Condensed matter physics
Condensed matter physics 

Phases · Phase transition 
Phase phenomena

Electronic phases

Electronic phenomena

Magnetic phases

Condensed matter physics is a branch of physics that deals with the physical properties of condensed phases of matter.^{[1]} Condensed matter physicists seek to understand the behavior of these phases by using physical laws. 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 atomic lattices, and the Bose–Einstein condensate found in cold atomic systems. The study of condensed matter physics involves measuring various material properties via experimental probes along with using techniques 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 identify themselves as condensed matter physicists,^{[2]} and the Division of Condensed Matter Physics is the largest division at the American Physical Society.^{[3]} The field overlaps with chemistry, materials science, and nanotechnology, and relates closely to atomic physics and biophysics. Theoretical condensed matter physics shares important concepts and techniques with theoretical particle and nuclear physics.^{[4]}
A variety of topics in physics such as crystallography, metallurgy, elasticity, magnetism, etc., were treated as distinct areas until the 1940s, when they were grouped together as solid state physics. Around the 1960s, the study of physical properties of liquids was added to this list, forming the basis for the new, related specialty of condensed matter physics.^{[5]} According to physicist Philip Warren Anderson, the term was coined by him and Volker Heine, when they changed the name of their group at the Cavendish Laboratories, Cambridge from "Solid state theory" to "Theory of Condensed Matter" in 1967,^{[6]} as they felt it did not exclude their interests in the study of liquids, nuclear matter and so on.^{[7]} Although Anderson and Heine helped popularize the name "condensed matter", it had been present in Europe for some years, most prominently in the form of a journal published in English, French, and German by SpringerVerlag titled Physics of Condensed Matter, which was launched in 1963.^{[8]} The funding environment and Cold War politics of the 1960s and 1970s were also factors that lead some physicists to prefer the name "condensed matter physics", which emphasized the commonality of scientific problems encountered by physicists working on solids, liquids, plasmas, and other complex matter, over "solid state physics", which was often associated with the industrial applications of metals and semiconductors.^{[9]} The Bell Telephone Laboratories was one of the first institutes to conduct a research program in condensed matter physics.^{[5]}
References to "condensed" state can be traced to earlier sources. For example, in the introduction to his 1947 "Kinetic Theory of Liquids",^{[10]} Yakov Frenkel proposed in the book, "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".
Contents
History
Classical physics
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, twentysix had metallic properties such as lustre, ductility and high electrical and thermal conductivity.^{[11]} This indicated that the atoms in Dalton's atomic theory were not indivisible as Dalton claimed, but had inner structure. 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.^{[12]}^{[notes 1]}
In 1823, Michael Faraday, then an assistant in Davy's lab, successfully liquefied chlorine and went on to liquefy all known gaseous elements, with the exception of nitrogen, hydrogen and oxygen.^{[11]} Shortly after, in 1869, Irish chemist Thomas Andrews studied the phase transition from a liquid to a gas and coined the term critical point to describe the condition where a gas and a liquid were indistinguishable as phases,^{[14]} and Dutch physicist Johannes van der Waals supplied the theoretical framework which allowed the prediction of critical behavior based on measurements at much higher temperatures.^{[15]}^{:35–38} By 1908, James Dewar and H. Kamerlingh Onnes were successfully able to liquefy hydrogen and then newly discovered helium, respectively.^{[11]}
Paul Drude in 1900 proposed the first theoretical model for a classical electron moving through a metallic solid.^{[4]} Drude's model described properties of metals in terms of a gas of free electrons, and was the first microscopic model to explain empirical observations such as the Wiedemann–Franz law.^{[16]}^{[17]}^{:27–29} However, despite the success of Drude's free electron model, it had one notable problem, in that it was unable to correctly explain the electronic contribution to the specific heat and magnetic properties of metals, as well as the temperature dependence of resistivity at low temperatures.^{[18]}^{:366–368}
In 1911, 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.^{[19]} The phenomenon completely surprised the best theoretical physicists of the time, and it remained unexplained for several decades.^{[20]} Albert Einstein, in 1922, said regarding contemporary theories of superconductivity that “with our farreaching ignorance of the quantum mechanics of composite systems we are very far from being able to compose a theory out of these vague ideas”.^{[21]}
Advent of quantum mechanics
Drude's classical model was augmented by Wolfgang Pauli, Arnold Sommerfeld, Felix Bloch and other physicists. Pauli realized that the free electrons in metal must obey the Fermi–Dirac statistics. Using this idea, he developed the theory of paramagnetism in 1926. Shortly after, Sommerfeld incorporated the Fermi–Dirac statistics into the free electron model and made it better able to explain the heat capacity. Two years later, Bloch used quantum mechanics to describe the motion of a quantum electron in a periodic lattice.^{[18]}^{:366–368} The mathematics of crystal structures developed by Auguste Bravais, Yevgraf Fyodorov and others was used to classify crystals by their symmetry group, and tables of crystal structures were the basis for the series International Tables of Crystallography, first published in 1935.^{[22]} Band structure calculations was first used in 1930 to predict the properties of new materials, and in 1947 John Bardeen, Walter Brattain and William Shockley developed the first semiconductorbased transistor, heralding a revolution in electronics.^{[4]}
In 1879, Edwin Herbert Hall working at the Johns Hopkins University discovered the development of a voltage across conductors transverse to an electric current in the conductor and magnetic field perpendicular to the current.^{[23]} This phenomenon arising due to the nature of charge carriers in the conductor came to be known as the Hall effect, but it was not properly explained at the time, since the electron was experimentally discovered 18 years later. After the advent of quantum mechanics, Lev Landau in 1930 developed the theory of landau quantization and laid the foundation for the theoretical explanation for the quantum hall effect discovered half a century later.^{[24]}^{:458–460}^{[25]}
Magnetism as a property of matter has been known in China since 4000BC.^{[26]}^{:1–2} However, the first modern studies of magnetism only started with the development of electrodynamics by Faraday, Maxwell and others in the nineteenth century, which included the classification of materials as ferromagnetic, paramagnetic and diamagnetic based on their response to magnetization.^{[27]} Pierre Curie studied the dependence of magnetization on temperature and discovered the Curie point phase transition in ferromagnetic materials.^{[26]} In 1906, Pierre Weiss introduced the concept of magnetic domains to explain the main properties of ferromagnets.^{[28]}^{:9} The first attempt at a microscopic description of magnetism was by Wilhelm Lenz and Ernst Ising through the Ising model that described magnetic materials as consisting of a periodic lattice of spins that collectively acquired magnetization.^{[26]} The Ising model was solved exactly to show that spontaneous magnetization cannot occur in one dimension but is possible in higherdimensional lattices. Further research such as by Bloch on spin waves and Néel on antiferromagnetism led to the development of new magnetic materials with applications to magnetic storage devices.^{[26]}^{:36–38,48}
Modern manybody physics
The Sommerfeld model and spin models for ferromagnetism illustrated the successful application of quantum mechanics to condensed matter problems in the 1930s. However, there still were several unsolved problems, most notably the description of superconductivity and the Kondo effect.^{[30]} After World War II, several ideas from quantum field theory were applied to condensed matter problems. These included recognition of collective modes of excitation 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 known as Landauquasiparticles.^{[30]} Landau also developed a mean field theory for continuous phase transitions, which described ordered phases as spontaneous breakdown of symmetry. The theory also introduced the notion of an order parameter to distinguish between ordered phases.^{[31]} Eventually in 1965, John Bardeen, Leon Cooper and John Schrieffer developed the socalled BCS theory of superconductivity, based on the discovery that arbitrarily small attraction between two electrons of opposite spin mediated by phonons in the lattice can give rise to a bound state called a Cooper pair.^{[32]}
The study of phase transition and the critical behavior of observables, known as critical phenomena, was a major field of interest in the 1960s.^{[34]} Leo Kadanoff, Benjamin Widom and Michael Fisher developed the ideas of critical exponents and scaling. These ideas were unified by Kenneth Wilson in 1972, under the formalism of the renormalization group in the context of quantum field theory.^{[34]}
The quantum Hall effect was discovered by Klaus von Klitzing in 1980 when he observed the Hall conductance to be integer multiples of a fundamental constant .(see figure) The effect was observed to be independent of parameters such as the system size and impurities.^{[33]} In 1981, theorist Robert Laughlin proposed a theory explaining the unanticipated precision of the integral plateau. It also implied that the Hall conductance can be characterized in terms of a topological invariable called Chern number.^{[35]}^{[36]}^{:69, 74} Shortly after, in 1982, Horst Störmer and Daniel Tsui observed the fractional quantum Hall effect where the conductance was now a rational multiple of a constant. Laughlin, in 1983, realized that this was a consequence of quasiparticle interaction in the Hall states and formulated a variational solution, known as the Laughlin wavefunction.^{[37]} The study of topological properties of the fractional Hall effect remains an active field of research.
In 1986, Karl Müller and Johannes Bednorz discovered the first high temperature superconductor, a material which was superconducting at temperatures as high as 50 Kelvin. It was realized that the high temperature superconductors are examples of strongly correlated materials where the electron–electron interactions play an important role.^{[38]} A satisfactory theoretical description of hightemperature superconductors is still not known and the field of strongly correlated materials continues to be an active research topic.
In 2009, David Field and researchers at Aarhus University discovered spontaneous electric fields when creating prosaic films of various gases. This has more recently expanded to form the research area of spontelectrics.^{[39]}
In 2012 several groups released preprints which suggest that samarium hexaboride has the properties of a topological insulator ^{[40]} in accordance with the earlier theoretical predictions.^{[41]} Since samarium hexaboride is an established Kondo insulator, i.e. a strongly correlated electron material, the existence of a topological surface state in this material would lead to a topological insulator with strong electronic correlations.
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 Band structure and the density functional theory. 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 techniques 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 hightemperature superconductivity, topological phases and gauge symmetries.
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.^{[32]} For example, a range of phenomena related to high temperature superconductivity are not well understood, although the microscopic physics of individual electrons and lattices is well known.^{[42]} Similarly, models of condensed matter systems have been studied where collective excitations behave like photons and electrons, thereby describing electromagnetism as an emergent phenomenon.^{[43]} Emergent properties can also occur at the interface between materials: one example is the lanthanumaluminatestrontiumtitanate interface, where two nonmagnetic insulators are joined to create conductivity, superconductivity, and ferromagnetism.
Electronic theory of solids
The metallic state has historically been an important building block for studying properties of solids.^{[44]} 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 thennewly discovered electrons. He was able to derive the empirical WiedemannFranz law and get results in close agreement with the experiments.^{[17]}^{: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.^{[17]}^{:101–103} In 1912, The structure of crystalline solids was studied by Max von Laue and Paul Knipping, when they observed the Xray diffraction pattern of crystals, and concluded that crystals get their structure from periodic lattices of atoms.^{[17]}^{:48}^{[45]} In 1928, Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, called the Bloch wave.^{[46]}
Calculating electronic properties of metals by solving the manybody wavefunction is often computationally hard, and hence, approximation techniques are necessary to obtain meaningful predictions.^{[47]} The 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 parameter. Later in the 1930s, Douglas Hartree, Vladimir Fock and John Slater developed the socalled Hartree–Fock wavefunction as an improvement over the Thomas–Fermi model. 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.^{[44]}^{:330–337} Finally in 1964–65, Walter Kohn, Pierre Hohenberg and Lu Jeu Sham proposed the density functional theory which gave realistic descriptions for bulk and surface properties of metals. The density functional theory (DFT) has been widely used since the 1970s for band structure calculations of variety of solids.^{[47]}
Symmetry breaking
Certain states of matter exhibit symmetry breaking, where the relevant laws of physics possess some 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.^{[48]}^{[49]}
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.^{[50]}
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. 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 nonthermal 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. Understanding the behavior of quantum phase transition is important in the difficult tasks of explaining the properties of rareearth magnetic insulators, hightemperature superconductors and other substances.^{[51]}
There are two classes of phase tranisitions: firstorder transitions and continuous transitions. For the continuous transitions, the two phases involved do not coexist at the transition temperature, also called critical point. Near the critical point, systems undergoes displays critical behavior, wherein several of their properties such as correlation length, specific heat and susceptibility diverge exponentially.^{[51]} These critical phenomena poses serious challenges to physicists because normal macroscopic laws are no longer valid in the region and novel ideas and methods has to be invented to find the new laws that can describe the system.^{[52]}^{:75ff}
The simplest theory that can describe continuous phase transitions is the Ginzburg–Landau theory, which works in the socalled mean field approximation. However, it can only roughly explain continuous phase transition for ferroelectrics and type I superconductors which invoves long range microscopic interactions. For other types of systems that involves short range interactions near the critical point, a better theory is needed.^{[53]}^{: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 techniques 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 techniques, together with powerful computer simulation, contribute greatly to the explanation of the critical phenomena associated with continuous phase transition.^{[52]}^{:11}
Experimental
Experimental condensed matter physics involves the use of experimental probes to try to discover new properties of materials. Experimental probes include effects of electric and magnetic fields, measurement of response functions, transport properties and thermometry.^{[54]} Commonly used experimental techniques include spectroscopy, with probes such as Xrays, infrared light and inelastic neutron scattering; study of thermal response, such as specific heat and measurement of transport via thermal and heat conduction.
Scattering
Several condensed matter experiments involve scattering of an experimental probe, such as Xray, optical photons, neutrons, etc., on constituents of a material. The choice of scattering probe depends on the observation energy scale of interest. Visible light has energy on the scale of 1 eV and is used as a scattering probe to measure variations in material properties such as dielectric constant and refractive index. Xrays 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.^{[55]}^{:33–34}
Neutrons can also probe atomic length scales and are used to study scattering off nuclei and electron spins and magnetization (as neutrons themselves have spin but no charge). Coulomb and Mott scattering measurements can be made by using electron beams as scattering probes.^{[55]}^{:33–34}^{[56]}^{:39–43} Similarly, positron annihilation can be used as an indirect measurement of local electron density.^{[57]} Laser spectroscopy is an excellent tool for studying the microscopic properties of a medium, for example, to study forbidden transitions in media with nonlinear optical spectroscopy.^{[52]} ^{:258–259}
External magnetic fields
In experimental condensed matter physics, external magnetic fields act as thermodynamic variables that control the state, phase transitions and properties of material systems.^{[58]} Nuclear magnetic resonance (NMR) is a technique by which external magnetic fields can be used to find resonance modes of individual electrons, thus giving information about the atomic, molecular and bond structure of their neighborhood. 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.^{[59]}^{:69}^{[60]}^{:185} Quantum oscillations is another experimental technique where high magnetic fields are used to study material properties such as the geometry of the Fermi surface.^{[61]} High magnetic fields will be useful in experimentally testing of the various theoretical predictions such as the quantized magnetoelectric effect, image magnetic monopole, and the halfinteger quantum Hall effect.^{[59]}^{:57}
Cold atomic gases
Cold atom trapping in optical lattices is an experimental tool commonly used in condensed matter as well as atomic, molecular, and optical physics. The technique involves using optical lasers to create an interference pattern, which acts as a "lattice", in which ions or atoms can be placed at very low temperatures. Cold atoms in optical lattices are used as "quantum simulators", that is, they act as controllable systems that can model behavior of more complicated systems, such as frustrated magnets.^{[62]} In particular, they are used to engineer one, two and threedimensional lattices for a Hubbard model with prespecified parameters, and to study phase transitions for antiferromagnetic and spin liquid ordering.^{[63]}^{[64]}
In 1995, a gas of rubidium atoms cooled down to a temperature of 170 nK was used to experimentally realize the Bose–Einstein condensate, a novel state of matter originally predicted by S. N. Bose and Albert Einstein, wherein a large number of atoms occupy a single quantum state.^{[65]}
Applications
Research in condensed matter physics has given rise to several device applications, such as the development of the semiconductor transistor,^{[4]} and laser technology.^{[52]} Several phenomena studied in the context of nanotechnology come under the purview of condensed matter physics.^{[66]}^{:111ff} Techniques such as scanningtunneling microscopy can be used to control processes at the nanometer scale, and have given rise to the study of nanofabrication.^{[67]}
In quantum computation, information is represented by quantum bits, or qubits. The qubits may decohere quickly before useful computation is completed. This serious problem must be solved before quantum computation may be realized. The superconducting Josephson junction qubits, the spintronic qubits using the spin orientation of magnetic materials, or the topological nonAbelian anyons from fractional quantum Hall states are a few of the promising approaches proposed in condensed matter physics to solve this problem.^{[67]}
Condensed matter physics also has important applications to biophysics, for example, the experimental technique of magnetic resonance imaging, which is widely used in medical diagnosis.^{[67]}
See also
Notes
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References
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Further reading
 Mudry, Christopher (2014). Lecture Notes on Field Theory in Condensed Matter Physics. World Scientific. ISBN 9789814449106.
 Khan, Abdul Qadeer (21 November 1998). "Dimensional Anistrophy in Condensed Matter Physics" (PDF). Seven National Symposium on Frontiers in Physics. 7. 7 (7). Retrieved 21 October 2012.
 P. M. Chaikin and T. C. Lubensky (2000). Principles of Condensed Matter Physics, Cambridge University Press; 1st edition, ISBN 0521794501
 Alexander Altland and Ben Simons (2006). Condensed Matter Field Theory, Cambridge University Press, ISBN 0521845084
 Michael P. Marder (2010). Condensed Matter Physics, second edition, John Wiley and Sons, ISBN 0470617985
 Lillian Hoddeson, Ernest Braun, Jürgen Teichmann and Spencer Weart, eds. (1992). Out of the Crystal Maze: Chapters from the History of Solid State Physics, Oxford University Press, ISBN 019505329X
 ↑ Taylor, Philip L. (2002). A Quantum Approach to Condensed Matter Physics. Cambridge University Press. ISBN 052177103X.
 ↑ "Condensed Matter Physics Jobs: Careers in Condensed Matter Physics". Physics Today Jobs. Archived from the original on 20090327. Retrieved 20101101.
 ↑ "History of Condensed Matter Physics". American Physical Society. Retrieved 27 March 2012.
 ↑ ^{4.0} ^{4.1} ^{4.2} ^{4.3} Cohen, Marvin L. (2008). "Essay: Fifty Years of Condensed Matter Physics". Physical Review Letters. 101 (25). Bibcode:2008PhRvL.101y0001C. doi:10.1103/PhysRevLett.101.250001. Retrieved 31 March 2012.
 ↑ ^{5.0} ^{5.1} Kohn, W. (1999). "An essay on condensed matter physics in the twentieth century" (PDF). Reviews of Modern Physics. 71 (2): S59. Bibcode:1999RvMPS..71...59K. doi:10.1103/RevModPhys.71.S59. Retrieved 27 March 2012.
 ↑ "Philip Anderson". Department of Physics. Princeton University. Retrieved 27 March 2012.
 ↑ "More and Different". World Scientific Newsletter. 33: 2. November 2011.
 ↑ "Physics of Condensed Matter". Retrieved 20 April 2015.
 ↑ Martin, Joseph D. (2015). "What's in a Name Change? Solid State Physics, Condensed Matter Physics, and Materials Science". Physics in Perspective. 17 (1): 3–32. doi:10.1007/s0001601401517. Retrieved 20 April 2015.
 ↑ Frenkel, J. (1947). Kinetic Theory of Liquids. Oxford University Press.
 ↑ ^{11.0} ^{11.1} ^{11.2} Goodstein, David; Goodstein, Judith (2000). "Richard Feynman and the History of Superconductivity" (PDF). Physics in Perspective. 2 (1): 30. Bibcode:2000PhP.....2...30G. doi:10.1007/s000160050035. Retrieved 7 April 2012.
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 ↑ "Understanding Emergence". National Science Foundation. Retrieved 30 March 2012.
 ↑ Levin, Michael; Wen, XiaoGang (2005). "Colloquium: Photons and electrons as emergent phenomena". Reviews of Modern Physics. 77 (3): 871–879. Bibcode:2005RvMP...77..871L. arXiv:condmat/0407140 . doi:10.1103/RevModPhys.77.871.
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 ↑ Siegel, R. W. (1980). "Positron Annihilation Spectroscopy". Annual Review of Materials Science. 10: 393–425. Bibcode:1980AnRMS..10..393S. doi:10.1146/annurev.ms.10.080180.002141.
 ↑ Committee on Facilities for Condensed Matter Physics (2004). "Report of the IUPAP working group on Facilities for Condensed Matter Physics : High Magnetic Fields" (PDF). International Union of Pure and Applied Physics.
The magnetic field is not simply a spectroscopic tool but is a thermodynamic variable which, along with temperature and pressure, controls the state, the phase transitions and the properties of materials.
 ↑ ^{59.0} ^{59.1} Committee to Assess the Current Status and Future Direction of High Magnetic Field Science in the United States; Board on Physics and Astronomy; Division on Engineering and Physical Sciences; National Research Council (25 November 2013). High Magnetic Field Science and Its Application in the United States:: Current Status and Future Directions. National Academies Press. ISBN 9780309286343.
 ↑ Moulton, W. G. and Reyes, A. P. (2006). "Nuclear Magnetic Resonance in Solids at very high magnetic fields". In Herlach, Fritz. High Magnetic Fields. Science and Technology. World Scientific. ISBN 9789812774880.
 ↑ DoironLeyraud, Nicolas; et al. (2007). "Quantum oscillations and the Fermi surface in an underdoped highTc superconductor". Nature. 447 (7144): 565–568. Bibcode:2007Natur.447..565D. PMID 17538614. arXiv:0801.1281 . doi:10.1038/nature05872.
 ↑ Buluta, Iulia; Nori, Franco (2009). "Quantum Simulators". Science. 326 (5949): 108–11. Bibcode:2009Sci...326..108B. PMID 19797653. doi:10.1126/science.1177838.
 ↑ Greiner, Markus; Fölling, Simon (2008). "Condensedmatter physics: Optical lattices". Nature. 453 (7196): 736–738. Bibcode:2008Natur.453..736G. PMID 18528388. doi:10.1038/453736a.
 ↑ Jaksch, D.; Zoller, P. (2005). "The cold atom Hubbard toolbox". Annals of Physics. 315 (1): 52–79. Bibcode:2005AnPhy.315...52J. arXiv:condmat/0410614 . doi:10.1016/j.aop.2004.09.010.
 ↑ Glanz, James (October 10, 2001). "3 Researchers Based in U.S. Win Nobel Prize in Physics". The New York Times. Retrieved 23 May 2012.
 ↑ Committee on CMMP 2010; Solid State Sciences Committee; Board on Physics and Astronomy; Division on Engineering and Physical Sciences, National Research Council (21 December 2007). CondensedMatter and Materials Physics:: The Science of the World Around Us. National Academies Press. ISBN 9780309134095.
 ↑ ^{67.0} ^{67.1} ^{67.2} Yeh, NaiChang (2008). "A Perspective of Frontiers in Modern Condensed Matter Physics" (PDF). AAPPS Bulletin. 18 (2). Retrieved 31 March 2012.
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