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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
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vale
Verdazyl-based K. Mukai
Spin Peierls
pseudog
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spin li a range of exotic phasesductivity
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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?
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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