Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Statistical mechanics and thermodynamics → Quantum statistical mechanics ↓


Quantum phase transition


Quantum phase transitions are fascinating phenomena that occur in quantum statistical mechanics, particularly when systems undergo transformations between different phases due to quantum fluctuations at absolute zero temperature. Unlike classical phase transitions that occur at finite temperatures and are induced by thermal fluctuations, quantum phase transitions are induced by quantum fluctuations arising from the Heisenberg uncertainty principle. Let's look at quantum phase transitions in more detail.

Understanding phases and phase transitions

In physics, a phase refers to a different state of matter that has similar physical properties. Classic examples include the solid, liquid, and gas phases. When a system changes from one phase to another, it undergoes a phase transition. For example, ice melting into water is a phase transition from solid to liquid.

These transitions are usually marked by changes in thermodynamic parameters such as temperature or pressure. However, quantum phase transitions occur at absolute zero and are driven by changes in another parameter, usually the strength of the quantum interaction, denoted as g.

Visualization of phase transitions

Phase A Phase B g (control parameter) State

In the above diagram, two different phases, A and B, are represented by blue and red circles. As we change the control parameter g, the system transitions from phase A to phase B. This illustrates the idea of phase transition mediated by a change in g.

Types of quantum phase transitions

Quantum phase transitions are generally classified based on the change in the properties of the system and the nature of the order parameter. There are two primary types:

  1. First-order transitions: Characterized by a discontinuous change in the order parameter. Similar to classical first-order transitions, these involve latent heat, even at zero temperature.
  2. Continuous transitions: Also known as second-order or critical transitions, these are marked by a continuous change in the order parameter, although some of its derivatives may be discontinuous. They are associated with divergent correlation lengths and critical exponents that describe universal behavior.

Consider the transition between the ferromagnetic and paramagnetic states at absolute zero. In a classical ferromagnet, thermal fluctuations at finite temperature cause this transition. However, near absolute zero, quantum fluctuations dominate, leading to a quantum phase transition, which is represented by a change in external parameters such as magnetic field or pressure.

Order parameter example

Order parameter variation g (control parameter) Order Parameters

This diagram models the behavior of the order parameter as a function of the control parameter g. The green path shows how the order parameter changes continuously, which is typical for second-order transitions. The small red dot indicates a discontinuity, similar to a first-order transition.

Mathematical framework

The mathematical treatment of quantum phase transitions involves both quantum mechanics and statistical physics. Here are the main components:

Hamiltonian dynamics

The dynamics of a quantum system is governed by a Hamiltonian H, an operator representing the total energy of the system. For a system at its quantum critical point, the Hamiltonian takes on a critical form.

H(g) = H_0 + gH_1

Here, H_0 is the Hamiltonian at criticality where quantum fluctuations dominate, and H_1 represents the perturbing effect that changes the state of the system. The task is to resolve the ground state and elementary excitations of this Hamiltonian.

Quantum Heisenberg model

Consider the Heisenberg model, which is often used to study magnetism in quantum systems. The Hamiltonian for a one-dimensional chain of spins is given as:

H = -J ∑ (S_i.S_(i+1)) - g ∑ (S_i^z)

Here, J is the exchange interaction between adjacent spins, and g is the strength of the external magnetic field. By changing g, we can induce a transition from the ordered phase to the disordered phase.

Experimental observations

Quantum phase transitions have been observed in a variety of materials and experimental setups. Cold atomic gases, quantum magnets, and superconductors are the major systems where these transitions are studied. For example:

  • Superconductivity: Transition between a superconducting and insulating state at zero temperature.
  • Quantum Hall effect: The quantum Hall plateau structure changes when external conditions change.
  • Graphene: Investigating the effect of electron correlation in two-dimensional sheets of graphene.

Simple experiment setup

Consider a setup using cold atoms trapped in an optical lattice, where you can control the interaction strength and the hopping parameter using the laser intensity:

Laser Intensity --> Interaction Strength
Lattice Depth --> Hopping Coefficient
Temperature --> Initialise to Nearly Absolute Zero

By fine-tuning the laser parameters, the researchers could observe the transition between the Mott insulating state and the superfluid state, an example of a quantum phase transition.

Theoretical implications and applications

Quantum phase transitions go beyond a theoretical curiosity; they provide insight into modern technological applications:

  • Quantum computing: Quantum state manipulation and error correction can be enhanced using quantum critical points.
  • Materials Science: Designing new materials with desirable electrical, magnetic and optical properties by adjusting control parameters.
  • Fundamental physics: understanding low-dimensional systems, non-equilibrium dynamics, and entanglement properties.

Conclusion

Quantum phase transitions are important phenomena in quantum statistical mechanics, providing profound insights into quantum behavior at absolute zero. By exploring various aspects of quantum phase transitions – mathematical descriptions, physical realizations, and potential applications – scientists can move toward new discoveries and technologies. Although complex, these transitions fundamentally enhance our understanding of quantum mechanics and statistical phenomena, opening new avenues in physics exploration.


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Quantum phase transition

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