7.2 Linear response theory

Foundations of Linear Response

Linear response theory describes how a system reacts when you nudge it slightly away from equilibrium. The central assumption is simple: if the perturbation is small enough, the system's response is proportional to the applied force. This proportionality lets you extract macroscopic transport properties (conductivity, susceptibility, diffusion coefficients) directly from the equilibrium fluctuations of the unperturbed system.

The theory is one of the most broadly useful tools in statistical mechanics. It connects microscopic correlation functions to experimentally measurable quantities across condensed matter, optics, and beyond.

Equilibrium vs. Non-equilibrium Systems

Equilibrium systems have macroscopic properties that remain constant in time; all thermodynamic fluxes vanish and detailed balance holds. Non-equilibrium systems, by contrast, exhibit time-dependent changes driven by external forces, gradients, or boundary conditions.

Linear response theory lives right at the boundary. It treats systems that are just barely pushed out of equilibrium, so the deviation from equilibrium is small enough that you can keep only the first-order (linear) term in the perturbation. This is what makes the math tractable while still capturing real physics.

Small Perturbations and Linearization

Suppose a system in equilibrium is subjected to a weak external field $F(t)$. The perturbation to the Hamiltonian takes the general form:

$H'(t) = -A \cdot F(t)$

where $A$ is the observable that couples to the field. The key assumption is that the induced change in any observable $\langle B \rangle$ is linear in $F$:

$\delta \langle B(t) \rangle = \int_{-\infty}^{t} \chi_{BA}(t - t') \, F(t') \, dt'$

Here $\chi_{BA}(t - t')$ is the response function (or generalized susceptibility). The integral is a convolution, which means you can move to Fourier space and write $\delta \langle B(\omega) \rangle = \chi_{BA}(\omega) \, F(\omega)$. This frequency-domain picture is where tools like Fourier transforms and Green's functions become especially powerful.

The upper limit of integration enforces causality: the response at time $t$ depends only on the perturbation at earlier times $t' \leq t$, so $\chi_{BA}(\tau) = 0$ for $\tau < 0$.

Time-Dependent Perturbations

When the external field oscillates or varies in time, the response function becomes frequency-dependent. This is how you analyze:

• AC conductivity $\sigma(\omega)$: the current response to an oscillating electric field
• Dielectric relaxation: how polarization lags behind an applied field
• Resonance phenomena: peaks in the imaginary part of $\chi(\omega)$ at characteristic frequencies

The frequency-dependent response naturally encodes relaxation times. A system with a single relaxation time $\tau$ gives a Debye-type susceptibility $\chi(\omega) = \chi_0 / (1 - i\omega\tau)$, while more complex systems show broadened or multi-peaked spectra.

Kubo Formula

The Kubo formula is the central result of linear response theory. It gives you a concrete recipe: compute an equilibrium time-correlation function, and you get the non-equilibrium response function.

Derivation and Significance

The derivation starts from the time-dependent perturbation to the Hamiltonian $H' = -A \cdot F(t)$ and uses the interaction picture of quantum mechanics. Working to first order in $F$, you arrive at the Kubo formula for the response function:

$\chi_{BA}(t) = \frac{i}{\hbar} \theta(t) \langle [B(t), A(0)] \rangle_0$

where:

• $\theta(t)$ is the Heaviside step function (enforcing causality)
• $[B(t), A(0)]$ is the commutator of the Heisenberg-picture operators
• $\langle \cdots \rangle_0$ denotes an average over the unperturbed equilibrium ensemble

The significance is profound: you never need to solve the full non-equilibrium problem. All the information about how the system responds to a perturbation is already encoded in the equilibrium fluctuations of the relevant observables.

For classical systems, the commutator is replaced by a Poisson bracket, and the formula becomes:

$\chi_{BA}(t) = -\beta \, \theta(t) \, \frac{d}{dt} \langle B(t) \, A(0) \rangle_0$

where $\beta = 1/(k_B T)$.

Time Correlation Functions

The correlation function $C_{BA}(t) = \langle B(t) \, A(0) \rangle_0$ measures how the spontaneous fluctuation of $A$ at time zero is statistically related to the value of $B$ at a later time $t$.

• At $t = 0$, the correlation is typically largest (equal to $\langle BA \rangle$).
• As $t \to \infty$, correlations decay to $\langle B \rangle \langle A \rangle$ (fluctuations become uncorrelated).
• The rate of decay reveals the system's relaxation time.

These correlation functions can be computed analytically for simple models, measured experimentally (e.g., via scattering), or evaluated numerically through molecular dynamics simulations. They also reveal memory effects: if correlations decay slowly, the system "remembers" its past state for a long time.

Fluctuation-Dissipation Theorem

The fluctuation-dissipation theorem (FDT) is one of the deepest results in statistical physics. It states that the way a system dissipates energy when driven out of equilibrium is directly related to how it fluctuates spontaneously at equilibrium.

In frequency space, the theorem takes the form:

$\text{Im} \, \chi(\omega) = \frac{\omega}{2 k_B T} \, S(\omega)$

where $S(\omega)$ is the spectral density of equilibrium fluctuations (the Fourier transform of the correlation function) and $\text{Im} \, \chi(\omega)$ is the dissipative part of the response.

Concrete examples of the FDT at work:

• Johnson-Nyquist noise: voltage fluctuations across a resistor are proportional to its resistance and temperature
• Brownian motion: Einstein's relation $D = k_B T / \gamma$ connects the diffusion coefficient (fluctuations) to the drag coefficient (dissipation)
• Magnetic susceptibility: spontaneous spin fluctuations determine the linear magnetic response

The FDT bridges microscopic reversibility (time-reversal symmetry of the underlying dynamics) with macroscopic irreversibility (entropy production, dissipation).

Response Functions

Response functions are the mathematical objects that encode everything about how a system reacts to external perturbations within the linear regime. They connect theoretical calculations to what you actually measure in the lab.

Susceptibility and Generalized Susceptibility

The static susceptibility $\chi_0$ measures the response to a constant applied field. For example, the magnetic susceptibility $\chi_m = \partial M / \partial H$ tells you how much magnetization $M$ you get per unit applied field $H$.

The generalized susceptibility $\chi(\omega)$ extends this to frequency-dependent perturbations. It is complex-valued:

$\chi(\omega) = \chi'(\omega) + i \chi''(\omega)$

• $\chi'(\omega)$ (real part): the reactive or in-phase response, related to energy storage
• $\chi''(\omega)$ (imaginary part): the dissipative or out-of-phase response, related to energy absorption

Physical examples:

• Electric susceptibility in dielectrics: relates polarization to applied electric field
• Magnetic susceptibility in paramagnets and ferromagnets: relates magnetization to applied magnetic field
• Mechanical compliance: relates strain to applied stress

Green's Functions in Linear Response

A Green's function $G(t - t')$ describes how a disturbance created at time $t'$ propagates through the system and affects it at time $t$. In linear response, the retarded Green's function is essentially the response function itself.

Green's functions are powerful because they let you solve inhomogeneous linear differential equations systematically. Once you know $G$, the response to any perturbation $F(t)$ is obtained by convolution. They appear throughout:

• Electronic structure calculations in solid-state physics (single-particle Green's functions)
• Many-body perturbation theory (Feynman diagrams are built from Green's functions)
• Quantum field theory

Kramers-Kronig Relations

The real and imaginary parts of any causal response function are not independent. The Kramers-Kronig relations connect them:

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

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

where $\mathcal{P}$ denotes the Cauchy principal value. These relations follow from two physical requirements:

1. Causality: the response cannot precede the perturbation
2. Analyticity: $\chi(\omega)$ is analytic in the upper half of the complex frequency plane

Practically, Kramers-Kronig relations are extremely useful. If you measure only the absorption spectrum ($\chi''$) in an experiment, you can reconstruct the full complex response function without additional measurements. They are widely used in optical spectroscopy, electrical impedance analysis, and mechanical vibration studies.

Applications in Statistical Mechanics

Electrical Conductivity

The electrical conductivity $\sigma(\omega)$ is obtained from the Kubo formula using the current-current correlation function:

$\sigma(\omega) = \frac{1}{k_B T} \int_0^{\infty} dt \, e^{i\omega t} \langle j(t) \cdot j(0) \rangle_0$

where $j$ is the current density operator. This single formula underlies:

• The Drude model: treating electrons as free particles with a scattering time $\tau$ gives $\sigma(\omega) = \sigma_0 / (1 - i\omega\tau)$, where $\sigma_0 = ne^2\tau/m$
• Frequency-dependent conductivity in AC circuits and optical measurements
• The quantum Hall effect in 2D electron systems, where the Hall conductivity is quantized in units of $e^2/h$

Magnetic Susceptibility

The magnetic susceptibility is calculated from spin-spin correlation functions:

$\chi_m(\omega) \sim \int_0^{\infty} dt \, e^{i\omega t} \langle S(t) \cdot S(0) \rangle_0$

This framework explains:

• Paramagnetism and diamagnetism: the sign and magnitude of the static susceptibility
• Curie-Weiss law: $\chi = C/(T - T_c)$ near a ferromagnetic transition, where diverging susceptibility signals a phase transition
• NMR spectroscopy: the spin relaxation times $T_1$ and $T_2$ are directly related to the spectral density of local field fluctuations

Optical Properties of Materials

Optical properties are governed by the dielectric function $\epsilon(\omega)$, which is related to the current-current or polarization-polarization correlation functions. From $\epsilon(\omega)$ you can extract:

• The refractive index $n(\omega)$ and absorption coefficient $\alpha(\omega)$
• Plasmon resonances in metals and semiconductors, where $\text{Re}[\epsilon(\omega)] = 0$
• Nonlinear optical effects (second-harmonic generation, four-wave mixing) arise when you go beyond the linear response regime

Linear Response in Quantum Systems

Quantum Mechanical Perturbation Theory

The quantum derivation of linear response starts from the time-dependent Schrödinger equation (or equivalently, the von Neumann equation for the density matrix). You expand the state in powers of the perturbation strength and keep only the first-order term.

The perturbed state to first order is:

$|\psi(t)\rangle = |\psi_0(t)\rangle + |\psi_1(t)\rangle$

where $|\psi_1(t)\rangle$ is proportional to the perturbation strength. This first-order correction is what gives rise to the Kubo formula. Higher-order terms correspond to nonlinear response.

Fermi's Golden Rule

Fermi's golden rule gives the transition rate from an initial state $|i\rangle$ to a final state $|f\rangle$ due to a perturbation $H'$:

$\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle f | H' | i \rangle|^2 \, \rho(E_f)$

where $\rho(E_f)$ is the density of final states at energy $E_f$. This result is derived from time-dependent perturbation theory in the long-time limit and is a key ingredient in calculating:

• Photon absorption and emission rates
• Electron scattering rates in solids (determining resistivity)
• Decay rates of unstable particles and excited states

Density Matrix Formulation

The density matrix $\rho$ provides a more general description than the wavefunction, especially for mixed states (statistical mixtures) and open quantum systems (systems coupled to an environment).

In linear response, you write $\rho(t) = \rho_0 + \delta\rho(t)$, where $\rho_0$ is the equilibrium density matrix and $\delta\rho$ is the first-order correction. The expectation value of any observable $B$ is then:

$\delta\langle B(t)\rangle = \text{Tr}[B \, \delta\rho(t)]$

This formulation is essential for:

• Quantum optics and laser theory (where photon number states are naturally mixed)
• Quantum information (decoherence, entanglement dynamics)
• The quantum-to-classical transition (how off-diagonal elements of $\rho$ decay)

Experimental Techniques

Spectroscopy Methods

Spectroscopy probes the frequency-dependent response of a material to electromagnetic radiation. Different frequency ranges access different excitations:

• Infrared and Raman spectroscopy: molecular vibrations (meV to hundreds of meV)
• X-ray absorption spectroscopy (XAS): electronic structure and oxidation states (keV range)
• NMR spectroscopy: local magnetic environments and spin dynamics (MHz range, corresponding to $\mu$eV)

In each case, the measured absorption or scattering cross-section is directly related to the imaginary part of the appropriate response function $\chi''(\omega)$.

Transport Measurements

Transport experiments measure how efficiently a material conducts charge, heat, or spin in response to applied gradients:

• Four-point probe: measures electrical resistivity while eliminating contact resistance
• Hall effect measurements: determine carrier concentration and mobility by applying a perpendicular magnetic field
• Thermal conductivity measurements: probe phonon and electron contributions to heat transport

These measurements provide the DC ($\omega = 0$) limit of the corresponding Kubo formula response functions.

Scattering Experiments

Scattering experiments access the dynamic structure factor $S(\mathbf{q}, \omega)$, which is the space-time Fourier transform of density-density (or spin-spin) correlation functions:

• Neutron scattering: probes magnetic structures and excitations (magnons, phonons) because neutrons carry a magnetic moment
• X-ray diffraction: determines crystal structures from elastic scattering; inelastic X-ray scattering probes phonon dispersions
• Electron energy loss spectroscopy (EELS): measures electronic excitations like plasmons and interband transitions

The connection to linear response is direct: the scattering cross-section is proportional to $S(\mathbf{q}, \omega)$, which is related to $\chi''(\mathbf{q}, \omega)$ through the fluctuation-dissipation theorem.

Limitations and Extensions

Nonlinear Response Theory

When the perturbation is not small, the linear approximation breaks down and higher-order terms matter. The response is then expanded as:

$\delta\langle B\rangle = \chi^{(1)} F + \chi^{(2)} F^2 + \chi^{(3)} F^3 + \cdots$

The higher-order susceptibilities $\chi^{(n)}$ involve $(n+1)$-point correlation functions, which are progressively harder to compute. Nonlinear response is central to:

• Nonlinear optics: second-harmonic generation ($\chi^{(2)}$), four-wave mixing ($\chi^{(3)}$)
• Chaos and bifurcation theory: where small changes in parameters produce qualitatively different behavior

Strong Perturbations

For strong driving fields, perturbation theory itself fails and you need non-perturbative methods:

• Keldysh formalism: a Green's function technique designed for genuine non-equilibrium situations, using a closed time contour
• Floquet theory: treats periodically driven systems by mapping the time-dependent problem onto an effective time-independent one in an extended Hilbert space
• Dynamical mean-field theory (DMFT): handles strongly correlated electron systems by mapping the lattice problem onto a self-consistent impurity problem

Time-Dependent Density Functional Theory

TDDFT extends ground-state density functional theory to excited states and time-dependent phenomena. The central quantity is the time-dependent electron density $n(\mathbf{r}, t)$, and the linear response version of TDDFT gives access to:

• Optical absorption spectra of molecules and solids
• Charge transfer dynamics in chemical and biological systems
• Ultrafast electronic processes on femtosecond timescales

TDDFT is computationally much cheaper than solving the full many-body Schrödinger equation, making it the workhorse method for excited-state calculations in large systems.

Numerical Methods

Monte Carlo Simulations

Monte Carlo methods use random sampling to evaluate statistical averages and correlation functions. For linear response applications:

• Metropolis algorithm: generates equilibrium configurations weighted by the Boltzmann distribution, from which you compute static correlation functions and susceptibilities
• Kinetic Monte Carlo: simulates the time evolution of stochastic processes, useful for reaction rates and diffusion
• Quantum Monte Carlo: handles many-body quantum systems, though the fermion sign problem limits applicability to certain systems

Molecular Dynamics Approaches

Molecular dynamics (MD) integrates Newton's equations (or the quantum equations of motion) for a system of interacting particles. For linear response, the key advantage is that MD directly produces time correlation functions $\langle A(t) B(0) \rangle$ from equilibrium trajectories.

• Classical MD: suitable for atomic and molecular systems with empirical force fields
• Ab initio MD (e.g., Car-Parrinello): computes forces from electronic structure on the fly, capturing bond breaking and charge transfer
• Dissipative particle dynamics: a coarse-grained method for mesoscale phenomena (polymers, membranes)

Density Matrix Renormalization Group

The DMRG method is the most accurate numerical technique for strongly correlated one-dimensional quantum systems. It works by systematically truncating the Hilbert space while retaining the most important states (those with the largest eigenvalues of the reduced density matrix).

Extensions relevant to linear response:

• Time-dependent DMRG (tDMRG): simulates real-time dynamics after a quench or perturbation, giving access to dynamical correlation functions
• Matrix product states (MPS): the mathematical framework underlying DMRG, also used in quantum information
• Tensor network methods: generalize MPS to two and three dimensions, though computational cost grows significantly