🔬Condensed Matter Physics Unit 8 Review

Kramers-Kronig (KK) relations connect the real and imaginary parts of any complex response function, and they do so purely from the requirement of causality. If you know the absorption spectrum of a material across all frequencies, you can compute its refractive index without any additional measurements. That's what makes these relations so useful in condensed matter physics: they let you extract complete optical information from incomplete experiments.

Complex Response Functions

When a material is subjected to an external perturbation (an electric field, magnetic field, or mechanical stress), its response is described by a frequency-dependent complex function. The two parts of this function carry distinct physical meaning:

The real part describes the in-phase response, representing energy storage (think of a spring stretching and returning energy).
The imaginary part describes the out-of-phase response, representing energy dissipation (think of friction converting energy to heat).

Common examples include the complex refractive index $\tilde{n}(\omega) = n(\omega) + i\kappa(\omega)$ , the complex dielectric function $\varepsilon(\omega) = \varepsilon'(\omega) + i\varepsilon''(\omega)$ , and the complex magnetic susceptibility. Each of these characterizes how a material interacts with electromagnetic fields at different frequencies.

Causality Principle

The physical foundation of KK relations is simple: an effect cannot precede its cause. A material cannot respond to a light pulse before that pulse arrives. This seemingly obvious statement has deep mathematical consequences.

In the time domain, causality means the response function $\chi(t)$ is zero for $t < 0$ . When you Fourier transform this constraint into the frequency domain, it forces $\chi(\omega)$ to be analytic in the upper half of the complex frequency plane. That analyticity is what creates the rigid link between the real and imaginary parts.

Dispersion Relations

Because causality forces the response function to be analytic, the real and imaginary parts are not independent. Knowing one across all frequencies completely determines the other. These mathematical relationships are called dispersion relations, and the KK relations are the specific form they take for physical response functions.

This is not just an abstract result. It means you can measure only the reflectance of a material (which depends on both $n$ and $\kappa$ ) and use KK analysis to separate the two contributions.

Mathematical Formulation

Hilbert Transform

The mathematical backbone of KK relations is the Hilbert transform, an integral transform that maps a function to its "conjugate function." For a function $f(t)$ , the Hilbert transform is:

$\mathcal{H}[f](t) = \frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{f(t')}{t' - t} \, dt'$

The real and imaginary parts of any causal response function are Hilbert transform pairs of each other. This is the core mathematical statement behind KK relations.

Principal Value Integral

The integrals in the KK relations have a singularity where $\omega' = \omega$ (the denominator goes to zero). The Cauchy principal value, denoted $P$ , handles this by symmetrically excluding a small interval around the singularity and taking the limit as that interval shrinks to zero:

$P \int_{-\infty}^{\infty} \frac{f(\omega')}{\omega' - \omega} \, d\omega' = \lim_{\delta \to 0^+} \left[ \int_{-\infty}^{\omega - \delta} + \int_{\omega + \delta}^{\infty} \right] \frac{f(\omega')}{\omega' - \omega} \, d\omega'$

The symmetric exclusion ensures that the divergent contributions from either side of the singularity cancel, leaving a finite result.

Kramers-Kronig Equations

For a complex susceptibility $\chi(\omega) = \chi'(\omega) + i\chi''(\omega)$ , the KK relations are:

$\chi'(\omega) = \frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{\chi''(\omega')}{\omega' - \omega} \, d\omega'$

$\chi''(\omega) = -\frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{\chi'(\omega')}{\omega' - \omega} \, d\omega'$

These are derived by applying Cauchy's integral theorem to $\chi(\omega)$ along a contour in the upper half-plane, then using the fact that $\chi(\omega) \to 0$ as $|\omega| \to \infty$ (the contribution from the semicircular arc vanishes).

Because physical response functions obey crossing symmetry ( $\chi(-\omega) = \chi^*(\omega)$ for real-valued time-domain responses), these can be rewritten as integrals over positive frequencies only:

$\chi'(\omega) = \frac{2}{\pi} P \int_{0}^{\infty} \frac{\omega' \chi''(\omega')}{\omega'^2 - \omega^2} \, d\omega'$

$\chi''(\omega) = -\frac{2\omega}{\pi} P \int_{0}^{\infty} \frac{\chi'(\omega')}{\omega'^2 - \omega^2} \, d\omega'$

This positive-frequency form is the version most commonly used in practice, since experiments only measure at positive frequencies.

Physical Significance

Connection to Causality

The KK relations are not just a mathematical convenience; they are a direct consequence of the physical requirement that materials cannot respond before being stimulated. Any proposed model for a response function that violates KK relations would imply acausal (and therefore unphysical) behavior. This makes KK relations a consistency check: if your measured or calculated $\chi'$ and $\chi''$ don't satisfy them, something is wrong with the data or the model.

Real vs. Imaginary Parts

The interdependence of $\chi'$ and $\chi''$ has a concrete physical consequence: you cannot have absorption without dispersion, and vice versa. Near an absorption resonance (a peak in $\chi''$ ), the real part $\chi'$ undergoes a rapid change known as anomalous dispersion. This is not a coincidence but a direct result of the KK relations.

For the complex refractive index $\tilde{n} = n + i\kappa$ :

$n(\omega)$ (refractive index) governs how much the phase velocity changes.
$\kappa(\omega)$ (extinction coefficient) governs how strongly light is absorbed.

KK relations tell you that a sharp absorption feature in $\kappa$ necessarily produces a characteristic S-shaped feature in $n$ at the same frequency.

Complex response functions, Frontiers | Better Resolved Low Frequency Dispersions by the Apt Use of Kramers-Kronig Relations ...

Time-Domain Interpretation

The connection between real/imaginary parts and even/odd symmetry provides useful intuition:

The real part $\chi'(\omega)$ is the Fourier transform of the even part of the time-domain response.
The imaginary part $\chi''(\omega)$ is the Fourier transform of the odd part of the time-domain response.

Since causality forces $\chi(t) = 0$ for $t < 0$ , the even and odd parts are not independent. They must combine to produce exact cancellation at negative times. That constraint is what the KK relations express in the frequency domain.

Applications in Condensed Matter

Optical Properties of Materials

The most common application of KK relations in condensed matter is extracting the full complex optical response from reflectance measurements. A typical workflow:

Measure the normal-incidence reflectance $R(\omega)$ over as wide a frequency range as possible.
Compute $\ln R(\omega)$ and note that the complex reflectivity is $r(\omega) = \sqrt{R(\omega)} \, e^{i\theta(\omega)}$ .
Use a KK relation to obtain the phase $\theta(\omega)$ from $\ln\sqrt{R(\omega)}$ : $\theta(\omega) = -\frac{2\omega}{\pi} P \int_0^{\infty} \frac{\ln\sqrt{R(\omega')}}{\omega'^2 - \omega^2} \, d\omega'$
Combine $R(\omega)$ and $\theta(\omega)$ to extract $n(\omega)$ and $\kappa(\omega)$ , and from those, the full dielectric function.

This approach is widely used in infrared and optical spectroscopy of metals, semiconductors, and insulators.

Dielectric Response Functions

KK relations applied to the complex dielectric function $\varepsilon(\omega) = \varepsilon'(\omega) + i\varepsilon''(\omega)$ connect the polarization response (energy storage in $\varepsilon'$ ) to the absorption response (energy loss in $\varepsilon''$ ). This is central to studying:

Phonons: Infrared-active lattice vibrations produce peaks in $\varepsilon''$ and corresponding features in $\varepsilon'$ .
Plasmons: Collective oscillations of free electrons, identified where $\varepsilon'(\omega) = 0$ and $\varepsilon''$ is small.
Excitons: Bound electron-hole pairs that create sharp absorption features below the band gap.

Electron Energy Loss Spectroscopy

In EELS, the measured quantity is the loss function $-\text{Im}[1/\varepsilon(\omega)]$ . KK analysis allows you to extract the full complex $\varepsilon(\omega)$ from this single measured quantity:

Measure the loss function from the EELS spectrum.
Apply KK transformation to obtain $\text{Re}[1/\varepsilon(\omega)]$ .
Invert to get both $\varepsilon'(\omega)$ and $\varepsilon''(\omega)$ .

This technique provides detailed information about interband transitions, surface plasmons, and core-level excitations.

Experimental Considerations

Measurement Limitations

KK relations formally require knowledge of the response function from $\omega = 0$ to $\omega = \infty$ . No experiment covers this full range. Typical optical measurements might span from the far-infrared (~10 cm $^{-1}$ ) to the ultraviolet (~50,000 cm $^{-1}$ ), leaving large gaps at both ends. Noise and systematic errors (baseline drift, detector nonlinearity) further degrade the reliability of KK-transformed data, especially far from the measured range.

Extrapolation Techniques

To handle the finite measurement range, you need to extrapolate the data at both high and low frequencies before performing the KK integral. Common approaches:

Low-frequency: For metals, a Drude model extrapolation ( $R \to 1$ as $\omega \to 0$ ). For insulators, $R$ is assumed roughly constant below the lowest measured frequency.
High-frequency: Power-law decay, typically $R(\omega) \propto \omega^{-s}$ with $s$ between 2 and 4, guided by free-electron or x-ray scattering behavior.
Model-based: Fit a physical model (Drude-Lorentz oscillators) to the measured data and use the model for extrapolation.

The choice of extrapolation has the largest impact on the KK-transformed result near the edges of the measured range and progressively less impact far from the edges.

Data Analysis Methods

Numerical implementation of KK transforms on discrete data requires care:

Direct integration: Evaluate the principal value integral numerically, using techniques like subtracting the singularity analytically before integrating.
FFT-based methods: Exploit the connection between KK relations and Hilbert transforms, which can be computed efficiently via fast Fourier transforms.
Regularization: Apply smoothing or constraints to suppress noise amplification during the transformation.

Several software packages exist for KK analysis (e.g., RefFIT, KKcalc), and many research groups maintain in-house codes tailored to their specific spectroscopic techniques.

Extensions and Generalizations

Complex response functions, ARS - Application of Kramers-Kronig transformations to increase the bandwidth of small antennas

Generalized Kramers-Kronig Relations

The standard KK relations assume a scalar, linear, isotropic response. Extensions exist for more complex situations:

Anisotropic materials: Tensor KK relations apply to each independent component of the dielectric tensor separately.
Magneto-optical effects: Off-diagonal components of the dielectric tensor (responsible for Faraday and Kerr effects) also satisfy KK relations.
Nonlinear optics: Modified KK relations can be formulated for nonlinear susceptibilities, though additional assumptions are needed.

Sum Rules

Integrating the KK relations over all frequencies yields sum rules, which are powerful constraints on response functions. These relate integrals of the response to static or fundamental properties of the material.

For example, the f-sum rule (or Thomas-Reiche-Kuhn sum rule) for the dielectric function states:

$\int_0^{\infty} \omega \, \varepsilon''(\omega) \, d\omega = \frac{\pi}{2} \omega_p^2$

where $\omega_p = \sqrt{n_e e^2 / \varepsilon_0 m_e}$ is the plasma frequency and $n_e$ is the electron density. This means the total integrated absorption is fixed by the number of electrons, regardless of how the absorption is distributed across frequencies.

Sum rules serve as consistency checks on experimental data and theoretical models. If the integrated absorption doesn't match the known electron density, either the data has errors or the frequency range is insufficient.

Limitations and Challenges

Finite Frequency Range

This is the single biggest practical challenge. Since KK integrals run from zero to infinity, truncation errors are unavoidable. Several strategies help mitigate this:

The anchor point method uses independently measured values of $n$ or $\varepsilon$ at specific frequencies to constrain the KK transformation.
Maximum entropy methods use statistical inference to find the most probable extrapolation consistent with the measured data.
Combining data from multiple experimental techniques (e.g., infrared reflectance + ellipsometry + x-ray scattering) extends the effective frequency range.

Kramers-Kronig vs. Direct Measurements

KK analysis is an indirect method. Techniques like spectroscopic ellipsometry can measure both $n$ and $\kappa$ (or $\varepsilon'$ and $\varepsilon''$ ) simultaneously and directly, without needing KK transformation. When direct measurements are available, they are generally preferred because they avoid extrapolation uncertainties.

That said, KK analysis remains valuable when only a single measured quantity is available (like reflectance or the EELS loss function), or as a cross-check on directly measured data.

Numerical Implementation Issues

Practical challenges in computing KK transforms include:

Singularity handling: The principal value integral requires careful treatment at $\omega' = \omega$ . Common approaches include analytic subtraction of the singular part or using specialized quadrature rules.
Discretization errors: Converting continuous integrals to sums over discrete data points introduces errors, especially if the data spacing is coarse near sharp spectral features.
Error propagation: Noise in the input data gets amplified by the KK transformation, particularly at frequencies far from strong spectral features. Regularization or smoothing can help, but at the cost of spectral resolution.

Linear Response Theory

KK relations apply specifically to linear response functions, where the output is proportional to the input. Linear response theory, developed by Kubo, provides the formal framework connecting microscopic quantum mechanics to macroscopic response functions like $\varepsilon(\omega)$ and $\sigma(\omega)$ . KK relations are built into this framework from the start, since the Kubo formulas automatically produce causal response functions.

Green's Functions

In many-body physics, single-particle and two-particle Green's functions describe how excitations propagate through a material. These Green's functions are analytic in the appropriate half-plane (upper or lower, depending on convention), and their real and imaginary parts satisfy KK-type relations. The imaginary part of the single-particle Green's function gives the spectral function, which is directly related to photoemission spectra.

Fluctuation-Dissipation Theorem

The fluctuation-dissipation theorem (FDT) connects the imaginary part of a response function to equilibrium fluctuations in the system. Combined with KK relations, this creates a powerful triad: equilibrium fluctuations determine the dissipative (imaginary) part of the response, and KK relations then determine the reactive (real) part. Together, FDT and KK relations mean that equilibrium thermal fluctuations contain all the information needed to predict a material's complete linear response.

🔬Condensed Matter Physics Unit 8 Review