* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Prof. Makarova Lecture 1 - pcam
Lorentz force wikipedia , lookup
Field (physics) wikipedia , lookup
Nuclear physics wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Magnetic field wikipedia , lookup
Quantum entanglement wikipedia , lookup
Fundamental interaction wikipedia , lookup
Hydrogen atom wikipedia , lookup
EPR paradox wikipedia , lookup
Old quantum theory wikipedia , lookup
Magnetic monopole wikipedia , lookup
Bell's theorem wikipedia , lookup
Photon polarization wikipedia , lookup
State of matter wikipedia , lookup
History of quantum field theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum vacuum thruster wikipedia , lookup
Neutron magnetic moment wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Electromagnet wikipedia , lookup
Spin (physics) wikipedia , lookup
Electromagnetism wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Superconductivity wikipedia , lookup
A gentle introduction to lowD magnetism “Few subjects in science are more difficult to understand than magnetism.” Encyclopedia Britannica, 15th Edition, 1989. Magnetism • Complications • Difficulties • Confusions Recommended literature H.‐J. Mikeska and A.K. Kolezhuk, One‐Dimensional Magnetism, Lect. Notes Phys. 645, 1–83 (2004) Christopher P. Landee , Mark M. Turnbull. Review: A gentle introduction to magnetism: units, fields, theory, and experiment. Journal of Coordination Chemistry 2014; 67: 375. https://gravityandlevity.wordpress.com/ Quantum nature of magnetism DIFFICULTIES Magnetism is a difficult subject • This fact is pretty well illustrated by how hard it is to predict whether a given material will make a good magnet or not. • If someone tells you the chemical composition of some material and then asks you “will it be magnetic?”, you (along with essentially all physicists) will have a hard time answering. • Whether a material behaves like a magnet usually depends very sensitively on things like the crystal structure of the material, the valence of the different atoms, and what kind of defects are present. • Subtle changes to any of these things can make the difference between having a strong magnet and having an inert block. • Consequently, a whole scientific industry has grown up around the question “will X material be magnetic?”, and it keeps thousands of scientists gainfully employed. Interpreting magnetic data COMPLICATIONS Complications • Over the past several decades, a significant number of research groups have begun inserting magnetic data into their work. • There are a variety of reasons for this development: the greater access to SQUID magnetometers with user‐friendly software; • Increased collaborations between chemists and physicists; and the greater ability to form international collaborations. • However, as is frequently the case, many of these scientists are not simply new to collecting magnetic data; they are also new to the field of magnetism and especially to understanding the influence of magnetic interactions The only evidence of interactions! Units and fields CONFUSIONS A gentle introduction to magnetic fields and units Units • The SI system is universally used in education and employs the mechanical and electrical units with which we are all familiar: joules, watts, newtons, volts, ohms, farads, etc. • Unfortunately, much of the research in magnetism is still reported in the older CGS (centimeters, grams, seconds) system of units • Converting from one system to the other is an endless source of confusion. Fields (SI) B magnetic flux density (Tesla) H magnetic field (A/m) The term "magnetic field" is historically reserved for H M magnetization (A/m) Magnetic field is NOT measured in Tesla! Fields (CGS) B magnetic flux density (Gauss) H magnetic field (Oersted) M magnetization (emu/cm ) 3 [Gauss] = [Oersted] = [emu/cm3] B‐field is the magnetic flux density The B‐field is also known as the magnetic induction • B is defined in terms of the force on a moving charge • (N) for force, Coulombs (C) for charge, m s−1 for velocity, and Tesla (T) for B. • One Tesla is thus equivalent to one N A−1 m−1 http://www.a‐levelphysicstutor.com Flux Flux density The B‐field is related to the two other magnetic fields, H and M B = µ0 (H + M) µ0 = 4π × 10−7 T m A‐1 •The H‐field, magnetic field, arises from electrical currents passing through wires, H =I /2r •Consequently the units for the H‐field and M‐field are amperes per meter, A m−1 •M is the volume magnetization The volume magnetization M equals the vector sum of all the magnetic moments per cubic meter. Magnetic moments arise from circulating currents (whether quantum or free currents) and are equal to the product of the current in the loop times the area of the loop; Moments have units of [A m2]. The units of the volume magnetization are [A m2 per cubic meter], or A m−1, the same as that of the magnetic field H Magnetization Scientists rarely know the volumes of their samples, but they can measure the masses. M per unit mass is volume magnetization by dividing M by the sample’s density One can then obtain the magnetization per mole by multiplying the mass magnetization by the formula weight (FW) Magnetic susceptibility χ The volume susceptibility is defined as the ratio of the sample’s volume magnetization to the applied H‐field, in the limit of a vanishingly small field Nevertheless, in the laboratory, we measure the magnetization per unit of mass Even 1/1000000 part of ferromagnetic impurity will lead to a positive iniitial magnetic susceptibility Connection between magnetic units and energy Connection between SI magnetic units and energy • In a B‐field, a magnetic moment experiences a torque that tends to align the moment parallel to the field. • The energy U required to rotate the moment away from the field direction is given as U = ‐ µ B the Zeeman equation Therefore, the SI unit of moment μ (A m2) is also equal to the ratio of energy to field, one joule per tesla: 1 A m2 = 1 J T−1. Magnetic fields and units in the CGS system In CGS, µ0 is set equal to unity, which makes B and H, and M numerically equal to one another, but each have different unit names Gauss, Oersted, and emu/cm3 B=H+4πM Confusion! The unit of the B‐field is the gauss (G, 1 G = 10−4 T) The unit of the H‐field is the oersted (Oe, 1 Oe = 103/4πAm−1 = 79.6 A m−1 Example: The Earth field is 0.5 Gauss or 0.5 Oe. Convert to SI 0.5 Gauss = 50 µT [B fields] 0.5 Oersted = 39.8 A/m [H fields] unpleasant! It is a current practice to report H in Tesla. If we say 100 milliTesla (mT), we really mean µ0H=100 mT. However, this is rarely noted. B = µ0 (H + M) B is measured in Tesla H is NOT measured in Tesla If one says that the SQUID produces magnetic field … say 5 Tesla, one keeps in mind that µ0H = 5 Tesla Standard mistakes in publications M (emu/g) H is not measured in Tesla µ0H is measured in Tesla H (T) emu/g are measured vs Oersted Magnetization in CGS Magnetization has units of emu/cm3. This is unfortunate nomenclature for it leads people to believe that the emu is the cgs‐emu unit of magnetic moment; it is not! This use of emu simply means that the magnetization is given in the emu system, with the cm3 identifying the magnetization as the volume magnetization If not the emu, what is the cgs unit of magnetic moment? It is identified from the Zeeman equation, U = ‐ µ B the Zeeman equation The cgs moment has units of ergs G−1 and is smaller than the SI unit of moment, the A m2, by exactly 1000. The cgs volume magnetization therefore has units of erg /G cm3, usually expressed as emu/cm3. B = H + 4πM The cgs volume magnetization has units of erg G−1 cm−3, usually expressed as emu cm−3. Volume magnetization has the units of oersteds, 1 Oe = 1 erg G−1 cm−3 = 1 emu cm−3. However, it is also seen that 4πM has the unit of gauss 1 Oe = 1 erg G−1 cm−3 = 1 emu cm−3 = [1 Gauss] In the cgs system, both the free currents in wires and the bound currents in magnetized matter produce magnetic flux lines but the free currents are weighted more heavily. This difference does not occur in the SI system. Now we compare definitions of moments (as a product of currents times area) from the cgs and SI systems Does 1 A cm2 = 10−3Am2? Clearly not, because 1 m2 = 104 cm2. The inequality arises because the cgs‐emu unit of current is not the ampere, but the abampere, equal to 10 amperes A gentle introduction to paramagnetism Paramagnetism Paramagnetic solid : thermal agitation (kT) larger than the interaction Origin of Magnetism I am an electron • rest mass me, • charge e-, • magnetic moment μB Picture: J.Barandiaran μtotal = μorbital + μspin Paramagnetism Magnetic behavior of any current loop arises from a magnetic moment μ that equals the product of the current and the area A of the loop, μ = IA. µ = qvR /2 The angular momentum of an object moving about a point is given as the vector product L=R×mv µ = qL/2m Bohr magneton µ = qL/2m electron µ = ‐ eL/2me Orbital angular momentum at the atomic level is quantized and appears as an integer multiple of Planck’s constant h : Bohr magneton eħ/2me Bohr magneton eħ/2me This ratio of basic units is one of the fundamental constants in science and is given the symbol μB and named the Bohr magneton. It has the very small value of 9.274 × 10−24 J /T 9.274 10−21 erg/ G 0.579 × 10‐8 eV/G Very small energy scale! If a laboratory SQUID generates the field 10 000 Oe, µBH = 0.06 meV Similar rules apply for the intrinsic, or spin angular momentum of the electron An electron has an angular momentum S = ħ/2 Consequently, each electron also has an intrinsic magnetic moment µe = ‐ g µB S / ħ where the negative sign arises from the electron’s negative charge. The constant g (2.0023) is the electron g‐factor and reflects the different proportionality between spin and orbital angular momentum in creating magnetic moments. Magnetization of a mole of independent spins The spin S = ½ case: ms = ± ½ and μz = – gμBms U = ‐ µ B the Zeeman equation The moments parallel to the field (ms = – ½ ) have their energies lowered as the field increases, while those antiparallel experience the opposite effect The molar magnetization Mmol equals the product of the net number of spins aligned with the field with the moment of each spin Mmol = µe (N+ ‐ N‐) where N+ + N‐ = Avogadro number The values of N+ and N– are related through the Boltzman•n relation N‐ /N+ = exp (‐ Δ /kBT) Zero T As T Æ 0, the exponential terms vanish, and N+ and N– reach their limiting values of NA and zero, respectively, and the molar magnetization reaches its saturation value, Msat. Temperature dependence of M Hyperbolic tangent tanh(1) = 0.761, tanh(2) = 0.964, and tanh(3) = 0.995. µB/kB = 0.6717 If B = 1 tesla, T = 1 K, then BµB/kBT ≅ 1 The final expression for the magnetization for a mole of S = ½ moments If the experimental data can fit to this expression with a reasonable value for g (2.10 ± 0.05), that means that no significant exchange interactions are present. For example, in a 5 T (50,000 Oe) field at 1.8 K, the argument equals 1.87(g/2). For a copper(II) sample with 〈g〉 = 2.10, the argument equals 1.96, enough to reach 95% of the saturation field. Magnetic susceptibility of a mole of independent spins: Curie’s law Demagnetization corrections to experimental susceptibilities For an unmagnetized sample, the external field H created by a solenoid or electromagnet equals the internal field. However, once the sample becomes magnetized, H is reduced by an amount which depends both on the shape of the sample and its magnetization, The true susceptibility depends on the value of Hint but the measured susceptibility is based on the value of the external H‐field, Hext For example, take a long thin needle shaped grain. The demagnetizing field will be less if the magnetization is along the long axis than if is along one of the short axes A gentle introduction to exchange interactions Magnetic ordering one of the less well-developed theories in solid state physics Magnetic ordering one of the less well-developed theories in solid state physics What if the spins are not independent but interacting? The first successful solution of a magnetic many‐body problem was obtained by Pierre Weiss in 1907 He called this new field the “molecular field” because it arose from the other molecules in the sample. Hw H (applied) HW = wM w=Weiss or molecular field coefficient An interaction between moments modifies Curie’s Law by subtracting the Weiss temperature θ from the sample temperature T J.H. Van Vleck (Nobel Prise 1974) “The Curie Weiss Law; the most overused equation in magnetism”, Physica, 69, 177 (1973). • Curie’s law and the Curie–Weiss law are derived by assuming that the arguments of the hyperbolic tangent were very small, i.e. at the limit of low fields/high temperatures. • The laws are only valid for temperature‐independent moments if the populations of these levels do not change with temperature. • The Curie–Weiss temperature θ (also known as the Weiss constant or Curie–Weiss constant) is incorrectly referred to as a magnetic ordering temperature • The existence of a finite θ does not imply the existence of ordering Weiss theory is a good phenomenological theory of magnetism, But does not explain the source of large Weiss field. Exchange interactions Magnetism exists because the orientation of a moment affects the orientations of other moments. This interaction is an electrostatic one at the atomic level, where the rules of quantum mechanics apply. Evaluating the electrostatic energy involves the exchange terms in the symmetric or antisymmetric wave functions, so this interaction energy is often called the exchange energy. The literature on exchange interactions is broad and deep So what is it that makes electron align their spins with each other? This is the essential puzzle of magnetism North‐to‐South attraction? No 1 Å Æ 1 Tesla; 3 nm Æ less than Earth’s The magnetic field created by one electron: 1 Å is appr. 10 000 Oe But the strength of that field decays quickly with distance, as r‐3 . At the distance 3 nm the magnetic field produced by one electron is weaker than 1 Oe So what is it that makes electron align their spins with each other? This is the essential puzzle of magnetism North‐to‐South attraction? No Magnetic forces between electrons are too weak. Even at about 1 Å of separation, the energy of magnetic interaction between two electrons is less than 0.0001 eV. There are really only two kinds of energies that matter: the electric repulsion between electrons the huge kinetic energy that comes from quantum motion Electrons are little magnets Electrons within a solid material are feeling enormous repulsive forces due to their electric charges, Electrons are flying around at speeds of millions of miles per hour. They don’t have time to worry about the magnetic forces that are being applied. In cases where electrons decide to align their spins with each other, it must be because it helps them reduce their enormous electric repulsion, and not because it has anything to do with the little magnetic forces. Same‐spin electrons avoid each other • Since electrons are strongly repulsive, they want to avoid each other as much as possible. • The simplest way to do this would be to just stop moving, but this is prohibited by the rules of quantum mechanics. • Instead of stopping, electrons try to find ways to avoid running into each other. • One clever method is to take advantage of the Pauli exclusion principle (no two electrons can have the same state at the same time) No two electrons can simultaneously have the same spin and the same location. No two electrons can simultaneously have the same spin and the same location In a magnet, electrons point their spins in the same direction, not because of any magnetic field‐based interaction, but in order to guarantee that they avoid running into each other. By not running into each other, the electrons can save a huge amount of repulsive electric energy. This saving of energy by aligning spins is what we call the “exchange interaction”. Probability that a given pair of electrons will find themselves with a separation r For electrons with opposite spin (in a metal), this probability distribution looks pretty flat: electrons with opposite spin are free to run over each other, and they do. But electrons with the same spin must never be at the same location at the same time, and thus the probability distribution must go to zero at r=0; it has a “hole” in it at small r . The size of that hole is given by the typical wavelength of electron states: the Fermi wavelength. This result makes some sense: after all, the only meaningful way to interpret the statement “two electrons can’t be in the same place at the same time” is to say that “two electrons can’t be within a distance of each other, where is the electron size”. And the only meaningful definition of the electron size is the electron wavelength. https://gravityandlevity.files.wordpress.com/2015/04/exchange_probabilitie s.png The probability is called the “pair distribution function“, which people spend a lot of time calculating for electron systems So why isn’t everything magnetic? 1. Electrons themselves are tiny magnets. 2. Electrons like to point their spins in the same direction, because this saves them a lot of electric repulsion energy. Why doesn’t everything become a magnet? The answer is that there is an additional cost that comes when the electrons align their spins. Specifically, electrons that align their spins are forced into states with higher kinetic energy. Quantum particle in a box. Every allowable state for an electron can hold only two electrons. But if you start forcing all electrons to have the same spin, then each energy level can only hold one electron, and a bunch of electrons get forced to sit in higher energy levels. When you give electrons the same spin, and thereby force them to avoid each other, you are confining by constraining their wavefunctions to not overlap with each other. Extra confinement means that their momentum has to go up (by the Heisenberg uncertainty principle), and so they start moving faster. Magnetism comes from the exchange interaction Aligning the electron spins means that the electrons have to acquire a larger kinetic energy. So when you try to figure out whether the electrons actually will align their spins, you have to weigh the benefit (having a lower interaction energy) against the cost (having a higher kinetic energy). A quantitative weighing of these two factors can be difficult But the basic driver of magnetism is really as simple as this: like‐spin electrons do a better job of avoiding each other, and when electrons line up their spins they make a magnet. A gentle introduction to LowD magnetism Interactions Direct exchange between the compass arrows. Antiparallel Parallel Magnetic interactions are directional Magnetic interactions are directional H = J Si•Sj where Si is the spin operator located at the lattice site i and J denotes the strength of exchange interaction For such a simple equation, a great deal of confusion appears in the literature ! To avoid confusion, it is necessary for every scientific article to display the Hamiltonian Ising model Interaction favors alignment of neighboring magnetic moments along a preferred direction, and thus lowers the energy of the system, making a negative contribution ‐J. What about thermal fluctuations? kT ∼ J Ising answered this question Ernst Ising 1900‐1998 In Hamburg University Ising was a PhD student of Wilhelm Lenz who have him the topic „Beitrag zur Theorie des Ferro‐ und Paramagnetismus“. Defended in 1924. Ising has only one publication Ising, E.: Beitrag zur Theorie des Ferromagnetismus Zeitschrift für Physik, Bd. 31, S. 253‐258, 1925 The thermal behavior of a linear and consisting of elementary magnets body is examined. In contrast to the Weiss theory of ferromagnetism, no molecular field, but only one (non‐ magnetic) interaction of adjacent elementary magnets adopted. It is shown that such a model still has no ferromagnetic properties and this statement is extended also on the three‐dimensional model. Ising did not take part in the scientific life, and only in 1949 learned that he was very famous He always asked to call his work ”Lenz‐Ising model” Ising model • Ising considered linear chain of magnetic moments with the interaction of the nearest neighbors. He showed that the spontaneous magnetization can not be explained within the framework of a one‐dimensional model. • Extended to 3D case ‐ erroneously! • Heisenberg W. // Zeitschrift f. Physik. 1928. Bd.49. S.619‐636. : Ising showed that even sufficiently large force between any two adjacent atoms in the chain does not explain the appearance of ferromagnetism. • Peierls R. // On the Ising model of ferromagnetism, Proc. Cambridge Phil. Soc. 1936. V.32. P.477‐481 coined the term “Ising model”. In 1D case ferromagnetism exists, but Tc = 0. • Kramers H.A., Wannier G.H. // Phys. Rev. 1941. V.60. P.252‐262. Onsager L. // Phys. Rev. 1944. V.65. P.117‐149 found the solution for the 2D model (Tc = 2.27 J) LowD magnetism LowD magnetism is not related to 2D lattices (on surfaces) or 2D electron gases (in quantum wells) or quantum dots LowD magnetism Magnets in restricted dimensions Exchange interactions which lead to magnetic coupling are much stronger in one or two spatial directions than in the remaining ones. Real materials are 3D but behave effectively as low‐dimensional systems if the dominant exchange interactions are intra‐chain (1D) or intra‐planar (2D) LowD magnetism The interest in low‐dimensional magnets developed into a field of its own because these materials provide: •a unique possibility to study ground and excited states of quantum models, •possible new phases of matter •interplay of quantum fluctuations and thermal fluctuations. LowD magnetism can be traced back 90 years ago: • In 1925 Ernst Ising followed a suggestion of his academic teacher Lenz and investigated the 1D version of the model which is now well known under his name in an effort to provide a microscopic justification for Weiss’ molecular field theory of cooperative behavior in magnets. • In 1931 Hans Bethe wrote his famous paper entitled ’Zur Theorie der Metalle. I.Eigenwerte und Eigenfunktionen der linearen Atomkette’ describing the ’Bethe ansatz’ method to find the exact quantum mechanical ground state of the antiferromagnetic Heisenberg model for the 1D case • Both papers were actually not to the complete satisfaction of their authors: The 1D Ising model failed to show any spontaneous order whereas Bethe did not live up to the expectation expressed in the last sentence of his text: ’in a subsequent publication the method is to be extended to cover 3D lattices’. For the first 40 years LowD magnetism was an exclusively theoretical field Theorists were attracted by the chance to find interesting exact results without having to deal with the hopelessly complicated case of models in 3D. They succeeded in extending the solution of Ising’s (classical) model to 2D (which, as Onsager showed, did exhibit spontaneous order) Succeeded in calculating excitation energies, correlation functions and thermal properties for the quantum mechanical 1D Heisenberg An important characteristic of low‐dimensional magnets is the absence of long range order in models with a continuous symmetry at any finite temperature as stated in the theorem of Mermin and Wagner, and sometimes even the absence of long range order in the ground state (Coleman). Question: Does a lower dimension (e.g., 2D instead of 3D), i.e. ”less neighbouring spins” change the ordering behaviour ? Answer: Yes. Fundamental statement ”At any non‐zero temperature, a one‐ or two‐dimensional isotropic spin‐S Heisenberg model with finite‐range exchange interaction can be neither ferromagnetic nor antiferromagnetic.” Mermin, N. D. & Wagner, H. (1966), "«Absence of Ferromagnetism or Antiferromagnetism in One‐ or Two‐ Dimensional Isotropic Heisenberg Models (note the assumptions ”isotropic” and ”finite‐range.) Faddeev and Takhtajan revealed the spinon nature of the excitation spectrum of the spin‐½ antiferromagnetic chain Spinons are one of three quasiparticles that electrons are able to split into during the process of spin–charge separation. Spinon (a spin ½ excitation) carries the spin of the electron, the orbiton carries the orbital location and the holon carries the charge. Magnons are the elementary particles of spin waves. Magnons have a spin equal to one and therefore obey Bose‐Einstein statistics Haldane discovered the principal difference between chains of integer and half‐integer spins MAGNETIC INTERACTIONS IN FLATLAND Exchange interactions: the S = 1/2 Heisenberg Dimer H = J Si•Sj Two spins, S1 and S2, interact according to the Hamiltonian H = JSi□Sj The spins can couple to form an Stot = 0 (singlet state) and an Stot = 1 triplet state, where Stot is the total spin for the system. The energies of the states depend on their mutual orientation. S1•S2 = ½ (Stot•Stot-S1•S1-S2•S2) = ½ (Stot(Stot+1)- ¾ - ¾) N‐ /N+ = exp (‐ Δ /kBT) ??? (paramagnetism) Nex/Ngr = exp (‐J/T) /(1+3exp (‐J/T) Interactions of spins on a 2D plane Create unpaired electrons Let them interact Get two possibilities: Align parallel Align antiparallel S = ½+ ½ = 1 S = ½- ½ = 0 Magnetic susceptibility S=0 χ = 0! J Transition probability: ’ Dimensionless magnetic susceptibility for lowD systems: Broad max Exp decay at low T Model calculation for J = 400 K where C is the Curie Constant 0,6 χ J/C χJC = χmolarJ/C 0,4 χmaxJ/C ∼ 0,2 0,0 0 100 200 T 300 0.58770511 at Tmax/J = 0.6408510 Temperature dependence of χ for AFM and FM dimers The normalized molar susceptibilities (χmol/C) of the ferro‐and antiferro‐ magnetic Heisenberg S = 1/2 dimer are plotted as a function of relative temperature kBT/|J|. (Figure 9 from Landee&Turnbull) Plotting χT versus T: an excellent method to check the presence of AF or FM interactions! Very Important Note! Paramagnetism: Model calculations for Npara = 1019 AF interacting dimers. Model calculations for Ndimer = 1021 Total χ 2 1 0 0 100 200 T, K 300 400 J Exchange coupled spin networks Magnetic susceptibilities of exchange coupled spin networks Alternating chain Uniform chain Dimers Ladder 2,5 -3 emu/molOe 2,0 1,5 1,0 χ, 10 Theoretical models describing possible arrangement of S= ½ spins: •The Heisenberg chain with isotropic antiferromagnetically coupled spins (Bonner‐ Fisher), • Dimerised chain (Bleaney‐Bowers), • Alternating chain (Hatfield), • Spin ladder (Troyer‐ Tsunetsugu‐Würtz). 0,5 2 2 χmaxJ /NA g μB Tmax/J = 0.64 0,0 0,0 0,2 0,4 T/J 0,6 0,8 1,0 EXPERIMENTAL 1D AND 2D MAGNETISM Experimental 1D and 2D magnetism • It was only around 1970 when it became clear that the one‐ and two‐dimensional might also be relevant for real materials • Magnets in restricted dimensions have a natural realization: Real bulk crystals with exchange interactions much stronger in one or two spatial directions than in the remaining ones. • Most studies of lowD magnetism concentrate on Сu or Ni compounds which realize spin‐½ or 1 correspondingly. Low Dimensional Quantum Magnets World‐wide progress in material design has provided a multitude of materials in which selected degrees of freedom like spins, orbitals, or charges are dispersing effectively only along one or two dimensions like in the spin‐1/2 Heisenberg antiferromagnet chains (HAFC) shown for copper pyrazine dinitrate. Technical University of Braunschweig http://www.fkt.tu‐bs.de/brenig/research.shtml Nd2BaNiO5 ‐ a Haldane spin chain The main ingredient of this material is Ni (S=1) ions arranged in chains, making it a near perfect example of 1D Heisenberg antiferomagnet. Brookhaven National Laboratory http://www.cmth.bnl.gov/~maslov/low‐ d_magnetism.htm It was predicted by Haldane that 1D antiferromagnet with integer (as opposed to half‐integer) spin will be in a quantum‐disordered state, where the long range antiferromagnetic order is completely destroyed by quantum fluctuations. Spin ladder ”telephone number compound” Sr14Cu24O41 An electronically low‐dimensional transition metal oxide, the spin‐ladder compound (Sr,Ca,La)14Cu24O42. This material is highly anisotropic and contain localized quantum spin‐½ magnetic moments which are located on the sites of the copper atoms. If biased by a temperature gradient the spin‐ladder compounds reveal the remarkable phenomenon of colossal spin heat‐transport due to thermal fluctuations of the localized magnetic moments. http://www.fkt.tu‐bs.de/brenig/research.shtml in Organic Radical Crystals Thiazyl-based F. Palacio Nitroxide-based Y. Hosokoshi Spin gap Spin ladder crystals d n o b e c n vale Verdazyl-based K. Mukai Spin Peierls pseudog apped me tals spin li a range of exotic phasesductivity quids on c r e p u s l a onvention unc Spin gap • The excitation spectrum of three‐dimensional magnets does not have a gap. Spin gap is an essentially low‐D phenomenon and has a purely quantum origin; its value is determined by the strength of exchange interactions between the spins. • Spin gap is a phenomenon which, due to quantum fluctuations, destroys magnetically ordered ground state at low temperatures despite strong interactions between magnetic units Why do we need SPIN GAPPED MAGNETS It is interesting! • The physics of low‐dimension quantum antiferromagnets (AF) is intriguing and surprising. AF spin chains or ladders display exotic behavior such as spin liquid, spin glass, and spin ice states; magnetic orders, spin Peierls state, etc. depending on the value of the spin, the dimensionality of the material, the anisotropy, the strengths and signs of the magnetic couplings. • Quantum phase transitions have been extensively studied over the last decades both from a theoretical and an experimental point of view. • It is well known that a magnetic system can show a crossover from a long range ordered state to quasi 1D magnetic behavior or even high‐Tc superconductivity. Quantum spin liquid • At sufficiently low temperatures, condensed‐ matter systems tend to develop order. An exception are quantum spin‐liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. • Physicists started paying more attention to quantum spin liquids in 1987, when Nobel laureate Philip W. Anderson theorized that quantum spin liquid theory may relate to the phenomenon of high‐temperature superconductivity, Herbertsmithite ZnCu3(OH)6Cl2 • The QSL is a solid crystal, but its magnetic state is described as liquid: the magnetic orientations of the individual particles within it fluctuate constantly, resembling the constant motion of molecules within a true liquid. • There is no static order to magnetic moments, but there is a strong interaction between them, and due to quantum effects, they don’t lock in place T.‐H. Han, et al., Fractionalized excitations in the spin liquid state of a kagome lattice antiferromagnet. Nature, 492, Dec. 20, 2012, doi: 10.1038/nature11659. Quantum spin liquids are exotic ground states of frustrated quantum magnets, in which local moments are highly correlated but still fluctuate strongly down to zero temperature. Superconductivity without phonons Doping the ladder with holes destroys the spin singlets. Doping with one hole breaks three AF bonds, giving rise to an energy loss proportional to 3J. When two holes are introduced and move independently of each other, the total energy loss is 6J • Holes bind into pairs • Superconductivity Z. Hiroi, J. Solid State Chem. 123, 223 (1996). La12xSrxCuO2.5 as a Doped Spin‐Ladder Compound The holes tend to share the same rung forming hole bound‐ states in order to minimize the energy damage M. Sigrist, PRB, 49, 12058 (1994) Spin chains as quantum wires • Spin systems have recently been suggested as candidates for the realization of quantum computation and communication prootocls • The spin systems considered so far include Spin Gap AFM chains, the two‐chain spin ladder • S. Bose Quantum communication through spin chain dynamics: an introductory overview, Contemporary Physics, 48:1, 13‐30 (2007) Entanglement: at the heart of quantum communication and computation • Two particles are said to be entangled when the quantum state of each particle cannot be described independently, no matter how far apart in space and time the two particles are. • Photons immediately spring to mind when we talk about long‐distance entanglement. • But the spins at the ends of one‐dimensional magnetic chains can be entangled over large distances too • Any large‐scale future quantum computer would likely be a hybrid system consisting of optical and solid‐state components — optical components for long‐range communication and solid‐ state components for connecting several quantum processors or gates on small scales. • Networks or chains of spins could serve as solid‐state‐based channels for quantum information transfer Experimental realization of long‐distance entanglement between spins in antiferromagnetic quantum spin chains Sahling, S. et al. Nature Phys. http://dx.doi.org/10.1038/ nphys318 (2015). Sr14Cu24O41 Sahling et al demonstrated long‐range entanglement in Sr14Cu24O41 consisting of alternating spin‐ladder and spin‐chain layers. AF coupled dimerized Heisenberg spin ½ chains. The interaction energy between any two neighbouring spins is H = J(SA ・ SB) The energy is min if the two spins are oriented in opposite directions |SA↑SB↓> or |SA↓SB↑> However, neither of these states is an eigenstate of the Hamiltonian. The eigenstate that correctly captures the antiferromagnetic correlation is a superposition of these two 1/√2(|SA↑SB↓> – |SA↓SB↑>). This state is the singlet state and an entangled state, as it cannot be written in the form of a product of two states. The energy level diagram of the dimer as a function of the applied magnetic field For a critical magnetic field strength, ihe excited triplet state, |SA↑SB↑> crosses the singlet state and becomes a new ground state Thus, there is a change in the symmetry of the ground state at this critical field and the system undergone a quantum phase transition (not thermal) Entanglement can be probed by measurements of macroscopic properties such as magnetic susceptibility There are three types of interaction . It is known that two of them (J1 = 115 K, J2 = –13 K give rise to a pattern of up‐ up‐down‐down spins along the chain. The third is a ferromagnetic interaction between dimers coupled with an inter‐ modulation potential between the two sublattices. This interaction is what gives rise to two free spins separated by about 200–250 A. This can be thought of as an effective exchange interaction (Jeff ~ 2.7 K), which is mediated by th spin chain. This causes these free spins to become entangled below 2.1 K, forming dimers. Entanglement can be probed by measurements of macroscopic properties such as magnetic susceptibility Curie law accounts for the low‐temperature divergence of a The broad maxima is small fitted with the population of formula for The critical field is unpaired free spins. antiferromagnetic proportional spin dimers, yielding to the spin singlet‐to‐ a gap 2.3 K. triplet gap. M(H) curves show the occurrence of three magnetization plateaux developing This represents a fourth category of spin below 500 mK: one due to the unpaired dimer present in Sr14Cu24O41, and is free spins and two others developing above distinguished from the others by its small energy gap. In contrast, the other three types 2 and 3 T fitted with the dimer model, using a field‐dependent gap according to of dimers are frozen at low temperature. the Zeeman splitting for a triplet Entanglement can be probed by measurements of macroscopic properties such as magnetic susceptibility • Magnetization plateaux are the signatures of the disappearance of the spin dimers on forming a ferromagnetic state where the spins are no longer entangled • This spin chain provides the rudimentary constituent of a quantum computer. One can initialize the system by going to a very low temperature and applying a magnetic field pulse that is high enough to ensure the system is in a triplet state (5 T). • Once the pulsed field is withdrawn, the system can relax to an entangled singlet state. • In this way, one should be able to build a quantum gate capable of carrying out a quantum computational protocol by undergoing time evolution We are done! So what did we learn? So what did we learn? • • • • • Units and fields Paramagnetism Exchange Interactions Spin gap LowD magnetism Milestones in Low D Magnetism • 1925/31 Ernst Ising, Hans Bethe (Heisenberg chain) • 1944 Lars Onsager: 2D Ising modell • 1966 Mermin‐Wagner theorem: strong temperature fluctuations • 1983 Haldane conjecture: strong quantum fluctuations • 1986 High Tc superconductivity based on 2D AF’s • since 1990 quantum phase diagrams / magnetization plateaus / order from disorder / BEC of quantum magnets / quantum solitons • 21th century: quantum communications