4.1 Landau theory

Landau theory provides a powerful framework for understanding phase transitions in condensed matter systems. It uses symmetry principles and thermodynamic considerations to model complex physical phenomena, forming the foundation for understanding critical behavior and universality.

The theory expands the free energy in terms of an order parameter, which quantifies the degree of order in a system undergoing a phase transition. Symmetry considerations determine the allowed terms in the expansion, guiding the construction of the Landau free energy functional.

Fundamentals of Landau theory

• Landau theory provides a powerful framework for describing phase transitions in condensed matter systems
• Utilizes symmetry principles and thermodynamic considerations to model complex physical phenomena
• Forms the foundation for understanding critical behavior and universality in phase transitions

Free energy expansion

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• Expands the free energy as a power series in terms of an order parameter
• Truncates the expansion to include only symmetry-allowed terms
• Minimization of free energy determines equilibrium states
• Coefficients in the expansion depend on temperature and other control parameters
• Higher-order terms become important near critical points

Order parameter concept

• Quantifies the degree of order in a system undergoing a phase transition
• Takes on non-zero values in the ordered phase and vanishes in the disordered phase
• Examples include magnetization in ferromagnets and density difference in liquid-gas transitions
• Symmetry of the order parameter reflects the symmetry breaking in the phase transition
• Can be scalar ($\phi$), vector ($\vec{M}$), or tensor ($Q_{ij}$) quantities depending on the system

Symmetry considerations

• Determines the allowed terms in the free energy expansion
• Ensures the free energy remains invariant under symmetry operations of the high-temperature phase
• Guides the construction of the Landau free energy functional
• Explains the universality of critical behavior for systems with similar symmetries
• Predicts possible types of phase transitions based on symmetry arguments

Phase transitions in Landau theory

• Landau theory classifies phase transitions based on the behavior of the order parameter
• Provides a unified description of various types of phase transitions in condensed matter systems
• Enables prediction of thermodynamic properties near critical points

Second-order transitions

• Characterized by continuous change in the order parameter at the transition temperature
• Free energy expansion contains only even powers of the order parameter
• Examples include ferromagnetic transitions and superconducting transitions in zero magnetic field
• Order parameter grows as $\phi \propto (T_c - T)^\beta$ below the critical temperature
• Susceptibility diverges as $\chi \propto |T - T_c|^{-\gamma}$ near the critical point

First-order transitions

• Exhibit discontinuous jumps in the order parameter at the transition temperature
• Free energy expansion includes odd powers of the order parameter
• Examples include liquid-gas transitions and some structural phase transitions in crystals
• Characterized by latent heat and coexistence of phases at the transition point
• Hysteresis effects often observed due to metastable states

Tricritical points

• Occur at the intersection of lines of second-order and first-order phase transitions
• Require higher-order terms in the Landau free energy expansion
• Examples include the tricritical point in $^3He-^4He$ mixtures and in certain metamagnetic systems
• Exhibit unique critical exponents different from both first-order and second-order transitions
• Landau theory predicts mean-field tricritical exponents ($\beta = 1/4$, $\gamma = 1$)

Critical exponents

• Describe the power-law behavior of various physical quantities near critical points
• Provide a quantitative characterization of universality in phase transitions
• Play a crucial role in connecting theory with experimental observations

Mean-field approximation

• Assumes spatial fluctuations in the order parameter are negligible
• Predicts universal critical exponents independent of microscopic details
• Examples of mean-field exponents: $\beta = 1/2$, $\gamma = 1$, $\delta = 3$
• Becomes exact for systems with long-range interactions or in high dimensions
• Provides a good starting point for more sophisticated treatments

Universality classes

• Group systems with similar symmetries and dimensionality into classes with identical critical behavior
• Examples include the Ising universality class ($d=3$, $n=1$) and the XY universality class ($d=3$, $n=2$)
• Determined by the dimensionality of space and the symmetry of the order parameter
• Explain why seemingly different systems exhibit the same critical exponents
• Allow for the application of results from simple models to more complex real-world systems

Scaling relations

• Connect different critical exponents through mathematical identities
• Examples include Rushbrooke's identity: $\alpha + 2\beta + \gamma = 2$
• Reduce the number of independent critical exponents
• Provide consistency checks for experimental measurements and theoretical predictions
• Derived from the homogeneity of the free energy near the critical point

Applications of Landau theory

• Landau theory finds widespread use in various areas of condensed matter physics
• Provides a unified framework for understanding diverse phase transitions
• Enables predictions of critical behavior and phase diagrams in complex systems

Ferromagnetic transitions

• Describes the spontaneous magnetization below the Curie temperature
• Order parameter: magnetization $\vec{M}$
• Free energy expansion: $F = F_0 + a(T-T_c)M^2 + bM^4 - \vec{H} \cdot \vec{M}$
• Predicts critical exponents for magnetization, susceptibility, and specific heat
• Explains the emergence of domains and domain walls in ferromagnets

Superconducting transitions

• Models the transition from normal to superconducting state
• Order parameter: complex superconducting gap $\Delta = |\Delta|e^{i\phi}$
• Free energy expansion includes gradient terms to account for spatial variations
• Predicts the existence of two characteristic lengths: penetration depth and coherence length
• Explains the Meissner effect and the difference between type-I and type-II superconductors

Structural phase transitions

• Describes transitions involving changes in crystal symmetry
• Order parameter often related to atomic displacements or strain
• Examples include ferroelectric transitions and martensitic transformations
• Predicts the appearance of soft modes and central peaks in spectroscopic measurements
• Explains the coupling between different order parameters in multiferroic materials

Limitations of Landau theory

• While powerful, Landau theory has certain limitations in describing phase transitions
• Understanding these limitations is crucial for applying the theory appropriately
• Motivates the development of more advanced theoretical approaches

Fluctuations near critical point

• Landau theory neglects spatial fluctuations of the order parameter
• Fluctuations become increasingly important as the critical point is approached
• Lead to deviations from mean-field behavior in low-dimensional systems
• Cause the breakdown of Landau theory within the Ginzburg criterion
• Require more sophisticated techniques like the renormalization group to describe accurately

Breakdown of mean-field approach

• Mean-field approximation fails below the upper critical dimension
• Upper critical dimension: $d_c = 4$ for most systems with short-range interactions
• Leads to incorrect predictions of critical exponents in low-dimensional systems
• Examples: 2D Ising model, where exact solutions deviate significantly from mean-field predictions
• Necessitates the use of more advanced theoretical methods for accurate descriptions

Beyond Landau theory

• More advanced approaches include renormalization group methods and conformal field theory
• These techniques can accurately describe critical phenomena in low-dimensional systems
• Capture the effects of fluctuations and long-range correlations near critical points
• Provide a deeper understanding of universality and scaling in phase transitions
• Enable the calculation of non-classical critical exponents and scaling functions

Ginzburg-Landau theory

• Extends Landau theory to include spatial variations of the order parameter
• Particularly important for describing superconductors and other inhomogeneous systems
• Provides a bridge between microscopic theories and macroscopic phenomenology

Extension to spatially varying systems

• Introduces gradient terms in the free energy functional: $F = \int d^3r [f(\phi) + K(\nabla\phi)^2]$
• Allows for the description of interfaces, domain walls, and topological defects
• Enables the study of finite-size effects and boundary conditions on phase transitions
• Predicts the existence of coherence lengths and correlation functions
• Forms the basis for more advanced field-theoretic treatments of critical phenomena

Coherence length

• Characterizes the spatial scale over which the order parameter can vary
• Defined as $\xi = \sqrt{K/|a|}$ in the Ginzburg-Landau formalism
• Diverges at the critical point as $\xi \propto |T-T_c|^{-\nu}$
• Plays a crucial role in determining the properties of interfaces and defects
• Determines the range of validity of mean-field theory (Ginzburg criterion)

Critical fields in superconductors

• Ginzburg-Landau theory predicts the existence of two critical magnetic fields
• Lower critical field $H_{c1}$: onset of flux penetration in type-II superconductors
• Upper critical field $H_{c2}$: complete suppression of superconductivity
• Explains the mixed state in type-II superconductors (Abrikosov vortex lattice)
• Provides a framework for understanding high-temperature superconductors

Experimental validation

• Experimental techniques play a crucial role in validating Landau theory predictions
• Provide quantitative measurements of critical exponents and scaling relations
• Enable the classification of systems into universality classes

Neutron scattering techniques

• Probe the spatial correlations of the order parameter
• Measure the static structure factor $S(q)$ and dynamic susceptibility $\chi(q,\omega)$
• Reveal critical scattering and the divergence of correlation length
• Provide direct evidence for soft modes in structural phase transitions
• Enable the study of magnetic order and fluctuations in spin systems

Specific heat measurements

• Measure the temperature dependence of specific heat near critical points
• Reveal the critical exponent $\alpha$ associated with the specific heat singularity
• Provide evidence for the lambda-point transition in liquid helium
• Enable the detection of latent heat in first-order phase transitions
• Allow for the determination of the order of the transition

Susceptibility studies

• Measure the response of the system to external fields
• Reveal the critical exponent $\gamma$ associated with the divergence of susceptibility
• Provide information about the nature of the ordered phase
• Enable the detection of crossover phenomena between different universality classes
• Allow for the study of critical slowing down in dynamic susceptibility measurements

Computational methods

• Computational techniques complement analytical approaches in studying phase transitions
• Enable the investigation of complex systems beyond the reach of exact solutions
• Provide a bridge between theory and experiment in critical phenomena

Monte Carlo simulations

• Simulate the behavior of many-particle systems near critical points
• Employ importance sampling techniques to efficiently explore phase space
• Enable the calculation of thermodynamic averages and correlation functions
• Provide accurate estimates of critical exponents and universal amplitude ratios
• Allow for the study of finite-size effects and crossover phenomena

Renormalization group approach

• Provides a systematic way to handle fluctuations near critical points
• Explains the origin of universality in critical phenomena
• Enables the calculation of non-classical critical exponents
• Reveals the existence of fixed points and relevant/irrelevant operators
• Allows for the construction of epsilon expansions and other perturbative schemes

Finite-size scaling

• Analyzes the behavior of finite systems to extract information about the thermodynamic limit
• Enables the determination of critical exponents from simulations of finite systems
• Provides a way to estimate critical temperatures and other parameters
• Reveals universal scaling functions characterizing the critical behavior
• Allows for the study of crossover effects between different universality classes