4.5 Renormalization group

Renormalization group is a powerful tool in condensed matter physics for understanding complex systems across different scales. It applies mathematical techniques to analyze how physical systems behave under changes in scale or resolution, crucial for studying phase transitions and critical phenomena.

The concept of scale invariance is central to renormalization group theory. It describes systems that look similar at different scales, characterized by power-law behavior in correlation functions and observables. This leads to universality in critical phenomena, where different physical systems exhibit identical critical behavior near phase transitions.

Fundamentals of renormalization group

• Renormalization group provides a powerful framework for understanding complex systems across different scales in condensed matter physics
• Applies mathematical techniques to analyze how physical systems behave under changes in scale or resolution
• Crucial for studying phase transitions, critical phenomena, and universality in many-body systems

Concept of scale invariance

Top images from around the web for Concept of scale invariance

• 

  Frontiers | The Forest Fire Model: The Subtleties of Criticality and Scale Invariance View original

  Is this image relevant?

• 

  Renormalization Group [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Frontiers | The Forest Fire Model: The Subtleties of Criticality and Scale Invariance View original

  Is this image relevant?

• 

  Renormalization Group [The Physics Travel Guide] View original

  Is this image relevant?

1 of 2

Top images from around the web for Concept of scale invariance

• 

  Frontiers | The Forest Fire Model: The Subtleties of Criticality and Scale Invariance View original

  Is this image relevant?

• 

  Renormalization Group [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Frontiers | The Forest Fire Model: The Subtleties of Criticality and Scale Invariance View original

  Is this image relevant?

• 

  Renormalization Group [The Physics Travel Guide] View original

  Is this image relevant?

1 of 2

• Describes systems that look similar at different scales or magnifications
• Characterized by power-law behavior in correlation functions and observables
• Manifests in critical phenomena near phase transitions (critical opalescence in fluids)
• Leads to self-similarity in physical properties across multiple length scales

Universality in critical phenomena

• Different physical systems exhibit identical critical behavior near phase transitions
• Grouped into universality classes based on shared critical exponents
• Determined by fundamental properties (dimensionality, symmetry) rather than microscopic details
• Explains why diverse systems (ferromagnets, liquid-gas transitions) show similar critical behavior

Wilson's approach to renormalization

• Developed by Kenneth Wilson to tackle multi-scale problems in physics
• Introduces the concept of effective theories valid at different length scales
• Involves iterative coarse-graining procedures to eliminate short-distance degrees of freedom
• Leads to renormalization group flow equations describing how coupling constants change with scale

Renormalization group transformations

Real-space renormalization

• Performs coarse-graining directly in physical space
• Involves grouping neighboring spins or particles into blocks
• Defines new effective interactions between coarse-grained variables
• Particularly useful for lattice models (Ising model, percolation)
• Can be implemented numerically for complex systems

Momentum-space renormalization

• Applies renormalization group transformations in Fourier space
• Involves integrating out high-momentum (short-wavelength) modes
• Leads to effective theories for low-energy, long-wavelength physics
• Widely used in quantum field theory and many-body physics
• Allows for systematic perturbative expansions in coupling constants

Decimation and blocking techniques

• Decimation reduces degrees of freedom by eliminating every nth variable
• Preserves essential long-range physics while simplifying the system
• Blocking groups nearby variables into single effective variables
• Requires careful definition of block transformations to maintain important symmetries
• Can be applied iteratively to study behavior across multiple scales

Fixed points and critical exponents

Stable vs unstable fixed points

• Fixed points represent scale-invariant states under renormalization group transformations
• Stable fixed points attract nearby trajectories in parameter space
• Correspond to low-temperature ordered phases or high-temperature disordered phases
• Unstable fixed points repel nearby trajectories
• Critical points are typically unstable fixed points, describing phase transitions

Calculation of critical exponents

• Critical exponents characterize power-law behavior near phase transitions
• Obtained from linearized renormalization group equations near fixed points
• Related to eigenvalues of the linearized transformation matrix
• Determine scaling of various thermodynamic quantities (specific heat, susceptibility)
• Can be calculated perturbatively or numerically for different models

Universality classes

• Group physical systems with identical critical behavior
• Determined by symmetries, dimensionality, and range of interactions
• Examples include Ising universality class (uniaxial ferromagnets, liquid-gas transitions)
• XY model universality class (superfluid helium, superconductors)
• Heisenberg model universality class (isotropic ferromagnets)

Applications in condensed matter

Ising model and phase transitions

• Simplest model exhibiting a phase transition between ordered and disordered states
• Renormalization group explains critical behavior and universality in Ising systems
• Predicts exact critical exponents in 2D, approximate results in 3D
• Applies to diverse systems (binary alloys, lattice gas models)
• Demonstrates power of renormalization group in solving complex many-body problems

Kondo effect and scaling

• Describes anomalous behavior of magnetic impurities in metals
• Renormalization group reveals a non-trivial fixed point at low temperatures
• Explains scaling behavior of resistivity and specific heat
• Demonstrates importance of quantum fluctuations in condensed matter systems
• Leads to concept of asymptotic freedom in quantum chromodynamics

Quantum criticality

• Studies phase transitions at zero temperature driven by quantum fluctuations
• Renormalization group reveals interplay between quantum and thermal fluctuations
• Leads to novel scaling laws and universality classes
• Applies to quantum magnets, heavy fermion systems, and high-Tc superconductors
• Connects condensed matter physics with quantum field theory and string theory

Numerical renormalization methods

Monte Carlo renormalization group

• Combines Monte Carlo simulations with renormalization group ideas
• Allows study of complex systems where analytical approaches fail
• Involves generating configurations at different scales using Monte Carlo methods
• Extracts critical exponents and scaling functions from numerical data
• Particularly useful for studying disordered systems and complex phase diagrams

Density matrix renormalization group

• Powerful numerical technique for studying one-dimensional quantum systems
• Based on systematic truncation of Hilbert space using density matrices
• Provides highly accurate results for ground state properties and excitations
• Extends renormalization group ideas to quantum many-body problems
• Applications include spin chains, quantum impurity models, and topological phases

Functional renormalization group

• Extends renormalization group to continuous fields and non-perturbative regimes
• Formulates flow equations for entire correlation functions or effective actions
• Allows treatment of strongly correlated electron systems and critical phenomena
• Provides a bridge between perturbative and non-perturbative approaches
• Applications include Fermi liquid instabilities, quantum criticality, and frustrated magnets

Limitations and extensions

Non-perturbative effects

• Renormalization group often relies on perturbative expansions in coupling constants
• Breaks down for strongly coupled systems or non-analytic behavior
• Requires development of non-perturbative techniques (conformal bootstrap, exact renormalization group)
• Challenges include treatment of bound states and topological effects
• Motivates development of dualities and non-perturbative methods in quantum field theory

Conformal field theory connection

• Describes scale-invariant systems with additional symmetry (conformal invariance)
• Provides exact solutions for critical exponents in two dimensions
• Connects renormalization group with string theory and AdS/CFT correspondence
• Applies to quantum critical points and boundary critical phenomena
• Leads to powerful classification schemes for universality classes

Renormalization in disordered systems

• Extends renormalization group to systems with quenched disorder
• Introduces concept of replica symmetry breaking and distribution of fixed points
• Applies to spin glasses, random field models, and localization phenomena
• Reveals novel universality classes and non-trivial scaling behavior
• Connects condensed matter physics with statistical mechanics of complex systems

Experimental verification

Critical exponents measurements

• Precise measurements of critical exponents near phase transitions
• Techniques include neutron scattering, specific heat measurements, and susceptibility studies
• Verifies predictions of renormalization group theory for various universality classes
• Challenges include sample purity, finite-size effects, and crossover phenomena
• Provides stringent tests of theoretical predictions and computational methods

Scaling relations in experiments

• Experimental verification of scaling laws predicted by renormalization group
• Includes data collapse techniques to reveal universal scaling functions
• Studies scaling of correlation lengths, susceptibilities, and order parameters
• Applies to diverse systems (superconductors, liquid crystals, polymers)
• Demonstrates power of renormalization group in unifying disparate phenomena

Universality across different systems

• Experimental confirmation of universality in critical phenomena
• Compares critical behavior across vastly different physical systems
• Examples include liquid-gas transitions and ferromagnetic ordering
• Verifies predictions of identical critical exponents within universality classes
• Demonstrates fundamental unity of physics across different scales and materials