chapter this is used only as an illustration of the STM dictionary on a realistic multiorbital basis: the impurity response is computed on the fixed converged state rather than from a fully self-consistent impurity recalculation.

Local density of states on the aligned LaNiGa $_{2}$ singlet-control basis, comparing the clean system with the impurity site and two nearby points for a scalar impurity of strength $V = 1.21$ . The impurity redistributes spectral weight most strongly near the coherence features, while the spectrum several unit cells away already relaxes toward the clean reference.

Friedel oscillations

Impurities in the quasiparticle sea produce oscillatory modulations of the local density of states, known as Friedel oscillations. When resolved by scanning tunnelling microscopy, these oscillations encode information about quasiparticle dispersion, lifetime, and pairing structure, and therefore provide a direct benchmark for the theory. The Davis-group FT-STS studies of Bi2212 are a canonical experimental realization of this logic, resolving impurity-induced modulations in real space and then identifying the associated scattering wavevectors in Fourier space. [9, 10, 11]

Linecut

For inhomogeneous samples it is often useful to compare spectra along a prescribed path rather than only at isolated points. A linecut is obtained by recording the density of states as the tip moves along a line, for example radially away from an impurity.

Linecuts of the impurity-induced local density of states for square lattices of increasing side length $N$ , shown along the straight and diagonal directions from the impurity and also rescaled by the percentage distance to the next impurity image. The largest-system analytical Friedel fit is overlaid as a dashed black curve. This makes the finite-size and direction dependence of the STM linecut explicit while preserving the common oscillation period set by the underlying Fermi wavevector.

The same linecut observable also separates two physically distinct controls. Moving the chemical potential upward through the band changes the Fermi wavevector and therefore shortens the Friedel period, while changing the impurity strength primarily alters the near-impurity amplitude and phase shift, with much weaker effect on the far-field wavelength. These control scans are useful because they distinguish band-structure information from the local scattering potential in the same native benchmark.

Linecuts of the impurity-induced LDOS for fixed impurity strength $V = 1.2$ and varying chemical potential $μ$ . As $μ$ moves upward in the band, the oscillation period decreases, reflecting the larger underlying Fermi wavevector.

Linecuts of the impurity-induced LDOS for fixed chemical potential $μ = - 3.8$ and varying impurity strength $V$ . The local response near the defect changes strongly with $V$ , but the far-field oscillation period remains set mainly by the band filling rather than by the impurity amplitude itself.

The same linecut construction can then be reused as a projected orthorhombic material-basis diagnostic. On the compact four-Wannier LaNiGa $_{2}$ singlet-control calculation, the impurity-induced linecut remains the same STM observable, but the multiorbital band structure and gap anisotropy determine which coherence features dominate along different directions.

Topography

To probe the integrated density of states, one may perform a topographic measurement in which the tip height is adjusted to maintain constant current at fixed bias voltage:

IDOS (r; V) \propto \int_{0}^{e V} d ω N (r, ω) (T \approx 0, e V > 0), (8.38)

Density of states map

A density-of-states map is obtained by scanning the surface at fixed energy and recording the resulting spatial pattern of (N(\mathbf r,\omega)). Already at the normal-state level, the impurity geometry strongly reshapes this STM image. A single defect produces approximately concentric Friedel rings, a defect pair introduces directional interference fringes, and a circular corral reorganizes the oscillatory weight into a partially confined standing-wave pattern. The figure below is regenerated from the native QTT tight-binding benchmark rather than from the older chapter-local scripts, so all three maps now come from the same canonical code path used elsewhere in the methodology.

Normal-state LDOS maps for three impurity geometries on the same (43\times 43) square-lattice benchmark with (\mu/t=-3.46), (V/t=0.1), and (\eta/t=0.05): a single impurity, an impurity pair displaced by (\pm(3,3)) from the centre, and a fifteen-site circular corral of radius (14.5a). These are the real-space scattering patterns from which later Fourier-space QPI maps are constructed.

On the compact LaNiGa $_{2}$ material basis the same real-space map can be examined at a finite bias where the coherence structure carries substantial spectral weight. The plotted quantity is the impurity-induced modulation (\delta N(\mathbf r,\omega)=N_V(\mathbf r,\omega)-N_{V=0}(\mathbf r,\omega)), so the homogeneous clean contribution has been removed before plotting. The resulting pattern is no longer a simple textbook Friedel ring pattern, but it still visualizes directly where the impurity perturbs the local tunnelling spectrum in real space. The diagonal anisotropy is not put in by hand: it follows from evaluating the scalar impurity response in the projected multiorbital Wannier basis, whose retained two-dimensional plane and orbital embedding do not impose square-lattice (C_4) symmetry. Numerically the regenerated maps are (C_2)-symmetric to precision but have a finite (C_4) residual, so the panel should be read as a material-basis diagnostic rather than a (C_4) benchmark.

Finite-bias impurity-induced LDOS modulation for the LaNiGa $_{2}$ singlet-control state at (\omega=-2.6) with impurity strength (V=1.21), computed by subtracting the matched clean (V=0) reference map. Because this clean-reference signal is strongly concentrated near the defect, the color scale shows (\log_{10}(|\delta N(\mathbf r,\omega)|+\epsilon)) rather than the raw signed value. The dominant spectral rearrangement is localized near the defect and extends anisotropically along a preferred diagonal direction in the projected material basis.

Bogoliubov quasiparticle interference

The tunnelling conductance may also be analysed in momentum space through the Fourier transform of the impurity-induced modulation,

δ N (r, ω) \equiv N (r, ω) - N_{0} (ω), g (q, ω) \equiv FT [δ N (r, ω)], FT - STS (q, ω) = ∣ g (q, ω) ∣. (8.13)

This quantity is the Bogoliubov quasiparticle-interference amplitude. The underlying real-space oscillations are Friedel oscillations, with characteristic wavelength (\lambda_\text{Friedel}=\frac{1}{2}\lambda_\text{Fermi}) arising from quasiparticle scattering at the Fermi surface³. The resulting interference pattern therefore provides a momentum-space characterisation of the relevant parts of the Fermi surface. In practice, the early cuprate FT-STS/QPI experiments provide the direct template for this dictionary from (\delta N(\mathbf r,\omega)) to dispersing scattering vectors on the underlying Fermi surface. [10, 11]

Schematic construction of quasiparticle interference from STM data. An impurity first generates a real-space modulation in the local density of states measured by the differential tunnelling conductance. Subtracting the homogeneous background isolates $δ N (r, ω)$ , and a two-dimensional Fourier transform then yields the FT-STS amplitude $∣ g (q, ω) ∣$ . Enhanced features in the resulting $q$ -space map correspond to impurity scattering vectors $q = k^{'} - k$ between equal-energy states.

Quasiparticle-interference map for the single-impurity square-lattice benchmark at impurity coupling strength $V = 1.2$ and chemical potential $μ = - 2.2$ . This is a direct example of the momentum-space FT-STS observable that the present code can generate from the impurity-induced modulation of the local density of states.

The same Fourier-space construction can also be carried out on the LaNiGa $_{2}$ singlet-control state. Here the Fourier transform is taken from the same clean-reference modulation map used in the real-space panel. At finite bias, the resulting FT-STS map exhibits extended diagonal weight rather than a featureless isotropic ring, reflecting the twofold anisotropy of the projected material-specific scattering channels.

Finite-bias FT-STS map for the LaNiGa $_{2}$ singlet-control state at (\omega=-2.6) with impurity strength (V=1.21), obtained from the clean-reference (\delta N(\mathbf r,\omega)) map. Percentile contrast limits are used only for visibility of the finite-bias modulation. The dominant QPI intensity lies on broad diagonal streaks in (\mathbf q)-space, illustrating how the same impurity-scattering formalism carries over from the square-lattice benchmark to a projected multiorbital material basis.

The particle-hole symmetry of interband scattering interference patterns depends on the relative sign of the energy gap on those bands. The energy (anti)symmetrised $ρ_{\pm} (q, ω)$ phase-resolved Bogoliubov scattering interference amplitudes [13] are defined as

ρ_{\pm} (q, ω) \equiv ℜ [g (q, + ω) \pm g (q, - ω)],

which at the $q$ for interband scattering, have distinct properties depending on the relative sign of the two gaps.

Scanning Josephson tunnelling microscopy

Cooper pair tunnelling is observable via Josephson scanning tunnelling microscopy (JSTM). Cooper-pair tunnelling in Josephson STM is controlled by the local anomalous propagator. A useful theoretical quantity is the anomalous spectral weight (ADOS), defined componentwise by

ADOS_{σ τ}^{mn} (r, ω) \equiv - \frac{1}{π} ℑ F_{σ τ}^{R, mn} (ω; r, r) . (8.40)

The Davis-group SJTM experiments on Bi2212 provide a concrete example in which locally resolved pair tunnelling and its Fourier analysis were used to detect a Cooper-pair density wave. [14]

Gap map

The analogue of the normal density-of-states map in this setting is the Cooper-pair map, or gap map, obtained from the spatially resolved Cooper-pair tunnelling conductance.

Cooper pair interference

Fourier transformation of the gap map yields a Cooper-pair interference pattern, which encodes the momentum dependence of the gap structure.

Anaxagoras derives the Inseparability Principle from his No-Least principle –‘properties such as hot and cold, like large and small, exist on a scale of intensity with no upper or lower limit such that “there are no extremes to the degree of intensity of an opposite”‘[15]. It is interesting to note that in closed systems (systems without boundary), such as the electron orbital motion of the atom, Anaxagoras’ Inseparability Principle does not hold, and quantisation can occur. ↩︎
lead and lead-iridium are also used ↩︎
Friedel oscillations are defined at zero applied bias $ω$ . Pedantically, at nonzero bias the oscillations are technically not Friedel oscillations as we are probing quasiparticles away from the Fermi surface. ↩︎

References

G. Binnig and H. Rohrer, Scanning tunneling microscopy—from birth to adolescence, Rev. Mod. Phys., vol. 59, no. 3, pp. 615–625, 1987. doi:10.1103/RevModPhys.59.615 (↩︎)
G. Green, An essay on the application of mathematical analysis to the theories of electricity and magnetism. Nottingham: T. Wheelhouse, 1828. [Online]. Available: https://arxiv.org/abs/0807.0088 (↩︎)
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Green’s Functions and Physical Observables

Nambu—Gor’kov Green’s function

Matrix structure

Symmetry relations

From Green’s functions to experimental observables

Single-particle observables (ARPES/STM)

Linear-response observables (Kubo dictionary)

Field and current probes (real-space equilibrium observables)

Observable dictionary (experiment (\leftrightarrow) correlator)

Retarded and advanced Green’s functions

Density of states

Local density of states (LDOS)

(Global) density of states

Anomalous density of states

Equal-time observables (real-frequency representation)

Normal density matrix

Anomalous (pair) density

Equal-time observables (Matsubara representation)

Normal density matrix

Anomalous (pair) density

Interpretation

Scanning tunnelling microscopy

History

Theory

Realisation

Experimental observables: topographic maps, spectral maps and quasiparticle interference

Density of states

Friedel oscillations

Linecut

Topography

Density of states map

Bogoliubov quasiparticle interference

Scanning Josephson tunnelling microscopy

Gap map

Cooper pair interference

References

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