Complexity, Symmetry and Strong Correlations

Home , Lev Landau

Jak Chakhalian COMPLEXITY, SYMMETRY AND Solid State Physics 601 STRONG CORRELATIONS Fall 2017 Phenomena Emerging from Complexity APPROACH TO COMPLEX PHENOMENA

reductionism and emergence 3 Exploring the Unit materials which deform under stress, like polymers, liquids, colloids, and granular materials. The written text for this unit focuses on solid state physics, whereas the video touches on both solid state and soft condensed matter. In today’s session, we will focus on emergence in condensed matter physics: both solid-state and soft condensed matter. By some accounts, all of condensed matter physics can be considered emergent.

The simplest definition of emergence is the interaction of individual pieces, following simple rules, which leads to collective behavior. There is no leader, or top-down control in such systems—the collective behavior comes from the bottom-up interaction of many individuals. In flocking, for example, each bird is following simple rules: Stay close to your neighbors, but not too close, and avoid predators. From these rules comes surprisingly coherent behavior of the flock as a whole. (Note: In particular, non-linear interactions, in which the character of the interaction changes with some parameter, like distance, lead to surprising behavior. In this way, complex behavior is not simply the additive sum of many individual interactions.) WHAT IS EMERGENCE ? Here is a definition of emergence from the National Academies: 3

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 phenomena lies in the complex interactions between many particles that result in the diverse behavior and often unpredictable collective motion of many particles.

Emergence relates to several other key concepts: Condensed Matter and Materials Physics: The Science of the World Around Us, NAS, National Academies Press (2007). Complex systems are systems with many interacting parts. Thus, emergence is essentially the description of the observable phenomena that arise from complex systems. Self-organization is an often-studied feature of complex systems. Complex adaptive systems are a subset of complex systems in which feedback is critical. 4

Phase transitions are the transition from one phase of matter to another. This can be a familiar state of matter (such as solid and liquid) or other types of phases (such as paramagnetic and diamagnetic). When a material changes phase sharply with the varying of some external parameter (as when ice freezes at a clearly defined temperature and pressure), this is called a first order phase transition. The properties of materials on either side of a phase transition are emergent. Phase transitions are a key sign of emergent behavior.

Phase transitions usually involve symmetry breaking—a change in the symmetry of the constituents. For example, the molecules in a liquid are randomly arranged, and thus very symmetric. The molecules in an ice crystal are very orderly. Since disorder is more symmetric than order (you can look at a liquid from any angle and it will look the same), this is an example of symmetry breaking. Since symmetry breaking would not be predicted from fundamental laws, it is seen as a key aspect of systems that cannot be explained through reductionist means.

At a critical point, there is no clear boundary between two phases—the transition between two phases is continuous (a second-order phase transition). For

3 Condensed Matter and Materials Physics: The Science of the World Around Us, National Academies Press (2007).

Unit 8 -167- Physics for the 21st Century JStatPhys(2009)137:777–797 DOI 10.1007/s10955-009-9814-1

More is the Same; Phase Transitions and Mean Field COMPLEXITY Theories

Leo P. Kadanoff

JStatPhys(2009)137:777–797 DOI 10.1007/s10955-009-9814-1 “Infinitely more is different.”

Received: 15 June 2009 / Accepted: 27 August 2009 / Published online: 12 September 2009 © Springer Science+Business Media, LLC 2009

MoreAbstract This is the paper isSame; the first in Phase a series that Transitions will look at the theory and of phase Mean transitions Field Theoriesfrom the perspectives of physics and the philosophy of science. The series will consider a group of related concepts derived from condensed matter and statistical physics. The key technical ideas go under the names of “singularity”, “order parameter”, “mean field the- Summary of theory”, paper: “variational method”, “correlation length”, “universality class”, “scale changes”, and Leo“renormalization”. P. Kadanoff The first four of these will be considered here. A dialog in Paris inIn a1920 less technical vein, the question here is how can matter, ordinary matter, support adiversityofforms.Weseethisdiversityeachtimeweobserveiceincontactwithliquid water or see water vapor (steam) come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. Received:A phase 15 transition June 2009 is a /change Accepted: from 27 one August behavior 2009 to another. / Published A first online: order phase 12 September transi- 2009 Nobel prize, 1977 ©tion Springer involves Science+Business a discontinuous jump Media, in some LLC statistical 2009 variable. The discontinuous5 property is called the order parameter. Each phase transition has its own order parameter. The possible order parameters range over a tremendous variety of physical properties. These properties include the density of a liquid-gas transition, the magnetization in a ferromagnet, the size of Abstractaconnectedclusterinapercolationtransition,andacondensatewavefunctioninasuper-This paper is the first in a series that will look at the theory of phase transitions fromfluid or the superconductor. perspectives A continuous of physics transition and occurs the philosophy when the discontinuity of science. in the The jump series will consider a groupapproaches of related zero. This concepts article is about derived statistical from mechanics condensed and the matter development and of statistical mean physics. The key technicalfield theory as ideas a basis go for under a partial the understanding names of of phase “singularity”, transition phenomena. “order parameter”, “mean field the- Much of the material in this review was first prepared for the Royal Netherlands Acad- ory”,emy of “variational Arts and Sciences method”, in 2006. It“correlation has appeared in length”, draft form “universality on the authors’ class”, web site “scale changes”, and “renormalization”.(http://jfi.uchicago.edu/~leop/ The)sincethen. first four of these will be considered here. TheIn a title less of this technical article is a vein,hommage theto Philipquestion Anderson here and is his how essay can “More matter, is Different” ordinary matter, support (Sci. New Ser. 177(4047):393–396, 1972;N.-P.OngandR.Bhatt(eds.)MoreisDifferent: adiversityofforms.WeseethisdiversityeachtimeweobserveiceincontactwithliquidFifty Years of Condensed Matter Physics, Princeton Series in Physics, Princeton University water or see water vapor (steam) come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walk- L.P. Kadanoff (!) ingThe James on liquid Franck Institute, water The is University proverbially of Chicago, forbidden Chicago, USA to ordinary humans. These differences have beene-mail: [email protected] to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in some statistical variable. The discontinuous property is called the order parameter. Each phase transition has its own order parameter. The possible order parameters range over a tremendous variety of physical properties. These properties include the density of a liquid-gas transition, the magnetization in a ferromagnet, the size of aconnectedclusterinapercolationtransition,andacondensatewavefunctioninasuper- fluid or superconductor. A continuous transition occurs when the discontinuity in the jump approaches zero. This article is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of phase transition phenomena. Much of the material in this review was first prepared for the Royal Netherlands Acad- emy of Arts and Sciences in 2006. It has appeared in draft form on the authors’ web site (http://jfi.uchicago.edu/~leop/)sincethen. The title of this article is a hommage to Philip Anderson and his essay “More is Different” (Sci. New Ser. 177(4047):393–396, 1972;N.-P.OngandR.Bhatt(eds.)MoreisDifferent: Fifty Years of Condensed Matter Physics, Princeton Series in Physics, Princeton University

L.P. Kadanoff (!) The James Franck Institute, The University of Chicago, Chicago, USA e-mail: [email protected] 40 YEARS LATER …

Robert B. Laughlin Nobel prize 1986 The low-energy excited quantum states of these systems [crystalline solids] are particles in exactly the same sense that the electron in the vacuum of quantum electrodynamics is a particle … Yet they are not elementary, and, as in the case of sound, simply do not exist outside the context of the stable state of matter in which they live. 6 COMPLEXITY IN SOLIDS

0 1 2 3 4 100 100 100 100 100 multiplier for the number Hydrogen Niobium Gallium Uranium Cuprate of compounds Arsenide Alloys Perovskites e.g.UPd2Al3, UNi2Al3

Atomic physics Super conductivity BCS type Emerging complexity Skyrmions and Fractional Heavy electrons, Quantum spin-charge separation, Hall effect High temperature Kondo physics superconductivity delocalized p-electrons

d-electrons A masterplan for designing complex materials strongly localized f-electrons

8 PEROVSKITE COMPLEX OXIDES A =La, Ce, Pr …. Tm =Fe, Ru, Ir … e.g. CaTiO3. NdNiO3, SrMnO3 AMO3 perovskite unit cell

A = + Tm

eg O MO6 AO12 rigid soft 5 distinct atomic orbital shapes on M

9 Metals and Insulators. Mott insulators

Metallic behavior expected

Kinetic and Interaction Energy The Mott transition in the Hubbard model Fig: Pickett, RMP 61, 433 (1989) STRONG CORRELATIONS - MOTT INSULATORSIntra-orbital Atomic lattice with a Kineticsingle orbital energy per site and average occupancy 1 half filling V repulsion Atomic lattice with a single orbital per site and average occupancy 1 (half filling) odd

Electron counting loses coulomb energy, U metal

} Nevill Francis Mott

Hopping Double occupancy La2CuO4: 2 La (57x2)+Cu (29) + 4E O (4x8)=175 electrons saves energy t costs energy U Nobel prize 1977 even but LaHopping2CuO 4 is a strongDouble insulator occupancy Intra ! -orbital savesTight energy-binding t (hoppingcosts energy) U insulator gains kinetic energy, t repulsion E. Bascones [email protected] U >> t electrons localize: Mott insulator For U >> t electrons localize: Mott insulator

Small U/t Increasing U/t Large U/t

E. Bascones [email protected] Insulator Metal 10

Mott transition

E. Bascones [email protected] COMPLEXITY IN CORRELATED MATERIALS

local entanglement of lattice, charge, spin, and orbital degrees of freedom defines multiple closely spaced energy landscape with meta-stable ground states Spin

Coulomb repulsion Hund’s electron exchange crystal spin feld

orbital lattice elastic strain orbital bandwidth

Fig. J.M. Rondinelli and N.A. Spaldin, Adv. Mater. 11 (2011) Figure 1.1: Competing energy scales (listed around the circle) and available couplings (denoted by the linked edges of the tetrahedron) that give rise to the diverse functionalities like ferroelectricity, colossal magnetoresistance, and superconductivity in transition metal oxides. complete control of the correlated electronic phases for innovative technologies has yet to rival the precision found in the semiconductor industry. If such design and processing can be achieved, an oxides electronics industry [] could establish revolutionary technologies for communications, computing and information networks through smart transistors, optical switches [, ], and powerless logic components [, ]. e experimental identication that new functionalities can emerge at oxide- oxide heterostructures was borne from early advances in making well-dened single termination substrate surfaces for lattice matched growth [, , ]. e combination of high quality substrate materials with layer-by-layer thin lm growth processes, like pulsed laser deposition (PLD) [] and molecular beam epitaxy (MBE) [], along with reective high-energy electron diraction (RHEED) characterization and x-ray techniques at the atomic scale [, ] made it possible to avoid bulk thermodynamic and kinetic restrictions on material synthesis to stabilize new electronic and structural phases [, ]. For example, fabrication of epitaxial oxide heterostructures, superlattices and nanopillars which show remarkable behavior is now routine: the interface between two non-magnetic insulators LaAlO/SrTiO provides metallic conductivity [], shows magnetic behavior [], and is even reported to display two-dimensional superconductivity []. Preliminary attempts to control the two-dimensional electron gases at such interfaces shows that dynamic circuit design may be possible []. Basic materials understanding of the interface properties, however, is necessary to harness this technique for innovative design of new logic and processing architectures. Despite much of this experimental progress, many of these phenomena still remain to be understood theoretically at a microscopic level. Furthermore REVIEW Complexity in Strongly Correlated Electronic Systems Elbio Dagotto

ly accepted. A remarkable cross-fertilization A wide variety of experimental results and theoretical investigations in recent years between theory and experiments has led to have convincingly demonstrated that several transition metal oxides and other considerable progress in unraveling the role of materials have dominant states that are not spatially homogeneous. This occurs in these inhomogeneities. Theoretical investiga- cases in which several physical interactions—spin, charge, lattice, and/or orbital—are tions (4 )predictedthat,inabroadregionof simultaneously active. This phenomenon causes interesting effects, such as colossal parameter space, the ground state is actually a magnetoresistance, and it also appears crucial to understand the high-temperature nanoscale mixture of phases, particularly in the superconductors. The spontaneous emergence of electronic nanometer-scale structures presence of quenched disorder (10–12 ), name- in transition metal oxides, and the existence of many competing states, are properties ly, when random ‘‘frozen’’ deviations from the often associated with complex matter where nonlinearities dominate, such as soft perfectly uniform system are incorporated in materials and biological systems. This electronic complexity could have potential the study. Many experimental results are consequences for applications of correlated electronic materials, because not only indeed in agreement with the basic notion that charge (semiconducting electronic), or charge and spin (spintronics) are of relevance, the relevant phases are not homogeneous; but in addition the lattice and orbital degrees of freedom are active, leading to giant these results also provide information crucial responses to small perturbations. Moreover, several metallic and insulating phases to understanding the CMR effect (4 , 5 , 13, 14 ). compete, increasing the potential for novel behavior. Some of the general theoretical ideas are summarized in the schematic phase diagram aterials in which the electrons are What are the implications of these and other (Fig. 2A) (10), which has been experimentally strongly correlated display a broad results reviewed below? It will be argued that confirmed (15 , 16 )(Fig.2B). range of interesting phenomena, in- the current status of correlated electrons inves-R EVIEWIn the clean limit without quenched dis- M superconducting (SC) state competition in This suggests that when phases compete, the may be sufficient to do the job. Calculations cluding colossal magnetoresistance (CMR), tigations must be considered in the broader order, the two key competing states in man- cuprates (17 )andmanyothercases(18 ). effect of (typically small amounts of) quenched without explicit disorder incorporating strain where enormous variations in resistance are context of complexity. In his pioneering article Calculationsganites, that ferromagnetic incorporate the effects (FM) of metallicdisorder results and in dramatic properties that are effects (9 ), or within a phenomenological produced by small magnetic field changes, and (1), Anderson wrote that Bthe ability to reducephaseantiferromagnetic competition and quenched (AF) disorder insulating have very (AFI), different are from those of a slightly impure Ginzburg-Landau theory, also lead to in- high-temperature superconductivity (HTSC). everything to simple fundamental laws doesbeenknown able to to reproduce be separated the huge by magneto- a first-ordermaterial transition (10 , 11, 19 , 20 ). Disorder in the re- homogeneous patterns (23 ). Although the resistance observed experimentally (10 , 11); gime of phase competition is not a mere per- discussion on the details of the origin of the An important characteristic of these materials not imply the ability to start from those lawsthis suggests(4 , 5 ). that However, the CMR once effect the would inevitable not turbation; quenched it alters qualitatively the properties inhomogeneities is still fluid, their crucial is the existence of several competing states, as and reconstruct the universe.[ In complexoccurdisorder without iseither included competing in the states calculation, or of the arising, material. relevance to understanding the manganites, as exemplified by the complicated phase diagrams systems (2 ), the properties of a few particlesquenchedfor example, disorder and from interactions the lattice-distorting necessary How chem- strong should the disorder be to induce originally predicted by theory (4 , 5 ), is by now to nucleate clusters. This is in agreement with the inhomogeneous patterns discussed here? widely accepted. that transition metal oxides (TMOs) presentR EVIEWare not sufficient to understand large aggre- ical doping procedure, nonstatistical fluctua- R EVIEW experiments for Re0.5Ba0.5MnO3 (where Re is Are there other alternatives? Studies incorpo- Cuprates. In the HTSC context, the (Fig. 1). The understanding of these oxides has gates when these particles strongly interact.arareearthelement)(tions of dopant16 ), whichdensity can or be pre- strainrating fields, long-range the effects, such as Coulombic La Sr CuO (LSCO) phase diagram is charge checkerboard, or glassy patterns). All plexity briefly reviewed in the introduction compete and2 percolativejx x 4 physics, as when charge checkerboard,dramatically or challengedglassy patterns). our view All of solids.plexity In Rather, briefly in reviewed such systems, in which the introduction are not merelyparedcompeteregion both in ordered in which and and disordered percolative the two forms states for physics, areforces nearly (21 as)orcooperativeoxygenoctahedra de- when usually considered the universal diagram for these characteristics are hallmarks of complex the Re-Baas well distribution. as the Remarkably, oxide only results the discusseddistortions (11 in), suggest the thatonly very a weak narrowcuprates. channel However, exists some investigations for electrical sug- these characteristicsfact, after are one hallmarks of the largest of complex research effortsas wellcomplicated, as the oxide one results expects discussed emergence, in namely the latteronlygenerate was found a narrow (that to exhibit is, channel they CMR can (Fig. coexist) exists 2D). disorder, is for dramat- electrical even infinitesimal disorder (21, 22), gest otherwise (17 , 24 ). For example, only systems, showing sensitivity to details as they previous section, it is natural to wonder conductivity through a material, is at work. systems, showingever sensitivity in physics, involvingto details hundreds as they of scien-previousthe section, generation it of properties is natural that todo not wonder preexist conductivityically modified. through In this regime, a material, there is is still at a work. occur in nonlineartists, chaotic even basic systems. properties of the HTSCoccurwhether inin nonlinear these a system systems_ chaoticsconstituents.Thisconceptis can systems. be considered as Fig. 1.whetherlocalPhaseAnother tendency diagrams these argument toward systems either can can FM be or be AFI found considered short- in the as Another argument can be found in the cuprates, such as the pairing mechanism, linearSomecontrary additional to the issues philosophy should of reductionism, be remarked theof representativespecialdistance materi- correlations. cases of complex However, matter. globally nei- Although known properties of traditional soft condensed Some additional issues should be remarked special cases of complex matter. Although als ofknown the strongly properties cor- of traditional soft condensed resistivity, and pseudogap phase, are stillupon: only (i)traditional Although physics the most hallmark. complex Complex behavior systemsrelatedcomplexityther electron of the family two is states natural dominates when (Fig. associated 2C). A with matter, which is a phase of matter between a upon: (i) Althoughpoorly the understood. most complex In the early behavior days of HTSC,complexityspontaneously is natural tend when to form associated structures with (self-(notationsmatter,mixed are standard glassy which region is a isphase generated of matter between between the a in cuprates appears in the underdoped regime, insoft cuprates matter appears (literally in the soft, underdoped for example, regime, and detailssimplesoft can matterbe fluid found and (literally a regular soft,solid crystal. for example, In soft simple fluid and a regular solid crystal. In soft it was expected that suitably modified theoriesdynamicorganization), electronic inhomogeneity and these structures and vary compe- widelyin thepolymerstrue original critical refer- and temperatures, liquid crystals), the Curie orit seemsNe´el out matter, large groups of atoms form regular dynamic electronicof ordinary inhomogeneity metals would and explain compe- the unusualpolymersin andsize and liquid scales. crystals), Exceptional it seems events out areences).matter,temperatures (A)Temperature large in groups this case, of and atoms a remnant form of the regular tition among the many degrees of freedom versusof hole place density phase in the context of hard materials. But patterns as in a solid, but when several of these tition amongproperties the many of degreesthe cuprate of_snormalstate.How- freedom of placeimportant, in the context as when of hard the last materials. metallic But linkdiagrampatternsclean-limit of bilayer as man- transition,in a solid,T but*. In when this regime, several per- of these could also underlie the superconductivity even ganitesthe (74), several including simultaneously active degrees of large groups are considered together a complex could also underlieever, important the superconductivity experimental results even gatheredthe severalcompletes simultaneously a percolative active network. degrees The average of severallargeturbations types of groups antiferro- such are as considered small magnetic together fields a complex can in recent years have revealed an unexpectedat optimalbehavior doping. is of Are no the relevance inhomogeneities for this phenome- and magneticfreedomhave (AF) dramaticphases, may a conspireconsequences, to provide because a they soft elec- fluid behavior emerges. Typical examples are at optimal doping. Are the inhomogeneities and freedom may conspire to provide a soft elec- ferromagneticfluid behavior (FM) emerges. Typical examples are complexity atproperty the root of of oxides: the superconducting Many TMOs are inho-complexitytronic componentnon, andat the often root to only transition of a few the rare superconducting events metal dominate. oxide phase,polymers:troniconly and even need a component glob- to in align each the large randomly to molecule transition oriented there metal mag- is atomic oxide polymers: in each large molecule there is atomic mogeneous at the nanoscale (and sometimesphase, orEvidence are they is accumulating unrelated? The that discussion TMOs andally disorderedcompounds,netic momentsregion at soft of not preformed in the nanosize physical FM hardness regularity, but an ensemble of them has a phase, orR areEVIEW they unrelated? The discussion compounds, soft not in the physical hardness x 0regularity,0.75. (B)Generic but an ensemble of them has a continues. (ii) InteractionsatR evenEVIEW longer can length also generatescales). This in- explainscontinues.sense butrelated (ii) denoting Interactions materials the have existence can properties also generate of similar a mul- in- tophasevarietysenseclusters diagram for but ofHTSC.to render fluiddenoting the phases system the ( existence48 globally). This ferro- of variable a mul- variety of fluid phases (48 ). This variable superconductingwhy the (SC) early state theories competition based in onThis homoge- suggests that whenstandard phases complex compete, systems, the may and be sufficient several to results do theSG job. standsmagnetic. Calculations for spin glass. A concomitant percolation induces homogeneouscuprates patternssuperconducting (17 )andmanyothercases( (17 , (SC)29 , state44 competition18),). and the ineffectThis ofhomogeneoustiplicity (typically suggests small that of when amounts nearly phases patterns of) compete,degenerate quenched (17 the, without29may, conformations44 be explicit), sufficient and disorder to the do theincorporating(C)Phasediagramof job.behaviortiplicity Calculations strain ofis also nearly present degenerate in some conformations TMOs: in behavior is also present in some TMOs: in single layered ruthenates Calculationsneouscuprates systems (that17 )andmanyothercases( incorporate were notthe effects successful18 of). disorder andeffect raises results of (typically in dramaticmust small be properties amounts reexamined of) that quenched are in thiseffectswithout broader (9 ), explicit orframework. within disorder a incorporating phenomenologicalmetallicity strain in the compounds. The fragility of combination of theseCalculations interactions that incorporate with the quenched effects of disordercombinationof the results electronic in dramatic of these properties component interactions that are thateffects with can (9 ), quenched be or within eas- a(75 phenomenological, manganites,76of), evolving the electronic from several component experimental that can investiga- be eas- manganites, several experimental investiga- phase competitionhopes that and a quenched novel avenuedisorder have for progressvery different has from those of a slightly impure Ginzburg-Landau theory,asuperconducting(SC) alsothe lead state to in- shown in Fig. 2C implies that several phase competition and quenched disorder have very different from those of a slightly impure Ginzburg-Landau theory, also lead to in- disorder maybeen be ableopened. at to the reproduce heart the of huge the magneto- complexitymaterialdisorderily (10 , modified11, 19Nanostructures, may20 ). Disorder bybe at external in the the heart re-in perturbations. Manganiteshomogeneous of the complexity patterns Con- (23state).tions Althoughily atperturbationsx 0 modified2toanAF have the found besides by external evidence magnetic perturbations. fields for Jahn-Teller should Con- tions have found evidence for Jahn-Teller resistancebeen observed able to experimentally reproduce the huge(10 , 11 magneto-); gimematerial of phase (10 competition, 11, 19 , 20 is). not Disorder a mere inper- the re- discussionhomogeneoush on the patterns details of (23insulator the). Although origin at x of0 the0(x h in cuprates. resistance observed experimentally (10 , 11); gimeinventional of cuprates. phaseand competition soft Cuprates matter is not a mere is classical per- discussion [( / on) the0 details0], of theorderedventional origin of the small soft regions matter is(i.e., classical with a [( particular/2p) 0 0], ordered small regions (i.e., with a particular this suggests that the CMR effect would not turbation; it alters qualitatively the properties inhomogeneities2p is still fluid,controlsinduce their the bandwidth crucial dramatic changes, including pressure, this suggests that the CMR effectEMERGING would not turbation; it altersCOMPLEXITY qualitatively the properties inhomogeneities FROM is STRONG stillrather fluid,than theirR the crucialEVIEW carrier CORRELATIONS Anotheroccur unexpected without either property competing of the states cuprates or of the material.butAnother in theManganites. electronic unexpectedThe systems Mn property oxidesrelevance described calledof the to understanding manganites cuprates here theform manganites,butstrain, in of and the aslattice electric electronic distortions) fields ( systems4 , 5 ). in Moreover, the described state the above here form of lattice distortions) in the state above Departmentoccur without of Physics, either competing University states of Tennessee or of the (UT), material. relevance to understandingdensity). the manganites, Ruthenates as is the giantquenched proximity disorder effect. and interactions This phenomenon necessary Howisquantum strong the should giant(3–9 the effects proximity), disorder especially are be to important. induce effect. thoseoriginally This displaying phenomenon predicted the by CMR theoryare (4 believedthe,quantumdiscussion5 ), is FM toby be now clean ordering centered effects on are temperature Fig. important. 2, A to C, (4, is inde-5 ). As a the FM ordering temperature (4, 5 ). As a Knoxville,quenched disorder TN 37996–1200, and interactions USA. necessary Condensed MatterHow strong should the disorder be to induce originally predicted by theory (4 , 5 ), is by now to nucleateto nucleate clusters. clusters. This is This in agreement is in agreement with withthe inhomogeneousthe inhomogeneouseffect, patterns patternsComplexity are discussed an discussed important here? here? oxidewidelywidely in accepted.family Strongly accepted. in which Correlatedmetalspendent at least at of large the details of the competing states has a longexperiments story,Sciences but for Re recently Division,Ba MnO Oak it has(where Ridge been Re National is veryAre Laboratory, therehas otherTMOs a alternatives? long are story, soft Studies but in incorpo- the recently sense describeditCuprates. has beenIn above the very HTSCx,thusprovidingasystem, context,TMOs thethese are small soft in Jahn-Teller the sense described clusters, along above system, these small Jahn-Teller clusters, along experiments0.5 for Re0.50.5Ba0.53 MnO3 (where Re is Are there other alternatives? Studies incorpo- Cuprates. In the HTSC context, the arareearthelement)(Oak Ridge, TN16 37831–6393,), which can USA. be pre- rating long-range effects,the presence such as Coulombic of inhomogeneousLa Sr CuO states(LSCO) is wide- phasefamilyand of diagram oxides should where is be valid for the AFI versus carefully studiedarareearthelement)( by using atomically16 ), which can smooth be pre- ratingcarefullyas long-range already studiedeffects, proposed such by as in Coulombic using theElectronic HTSC atomicallyLa2j2xj contextxSrx xCuO4 Systems smooth4 (LSCO) (44). phasewithas diagram already the is magnetic proposed inclusters the HTSC present context in (the44). with the magnetic clusters present in the pared both in ordered and disordered forms for forces (21)orcooperativeoxygenoctahedra usually considered the universalcompetition diagram and for com- pared both in ordered and disordered forms for forces (21)orcooperativeoxygenoctahedra usuallyElbio considered Dagotto the universalplexity can diagramsame be studied for phase region, generate a collective same phase region, generate a collective the Re-Bathe distribution. Re-Ba distribution. Remarkably, Remarkably, only the only thedistortionsdistortions (11), (11 suggest), suggest that that very very weak weak cuprates.cuprates. However, However, some some investigationswith investigations less quenched sug- dis- latter waslatter found was to found exhibit to exhibit CMR CMR (Fig. 2D).(Fig. 2D).disorder,disorder, evenwww.sciencemag.org even infinitesimal infinitesimal disorder disorder (21 (, SCIENCE2122, ),22), gestgest otherwiseVOL otherwise 309 (17 (17, 8,2424 JULY).). For For 2005ly example, example, accepted.behavior A only remarkable cross-fertilization that is different from the257 be- A wide variety of experimental results and theoretical investigations in recent yearsorder than in Mn ox- behavior that is different from the be- ides.between (D)Phasediagram theory and experiments has led to have convincingly demonstrated that several transition metal oxides and other considerablehavior progress in unraveling of the the role system’s of individual parts, havior of the system’s individual parts, materials have dominant states that are not spatially homogeneous. This occurs inof Cothese oxides inhomogeneities. (77), with Theoretical investiga- Fig. 1. PhaseFig. 1. diagramsPhase diagrams d-wave SC gapcases in which several physical interactions—spin, charge, lattice, and/or orbital—areSC, charge-orderedtions (4 )predictedthat,inabroadregionofand (CO), in this temperaturecheckerboard range colossal and in this temperature range colossal of representativeof representative materi- materi- simultaneously active. This phenomenon causes interesting effects, such as colossalandparameter magnetic space, regimes. the ground state is actually a als of the strongly cor- magnetoresistance, and it also appears crucial to understand the high-temperature nanoscale mixture of phases, particularly in the als of the strongly cor- superconductors. The spontaneous emergence of electronic nanometer-scale structures(E)Phasediagramofmagnetoresistance occurs (4, 5 ). Also magnetoresistance occurs (4, 5 ). Also related electron family distribution presence of quenched disorder (10–12 ), name- related electron family in transition metal oxides, and the existence of many competing states, are propertiesthely, organic when randomk-(BEDT- ‘‘frozen’’ deviations from the charge order (notations(notations are standard are standard often associated with complex matter where nonlinearities dominate, such as softTTF) Cu[N(CN)cuprates]Cl salt may behave as electron liquid cuprates may behave as electron liquid and details can be found 2perfectly uniform2 system are incorporated in and details can be found materials and biological systems. This electronic complexity could have potential(57).the The study. hatched Many experimental re- results are in the original refer- consequences for applications of correlated electronic materials, because not only indeed in agreementcrystals, with the basicintermediate notion that between electron crystals, intermediate between electron in the original refer- charge (semiconducting electronic), or charge and spin (spintronics) are of relevance,gion denotes the co- ences). (A)Temperature existencethe relevant of metal phases and are not homogeneous; ences). (Aversus)Temperature hole density phase but in addition the lattice and orbital degrees of freedom are active, leading to giant these resultsliquid also provide and information electron crucial crystal (44). Softness liquid and electron crystal (44). Softness responses to small perturbations. Moreover, several metallic and insulating phasesinsulator phases. (F) versus holediagram density of phase bilayer man- to understanding the CMR effect (4 , 5 , 13, 14 ). compete, increasing the potential for novel behavior. Schematic phasein the dia- manganite context has also been diagram ofganites bilayer (74 man-), including Some of the general theoretical ideas are in the manganite context has also been ganites (74), including gramsummarized of the Ce-basedin the schematic phase diagram several types of antiferro- aterials in which the electrons are What are the implications of these and other (Fig. 2A) (recently10), which has been discussed experimentally (23 ). Once these con- several typesmagnetic of antiferro- (AF) phases, a heavy fermion materials recently discussed (23 ). Once these con- strongly correlated display a broad results reviewed below? It will be argued that(51).confirmed (15 , 16 )(Fig.2B). magneticferromagnetic (AF) phases, a (FM) M range of interesting phenomena, in- the current status of correlated electrons inves- In thecepts clean limit are without accepted, quenched dis- then the long history cepts are accepted, then the long history ferromagneticphase, and (FM) even a glob- cluding colossal magnetoresistance (CMR), tigations must be considered in the broader order, the two key competing states in man- phase, andally even disordered a glob- region at where enormous variations in resistance are context of complexity. In his pioneering article ganites, ferromagneticof soft-matter (FM) metallic and investigations suggests of soft-matter investigations suggests x 0 0.75. (B)Generic produced by small magnetic field changes, and (1), Anderson wrote that Bthe ability to reduce antiferromagnetic (AF) insulating (AFI), are ally disordered region at high-temperature superconductivity (HTSC). everything to simple fundamental laws does known to bethat separated it by is a first-order natural transition to expect new kinds of x 0 0.75.phase (B)Generic diagram for HTSC. An important characteristic of these materials not imply the ability to start from those laws (4 , 5 ). However, once the inevitable quenched that it is natural to expect new kinds of Fig. 3. Examplesphase diagramSG of stands for inhomogeneous HTSC. for spin glass. T. Hanaguri states et in al., HTSC Nature materials. 430,is the existence1001 of ((2004) severalA) Schematic competing states, as perfectand reconstruct stripes the universe. (35[ In) complex (circlesdisorder isorganized included in the calculation, behavior. arising, In complex systems, SG stands(C for)Phasediagramof spin glass. Fig.exemplified 3. Examples by the complicated of phase inhomogeneous diagrams systems (2 ), the properties states of a few in particles HTSCfor materials. example, from the lattice-distorting (A) Schematic chem-M. Vershinin perfect et al., stripes Science (35 303,) (circles 1995 organized behavior. In complex systems, are holes;( arrows,C)Phasediagramofsingle spins). layered ruthenates Real systems may present morethat transition dynamical metal oxides stripes (TMOs) present (29are). not (B sufficient) d-wave to understand SC large gap aggre- real-ical dopingrandomness procedure, nonstatistical and fluctua- determinism are simul- Colossal magnetoresistance(Fig. 1). The (CMR) understanding of these oxides has gatesHigh when theseTc particlessuperconducting strongly interact. tions ofcuprates dopant density or strainOrganic fields, the(2004) 1D salts randomness and determinism are simul- single layered(75, 76 ruthenates), evolving from are holes; arrows, spins). Real systems may present258 more dynamical stripes (298 JULY). (B 2005) d-wave VOL 309 SC gapSCIENCE real- www.sciencemag.org space distributionasuperconducting(SC) obtained by using STM techniquesdramatically (37 challenged). Inhomogeneities our view of solids. In Rather, at in such the systems, nanoscale which are not merely areregion in whichtaneously the two states are relevant, nearly de- and these are ideas (75, 76), evolving from fact, after one of the largest research efforts complicated,e.g. YBa one expects2Cu emergence,3O7-x namely generate (that is, they can coexist)e.g. is dramat- k-(BEDT- TTF) Cu[N(CN) ]Cl 0 manganites space distribution obtained by using STM techniques (37). Inhomogeneities at the2 nanoscale2 are taneously relevant, and these are ideas observedasuperconducting(SC) (patches).state at x The2toanAF entire frame is 560 A˚ by 560ever in physics, A˚.( involvingC) Recently hundreds of scien- unveiledthe generationcharge-order of properties that do not preexist stateically modified. In this regime, there is still a insulator at x 0 0(x ˚ compatible˚ with recent correlated elec- state at x 0 2toanAF observedtists, even basic (patches). properties of the The HTSC entirein a system frame_sconstituents.Thisconceptis is 560 A bylocal tendency 560 toward A.( eitherC) FM Recently or AFI short- unveiled charge-order state compatible with recent correlated elec- (checkerboard)controls in Na-doped the bandwidth cuprates (40, 41). cuprates, such as the pairing mechanism, linear contrary to the philosophy of reductionism, the distance correlations.trons investigationsHowever, globally nei- (10 , 11, 15 , 19 , 20 ). insulatorrather at x than0 0( thex carrier (checkerboard)resistivity, and pseudogap inphase, Na-doped are still only traditional cuprates physics hallmark. (40 Complex, 41 systems). ther of the two states dominates (Fig. 2C). A trons investigations (10 , 11, 15 , 19 , 20 ). controlsdensity). the bandwidth Ruthenates poorlyExciting understood. In the early physics days of HTSC, spontaneously but tendhard to form structuresto realize (self- mixedEach glassy in region complexbulk is generated betweenmaterials situation the in correlated elec- rather than the carrier it was expected that suitably modified theories organization), and these structures vary widely true critical temperatures, the Curie or Ne´el 12 are believed to be clean of ordinary metals would explain the unusual in size and scales. Exceptional events are temperatures in this case, and a remnant of the Each complex situation in correlated elec- films madedensity). ofmetals HTSC Ruthenates at least compounds at large in S-N-S Theyproperties are of the also cuprate complex,_snormalstate.How- becauseimportant, several as when the lasteffects metallic link tronsclean-limit may transition, leadT*. In this to regime, a unique per- state. Some materials are believedx,thusprovidinga to be clean filmsever, important made experimental of HTSC results gathered compoundscompletes a percolative in network. S-N-S The average turbationsThey such are as small also magnetic complex, fields can because several effects trons may lead to a unique state. Some materials trilayer junctionsmetals atfamily least (S isof at oxides forlarge superconductor where and N becomein recent years simultaneously have revealed an unexpected importantbehavior is of no relevance and for this pre- phenome- mayhave dramatic have consequences, stripes, because others they may have patches, trilayerproperty junctionsof oxides: Many TMOs (S is are for inho- superconductornon, and often only a few rare events and dominate. N onlybecome need to align the simultaneously randomly oriented mag- important and pre- may have stripes, others may have patches, equals normalx,thusprovidingacompetition metal) ( and45 ). com- The big paradox vent a simple physical description. More specif- some may have phase separation at nanoscales, family ofplexity oxides can where be studied mogeneous at the nanoscale (and sometimes Evidence is accumulating that TMOs and netic moments of preformed nanosize FM equalsat even longer normal length scales). metal) This explains (45 ).related The materials big have properties paradox similar to clustersvent to a render simple the system physical globally ferro- description. More specif- some may have phase separation at nanoscales, here is thatcompetition thewith trilayer andless quenched com- behaves dis- as a Josephson ically,why the consider early theories based one on ofhomoge- thestandard popular complex systems,definitions and several results andmagnetic. others A concomitant may percolation have induces mesoscale phase separa- plexity canorder be than studied in Mn ox- hereneous is systems that were the not successful trilayer and raises behavesmust be reexamined as a in Josephson this broader framework. metallicityically, in the consider compounds. The one fragility of of the popular definitions and others may have mesoscale phase separa- junction forwith N less barriersides. quenched (D)Phasediagram dis- a hundred times thicker ofhopes complexity that a novel avenue forrecently progress has discussed in (47 ): tion;the state shown the in Fig. number 2C implies that of several states in competition and Nanostructures in Manganites order thanof Co in oxides Mn ox- (77), with junctionopened. for N barriers a hundred times thicker perturbationsof complexity besides magnetic fields recently should discussed in (47 ): tion; the number of states in competition and that the coherenceSC, charge-ordered length, (CO),x.Then,thenormal ‘‘I randomness and determinismand Cuprates are both theirinduce dramatic nature changes, can including lead pressure, to enormous possibilities. ides. (D)Phasediagram Manganites. The Mn oxides called manganites strain, and electric fields (4 , 5 ). Moreover, the and magnetic regimes. Department of Physics, University of Tennessee (UT), of Co oxides (77), with that the coherence length, (x3–9.Then,thenormal), especially those displaying the CMR discussion‘‘I centeredrandomness on Fig. 2, A to C, is andinde- determinism are both their nature can lead to enormous possibilities. state cannot be(E)Phasediagramof featureless, it must already relevantKnoxville, TN 37996–1200, to the USA. Condensed system’s Matter overall behavior. This is exciting for applications but frustrating SC, charge-ordered (CO), stateSciences cannot Division, Oak Ridge be National featureless, Laboratory, effect, are it an importantmust oxide already family in which pendentrelevant of the details to of the thecompeting system’s states overall behavior. This is exciting for applications but frustrating contain a tendencythe organic towardk-(BEDT- superconductivity, SuchOak Ridge, [complex] TN 37831–6393, USA. systems existthe presence on of inhomogeneous the edge states of is wide- forand should those be valid with for a the reductionist AFI versus soul. What is likely and magneticTTF) Cu[N(CN) regimes.]Cl salt (E)Phasediagramof2 2 contain a tendency toward superconductivity, Such [complex] systems exist on the edge of for those with a reductionist soul. What is likely which could be(57 in). The the hatched form re- of nanoscale SC chaos—they may exhibitwww.sciencemag.org almost SCIENCE regular VOL 309 be- 8 JULY 2005is that new general concepts257 and paradigms will the organicgion denotesk-(BEDT- the co- which could be in the form of nanoscale SC chaos—they may exhibit almost regular be- is that new general concepts and paradigms will islands (17TTF))orphase-fluctuatinghomogeneous2Cu[N(CN)existence2]Cl of salt metal and havior, but also can change dramatically and emerge as guiding qualitative principles in (57). Theinsulator hatched phases. re- (F) islands (17 )orphase-fluctuatinghomogeneous havior, but also can change dramatically and emerge as guiding qualitative principles in states (46 ).gion This denotesSchematic proposal the phase co- leads dia- to an exciting stochastically in time and/or space as a result the study of complex oxides. It will be dif- existencegram of metal of the and Ce-based states (46 ). This proposal leads to an exciting stochastically in time and/or space as a result the study of complex oxides. It will be dif- prediction:insulator Underheavy phases. the fermion proper (F materials) perturbation, the of small changes in conditions.’’ This defini- ficult to predict the precise shape of the nano- state withSchematic preformed(51). phase dia- SC clusters should prediction:tion is satisfied Under by the manganites, proper perturbation, in which the a patternsof small and changes the phases in conditions.’’ in competition This unless defini- ficult to predict the precise shape of the nano- gram of the Ce-based present aheavy gigantic fermion susceptibility materials toward super- statesmall magnetic with preformed field produces SC clusters a drastic change should detailedtion is satisfiedcalculations by are manganites, performed. in But which the a patterns and the phases in competition unless conductivity(51). (17 ). This is the analog of CMR presentin transport a gigantic properties, susceptibility and it may toward apply super- to existencesmall magnetic of some field patterns, produces as a well drastic as change giant detailed calculations are performed. But the in Mn oxides but translated into Cu-oxide conductivityunderdoped cuprates (17 ). This as is well the ( analog17 , 45 ). of When CMR responsesin transport to selected properties, external and it perturbations, may apply to existence of some patterns, as well as giant language. In general, theory predicts that giant inphases Mn compete, oxides but general translated arguments into Cu-oxide suggest willunderdoped be predictable. cuprates Certainly as well the (17 highest, 45 ). When de- responses to selected external perturbations, 258 8 JULY 2005 VOL 309 SCIENCE www.sciencemag.org responses to external perturbations should be language.that large In responses general, theory to weak predicts perturbations that giant greephases of complexity compete, general is expected arguments when suggest many will be predictable. Certainly the highest de- far more common than previously anticipated. responsesshould be to far external more common perturbations than previously should be degreesthat large of freedom responses are toactive weak simultaneously perturbations gree of complexity is expected when many 258 8 JULY 2005 VOL 309 SCIENCE www.sciencemag.org farbelieved more ( common4, 5 , 17 ). thanAlthough previously the basic anticipated. rules for andshould when be many far more phases common with different than previously proper- degrees of freedom are active simultaneously The Case for Complexity in electrons (i.e., the Hamiltonians) are decep- tiesbelieved are in (4 competition., 5 , 17 ). Although the basic rules for and when many phases with different proper- Correlated Electron Systems Thetively Case simple for (nearest-neighbor Complexity in carrier hopping, electronsTheory, (i.e., phenomenology, the Hamiltonians) computer are simu- decep- ties are in competition. Are TMOs examples of complex matter? CorrelatedCoulomb or Electronphononic interactions, Systems etc.), the lations.tively simpleHow can (nearest-neighbor we make further carrier progress hopping, in Theory, phenomenology, computer simu- Considering the general properties of com- Areoutcomes TMOs are examples highly nontrivial of complex when matter? phases thisCoulomb context? or phononic Investigations interactions, involving etc.), the the lations. How can we make further progress in Considering the general properties of com- outcomes are highly nontrivial when phases this context? Investigations involving the 260 8 JULY 2005 VOL 309 SCIENCE www.sciencemag.org 260 8 JULY 2005 VOL 309 SCIENCE www.sciencemag.org Broken Symmetry and Emerging Phases Exploring the Unit materials which deform under stress, like polymers, liquids, colloids, and granular materials. The written text for this unit focuses on solid state physics, whereas the video touches on both solid state and soft condensed matter. In today’s session, we will focus on emergence in condensed matter physics: both solid-state and soft condensed matter. By some accounts, all of condensed matter physics can be considered emergent.

Here is a definition of emergence from the National Academies: 3

Emergence relates to several other key concepts:

Complex systems are systems with many interacting parts. Thus, emergence is essentially the description of the observable phenomena that arise from complex systems. Self-organization is an often-studied feature of complex systems. Complex adaptive systems are a subset of complex systems in which feedback is critical.

Phase transitions are the transition from one phase of matter to another. This can SIGNS be a familiar OF EMERGENT state of matter BEHAVIOR (such as solid and liquid) or other types of phases (such as paramagnetic and diamagnetic). When a material changes phase sharply with the varying of some external parameter (as when ice freezes at a clearly defined temperature and pressure), this is called a first order phase transition. The properties of materials on either side of a phase transition are emergent. Phase transitions are a key sign of emergent behavior.

Phase transitions usually involve symmetry breaking—a change in the symmetry of the constituents. For example, the molecules in a liquid are randomly arranged, and thus very symmetric. The molecules in an ice crystal are very orderly. Since disorder is more symmetric than order (you can look at a liquid Condensedfrom any Matter angle and Materials and it Physics: will look The theScience same), of the World this Around is an Us, example National ofAcademies symmetry Press (2007). breaking. Since symmetry breaking would not be predicted from fundamental laws, it is seen as a key aspect of systems that cannot be explained through reductionist means.

At a critical point, there is no clear boundary between two phases—the transition between two phases is continuous (a second-order phase transition). For

3 Condensed Matter and Materials Physics: The Science of the World Around Us, National 14 Academies Press (2007).

Unit 8 -167- Physics for the 21st Century SYMMETRY IN SPACE - INVERSION, ROTATION, TRANSLATION Most fundamental idea - central to many modern sciences

In antiquity synonymous with harmony and beauty / inversion symmetry, translational symmetry /

Symmetry in art / rotational symmetry/

Symmetry in biology

images from google.com 15 SYMMETRY IN TIME OR TIME INVERSION

A canon "Quaerendo invenietis" based on retrogression

J.S. Bach, The Musical Offering, BWV 1079 16 BROKEN SYMMETRY

Left vs. Right

Arrow of time - past vs. future

images from google.com 17 CHIRALITY OR BROKEN SYMMETRY OF LIFE molecules of life are chiral ! sugars DNA

левый и правый кристаллы винной кислоты “Я не мог себе представить, каким образом tartaricПрирода могла создавать acid правое in old соединение wine не создавая одновременно левого, так как те же самые силы, действующие в момент создания молекулы правой винной кислоты должны были бы, мне казалось, привести и к образованию левой молекулы. Почему лишь только правые или левые изомеры? <...>

Несомненно, имеются причины, обуславливающие эти удивительные проявления молекулярных сил. Без сомнения, точное определение их представляет большие трудности. <...> Но я считаю необходимым сделать вывод о наличии диссимметрических сил в момент образования органических Louis Pasteur, 1822-1895 натуральных соединений.“ Луи Пастер , 1860г. LouisFrance Pasteur 1822-1895

images from google.com c.1850 18 BROKEN SYMMETRY = EMERGENCE 0F A NEW PHASE

iron Solid Argon melting Tc

Solid Ar Ar

Liquid Ar gas Decreasing Temperature or Energy 0 =0 ≠ A tri-critical point of Ar

19 images from google.com (BROKEN) SYMMETRY AND ORDER PARAMETER

liquid metal magnetic insulator symmetry higher disordered phase disordered

critical temperature Tc Temperature or Energy or Temperature

symmetry ordered phase ordered lower

solid insulator superconductor Broken symmetry means the appearance of an ordered phase with a non-zero order parameter 20 WHERE ARE THE NEW PHASES OF MATTER? Landau “recipes” for getting new phases and states:

The sudden disappearance of a certain symmetry in the high-symmetry phase leads to the occurrence of a phase transition into a new phase Lev Landau with lower symmetry. Nobel prize 1962

If we assume control over symmetry breaking, we may end up with new designer phases 21