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Interacting Dirac Matter Interacting Dirac Matter SAIKAT BANERJEE Doctoral Thesis Stockholm, Sweden 2018 TRITA-SCI-FOU 2018:22 ISSN xxxx-xxxx KTH School of Engineering Sciences ISRN KTH/xxx/xx--yy/nn--SE SE-100 44 Stockholm ISBN 978-91-7729-821-2 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av Doktorsexamen i teoretisk fysik Onsdag den 13 Juni 2018 kl 13.00 i Huvudbyggnaden, våningsplan 3, AlbaNova, Kungl Tekniska Högskolan, Roslagstullsbacken 21, Stockholm, FA-32, A3: 1077. © Saikat Banerjee, June 2018 Tryck: Universitetsservice US AB iii Sammanfattning Kondenserade materiens fysik är den mikroskopiska läran om materia i kon- denserad form, såsom fasta och flytande material. Egenskaperna hos fasta material uppstår som en följd av samverkan av ett mycket stort antal atomkärnor och elek- troner. Utgångspunkten för modellering från första-principer av kondenserad mate- ria är Schrödingerekvationen. Denna kvantmekaniska ekvation beskriver dynamik och egentillstånd för atomkärnor och elektroner. Då antalet partiklar i en solid är mycket stort, av storleksordningen 1026, är en direkt behandling av mångpartikel- problemet ej möjlig. Kännedom om enskilda elektroners tillstånd har vanligen liten relevans då det är det kollektiva beteendet av elektroner och atomkärnor som ger upphov till en solids makroskopiska och termodynamiska egenskaper. Det kanske enklaste sättet att beskriva elektrontillstånd i en solid utgår från att vi kan beakta N antal icke växelverkande elektroner i en volym V . För en enskild elektron i en potential följer av Schrödingerekvationen att ett antal diskreta energitillstånd finns tillgängliga medan det för det termodynamiskt stora antalet elektroner i en solid finns ett kontinuum av energitillstånd i form av bandstruktur. Centrala begrepp är Ferminivån vilken är den energinivå till vilken elektrontillstånd finns ockuperade, samt Fermiytan vilken är den sfär i reciproka rymden inom vilken de ockuperade tillstånden befinner sig. I ett riktigt material växelverkar elektronerna och teorin för icke växelverkande partiklar behöver kompletteras. Ett viktigt steg i den riktningen togs av Lev Landau som formulerade teorin för Fermivätskor. Enligt denna teori kan lågen- ergiexcitationerna hos ett oändligt stort antal växelverkande fermioner fortfarande beskrivas av ett oändligt stort antal icke växelverkande fermioner. Dessa icke växelverkande partiklarna skiljer sig något från de ursprungliga elektronerna och benämns därför kvasipartiklar, ett koncept som äger stor tillämpbarhet för model- lering och beskrivning av flertalet vanliga metaller. Fermiytan för ett d-dimensionellt material är en d − 1-dimensionell mångfald vars topologi utgör en begränsning för kvasipartiklarnas spektrum, vilket för många solider kan beskrivas av en Schrödingerekvation för en elektron med effektiv massa, avvikande från elektronmassan för en fri elektron, i en effektiv potential. År 2004 gjorde Andre Geim och Konstantin Novoselov upptäckten att det går att framställa mycket tunna filmer av kol på ett förbluffande enkelt sätt. En bit vanlig kontorstejp fästs på kolmaterialet grafit, och då tejpen avlägsnas exfolieras ett monoatomärt lager med kol. Detta material kom att benämnas grafen och besitter en rad intressanta elektriska och mekaniska egenskaper, däribland kvasi- partiklar med linjär dispersion. Till skillnad från de kvasipartiklar som beskrivs av Schrödingerekvationen kan den linjära dispersionen modelleras som lösningen till en Diracekvation i vilken ljushastigheten i detta sammanhang ersätts med Fermi- hastigheten som i grafen antar värden av storleksordning 106 meter per sekund. Den karakteristiska egenskapen hos ett Diracmaterial är att noderna hos dess spektrum skyddas av olika slags symmetri, till exempel för grafen i form av tidsreversions- och inversionssymmetri. iv För fermioner gäller Fermi-Dirac-statistik och Pauli-principen som uttrycker att ett enskilt fermiontillstånd endast kan vara ockuperat av noll eller en fermion, till skillnad från bosoner vars tillstånd kan besättas av i princip godtyckligt många partiklar. Ett extremfall är så kallade Bose-Einstein-kondensat i vilket en my- cket stor mängd bosoner kondenserar till ett och samma kvanttillstånd. I en en- partikelbeskrivning av Diracliknande kvasipartiklar är statistiken av underordnad betydelse och vi kan därmed förvänta oss att det utöver fermioniska Dirackvasipar- tiklar även kan finnas bosoniska Dirackvasipartiklar. Temat för denna avhandling är att studera egenskaper hos fermioniska och bosoniska kvasipartiklar med Diracdispersion. Jag har studerat hur bosoniska kvasipartiklar kan förekomma i en bikakestruktur av supraledande öar, samt i den ferromagnetiska isolatorn CrBr3. För det senare materialet har jag analyserat hur yttillstånd kan uppstå i ett bosoniskt Diracmaterial, och hur dessa tillstånd skiljer sig från bulktillstånden. Avvikelser från effektiv enpartikelbeskrivning är temat för mina studier på växelverkande Diracmaterial, projekt inom vilka jag har under- sökt hur Coulomb-, Hubbard- och Heisenbergväxelverkan samt elektromagnetisk dipolväxelverkan påverkar spektra och tillståndstätheter hos Diracmaterial. I take this opportunity to dedicate my thesis to my mother and father. I would not have come this far in my life without their continuous support throughout my life. vi Abstract The discovery of graphene in 2004 has led to a surge of activities focused on the theoretical and experimental studies of materials hosting linearly dis- persive quasiparticles during the last decade. Rapid expansion in the list of materials having similar properties to graphene has led to the emergence of a new class of materials known as the Dirac materials. The low energy quasi- particles in this class of materials are described by a Dirac-like equation in contrast to the Schrödinger equation which governs the low energy dynamics in any conventional materials such as metals. The Dirac fermions, as we call these low-energy quasiparticles, in a wide range of materials ranging from the d-wave superconductors, graphene to the surface states of topological insula- tors share the common property. The particles move around as if they have lost their mass. This feature results in a completely new set of physical ef- fects consisting of various transport and thermodynamic quantities, that are absent in conventional metals. This thesis is devoted to studying the properties of bosonic analogs of the commonly known Dirac materials [1, 2] where the quasiparticle are fermionic. In chapter one, we discuss the microscopic origin of the Dirac equation in several fermionic and bosonic systems. We observe identical features of the Dirac materials with quasiparticles of either statistics when the interparticle interaction is absent. Dirac materials with both types of quasiparticles possess the nodal excitations that are described by an effective Dirac-like equation. The possible physical effects due to the linear dispersions in fermionic and bosonic Dirac materials are also outlined. In chapter two, we propose a system of superconducting grains arranged in honeycomb lattice as a realization for Bosonic Dirac Materials (BDM). The underlying microscopic dynamics, which give rise to the emergence of Dirac structure in the spectrum of the collective phase oscillations, is discussed in detail. Similarities and differences of BDM systems to the conventional Dirac materials with fermionic quasiparticles are also mentioned. Chapter three is dedicated to the detailed analysis of the interaction ef- fects on the stability and renormalization of the conical Dirac band structure. We find that the type of interaction dictates the possible fate of renormal- ized Dirac cone in both fermionic and bosonic Dirac materials. We study interaction effects in four different individual systems : (a) Dirac fermions in graphene interacting via Coulomb interactions, (b) Dirac fermions subjected to an onsite Hubbard repulsion, (c) Coulomb repulsion in charged Cooper pairs in honeycomb lattice and (d) Dirac magnons interacting via Heisenberg exchange interaction. The possibility of interaction induced gap opening at the Dirac nodal point described is also discussed in these cases. Chapter four mainly concerns the study of a related topic of the synthetic gauge fields. We discuss the possibility of Landau quantization in neutral particles. Possible experimental evidence in toroidal cold atomic traps is also mentioned. A connection to Landau levels in case of magnons is also described. We finally conclude our thesis in chapter five and discuss the possible future directions that can be taken as an extension for our works in interacting Dirac materials. vii Preface The work presented in this thesis was carried out at the group of Condensed Matter Physics in Nordic Institute for Theoretical Physics (Nordita) and Royal Institute of Technology (KTH), Stockholm, Sweden. I take this opportunity to express my sincere gratitude to the academic members in the Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, New Mexico, USA where part of my thesis work was performed. I also am grateful to the academic members at the University of Connecticut, Storrs, USA for their generous cooperation and help during my visit. List of papers included in the thesis Paper I S. Banerjee, J. Fransson, A.M. Black-Schaffer, H. Ågren, and A. V. Balatsky; Granular superconductor in a honeycomb lattice as a realization of bosonic Dirac material, Phys. Rev. B 93, 134502, 2016 Paper II S. S.
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