Editing Condensed matter physics (section)

==Theoretical==
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 [[Electronic band structure|band structure]] and the [[density functional theory]]. Theoretical models have also been developed to study the physics of [[phase transition]]s, such as the [[Ginzburg–Landau theory]], [[critical exponent]]s 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 phase]]s, and [[gauge symmetry|gauge symmetries]].

===Emergence===
{{Main|Emergence}}

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.<ref name=coleman>{{cite book|last=Coleman |first=Piers |title=Introduction to Many Body Physics |year=2016 |publisher=Cambridge University Press |isbn=978-0-521-86488-6|url=http://www.cambridge.org/us/academic/subjects/physics/condensed-matter-physics-nanoscience-and-mesoscopic-physics/introduction-many-body-physics}}</ref><ref name=":0" /> For example, a range of phenomena related to high temperature superconductivity are understood poorly, although the microscopic physics of individual electrons and lattices is well known.<ref name=nsf-emergence>{{cite web|title=Understanding Emergence|url=https://www.nsf.gov/news/overviews/physics/physics_q01.jsp|publisher=National Science Foundation|access-date=30 March 2012}}</ref> Similarly, models of condensed matter systems have been studied where [[collective excitation]]s behave like [[photon]]s and [[electron]]s, thereby describing [[electromagnetism]] as an emergent phenomenon.<ref name=levin-rmp>{{cite journal|last=Levin|first=Michael|author2=Wen, Xiao-Gang |title=Colloquium: Photons and electrons as emergent phenomena|journal=Reviews of Modern Physics|year=2005|volume=77|issue=3|doi=10.1103/RevModPhys.77.871|arxiv= cond-mat/0407140 |bibcode= 2005RvMP...77..871L |pages=871–879 |s2cid=117563047}}</ref> Emergent properties can also occur at the interface between materials: one example is the [[lanthanum aluminate-strontium titanate interface]], where two band-insulators are joined to create conductivity and [[superconductivity]].

===Electronic theory of solids===
{{Main|Electronic band structure}}

The metallic state has historically been an important building block for studying properties of solids.<ref name="AshcroftMermin1976"/> The first theoretical description of metals was given by [[Paul Drude]] in 1900 with the [[Drude model]], which explained electrical and thermal properties by describing a metal as an [[ideal gas]] of then-newly discovered [[electron]]s.  He was able to derive the empirical [[Wiedemann-Franz law]] and get results in close agreement with the experiments.<ref name=Hoddeson-1992/>{{rp|90–91}}  This classical model was then improved by [[Arnold Sommerfeld]] who incorporated the [[Fermi–Dirac statistics]] of electrons and was able to explain the anomalous behavior of the [[specific heat]] of metals in the [[Wiedemann–Franz law]].<ref name=Hoddeson-1992/>{{rp|101–103}}  In 1912, The structure of crystalline solids was studied by [[Max von Laue]] and Paul Knipping, when they observed the [[X-ray diffraction]] pattern of crystals, and concluded that crystals get their structure from periodic [[lattice model (physics)|lattices]] of atoms.<ref name=Hoddeson-1992/>{{rp|48}}<ref>{{cite journal|last=Eckert|first=Michael|title=Disputed discovery: the beginnings of X-ray diffraction in crystals in 1912 and its repercussions|journal=Acta Crystallographica A|year=2011|volume=68|issue=1|doi=10.1107/S0108767311039985|pmid=22186281|url=http://journals.iucr.org/a/issues/2012/01/00/wx0005/index.html|bibcode= 2012AcCrA..68...30E|pages=30–39|doi-access=free}}</ref>  In 1928, Swiss physicist [[Felix Bloch]] provided a wave function solution to the [[Schrödinger equation]] with a [[Periodic function|periodic]] potential, known as [[Bloch's theorem]].<ref name=han-2010>{{cite book|last=Han|first=Jung Hoon|title=Solid State Physics|year=2010|publisher=Sung Kyun Kwan University|url=http://manybody.skku.edu/Lecture%20notes/Solid%20State%20Physics.pdf|url-status=dead|archive-url=https://web.archive.org/web/20130520224858/http://manybody.skku.edu/Lecture%20notes/Solid%20State%20Physics.pdf|archive-date=2013-05-20}}</ref>

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.<ref name=perdew-2010>{{cite journal|last=Perdew|first=John P.|author2=Ruzsinszky, Adrienn|author2-link=Adrienn Ruzsinszky |title=Fourteen Easy Lessons in Density Functional Theory|journal=International Journal of Quantum Chemistry|year=2010|volume=110|pages=2801–2807|url=http://www.if.pwr.wroc.pl/~scharoch/Abinitio/14lessons.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.if.pwr.wroc.pl/~scharoch/Abinitio/14lessons.pdf |archive-date=2022-10-09 |url-status=live|access-date=13 May 2012|doi=10.1002/qua.22829|issue=15|doi-access=free}}</ref> The [[Thomas–Fermi model|Thomas–Fermi theory]], developed in the 1920s, was used to estimate system energy and electronic density by treating the local electron density as a [[Variational method|variational parameter]]. Later in the 1930s, [[Douglas Hartree]], [[Vladimir Fock]] and [[John C. Slater|John Slater]] developed the so-called [[Hartree–Fock method|Hartree–Fock wavefunction]] as an improvement over the Thomas–Fermi model. The Hartree–Fock method accounted for [[Exchange symmetry|exchange statistics]] of single particle electron wavefunctions.  In general, it is very difficult to solve the Hartree–Fock equation.  Only the free electron gas case can be solved exactly.<ref name="AshcroftMermin1976">{{cite book|author1=Neil W. Ashcroft|author2=N. David Mermin|title=Solid state physics|year=1976|publisher=Saunders College|isbn=978-0-03-049346-1}}</ref>{{rp|330–337}} Finally in 1964–65, [[Walter Kohn]], [[Pierre Hohenberg]] and [[Lu Jeu Sham]] proposed the [[density functional theory]] (DFT) which gave realistic descriptions for bulk and surface properties of metals. The density functional theory has been widely used since the 1970s for band structure calculations of variety of solids.<ref name=perdew-2010 />

===Symmetry breaking===
{{Main|Symmetry breaking}}

Some states of matter exhibit ''symmetry breaking'', where the relevant laws of physics possess some form of [[Symmetry (physics)|symmetry]] that is broken. A common example is [[crystal|crystalline solid]]s, which break continuous [[translational symmetry]]. Other examples include magnetized [[ferromagnetism|ferromagnets]], which break [[rotational symmetry]], and more exotic states such as the ground state of a [[BCS theory|BCS]] [[superconductor]], that breaks [[U(1)]] phase rotational symmetry.<ref>{{cite web |url = https://www.nobelprize.org/nobel_prizes/physics/laureates/2008/nambu-lecture.html |title= Spontaneous Symmetry Breaking in Particle Physics: a Case of Cross Fertilization |last= Nambu |first= Yoichiro |date= 8 December 2008 |website= Nobelprize.org }}</ref><ref>{{cite journal |last=Greiter |first=Martin |arxiv=cond-mat/0503400 |title=Is electromagnetic gauge invariance spontaneously violated in superconductors? |date=16 March 2005 |doi=10.1016/j.aop.2005.03.008 |volume=319 |issue=2005 |journal=Annals of Physics |pages=217–249 |bibcode=2005AnPhy.319..217G |s2cid=55104377 }}</ref>

[[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 [[boson]]s. For example, in crystalline solids, these correspond to [[phonon]]s, which are quantized versions of lattice vibrations.<ref name=goldstone>{{cite journal|last=Leutwyler|first=H.|title=Phonons as Goldstone bosons|journal= Helv. Phys. Acta  |volume=70|issue=1997|year=1997|arxiv=hep-ph/9609466|bibcode= 1996hep.ph....9466L|pages=275–286}}</ref>

===Phase transition===
{{Main|Phase transition}}

Phase transition refers to the change of phase of a system, which is brought about by change in an external parameter such as [[temperature]], [[pressure]], or [[molar composition]]. In a single-component system, a classical phase transition occurs at a temperature (at a specific pressure) where there is an abrupt change in the  order of the system. For example, when ice melts and becomes water, the ordered hexagonal crystal structure of ice is modified to a hydrogen bonded, mobile arrangement of water molecules.

In [[quantum phase transition]]s,  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 fluctuation]]s originating from the [[Heisenberg uncertainty principle]]. Here, the different quantum phases of the system refer to distinct [[ground state]]s 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.<ref name=Vojta2003/>

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 (thermodynamics)|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.<ref name=Vojta2003>{{cite journal |last=Vojta |first=Matthias |arxiv= cond-mat/0309604 |title=Quantum phase transitions |year=2003 |doi=10.1088/0034-4885/66/12/R01 |volume=66 |issue=12 |journal=Reports on Progress in Physics |pages=2069–2110|bibcode= 2003RPPh...66.2069V |citeseerx=10.1.1.305.3880 |s2cid=15806867 }}</ref>  These critical phenomena present serious challenges to physicists because normal [[Macroscopic scale|macroscopic]] laws are no longer valid in the region, and novel ideas and methods must be invented to find the new laws that can describe the system.<ref name=NRC1986>{{cite book |title = Condensed-Matter Physics, Physics Through the 1990s |publisher=National Research Council|year=1986 |url = http://www.nap.edu/catalog/626/an-overview-physics-through-the-1990s |isbn=978-0-309-03577-4 |doi=10.17226/626}}</ref>{{rp|75ff}}

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.  For other types of systems that involves short range interactions near the critical point, a better theory is needed.<ref name="University1989">{{cite book|author=Malcolm F. Collins Professor of Physics McMaster University|title=Magnetic Critical Scattering|publisher=Oxford University Press, USA|isbn=978-0-19-536440-8|date=1989-03-02}}</ref>{{rp|8–11}}

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.  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.<ref name=NRC1986/>{{rp|11}}