Spin ice

{{Short description|Magnetic material under special conditions}} thumb|upright=1|'''Figure 1.''' The arrangement of hydrogen atoms (black circles) about oxygen atoms (open circles) in ice. Two hydrogen atoms (bottom ones) are ''close'' to the central oxygen atom while two of them (top ones) are ''far'' and closer to the two other (top left and top right) oxygen atoms.

A '''spin ice''' is a magnetic substance that does not have a single minimal-energy state. It has magnetic moments (i.e. "spin") as elementary degrees of freedom which are subject to frustrated interactions. By their nature, these interactions prevent the moments from exhibiting a periodic pattern in their orientation down to a temperature much below the energy scale set by the said interactions. Spin ices show low-temperature properties, residual entropy in particular, closely related to those of common crystalline water ice.<ref name="review">{{cite journal |last1=Bramwell|first1=S. T.|title=Spin Ice State in Frustrated Magnetic Pyrochlore Materials|year=2001|last2=Gingras|first2=M. J. P.|journal=Science|volume=294|issue=5546|pages=1495–1501 |arxiv=cond-mat/0201427|bibcode=2001Sci...294.1495B|doi=10.1126/science.1064761|pmid=11711667|s2cid=9402061}}</ref> The most prominent compounds with such properties are dysprosium titanate (Dy₂Ti₂O₇) and holmium titanate (Ho₂Ti₂O₇). The orientation of the magnetic moments in spin ice resembles the positional organization of hydrogen atoms (more accurately, ionized hydrogen, or protons) in conventional water ice (see figure 1).

Experiments have found evidence for the existence of deconfined magnetic monopoles in these materials,<ref name=":0">{{Cite journal|last1=Castelnovo|first1=C.|last2=Moessner|first2=R.|last3=Sondhi|first3=S. L. |date=2008-01-03|title=Magnetic monopoles in spin ice|journal=Nature|language=en|volume=451|issue=7174|pages=42–45 |doi=10.1038/nature06433 |pmid=18172493 |issn=0028-0836|bibcode=2008Natur.451...42C|arxiv=0710.5515|s2cid=2399316}}</ref><ref name=":1">{{Cite journal|last=Tchernyshyov|first=Oleg|date=2008-01-03|title=Magnetism: Freedom for the poles |journal=Nature|language=en|volume=451|issue=7174|pages=22–23|doi=10.1038/451022b|pmid=18172484|issn=0028-0836|bibcode=2008Natur.451...22T|s2cid=30259694}}</ref><ref name="MonopoleReview">{{cite journal |last=Gingras |first=M.J.P. |year=2009 |title=Observing Monopoles in a Magnetic Analog of Ice |journal=Science |volume=326 |issue=5951 |pages=375–376 |doi=10.1126/science.1181510 |pmid=19833948|arxiv=1005.3557 |s2cid=31038263 }}</ref> with properties resembling those of the hypothetical magnetic monopoles postulated to exist in vacuum.

== Technical description == In 1935, Linus Pauling noted that the hydrogen atoms in water ice would be expected to remain disordered even at absolute zero. That is, even upon cooling to zero temperature, water ice is expected to have residual entropy, ''i.e.'', intrinsic randomness. This is due to the fact that the hexagonal crystalline structure of common water ice contains oxygen atoms with four neighboring hydrogen atoms. In ice, for each oxygen atom, two of the neighboring hydrogen atoms are near (forming the traditional H₂O molecule), and two are further away (being the hydrogen atoms of two neighboring water molecules). Pauling noted that the number of configurations conforming to this "two-near, two-far" ice rule grows exponentially with the system size, and, therefore, that the zero-temperature entropy of ice was expected to be extensive.<ref name="Pauling 1935 pp. 2680–2684">{{cite journal | last=Pauling | first=Linus | title=The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement | journal=Journal of the American Chemical Society | publisher=American Chemical Society (ACS) | volume=57 | issue=12 | year=1935 | issn=0002-7863 | doi=10.1021/ja01315a102 | pages=2680–2684| bibcode=1935JAChS..57.2680P }}</ref> Pauling's findings were confirmed by specific heat measurements, though pure crystals of water ice are particularly hard to create. thumb|'''Figure 2.''' Portion of a pyrochlore lattice of corner-linked tetrahedra. The magnetic ions (dark blue spheres) sit on a network of tetrahedra linked at their vertices. The other atoms (e.g. Ti and O) making the pyrochlore crystal structure are not displayed. The magnetic moments (light blue arrows) obey the two-in, two out spin ice rule over the whole lattice. The system is thus in a ''spin ice state.''|upright=1.5|left Spin ices are materials that consist of regular corner-linked tetrahedra of magnetic ions, each of which has a non-zero magnetic moment, often abridged to "spin", which must satisfy in their low-energy state a "two-in, two-out" rule on each tetrahedron making the crystalline structure (see figure 2). This is highly analogous to the two-near, two far rule in water ice (see figure 1). Just as Pauling showed that the ice rule leads to an extensive entropy in water ice, so does the two-in, two-out rule in the spin ice systems – these exhibit the ''same'' residual entropy properties as water ice. Be that as it may, depending on the specific spin ice material, it is generally much easier to create large single crystals of spin ice materials than water ice crystals. Additionally, the ease to induce interaction of the magnetic moments with an external magnetic field in a spin ice system makes the spin ices more suitable than water ice for exploring how the residual entropy can be affected by external influences.

While Philip Anderson had already noted in 1956<ref name=":2">{{cite journal | last=Anderson | first=P. W. | title=Ordering and Antiferromagnetism in Ferrites | journal=Physical Review | publisher=American Physical Society (APS) | volume=102 | issue=4 | date=15 May 1956 | issn=0031-899X | doi=10.1103/physrev.102.1008 | pages=1008–1013| bibcode=1956PhRv..102.1008A }}</ref> the connection between the problem of the frustrated Ising antiferromagnet on a (pyrochlore) lattice of corner-shared tetrahedra and Pauling's water ice problem, real spin ice materials were only discovered forty years later.<ref name="Harris Bramwell McMorrow Zeiske pp. 2554–2557">{{cite journal | last1=Harris | first1=M. J. | last2=Bramwell | first2=S. T. | last3=McMorrow | first3=D. F. | last4=Zeiske | first4=T. | last5=Godfrey | first5=K. W. | title=Geometrical Frustration in the Ferromagnetic Pyrochlore Ho₂Ti₂O₇ | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=79 | issue=13 | date=29 September 1997 | issn=0031-9007 | doi=10.1103/physrevlett.79.2554 | pages=2554–2557| url=https://www.pure.ed.ac.uk/ws/files/11821795/Geometrical_Frustration_in_the_Ferromagnetic_Pyrochlore_Ho_2_Ti_2_O_7.pdf | bibcode=1997PhRvL..79.2554H | hdl=20.500.11820/f7958ee9-5fb1-4965-8ab3-70c50216943c | s2cid=121002411 | hdl-access=free }}</ref> The first materials identified as spin ices were the pyrochlores Dy₂Ti₂O₇ (dysprosium titanate), Ho₂Ti₂O₇ (holmium titanate). In addition, compelling evidence has been reported that Dy₂Sn₂O₇ (dysprosium stannate) and Ho₂Sn₂O₇ (holmium stannate) are spin ices.<ref name="Matsuhira Hinatsu Tenya Amitsuka pp. 1576–1582">{{cite journal | last1=Matsuhira | first1=Kazuyuki | last2=Hinatsu | first2=Yukio | last3=Tenya | first3=Kenichi | last4=Amitsuka | first4=Hiroshi | last5=Sakakibara | first5=Toshiro | title=Low-Temperature Magnetic Properties of Pyrochlore Stannates | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=71 | issue=6 | date=15 June 2002 | issn=0031-9015 | doi=10.1143/jpsj.71.1576 | pages=1576–1582| bibcode=2002JPSJ...71.1576M }}</ref> These four compounds belong to the family of rare-earth pyrochlore oxides. CdEr₂Se₄, a spinel in which the magnetic Er³⁺ ions sit on corner-linked tetrahedra, also displays spin ice behavior.<ref>{{Cite journal|last1=Lago|first1=J.|last2=Živković|first2=I.|last3=Malkin|first3=B. Z.|last4=Rodriguez Fernandez|first4=J.|last5=Ghigna|first5=P.|last6=Dalmas de Réotier|first6=P.|last7=Yaouanc|first7=A.|last8=Rojo|first8=T.|date=2010-06-15|title=CdEr₂Se₄: A New Erbium Spin Ice System in a Spinel Structure|journal=Physical Review Letters|volume=104|issue=24|article-number=247203|doi=10.1103/PhysRevLett.104.247203|pmid=20867332|bibcode=2010PhRvL.104x7203L|url=http://infoscience.epfl.ch/record/151953|hdl=10902/28560|hdl-access=free}}</ref>

Spin ice materials are characterized by a random disorder in the orientation of the moment of the magnetic ions, even when the material is at very low temperatures. Alternating current (AC) magnetic susceptibility measurements find evidence for a dynamic freezing of the magnetic moments as the temperature is lowered somewhat below the temperature at which the specific heat displays a maximum. The broad maximum in the heat capacity does not correspond to a phase transition. Rather, the temperature at which the maximum occurs, about 1{{nbsp}}K in Dy₂Ti₂O₇, signals a rapid change in the number of tetrahedra where the two-in, two-out rule is violated. Tetrahedra where the rule is violated are sites where the aforementioned monopoles reside. Mathematically, spin ice configurations can be described by closed Eulerian paths.<ref name=":sj">{{Cite journal|last=Schrijver|first=A.|title=Bounds on the number of Eulerian Orientations|year=1983|volume=3|journal=Combinatorica|pages=375–380|number=3|doi=10.1007/BF02579193 |s2cid=13708977 |url=https://ir.cwi.nl/pub/10053 }}</ref><ref name=":sj2">{{Cite journal|last1=Caravelli|first1=F.|last2=Saccone|first2=M.|last3=Nisoli|first3=C.|title=On the Degeneracy of Spin Ice Graphs, and its Estimate via the Bethe Permanent |year=2021|volume=477|journal=Proc. R. Soc. A|number=20210108|article-number=20210108 |doi=10.1098/rspa.2021.0108 |s2cid=231728393 |doi-access=free|arxiv=2101.12280|bibcode=2021RSPSA.47710108C }}</ref>

== Magnetic monopoles == thumb|'''Figure 3.''' The orientation of the magnetic moments (light blue arrows) considering a single tetrahedron within the spin ice state, as in figure 2. Here, the magnetic moments obey the two-in, two-out rule: there is as much "magnetization field" going in the tetrahedron (bottom two arrows) as there is going out (top two arrows). The corresponding magnetization field has zero divergence. There is therefore no sink or source of the magnetization inside the tetrahedron, or no ''monopole''. If a thermal fluctuation caused one of the bottom two magnetic moments to flip from "in" to "out", one would then have a 1-in, 3-out configuration; hence an "outflow' of magnetization, hence a positive divergence, that one could assign to a positively charged monopole of charge +''Q''. Flipping the two bottom magnetic moments would give a 0-in, 4-out configuration, the maximum possible "outflow" (i.e. divergence) of magnetization and, therefore, an associated monopole of charge +2''Q''.|upright=1.5 Spin ices are geometrically frustrated magnetic systems. While frustration is usually associated with triangular or tetrahedral arrangements of magnetic moments coupled via antiferromagnetic exchange interactions, as in Anderson's Ising model,<ref name=":2" /> spin ices are frustrated ferromagnets. It is the very strong local magnetic anisotropy from the crystal field forcing the magnetic moments to point either in or out of a tetrahedron that renders ferromagnetic interactions frustrated in spin ices. Most importantly, it is the long-range magnetostatic dipole–dipole interaction, and ''not'' the nearest-neighbor exchange, that causes the frustration and the consequential two-in, two-out rule that leads to the spin ice phenomenology.<ref name="den Hertog Gingras pp. 3430–3433">{{cite journal | last1=den Hertog | first1=Byron C. | last2=Gingras | first2=Michel J. P. | title=Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=84 | issue=15 | date=10 April 2000 | issn=0031-9007 | doi=10.1103/physrevlett.84.3430 | pages=3430–3433| pmid=11019107 | arxiv=cond-mat/0001369 | bibcode=2000PhRvL..84.3430D | s2cid=45435198 }}</ref><ref name="Isakov Moessner Sondhi p. ">{{cite journal | last1=Isakov | first1=S. V. | last2=Moessner | first2=R. | last3=Sondhi | first3=S. L. | title=Why Spin Ice Obeys the Ice Rules | journal=Physical Review Letters | volume=95 | issue=21 | date=14 November 2005 | issn=0031-9007 | doi=10.1103/physrevlett.95.217201 | article-number=217201| pmid=16384174 | arxiv=cond-mat/0502137 | bibcode=2005PhRvL..95u7201I | s2cid=30364648 }}</ref>

For a tetrahedron in a two-in, two-out state, the magnetization field is divergent-free; there is as much "magnetization intensity" entering a tetrahedron as there is leaving (see figure 3). In such a divergent-free situation, there exists no source or sink for the field. According to Gauss' theorem (also known as Ostrogradsky's theorem), a nonzero divergence of a field is caused, and can be characterized, by a real number called "charge". In the context of spin ice, such charges characterizing the violation of the two-in, two-out magnetic moment orientation rule are the aforementioned monopoles.<ref name=":0" /><ref name=":1" /><ref name="MonopoleReview" />

In Autumn 2009, researchers reported experimental observation of low-energy quasiparticles resembling the predicted monopoles in spin ice.<ref name=":0" /> A single crystal of the dysprosium titanate spin ice candidate was examined in the temperature range of 0.6–2.0{{nbsp}}K. Using neutron scattering, the magnetic moments were shown to align in the spin ice material into interwoven tube-like bundles resembling Dirac strings. At the defect formed by the end of each tube, the magnetic field looks like that of a monopole. Using an applied magnetic field, the researchers were able to control the density and orientation of these strings. A description of the heat capacity of the material in terms of an effective gas of these quasiparticles was also presented.<ref>{{cite web |url=https://www.sciencedaily.com/releases/2009/09/090903163725.htm |title=Magnetic Monopoles Detected In A Real Magnet For The First Time |publisher=Science Daily |date=2009-09-04 |access-date=2009-09-04 }}</ref><ref>{{cite journal |title=Dirac Strings and Magnetic Monopoles in Spin Ice Dy₂Ti₂O₇ |author1=D.J.P. Morris |author2=D.A. Tennant |author3=S.A. Grigera |author4=B. Klemke |author5=C. Castelnovo |author6=R. Moessner |author7=C. Czternasty |author8=M. Meissner |author9=K.C. Rule |author10=J.-U. Hoffmann |author11=K. Kiefer |author12=S. Gerischer |author13=D. Slobinsky |author14=R.S. Perry |name-list-style=amp |journal=Science |date=2009-09-03 |doi=10.1126/science.1178868 |pmid=19729617 |bibcode=2009Sci...326..411M |volume=326 |issue=5951 |pages=411–4 |arxiv = 1011.1174 |s2cid=206522398 }}</ref>

The effective charge of a magnetic monopole, ''Q'' (see '''figure 3''') in both the dysprosium and holmium titanate spin ice compounds is approximately {{nowrap|''Q'' {{=}} {{val|5|u=''μ''_BÅ⁻¹}}}} (Bohr magnetons per angstrom).<ref name=":0" /> The elementary magnetic constituents of spin ice are magnetic dipoles, so the emergence of monopoles is an example of the phenomenon of fractionalization.

The microscopic origin of the atomic magnetic moments in magnetic materials is quantum mechanical; the Planck constant enters explicitly in the equation defining the magnetic moment of an electron, along with its charge and its mass. Yet, the magnetic moments in the dysprosium titanate and the holmium titanate spin ice materials are effectively described by ''classical'' statistical mechanics, and not quantum statistical mechanics, over the experimentally relevant and reasonably accessible temperature range (between 0.05{{nbsp}}K and 2{{nbsp}}K) where the spin ice phenomena manifest themselves. Although the weakness of quantum effects in these two compounds is rather unusual, it is believed to be understood.<ref>{{Cite journal|last1=Rau|first1=Jeffrey G.|last2=Gingras|first2=Michel J. P.|year=2015|title=Magnitude of quantum effects in classical spin ices|journal=Physical Review B|volume=92|issue=14|article-number=144417|doi=10.1103/PhysRevB.92.144417|bibcode=2015PhRvB..92n4417R|arxiv=1503.04808|s2cid=119153613 }}</ref> There is current interest in the search of quantum spin ices,<ref name=":3">{{Cite journal |last1=Gingras |first1=M. J. P.|last2=McClarty|first2=P. A.|date=2014-01-01|title=Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets|journal=Reports on Progress in Physics|language=en|volume=77|issue=5|article-number=056501|doi=10.1088/0034-4885/77/5/056501|pmid=24787264|issn=0034-4885|bibcode=2014RPPh...77e6501G|arxiv=1311.1817|s2cid=23594100}}</ref> materials in which the laws of quantum mechanics now become needed to describe the behavior of the magnetic moments. Magnetic ions other than dysprosium (Dy) and holmium (Ho) are required to generate a quantum spin ice, with praseodymium (Pr), terbium (Tb) and ytterbium (Yb) being possible candidates.<ref name=":3" /><ref>{{Cite journal|last1=Rau|first1=Jeffrey G.|last2=Gingras|first2=Michel J.P.|date=2019-03-10|title=Frustrated Quantum Rare-Earth Pyrochlores|journal=Annual Review of Condensed Matter Physics|language=en|volume=10|issue=1|pages=357–386|doi=10.1146/annurev-conmatphys-022317-110520|arxiv=1806.09638|bibcode=2019ARCMP..10..357R |s2cid=85498113|issn=1947-5454}}</ref> One reason for the interest in quantum spin ice is the belief that these systems may harbor a ''quantum spin liquid'',<ref>{{Cite journal|author1-link=Leon Balents|last=Balents|first=Leon|date=2010-03-10|title=Spin liquids in frustrated magnets|journal=Nature|language=en|volume=464|issue=7286|pages=199–208|doi=10.1038/nature08917|pmid=20220838|issn=0028-0836|bibcode=2010Natur.464..199B|s2cid=4408289}}</ref> a state of matter where magnetic moments continue to wiggle (fluctuate) down to absolute zero temperature. The theory<ref>{{Cite journal|last1=Hermele|first1=Michael|last2=Fisher|first2=Matthew P. A.|last3=Balents|first3=Leon|author3-link=Leon Balents|date=2004-02-12|title=Pyrochlore photons: The U(1) spin liquid in a S=1/2 three-dimensional frustrated magnet|journal=Physical Review B|volume=69|issue=6|article-number=064404|doi=10.1103/PhysRevB.69.064404|bibcode=2004PhRvB..69f4404H|arxiv=cond-mat/0305401|s2cid=28840838}}</ref> describing the low-temperature and low-energy properties of quantum spin ice is akin to that of vacuum quantum electrodynamics, or QED. This constitutes an example of the idea of emergence.<ref>{{Cite journal|last1=Rehn|first1=J.|last2=Moessner|first2=R.|date=2016-05-19 |title=Maxwell electromagnetism as an emergent phenomenon in condensed matter|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=374 |issue=2075 |article-number=20160093 |arxiv=1605.05874 |doi=10.1098/rsta.2016.0093 |pmid=27458263 |bibcode=2016RSPTA.37460093R|s2cid=206159482}}</ref>

== Artificial spin ices ==

Artificial spin ices are metamaterials consisting of coupled nanomagnets arranged on periodic and aperiodic lattices. <ref name="SkjærvøMarrowsStamps2019">{{cite journal | last1 = Skjærvø | first1 = Sandra H. | last2 = Marrows | first2 = Christopher H. | last3 = Stamps | first3 = Robert L. | last4 = Heyderman | first4 = Laura J. | title = Advances in artificial spin ice | journal = Nature Reviews Physics | date = 8 November 2019 | volume = 2 | issue = 1 | pages = 13–28 | eissn = 2522-5820 | doi = 10.1038/s42254-019-0118-3 | pmid = | s2cid = 207959741 | url = https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A27297}}</ref> These systems have enabled the experimental investigation of a variety of phenomena such as frustration, emergent magnetic monopoles, and phase transitions. In addition, artificial spin ices show potential as reprogrammable magnonic crystals and have been studied for their fast dynamics. A variety of geometries have been explored, including quasicrystalline systems and 3D structures, as well as different magnetic materials to modify anisotropies and blocking temperatures.

For example, polymer magnetic composites comprising 2D lattices of droplets of solid-liquid phase change material, with each droplet containing a single magnetic dipole particle, form an artificial spin ice above the droplet melting point, and, after cooling, a spin glass state with low bulk remanence. Spontaneous emergence of 2D magnetic vortices was observed in such spin ices, which vortex geometries were correlated with the external bulk remanence. <ref name="KayaIserivan der Wijngaart2022">{{cite journal | last1 = Kaya | first1 = Kerem | last2 = Iseri | first2 = Emre | last3 = van der Wijngaart | first3 = Wouter | title = Soft metamaterial with programmable ferromagnetism | journal = Microsystems & Nanoengineering | date = 6 December 2022 | volume = 8 | issue = 1 | page = 127 | eissn = 2055-7434 | doi = 10.1038/s41378-022-00463-2 | pmid = 36483621| pmc = 9722694 | bibcode = 2022MicNa...8..127K | url = }}</ref>

Future work in this field includes further developments in fabrication and characterization methods, exploration of new geometries and material combinations, and potential applications in computation,<ref name=":sp">{{Cite journal|last1=Caravelli|first1=F.|last2=Nisoli|first2=C.|title=Logical gates embedding in artificial spin ice|journal=New J. Phys.|volume=22|number= 103052|year=2020|page=103052 |doi=10.1088/1367-2630/abbf21 |s2cid=216056260 |doi-access=free|arxiv=1810.09190|bibcode=2020NJPh...22j3052C }}</ref> data storage, and reconfigurable microwave circuits.<ref name=":Lh">{{Cite journal|last1=Heyderman|first1=L.J.|title=Spin ice devices from nanomagnets|journal=Nature Nanotechnology|volume=17|pages=435–436|year=2022|issue=5 |doi=10.1038/s41565-022-01088-2 |pmid=35513585 |bibcode=2022NatNa..17..435H |s2cid=248530509 |url=https://www.nature.com/articles/s41565-022-01088-2|url-access=subscription}}</ref> In 2021 a study demonstrated neuromorphic reservoir computing using artificial spin ice, solving a range of computational tasks using the complex magnetic dynamics of the artificial spin ice.<ref name=":res">{{Cite journal|last1=Gartside|first1=J. C.|last2=Stenning|first2=K. D.|last3=Vanston|first3=A.|last4=Holder|first4=H.H.|last5=Arroo|first5=D.M.|last6=Dion|first6=T.|last7=Caravelli|first7=F.|last8=Kurebayashi|first8=H.|last9=Branford|first9=W. R. |date=2022-04-04|title=Reconfigurable training and reservoir computing in an artificial spin-vortex ice via spin-wave fingerprinting|journal=Nature Nanotechnology|language=en|volume=17|issue=5 |pages=460–469|doi=10.1038/s41565-022-01091-7 |pmid=35513584 |arxiv=2107.08941 |bibcode=2022NatNa..17..460G |s2cid=246431279 |url=https://www.nature.com/articles/s41565-022-01091-7 }}</ref> In 2022, another studied achieved an artificial kagome spin ice which could potentially be used in the future for novel high-speed computers with low power consumption.<ref>{{Cite web |date=2022-04-04 |title=A look into the magnetic future {{!}} Our Research {{!}} Paul Scherrer Institut (PSI) |url=https://www.psi.ch/en/media/our-research/a-look-into-the-magnetic-future |access-date=2022-04-10 |website=www.psi.ch |language=en}}</ref>

== See also == * Lieb's square ice constant * Spin glass * Magnetic monopole * Magnetricity

== References == {{reflist|40em}}

{{magnetic states}} {{authority control}}

{{DEFAULTSORT:Spin Ice}} Category:Magnetic ordering Category:Condensed matter physics Category:Ice