Quantum energy inequalities in two dimensions

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The_electronic_properties_of_graphene

The electronic properties of grapheneA.H.Castro NetoDepartment of Physics,Boston University,590Commonwealth Avenue,Boston,Massachusetts02215,USAF.GuineaInstituto de Ciencia de Materiales de Madrid,CSIC,Cantoblanco,E-28049Madrid,SpainN.M.R.PeresCenter of Physics and Department of Physics,Universidade do Minho,P-4710-057,Braga,PortugalK.S.Novoselov and A.K.GeimDepartment of Physics and Astronomy,University of Manchester,Manchester,M139PL,United Kingdom͑Published14January2009͒This article reviews the basic theoretical aspects of graphene,a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations.The Dirac electrons can be controlled by application of external electric and magneticﬁelds,or by altering sample geometry and/or topology.The Dirac electrons behave in unusual ways in tunneling,conﬁnement,and the integer quantum Hall effect.The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers.Edge͑surface͒states in graphene depend on the edge termination͑zigzag or armchair͒and affect the physical properties of nanoribbons.Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties.The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.DOI:10.1103/RevModPhys.81.109PACS number͑s͒:81.05.Uw,73.20.Ϫr,03.65.Pm,82.45.MpCONTENTSI.Introduction110II.Elementary Electronic Properties of Graphene112A.Single layer:Tight-binding approach1121.Cyclotron mass1132.Density of states114B.Dirac fermions1141.Chiral tunneling and Klein paradox1152.Conﬁnement and Zitterbewegung117C.Bilayer graphene:Tight-binding approach118D.Epitaxial graphene119E.Graphene stacks1201.Electronic structure of bulk graphite121F.Surface states in graphene122G.Surface states in graphene stacks124H.The spectrum of graphene nanoribbons1241.Zigzag nanoribbons1252.Armchair nanoribbons126I.Dirac fermions in a magneticﬁeld126J.The anomalous integer quantum Hall effect128 K.Tight-binding model in a magneticﬁeld128 ndau levels in graphene stacks130 M.Diamagnetism130 N.Spin-orbit coupling131 III.Flexural Phonons,Elasticity,and Crumpling132 IV.Disorder in Graphene134A.Ripples135B.Topological lattice defects136C.Impurity states137D.Localized states near edges,cracks,and voids137E.Self-doping138F.Vector potential and gaugeﬁeld disorder1391.Gaugeﬁeld induced by curvature1402.Elastic strain1403.Random gaugeﬁelds141G.Coupling to magnetic impurities141H.Weak and strong localization142I.Transport near the Dirac point143J.Boltzmann equation description of dc transport indoped graphene144 K.Magnetotransport and universal conductivity1451.The full self-consistent Born approximation͑FSBA͒146 V.Many-Body Effects148A.Electron-phonon interactions148B.Electron-electron interactions1501.Screening in graphene stacks152C.Short-range interactions1521.Bilayer graphene:Exchange1532.Bilayer graphene:Short-range interactions154D.Interactions in high magneticﬁelds154VI.Conclusions154 Acknowledgments155 References155REVIEWS OF MODERN PHYSICS,VOLUME81,JANUARY–MARCH20090034-6861/2009/81͑1͒/109͑54͒©2009The American Physical Society109I.INTRODUCTIONCarbon is the materia prima for life and the basis of all organic chemistry.Because of the ﬂexibility of its bond-ing,carbon-based systems show an unlimited number of different structures with an equally large variety of physical properties.These physical properties are,in great part,the result of the dimensionality of these structures.Among systems with only carbon atoms,graphene—a two-dimensional ͑2D ͒allotrope of carbon—plays an important role since it is the basis for the understanding of the electronic properties in other allotropes.Graphene is made out of carbon atoms ar-ranged on a honeycomb structure made out of hexagons ͑see Fig.1͒,and can be thought of as composed of ben-zene rings stripped out from their hydrogen atoms ͑Pauling,1972͒.Fullerenes ͑Andreoni,2000͒are mol-ecules where carbon atoms are arranged spherically,and hence,from the physical point of view,are zero-dimensional objects with discrete energy states.Fullerenes can be obtained from graphene with the in-troduction of pentagons ͑that create positive curvature defects ͒,and hence,fullerenes can be thought as wrapped-up graphene.Carbon nanotubes ͑Saito et al.,1998;Charlier et al.,2007͒are obtained by rolling graphene along a given direction and reconnecting the carbon bonds.Hence carbon nanotubes have only hexa-gons and can be thought of as one-dimensional ͑1D ͒ob-jects.Graphite,a three dimensional ͑3D ͒allotrope of carbon,became widely known after the invention of the pencil in 1564͑Petroski,1989͒,and its usefulness as an instrument for writing comes from the fact that graphite is made out of stacks of graphene layers that are weakly coupled by van der Waals forces.Hence,when one presses a pencil against a sheet of paper,one is actually producing graphene stacks and,somewhere among them,there could be individual graphene layers.Al-though graphene is the mother for all these different allotropes and has been presumably produced every time someone writes with a pencil,it was only isolated 440years after its invention ͑Novoselov et al.,2004͒.The reason is that,ﬁrst,no one actually expected graphene to exist in the free state and,second,even with the ben-eﬁt of hindsight,no experimental tools existed to search for one-atom-thick ﬂakes among the pencil debris cov-ering macroscopic areas ͑Geim and MacDonald,2007͒.Graphene was eventually spotted due to the subtle op-tical effect it creates on top of a chosen SiO 2substrate ͑Novoselov et al.,2004͒that allows its observation with an ordinary optical microscope ͑Abergel et al.,2007;Blake et al.,2007;Casiraghi et al.,2007͒.Hence,graphene is relatively straightforward to make,but not so easy to ﬁnd.The structural ﬂexibility of graphene is reﬂected in its electronic properties.The sp 2hybridization between one s orbital and two p orbitals leads to a trigonal planar structure with a formation of a ␴bond between carbon atoms that are separated by 1.42Å.The ␴band is re-sponsible for the robustness of the lattice structure in all allotropes.Due to the Pauli principle,these bands have a ﬁlled shell and,hence,form a deep valence band.The unaffected p orbital,which is perpendicular to the pla-nar structure,can bind covalently with neighboring car-bon atoms,leading to the formation of a ␲band.Since each p orbital has one extra electron,the ␲band is half ﬁlled.Half-ﬁlled bands in transition elements have played an important role in the physics of strongly correlated systems since,due to their strong tight-binding charac-ter,the Coulomb energies are large,leading to strong collective effects,magnetism,and insulating behavior due to correlation gaps or Mottness ͑Phillips,2006͒.In fact,Linus Pauling proposed in the 1950s that,on the basis of the electronic properties of benzene,graphene should be a resonant valence bond ͑RVB ͒structure ͑Pauling,1972͒.RVB states have become popular in the literature of transition-metal oxides,and particularly in studies of cuprate-oxide superconductors ͑Maple,1998͒.This point of view should be contrasted with contempo-raneous band-structure studies of graphene ͑Wallace,1947͒that found it to be a semimetal with unusual lin-early dispersing electronic excitations called Dirac elec-trons.While most current experimental data in graphene support the band structure point of view,the role of electron-electron interactions in graphene is a subject of intense research.It was P .R.Wallace in 1946who ﬁrst wrote on the band structure of graphene and showed the unusual semimetallic behavior in this material ͑Wallace,1947͒.At that time,the thought of a purely 2D structure was not reality and Wallace’s studies of graphene served him as a starting point to study graphite,an important mate-rial for nuclear reactors in the post–World War II era.During the following years,the study of graphite culmi-nated with the Slonczewski-Weiss-McClure ͑SWM ͒band structure of graphite,which provided a description of the electronic properties in this material ͑McClure,1957;Slonczewski and Weiss,1958͒and was successful in de-scribing the experimental data ͑Boyle and Nozières 1958;McClure,1958;Spry and Scherer,1960;Soule et al.,1964;Williamson et al.,1965;Dillon et al.,1977͒.From 1957to 1968,the assignment of the electron and hole states within the SWM model were oppositetoFIG.1.͑Color online ͒Graphene ͑top left ͒is a honeycomb lattice of carbon atoms.Graphite ͑top right ͒can be viewed as a stack of graphene layers.Carbon nanotubes are rolled-up cylinders of graphene ͑bottom left ͒.Fullerenes ͑C 60͒are mol-ecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice.From Castro Neto et al.,2006a .110Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009what is accepted today.In1968,Schroeder et al.͑Schroeder et al.,1968͒established the currently ac-cepted location of electron and hole pockets͑McClure, 1971͒.The SWM model has been revisited in recent years because of its inability to describe the van der Waals–like interactions between graphene planes,a problem that requires the understanding of many-body effects that go beyond the band-structure description ͑Rydberg et al.,2003͒.These issues,however,do not arise in the context of a single graphene crystal but they show up when graphene layers are stacked on top of each other,as in the case,for instance,of the bilayer graphene.Stacking can change the electronic properties considerably and the layering structure can be used in order to control the electronic properties.One of the most interesting aspects of the graphene problem is that its low-energy excitations are massless, chiral,Dirac fermions.In neutral graphene,the chemical potential crosses exactly the Dirac point.This particular dispersion,that is only valid at low energies,mimics the physics of quantum electrodynamics͑QED͒for massless fermions except for the fact that in graphene the Dirac fermions move with a speed v F,which is300times smaller than the speed of light c.Hence,many of the unusual properties of QED can show up in graphene but at much smaller speeds͑Castro Neto et al.,2006a; Katsnelson et al.,2006;Katsnelson and Novoselov, 2007͒.Dirac fermions behave in unusual ways when compared to ordinary electrons if subjected to magnetic ﬁelds,leading to new physical phenomena͑Gusynin and Sharapov,2005;Peres,Guinea,and Castro Neto,2006a͒such as the anomalous integer quantum Hall effect ͑IQHE͒measured experimentally͑Novoselov,Geim, Morozov,et al.,2005a;Zhang et al.,2005͒.Besides being qualitatively different from the IQHE observed in Si and GaAlAs͑heterostructures͒devices͑Stone,1992͒, the IQHE in graphene can be observed at room tem-perature because of the large cyclotron energies for “relativistic”electrons͑Novoselov et al.,2007͒.In fact, the anomalous IQHE is the trademark of Dirac fermion behavior.Another interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials due to the so-called Klein paradox,that is,the fact that Dirac fermions can be transmitted with probability1through a classically forbidden region͑Calogeracos and Dombey, 1999;Itzykson and Zuber,2006͒.In fact,Dirac fermions behave in an unusual way in the presence of conﬁning potentials,leading to the phenomenon of Zitter-bewegung,or jittery motion of the wave function͑Itzyk-son and Zuber,2006͒.In graphene,these electrostatic potentials can be easily generated by disorder.Since dis-order is unavoidable in any material,there has been a great deal of interest in trying to understand how disor-der affects the physics of electrons in graphene and its transport properties.In fact,under certain conditions, Dirac fermions are immune to localization effects ob-served in ordinary electrons͑Lee and Ramakrishnan, 1985͒and it has been established experimentally that electrons can propagate without scattering over large distances of the order of micrometers in graphene͑No-voselov et al.,2004͒.The sources of disorder in graphene are many and can vary from ordinary effects commonly found in semiconductors,such as ionized impurities in the Si substrate,to adatoms and various molecules ad-sorbed in the graphene surface,to more unusual defects such as ripples associated with the soft structure of graphene͑Meyer,Geim,Katsnelson,Novoselov,Booth, et al.,2007a͒.In fact,graphene is unique in the sense that it shares properties of soft membranes͑Nelson et al.,2004͒and at the same time it behaves in a metallic way,so that the Dirac fermions propagate on a locally curved space.Here analogies with problems of quantum gravity become apparent͑Fauser et al.,2007͒.The soft-ness of graphene is related with the fact that it has out-of-plane vibrational modes͑phonons͒that cannot be found in3D solids.Theseﬂexural modes,responsible for the bending properties of graphene,also account for the lack of long range structural order in soft mem-branes leading to the phenomenon of crumpling͑Nelson et al.,2004͒.Nevertheless,the presence of a substrate or scaffolds that hold graphene in place can stabilize a cer-tain degree of order in graphene but leaves behind the so-called ripples͑which can be viewed as frozenﬂexural modes͒.It was realized early on that graphene should also present unusual mesoscopic effects͑Peres,Castro Neto, and Guinea,2006a;Katsnelson,2007a͒.These effects have their origin in the boundary conditions required for the wave functions in mesoscopic samples with various types of edges graphene can have͑Nakada et al.,1996; Wakabayashi et al.,1999;Peres,Guinea,and Castro Neto,2006a;Akhmerov and Beenakker,2008͒.The most studied edges,zigzag and armchair,have drastically different electronic properties.Zigzag edges can sustain edge͑surface͒states and resonances that are not present in the armchair case.Moreover,when coupled to con-ducting leads,the boundary conditions for a graphene ribbon strongly affect its conductance,and the chiral Dirac nature of fermions in graphene can be used for applications where one can control the valleyﬂavor of the electrons besides its charge,the so-called valleytron-ics͑Rycerz et al.,2007͒.Furthermore,when supercon-ducting contacts are attached to graphene,they lead to the development of supercurrentﬂow and Andreev pro-cesses characteristic of the superconducting proximity effect͑Heersche et al.,2007͒.The fact that Cooper pairs can propagate so well in graphene attests to the robust electronic coherence in this material.In fact,quantum interference phenomena such as weak localization,uni-versal conductanceﬂuctuations͑Morozov et al.,2006͒, and the Aharonov-Bohm effect in graphene rings have already been observed experimentally͑Recher et al., 2007;Russo,2007͒.The ballistic electronic propagation in graphene can be used forﬁeld-effect devices such as p-n͑Cheianov and Fal’ko,2006;Cheianov,Fal’ko,and Altshuler,2007;Huard et al.,2007;Lemme et al.,2007; Tworzydlo et al.,2007;Williams et al.,2007;Fogler, Glazman,Novikov,et al.,2008;Zhang and Fogler,2008͒and p-n-p͑Ossipov et al.,2007͒junctions,and as“neu-111Castro Neto et al.:The electronic properties of graphene Rev.Mod.Phys.,V ol.81,No.1,January–March2009trino”billiards ͑Berry and Modragon,1987;Miao et al.,2007͒.It has also been suggested that Coulomb interac-tions are considerably enhanced in smaller geometries,such as graphene quantum dots ͑Milton Pereira et al.,2007͒,leading to unusual Coulomb blockade effects ͑Geim and Novoselov,2007͒and perhaps to magnetic phenomena such as the Kondo effect.The transport properties of graphene allow for their use in a plethora of applications ranging from single molecule detection ͑Schedin et al.,2007;Wehling et al.,2008͒to spin injec-tion ͑Cho et al.,2007;Hill et al.,2007;Ohishi et al.,2007;Tombros et al.,2007͒.Because of its unusual structural and electronic ﬂex-ibility,graphene can be tailored chemically and/or struc-turally in many different ways:deposition of metal at-oms ͑Calandra and Mauri,2007;Uchoa et al.,2008͒or molecules ͑Schedin et al.,2007;Leenaerts et al.,2008;Wehling et al.,2008͒on top;intercalation ͓as done in graphite intercalated compounds ͑Dresselhaus et al.,1983;Tanuma and Kamimura,1985;Dresselhaus and Dresselhaus,2002͔͒;incorporation of nitrogen and/or boron in its structure ͑Martins et al.,2007;Peres,Klironomos,Tsai,et al.,2007͓͒in analogy with what has been done in nanotubes ͑Stephan et al.,1994͔͒;and using different substrates that modify the electronic structure ͑Calizo et al.,2007;Giovannetti et al.,2007;Varchon et al.,2007;Zhou et al.,2007;Das et al.,2008;Faugeras et al.,2008͒.The control of graphene properties can be extended in new directions allowing for the creation of graphene-based systems with magnetic and supercon-ducting properties ͑Uchoa and Castro Neto,2007͒that are unique in their 2D properties.Although the graphene ﬁeld is still in its infancy,the scientiﬁc and technological possibilities of this new material seem to be unlimited.The understanding and control of this ma-terial’s properties can open doors for a new frontier in electronics.As the current status of the experiment and potential applications have recently been reviewed ͑Geim and Novoselov,2007͒,in this paper we concen-trate on the theory and more technical aspects of elec-tronic properties with this exciting new material.II.ELEMENTARY ELECTRONIC PROPERTIES OF GRAPHENEA.Single layer:Tight-binding approachGraphene is made out of carbon atoms arranged in hexagonal structure,as shown in Fig.2.The structure can be seen as a triangular lattice with a basis of two atoms per unit cell.The lattice vectors can be written asa 1=a 2͑3,ͱ3͒,a 2=a2͑3,−ͱ3͒,͑1͒where a Ϸ1.42Åis the carbon-carbon distance.Thereciprocal-lattice vectors are given byb 1=2␲3a͑1,ͱ3͒,b 2=2␲3a͑1,−ͱ3͒.͑2͒Of particular importance for the physics of graphene are the two points K and K Јat the corners of the graphene Brillouin zone ͑BZ ͒.These are named Dirac points for reasons that will become clear later.Their positions in momentum space are given byK =ͩ2␲3a ,2␲3ͱ3aͪ,K Ј=ͩ2␲3a ,−2␲3ͱ3aͪ.͑3͒The three nearest-neighbor vectors in real space are given by␦1=a 2͑1,ͱ3͒␦2=a 2͑1,−ͱ3͒␦3=−a ͑1,0͒͑4͒while the six second-nearest neighbors are located at ␦1Ј=±a 1,␦2Ј=±a 2,␦3Ј=±͑a 2−a 1͒.The tight-binding Hamiltonian for electrons in graphene considering that electrons can hop to both nearest-and next-nearest-neighbor atoms has the form ͑we use units such that ប=1͒H =−t͚͗i ,j ͘,␴͑a ␴,i †b ␴,j +H.c.͒−t Ј͚͗͗i ,j ͘͘,␴͑a ␴,i †a ␴,j +b ␴,i †b ␴,j +H.c.͒,͑5͒where a i ,␴͑a i ,␴†͒annihilates ͑creates ͒an electron with spin ␴͑␴=↑,↓͒on site R i on sublattice A ͑an equiva-lent deﬁnition is used for sublattice B ͒,t ͑Ϸ2.8eV ͒is the nearest-neighbor hopping energy ͑hopping between dif-ferent sublattices ͒,and t Јis the next nearest-neighbor hopping energy 1͑hopping in the same sublattice ͒.The energy bands derived from this Hamiltonian have the form ͑Wallace,1947͒E ±͑k ͒=±t ͱ3+f ͑k ͒−t Јf ͑k ͒,1The value of t Јis not well known but ab initio calculations ͑Reich et al.,2002͒ﬁnd 0.02t Շt ЈՇ0.2t depending on the tight-binding parametrization.These calculations also include the effect of a third-nearest-neighbors hopping,which has a value of around 0.07eV.A tight-binding ﬁt to cyclotron resonance experiments ͑Deacon et al.,2007͒ﬁnds t ЈϷ0.1eV.FIG.2.͑Color online ͒Honeycomb lattice and its Brillouin zone.Left:lattice structure of graphene,made out of two in-terpenetrating triangular lattices ͑a 1and a 2are the lattice unit vectors,and ␦i ,i =1,2,3are the nearest-neighbor vectors ͒.Right:corresponding Brillouin zone.The Dirac cones are lo-cated at the K and K Јpoints.112Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009f ͑k ͒=2cos ͑ͱ3k y a ͒+4cosͩͱ32k y a ͪcosͩ32k x a ͪ,͑6͒where the plus sign applies to the upper ͑␲*͒and the minus sign the lower ͑␲͒band.It is clear from Eq.͑6͒that the spectrum is symmetric around zero energy if t Ј=0.For ﬁnite values of t Ј,the electron-hole symmetry is broken and the ␲and ␲*bands become asymmetric.In Fig.3,we show the full band structure of graphene with both t and t Ј.In the same ﬁgure,we also show a zoom in of the band structure close to one of the Dirac points ͑at the K or K Јpoint in the BZ ͒.This dispersion can be obtained by expanding the full band structure,Eq.͑6͒,close to the K ͑or K Ј͒vector,Eq.͑3͒,as k =K +q ,with ͉q ͉Ӷ͉K ͉͑Wallace,1947͒,E ±͑q ͒Ϸ±vF ͉q ͉+O ͓͑q /K ͒2͔,͑7͒where q is the momentum measured relatively to the Dirac points and v F is the Fermi velocity,given by v F =3ta /2,with a value v F Ӎ1ϫ106m/s.This result was ﬁrst obtained by Wallace ͑1947͒.The most striking difference between this result and the usual case,⑀͑q ͒=q 2/͑2m ͒,where m is the electron mass,is that the Fermi velocity in Eq.͑7͒does not de-pend on the energy or momentum:in the usual case we have v =k /m =ͱ2E /m and hence the velocity changes substantially with energy.The expansion of the spectrum around the Dirac point including t Јup to second order in q /K is given byE ±͑q ͒Ӎ3t Ј±vF ͉q ͉−ͩ9t Јa 24±3ta 28sin ͑3␪q ͉͒ͪq ͉2,͑8͒where␪q =arctanͩq x q yͪ͑9͒is the angle in momentum space.Hence,the presence of t Јshifts in energy the position of the Dirac point and breaks electron-hole symmetry.Note that up to order ͑q /K ͒2the dispersion depends on the direction in mo-mentum space and has a threefold symmetry.This is the so-called trigonal warping of the electronic spectrum ͑Ando et al.,1998,Dresselhaus and Dresselhaus,2002͒.1.Cyclotron massThe energy dispersion ͑7͒resembles the energy of ul-trarelativistic particles;these particles are quantum me-chanically described by the massless Dirac equation ͑see Sec.II.B for more on this analogy ͒.An immediate con-sequence of this massless Dirac-like dispersion is a cy-clotron mass that depends on the electronic density as its square root ͑Novoselov,Geim,Morozov,et al.,2005;Zhang et al.,2005͒.The cyclotron mass is deﬁned,within the semiclassical approximation ͑Ashcroft and Mermin,1976͒,asm *=12␲ͫץA ͑E ͒ץEͬE =E F,͑10͒with A ͑E ͒the area in k space enclosed by the orbit andgiven byA ͑E ͒=␲q ͑E ͒2=␲E 2v F2.͑11͒Using Eq.͑11͒in Eq.͑10͒,one obtainsm *=E Fv F2=k Fv F.͑12͒The electronic density n is related to the Fermi momen-tum k F as k F2/␲=n ͑with contributions from the two Dirac points K and K Јand spin included ͒,which leads tom *=ͱ␲v Fͱn .͑13͒Fitting Eq.͑13͒to the experimental data ͑see Fig.4͒provides an estimation for the Fermi velocity andtheFIG.3.͑Color online ͒Electronic dispersion in the honeycomb lattice.Left:energy spectrum ͑in units of t ͒for ﬁnite values of t and t Ј,with t =2.7eV and t Ј=−0.2t .Right:zoom in of the energy bands close to one of the Diracpoints.FIG.4.͑Color online ͒Cyclotron mass of charge carriers in graphene as a function of their concentration n .Positive and negative n correspond to electrons and holes,respectively.Symbols are the experimental data extracted from the tem-perature dependence of the SdH oscillations;solid curves are the best ﬁt by Eq.͑13͒.m 0is the free-electron mass.Adapted from Novoselov,Geim,Morozov,et al.,2005.113Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009hopping parameter as v F Ϸ106ms −1and t Ϸ3eV,respec-tively.Experimental observation of the ͱn dependence on the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene ͑No-voselov,Geim,Morozov,et al.,2005;Zhang et al.,2005;Deacon et al.,2007;Jiang,Henriksen,Tung,et al.,2007͒—the usual parabolic ͑Schrödinger ͒dispersion im-plies a constant cyclotron mass.2.Density of statesThe density of states per unit cell,derived from Eq.͑5͒,is given in Fig.5for both t Ј=0and t Ј 0,showing in both cases semimetallic behavior ͑Wallace,1947;Bena and Kivelson,2005͒.For t Ј=0,it is possible to derive an analytical expression for the density of states per unit cell,which has the form ͑Hobson and Nierenberg,1953͒␳͑E ͒=4␲2͉E ͉t 21ͱZ 0F ͩ␲2,ͱZ 1Z 0ͪ,Z 0=Άͩ1+ͯE t ͯͪ2−͓͑E /t ͒2−1͔24,−t ഛE ഛt4ͯE t ͯ,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,Z 1=Ά4ͯE t ͯ,−t ഛE ഛtͩ1+ͯE tͯͪ2−͓͑E /t ͒2−1͔24,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,͑14͒where F ͑␲/2,x ͒is the complete elliptic integral of theﬁrst kind.Close to the Dirac point,the dispersion is ap-proximated by Eq.͑7͒and the density of states per unit cell is given by ͑with a degeneracy of 4included ͒␳͑E ͒=2A c ␲͉E ͉v F2,͑15͒where A c is the unit cell area given by A c =3ͱ3a 2/2.It is worth noting that the density of states for graphene is different from the density of states of carbon nanotubes ͑Saito et al.,1992a ,1992b ͒.The latter shows 1/ͱE singu-larities due to the 1D nature of their electronic spec-trum,which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective,graphene nanoribbons,which also have momentum quantization perpendicular to the ribbon length,have properties similar to carbon nano-tubes.B.Dirac fermionsWe consider the Hamiltonian ͑5͒with t Ј=0and theFourier transform of the electron operators,a n =1ͱN c͚ke −i k ·R na ͑k ͒,͑16͒where N c is the number of unit ing this transfor-mation,we write the ﬁeld a n as a sum of two terms,coming from expanding the Fourier sum around K Јand K .This produces an approximation for the representa-tion of the ﬁeld a n as a sum of two new ﬁelds,written asa n Ӎe −i K ·R n a 1,n +e −i K Ј·R n a 2,n ,b n Ӎe −i K ·R n b 1,n +e −i K Ј·R n b 2,n ,͑17͒ρ(ε)ε/tρ(ε)ε/tFIG.5.Density of states per unit cell as a function of energy ͑in units of t ͒computed from the energy dispersion ͑5͒,t Ј=0.2t ͑top ͒and t Ј=0͑bottom ͒.Also shown is a zoom-in of the density of states close to the neutrality point of one electron per site.For the case t Ј=0,the electron-hole nature of the spectrum is apparent and the density of states close to the neutrality point can be approximated by ␳͑⑀͒ϰ͉⑀͉.114Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009where the index i =1͑i =2͒refers to the K ͑K Ј͒point.These new ﬁelds,a i ,n and b i ,n ,are assumed to vary slowly over the unit cell.The procedure for deriving a theory that is valid close to the Dirac point con-sists in using this representation in the tight-binding Hamiltonian and expanding the opera-tors up to a linear order in ␦.In the derivation,one uses the fact that ͚␦e ±i K ·␦=͚␦e ±i K Ј·␦=0.After some straightforward algebra,we arrive at ͑Semenoff,1984͒H Ӎ−t͵dxdy ⌿ˆ1†͑r ͒ͫͩ3a ͑1−i ͱ3͒/4−3a ͑1+i ͱ3͒/4ͪץx +ͩ3a ͑−i −ͱ3͒/4−3a ͑i −ͱ3͒/4ͪץy ͬ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒ͫͩ3a ͑1+i ͱ3͒/4−3a ͑1−i ͱ3͒/4ͪץx +ͩ3a ͑i −ͱ3͒/4−3a ͑−i −ͱ3͒/4ͪץy ͬ⌿ˆ2͑r ͒=−i v F͵dxdy ͓⌿ˆ1†͑r ͒␴·ٌ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒␴*·ٌ⌿ˆ2͑r ͔͒,͑18͒with Pauli matrices ␴=͑␴x ,␴y ͒,␴*=͑␴x ,−␴y ͒,and ⌿ˆi†=͑a i †,b i †͒͑i =1,2͒.It is clear that the effective Hamil-tonian ͑18͒is made of two copies of the massless Dirac-like Hamiltonian,one holding for p around K and the other for p around K Ј.Note that,in ﬁrst quantized lan-guage,the two-component electron wave function ␺͑r ͒,close to the K point,obeys the 2D Dirac equation,−i v F ␴·ٌ␺͑r ͒=E ␺͑r ͒.͑19͒The wave function,in momentum space,for the mo-mentum around K has the form␺±,K ͑k ͒=1ͱ2ͩe −i ␪k /2±e i ␪k /2ͪ͑20͒for H K =v F ␴·k ,where the Ϯsigns correspond to the eigenenergies E =±v F k ,that is,for the ␲*and ␲bands,respectively,and ␪k is given by Eq.͑9͒.The wave func-tion for the momentum around K Јhas the form␺±,K Ј͑k ͒=1ͱ2ͩe i ␪k /2±e −i ␪k /2ͪ͑21͒for H K Ј=v F ␴*·k .Note that the wave functions at K and K Јare related by time-reversal symmetry:if we set the origin of coordinates in momentum space in the M point of the BZ ͑see Fig.2͒,time reversal becomes equivalent to a reﬂection along the k x axis,that is,͑k x ,k y ͒→͑k x ,−k y ͒.Also note that if the phase ␪is rotated by 2␲,the wave function changes sign indicating a phase of ␲͑in the literature this is commonly called a Berry’s phase ͒.This change of phase by ␲under rotation is char-acteristic of spinors.In fact,the wave function is a two-component spinor.A relevant quantity used to characterize the eigen-functions is their helicity deﬁned as the projection of the momentum operator along the ͑pseudo ͒spin direction.The quantum-mechanical operator for the helicity has the formhˆ=12␴·p ͉p ͉.͑22͒It is clear from the deﬁnition of h ˆthat the states ␺K͑r ͒and ␺K Ј͑r ͒are also eigenstates of h ˆ,h ˆ␺K ͑r ͒=±12␺K͑r ͒,͑23͒and an equivalent equation for ␺K Ј͑r ͒with inverted sign.Therefore,electrons ͑holes ͒have a positive ͑negative ͒helicity.Equation ͑23͒implies that ␴has its two eigen-values either in the direction of ͑⇑͒or against ͑⇓͒the momentum p .This property says that the states of the system close to the Dirac point have well deﬁned chiral-ity or helicity.Note that chirality is not deﬁned in regard to the real spin of the electron ͑that has not yet ap-peared in the problem ͒but to a pseudospin variable as-sociated with the two components of the wave function.The helicity values are good quantum numbers as long as the Hamiltonian ͑18͒is valid.Therefore,the existence of helicity quantum numbers holds only as an asymptotic property,which is well deﬁned close to the Dirac points K and K Ј.Either at larger energies or due to the presence of a ﬁnite t Ј,the helicity stops being a good quantum number.1.Chiral tunneling and Klein paradoxIn this section,we address the scattering of chiral elec-trons in two dimensions by a square barrier ͑Katsnelson et al.,2006;Katsnelson,2007b ͒.The one-dimensional scattering of chiral electrons was discussed earlier in the context on nanotubes ͑Ando et al.,1998;McEuen et al.,1999͒.We start by noting that by a gauge transformation the wave function ͑20͒can be written as115Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009。

用Mathematica计算氢原子二级斯塔克效应

关键词氢原子；斯塔克效应；微扰论；二级修正；Mathematica
I
2002 级物理学专业毕业论文
英文摘要
Abstract
In Quantum Mechanics, the energy-levels of Hydrogen atom are usually degeneration. The degeneration energy-levels of Hydrogen atom which is put into lower uniform electric field are split. Such these systems were dealt with the degeneration perturbation theory. Because of degeneration energy-levels of Hydrogen atom wasn’t completely removed in the first order Stark-effect. In the paper, we have calculated the second order Stark-effect with Mathematica.It was found that the degeneration energy-levels were fully removed, but not completely.
通过实验观察发现，处于均匀弱电场中的第一激发态氢原子所产生的谱线发生了分裂现象，由于这种现象最初是由德国物理学家斯塔克（Johnnes Stark,1874-1957）观测到，所以被人们称为氢原子的斯塔克效应[2]。利用简并微扰论计算氢原子的一级斯塔克效应[3~4]发现，原来简并的能级在外电场作用下分裂为三个能级，一个在原来的上面，另一个在原来的下面，而且能量差是相等的。这样，没有外电场时的一条谱线，在外电场中就分裂成三条；它们的频率一条比原来稍小，一条比原来稍大，另一条与原来相等，且上下两条谱线关于中间那条谱线对称。由于第一激发态氢原子的简并度为 4，根据目前的理论推断，能级最终应该分裂成四条，所以能级简并没有完全消除。

那神奇的纳米时代英语作文

那神奇的纳米时代英语作文The Miraculous Nanotech Era.As technology relentlessly advances, we stand on the cusp of a remarkable era: the Nanotech Era. This transformative field involves manipulating matter at the atomic and molecular scale, unlocking unprecedented possibilities for innovation and progress.The Essence of Nanotechnology.Nanotechnology operates on an incredibly small scale. A nanometer is one billionth of a meter, roughly the size of a few atoms. At this scale, matter exhibits unique properties and behaviors that differ significantly from its macroscopic counterparts. These unique characteristics stem from quantum effects, the laws that govern the subatomic world.Revolutionary Applications.Nanotechnology has far-reaching applications across diverse industries, paving the way for groundbreaking advancements in healthcare, energy, electronics, manufacturing, and more.In Medicine:Targeted Drug Delivery: Nanoparticles can be designed to deliver drugs directly to diseased cells, enhancing efficacy and minimizing side effects.Tissue Engineering: Nanomaterials can be used to grow and repair damaged tissues, offering hope for treating degenerative diseases.Diagnostics: Nanosensors can detect minute amounts of biomarkers, enabling early diagnosis and personalized treatment.In Energy:Solar Energy Harvesting: Nanoengineered materials can improve the efficiency of solar cells, capturing more sunlight and generating more electricity.Fuel Cells: Nanocatalysts can enhance the performance and durability of fuel cells, providing a cleaner and more efficient energy source.Batteries: Nanomaterials can lead to the development of higher-capacity, longer-lasting batteries for portable devices and electric vehicles.In Electronics:Miniaturization: Nanotechnology allows for the creation of smaller, more powerful electronic devices with enhanced capabilities.Advanced Computing: Nanomaterials can enable faster processing speeds and increased memory capacity in computers.Nanophotonics: Nanostructures can manipulate light in novel ways, opening up possibilities for optical computing and ultra-high-speed data transmission.In Manufacturing:Lightweight Materials: Nanomaterials can be engineered to create lightweight, ultra-strong materials for aerospace and automotive applications.Self-Cleaning Surfaces: Nanoparticles can impart self-cleaning properties to surfaces, reducing the need for harsh chemicals and detergents.Antimicrobial Textiles: Nanomaterials can be incorporated into textiles to provide antimicrobial protection, preventing the growth of bacteria and viruses.Challenges and Ethical Considerations.While nanotechnology holds immense promise, it also presents challenges and ethical considerations that need tobe carefully addressed.Environmental Impact: The potential environmental impact of nanomaterials requires thorough assessment and responsible disposal practices.Health and Safety: The health and safety implications of nanomaterials must be fully understood and managed to ensure their safe use.Ethical Responsibility: The rapid advancement of nanotechnology raises ethical concerns about its potential use for malicious purposes or exacerbation of existing inequalities.Conclusion.The Nanotech Era presents both unparalleled opportunities and challenges. By embracing the responsible development and application of nanotechnology, we have the potential to unlock transformative solutions to some of the world's most pressing problems. From revolutionizinghealthcare to addressing the energy crisis, nanotechnology holds the key to shaping a brighter and more sustainable future.。

量子点具有量子力学的

量子点具有量子力学的英文回答：Quantum dots exhibit quantum mechanical effects due to their nanoscale dimensions. These effects include:Quantization of energy levels: The energy levels of electrons in quantum dots are discrete, meaning they can only occupy certain specific energies. This is in contrast to the continuous energy levels of electrons in bulk materials.Tunable bandgap: The bandgap of a quantum dot is the energy difference between the valence band and the conduction band. The bandgap of a quantum dot can be tuned by changing the size of the dot. This allows quantum dots to be used in a variety of optoelectronic applications.Enhanced optical properties: Quantum dots have enhanced optical properties, such as high photoluminescenceefficiency and narrow emission spectra. These properties make quantum dots ideal for use in applications such as light-emitting diodes (LEDs), lasers, and solar cells.中文回答：量子点由于其纳米尺度的尺寸而表现出量子力学效应。

光子阻塞效应

学号：201105774题目名称: 强耦合下的光子阻塞效应研究题目类型: 研究论文学生姓名: 董昌瑞院（系）: 物理与光电工程学院专业班级: 物理11102班指导教师: 邹金花辅导教师: 邹金花时间: 2015年1月至2015年6月目录毕业论文任务书` (I)指导教师评审意见 (VIII)评阅教师评语 (IX)答辩记录及成绩评定 (X)中文摘要 (XI)外文摘要 (XII)1引言 (1)2 基础理论知识 (1)2.1 光力振子系统 (1)2.2二能级原子与光场相互作用的全量子理论 (2)2.3光场关联函数 (5)2.4 光子计数统计 (8)3 模型方程与结果分析 (10)3.1模型方程 (10)3.2 方程分析 (12)4总结与展望 (14)参考文献 (14)致谢 (16)毕业论文任务书`院（系）物理与光电工程学院专业物理班级物理11102 学生姓名董昌瑞指导教师/职称邹金花/副教授1.毕业论文(设计)题目：强耦合下的光子阻塞效应研究2.毕业论文(设计)起止时间： 2015 年1月1 日～2015 年 6月10 日3．毕业论文(设计)所需资料及原始数据（指导教师选定部分）[1] A Ridolfo, M Leib, S Savasta, M J Hartmann. Photon Blockade in the Ultrastrong CouplingRegime [J]. Phys. Rev. Lett., 2012, 109: 193602-1~193602-5[2] Jieqiao Liao, C K Law. Cooling of a mirror in cavity optomechanics with a chirped pulse [J]. Phys. Rev. A, 2011, 84: 053838-1~053838-6[3] P Komar, S D Bennett, K Stannigel, S J M Habraken, P Rabl, P Zoller, M D Lukin. Single-photon nonlinearities in two-mode optomechanics [J]. Phys. Rev. A, 2013, 87: 013839-1~013839-10[4] T Ramos, V Sudhir, K Stannigel, P Zoller, T Kippenbrg. Nonlinear quantum optomechanics viaindividual intrinsic two-level defects [J]. Phys. Rev. Lett., 2013, 110: 193602-1~193602-5 [5] G Anetsberger, O Arcizet, Q P Unterreithmeier, R Riviere, A Schliesser, E M Weig, J P Kotthaus,T Kippenberg. Near-field cavity optomechanics with nanomechanical oscillators [J]. Nat. Phys., 2009, 5: 909~914[6] S J M Habraken, W Lechner, P Zoller. Resonances in dissipative optomechanics withnanoparticles: Sorting, speed rectification, and transverse coolings [J]. Phys. Rev. A, 2013, 87: 053808-1~053808-8[7] K Qu, G S Agarwal. Fano resonances and their control in optomechanics [J]. Phys. Rev. A, 2013,87: 063813-1~063813-7[8] A Nunnenkamp, K Borkje, S M Girvin. Cooling in the single-photon strong-coupling regime ofcavity optomechanics [J]. Phys. Rev. A, 2012, 85: 051803-1~051803-4[9] Y C Liu, Y F Xiao, X S Luan, C W Wong. Dynamic Dissipative Cooling of a MechanicalResonator in Strong Coupling Optomechanics [J]. Phys. Rev. A, 2013, 110: 153606-1~153606-5[10] A Nunnekamp, K Borkie, S M Girvin. Single-photon optomechanics [J]. Phys. Rev. Lett., 2011,107: 063602-1~063602-5[11] J M Dobrindt, I Wilson-Rae, T J Kippenbeg. Parametric Normal-Mode Splitting in CavityOptomechanics [J]. Phys. Rev. Lett., 2008, 101: 263602-1~263602-4[12]樊菲菲. 光力振子与原子间量子纠缠和振子压缩的研究[D]. 华中师范大学，2014[13] 张文慧. 光机械腔系统的动力学行为[D]. 华中师范大学，2014[14]詹孝贵. 腔光机械系统中电磁诱导透明及其相关现象的理论研究[D]. 华中科技大学，20134．毕业论文(设计)应完成的主要内容在阅读大量文献的基础上，完成开题报告，并通过开题答辩。

量子信息和量子纠缠理论

Multipartite Schmidt-correlated State
Fully separable
PPT
Fully separable (maximally entangled)
~ 1 (N)
M.J. Zhao, S.M. Fei and Z.X. Wang, Phys. Lett. A 372(2008)2552
S. Albeverio, S.M. Fei, Phys. Lett. A 276(2000)8 S. Albeverio, S.M. Fei and W.L. Yang, Comm. Theor. Phys. 38 (2002) 301
S. Albeverio, S.M. Fei and W.L. Yang, Phys. Rev. A 66 (2002) 012301 M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A 60, 1888 (1999)
Separable! Separable！
Separability of mixed states: no general criteria a) Peres (PPT) criterion:
Peres PRL 77, 1413 (1996)
2x2, 2x3:
PPT
Separable
Horodeckis, Phys. Lett. A 223,1 (1996)
Caltech (Kimble et al)
/~qoptics/teleport.html
Nature 390(1997) 575
Science 282(1998) 706
Wigner functions before &after

杨泰勒效应

HS: d7
Energy level splitting of d-orbitals in octahedron
Cu: 1s2 2s2 2p6 3s2 3p6 3d10 4s1 Cu2+: 1s2 2s2 2p6 3s2 3p6 3d9 dz2，dx -y dz2，dx -y
2 2 2
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d9
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d9
z
y x x z
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Energy level splitting of d-orbitals in octahedron
Octahedral with shorter z Octahedral with longer z
d z2 dx -y
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2
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2
d z2 dxy
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Energy level splitting of d-orbitals in octahedron
Ni: 1s2 2s2 2p6 3s2 3p6 3d10 Ni2+: 1s2 2s2 2p6 3s2 3p6 3d8
Filling types with spherically symmetric electron cloud
d0
Empty
HS d5
Half filled
d10
Full filled
No Jahn-Teller effect
Filling types with octahedrally symmetric electron cloud

关于quantum的雅思阅读理解 (2)

关于quantum的雅思阅读理解引言概述：Quantum physics, also known as quantum mechanics, is a branch of physics that deals with the behavior of matter and energy at the smallest scales. Understanding quantum concepts is crucial for advancements in various fields, including technology, medicine, and communication. In this article, we will delve into the topic of quantum physics and its significance in IELTS reading comprehension.正文内容：1. Quantum Theory and Its Principles1.1 Wave-Particle Duality: Quantum theory proposes that particles, such as electrons and photons, exhibit both wave-like and particle-like behavior. This principle challenges classical physics, where particles were considered solely as particles or waves.1.2 Superposition: According to quantum theory, particles can exist in multiple states simultaneously. This concept is known as superposition, and it allows for the potential of quantum computing and cryptography.1.3 Quantum Entanglement: Quantum entanglement refers to the phenomenon where two or more particles become correlated in such a way that the state of one particle is instantly linked to the state of another, regardless of the distance between them. This principle has implications for secure communication and quantum teleportation.2. Applications of Quantum Physics2.1 Quantum Computing: Quantum computers utilize the principles of superposition and entanglement to perform complex calculations at an exponential speed compared to classical computers. This technology has the potential to revolutionize fields such as cryptography, optimization problems, and drug discovery.2.2 Quantum Communication: Quantum communication involves the transmission of information using quantum states. Quantum encryption ensures secure communication by exploiting the principles of entanglement and uncertainty. This technology has the potential to protect sensitive information from hacking.2.3 Quantum Sensing: Quantum sensors utilize the unique properties of quantum particles to measure physical quantities with unprecedented precision. This has applications in fields such as navigation, medical imaging, and environmental monitoring.3. Challenges in Quantum Physics3.1 Measurement Problem: The act of measuring a quantum system can disturb its state, leading to the collapse of the superposition. This measurement problem raises questions about the nature of reality and the role of the observer in quantum physics.3.2 Quantum Decoherence: Quantum systems are highly sensitive to their surroundings, which can cause decoherence. This phenomenon disrupts the delicate quantum states and poses challenges for maintaining coherence in quantum technologies.3.3 Quantum Interpretations: The interpretation of quantum mechanics is still a subject of debate among physicists. Different interpretations, such as the Copenhagen interpretation and the Many-Worlds interpretation, offer different explanations for the behavior of quantum systems.4. Quantum Physics in IELTS Reading Comprehension4.1 Vocabulary: Familiarity with quantum-related terms and concepts is essential for understanding reading passages that discuss quantum physics. Being well-versed in terms like superposition, entanglement, and decoherence will aid in comprehending the content.4.2 Inference: IELTS reading passages often require candidates to make inferences based on the information provided. Understanding the principles and applications of quantum physics will enable candidates to make accurate inferences when encountering quantum-related texts.4.3 Critical Analysis: IELTS reading tests candidates' ability to critically analyze information. Being knowledgeable about the challenges and interpretations in quantum physics will help candidates evaluate the validity and implications of the given information.总结：In conclusion, quantum physics plays a crucial role in various scientific and technological advancements. Understanding the principles of quantum theory, its applications, and the challenges it poses is essential for comprehending quantum-related passages in IELTS reading comprehension. By familiarizing oneself with quantum vocabulary, making accurate inferences, and critically analyzing information, candidates can enhance their performance in this aspect of the IELTS examination.。

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ten, they made use of ﬂat spacetime QEIs, and the validity of their conclusions depends on the assumption that quantum ﬁelds in curved spacetimes are subject to the same restrictions as those in ﬂat spacetimes, at least on sampling timescales short in comparison with natural geometric scales. We will refer to this as the ‘usual assumption’: one of our aims is to establish its general validity in arbitrary two-dimensional globally hyperbolic spacetimes. Our main tool will be the QEI established for the massless scalar ﬁeld by Flanagan [5] (based on an earlier argument due to Vollick [6]). As this was originally proved only for those two-dimensional spacetimes which are globally conformal to Minkowski space, we begin by obtaining a generalisation to an arbitrary twodimensional globally hyperbolic spacetime (M, gab ). The same argument applies to spacetimes with boundaries. We make use of two observations, the ﬁrst of which is that any point p ∈ M has a ‘diamond neighbourhood’ of the form D = int(J + (q ) ∩ J − (r)) such that (D, gab |D ) (considered as a spacetime in its own right) is globally conformal to the whole of Minkowski space. This may be seen by introducing null coordinates (u, v ) in a neighbourhood of p so that the metric takes the form ds2 = e2σ du dv for some smooth function σ . We may assume, without loss of generality, that this neighbourhood contains a diamond neighbourhood D of p corresponding to coordinate ranges |u| < u0 , |v | < v0 , say, and by reparametrising U = tan(πu/(2u0 )), V = tan(πv/(2v0 )), we see that (D, gab |D ) is conformal to the whole of Minkowski space. Furthermore, we may choose a smooth partition of unity χα on M such that each χα is supported within some diamond region Dα of the above type (and only ﬁnitely many χα are nonzero at any point of M ) [16]. Our second observation is that any state of the ﬁeld on (M, gab ) induces a state of the ﬁeld on each (Dα , gab |Dα ) whose n-point functions are simply the restrictions to Dα of the n-point functions on M . Now the renormalised stress-energy tensor at any point p ∈ Dα is obtained from derivatives of the diﬀerence between the two-point function and the Hadamard parametrix. Since the latter is determined by the local geometry alone, and therefore
Quantum energy inequalities in two dimensions
Christopher J. Fewster∗
Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK (Dated: February 7, 2008) Quantum energy inequalities (QEIs) were established by Flanagan for the massless scalar ﬁeld on two-dimensional Lorentzian spacetimes globally conformal to Minkowski space. We extend his result to all two-dimensional globally hyperbolic Lorentzian spacetimes and use it to show that ﬂat spacetime QEIs give a good approximation to the curved spacetime results on sampling timescales short in comparison with natural geometric scales. This is relevant to the application of QEIs to constrain exotic spacetime metrics.
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∞ −∞
G′ (τ )2 dτ (1) G(τ )
for all Hadamard states ω , where γ is the worldline of an inertial observer parametrised by proper time τ with two-velocity ua , and G is any smooth, nonnegative sampling function of compact support [i.e., vanishing outside a compact interval]. The right-hand side is large and negative if G is tightly peaked, but small if it is broadly spread. Thus the magnitude and duration are constrained by a relationship reminiscent of the uncertainty relations: in d-dimensional Minkowski space, the energy density can drop below ρ0 < 0 for a time τ0 only d if |ρ0 |τ0 < κd for some (small) constant κd (κ2 = π/6, for example). Many exotic spacetimes (wormholes, warp drives, etc) entail violations of the weak energy condition and it has often been suggested that quantum ﬁelds might provide the necessary distributions of stress-energy. Quantum energy inequalities provide a quantitative check on such proposals and have been used to argue that exotic spacetimes are tightly constrained [3, 4]. As no curved spacetime QEIs were available when these references were writ-
PACS numbers: 04.62.+v Keywords: Energy conditions, Quantum inequalities
arXiv:gr-qc/0411114v1 24 Nov 2004
Classically, the massless (minimally coupled) free scalar ﬁeld obeys the weak energy condition: it displays a nonnegative energy density to all observers at all points in spacetime. Its quantised sibling is quite diﬀerent, however, admitting unboundedly negative energy densities at individual spacetime points. Violations of the energy conditions are cause for concern, and a considerable eﬀort has been expended, beginning with the work of Ford [1], in trying to understand what constraints quantum ﬁeld theory might place on such eﬀects. It turns out that averages of the energy density along, for example, timelike curves obey state-independent lower bounds called quantum inequalities, or quantum energy inequalities (QEIs). In two-dimensional Minkowski space, for example, the massless free ﬁeld obeys the QEI bound [2] Tab (γ (τ )) ω ua ub G(τ )dτ ≥ −