Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
2 How Do Complex Phenomena Emerge from Simple Ingredients? Most materials are made of simple, well-understood constituents, and yet the ag- gregate behaviors of materials are stunningly diverse and often deeply mysteriousâa direct result of the complexity of large systems. Just as a crowd can act in ways un- characteristic of any individual within it, surprising emergent phenomena are also seen in collections of electrons, molecules, and even familiar objects such as grains of sand. For example, sand can be poured like water from a bucket, but unlike any liquid, it also supports the weight of a person walking on the beach. In the fractional quantum Hall state, a bizarre liquid state of electrons, an added electron will break up into new particles, each of which carries a precise fraction of the charge of the original electron. In a superconductor, an electrical current can flow indefinitely without decaying. These are impossible feats for individual grains of sand or indi- vidual electrons. The relationship between the properties of the individual and the behavior of the whole is very subtle and difficult to uncover and lies at the heart of condensed-matter and materials physics (CMMP). The challenge is to understand how collective phenomena emerge, to discover new ones, and to determine which microscopic details are unimportant and which are essential. Emergent Phenomena: Beautiful and Useful Twentieth-century physicists created a spectacularly successful understanding of the structure of atoms and molecules, the interaction of subatomic particles with light, and a unified description of all fundamental forces in nature but gravity. Quantum mechanics and quantum electrodynamics, the most successful quantita- 30
How Do Complex Phenomena Emerge from Simple Ingredients? 31 tive theories developed by humankind, allow for extraordinarily accurate calcula- tions of the properties of individual and small collections of particles. Nature, however, confronts us with materials consisting of unimaginably large numbers of particles. For example, there are many more electrons in a copper penny than there are stars in the known universe. It is therefore not surprising that condensed-matter and materials physicists regularly discover phenomena that nei- ther were foreseen nor are easily understood. These phenomena emerge as collective aspects of the material at hand. Emergent phenomena are properties of a system of many interacting parts that are not properties of the individual microscopic con- stituents. It is often not readily possible to understand such collective properties in terms of the motion of individual constituent particles. Emergent phenomena occur at all scales, from the microscopic to the everyday to the astronomical, and from the precincts of quantum mechanics to the world known to Newton and Maxwell. The infinite diversity of emergent phenomena ensures that the beauty, excitement, and deep practical utility of condensed-matter and materials physics comprise an inexhaustible resource. Emergent phenomena are not merely academic curiosities. Some, like the emergence of life from biomolecules, define our very existence. Others, like the regular arrangements of atoms in crystals, are simply so familiar that we rarely even pause to wonder at them anymore. There are countless examples of this kind. At the same time, the discovery and study of emergent phenomena often lead to immensely important practical applications. Superconductivity, discovered almost 100 years ago, is a good example. While Dutch physicist Kamerlingh Onnes did envision producing magnetic fields using solenoids wound from superconduct- ing wire, he could never have foreseen superconducting magnets big enough to surround a human, nor that such a magnet would be the heart of a technological marvel (magnetic resonance imaging; see Figure 1.1 in Chapter 1) that would revolutionize medicine. Looking ahead, one can imagine that the recently discov- ered high-temperature superconductors, which have so far seen limited applica- tion, might ultimately play a major role in reducing world energy consumption by allowing lossless transmission of electrical power over long distances. Unlike superconductors, which took many decades to see large-scale application, there is the very recent dramatic example of giant magnetoresistive materials, which came to dominate hard disk data storage in just a few years. Liquid crystalline materials, in which large numbers of asymmetric molecules in solution exhibit a dizzying variety of emergent phases, are used in everyday electronics like cellular telephones and laptop computers. Jamming of granular materials (discussed below), perhaps unfamiliar to the average citizen, is an emergent phenomenon with real economic consequences in the mining, pharmaceutical, and other industries. And the list goes on. Emergent phenomena are so widespread that a comprehensive review is both
32 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s impossible and inappropriate. The Committee on CMMP 2010 fully expects that much of the most significant research in the coming decade, as was true of past decades, will be triggered by the discovery of new emergent phenomena that are unlikely to be anticipated in any present list of âmost importantâ problems. Here, a few examples are discussed to illustrate the quest, which underlies much of CMMP, to understand the relation between the properties of the âmicroscopicâ constituents of matter and the macroscopic behavior of the whole. Superconductivity, a century-old phenomenon, is discussed first because it is both an extraordinarily dramatic example of emergence and one of the most ac- tive fields of research in CMMP today. That example is followed by more general discussions of current trends in research on Fermi and non-Fermi liquids, on quantum Hall effect systems, and on critical phenomena and universality in clas- sical and quantum-phase transitions. Emergence in ultracold atomic gases and in granular matter round out the list of case studies. (Further discussion of important emergent phenomena in CMMP can be found elsewhere in this report, especially in Chapter 4 on the physics of life and Chapter 5 on systems far from equilibrium.) Following the examples are some brief remarks on how to realize the full potential of emergence. Superconductivity: An Illustrative Example and a Frontier of Research In many materials, exotic and unexpected phenomena emerge from strong interactions between the constituent particles at the microscopic level. Quantum mechanics often plays a key role and renders the phenomena particularly puzzling and counterintuitive. Superconductivity, the property of certain materials to carry electrical currents without any dissipation of energy, is the quintessential example of such a quantum emergent phenomenon. First discovered in mercury in 1911 by Kamerlingh Onnes and his graduate student Holst in their pioneering experiments on the properties of matter near the absolute zero of temperature, superconductiv- ity was utterly unheralded and resisted explanation for nearly 50 years. Nowadays, CMMP researchers have a good understanding of the phenomenon in mercury and other similar metals where the transition to the superconducting state occurs at very low temperature. But nature is much more resourceful, and this comfortable situation was radically upset just 20 years ago with the discovery of a new class of superconductors having much higher transition temperatures. No accepted theory of high-temperature superconductivity has yet been developed, in spite of immense effort and the application of the most sophisticated tools of theoretical physics by large numbers of researchers across the globe. Unlike the low-temperature metallic superconductors, these new materials are not completely understood even in their normal, non-superconducting, states. Indeed, high-temperature superconductors
How Do Complex Phenomena Emerge from Simple Ingredients? 33 highlight one of the broadest and most important current problems in all physics, the strongly interacting quantum many-body problem. All materials conduct electric current to some extent, but the amount of force that must be applied to maintain a current varies greatly from material to material. The resistance of a material quantifies how large a voltage must be applied to ob- tain a given amount of current flow. Put another way, to maintain a given amount of current consumes power in proportion to the resistance. In a superconductor, the resistance is precisely zero, so a persistent current can flow, forever, around a superconducting ring without need of a battery or a generator! In an attempt to be quantitative about the meaning of âforever,â measurements have been carried out to try to detect the rate of decay of persistent currents in superconducting rings. In these experiments, the current in a ring is measured very accurately at an initial time and then again a long time later. Despite the extreme accuracy of these measurements, no decrease in the current is detected. Even if the current were decaying at the fastest rate possible consistent with the accuracy of the measurement, the current would not decay within the age of the universe. Superconductivity appears when the temperature is reduced below a critical value. What this means is that the resistance of a metal, such as the mercury stud- ied by Kamerlingh Onnes, has a non-zero value at a temperature just a fraction of a degree above the critical temperature. However, at any temperature below the critical temperature, the resistance is zero. It is the same piece of metal both above and below the critical temperature, and the same electrons are carrying the current. At the critical temperature, something subtle but spectacular happens in the organization of the vast number of electrons in the metal that causes them to form a superconducting state. Normally, scientists think of quantum mechanics as the set of physical prin- ciples that govern the motion of small numbers of microscopic particlesâatoms and electrons and nuclear matter. Quantum mechanics is usually only indirectly seen in the properties of macroscopic matterâobjects large enough to hold in oneâs hand. However, superconductors have many counterintuitive properties that reflect their underlying quantum nature. The existence of a persistent current is a concrete demonstration of quantum mechanics at a macroscopic scale. Since cur- rents produce magnetic fields, it is perhaps not surprising that the magnetic prop- erties of superconductors are likewise unprecedented. Indeed, in 1933 Meissner and Oschenfeld discovered that superconductors entirely expel (small) magnetic fields from their interior. In fact, in a superconducting quantum interference device (SQUID), a relation exists between the frequency of current oscillations and an applied voltage that only involves the charge of the electron and the fundamental constant of quantum mechanics, Planckâs constant. This relation has been found experimentally to be universal to a precision of better than 3 parts in 1019, which means that clocks based on two independent Josephson junctions kept at the same
34 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s voltage would differ from each other by no more than one-tenth of a second over a time interval equal to the age of the universe! It is inconceivable that any of the founders of quantum mechanics could have foreseen that a measurement of mac- roscopic quantities would depend directly, and with such precision, on the laws of quantum mechanics. Finally, macroscopic quantum phenomena are certainly not limited to superconductors. The ordinary magnet holding up a note on a kitchen refrigerator offers a dramatic everyday example: A steady magnetic field is present without any battery to keep currents flowing. This macroscopic quantum phenom- enon is just as spectacular as the persistent currents in a superconductor. One of the great triumphs of 20th-century CMMP is the microscopic theory of superconductivity in ordinary metals, the Bardeen-Cooper-Schrieffer (BCS) theory. Not surprisingly, since superconductivity involves only a subtle, low-Âtemperature change in the properties of the electrons in the metal, the BCS theory is based on the equally successful Fermi liquid theory of the properties of normal metals. (Fermi liquid theory and its breakdown are discussed below.) Moreover, hundreds of dif- ferent metals that are BCS superconductors have been identified, although mostly with superconducting transition temperatures less than 10 K. While the theory has rarely led to the prediction of new superconductors, it has provided qualitative guidance for the search for new, low-temperature superconductors. Starting with the 1986 discovery of superconductivity at 30 K in La2âxBaxCuO4 by Bednorz and Mueller, a new class of materials, now known as high-temperature superconductors, became the focus of intensive research. The highest supercon- ducting transition temperature found to date in these materials is approximately 150 K in HgBa2Ca2Cu3O8+x under pressureâroughly 10 times higher than any previously known superconductor. The microscopic interactions responsible for the transition to the superconducting state are different from those in BCS super- conductors. Moreover, the so-called normal state observed at temperatures above the superconducting transition is very different from that of a normal metal and is not well understood. The high-temperature superconductors belong to a large class of synthetic materials (i.e., they do not appear in nature) known as highly correlated electronic materials (see Figure 2.1). Research in this area in the past two decades has been rich in discovery and in producing challenges to the entire quantum theory of solids. These new materials exhibit a startling array of emergent phenomena: fer- romagnetism and antiferromagnetism, orbital ordering and long-period charge ordering, giant and colossal magnetoresistance, new types of superconductivity with new forms of broken symmetry, and all sorts of fluctuation phenomena over unprecedentedly wide ranges of temperature and material parameters. Obtaining a well-founded qualitative understanding of the normal state, at the level of the Fermi liquid theory of simple metals, is among the most challenging and most profound problems facing CMMP. Clearly, understanding the mechanism of high-
How Do Complex Phenomena Emerge from Simple Ingredients? 35 FIGURE 2.1â Local electronic structure of a highly correlated electronic solid visualized with sub- atomic-scale resolution using a scanning tunneling microscope. This is not an Abstract Impressionist painting. It is a self-organized structure âseenâ on the smooth, cleaved surface of a crystal of the high-temperature superconductor Bi2Sr2CaCu2O8+Î´. The patterns represent changes in the electronic structure that are pinned in a highly organized but ultimately random (âglassyâ) pattern. The detailed information concerning the organized structures of electrons in solids that this kind of experiment provides has opened unprecedented opportunities to explore the ultimate connections between micro- scopic physics and the emergent properties of materials. SOURCE: Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J.C. Davis, âAn Intrinsic Bond-Centered Electronic Glass with Unidirectional Domains in Underdoped Cuprates,â Science 315, 1380-1385 (2007). Reprinted with permission from the American Association for the Advancement of Science. temperature superconductivity at a level that can provide qualitative guidance for the search for other, possibly even higher-temperature superconductors is a prob- lem of enormous importance. It is surely not an accident that so many other sorts of emergent states (ordered phases) occur in this class of materialsâunderstanding the relation between the various types of ordered phases of these materials and understanding how they relate to the properties of the non-Fermi liquid normal
36 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s phase are problems that will occupy much of the focus of CMMP in the coming decade. Fermi Liquids and Non-Fermi Liquids If one could ignore the interactions between electrons in a solid, the proper- ties of the material could be derived from a theory that treats only one electron at a time. While this might seem to be an absurd assumption, since electrons are charged and repel one another strongly, such single-electron theories often work remarkably well. Quantum mechanics is essential for understanding why this is so. All electrons are intrinsically identical to one another, just as are all protons, all neutrons, and so forth. Quantum mechanics sets very stringent rules for the behavior of systems containing many identical particles. If the positions of two identical particles are interchanged, quantum mechanics naturally insists that there be no observable consequence. Except in certain rare cases to be described below, there are only two ways in which the quantum wave function of the material can satisfy this requirement: either (1) nothing at all happens to the wave function upon interchange of two particles, or (2) it changes its sign. Particles for which the wave function changes sign are called fermions, while those for which the sign is preserved upon interchange are known as bosons. Electrons are fermions, and thus a many-electron wave function changes sign when two are interchanged. This property underlies the Pauli exclusion principle, which high school chemistry students are usually told means that no two electrons can be in the same place at the same time. More precisely, two electrons are forbid- den from occupying the same quantum state. Despite their vagueness, these state- ments make it easy to see why the Pauli principle has such vast significance for the theory of materials. If no two electrons can be in the same place at the same time, they rarely get so close together that their mutual repulsion is extremely strong. If no two can occupy the same quantum state, then at low temperatures the many electrons in a material are forced to sequentially occupy higher and higher energy levels, forming a âFermi sea.â In some circumstances, interactions between electrons are not strong enough to disturb any but the relatively few levels that are near the surface of this sea. In effect, the Pauli principle converts what at first appears to be a hopelessly strongly interacting system into a more weakly interacting one. In a nutshell, this is why scientists understand the properties of simple metals as well as they do. Remarkably, a more sophisticated version of the âFermi liquidâ picture just described often works very well even when the repulsive interactions between electrons are fairly strong (compared to the average kinetic energy of electrons). In these cases, the properties of the material can be described in terms of new enti- ties, known as quasi-particles, which behave in much the same way as the original
How Do Complex Phenomena Emerge from Simple Ingredients? 37 particlesâthat is, electronsâexcept that they do not interact strongly with one another (see Figure 2.2). One can think of these quasi-particles as consisting of an electron plus a disturbance among the other electrons around it. In some cases, the quasi-particles behave so similarly to the original electrons that it is hard to remember that they are distinct objects. In other cases, the quasi-particle behaves like an electron with strongly modified propertiesâfor instance, in the âheavy fermion materials,â metal alloys containing heavy elements such as uranium and cerium, the quasi-particle can be as much as a thousand times heavier than an electron. A system in which the properties of a dense electron fluid can be related to those of a gas of weakly interacting quasi-particles is called a Fermi liquid. The very successful theory of normal metals, as well as the equally successful theory of simple semiconductors, is based on Fermi liquid theory. However, in many electronically interesting solids, over a wide range of tem- peratures, pressures, and compositions, the Fermi liquid description fails badly. This class includes many of the most interesting materials that have been Âdiscovered a b 4 *1/3 Amplitude (nâ¦cm) 2 0.2 0.4 0.6 0.8 1.0 1.2 Temperature (K) 0 2 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 1 / B (T 1 ) FIGURE 2.2â Signatures of the Fermi liquid state in Sr2RhO4, a strongly correlated material. (Left) A map of the Fermi surface (constant energy surface at the Fermi level) in reciprocal space revealed by high-resolution angle-resolved photoemission experiments. (Right) The resistance as a function of magnetic field at temperatures close to absolute zero; these âquantum oscillationsâ reflect the shape of the Fermi surface. Despite the fact that this material is structurally extremely similar to cuprate high- 2-2 a, b temperature superconductors (such as La2CuO4+Î´), these experiments unambiguously demonstrate that at low-enough energies, the electronic properties of Sr2RhO4 are perfectly represented by a gas of essentially non-interacting, electron-like âquasi-particles.â SOURCE: Reprinted with permission from F. Baumberger, N.J.C. Ingle, W. Meevasana, K.M. Shen, D.H. Lu, R.S. Perry, A.P. Mackenzie, Z. Hussain, D.J. Singh, and Z.-X. Shen, âFermi Surface and Quasiparticle Excitations of Sr 2RhO4,â Phys. Rev. Lett. 96, 246402 (2006). Copyright 2006 by the American Physical Society.
38 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s and studied in the past two decades, among them the high-temperature super- conductors and quantum Hall effect systems. There are many well-understood qualitative reasons why Fermi liquid theory is less successful in these new materials. More importantly, the experimental evidence that the behavior of these systems is incompatible with Fermi liquid theory is clear and incontrovertible. This evidence ranges from direct tests of the quasi-particle hypothesis, such as angle-resolved photoemission measurements that can directly âseeâ a quasi-particle if it exists, to indirect tests, such as a measured metallic resistivity that exceeds the maximum that is consistent with the quantum motion of independent quasi-particles. Lack- ing today is a conceptually clear and computationally tractable framework for understanding the properties of a ânon-Fermi liquidâ of electrons. Some of the essential ingredients in an understanding of a non-Fermi liquid are clear. The basic objects that move are no longer electrons, but more likely large clusters of electrons moving in concert. The building blocks of a theory of such a state are thus very different from the quasi-particles of Fermi liquid theory. A better intuitive picture may come from envisaging a fluid made up of pieces of melted electron crystals, or magnets, or superconductors, not just individual quasi-particles. Correspondingly, the properties of such a system are not readily inferred from the properties of individual electrons. For instance, a fluid of par- tially ordered magnets can have magnetic properties intermediate between those of a magnet and a normal metal (see Figure 2.3). Such behavior is most readily addressed in the proximity of a âquantum critical pointâ separating two distinct phases, as discussed below. However, this is just the tip of the iceberg. The broad occurrence of non-Fermi liquid phenomena suggests that it is related to new quantum phases, or at least to extremely new regimes of matter. Correlated motion of many particles is difficult to characterize and still more difficult to understand. However, the non-Fermi liquid character exhibited by a rapidly increasing number of interesting materials imbues the problem with an immediacy and focus that is compelling. Moreover, various new theoretical ideas, new experimental discoveries, and methodological advances in theory and experiment (some of which are discussed in Chapter 11) give hope that substantial advances in understanding are occurring. One set of new ideas involves the existence of broken-symmetry quantum phases possessing âhiddenâ (i.e., hard to detect) types of order. For example, interest has recently focused on a class of states with complex patterns of spontaneously generated persistent currents. No such state has yet been unambiguously identified in a real material; conversely, even if such a state occurs, it would be very difficult to detect by most conventional measurements. Another interesting class of states with hidden order is electronic analogues of the classical liquid crystalline states that occur in complex fluids. In a simple liquid, the particles can flow from one point to another, all points in space are
How Do Complex Phenomena Emerge from Simple Ingredients? 39 a b FIGURE 2.3â Calculated magnetic structure factor for a âstripe-orderedâ antiferromagnet. Strong interactions between electrons can lead to complex ordered states with particle-like excitations that look nothing like those of an electron. One such state, which has been directly identified in neutron 2-3 a, b scattering experiments on a number of transition metal oxides, including some high-temperature su- perconductors (such as La2CuO4+Î´), is a unidirectional incommensurate antiferromagnet, or âstriped phase,â shown schematically (right panel). The emergent particle-like excitations that occur in such a state are charge-neutral âspin-waves,â whose spectrum is calculated here (left panels) and which are valid deep in the ordered phase. SOURCE: (Left) Reprinted with permission from D.X. Yao, E.W. Carlson, and D.K. Campbell, âMagnetic Excitations of Stripes and Checkerboards in the Cuprates,â Phys. Rev. B 73, 224525 (2006). Copyright 2006 by the American Physical Society. (Right) Steven A. Kivelson, Stanford University. equivalent, and all directions are the same. In a crystal, there is a lattice on which the atoms or molecules are localized in a pattern that repeats periodically through space, so that different points within the unit cell are different from each other, and different directions, relative to the axes of symmetry defined by the crystal- line order, are distinct from each other. Liquid crystalline states exhibit patterns of symmetry breaking intermediate between those of a simple liquid and a crystal.
40 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s For instance, a ânematicâ is a uniform fluid state in which one spatial direction is distinguished from the other two. Traditionally, a nematic is described as consist- ing of a fluid of cigar-shaped molecules, in which the molecules can flow, but they are preferentially oriented along one direction. At first sight, a nematic electron fluid seems improbable, since electrons are point-like, not cigar-shaped. However, in some circumstances it is legitimate to think of a highly correlated electron fluid as consisting of melted fragments of an appropriate electron crystal; these fragments, in turn, can play the role of the cigar-shaped molecules in the classical nematic. An electron nematic phase is also difficult to detect for various technical reasons, including the fact that crystalline imperfections can mask its occurrence on Âmacroscopic scales and that it can be hard to distinguish from a more conven- tional strain-driven change in the crystal structure of the host material. However, as discussed below, strong evidence for an electronic nematic phase has recently been found in extremely high mobility quantum Hall devices. Moreover, evidence of the existence of such phases has recently been found in a number of interesting highly correlated materials, including Sr3Ru2O7 and certain of the high-Âtemperature superÂ conductors (e.g., La2âxBaxCuO4). A still more revolutionary circle of ideas, built around the notion of âfractional- izedâ phases, has been the focus of increasing attention in recent years. The key idea here is that sharp distinctions can exist between distinct quantum phases of matter that have nothing to do with distinct patterns of symmetry breaking. Rather, these phases are characterized by an abstract form of orderâso-called topological order. Because the order is so subtle, it is difficult to establish experimentally where such phases occur in nature. The most directly experimentally accessible characteristic of these phases is the existence of new types of quasi-particles that behave like a âfractionâ of an electron. For instance, in one of the best theoretically characterized of these phases, there are no quasi-particles that carry both the charge and spin of ordinary electrons. Instead, two new and very strange kinds of particles appear, one of which carries only spin and another of which carries only charge. In other sys- tems, the quasi-particles carry a specific fractionâfor example, one-thirdâof the electron charge. Under some circumstances, these quasi-particles are also believed to possess âfractionalâ quantum statistics. In other words, the wave function of two such identical particles neither preserves nor changes its sign when the particles are interchanged but instead is multiplied by a complex number (e.g., the cube root of â1!). Such particles are neither bosons nor fermions; they are called anyons. At present, fractionalization is an established experimental fact only in the fractional quantum Hall state, as explained in the next section. Evidence suggestive of the existence of fractionalized phases has been reported in the past few years in a number of strongly correlated materials that exhibit particularly unusual proper- ties. Moreover, the search for fractionalized states has gained added impetus from the realization, discussed in Chapter 7, that such phases might produce uniquely
How Do Complex Phenomena Emerge from Simple Ingredients? 41 favorable structures for the construction of a quantum computer. However, it is an issue for the coming decade to discover which of these phases actually exist in nature, and where (i.e., in what materials). Quantum Hall Systems and the Discovery of New Quantum States of Matter The concepts of fractionalization and non-Fermi liquids have roots in elec- tronic materials that are less than three-dimensional. For example, it has been known since the 1960s that a disorder-free, one-dimensional electron system can- not be a Fermi liquid. Unlike electrons in two and three dimensions, the distribu- tion in momentum space of one-dimensional electrons does not have a discontinu- ity at the Fermi momentum, even at zero temperature. Experimental evidence for such âLuttingerâ liquids has accumulated in recent years from studies of electrical conduction in carbon nanotubes and along the edges of two-dimensional electron systems in a high magnetic field, as well as from more complex experiments (such as angle-resolved photoemission and optical conductivity) performed on various other âquasi-one-dimensionalâ materials in which electrons can move readily only in one direction. Fractionalization is already well known in two-dimensional electron systems (see Figure 2.4). A large magnetic field perpendicular to the two-dimensional plane breaks up the otherwise-continuous distribution of electron energies into a ladder of discrete states known as Landau levels. Since the energy of an electron in a cir- cular cyclotron orbit does not depend on where the orbit is located, each Landau level can hold many electrons, the total number being proportional to the mag- netic field. Consequently, at sufficiently high magnetic fields, all two-dimensional electrons in a given sample can fit into the lowest Landau level and have precisely the same kinetic energy. With no variation in the kinetic energy, the Coulomb interaction between electrons completely dominates the physics in clean samples. In a real sense this is the most strongly interacting quantum system imaginable. It is in this regime that Tsui, Stormer, and Gossard discovered the famous fractional quantum Hall effect (FQHE) in 1982. They observed that when the lowest Landau level is one-third filled, a wholly unexpected quantized plateau in the Hall resis- tance appears, signaling the opening of an energy gap. Shortly thereafter, Laughlin explained the effect as the emergence of a new state of matter, one driven entirely by strong Coulomb correlations among the electrons. Laughlinâs theory was the âD.C. Tsui, H.L. Stormer, and A.C. Gossard, âTwo-Dimensional Magnetotransport in the Extreme Quantum Limit,â Phys. Rev. Lett. 48, 1559 (1982). âR.B. Laughlin, âAnomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Frac- tionally Charged Excitations,â Phys. Rev. Lett. 50, 1395 (1983).
42 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s FIGURE 2.4â Quantum phases and phase transitions in a two-subband quantum Hall device. Shown here is a contour map of the resistivity of a two-dimensional electron gas that has been trapped at the interface between two semiconductors. The y-axis is the gate voltage applied across the sample (which changes the density of electrons), and the x-axis is the strength of an applied magnetic field. The dark regions, where the resistivity becomes vanishingly small as the temperature tends toward absolute zero, are various quantum Hall effect states, where the integers label the value of the quan- tized Hall conductance in units of the quantum of conductance. The bright regions mark the points at which quantum phase transitions occur between the different phases. The fact that the resistance neither vanishes nor diverges as the temperature tends to zero along the quantum critical lines is a tangible reflection of the existence of quantum fluctuations at all length scales. SOURCE: Reprinted with permission from X.C. Zhang, D.R. Faulhaber, and H.W. Jiang, âMultiple Phases with the Same Quantized Hall Conductance in a Two-Subband System,â Phys. Rev. Lett. 95, 216801 (2005). Copyright 2005 by the American Physical Society. first to imply the existence of fractionalized elementary particles within a strongly correlated electron system. These new particles, which carry precisely one-third the charge of an ordinary electron, are neither bosons nor fermions, possessing instead the bizarre anyonic exchange statistics mentioned above. Very strong experimental evidence for the fractional charge of these particles now exists, and experimental proof of fractional statistics is currently being hotly pursued. The FQHE at one-third filling of the lowest Landau level turned out to be only one member of a large family of similar correlated phases of two-dimensional elec-
How Do Complex Phenomena Emerge from Simple Ingredients? 43 trons. Over the past 25 years, many dramatic experimental observations have been made, and a sophisticated and unified theoretical understanding of much of FQHE phenomenology has been developed. The field remains vibrant, with the pace of new discoveries paralleling the steadily increasing quality of the semiconductor het- erostructure samples grown by molecular beam epitaxy. Most recently, interest has focused on an FQHE that appears at one-half filling of the first excited Landau level (the so-called 5/2-state). This fragile state is expected to possess an even stranger form of exchange statistics in which the outcome of multiple interchanges of pairs of particles depends on the order in which the interchanges occur. Observation of such ânon-Abelianâ statistics would have deep fundamental significance for phys- ics and possible impact on schemes for quantum computation, since non-Abelian systems are anticipated to be especially insensitive to the kinds of disturbances that ordinarily disrupt quantum coherence. The steady increase in sample quality has also led to the recent discovery of new electronic phases outside the FQHE paradigm. At modest magnetic fields, where several Landau levels are occupied, collective states emerge that are reminiscent of both classical liquid crystals and pinned charge density waves. For example, near one-half filling of highly excited Landau levels, electrical conduction in the two-dimensional system spontaneously becomes extremely anisotropic at very low temperature (below about 150 mK). Strikingly, this anisotropy disappears on moving slightly away from one-half filling and is absent altogether at both very low and very high magnetic fields. The effect is widely believed to reflect a stripe- like pattern of charge density modulation in the two-dimensional system. While quantum and thermal fluctuations destroy the long-range order of the stripes, local order persists. In effect, the two-dimensional electron system is broken up into a collection of striped domains. Above about 150 mK, these domains are ap- parently randomly oriented, and the net resistivity of the system is isotropic. At lower temperatures, orientational order sets in and the resistivity rapidly becomes anisotropic. As discussed above, this situation is highly analogous to the isotropic- to-nematic phase transition in classical liquid crystals (see Figure 2.5). In this case, the local stripe domains of electrons play the role of the funny-shaped molecules in the liquid crystal. That a system of point-like electrons would emulate a liquid crystal is one of the most dramatic examples of emergence in recent years. Excitingly, the quantum Hall effect has recently been observed in graphene (single atomic layers of graphite). This is especially interesting since graphene is a semimetal whose low-energy band structure precisely mimics the dynamics of massless relativistic particles, the so-called Dirac fermions familiar to high-energy physicists. In other words, the kinetic energy of electrons (or holes) in graphene is directly proportional to their momentum, rather than its square. This aspect (and others) of the graphene band structure creates a spectrum of quantum Hall states that is distinct from that found in conventional two-dimensional electronic
44 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s FIGURE 2.5â Quantum and classical nematic liquid crystals. The background image is a polarization microscope image of a classical nematic liquid crystal, similar to those found in cellular telephones, computer displays, or wristwatches. The graph and the insets describe the development of quantum liquid crystalline behavior in a collection of electrons moving on a plane surface of a nematic liquid crystal in the presence of a perpendicular magnetic field. The red and blue traces show how the electri- cal resistance of the system, measured in two mutually perpendicular directions, becomes extremely anisotropic at temperatures close to absolute zero. This anisotropy is believed to arise from the spon- taneous orientational ordering of small elongated clumps of electrons, as suggested by the black rods in the lower insets. SOURCES: (Graph) J.P. Eisenstein, âTwo-Dimensional Electrons in Excited Landau Levels: Evidence for New Collective States,â Solid State Commun. 117, 123-131 (2001). (Background image) Photograph by Oleg D. Lavrentovich, Liquid Crystal Institute, Kent State University. materials (e.g., in silicon inversion layers or gallium arsenide quantum wells). To date, only integer, as opposed to fractional, quantized Hall states have been detected in graphene. This suggests that the quality of the current samples is not high enough for the subtle, many-particle correlations of the FQHE to hold sway
How Do Complex Phenomena Emerge from Simple Ingredients? 45 over the disordering effects of impurities. The intensity of research on graphene is now enormous. There is every reason to believe that significant advances, including the discovery of new emergent phenomena and the development of more effective means for creating, manipulating, and employing clean graphene films, will occur in the coming decade. Clearly, the nature of correlated motion of strongly interacting particles is a conceptually deep problem with broad implications in areas of condensed-matter and materials physics and in other areas of physics as well. The quest to understand the emergent behaviors produced by these correlations is one of the central issues in physics today. Moreover, with the spirit of the past as a prologue, there is every reason to believe that some of these new emergent phenomena will be the basis of future technologies of profound importance. Critical Phenomena and Universality Generally, the phases of matter are well defined in the sense that many of the properties of a given material depend primarily on the materialâs state (for example, solid, liquid, or gas), and not so much on the substance itself. All liquids behave in many familiar ways; as elementary school students learn, a liquid has a fixed volume but takes the shape of its container; it flows; sound can propagate through it; and so forth. These and many other features are common to liquid water, gasoline, al- cohol, and liquid helium. A metalâbe it copper or silver or an organic metal made largely of carbon and hydrogenâis shiny and conducts electricity readily. Indeed, an organic metal and an organic insulator, even though they may be made of es- sentially the same constituent elements, have physical properties that share many fewer features than do an organic metal and copper. This universal character of the phases of matter generally holds all the way from the nanometer to the centimeter scale and beyond. Phase transitions can occur as a function of temperature, or pressure, or magnetic field, or composition, and so forth. When water freezes, the water at temperatures just below the freezing point is a solid (ice), which behaves pretty much in the same way as ice that is far colder. The water just above the freezing point is a liquid, and similar to liquid water at higher temperatures. The change in the behavior is highly discontinuous across the freezing point. This is an example of a first-order transition. Some other transitions, such as the transition from a paramagnetic to a fer- romagnetic phase, occur in a much more mysterious manner, by a âcontinuous transition.â Close to the critical temperature, it becomes increasingly difficult to tell which side of the transition the system is on. On length scales that get increasingly large in proximity to the transition, the system cannot decide which phase behavior to exhibit, and so it exhibits a new, intermediate behaviorâcritical
46 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s b Â ehaviorâthat is different from the behavior of either of the phases themselves. At a critical point, there are fluctuations on all length scales from the microscopic to the macroscopic. The broad distribution of length scales at a critical pointâor technically, the scale invarianceâis a spectacular phenomenon. Most phenomena in nature have a characteristic size. Atoms are all (within a factor of two) a couple of angstroms in diameter, and people are typically 5 to 6 feet tall. In a piece of ice, when atoms rearrange (i.e., flow on a microscopic scale), typically only a few atoms move at a time. However, near a ferromagnetic critical point, collective motions occur involv- ing reorientation of the magnetic dipoles of small groups of a few electrons and enormous patches of millions or billions of electrons. The ârenormalization groupâ theory of critical phenomena in classical systems undergoing a continuous phase transition, which was developed throughout the last three decades of the 20th century, is one of the most significant contributions of CMMP to science. It provides an understanding of how scale invariance at a critical point arises from simple microscopic interactions. Because physics near a critical point occurs on such a broad range of length scales, much of the detailed information about the microscopic constituents of the material is averaged out. In a quantifiable sense, the behavior of systems near a critical point is âuniversalââthat is, precisely the same for different systems. Not only does the magnetization grow with decreasing temperature in precisely the same way in ferromagnets made of pure iron or of neodymium alloys, but it grows in exactly the same way that the concentration difference grows near the critical point for phase separation of a mixture of water and oil. Indeed, renormalization group theory offers a top-down perspective for un- derstanding condensed matter that is opposite to the usual bottom-up reductionist approach, which focuses on the identification of a small number of elementary building blocks. Since the behavior of the system is independent of what material is being studied, there is not a unique route from the microscopic understanding of the laws of quantum mechanics to the macroscopic properties of a system near its critical point. Rather, the âanswerâ is largely independent of the âquestion.â So powerful is the notion of universality that the solution of even a vastly simplified abstract mathematical model problem, so long as it respects certain symmetries and constraints of the real world, can be used to obtain a precise and quantitative understanding of experimental observations in the complex real world! This is the most precisely understood realization of the more general notion of emergence. As with any revolutionary change, the full implications of the renormalization group approach continue to reverberate. When continuous phase transitions occur at zero temperature, quantum mechanics on a macroscopic scale becomes impor- tant. Here, quantum mechanics intertwines dynamics and thermodynamics in a way that they never are in finite temperature (âclassicalâ) phase transitions. The
How Do Complex Phenomena Emerge from Simple Ingredients? 47 naÃ¯ve extension of the successful theory of classical phase transitions has already been shown to produce results that are in conflict with experiment for quantum phase transitions. It remains unclear whether minor modifications of the classical approach or much more fundamental changes are needed to address these difficul- ties. Transitions occur, as well, in non-equilibrium, âdrivenâ Âdynamical systems, of which the best known is the transition from laminar flow to turbulence. Many driven dynamical systems exhibit phenomena on a broad range of scalesâexhibitingÂ an approximate form of scale invariance. Much beautiful work has already been done applying renormalization group ideas to this broad class of systems, but it is clearly just the tip of the iceberg, as discussed in Chapter 5. ÂAnother vast area in which many related open problems exist is systems with quenched disorderâin which there are degrees of freedom, such as the locations of impurity atoms, which are not in thermal equilibrium. Phase transitionsâeven classical phase Âtransitionsâin the presence of quenched disorder are not fully underÂstood, and where quenched disorder and quantum phase transitions intersect, there is a growing understanding that entirely new conceptual tools are needed. Emergence in UltraCold Atomic Gases The rapidly dissolving boundary between conventional atomic physics and CMMP has focused yet another spotlight on the phenomenon of emergence. The convergence of these two fields began about 10 years ago when, upon cooling dilute gases of atoms (e.g., 87Rb) trapped in magnetic bottles into the nano-kelvin tem- perature range, atomic physicists succeeded in directly witnessing the phenomenon of Bose-Einstein condensation (BEC). In a BEC, quantum uncertainty obscures the identity of the individual atoms and instead endows the entire ensemble with a single coherent wave function. While Bose-Einstein condensation had been long known to CMMP physicists to be responsible for the phenomenon of superfluidity in liquid helium, its unambiguous observation in a wide variety of highly control- lable atomic systems was a watershed event in physics. Since the initial observations of BEC, the field of ultracold atomic gases has expanded dynamically in both experiment and theory. Particularly dramatic among many exciting developments has been the observation of Cooper pairing in cold fermionic systems and the detection of the superfluid-to-insulator transition among cold bosonic atoms held in an optical lattice potential. These two examples illustrate the power of the ultracold atom field to study classic condensed-matter phenomena in an extremely controllable way. Moreover, ultracold atoms in spe- cially tailored optical lattices, as described in Chapter 8, may be used to simulate âNational Research Council, Controlling the Quantum World: The Science of Atoms, Molecules, and Photons, Washington, D.C.: The National Academies Press, 2007.
48 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s models of some of the most significant outstanding problems in condensed-matter physics, including high-temperature superconductivity and related strong correla- tion phenomena. These optical lattice-based systems, acting as âanalog quantum computers,â may provide solutions to problems that have been found to be essen- tially unsolvable using conventional digital computers. Most importantly, ultracold atoms offer the prospect of discovery of wholly new and highly exotic states of condensed matter that have no roots in traditional material systems. Emergence in Classical Condensed-Matter SYSTEMS A number of examples of emergence in classical condensed-matter systems illustrate the scope and unity of the concepts underlying the study of emergence, where neither quantum mechanics nor even conventional thermal physics plays any direct role in determining the emergent behavior (see Figure 2.6). A few examples are described below. Granular matter, like other forms of condensed matter, is made up of an enor- mously large number of simple constituentsâthe individual grains, for instance, of sand or wheat. The difference is, however, that the grains themselves are very large compared with the atoms and small molecules that make them up. Since the FIGURE 2.6â A wide variety of regular patterns spontaneously emerge in many systems driven away from equilibrium. There is a growing understanding of the variety and complexity of the patterns in terms of the nonlinear interaction and competition between spatial modes that become unstable ow- ing to the drive away from equilibrium. The three panels in this figure show results of simulations of a simple partial differential equation that captures these effectsâthe different patterns are given by changing parameters. Patterns remarkably similar to these are seen in experimental systems such as boiling water, vertically shaken fluids, and granular flows. SOURCE: Ron Lifshitz, Tel Aviv University. Based on R. Lifshitz and D.M. Petrich, âTheoretical Model for Faraday Waves with Multiple-Frequency Forcing,â Phys. Rev. Lett. 79, 1261-1264 (1997).
How Do Complex Phenomena Emerge from Simple Ingredients? 49 characteristic energy scale that characterizes the quantum motion of a collection of particles decreases rapidly with the increasing size of the particle, even at the lowest accessible temperatures quantum effects in granular systems are negligible. Conversely, even at room temperature or above, thermal effects are also negligible: the interaction energy between pairs of grains increases with size (roughly in pro- portion to the surface area) and so is always large compared with the temperature. Put another way, the ratio of the entropic and interaction contributions to the free energy at any fixed temperature decreases rapidly with the size of the grains. Thus, granular matter effectively presents scientists with a problem in which temperature can also be ignored, so the collective phenomena typically involve nonequilibrium physics, as discussed in Chapter 5. Nevertheless, emergent phenomena occur in granular matter in much the same manner that they occur in conventional fluids and solids, although often with new and fascinating wrinkles. When a granular material is fluidized, by shaking it vigorously or by blowing gas through it, the resulting non-equilibrium state has many features in common with a simple gasâfor example, the individual grains exhibit a distribution of speeds that is very similar to that of molecules in a gas at an âeffective temperature,â which depends on how violently the granular matter is shaken. A dense granular material, like sand on a beach, shares many properties with a solid, including its ability to support a person. However, unlike a simple solid, where the strain is fairly smoothly distributed throughout the material, in a compressed granular system the strains can be highly irregularly distributed along lines of force (Figure 2.7). The principles that govern the distribution of strain in dense granular matter, and which features are universal (i.e., do not depend on the nature [size, shape, hardness] of the individual grains), are at present an area of active research. It is similarly not fully understood to what extent the motion of broader classes of driven granular systems can be related to properties of a related equilibrium system at an effective temperature. Jamming is a phenomenon in granular materials that, when better understood, may shed light on a broad class of phenomena in condensed-matter systems. At low density, it is clearly easy for granular matter to flowâeach grain simply moves in the general direction of the flow and is occasionally scattered when it collides with another grain. This is a classical analogue of the motion of quasi-particles in a Fermi liquid. However, when hard grains, such as grains of sand, reach a critical density, there is simply no room for the grains to flow. Every grain is jammed in a cage of other grains from which it cannot escape. Here, the strong correlations between grains entirely quench the free motion of the individual grains. Since there is no balance of quantum and classical energies, no competing tendencies of energy and entropy, this jamming transition may be, in some sense, the simplest model problem in the physics of strong correlations. Jamming is, in a sense, a purely geometric phenomenon.
50 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s FIGURE 2.7â A collapsing grain silo provides a dramatic example of the unexpected behavior of granu- lar materials. If the grains are flowing in some regions and jammed in others within the silo, there can be large variations of the stress on the silo walls, leading to disaster. SOURCE: J.M.Rotter, University of Edinburgh, and J.W. Carson, Jenike and Johanson, Inc.
How Do Complex Phenomena Emerge from Simple Ingredients? 51 Realizing the Full Potential of Emergence Emergent phenomena in condensed matter are often discovered serendipi- tously. The discovery of superconductivity by Kamerlingh Onnes in 1911 was certainly accidental. The discovery of the fractional quantum Hall effect by Tsui, Stormer, and Gossard in 1982 was similarly unanticipated. These two great discov- eries, decades apart in time, have some important similarities that offer insights into how to increase the odds for the discovery of emergent phenomena. For example, both experiments were part of a program of curiosity-driven, âblue-skyâ research, and both fundamentally altered the landscape of CMMP. At the same time, both discoveries are intimately connected to the technological side of CMMP. With su- perconductivity, that connection lay in the future applications of the phenomenon itself. Conversely, the discovery of the fractional quantum Hall effect was enabled by technical advances in semiconductor crystal growth critical to the development of high-speed transistors for telecommunications. These two discoveries beauti- fully illustrate the inseparability of the applied and fundamental sides of CMMP. They also illustrate the need to maintain a robust funding base for the pursuit of curiosity-driven CMMP research and the ready availability of the exotic materials that enable such great discoveries. Both of these issues figure prominently in the recommendations of this report. Superconductivity and the fractional quantum Hall effect were discovered by investigating the properties of matter under extreme conditions. Kamerlingh Onnes, having recently succeeded in liquefying helium, was studying the resistivity of metals cooled to near absolute zero. Tsui, Stormer, and Gossard, also examining electrical conduction, were subjecting their ultrapure semiconductor samples to the highest-available magnetic fields. The fruitfulness of high magnetic fields and low temperatures of course remains fully appreciated, with the National High Magnetic Field Laboratory in Tallahassee, Florida, the most visible evidence. Beyond high magnetic fields and low temperatures, CMMP researchers regularly subject their samples to high pressures, intense electromagnetic fields, dimensional confine- ment, and other extreme conditions in search of understanding and, best of all, surprises. Discoveries of emergent phenomena sometimes have the appearance of mere lucky breaks in an otherwise random walk. This is no truer of superconductivity or the fractional quantum Hall effect than it is of Edisonâs discovery of the correct fila- ment material for incandescent light bulbs. Kamerlingh Onnes was naturally trying to understand the conduction of electricity in metals in the context of the Drude theory advanced just a few years before. Similarly, Tsui, Stormer, and Gossard were guided in part by theoretical predictions that electron gases would freeze directly into a crystalline solid at high enough magnetic field. Both examples illustrate the
52 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s close interaction between theory and experiment that characterizes CMMP; each informs and guides the other. Conclusions Emergent phenomena in condensed-matter and materials physics are those that cannot be understood with models that treat the motions of the individual particles within the material independently. Instead, the essence of emergent phe- nomena lies in the complex interactions between many particles that result in the diverse behavior and often unpredictable collective motion of many particles. It is wonderful and exciting that the well of such phenomena is infinitely deep; CMMP researchers will never run short of mysteries to solve and phenomena to exploitâthey are out there for the inquisitive to find. Emergent phenomena beautifully illustrate the inseparability of the fundamen- tal and applied research in CMMP. In some cases, the application of an emergent phenomenon is nearly immediate; in other cases it takes decades to occur; and in still others it may never occur. At the same time, technical advances in one area of CMMP can enable the discovery of an exotic phenomenon in a seemingly remote area of the field. The nationâs CMMP community has historically been extraordinarily success- ful at discovering, understanding, and applying emergent phenomena. In terms of opportunity, the future is extremely bright. Ever-more-complex materials are being synthesized and ever-more-sophisticated tools are being developed for their study. The explosion of research on nanoscale systems and the rapidly dissolving boundaries between CMMP and other scientific disciplines will surely lead to new vistas in emergent phenomena. The challenge is to make sure that U.S. research- ers have access to the best new materials and tools and the time and resources to make the most of them. The paths between discovery, understanding, and applications of scientific research are obscure and unpredictable. They are full of sharp turns, dead ends, and unexpected forks in the road. But they also can lead to beautiful places that no one knew existed. Robert Frost had it right: It is important to take the road less traveled, for that will make all the difference.