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131-464: In condensed-matter physics , Chern–Simons theory describes the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial . Particularly, Chern–Simons theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as

262-575: A Hubbard model with pre-specified parameters, and to study phase transitions for antiferromagnetic and spin liquid ordering. 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 one quantum state . Research in condensed matter physics has given rise to several device applications, such as

393-454: A stationary point (usually, a minimum) of the action. Action has the dimensions of [energy]  ×  [time] , and its SI unit is joule -second, which is identical to the unit of angular momentum . Several different definitions of "the action" are in common use in physics. The action is usually an integral over time. However, when the action pertains to fields , it may be integrated over spatial variables as well. In some cases,

524-802: A G Wess–Zumino–Witten model on the boundary. In addition the U ( N ) and SO( N ) Chern–Simons theories at large N are well approximated by matrix models . In 1982, S. Deser , R. Jackiw and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the Einstein–Hilbert action in gravity theory is modified by adding the Chern–Simons term. ( Deser, Jackiw & Templeton (1982) ) In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions ( Jackiw & Pi (2003) ) and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy. The four-dimensional case

655-457: A body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a geodesic . Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations , which are derived from the action principle. An example is Noether's theorem , which states that to every continuous symmetry in

786-424: A certain value. The phenomenon completely surprised the best theoretical physicists of the time, and it remained unexplained for several decades. Albert Einstein , in 1922, said regarding contemporary theories of superconductivity that "with our far-reaching ignorance of the quantum mechanics of composite systems we are very far from being able to compose a theory out of these vague ideas." Drude's classical model

917-431: A closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve. He later studied the theory in more detail together with Witten and Masahito Yamazaki, demonstrating how the gauge theory could be related to many notions in integrable systems theory, including exactly solvable lattice models (like

1048-565: A field of study 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, as they felt it better included their interest in liquids, nuclear matter , and so on. Although Anderson and Heine helped popularize the name "condensed matter", it had been used in Europe for some years, most prominently in

1179-411: A generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called equations of motion . Action is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system

1310-573: A loop K in M , one may define the Wilson loop W R ( K ) {\displaystyle W_{R}(K)} by where A is the connection 1-form and we take the Cauchy principal value of the contour integral and P exp {\displaystyle {\mathcal {P}}\exp } is the path-ordered exponential . Consider a link L in M , which is a collection of ℓ disjoint loops. A particularly interesting observable

1441-414: A massive photon if this term is added to the action of Maxwell's theory of electrodynamics . This term can be induced by integrating over a massive charged Dirac field . It also appears for example in the quantum Hall effect . The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in

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1572-552: A material. 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 the 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 and crystal structure. Neutrons can also probe atomic length scales and are used to study

1703-451: 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. This indicated that

1834-571: A metal as an ideal gas of then-newly discovered electrons . He was able to derive the empirical Wiedemann-Franz law and get results in close agreement with the experiments. 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 . In 1912, The structure of crystalline solids

1965-531: A path. Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation . Thus, there are two distinct approaches for formulating dynamical models. Hamilton's principle applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and gravitational fields . Hamilton's principle has also been extended to quantum mechanics and quantum field theory —in particular

2096-473: A periodic lattice of spins that collectively acquired magnetization. The Ising model was solved exactly to show that spontaneous magnetization can occur in one dimension and it is possible in higher-dimensional lattices. Further research such as by Bloch on spin waves and Néel on antiferromagnetism led to developing new magnetic materials with applications to magnetic storage devices. The Sommerfeld model and spin models for ferromagnetism illustrated

2227-399: A physical situation there corresponds a conservation law (and conversely). This deep connection requires that the action principle be assumed. In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate

2358-413: A range of phenomena related to high temperature superconductivity are understood poorly, although the microscopic physics of individual electrons and lattices is well known. Similarly, models of condensed matter systems have been studied where collective excitations behave like photons and electrons , thereby describing electromagnetism as an emergent phenomenon. Emergent properties can also occur at

2489-528: A rational multiple of the constant e 2 / h {\displaystyle e^{2}/h} . Laughlin, in 1983, realized that this was a consequence of quasiparticle interaction in the Hall states and formulated a variational method solution, named the Laughlin wavefunction . The study of topological properties of the fractional Hall effect remains an active field of research. Decades later,

2620-427: A solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots. It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R ). This problem is also open in physics level. In

2751-418: A stack of D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory. Chern–Simons theories are related to many other field theories. For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a two-dimensional conformal field theory known as

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2882-498: A state can be defined on any surface. Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. Witten has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with

3013-478: Is S = − m c 2 ∫ C d τ . {\displaystyle S=-mc^{2}\int _{C}\,d\tau .} If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t 1 to t 2 , then the action becomes S = ∫ t 1 t 2 L d t , {\displaystyle S=\int _{t1}^{t2}L\,dt,} where

3144-452: Is a gauge theory , which means that a classical configuration in the Chern–Simons theory on M with gauge group G is described by a principal G -bundle on M . The connection of this bundle is characterized by a connection one-form A which is valued in the Lie algebra g of the Lie group G . In general the connection A is only defined on individual coordinate patches , and

3275-541: Is a mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensions of energy  ×  time or momentum  ×  length , and its SI unit is joule -second (like the Planck constant h ). Introductory physics often begins with Newton's laws of motion , relating force and motion; action

3406-523: Is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action , an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action similar to

3537-530: Is a two-dimensional manifold and C {\displaystyle C} is a complex curve is S = ∫ M ω ∧ C S ( A ) {\displaystyle S=\int _{M}\omega \wedge CS(A)} where ω {\displaystyle \omega } is a meromorphic one-form on C {\displaystyle C} . The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to

3668-399: Is an established Kondo insulator , i.e. a strongly correlated electron material, it is expected that the existence of a topological Dirac surface state in this material would lead to a topological insulator with strong electronic correlations. Theoretical condensed matter physics involves the use of theoretical models to understand properties of states of matter. These include models to study

3799-477: Is an evolution for which the action S [ q ( t ) ] {\displaystyle {\mathcal {S}}[\mathbf {q} (t)]} is stationary (a minimum, maximum, or a saddle point ). This principle results in the equations of motion in Lagrangian mechanics . In addition to the action functional, there is another functional called the abbreviated action . In the abbreviated action,

3930-432: Is an important concept in modern theoretical physics . Various action principles and related concepts are summarized below. In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). Maupertuis's principle uses the abbreviated action between two generalized points on

4061-483: Is an important step in the theory of characteristic classes in differential geometry . Given a flat G - principal bundle P on M there exists a unique homomorphism, called the Chern–Weil homomorphism , from the algebra of G -adjoint invariant polynomials on g (Lie algebra of G ) to the cohomology H ∗ ( M , R ) {\displaystyle H^{*}(M,\mathbb {R} )} . If

Chern–Simons theory - Misplaced Pages Continue

4192-1039: Is defined as the integral of the generalized momenta, p i = ∂ L ( q , t ) ∂ q ˙ i , {\displaystyle p_{i}={\frac {\partial L(q,t)}{\partial {\dot {q}}_{i}}},} for a system Lagrangian L {\displaystyle L} along a path in the generalized coordinates q i {\displaystyle q_{i}} : S 0 = ∫ q 1 q 2 p ⋅ d q = ∫ q 1 q 2 Σ i p i d q i . {\displaystyle {\mathcal {S}}_{0}=\int _{q_{1}}^{q_{2}}\mathbf {p} \cdot d\mathbf {q} =\int _{q_{1}}^{q_{2}}\Sigma _{i}p_{i}\,dq_{i}.} where q 1 {\displaystyle q_{1}} and q 2 {\displaystyle q_{2}} are

4323-465: Is described by the Wess–Zumino–Witten (WZW) model on N at level k . To canonically quantize Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a Hilbert space . There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a Cauchy surface , in fact,

4454-443: Is either a constant or varies very slowly; hence, the variable J k is often used in perturbation calculations and in determining adiabatic invariants . For example, they are used in the calculation of planetary and satellite orbits. When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper time τ {\displaystyle \tau }

4585-752: Is especially ideal for the study of phase changes at extreme temperatures above 2000 °C due to the temperature independence of the method. Ultracold atom trapping in optical lattices is an experimental tool commonly used in condensed matter physics, and in atomic, molecular, and optical physics . The method involves using optical lasers to form 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 . In particular, they are used to engineer one-, two- and three-dimensional lattices for

4716-400: Is its "angle" w k , for reasons described more fully under action-angle coordinates . The integration is only over a single variable q k and, therefore, unlike the integrated dot product in the abbreviated action integral above. The J k variable equals the change in S k ( q k ) as q k is varied around the closed path. For several physical systems of interest, J k

4847-503: Is just the abbreviated action . A variable J k in the action-angle coordinates , called the "action" of the generalized coordinate q k , is defined by integrating a single generalized momentum around a closed path in phase space , corresponding to rotating or oscillating motion: J k = ∮ p k d q k {\displaystyle J_{k}=\oint p_{k}\,dq_{k}} The corresponding canonical variable conjugate to J k

4978-458: Is modified to a hydrogen bonded, mobile arrangement of water molecules. 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

5109-494: Is obtained from the action functional S {\displaystyle {\mathcal {S}}} by fixing the initial time t 0 {\displaystyle t_{0}} and the initial endpoint q 0 , {\displaystyle q_{0},} while allowing the upper time limit t {\displaystyle t} and the second endpoint q {\displaystyle q} to vary. The Hamilton's principal function satisfies

5240-463: Is part of a completely equivalent alternative approach with practical and educational advantages. However, the concept took many decades to supplant Newtonian approaches and remains a challenge to introduce to students. For a trajectory of a ball moving in the air on Earth the action is defined between two points in time, t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} as

5371-410: Is that for which the action is minimized , or more generally, is stationary . In other words, the action satisfies a variational principle: the principle of stationary action (see also below). The action is defined by an integral , and the classical equations of motion of a system can be derived by minimizing the value of that integral. The action principle provides deep insights into physics, and

Chern–Simons theory - Misplaced Pages Continue

5502-555: Is the ℓ -point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the fundamental representation of G . One may form a normalized correlation function by dividing this observable by the partition function Z ( M ), which is just the 0-point correlation function. In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known knot polynomials . For example, in G  =  U ( N ) Chern–Simons theory at level k

5633-741: Is the evolution q ( t ) of the system between two times t 1 and t 2 , where q represents the generalized coordinates . The action S [ q ( t ) ] {\displaystyle {\mathcal {S}}[\mathbf {q} (t)]} is defined as the integral of the Lagrangian L for an input evolution between the two times: S [ q ( t ) ] = ∫ t 1 t 2 L ( q ( t ) , q ˙ ( t ) , t ) d t , {\displaystyle {\mathcal {S}}[\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt,} where

5764-469: Is the first Pontryagin number and s ( M ) is the section of the normal orthogonal bundle P . Moreover, the Chern–Simons term is described as the eta invariant defined by Atiyah, Patodi and Singer. The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The action integral ( path integral ) of the field theory in physics

5895-666: Is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is This variation gives the Cotton tensor Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action. In 2013 Kenneth A. Intriligator and Nathan Seiberg solved these 3d Chern–Simons gauge theories and their phases using monopoles carrying extra degrees of freedom. The Witten index of

6026-432: Is viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on M . These explain why the Chern–Simons theory is closely related to topological field theory . Chern–Simons theories can be defined on any topological 3-manifold M , with or without boundary. As these theories are Schwarz-type topological theories, no metric needs to be introduced on M . Chern–Simons theory

6157-413: The Lagrangian is L = − m c 2 1 − v 2 c 2 . {\displaystyle L=-mc^{2}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.} Physical laws are frequently expressed as differential equations , which describe how physical quantities such as position and momentum change continuously with time , space or

6288-508: The Max Planck Institute for Solid State Research , physics professor Manuel Cardona, it was Albert Einstein who created the modern field of condensed matter physics starting with his seminal 1905 article on the photoelectric effect and photoluminescence which opened the fields of photoelectron spectroscopy and photoluminescence spectroscopy , and later his 1907 article on the specific heat of solids which introduced, for

6419-468: The Planck constant , quantum effects are significant. In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion ), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy , times the duration for which it has that amount of energy. More formally, action

6550-482: The Springer-Verlag journal Physics of Condensed Matter , launched in 1963. The name "condensed matter physics" emphasized the commonality of scientific problems encountered by physicists working on solids, liquids, plasmas, and other complex matter, whereas "solid state physics" was often associated with restricted industrial applications of metals and semiconductors. In the 1960s and 70s, some physicists felt

6681-467: The calculus of variations . The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field . Maxwell's equations can be derived as conditions of stationary action . The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle . The trajectory (path in spacetime ) of

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6812-471: The curvature form or field strength . It also transforms in the adjoint representation. The action S of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form The constant k is called the level of the theory. The classical physics of Chern–Simons theory is independent of the choice of level k . Classically the system is characterized by its equations of motion which are

6943-464: The level of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime. It is also the central mathematical object in theoretical models for topological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes

7074-451: The molecular car , molecular windmill and many more. 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 computing may be realized. To solve this problem, several promising approaches are proposed in condensed matter physics, including Josephson junction qubits, spintronic qubits using

7205-456: The n -point correlation functions of gauge-invariant operators. The most often studied class of gauge invariant operators are Wilson loops . A Wilson loop is the holonomy around a loop in M , traced in a given representation R of G . As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to irreducible representations R . More concretely, given an irreducible representation R and

7336-661: The path integral , which gives the probability amplitudes of the various outcomes. Although equivalent in classical mechanics with Newton's laws , the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman 's path integral formulation , where it arises out of destructive interference of quantum amplitudes. The action principle can be generalized still further. For example,

7467-492: The path integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the final probability amplitude adds all paths using their complex amplitude and phase. Hamilton's principal function S = S ( q , t ; q 0 , t 0 ) {\displaystyle S=S(q,t;q_{0},t_{0})}

7598-452: 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, 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. By 1908, James Dewar and Heike Kamerlingh Onnes were successfully able to liquefy hydrogen and

7729-518: The quantum Hall effect which was discovered half a century later. Magnetism as a property of matter has been known in China since 4000 BC. 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 classifying materials as ferromagnetic , paramagnetic and diamagnetic based on their response to magnetization. Pierre Curie studied

7860-737: The six-vertex model or the XXZ spin chain ), integrable quantum field theories (such as the Gross–Neveu model , principal chiral model and symmetric space coset sigma models ), the Yang–Baxter equation and quantum groups such as the Yangian which describe symmetries underpinning the integrability of the aforementioned systems. The action on the 4-manifold M = Σ × C {\displaystyle M=\Sigma \times C} where Σ {\displaystyle \Sigma }

7991-436: The spin orientation of magnetic materials, and the topological non-Abelian anyons from fractional quantum Hall effect states. Condensed matter physics also has important uses for biomedicine . For example, magnetic resonance imaging is widely used in medical imaging of soft tissue and other physiological features which cannot be viewed with traditional x-ray imaging. Action integral In physics , action

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8122-459: The uncertainty principle and the de Broglie wavelength . Whenever the value of the action approaches the Planck constant, quantum effects are significant. Pierre Louis Maupertuis and Leonhard Euler working in the 1740s developed early versions of the action principle. Joseph Louis Lagrange clarified the mathematics when he invented the calculus of variations . William Rowan Hamilton made

8253-407: 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 more comprehensive specialty of condensed matter physics. The Bell Telephone Laboratories was one of the first institutes to conduct a research program in condensed matter physics. According to the founding director of

8384-407: The 1970s for band structure calculations of variety of solids. 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

8515-618: The Hamilton–Jacobi equation, a formulation of classical mechanics . Due to a similarity with the Schrödinger equation , the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics . In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using

8646-617: The Thomas–Fermi model. The Hartree–Fock method accounted for 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. 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

8777-410: The above formula where C is a (2 k  − 1)-dimensional cycle on M . This invariant is called Chern–Simons invariant . As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS( M ) is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as where p 1 {\displaystyle p_{1}}

8908-470: The action an input to the powerful stationary-action principle for classical and for quantum mechanics . Newton's equations of motion for the ball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where Newton's laws are difficult to apply. Replace the ball with an electron: classical mechanics fails but stationary action continues to work. The energy difference in

9039-440: The action is integrated along the path followed by the physical system. The action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system: S = ∫ t 1 t 2 L d t , {\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,dt,} where

9170-400: The actions of all ten- and eleven-dimensional supergravity theories. If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds n Majorana fermions then, due to the parity anomaly , when integrated out they lead to a pure Chern–Simons theory with a one-loop renormalization of the Chern–Simons level by − n /2, in other words

9301-480: The aforementioned topological band theory advanced by David J. Thouless and collaborators was further expanded leading to the discovery of topological insulators . In 1986, Karl Müller and Johannes Bednorz discovered the first high temperature superconductor , La 2-x Ba x CuO 4 , which is superconducting at temperatures as high as 39 kelvin . It was realized that the high temperature superconductors are examples of strongly correlated materials where

9432-665: The atoms in John 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. In 1823, 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 . Shortly after, in 1869, Irish chemist Thomas Andrews studied

9563-400: The base M . More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation. If M has a boundary N then there is additional data which describes a choice of trivialization of the principal G -bundle on N . Such a choice characterizes a map from N to G . The dynamics of this map

9694-404: The behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics , electromagnetism , statistical mechanics , and other physics theories to develop mathematical models and predict the properties of extremely large groups of atoms. The diversity of systems and phenomena available for study makes condensed matter physics

9825-458: The canonical framing the above phase is the exponential of 2π i /( k  +  N ) times the linking number of L with itself. "The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?" See section 1.1 of this paper for the background and the history of this problem. Kauffman submitted

9956-465: The case of Alexander polynomial, this problem is solved. In the context of string theory , a U ( N ) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold X arises as the string field theory of open strings ending on a D-brane wrapping X in the A-model topological string theory on X . The B-model topological open string field theory on the spacefilling worldvolume of

10087-413: 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

10218-400: The classical data. The linking number of a loop with itself enters into the calculation of the partition function, but this number is not invariant under small deformations and in particular, is not a topological invariant. This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzero normal vector at each point along which one deforms

10349-458: The critical behavior of observables, termed critical phenomena , was a major field of interest in the 1960s. Leo Kadanoff , Benjamin Widom and Michael Fisher developed the ideas of critical exponents and widom scaling . These ideas were unified by Kenneth G. Wilson in 1972, under the formalism of the renormalization group in the context of quantum field theory. The quantum Hall effect

10480-536: 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. 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

10611-466: The current. This phenomenon, arising due to the nature of charge carriers in the conductor, came to be termed the Hall effect , but it was not properly explained at the time because the electron was not experimentally discovered until 18 years later. After the advent of quantum mechanics, Lev Landau in 1930 developed the theory of Landau quantization and laid the foundation for a theoretical explanation of

10742-523: The dependence of magnetization on temperature and discovered the Curie point phase transition in ferromagnetic materials. In 1906, Pierre Weiss introduced the concept of magnetic domains to explain the main properties of ferromagnets. 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

10873-570: The development of the semiconductor transistor , laser technology, magnetic storage , liquid crystals , optical fibres and several phenomena studied in the context of nanotechnology . Methods such as scanning-tunneling microscopy can be used to control processes at the nanometer scale, and have given rise to the study of nanofabrication. Such molecular machines were developed for example by Nobel laureates in chemistry Ben Feringa , Jean-Pierre Sauvage and Fraser Stoddart . Feringa and his team developed multiple molecular machines such as

11004-507: 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 methods of quantum field theory and the renormalization group . Modern theoretical studies involve

11135-458: The electron–electron interactions play an important role. A satisfactory theoretical description of high-temperature superconductors is still not known and the field of strongly correlated materials continues to be an active research topic. In 2012, several groups released preprints which suggest that samarium hexaboride has the properties of a topological insulator in accord with the earlier theoretical predictions. Since samarium hexaboride

11266-413: The endpoints of the evolution are fixed and defined as q 1 = q ( t 1 ) {\displaystyle \mathbf {q} _{1}=\mathbf {q} (t_{1})} and q 2 = q ( t 2 ) {\displaystyle \mathbf {q} _{2}=\mathbf {q} (t_{2})} . According to Hamilton's principle , the true evolution q true ( t )

11397-765: 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 . Commonly used experimental methods include spectroscopy , with probes such as X-rays , infrared light and inelastic neutron scattering ; study of thermal response, such as specific heat and measuring transport via thermal and heat conduction . Several condensed matter experiments involve scattering of an experimental probe, such as X-ray , optical photons , neutrons , etc., on constituents of

11528-502: The extrema of the action with respect to variations of the field A . In terms of the field curvature the field equation is explicitly The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to be flat . Thus the classical solutions to G Chern–Simons theory are the flat connections of principal G -bundles on M . Flat connections are determined entirely by holonomies around noncontractible cycles on

11659-477: The first time, the effect of lattice vibrations on the thermodynamic properties of crystals, in particular the specific heat . Deputy Director of the Yale Quantum Institute A. Douglas Stone makes a similar priority case for Einstein in his work on the synthetic history of quantum mechanics . According to physicist Philip Warren Anderson , the use of the term "condensed matter" to designate

11790-494: 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

11921-469: The important notion of a quasiparticle. Soviet 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. Landau also developed a mean-field theory for continuous phase transitions, which described ordered phases as spontaneous breakdown of symmetry . The theory also introduced

12052-508: The input function is the path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path. The abbreviated action S 0 {\displaystyle {\mathcal {S}}_{0}} (sometime written as W {\displaystyle W} )

12183-399: The integrand L is called the Lagrangian . For the action integral to be well-defined, the trajectory has to be bounded in time and space. Most commonly, the term is used for a functional S {\displaystyle {\mathcal {S}}} which takes a function of time and (for fields ) space as input and returns a scalar . In classical mechanics , the input function

12314-461: The interface between materials: one example is the lanthanum aluminate-strontium titanate interface , where two band-insulators are joined to create conductivity and superconductivity . The metallic state has historically been an important building block for studying properties of solids. 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

12445-420: The invariant polynomial is homogeneous one can write down concretely any k -form of the closed connection ω as some 2 k -form of the associated curvature form Ω of ω . In 1974 S. S. Chern and J. H. Simons had concretely constructed a (2 k  − 1)-form df ( ω ) such that where T is the Chern–Weil homomorphism. This form is called Chern–Simons form . If df ( ω ) is closed one can integrate

12576-408: The kinetic energy (KE) minus the potential energy (PE), integrated over time. The action balances kinetic against potential energy. The kinetic energy of a ball of mass m {\displaystyle m} is ( 1 / 2 ) m v 2 {\displaystyle (1/2)mv^{2}} where v {\displaystyle v} is the velocity of the ball;

12707-438: 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. These critical phenomena present serious challenges to physicists because normal macroscopic laws are no longer valid in

12838-422: The level k theory with n fermions is equivalent to the level k  −  n /2 theory without fermions. Condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter , especially the solid and liquid phases , that arise from electromagnetic forces between atoms and electrons . More generally,

12969-433: The loop to calculate its self-linking number. This procedure is an example of the point-splitting regularization procedure introduced by Paul Dirac and Rudolf Peierls to define apparently divergent quantities in quantum field theory in 1934. Sir Michael Atiyah has shown that there exists a canonical choice of 2-framing, which is generally used in the literature today and leads to a well-defined linking number. With

13100-588: The many vacua discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua, supersymmetry was computed to be broken. These monopoles were related to condensed matter vortices . ( Intriligator & Seiberg (2013) ) The N  = 6 Chern–Simons matter theory is the holographic dual of M-theory on A d S 4 × S 7 {\displaystyle AdS_{4}\times S_{7}} . In 2013 Kevin Costello defined

13231-462: The many-body wavefunction is often computationally hard, and hence, approximation methods are needed to obtain meaningful predictions. 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 so-called Hartree–Fock wavefunction as an improvement over

13362-416: The more comprehensive name better fit the funding environment and Cold War politics of the time. References to "condensed" states can be traced to earlier sources. For example, in the introduction to his 1947 book Kinetic Theory of Liquids , 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

13493-1052: The most active field of contemporary physics: one third of all American physicists self-identify as condensed matter physicists, and the Division of Condensed Matter Physics is the largest division of the American Physical Society . These include solid state and soft matter physicists, who study quantum and non-quantum physical properties of matter respectively. Both types study a great range of materials, providing many research, funding and employment opportunities. The field overlaps with chemistry , materials science , engineering and nanotechnology , and relates closely to atomic physics and biophysics . 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., were treated as distinct areas until

13624-406: The motion of an electron in a periodic lattice. 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. Band structure calculations were first used in 1930 to predict

13755-446: The next big breakthrough, formulating Hamilton's principle in 1853. Hamilton's principle became the cornerstone for classical work with different forms of action until Richard Feynman and Julian Schwinger developed quantum action principles. Expressed in mathematical language, using the calculus of variations , the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to

13886-608: The normalized correlation function is, up to a phase, equal to times the HOMFLY polynomial . In particular when N  = 2 the HOMFLY polynomial reduces to the Jones polynomial . In the SO( N ) case, one finds a similar expression with the Kauffman polynomial . The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by

14017-433: The notion of an order parameter to distinguish between ordered phases. Eventually in 1956, John Bardeen , Leon Cooper and Robert Schrieffer developed the so-called 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 . The study of phase transitions and

14148-538: The potential energy is m g x {\displaystyle mgx} where g {\displaystyle g} is the gravitational constant. Then the action between t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} is The action value depends upon the trajectory taken by the ball through x ( t ) {\displaystyle x(t)} and v ( t ) {\displaystyle v(t)} . This makes

14279-575: The probe of these hyperfine interactions ), which couple the electron or nuclear spin to the local electric and magnetic fields. These methods are suitable to study defects, diffusion, phase transitions and magnetic order. Common experimental methods include NMR , nuclear quadrupole resonance (NQR), implanted radioactive probes as in the case of muon spin spectroscopy ( μ {\displaystyle \mu } SR), Mössbauer spectroscopy , β {\displaystyle \beta } NMR and perturbed angular correlation (PAC). PAC

14410-574: The properties of new materials, and in 1947 John Bardeen , Walter Brattain and William Shockley developed the first semiconductor -based transistor , heralding a revolution in electronics. In 1879, Edwin Herbert Hall working at the Johns Hopkins University discovered that a voltage developed across conductors which was transverse to both an electric current in the conductor and a magnetic field applied perpendicular to

14541-462: The quality of NMR measurement data. Quantum oscillations is another experimental method where high magnetic fields are used to study material properties such as the geometry of the Fermi surface . High magnetic fields will be useful in experimental testing of the various theoretical predictions such as the quantized magnetoelectric effect , image magnetic monopole , and the half-integer quantum Hall effect . The local structure , as well as

14672-504: The region, and novel ideas and methods must be invented to find the new laws that can describe the system. 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

14803-622: The 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. Similarly, positron annihilation can be used as an indirect measurement of local electron density. 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 . In experimental condensed matter physics, external magnetic fields act as thermodynamic variables that control

14934-587: The simple action definition, kinetic minus potential energy, is generalized and called the Lagrangian for more complex cases. The Planck constant , written as h {\displaystyle h} or ℏ {\displaystyle \hbar } when including a factor of 1 / 2 π {\displaystyle 1/2\pi } , is called the quantum of action . Like action, this constant has unit of energy times time. It figures in all significant quantum equations, like

15065-506: The simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model. The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory , and in particular WZW theory . In the 1940s S. S. Chern and A. Weil studied the global curvature properties of smooth manifolds M as de Rham cohomology ( Chern–Weil theory ), which

15196-476: The space of conformal blocks of the G WZW model at level k. For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory. The observables of Chern–Simons theory are

15327-658: The starting and ending coordinates. According to Maupertuis's principle , the true path of the system is a path for which the abbreviated action is stationary . When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables : S ( q 1 , … , q N , t ) = W ( q 1 , … , q N ) − E ⋅ t , {\displaystyle S(q_{1},\dots ,q_{N},t)=W(q_{1},\dots ,q_{N})-E\cdot t,} where

15458-414: The state, phase transitions and properties of material systems. Nuclear magnetic resonance (NMR) is a method by which external magnetic fields are used to find resonance modes of individual nuclei, thus giving information about the atomic, molecular, and bond structure of their environment. NMR experiments can be made in magnetic fields with strengths up to 60 tesla . Higher magnetic fields can improve

15589-415: The structure of the nearest neighbour atoms, can be investigated in condensed matter with magnetic resonance methods, such as electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR), which are very sensitive to the details of the surrounding of nuclei and electrons by means of the hyperfine coupling. Both localized electrons and specific stable or unstable isotopes of the nuclei become

15720-490: The subject deals with condensed phases of matter: systems of many constituents with strong interactions among them. More exotic condensed phases include the superconducting phase exhibited by certain materials at extremely low cryogenic temperatures , the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, the Bose–Einstein condensates found in ultracold atomic systems, and liquid crystals . Condensed matter physicists seek to understand

15851-450: 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 . After World War II , several ideas from quantum field theory were applied to condensed matter problems. These included recognition of collective excitation modes of solids and

15982-465: 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

16113-440: The then newly discovered helium respectively. Paul Drude in 1900 proposed the first theoretical model for a classical electron moving through a metallic solid. 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 . However, despite the success of Drude's model , it had one notable problem: it

16244-943: The time-independent function W ( q 1 , q 2 , ..., q N ) is called Hamilton's characteristic function . The physical significance of this function is understood by taking its total time derivative d W d t = ∂ W ∂ q i q ˙ i = p i q ˙ i . {\displaystyle {\frac {dW}{dt}}={\frac {\partial W}{\partial q_{i}}}{\dot {q}}_{i}=p_{i}{\dot {q}}_{i}.} This can be integrated to give W ( q 1 , … , q N ) = ∫ p i q ˙ i d t = ∫ p i d q i , {\displaystyle W(q_{1},\dots ,q_{N})=\int p_{i}{\dot {q}}_{i}\,dt=\int p_{i}\,dq_{i},} which

16375-405: The unanticipated precision of the integral plateau. It also implied that the Hall conductance is proportional to a topological invariant, called Chern number , whose relevance for the band structure of solids was formulated by David J. Thouless and collaborators. Shortly after, in 1982, Horst Störmer and Daniel Tsui observed the fractional quantum Hall effect where the conductance was now

16506-431: 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. For example,

16637-420: The values of A on different patches are related by maps known as gauge transformations . These are characterized by the assertion that the covariant derivative , which is the sum of the exterior derivative operator d and the connection A , transforms in the adjoint representation of the gauge group G . The square of the covariant derivative with itself can be interpreted as a g -valued 2-form F called

16768-460: 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 to explain the heat capacity. Two years later, Bloch used quantum mechanics to describe

16899-405: Was discovered by Klaus von Klitzing , Dorda and Pepper in 1980 when they observed the Hall conductance to be integer multiples of a fundamental constant e 2 / h {\displaystyle e^{2}/h} .(see figure) The effect was observed to be independent of parameters such as system size and impurities. In 1981, theorist Robert Laughlin proposed a theory explaining

17030-468: 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 lattices of atoms. In 1928, Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, known as Bloch's theorem . Calculating electronic properties of metals by solving

17161-415: Was unable to correctly explain the electronic contribution to the specific heat and magnetic properties of metals, and the temperature dependence of resistivity at low temperatures. 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

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