Advanced Statistical Mechanics

Define spontaneous symmetry breaking (SSB) and the role of the order parameter for discrete (Ising) and continuous (O(N)) symmetries.

Analyze the mean-field phase diagram of the O(N) model, identifying the XY (N=2) and Heisenberg (N=3) models as special cases.

State Goldstone's theorem regarding the emergence of massless modes upon the breaking of a continuous symmetry.

Prove Goldstone's theorem by analyzing the second derivative of the effective potential in the direction of a broken symmetry generator.

Identify the physical Goldstone modes in different systems: magnons in ferromagnets and phonons in crystals.

Derive the classical spin-wave dispersion relation for the Heisenberg ferromagnet.

Calculate the low-temperature correction to the magnetization in a 3D Heisenberg ferromagnet due to magnons (Bloch's T^(3/2) law).

Derive the nonlinear sigma model as the low-energy effective field theory for the Goldstone modes of the O(N) model.

Analyze infrared divergences in the propagator of the 2D nonlinear sigma model to motivate the absence of long-range order.

State and prove the Mermin-Wagner theorem, which forbids SSB of continuous symmetries in d ≤ 2 for systems with short-range interactions.

Discuss the physical implications of the Mermin-Wagner theorem for 2D magnets and thin superfluid films.

Explain the concept of quasi-long-range order, characterized by the algebraic decay of correlation functions.

Define the upper critical dimension (d_u=4) and lower critical dimension (d_l=2 for continuous symmetry) for the O(N) model.

Contrast the behavior of systems with discrete vs. continuous symmetries in low dimensions.

Define topological defects (vortices and anti-vortices) in the 2D XY model and compute the energy of an isolated vortex.

Derive the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature by balancing the energy and entropy of a free vortex.

Describe the BKT transition as a vortex-unbinding transition, distinguishing the low-T (bound pairs) and high-T (free plasma) phases.

Explain the concept of universal jump in the superfluid stiffness (helicity modulus) at the BKT transition.

Outline the RG flow equations for the BKT transition and interpret the flow diagram in the fugacity-stiffness plane.

Describe experimental signatures of the BKT transition in thin-film superfluid helium and 2D superconductors.

Define the standard critical exponents (α, β, γ, δ, ν, η) that characterize singular behavior near a continuous phase transition.

State the scaling hypothesis for the singular part of the free energy and the two-point correlation function.

Derive the Widom, Fisher, and Josephson scaling relations among the critical exponents from the scaling hypothesis.

Explain the concept of universality and universality classes in terms of shared critical exponents.

Demonstrate how to use data collapse of experimental or numerical data to verify scaling and extract critical exponents.

Formulate the principles of finite-size scaling for systems of finite linear extent L.

Define multicritical points (e.g., tricritical point) and describe how they modify the scaling picture.

Explain the conceptual basis of the Renormalization Group (RG) using Kadanoff's block-spin real-space transformation.

Define the RG transformation as a flow in the space of Hamiltonians (couplings) and identify fixed points with scale-invariant critical points.

Classify operators/couplings as relevant, irrelevant, or marginal based on their behavior under the RG flow.

Relate the critical exponents ν and y_h to the eigenvalues of the linearized RG flow around a non-trivial fixed point.

Explain universality as the convergence of RG flows from a large basin of attraction to the same fixed point.

Describe the role of the Wilson-Fisher fixed point in determining non-mean-field exponents for d < 4.

Outline the logic of the epsilon-expansion (ε = 4-d) as a perturbative tool for calculating critical exponents.

Derive the formula for the linear response of an observable to a weak, time-dependent external field using first-order perturbation theory.

Define the generalized dynamic susceptibility (response function) and express it as a retarded correlation function using the Kubo formula.

Derive the Kramers-Kronig relations connecting the real and imaginary parts of the frequency-dependent susceptibility.

State and derive the fluctuation-dissipation theorem, relating the dissipative part of the response function to equilibrium time-correlation functions.

Apply the fluctuation-dissipation theorem to derive the Einstein relation between diffusion and mobility.

Apply the fluctuation-dissipation theorem to derive the Johnson-Nyquist relation between electrical resistance and voltage noise.

State the Onsager reciprocal relations for coupled transport coefficients and explain their origin in microscopic time-reversal symmetry.

Describe the two-fluid model of superfluidity and the Landau criterion for dissipationless flow based on the excitation spectrum.

Connect Bose-Einstein condensation to the concept of off-diagonal long-range order (ODLRO) in the one-body density matrix.

Construct the Ginzburg-Landau (GL) free energy for a superconductor with a complex order parameter minimally coupled to a magnetic field.

Derive the two characteristic length scales of GL theory: the coherence length (ξ) and the magnetic penetration depth (λ).

Use the Ginzburg-Landau parameter κ = λ/ξ to classify superconductors into Type I and Type II.

Derive flux quantization in a superconducting ring from the single-valuedness of the GL order parameter.

Describe the structure of an Abrikosov vortex and the formation of a vortex lattice in Type II superconductors between H_c1 and H_c2.

Define the nematic liquid crystal phase and its tensorial order parameter Q_αβ.

Construct the Landau-de Gennes free energy for the nematic-isotropic transition and show how the cubic invariant leads to a first-order transition.

Write down the Frank-Oseen elastic free energy for splay, twist, and bend distortions of the nematic director field.

Model an ideal polymer as a Gaussian chain and derive its mean-square end-to-end distance and entropic elasticity.

Model a real polymer in a good solvent as a self-avoiding walk and use the Flory argument to estimate the swelling exponent ν.

Derive the Flory-Huggins free energy of mixing for a polymer solution using a mean-field lattice model.

Analyze the Flory-Huggins free energy to determine the phase diagram and identify the upper critical solution temperature.