In particular, they include the laws of quantum mechanics , electromagnetism and statistical mechanics. The most familiar condensed phases are solids and liquids while more exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature , the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose—Einstein condensate found in ultracold atomic systems. The study of condensed matter physics involves measuring various material properties via experimental probes along with using methods of theoretical physics to develop mathematical models that help in understanding physical behavior.

The diversity of systems and phenomena available for study makes condensed matter physics the most active field of contemporary physics: one third of all American physicists self-identify as condensed matter physicists, [1] and the Division of Condensed Matter Physics is the largest division at the American Physical Society. The theoretical physics of condensed matter shares important concepts and methods with that of particle physics and nuclear physics.

A variety of topics in physics such as crystallography , metallurgy , elasticity , magnetism , etc. Around the s, the study of physical properties of liquids was added to this list, forming the basis for the new, related specialty of condensed matter physics. References to "condensed" state can be traced to earlier sources. For example, in the introduction to his book Kinetic Theory of Liquids , [9] Yakov Frenkel proposed that "The kinetic theory of liquids must accordingly be developed as a generalization and extension of the kinetic theory of solid bodies.

As a matter of fact, it would be more correct to unify them under the title of 'condensed bodies'". One of the first studies of condensed states of matter was by English chemist Humphry Davy , in the first decades of the nineteenth century. Davy observed that of the forty chemical elements known at the time, twenty-six had metallic properties such as lustre , ductility and high electrical and thermal conductivity. Davy further claimed that elements that were then believed to be gases, such as nitrogen and hydrogen could be liquefied under the right conditions and would then behave as metals.

In , Michael Faraday , then an assistant in Davy's lab, successfully liquefied chlorine and went on to liquefy all known gaseous elements, except for nitrogen, hydrogen, and oxygen.

### Condensed Matter Physics II (Fall 2016)

Paul Drude in proposed the first theoretical model for a classical electron moving through a metallic solid. In , three years after helium was first liquefied, Onnes working at University of Leiden discovered superconductivity in mercury , when he observed the electrical resistivity of mercury to vanish at temperatures below a certain value. Pauli realized that the free electrons in metal must obey the Fermi—Dirac statistics. Using this idea, he developed the theory of paramagnetism in Shortly after, Sommerfeld incorporated the Fermi—Dirac statistics into the free electron model and made it better to explain the heat capacity.

Two years later, Bloch used quantum mechanics to describe the motion of an electron in a periodic lattice. In , Edwin Herbert Hall working at the Johns Hopkins University discovered a voltage developed across conductors transverse to an electric current in the conductor and magnetic field perpendicular to the current.

After the advent of quantum mechanics, Lev Landau in developed the theory of Landau quantization and laid the foundation for the theoretical explanation for the quantum Hall effect discovered half a century later. Magnetism as a property of matter has been known in China since BC. The Sommerfeld model and spin models for ferromagnetism illustrated the successful application of quantum mechanics to condensed matter problems in the s.

However, there still were several unsolved problems, most notably the description of superconductivity and the Kondo effect. These included recognition of collective excitation modes of solids and the important notion of a quasiparticle. Russian physicist Lev Landau used the idea for the Fermi liquid theory wherein low energy properties of interacting fermion systems were given in terms of what are now termed Landau-quasiparticles.

The theory also introduced the notion of an order parameter to distinguish between ordered phases. The study of phase transition and the critical behavior of observables, termed critical phenomena , was a major field of interest in the s. These ideas were unified by Kenneth G.

## Field Theories in Condensed Matter Physics

Wilson in , under the formalism of the renormalization group in the context of quantum field theory. It also implied that the Hall conductance can be characterized in terms of a topological invariable called Chern number. Laughlin, in , realized that this was a consequence of quasiparticle interaction in the Hall states and formulated a variational method solution, named the Laughlin wavefunction.

It was realized that the high temperature superconductors are examples of strongly correlated materials where the electron—electron interactions play an important role. In , David Field and researchers at Aarhus University discovered spontaneous electric fields when creating prosaic films [ clarification needed ] of various gases. This has more recently expanded to form the research area of spontelectrics. In several groups released preprints which suggest that samarium hexaboride has the properties of a topological insulator [39] in accord with the earlier theoretical predictions.

Theoretical condensed matter physics involves the use of theoretical models to understand properties of states of matter. These include models to study the electronic properties of solids, such as the Drude model , the Band structure and the density functional theory.

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Theoretical models have also been developed to study the physics of phase transitions , such as the Ginzburg—Landau theory , critical exponents and the use of mathematical methods of quantum field theory and the renormalization group. Modern theoretical studies involve the use of numerical computation of electronic structure and mathematical tools to understand phenomena such as high-temperature superconductivity , topological phases , and gauge symmetries.

Theoretical understanding of condensed matter physics is closely related to the notion of emergence , wherein complex assemblies of particles behave in ways dramatically different from their individual constituents. The metallic state has historically been an important building block for studying properties of solids. He was able to derive the empirical Wiedemann-Franz law and get results in close agreement with the experiments.

Calculating electronic properties of metals by solving the many-body wavefunction is often computationally hard, and hence, approximation methods are needed to obtain meaningful predictions. The Hartree—Fock method accounted for exchange statistics of single particle electron wavefunctions. In general, it's very difficult to solve the Hartree—Fock equation. Only the free electron gas case can be solved exactly. The density functional theory DFT has been widely used since the s for band structure calculations of variety of solids.

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Some states of matter exhibit symmetry breaking , where the relevant laws of physics possess some form of symmetry that is broken. A common example is crystalline solids , which break continuous translational symmetry. Other examples include magnetized ferromagnets , which break rotational symmetry , and more exotic states such as the ground state of a BCS superconductor , that breaks U 1 phase rotational symmetry. Goldstone's theorem in quantum field theory states that in a system with broken continuous symmetry, there may exist excitations with arbitrarily low energy, called the Goldstone bosons.

For example, in crystalline solids, these correspond to phonons , which are quantized versions of lattice vibrations.

Phase transition refers to the change of phase of a system, which is brought about by change in an external parameter such as temperature. Classical phase transition occurs at finite temperature when the order of the system was destroyed. For example, when ice melts and becomes water, the ordered crystal structure is destroyed. In quantum phase transitions , the temperature is set to absolute zero , and the non-thermal control parameter, such as pressure or magnetic field, causes the phase transitions when order is destroyed by quantum fluctuations originating from the Heisenberg uncertainty principle.

Here, the different quantum phases of the system refer to distinct ground states of the Hamiltonian matrix. Understanding the behavior of quantum phase transition is important in the difficult tasks of explaining the properties of rare-earth magnetic insulators, high-temperature superconductors, and other substances.

Two classes of phase transitions occur: first-order transitions and second-order or continuous transitions. For the latter, the two phases involved do not co-exist at the transition temperature, also called the critical point. Near the critical point, systems undergo critical behavior, wherein several of their properties such as correlation length , specific heat , and magnetic susceptibility diverge exponentially.

The simplest theory that can describe continuous phase transitions is the Ginzburg—Landau theory , which works in the so-called mean field approximation. However, it can only roughly explain continuous phase transition for ferroelectrics and type I superconductors which involves long range microscopic interactions.

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For other types of systems that involves short range interactions near the critical point, a better theory is needed. Near the critical point, the fluctuations happen over broad range of size scales while the feature of the whole system is scale invariant. Renormalization group methods successively average out the shortest wavelength fluctuations in stages while retaining their effects into the next stage.

## Quantum Field Theory for condensed matter FK Fysikum

Thus, the changes of a physical system as viewed at different size scales can be investigated systematically. The methods, together with powerful computer simulation, contribute greatly to the explanation of the critical phenomena associated with continuous phase transition. Experimental condensed matter physics involves the use of experimental probes to try to discover new properties of materials. Such probes include effects of electric and magnetic fields , measuring response functions , transport properties and thermometry.

Several condensed matter experiments involve scattering of an experimental probe, such as X-ray , optical photons , neutrons , etc. The choice of scattering probe depends on the observation energy scale of interest. Visible light has energy on the scale of 1 electron volt eV and is used as a scattering probe to measure variations in material properties such as dielectric constant and refractive index. X-rays have energies of the order of 10 keV and hence are able to probe atomic length scales, and are used to measure variations in electron charge density.

Neutrons can also probe atomic length scales and are used to study scattering off nuclei and electron spins and magnetization as neutrons have spin but no charge. Coulomb and Mott scattering measurements can be made by using electron beams as scattering probes. In experimental condensed matter physics, external magnetic fields act as thermodynamic variables that control the state, phase transitions and properties of material systems.

NMR experiments can be made in magnetic fields with strengths up to 60 Tesla. Higher magnetic fields can improve the quality of NMR measurement data. Ultracold atom trapping in optical lattices is an experimental tool commonly used in condensed matter physics, and in atomic, molecular, and optical physics.