🔬Condensed Matter Physics Unit 4 – Phase Transitions & Critical Phenomena

Key Concepts

• Phase transitions occur when a system undergoes a change in its physical properties (density, magnetization, or electrical conductivity) as a result of varying external conditions (temperature, pressure, or magnetic field)
• Critical phenomena refer to the behavior of a system near its critical point, where the distinction between two phases vanishes and the system exhibits scale invariance and universality
  • Scale invariance means that the system looks the same at different length scales near the critical point
  • Universality implies that systems with different microscopic details can exhibit the same critical behavior
• Order parameters quantify the degree of order in a system and distinguish between different phases (magnetization in ferromagnets or density difference between liquid and gas phases)
• Symmetry breaking occurs when a system transitions from a high-symmetry phase to a low-symmetry phase (liquid to solid or paramagnetic to ferromagnetic)
• Fluctuations become long-ranged and correlated near the critical point, leading to divergences in thermodynamic quantities (specific heat, susceptibility, or correlation length)
• Renormalization group theory provides a framework to understand the scale invariance and universality of critical phenomena by systematically coarse-graining the system and studying how its properties change under scale transformations

Types of Phase Transitions

• First-order phase transitions exhibit a discontinuous change in the order parameter and latent heat (melting, boiling, or sublimation)
  • They involve phase coexistence and hysteresis, where the system can exist in both phases simultaneously (liquid-gas coexistence or supercooling/superheating)
• Second-order phase transitions show a continuous change in the order parameter and no latent heat (ferromagnetic or superconducting transitions)
  • They are characterized by divergences in thermodynamic quantities and power-law behavior near the critical point
• Infinite-order phase transitions, such as the Kosterlitz-Thouless transition in 2D systems, involve a continuous change in the order parameter but no symmetry breaking
• Quantum phase transitions occur at absolute zero temperature and are driven by changes in quantum fluctuations rather than thermal fluctuations (superconductor-insulator transition or quantum Hall transitions)
• Topological phase transitions involve changes in the topological properties of the system without any symmetry breaking (topological insulator transitions)
• Glass transitions, observed in amorphous materials, are characterized by a dramatic slowdown of relaxation dynamics without any clear thermodynamic singularities

Thermodynamics of Phase Transitions

• Gibbs free energy $G = U - TS + PV$ determines the stability of phases, with the stable phase having the lowest $G$ at given temperature and pressure
• First-order transitions occur when the Gibbs free energy of two phases becomes equal, leading to a discontinuous change in the order parameter and latent heat
• Second-order transitions are characterized by a continuous change in the order parameter and divergences in the second derivatives of the Gibbs free energy (specific heat $C = -T(\partial^2 G/\partial T^2)_P$ or susceptibility $\chi = -(\partial^2 G/\partial h^2)_T$)
• Ehrenfest classification of phase transitions is based on the discontinuities in the derivatives of the Gibbs free energy
• Clausius-Clapeyron equation $dP/dT = \Delta S/\Delta V$ relates the slope of the phase boundary to the entropy and volume changes during a first-order transition
• Critical point is characterized by the vanishing of the first and second derivatives of the Gibbs free energy, leading to divergences in thermodynamic quantities and power-law behavior
• Scaling hypothesis states that the singular part of the Gibbs free energy near the critical point is a homogeneous function of the reduced temperature $t = (T - T_c)/T_c$ and the reduced field $h$, leading to universal scaling relations between critical exponents

Order Parameters and Symmetry Breaking

• Order parameters are physical quantities that distinguish between different phases and quantify the degree of order in the system
  • Examples include magnetization in ferromagnets, density difference between liquid and gas phases, or the complex amplitude of the superconducting order parameter
• Symmetry breaking occurs when the system transitions from a high-symmetry phase to a low-symmetry phase, with the order parameter acquiring a non-zero value
  • Ferromagnetic transition breaks the rotational symmetry, with the magnetization pointing in a specific direction
  • Liquid-solid transition breaks the translational and rotational symmetries, with the atoms arranging in a periodic lattice
• Landau theory describes phase transitions in terms of the order parameter and its symmetry, with the free energy expanded as a polynomial in the order parameter
• Ginzburg-Landau theory extends Landau theory to include spatial variations of the order parameter, allowing for the description of domain walls, vortices, or other topological defects
• Spontaneous symmetry breaking occurs when the system chooses a specific direction for the order parameter among equivalent possibilities, even in the absence of an external field
• Goldstone modes are low-energy excitations that arise from the spontaneous breaking of a continuous symmetry, such as spin waves in ferromagnets or phonons in solids

Critical Exponents and Universality

• Critical exponents characterize the power-law behavior of thermodynamic quantities near the critical point, such as the specific heat $C \sim |t|^{-\alpha}$, the order parameter $m \sim |t|^\beta$, or the correlation length $\xi \sim |t|^{-\nu}$
  • $t = (T - T_c)/T_c$ is the reduced temperature, measuring the distance from the critical point
• Universality means that systems with different microscopic details can exhibit the same critical behavior and share the same set of critical exponents
  • Systems are grouped into universality classes based on their dimensionality, symmetry of the order parameter, and range of interactions
• Scaling relations connect different critical exponents, such as $\alpha + 2\beta + \gamma = 2$ or $\nu d = 2 - \alpha$, where $d$ is the spatial dimension
• Hyperscaling relations involve the spatial dimension and are valid below the upper critical dimension, such as $2 - \alpha = \nu d$
• Renormalization group theory explains the universality of critical phenomena by studying how the system's properties change under scale transformations and identifying fixed points that correspond to different universality classes
• Experimental measurements of critical exponents provide a stringent test of theoretical predictions and help classify systems into universality classes

Landau Theory

• Landau theory is a phenomenological approach to describe phase transitions in terms of the order parameter and its symmetry
• Free energy is expanded as a polynomial in the order parameter $F = F_0 + a(T-T_c)\phi^2 + b\phi^4 + \cdots$, with the coefficients depending on temperature and other external parameters
  • $a$ and $b$ are phenomenological coefficients, with $a$ changing sign at the critical temperature $T_c$ and $b > 0$ for stability
• Minimizing the free energy with respect to the order parameter yields the equilibrium value of $\phi$ and the nature of the phase transition
  • First-order transitions occur when $a > 0$ and $b < 0$, with a discontinuous jump in $\phi$ and latent heat
  • Second-order transitions occur when $a$ changes sign and $b > 0$, with a continuous change in $\phi$ and diverging susceptibility
• Landau theory can be extended to include spatial variations of the order parameter (Ginzburg-Landau theory) or coupling to other fields (magnetoelastic coupling or multiferroics)
• Limitations of Landau theory include the neglect of fluctuations near the critical point and the assumption of a single order parameter
• Renormalization group theory provides a more rigorous framework to study critical phenomena and goes beyond the mean-field approximation of Landau theory

Experimental Techniques

• Specific heat measurements can detect the divergence or discontinuity associated with phase transitions, using techniques such as calorimetry or differential scanning calorimetry (DSC)
• Magnetic susceptibility measurements probe the response of the system to an external magnetic field and can identify magnetic phase transitions (Curie-Weiss law or superconducting transitions)
• Neutron scattering is a powerful technique to study the microscopic structure and dynamics of materials, providing information on the order parameter, correlation functions, and excitation spectra
  • Elastic neutron scattering measures the static structure factor and can detect long-range order or symmetry breaking
  • Inelastic neutron scattering probes the dynamic structure factor and can measure the dispersion relations of excitations (phonons, magnons, or superconducting gap)
• X-ray scattering is complementary to neutron scattering and is sensitive to the electron density distribution, allowing for the study of structural phase transitions or charge order
• Electrical transport measurements (resistivity, Hall effect, or magnetoresistance) can detect metal-insulator transitions, superconductivity, or quantum Hall effects
• Thermal expansion and magnetostriction measurements probe the coupling between the order parameter and the lattice, which can lead to anomalies near phase transitions
• Scanning probe techniques (STM, AFM, or MFM) provide real-space imaging of the local order parameter, allowing for the study of domain structures, vortices, or other topological defects

Applications in Real Systems

• Ferromagnetic materials (Fe, Ni, or Gd) exhibit a second-order phase transition from a paramagnetic to a ferromagnetic state, with critical behavior described by the Heisenberg universality class
• Superconductors undergo a second-order phase transition from a normal to a superconducting state, with the complex order parameter breaking the U(1) gauge symmetry (BCS theory or Ginzburg-Landau theory)
• Liquid-gas transitions are first-order and exhibit phase coexistence and critical opalescence near the critical point, with universal behavior described by the Ising universality class
• Structural phase transitions in solids, such as the cubic-tetragonal transition in SrTiO3 or the martensitic transition in shape-memory alloys, involve a change in the crystal symmetry and can be first-order or second-order
• Quantum phase transitions, such as the superconductor-insulator transition in thin films or the quantum Hall transitions in 2D electron gases, are driven by quantum fluctuations and exhibit critical behavior at zero temperature
• Topological phase transitions, such as the quantum spin Hall effect in HgTe/CdTe quantum wells or the topological insulator-to-metal transition in Bi2Se3, involve changes in the topological properties of the electronic band structure
• Universality and scaling behavior have been observed in a wide range of systems, from magnets and superconductors to liquid crystals and biological membranes, highlighting the power of the renormalization group approach to critical phenomena