Phase transitions and critical phenomena

Classify phase transitions as first-order or continuous based on the behavior of free energy derivatives.

Define order parameters for various physical systems (e.g., liquid-gas, magnetic, structural).

Explain the concept of spontaneous symmetry breaking and its fundamental role in phase transitions.

Derive thermodynamic response functions (susceptibility, specific heat, compressibility) from the appropriate free energy.

Describe the qualitative behavior of thermodynamic quantities near a critical point.

Derive the Van der Waals equation of state and provide a physical interpretation of its parameters.

Analyze Van der Waals isotherms, identifying the unphysical region and its relation to phase separation.

Apply the Maxwell equal-area construction to determine the liquid-gas coexistence curve.

Define and plot the spinodal and binodal curves on a phase diagram, explaining their physical significance.

Locate the critical point of the Van der Waals fluid and calculate the critical constants.

Derive the law of corresponding states and explain its connection to the concept of universality.

Extract the mean-field critical exponents (β, γ, δ) from the Van der Waals equation near the critical point.

Discuss the concepts of metastability and nucleation in the context of the Van der Waals model.

Formulate the Ising Hamiltonian for ferromagnetic and antiferromagnetic interactions on a lattice.

Solve the one-dimensional Ising model exactly using the transfer matrix method.

Demonstrate the absence of a finite-temperature phase transition in the one-dimensional Ising model.

Outline the transfer matrix formalism for the two-dimensional Ising model.

State Onsager's exact solution for the free energy of the 2D Ising model.

Identify the critical exponents from the exact 2D Ising model solution and contrast them with mean-field values.

Derive the mean-field equation for the Ising model using the molecular field approximation.

Solve the self-consistency equation for magnetization graphically and analytically near the critical point.

Calculate the full set of mean-field critical exponents (α, β, γ, δ) for the Ising model.

Derive the mean-field free energy using the Bogoliubov inequality (variational principle).

Apply mean-field theory to the classical Heisenberg model and compare its predictions with the Ising case.

Qualitatively discuss the role of fluctuations and dimensionality in the breakdown of mean-field theory.

Construct the Landau free energy expansion based on symmetry arguments for a given phase transition.

Analyze the φ⁴ theory to describe a continuous phase transition and derive the temperature dependence of the order parameter.

Demonstrate that Landau theory reproduces the standard mean-field critical exponents.

Use φ⁶ theory to model first-order transitions and locate the tricritical point.

Incorporate an external field into the Landau expansion and derive the equation of state near criticality.

Apply Landau theory to model a physical system, such as a cubic-to-tetragonal structural transition.

Define the order parameter correlation function and correlation length, and explain their behavior at criticality.

Relate the static susceptibility to the integral of the correlation function via the fluctuation-dissipation theorem.

Construct the Landau-Ginzburg free energy functional by including a spatial gradient term to penalize inhomogeneities.

Derive the Ornstein-Zernike form for the correlation function from the Landau-Ginzburg functional in the Gaussian approximation.

Calculate the structure and energy of a domain wall between two ordered phases.

Derive the Ginzburg criterion for the validity of mean-field theory.

Calculate the upper critical dimension for φ⁴ theory, above which mean-field exponents are exact.

Explain the role of the Landau-Ginzburg functional as a universal effective field theory for critical phenomena.

Define the standard set of critical exponents (α, β, γ, δ, ν, η) that characterize critical points.

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

Derive the scaling relations (Rushbrooke, Widom, Fisher, Josephson) that connect the critical exponents.

Verify that mean-field exponents satisfy the scaling relations, but 2D Ising exponents require a non-zero η.

Define universality classes and explain what physical properties (symmetry, dimension) determine them.

Explain the physical motivation for the renormalization group (RG) in terms of coarse-graining and scale invariance.

Perform a Kadanoff block-spin transformation for the 1D Ising model to derive an RG flow equation.

Define RG flow in parameter space, fixed points, and their connection to critical behavior and thermodynamic phases.

Linearize the RG transformation near a fixed point and relate the eigenvalues to critical exponents.

Classify fixed points as stable, unstable, or marginal and interpret their physical meaning.

Perform a real-space RG calculation (e.g., decimation) for the 2D Ising model on a square lattice.

Explain universality in the language of RG flows and basins of attraction.

Identify the limitations of real-space RG methods that motivate a field-theoretic approach.

Outline the mapping of the Ising model to a φ⁴ field theory via the Hubbard-Stratonovich transformation.

Set up the momentum-shell RG procedure for the Landau-Ginzburg-Wilson Hamiltonian.

Perform the momentum-shell integration and rescaling steps to derive the RG flow equations for φ⁴ theory at one-loop order.

Identify the Gaussian fixed point and the Wilson-Fisher fixed point from the flow equations.

Explain the concept of relevant, irrelevant, and marginal operators in the context of momentum-space RG.

Calculate the critical exponent ν at the Wilson-Fisher fixed point to first order in ε = 4-d.

Explain the logic of the ε-expansion as a controlled perturbative expansion around the upper critical dimension d=4.

Calculate the critical exponents η and γ to first order in ε.

Compare the predictions of the ε-expansion for d=3 with numerical results for the 3D Ising model.

Describe how the RG framework explains crossover phenomena between different critical behaviors.

Generalize the φ⁴ theory to the O(n) model to describe systems with n-component order parameters.

State the key results for special limits of the O(n) model, such as the self-avoiding walk (n→0) and the spherical model (n→∞).

State the Mermin-Wagner theorem and explain why continuous symmetries cannot be spontaneously broken in d≤2.

Describe the Kosterlitz-Thouless (KT) transition in the 2D XY model as a topological phase transition without long-range order.

Explain the role of vortex-antivortex pair unbinding in driving the KT transition.

Introduce dynamic critical phenomena and the concept of critical slowing down near a phase transition.

State the dynamic scaling hypothesis and define the dynamic critical exponent z.

Implement the Metropolis Monte Carlo algorithm to simulate the 2D Ising model.

Observe critical slowing down in simulations and explain its origin from the diverging correlation time.

Describe a cluster algorithm (e.g., Wolff) and explain how it mitigates critical slowing down.

Apply the histogram reweighting technique to extrapolate simulation results to nearby temperatures from a single run.

State the finite-size scaling ansatz for thermodynamic observables like magnetization and susceptibility.

Use the Binder cumulant method to accurately locate the critical temperature from simulations on different system sizes.

Extract critical exponent ratios (e.g., β/ν, γ/ν) by performing a data collapse using the finite-size scaling ansatz.