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Early promise Quantum materials: Arguably, it all began with supercon- ductivity. It was recognized in the 1950s Where many paths meet that the ability of some materials to con- duct electricity without resistance is a By Philip Ball phenomenon that demands a quantum explanation. The theory put forward by ll matter, in the end, must be explained Aside from the matter of their intrinsic John Bardeen, Leon Cooper, and Robert A by , which de- properties, however, what tends to unite Schrieffer (BCS) in 1957 invoked an scribes how atoms bind and electrons in- quantum materials is who is interested interaction between mobile conduction teract at a fundamental level. Typically, the in them. The community of researchers electrons, mediated by vibrations of the quantum behavior can be approximated that 20 years ago was grappling with crystal lattice that unites them into pairs, by a classical description, in which atoms high-temperature is called Cooper pairs. These pairs can be become balls that stick together in well- likely today to be pondering topologi- considered as quasiparticles, which—in GH¿QHGDUUDQJHPHQWVYLDVLPSOHIRUFHV cal insulators and Weyl semimetals. A contrast to electrons themselves—be- and vibrate much like balls on springs. somewhat distinct community that used long to the fundamental class of particles Sometimes, however, that is not so. In to focus around the 1980s on the exotic called bosons, which have integer values some materials, the quantum aspects as- 2D phenomenon called the quantum Hall of quantum spin. sert themselves tenaciously, and the only HIIHFWLVQRZ¿QGLQJFRPPRQFDXVH$QG This makes all the difference. Particles way to fully understand how the material quantum materials draw inspiration from with half-integer spin (fermions), like behaves is to keep the quantum in view. a third direction too, apparently unlikely lone electrons, cannot occupy the same Such substances are now grouped togeth- at first flush: particle physics, within quantum state. However, bosons can, and er under the banner of TXDQWXPPDWHULDOV which some unusual types of fundamental so Cooper pairs can condense into a quan- The US Department of Energy (DOE) particles proposed around the mid-to-late tum state in which all the electrons can be describes quantum materials as “solids WKFHQWXU\DUHQRZ¿QGLQJDQDORJXHV described by a single quantum wave func- with exotic physical properties, arising in the quasiparticles of condensed matter. tion. In effect, they become a collective from the quantum mechanical proper- This is not a matter of leaping onboard entity, which is then immune to the dis- ties of their constituent electrons, that the latest fancy-titled bandwagon. Rather, turbing effects of electron scattering from KDYHJUHDWVFLHQWL¿FDQGRUWHFKQRORJL- LWLVDUHÀHFWLRQRIDFRPPRQH[SHULHQFH the crystal lattice—the source of energy cal potential.” It’s a diverse class of ma- in physics, whereby concepts developed dissipation and electrical resistance. But terials—so much so that some question to explore one phenomenon turn out to this collective state can only be sustained whether the designation is meaningful. It be a subset of more general principles. DWORZWHPSHUDWXUHVWRRPXFKWKHUPDO includes well-advertised materials such Quantum materials reveal that proper- noise, and the Cooper pairs are broken, as superconductors and , along WLHVRQFHWKRXJKWWREHTXLUNVFRQ¿QHG and the material becomes a regular metal. with ones with less familiar names: top- WRH[RWLFFRQGLWLRQVDUHLQIDFWDVLJQL¿- Here, then, is a behavior dictated by ological insulators, Weyl semimetals, cant feature of the materials universe. quantum mechanical correlations be- quantum spin liquids, and spin ices. To turn those ideas into real materials, tween electrons: the trademark signature A collection of oddballs they may meanwhile, demands expertise from of many quantum materials. Yet BCS be, but quantum materials are both a RWKHU¿HOGVLQYROYLQJWKHVNLOOVRIV\Q- theory did not seem to work for the class treasure trove of interesting physics and thetic solid-state materials chemistry and of “high-temperature” superconductors a potential source of useful substances. crystal growth. This union of fundamental discovered by Georg Bednorz and K. And while the compositions and behav- physics and practical materials science is Alex Müller of IBM’s Zurich research iors of quantum materials cover a wide turning into one of the most vibrant areas laboratories in 1986, some of which spectrum, some common themes recur. of physical science today. Most of all, it proved to have superconducting transi- Many of them derive their properties from RIIHUVDSOD\JURXQGIRU¿QGLQJDQGH[- tion temperatures much higher than the reduced dimensionality, in particular from ploring new physics. But there could be boiling point of liquid nitrogen (77 K). FRQ¿QHPHQWRIHOHFWURQVWRWZRGLPHQ- SUDFWLFDOEHQH¿WVWRRQHZHOHFWURQLFGH- These materials are not metals but ceram- sional (2D) sheets. In many, magnetism vices, and, in particular, new possibilities ics: metal oxides with a characteristic lay- and electronic structure interact in curious for building quantum computers, which ered structure, of which the paradigmatic ways. In particular, they tend to be materi- would exploit the rules of quantum me- example was lanthanum copper oxides. als in which electrons cannot be consid- chanics to achieve computational power Despite several decades of work, ered as independent particles but act as far in excess of anything the classical there is still no fully satisfactory con- collective states dubbed quasiparticles. computers of today can attain. sensus view of how high-temperature

Philip Ball, [email protected]

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superconductors work. “The complexity of the materials has made arriving at sim- SOHPRGHOVGLI¿FXOW´VD\VSK\VLFLVW'DYLG Ceperley of the University of Illinois at Urbana-Champaign. All the same, most researchers agree that the electron cor- relations that must surely lie at the root of the phenomenon are related to both the Ȁ layered, quasi-2D crystal structure of the Ȁ' materials and to their magnetic behavior. Meanwhile, another strand of exotic quantum properties was evolving in a dif- ferent arena. In 1980, Klaus von Klitzing showed that at low temperatures, the Hall effect—an effect known for a century, in which a voltage across an electrical FRQGXFWRULVFUHDWHGE\WKHLQÀXHQFHRI DPDJQHWLF¿HOGRQWKHHOHFWURQSDWKV² can become quantized. That’s to say, the conductance of a thin, quasi-2D conduc- tor may change in stepwise jumps as the Figure 1. The band structure of graphene. The valence (blue/green) and conduction (red/yellow) VWUHQJWKRIDWUDQVYHUVHPDJQHWLF¿HOG bands touch at points K and K' in momentum space. is increased: the quantum Hall effect (QHE). Qualitatively, this quantization is the result of electrons in the 2D material Kissing cones EHLQJFRQ¿QHGWRGLVFUHWHORRSLQJRUELWV The connections between high-tempera- Konstantin Novoselov of The University not unlike those postulated for simple ture superconductors, the QHE, and 2D of Manchester in the UK were awarded theories of the quantum atom, but much electron gases are, in retrospect, clear the 2010 Nobel Prize in Physics. larger in size. These electron states owe enough: in particular, the reduced dimen- From a chemist’s point of view, the their stability to the topological properties sionality, interaction of electronic and extended network of conjugated bonds in of the electronic band structure in the ma- magnetic or electron-spin properties, and the graphene sheet creates pathways for terial. They are in some ways analogous the importance of electron correlations and mobile electrons that give graphene its to the topological defects that may appear quasiparticle descriptions. electrical conductivity. To the solid-state in crystal structures, such as screw dislo- These aspects all come together in physicist, more used to describing elec- cations, which, like the whorled crown of the much-vaunted “wonder material” tronic structure in terms of band theory, hairs on your head, are irreducible con- graphene, which is also perhaps the most the picture is more complex. The pecu- sequences of shape. These structures are familiar and celebrated quantum material liarity of graphene is that it has the fully said to be “topologically protected.” today. These 2D sheets of pure carbon, ¿OOHGYDOHQFHEDQGDQGHPSW\FRQGXFWLRQ Von Klitzing won the 1985 Nobel Prize the atoms united into hexagonal rings band of a semiconductor, but with a band- in Physics for his work (it went to Bednorz long familiar from the mundane parent gap between these states that falls to zero and Müller in 1987), and, subsequently, material graphite, have been proposed— at a particular value of electron momen- more exotic variants of the QHE were some would say hyped—as a fabric for the WXP7KDWLVWKHGH¿QLQJFKDUDFWHULVWLFRI discovered. Robert B. Laughlin, Horst next generation of electronic circuitry and a semimetal. L. Störmer, and Daniel C. Tsui won the touchscreens. That graphene should be a gapless 1998 physics Nobel for showing that un- While the real value of graphene in semiconductor has been recognized since der certain conditions, electrons in a 2D that applied arena remains to be seen, LWVHOHFWURQLFEDQGVWUXFWXUHZDV¿UVWFDOFX- material that are free to move more or less few physicists would dispute that, as a lated in 1947—well before graphene was without impediment—a 2D electron gas— playground for ideas in basic physics, recognized as a distinct substance rather can condense into quasiparticle states that graphene’s potential is already proven. than just the hypothetical 2D “unit” of also show the QHE, but act as if they are Indeed, it is really for elucidating the graphite. This band structure is typically electrons with fractional electric charge. strange and often surprising electronic described in terms of the so-called Fermi This behavior is called the fractional quan- properties of this material—and not for surface: loosely speaking, the surface of tum Hall effect (FQHE). It’s still not fully their convenient way of making it by the volume that electrons in the material understood, but again collectivity and to- stripping individual layers from graphite occupy in momentum space. For graphene, pology of the electron states hold the key. using Scotch tape—that Andre Geim and the Fermi surface sits right at the point

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where the conduction and valence bands Table I. Quasiparticles in quantum materials.

touch like two cones “kissing” at their Cooper pair apex, and so in theory it seems to have Two electrons may become coupled via vibrations of the underlying zero area, making graphene a semicon- atomic lattice in conventional (Bardeen–Cooper–Schrieffer) superconductors, producing ductor (see Figure 1). In fact, some of the a composite electron-pair quasiparticle, a boson with integer spin (0 or 1). As bosons, these quasiparticles can condense into a ground state described by a single wave electron density can leak into the conduc- function, producing the phenomenon of superconductivity. tion band at these points due to quantum Relativistic Dirac fermion effects, making graphene electrically con- All known fermionic (spin-½) particles, such as electrons, ductive even at zero temperature. are Dirac fermions, described (with their corresponding antiparticles) by Paul Dirac’s The shape of the Fermi surface relativistic quantum wave equation. Quasiparticle relativistic Dirac fermions, such as means that the mobile electrons interact many-electron excitations in Dirac semimetals, behave as though they have no mass to form collective quasiparticles (see and have the same speed at all energies. Table I) that act as though they have Weyl fermion These are massless fermions related to Dirac fermions. They have no mass.1 This enables them to achieve a chirality or handedness depending on whether their momentum is aligned or anti- YHU\KLJKVSHHGV²DVLJQL¿FDQWIUDFWLRQ aligned with their spin. Each Dirac fermion can be resolved into two Weyl fermions of of the speed of light—giving graphene opposite chirality. Never observed as fundamental particles, they are seen as electron a large electron mobility that could be quasiparticle excitations in Weyl semimetals. very useful for high-frequency elec- Laughlin quasiparticle tronic devices. The quasiparticles must The fractional quantum Hall effect (FQHE) in a 2D “electron be described by the relativistic form of gas” involves collective electron quasiparticles that behave as though they have a quantum mechanics devised by physi- fractional charge, such as 1/3, 2/5, or 3/7, the charge on a single electron. Physicist cist Paul Dirac in the late 1920s: They Robert Laughlin proposed this quasiparticle picture of the FQHE in 1983. are known as Dirac fermions, and they Majorana fermion Hypothetical particle proposed in 1937, which is its own anti- were in fact one of the predictions of particle. These might arise as quasiparticle excitations in superconductors. They are Dirac’s theory. Graphene is thus called potential ingredients for topological quantum computers. a 2D “Dirac semimetal.” Anyon This sort of relativistic behavior of Hypothetical particle proposed by Frank Wilczek in 1982, which can have particles is more familiar to particle phys- quantum statistics anywhere on a continuum between fermions (half-integer spin) and icists. Indeed, some quantum relativistic bosons (integer spin). They might appear as quasiparticles in quantum spin liquids, but have not been definitively identified yet. effects that are hard to achieve in particle physics turn out to be easier to arrange Skyrmion Originally predicted as an exotic kind of baryon (the constituents of most in graphene. One of them is called Klein ordinary matter) in 1962, skyrmions have been found to exist in the form of vortex-like tunneling, a variation on quantum tunnel- topological quasiparticle excitations of spins in some magnetic materials. ing of particles through barriers, in which, for massless relativistic particles, a barrier can become virtually transparent regard- less of its height and thickness. It is be- Dirac semimetals ought to exist: their cal- electron orbitals of sodium (the 3s or- cause of this effect, which was observed culations suggested that a form of bismuth bitals) and bismuth (the 6p orbitals). 2 in 2013, that the charge carriers in the R[LGHGHQRWHGȕ%L22 might display this Classically, we would expect these orbi- 2D structures can remain highly mobile property.3 The peculiar electronic proper- tals to “mix” into hybrid electronic states, even in the presence of potential scatter- ties required for this sort of behavior can which would change the band structure ing sites. On the other hand, this transpar- be predicted in principle from quantum and open up an energy gap: they would ency of barriers is a problem for trying to calculations of band structure, and in the become semiconductors (for a small gap) make graphene transistors (at least from same year, Xi Dai of the Beijing National or insulators. But that doesn’t happen single-layer graphene), because it means Laboratory for because certain symmetry properties of that the gates that control charge-carrier in China and his co-workers4 showed that the electron states forbid such mixing. motion in conventional transistors cannot WKHVHFRQGLWLRQVPLJKWEHVDWLV¿HGLQVR- As physicist Nai Phuan Ong of Princeton

be fully closed. dium bismuthate (Na3Bi). University says, “the electronic proper- There is no reason why Dirac semi- In a Dirac semimetal, there is no en- ties arise from states that are protected metals have to be 2D, although the band ergy gap between the conduction and by symmetry.” In particular, the conduct- structure and the Fermi surface become valence bands, but strictly speaking no ing states have time-reversal symmetry: rather tricky to picture in three dimen- overlap either: they touch at a single it makes no difference if one writes t or sions. In 2012, Charles Kane, of the point, where the apices of the “Dirac –t in their quantum wave functions. It’s

University of Pennsylvania, and co-work- cones” meet. In Na3Bi, these electron a little like the way the symmetry of per- ers predicted that three-dimensional (3D) states are comprised from the atomic mitted moves prevents bishops in chess

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from crossing onto squares of a different make the equivalent of these monopoles, conducting states persist. Such materials color: it can’t be done. namely the corresponding Weyl-fermion are said to be topological insulators, and While the Dirac cones in 2D graph- quasiparticles. “Weyl nodes must always they supply another key focus of current ene are symmetry-protected, there’s an come in pairs,” Ong explains. “Just like interest in quantum materials.11 After the extra degree of such protection in 3D PDJQHWLFPRQRSROHVLWLVGLI¿FXOWWR GLVFRYHU\RIWKH¿UVWWRSRORJLFDOLQVX- 12 Dirac semimetals such as Na3Bi, which pull the pair apart. But an advantage in lator, bismuth antimonide (BixSb1–x), comes from the topology of the electron condensed matter is that one can go to a variety of other candidates have been states—more akin to the prohibition of the surface of a crystal where such op- LGHQWL¿HGLQFOXGLQJELVPXWKDQGDQWL- transforming a left-handed glove into a erations—forbidden in the bulk—may mony selenides and tellurides. They are right-handed one. Because of this, 3D become feasible.” In that way, one might predicted to occur, for example, in a Dirac semimetals are said to be “topo- create a Weyl semimetal. While such ma- wide range of complex materials called logical semimetals.” terials would contain Weyl nodes in the Heusler compounds.13 In 2014, Dai collaborated with Yulin bulk, only at the surface is symmetry Topological insulators represent a Chen of the University of Oxford in the broken in a way that produces the gap- FRQÀXHQFHRIVHYHUDOWKHPHVFRQFHUQLQJ 5 UK to verify his prediction for Na3Bi. less “kissing” of Weyl cones. In 2015, the exotic electron behavior in materials Measurements using electron photoemis- Zahid Hasan at Princeton and his collab- with unusual topological features of their sion spectroscopy revealed that the en- orators predicted that tantalum arsenide EDQGVWUXFWXUH³7KH¿HOGRIWRSRORJLFDO ergy and momentum of electrons in the (TaAs) might act as a Weyl semimetal,8 insulators is an offspring of QHE phys- material have the characteristics expected DQGVKRUWO\WKHUHDIWHUWKH\YHUL¿HGH[- ics and quantum spin physics, with high- for a 3D Dirac semimetal. Independently perimentally, again using photoelectron temperature superconductivity sneaking and at the same time, two international spectroscopy, that this is so.9 in some genes,” Ong says. But research groups that both included researchers One of the reasons for a fundamen- on topological matter has now expanded from Princeton University described tal interest in these systems is that, as well beyond QHE physics, he adds. another 3D Dirac semimetal, cadmium analogues of the particle-physics context For one thing, it has reignited interest 6,7 arsenide (Cd3As2). LQZKLFKWKH\ZHUH¿UVWREVHUYHGWKH\ in a host of unusual hypothetical particles Dirac fermions can be regarded as the should exhibit some of the peculiarities (see Table I). Take skyrmions, a breed of combination of two related massless par- of such systems. Massless Weyl fermi- SDUWLFOH¿UVWSUHGLFWHGLQWKHVLQD ticles called Weyl fermions, which have ons were invoked in the 1960s to explain particle-physics context. Skyrmions can a chiral twist or handedness to them: in why particles called neutral pions decay be realized as quasiparticles in certain a Dirac fermion, the two twists cancel so fast. This decay can create an apparent magnetic materials, where they take the out. To resolve Dirac fermions into their nonconservation of charge in the chiral form of vortex-like quasiparticle excita- constituent Weyl fermions, one can split fermions, called the chiral anomaly. It was tions created by topological constraints WKHPZLWKDQDSSOLHGPDJQHWLF¿HOG,Q predicted in 1983 that this anomaly might on the orientation of the magnetic spins. HIIHFWWKH¿HOGGHVWUR\VWKHWLPHUHYHUVDO be found in material analogue systems, 7KH\KDYHEHHQLGHQWL¿HGLQWKHFKLUDO symmetry, because—for electron-based and in 2015, Ong and his Princeton col- magnet manganese silicide.14 10 quasiparticles—it aligns most of the laborators saw it in Na3Bi. Another exotic particle potentially electron spins to break the symmetry: re- realizable as a quasiparticle in quantum verse time, and they would have opposite Making new particles materials is the anyon. Originally hy- spin. In momentum space, this process in materials pothesized by theoretical physicist Frank corresponds to the splitting of the Dirac The 3D Dirac semimetals predicted Wilczek in 1982, the anyon is a strange in- cone into two Weyl cones. Formally, this by Kane in 2012 had another curious termediate between the two conventional is analogous to the splitting apart of the feature. They are electrical insulators, classes (like photons) and fermions (like SROHVRIDPDJQHWLF¿HOGWKHQRGHVRI but as Ong says, at surfaces different electrons). These are usually regarded each Weyl cone then correspond to a constraints may apply. In this case, the as two mutually exclusive classes (al- “monopole,” a hypothetical magnetic VXUIDFHHOHFWURQVPD\EHFRQ¿QHGWR though the theory called supersymmetry, entity with just an isolated north or south 2D conducting states: they are metallic. for which particle physicists have yet to pole. It’s not known how one might cre- What’s more, in these states electron ¿QGYDOLGDWLQJH[SHULPHQWDOVXSSRUWVXJ- ate a genuine magnetic monopole—chop scattering becomes negligible, purely JHVWVWKDWWKH\FDQEHXQL¿HGE\LQYRNLQJ a bar magnet in half and you create two because of the topological shape of the a particular kind of symmetry). Anyons, new poles at the new ends. But magnetic band structure. For this reason, the states KRZHYHUUHMHFWWKDWGLFKRWRP\WKH\FDQ monopoles are predicted to exist in some are topologically protected: even if there exhibit quantum statistics anywhere be- fundamental theories in particle physics, is a considerable amount of disorder or tween those of bosons and fermions. such as string theory. impurities at the material surface, which Anyon statistics are again a conse- In a quantum material that behaves might be expected to disrupt electrical quence of topological considerations. as a Dirac semimetal, however, you can conductivity in an ordinary material, the In effect, exchanging any two such

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particles—the operation that distin- they would be controlled and manipulated A consequence of quantum uncertain- guishes the exclusivity of fermions from to process information. However, making ty is that some properties of a quantum the sociability of bosons—involves a tan- them is still a challenge. Whereas anyons V\VWHPPD\ÀXFWXDWHHYHQDW]HURWHP- gling or “braiding” of their world-lines are predicted in 2D systems, Majorana fer- perature. Such behavior is predicted for in spacetime, which means the particles mions are zero-dimensional. In one view, a wide range of systems that demand a can acquire any arbitrary increment in the they are predicted to appear at the ends of quantum description, including supercon- phase of their quantum wave functions. a one-dimensional conducting nanowire ductors and heavy-fermion metals—in- For bosons and fermions, in contrast, this coupled to a superconducting electrode, in termetallic compounds with f orbital elec- 18 SKDVHLQFUHPHQWLVHLWKHURUʌVRWKH WKHSUHVHQFHRIDPDJQHWLF¿HOG Possible trons, such as CeCu6, CeAl3, and UPt3, two particles either superimpose or can- signatures of such quasiparticles have been in which the charge carriers are quasi- cel exactly. This odd property of anyons, reported for indium antimonide (InSb) and particles of strongly correlated electrons seemingly forbidden in ordinary particle arsenide (InAs) nanowires with supercon- with a collective mass much greater than physics, becomes possible for particles ducting contacts, although the evidence that of individual electrons. Such systems FRQ¿QHGLQWZRGLPHQVLRQVZKLFKVXJ- VWLOOIDOOVVKRUWRIGH¿QLWLYHSURRI19,20 have two possible states even at absolute gests that 2D quantum materials might The manifestation of such exotic zero: an orderly one (such as a magneti- be the place to look for the quasiparticle behaviors induced by topological con- cally ordered material), and one in which analogues of such objects. straints on the electronic degrees of free- TXDQWXPÀXFWXDWLRQVGLVUXSWWKHRUGHU In fact, the vortex-like, fractionally dom constitutes “the biggest intellectual The two states are separated by a quantum charged quasiparticles observed in the leap” that quantum materials represent, phase transition demarcating the point at )4+(VHHQLQWKLQ¿OPVRIPHWDOOLFPD- according to physicist Stephen Blundell ZKLFKÀXFWXDWLRQVGRPLQDWH7KLVWUDQVL- terial in the presence of a strong magnetic of the University of Oxford. It shows that tion is entirely analogous to the critical ¿HOGKDYHDQ\RQOLNHIHDWXUHVDV¿UVWUH- “what seemed to be a peculiarity of the points known in classical statistical me- SRUWHGLQDIUDFWLRQDOTXDQWXP+DOOÀXLG quantum Hall effect can be realized in chanics, for example in ferromagnets and in 2005,15 by observing quantum interfer- many materials in less stringently con- liquids and gases, where the critical point ence behavior between the quasiparticle trolled experimental systems,” he says. marks the end of any clear distinction be- states. Good candidates for hosting anyon “Topologically protected states have tween the two phases. The difference is quasiparticles more directly, however, the advantage that they are resistant to WKDWLQTXDQWXPFULWLFDOLW\WKHÀXFWXDWLRQV are the class of quantum materials called disorder and defects and therefore can are not thermal but driven by quantum quantum spin liquids (see p. 703). They persist in the sometimes messy condi- uncertainty. haven’t been seen yet, but the hunt is WLRQWKDWDIÀLFWVUHDOPDWHULDOV´%OXQGHOO Quantum criticality has emerged as a on.16 Aside from the fundamental interest, adds. “It’s not yet clear what they might rather general framework for understand- there is good practical reason for want- be useful for, but topological quantum ing quantum behavior in diverse materi- ing to make them, since a certain type computing is a possibility.” It was recent- als.22 It has been seen, for example, in the of anyon has been proposed as the quan- ly shown that, at least in one dimension, magnetic states of heavy-fermion systems 23 tum bit for making error-proof quantum symmetry-protected topological phases such as CeCu6. The surprising thing is computers called topological quantum of matter quite generally contain quantum that a quantum critical point does not go computers.17 “resources” that can be exploited for ef- DZD\DWWHPSHUDWXUHVDERYH]HURWKHUH- Similar to anyons, Majorana fermi- ¿FLHQWFRPSXWDWLRQ21 they can be consid- JLRQLQZKLFKTXDQWXPÀXFWXDWLRQVRYHU- ons are unusual hypothetical particles ered a “computational phase of matter.” whelm the system is predicted to broaden that acquire their properties from top- DW¿QLWHWHPSHUDWXUHV ological constraints on their “world- Quantum criticality  ³,WLVWKLVLQÀXHQFHWKDWHOHYDWHVTXDQ- lines”—the hypothetical trajectories One of the most celebrated features tum criticality from an intellectual ab- they can be considered to trace out of quantum theory is uncertainty: straction at absolute zero to a real-world WKURXJKVSDFHWLPH$JDLQ¿UVWLGHQWL- crudely, the idea that you cannot si- phenomenon that can profoundly change ¿HGDVDWKHRUHWLFDOSRVVLELOLW\LQSDU- multaneously know everything that is ¿QLWHWHPSHUDWXUHPDWHULDOSURSHUWLHV´ ticle physics, Majorana fermions have knowable in principle about a quan- according to physicists Piers Coleman the unusual property that, unlike familiar tum system. This notion, enshrined in DQG$QGUHZ6FKR¿HOG24 fermions such as electrons, they do not Werner Heisenberg’s uncertainty prin- Just as cosmic black holes distort the have an antiparticle counterpart because ciple, is not an expression of ignorance surrounding spacetime, quantum critical they are their own antiparticle. They too or clumsiness—it’s not that we can’t points distort the “fabric” of the phase emerge as a prediction of Dirac’s rela- measure properties carefully enough— diagram, creating a V-shaped region of tivistic quantum theory. EXWUDWKHUUHÀHFWVWKHIDFWWKDWVRPH “quantum critical matter” fanning out Like anyons, Majorana fermions properties of quantum systems can’t DW¿QLWHWHPSHUDWXUHVIURPWKHTXDQWXP could potentially serve as qubits for to- VLPXOWDQHRXVO\H[LVWLQDZHOOGH¿QHG critical point. Such a broadening regime pological quantum computers, in which way alongside others. of quantum criticality was reported in

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2014 in the magnetic material CoNb O , 2 6 Ru Cl in which the magnetic spins are coupled in one-dimensional chains.25 7KHÀXFWXDWLRQVRIDTXDQWXPFULWLFDO point can drive electrons to reorganize themselves close to the transition into new phases. For this reason, Coleman DQG6FKR¿HOGVD\³TXDQWXPFULWLFDOLW\ may be a highly effective catalyst for the formation of new stable types of mate- rial behavior, providing an important new route for the design and discovery of new classes of material.” One particularly important class of PDWHULDOVWKRXJKWWRIHHOWKHLQÀXHQFH of quantum criticality is high-tempera- ture superconductors.26 Nicholas Butch of the University of Maryland and co- workers have reported a signature of quantum criticality in the copper-oxide Figure 2. Honeycomb crystal structure of the candidate quantum spin liquid D-RuCl3. Image 27 courtesy of Arnab Banerjee, Oak Ridge National Laboratory. ceramic La2–xCexCuO4, with the su- perconducting phase separated from a conventional “Fermi liquid” metallic phase by a region of “non-Fermi liquid” effects dominate. In addition to super- Spin liquids and behavior at temperatures below the nor- conductivity and quantum critical points, quantum magnetism mal superconducting transition tempera- some heavy-fermion metals exhibit the One might say the same about another ture of around 10–17 K, where there are so-called : an interaction be- class of quantum materials, called quan- VWURQJÀXFWXDWLRQVRIWKHFRSSHUVSLQV tween a local magnetic moment created tum spin liquids. In the 1970s, Nobel The researchers concluded that quantum by f-shell electrons and the spins of the laureate physicist Philip W. Anderson FULWLFDOLW\SOD\VDVLJQL¿FDQWUROHLQVKDS- delocalized conduction electrons in the pointed out that some magnetic materi- ing the phase diagram of this and other material. This interaction can lead to the als might be geometrically incapable of electron-doped copper-oxide systems. opening up of a small bandgap, making aligning all their spins in a single, most There are two quantum critical points in the material insulating—a Kondo insula- VWDEOHPDQQHUWRIRUPDZHOOGH¿QHG this system, closely spaced in the phase WRU²DWYHU\ORZWHPSHUDWXUHV7KH¿UVW orderly magnetic phase. Imagine an an- VSDFHGH¿QHGE\WKHWHPSHUDWXUHGRSLQJ Kondo insulator to be recognized, samar- tiferromagnet—in which adjacent spins

OHYHODQGDSSOLHGPDJQHWLF¿HOG7KHLU ium hexaboride (SmB6), was discovered prefer to be oppositely oriented—on a tri- effects compete: one stabilizes the super- DOPRVW\HDUVDJR30 many others have angular lattice. Each spin has two nearest conducting phase, the other originates subsequently been found. neighbors in a triangle, but the antiparallel from the suppression of superconduc- In a further illustration of the deep DOLJQPHQWFDQQRWEHVDWLV¿HGIRUDOORI WLYLW\E\DPDJQHWLF¿HOGDQGEHWZHHQ connections between the cluster of ideas the trio. One possibility is that the spin WKHPWKH\GH¿QHWKHUHJLRQRISKDVH that underpin quantum materials, it was lattice freezes into a disordered “glassy” space where superconductivity pertains. recognized in 2010 that Kondo insulators state, but Anderson showed that quantum It’s also thought that quantum critical- may possess topologically protected me- PHFKDQLFVDOORZVWKHSRVVLELOLW\RIÀXF- ity plays a role in the “unconventional” tallic surface states: they are versions of tuating spins even at absolute zero. This superconductivity of some other systems topological insulators.31,32 This behavior state is called a quantum spin liquid, and with strongly correlated electrons, such ZDVFRQ¿UPHGH[SHULPHQWDOO\LQ6P%6 Anderson later suggested that it might be as iron pnictides28 and heavy-fermion two years later.33,34 “Topological Kondo connected to high-temperature supercon- metals.29 insulators tie together the nominally dis- ductivity in copper oxides.35 The latter materials are one of the best- SDUDWHVXE¿HOGVRIVWURQJHOHFWURQFRUUHOD- “A spin glass is a structure where studied systems in which strong correla- tions and topological electronic states,” spins freeze in an effectively disordered tions between electrons create quasipar- Butch says. These exotic materials might FRQ¿JXUDWLRQDWVRPHQRQ]HURWHPSHUD- ticle behavior, with the effective charge QHYHU¿QG³DSSOLFDWLRQV´LQWKHWHFKQR- ture,” explains Stephen Nagler of Oak carriers having many times the mass of an ORJLFDOVHQVHWKHLUYDOXHLVUDWKHUWRVKRZ Ridge National Laboratory (ORNL) in electron. Many have rich phase diagrams how various parts of the story of quantum Tennessee. “Spin liquids, on the other at low temperatures where quantum PDWHULDOV¿WWRJHWKHU hand, remain disordered down to zero

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temperature but do not freeze as such spikes in the thermodynamic quantities SURSHUWLHVDUHLQÀXHQFHGE\WRSRORJLFDO at a non-zero temperature.” In quantum OLNHVSHFL¿FKHDW$QGWKHH[FLWDWLRQVRI IDFWRUV

“When they are cooled down, they should in such materials. Quantum spin liquids as strained lead suboxide Pb2O. The elec- show no order even down to absolute zero, may have Dirac nodes like those of Dirac tron correlations that are essential to the but as you cool them down, all sorts of semimetals, and the electrons are expect- behavior of many of these quantum materi- minor effects that you can ignore at high ed to congregate into quasiparticles with als mean that one cannot understand them temperature start to come into play.” fractional charge, like those in the FQHE. by thinking about how electrons behave in Quantum spin liquids should appear “Mostly, quantum spin liquids are intel- isolation. The electronic behaviors are an in materials with particular kinds of mag- lectually interesting,” as opposed to hav- HPHUJHQWSURSHUW\PXFKDVLVWKHÀRFNLQJ netic structures (e.g., antiferromagnets ing obvious practical value, says physicist of birds or the mound building of termites. with triangular lattices). Nagler and his Steven Kivelson of Stanford University. The collective phenomena cannot be de- co-workers at ORNL have reported the “They are entirely new phases of quantum duced simply by adding up the behaviors possible signature of one such variety matter, generalizing notions developed in of individual constituents. In lieu of “quan- of a quantum spin liquid, revealed by the context of the FQHE and conventional tum materials,” says Butch, “personally I neutron scattering from both powder36 superconductors—both of which are states prefer the term emergence, and I’m not and single crystals37 in the alpha phase with considerable structure in common sure why it didn’t catch on.” The concept

RIUXWKHQLXPWULFKORULGH Į5X&O3) (see with spin liquids.” of emergent or collective behavior, he says, Figure 2), which has a layered quasi-2D Still, applications are not entirely out “was a foundation of strongly correlated hexagonal structure. This is not itself a of the picture. For example, the Kitaev electron research”—in superconductors spin liquid, but it is thought to be close model for quantum spin liquids predicts and heavy fermions, for example—“and WRVXFKDVWDWH²VSHFL¿FDOO\FORVHWRD that the electrons could form excited-state, it has been recognized as an important special theoretical case called a Kitaev topologically protected quasiparticles cor- concept in topological materials.” spin liquid, in which the spins exist on a responding to Majorana fermions, raising Perhaps this is the way to delimit the 2D honeycomb lattice that is frustrated, the possibility of using quantum spin liq- RWKHUZLVHDOPRVWLQGH¿QLWHVFRSHRITXDQ- meaning that there’s no way of aligning uids for topological quantum computing. tum materials. Some researchers would the spins so that each enjoys the most fa- But that’s a dim prospect at best. “Perhaps like to see the term used very broadly to vorable interactions with all its neighbors. it is possible that this type of research will cover all materials whose properties are The Kitaev spin liquid model was once one day lead to a useful technology for strongly dictated by quantum mechan- WKRXJKWWREHUDWKHUDUWL¿FLDOEXWLW¶VQRZ quantum computing or other applications,” ics—for example, quantum dots, particles

thought that not just RuCl3 but also some Nagler says, “but we are a long way off of matter so small that the energy states of iridate compounds might be good candi- from using quantum spin liquids in this the electrons are altered from their bulk dates for embodying it.38 What’s more, way.” However, he adds, “we are break- YDOXHVE\WKHTXDQWXPPHFKDQLFDOLQÀX- there is evidence that quantum spin liq- ing ground in elucidating the behavior of HQFHRIFRQ¿QHPHQW%XWWKRVHHIIHFWVFDQ uids exist in other geometrically frustrat- complex materials, and in the long run, I be understood by considering the states of ed triangular-lattice materials, including believe that knowledge is very likely to be single electrons, in contrast to the quasipar- the natural copper mineral herbertsmithite, useful in some fashion, perhaps one that ticles that typify most quantum materials. 39 40 Ca10Cr7O28 and YbMgGaO4. we have not yet thought of.”  $UHVXFKTXDQWXPFRQ¿QHPHQWHIIHFWV But the signature of a true quantum spin then to be considered one of the “exotic liquid is not very clear. When a material Emergence physical properties” included in the DOE enters this state, there is no change in the Today one can hardly pick up a journal that GH¿QLWLRQ"7KHORRVHZRUGLQJPLJKWPDNH symmetry of the system, and so one does publishes condensed-matter physics with- that a matter of personal preference. But not see the marked signatures typical of out seeing a rash of papers on quantum ma- this looseness is what recommends the other magnetic phase transitions, such as terials, particularly those whose electronic GH¿QLWLRQWRVRPHUHVHDUFKHUV³,W¶VDERXW

704 MRS BULLETIN ‡ VOLUME 42 ‡ OCTOBER 2017 ‡ www.mrs.org/bulletin NEWS & ANALYSIS MATERIALS NEWS

right,” Blundell says. “Of course, in a Q. Wu, H.Z. Tsai, W. Regan, A. Zettl, R.K. Kawakami, 22. S. Sachdev, B. Keimer, Phys. Today 64, 29 (2011). Science 340 sense, all materials are quantum, but I think S.G. Louie, L.S. Levitov, M.F. Crommie, , 23. H. von Löhneysen, T. Pietrus, G. Portisch, H.G. 734 (2013). Phys. what we have in mind when we talk about Schlager, A. Schröder, M. Sieck, T. Trappmann, 3. S.M. Young, S. Zaheer, J.C.Y. Teo, C.L. Kane, E.J. Rev. Lett. 72, 3262 (1994). a quantum material is those cases when Phys. Rev. Lett. 108 Mele, A.M. Rappe, , 140405 24. P. Coleman, A.J. Schofield, Nature 433, 226 VSHFL¿FPDWHULDOSURSHUWLHVGHULYHPRUH (2012). (2005). spectacularly from seemingly unusual 4. Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, 25. A.W. Kinross, M. Fu, T.J. Munsie, H.A. Dabkowska, Phys. Rev. B 85 quantum states.” H. Weng, X. Dai, Z. Fang, , 195320 G.M. Luke, S. Sachdev, T. Imai, Phys. Rev. X 4, Others are less sanguine. “I’m not (2012). 031008 (2014). 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