Charge-order transition in the extended Hubbard model on a two-leg ladder

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。

a r X i v :c o n d -m a t /0301048v 1 [c o n d -m a t .s t r -e l ] 6 J a n 2003Magnetic,orbital and charge ordering in the electron-doped manganitesTulika Maitra ∗1and A.Taraphder †2∗Max-Planck-Institut f¨u r Physik Komplexer Systeme N¨o thnitzer Str.3801187Dresden,Germany †Department of Physics &Meteorology and Centre for Theoretical Studies,Indian Institute of Technology,Kharagpur 721302India Abstract The three dimensional perovskite manganites R 1−x A x MnO 3in the range of hole-doping x >0.5are studied in detail using a double exchange model with degenerate e g orbitals including intra-and inter-orbital correlations and near-neighbour Coulomb repulsion.We show that such a model captures the observed phase diagram and orbital-ordering in the intermediate to large band-width regime.It is argued that the Jahn-Teller effect,considered to be crucial for the region x <0.5,does not play a major role in this region,particularly for systems with moderate to large band-width.The anisotropic hopping across the degenerate e g orbitals are crucial in understanding the ground state phases of this region,an observation emphasized earlier by Brink and Khomskii.Based on calculations using a realistic limit of finite Hund’s coupling,we show that the inclusion of interactions stabilizes the C-phase,the antiferromagnetic metallic A-phase moves closer to x =0.5while the ferromagnetic phase shrinks in agreement with recent observations.The charge ordering close to x =0.5and the effect of reduction of band-width are also outlined.The effect of disorder and the possibility of inhomogeneous mixture of competing states have been discussed.PACS Nos.75.30.Et,75.47.Lx,75.47.Gk I.Introduction The colossal magnetoresistive manganites have been investigated with renewed vigour inthe recent past mainly because of their technological import.It was soon realized that these systems have a rich variety of unusual electronic and magnetic properties involving almost all the known degrees of freedom in a solid,viz.,the charge,spin,orbital and lattice degrees of freedom[1,2,3].Of particular interest have been the systems R 1−x A x MnO 3,where Rand A stand for trivalent rare-earth(e.g.,La,Nd,Pr,Sm)and divalent alkaline-earth ions (Ca,Sr,Ba,Pb etc.)respectively.Around the region0.17<x<0.4,electrical transport properties of these systems generically show extreme sensitivity towards external magnetic field with a concomitant paramagnetic insulator(or poor metal)to ferromagnetic metal transition at fairly high temperatures[5,6,7].For a long time the dominant paradigm in the theory of this unusual magneticfield-dependence of transport has been the idea of double exchange(DE)[8]involving the localized core spins(the three t2g electrons at each Mn site)coupled to the itinerant electrons in the Jahn-Teller split e g level via strong Hund’s exchange.It has been realized recently that such a simplifying theoretical framework may not be adequate in explaining several other related features involving transport,electronic and magnetic properties[9,10,11,12,13].It was already known that the observed structural distortions and magnetic and orbital orders in these systems in the region x≃0.5require interactions not included in the DE model[14,15,16].Owing to the observation of colossal magnetoresistance(CMR)in the region x<0.5in the relatively narrow band-width materials[6]at high temperatures,much of the attention was centred around this region.Only in the last few years CMR effect has been observed in the larger band-width materials like Nd1−x Sr x MnO3[17,18]and Pr1−x Sr x MnO3[19,20] in the region x>0.5.If one counts the doping from the side x=1in R1−x A x MnO3where all Mn ions are in+4state,then doping by R y(y=1−x)introduces Mn+3ions carrying one electron in the e g orbitals.This region,therefore,is also called the electron-doped region. The charge,magnetic and orbital structures of the manganites in the electron-doped regime have already been found to be quite rich[2,4,21,22,23]and the coupling between all these degrees lead to stimulating physics[24].In the framework of the conventional DE model with one e g orbital,one would expect qualitatively similar physics for x∼0and x∼1[25].On the contrary,experiments reveal a very different and assymetric picture for the phase diagram between the regions x<0.5andx>0.5.The lack of symmetry about x=0.5manifests itself most clearly in the magnetic phase diagram of these manganites.It has now been shown quite distinctly[18,32,27,34] that the systems Nd1−x Sr x MnO3,Pr1−x Sr x MnO3,La1−x Sr x MnO3are antiferromagnetically ordered beyond x=0.5while one observes either a metallic ferromagnetic state[7]ora charge ordered state with staggered charge-ordering[29,30]in the approximate range0.25<x<0.5.This charge ordered insulating state can be transformed into a ferromagnetic metallic state[19,31]by the application of magneticfields.There are several different types of AFM phases with their characteristic dimensionality of spin ordering observed in this 1−x Sr x MnO3shows A-type antiferromagnetic ground state(in which ferromagnetically aligned xy-planes are coupled antiferromagnetically)in the range0.52<x<0.58.It also shows a sliver of FM phase[27]immediately above x=0.5. In Nd1−x Sr x MnO3[34,18],the A-type spin structure appears at x=0.5and is stable upto x=0.62while in Pr1−x Sr x MnO3[19,26],this region extends from x=0.48to x=0.6.In all these cases,the phase that abuts the A-type antiferromagnet(AFM)in the region of higher hole-doping(x)is the C-type AFM state,in which antiferromagnetically aligned planes are coupled ferromagnetically.The C-type AFM phase occupies largest part of the phase diagram in this region.For even larger x,the C-phase gives way to the three dimensional antiferromagnetic G-phase.The systematics of the phase diagram changes considerably(except close to x=1)in these systems as a function of the bandwidth.Recently Kajimoto et al.[28]have quite succinctly summarized the phase diagrams of various manganites of varying bandwidths across the entire range of doping.Their phase diagram is reproduced infig.1.The phase diagram changes considerably with changing bandwidth as shown in thefigure.We note that the narrow bandwidth compounds like Pr1−x Ca x MnO3,La1−x Ca x MnO3etc.exhibit a wide region of CE-type insulating charge-ordered state around x=0.5whereas the inter-mediate bandwidth material Nd1−x Sr x MnO3shows a conducting A-type antiferromagneticphase around x=0.5.As one moves towards the larger bandwidth compounds such as Pr1−x Sr x MnO3,La1−x Sr x MnO3,a small strip of ferromagnetic(F)metallic phase appears at x=0.5[27,28]followed by the A-type AFM state.In contrast with the narrow band-width manganites,the relatively wider bandwidth manganites generally show the following sequence of spin/charge ordering upon hole doping(in the entire range0≤x≤1):insulating A-type AFM→metallic FM→metallic A-type AFM→insulating C-type AFM andfinally insulating G-type AFM states.Clearly,the most important feature here is the absence of CE-type spin/charge ordering and the presence of a metallic A-type AFM state in these wider band-width compounds in the region close to x=0.5.It appears that the physics involved in the CE-type charge/spin ordering,important for the low band-width systems,is not quite as relevant in this case.In addition,it is also observed in the neutron diffraction studies that the metallic A-type AFM state is orbitally ordered[32,34]with predominant occupation of d x2−y2orbitals.The importance of orbital-ordering has been emphasized pre-viously in several other experimental[18,35,39,40]and theoretical[33,41,42,43,44,45,46] investigations.The crucial role of the e g orbitals and inter-orbital Coulomb interaction has been underlined by Takahashi and Shiba[50]from a study of the optical absorption spectra in the ferromagnetic metallic phase of the doped manganite La1−x Sr x MnO3.They point out that it is imperative to consider the transition between nearly degenerate and moder-ately interacting e g orbitals even in the hole-doped region in order to interpret the optical absorption spectra in La1−x Sr x MnO3.In a detailed observation carried out by Akimoto et al.[27]the electronic and magnetic properties of a heavily doped manganite R1−x Sr x MnO3with R=La1−z Nd z are studied by continuously changing the band-width.In this novel procedure they were able to control the band-width chemically by changing the average ionic radius by manipulating the ratio of La and Nd(i.e.,changing z).Substitution of the smaller Nd+3ions for the larger La+3 ions effectively reduces the one-electron band-width.By increasing z chemically,they wereable to go continuously from the large band-width system La1−x Sr x MnO3down to the intermediate band-width system Nd1−x Sr x MnO3.For z<0.5,there is a metallic FM phase in the region0.5<x<0.52.From x≥0.54to about x=0.58the ground state is A-type antiferromagnetic metallic irrespective of the value of z,i.e.from La1−x Sr x MnO3 (z=0)all the way down to Nd1−x Sr x MnO3(z=1).They believe that the key factor that stabilizes the A-type AFM metallic state in a wide range of z is the structure of the two e g orbitals(d x2−y2and d3z2−r2)and the anisotropic hopping integral between them.There is no signature of charge-ordering or CE-type ordering below z=0.5for any x.The charge-ordered(CO)insulating state appears above z=0.5and around x=0.5primarily due to the commensuration(between the lattice periodicity and hole concentration)effect in the low band-width systems.The ground state phase diagram for doped manganites in x−z plane(i.e.,doping versus band-width plane)is shown infig.2after Akimoto et al.[27].The general inferences from all these measurements are that the physics of the electron-doped region is very different from the hole-doped region.In this region,with decreasing band-width starting from La1−x Sr x MnO3down to Nd1−x Sr x MnO3,the F-phase shrinks,the A-phase and C-phase remain nearly unaffected.The A-phase disappears and the C-phase shrinks(with the possible growth of incommesurate charge order region as infig.1)rapidly in the low band-width systems like La1−x Ca x MnO3and Pr1−x Ca x MnO3.The G-phase at the low electron-doping region seems to remain unaffected all through.It has been seen [27,34,35]that the gradual build up of the AFM correlations in the electron-doped region is pre-empted by the orbital ordering in the A-and C-phases.The e g orbitals and the anisotropic hopping of electrons between them[16,63],must indeed play a significant role given the presence of orbital ordering in much of the phase diagram beyond x=0.5.It is also realized that the effect of lattice could be ignored in thefirst approximation for these moderate to large band-width systems in this region of doping.All these point to the fact that the interactions that play a dominant role in the elctron-doped region are different[36,37,38]from the ones that are considered crucial in the hole-doped side.There has been a large number of reports of charge ordering and inhomogeneous states [17,18,19,47,48,49,52,53]in the region x≃0.5.These states are quite abundant in the low band-width materials.The inhomogeneous states result primarily from the competing ground states[54](charge ordered/AFM and FM in this case)that lead to1st.order phase transitions with a discontinuity in the density as the chemical potential is varied.Such transitions are known to lead to phase separation in the canonical ensemble.Phase separation in this context has been dicussed in the literature for quite some time[51,55,56,57,58,61]. Such macroscopic phase separations are not stable against long range Coulomb interactions and tend to break up into microscopic inhomogeneities[55,59,60].There is also the well-known CE-type charge and spin ordering that has been seen at x=0.5in most of these systems[14,15,47]with low band-width.In both Nd1−x Sr x MnO3and P r1−x Sr x MnO3Kawano et al.[32]and Kajimoto et al. [28,34]have seenfinite temperature(T≃150K)first order transitions at x=0.5from a ferromagnetic metal to an antiferromagnetic A-phase which is insulating but has quite low resistivity(immediately away from x=0.5it becomes A-type metal).In a neutron diffraction study Kajimoto et al.[28]have also observed that close to the boundary of the FM and A-type AFM metallic phases of P r1−x Sr x MnO3,an unusual stripe-like charge-order appears along with this weaklyfirst order transition.This stripe-like charge-order is distinctly different from the staggered charge-ordering of the CE-type state.Very recently,an inhomegeneous mixture of micron-size antiferromagnetic grains(possibly charge-ordered)and similar sized ferromagnetic grains has been seen in electron diffraction and dark-field imaging in the low band-width system La1−x Ca x MnO3at x=0.5[62]without any evidence of the long-range CE or any other macroscopic ordering. The ground state energies of these different phases seem to be very close[67]in this region leading to a possiblefirst order phase transition and consequent phase seggregation.Apossible nanoscale phase separation between A-type AFM and ferromagnetic regions has recently been reported by Jirac et al.,[81]in the cintered ceramic samples of Pr0.44Sr56MnO3 doped with Cr(upto8percent).It is also observed that although both the ferromagnetic domains and A-type AFM host are independently metallic(though anisotropic for A-AFM), the resultant inhomohgeneous state is non-metallic.Almost all the experiments discussed above consider orbital ordering as the underlying reason for the various magnetic orders observed in the electron-doped regime.The anisotropy of the two e g orbitals and the nature of overlap integral between them[16,63]make the electronic bands low dimensional.Such anisotropic conduction in turn leads to anisotropic spin exchanges and different magnetic structures.In the A-phase the kinetic energy gain of the electrons is maximum when the orbitals form a2D band in the xy-plane and maximize the in-plane ferromagnetic exchange interaction.However,in the z-direction AFM super-exchange interaction dominates due to the negligible overlap of d x2−y2and d3z2−r2orbitals. In addition,the presence of charge ordering and inhomegeneous or phase separated states, particularly around the commensurate densities,are suggestive of the vital role of Coulomb interactions in the manganites.The absence of CE-phase in the moderate to large band-width materials imply that the role of Jahn-Teller or static lattice distortions may not be as crucial in the electron-doped regime even in the region close to x=0.5.A model,for the electron-doped systems,therefore,should have as its primary ingredients,the two e g orbitals at each Mn site and the anisotropy of hopping between them.In addition,the Coulomb interactions are present,and their effects on the charge,orbital and magnetic order are important[9,38,41,46,67].In the next section,we motivate a model recently proposed by Brink and Khomskii[36]for the electron-doped manganites and later extended by us [38]in order to take into account the effects of local Coulomb interactions present in these systems.We extend this model further in the present work,study the magnetic and orbital orders in more detail,investigate the possibility of charge-ordering and phase separation anddiscuss their consequences.In sections II and III we present our calculations and results and compare them with experimental literature.We conclude with a brief discussion on the implications of our results.II.a.Degenerate Double-Exchange ModelEvidently the physics of the region x>0.5is quite different from that in the x<0.5 for the manganites and one has to look at the electron-doped manganites from a different perspective.In order to pay due heed to the compelling experimental and theoretical evidence in support of the vital role of the orbitals,Brink and Khomskii[36](hereafter referred to as BK)have proposed a model for the electron-doped manganites that incorporates the e g orbitals and the anisotropic hopping between them.In the undoped LaMnO3compound each Mn ion has one electron and acts as a Jahn-Teller centre,the e g orbitals are split and the system is orbitally ordered.Thus for the lightly(hole-)doped system one can at the first approxmation ignore the orbital degree of freedom and apply a single band model like the conventional double exchange(DE)model to describe it.If,however,one proceeds from the opposite end and starts,for example,from the insulating CaMnO3compound where the empty e g orbitals of Mn+4ions are degenerate,then doping trivalent(La,Nd,Pr etc.)ions into CaMnO3results in adding electrons into the doubly degenerate e g manifold.In the doped manganites R1−x A x MnO3there are y=1−x number of electrons in the e g orbitals at each Mn site.Since each site has two e g orbitals,four electrons can be accommodated per site and hence the actualfilling(electron density)is y[we restrict ourselves to the region0.5≤x≤1.0(0.5≥y≥0) 8in the foregoing].Due to this low electron concentration and hence very few Jahn-Teller centres the e g band is mostly degenerate and the Jahn-Teller effect is negligible to a leadingapproximation.The neglect of Jahn-Teller effect is also justified from the experimental evidence presented above.The usual charge and spin dynamics of the conventional DE model then operate here too,albeit with an additional degree of freedom coming from the degenerate set of e g orbitals.This process has been described by BK as double exchange via degenerate orbitals.In order to capture the magnetic phases properly,the model has,in addition to the usual double exchange term,orbital degeneracy and the superexchange(SE)coupling between neighbouring t2g spins.At x=1(or y=0)end the e g band is completely empty and the physics is governed entirely by the antiferromagnetic exchange(superexchange)between the t2g spins at neighbouring sites.On doping,the band begins tofill up,the kinetic energy of electrons in the degenerate e g levels along with the attendant Hund’s coupling between t2g and e g spins begin to compete with the antiferromagnetic superexchange interaction leading to a rich variety of magnetic and orbital structures.The model used to describe the ground state properties of the electron-doped manganites contains the following termsH=J AF<ij>S i.S j−J HiS i.s i−<ij>σ,α,βtαβi,j c†i,α,σc j,β,σ(1)Thefirst term is the usual AF superexchange between t2g spins at nearest-neighbour sites,the second term represents the Hund’s exchange coupling between t2g and e g spins at each site and the third term stands for the hopping of electrons between the two orbitals [16,63,64](α,βtake values1and2for d x2−y2and d3z2−r2orbitals,corresponding to the choice of the phaseξi=0in Ref.[65]).The hopping matrix elements are determined by the symmetry of e g orbitals[16,63].Although similar in appearance to the conventional DE model the presence of orbital degeneracy together with the very anisotropic hopping matrix elements tαβij makes this model and its outcome very different from the conventional DE model of Zener[8,25,68,69]with a single non-degenerate orbital.In the manner often used in literature[8]BK treated the t2g spins quasi-classically andthe Hund’s coupling was set to infinity.At each site the spins were allowed to cant in the xz-plane leading to the effective hopping matrix elements[8]t xy=tcos(θxy/2)and t z=tcos(θz/2).Hereθxy is the angle between nearest neighbour t2g spins in the xy−plane andθz is the same in the z−direction.The superexchange energy per state then becomesE SE=J AF S20√3t xy(cosk x+cosk y)−83(cosk x+cosk y)−4t z3(cosk x+cosk y)−4t z3(cosk x−cosk y)2)1we have plotted the bands for the A-phase in the presence of a small canting infig.3a.Note that even in the presence of canting,there is almost no dispersion in the z-direction(Γ-Z and M-L directions).In the canted C-phase as well the band disperses little in the x and y directions and remains almost indistinguishable from the pure phase.The total energy is then obtained for a particularfilling by adding the superexchange contribution to the band energy.It is evident that the energy spectrum obtained depends on the underlying magnetic structure as well as the orbital-dependent(anisotropic)hopping ma-trix elements.This will lead to different anisotropic magnetic structures at different doping. The magnetic phase diagram in the(electron)doping y-t/J AF plane is then calculated by minimizing the total energy with respect toθxy andθz.The sequence of phases follows from the nature of the DOS modulated by the anisotropic overlap of orbitals as well as the DE mechanism.At very low doping(x∼1)BK get a stable A-type(canted)antiferromagnetic phase and on increasing the doping the systemfirst enters the C-phase and then depending on the value of t/J AF directly gets into the ferromagnetic phase or reenters the A-phase before becoming ferromagnetic at large doping.The presence of ferromagnetic phase at large doping and C-type antiferromagnetic order at the intermediate electron doping range is rightly captured in their model.Such a sequence of phases is indeed seen in the experimen-tal phase diagram in these systems.Quite remarkably the phase diagram has almost all the magnetic phases except the G-type antiferromagnetic one that is observed experimentally in these systems at low electron doping.The phase diagram of BK,unfortunately,has two major shortcomings in it.At very low electron-doping a canted A-type antiferromagnetic phase is obtained which is stable for all values of J AF whereas experimentally G-type anti-ferromagnetic phase is observed at this end.The stability of the G-phase around x→1is quite naturally expected on physial grounds.At the y=0(x=1)end there are no electrons in the e g band,the only interaction is the antiferromagnetic exchange between neighbouring t2g spins which should lead to the three dimensional G-type antiferromagnetic order.Theother problem is that of the limiting behaviour.When the antiferromagnetic exchange in-teraction is zero or very close to zero(i.e.t/J AF→∞)the system should be completely ferromagnetic,a feature which is also missed out in their phase diagram.It appears that the designation of the A-type ordering by BK was somewhat ambiguous and that might have led to the absence of the G-phase around x=1in their phase diagram. This is particularly relevant as the typical values of canting obtained by BK in their A-phase are quite large.In their convention for different spin ordering,they chose to designate A-phase whenθxy<θz.It is apparent,therefore,that by this convention,a spin ordering with both the anglesθxy andθz close toπbutθxy<θz,could be designated as a canted A-phase.On the other hand,from the structure of spin arrangements,it should be more appropriately called a canted G-phase.Although G-phase with such large canting has not been seen experimentally(there is hardly any evidence of significant canting in the region close to x=1).This ambiguity is easily resolved if in addition one considers orbital ordering which,however,was not included in their treatment.We discuss this in more detail later on with reference to our calculations.The limit of infinite Hund’s coupling which BK worked with is unphysical for the man-ganites considered[3,9,65,67].Typical values reported in the experiments[3,4,23]and various model studies[9,41,65]and LDA calculations[67,70]do not suggest the spin spilit-tings of the e g band in various manganites to be very large.These are typically comparable to(or slightly larger than)the e g band-width.The scale of Coulomb correlations are most likely to be even higher[3,9].The other serious consequence of using such large values of Hund’s coupling is that the predictions about low energy excitations(like optical spectra, specific heat,spinfluctuation energy scales)are going to be inaccurate.BK’s calculation, though,serves as a starting point for improved theories.Based on their phase diagram BK argue that the degeneracy of orbitals and the anisotropy of hopping are crucial and the lattice(including Jahn-Teller(JT)effect)is of secondaryimportance for the physics of electron-doped manganites.This was borne out by a more refined calculation by Pai.In a more realistic treatment of the spin degrees of freedom,Pai [37]considered the limit offinite J H in the same model and succeeded in recovering the G-and F-phases.II.b.Double exchange and correlationWe mentioned earlier that by all estimates the Coulomb correlations in these systems are large[24,73,70]and it is not obvious,therefore,that the phase diagram obtained by BK will survive once these are introduced in the model.Neither of the treatments of BK or Pai includes the interactions present in the system,namely the inter-and intra-orbital Coulomb interactions as well as the longer-range Coulomb interactions.Although for low doping the local correlations are expected to be ineffective,with increase in doping they preferentially enhance the orbital ordering[38].This affects the F-phase and alters the relative stability of the A-and C-phases.The longer-range part of the interactions would tend to localize the carriers and lead to charge ordering.It is,therefore,necessary to include them in the Hamiltonian and look for their effects on the phase diagram.In the present work we have incorporated the onsite inter-and intra-orbital as well as the nearest neighbour Coulomb interactions in the model Hamiltonian and studied how these terms affect the nature of magnetic phase diagram,orbital ordering and other properties of electron doped manganites.We also set out from the double exchange model with degenerate e g orbitals and the superexchange interaction between the neighbouring t2g spins.The addition of the correlation terms makes the model very different from the ones considered by BK and Pai. Besides,the physics of charge ordering is beyond the scope of the models earlier considered.The model Hamiltonian we consider consists of two parts,thefirst part is the same as the Hamiltonian in eqn.(1)we discussed in the previous section.The second part,which is the interaction part,has onsite inter-and intra-orbital interaction and the nearest neighbourCoulomb interaction terms in it.The total Hamiltonian is thereforeH=H1+H intH1is the same as in eqn.(1)andH int=Uiαˆn iα↑ˆn iα↓+U′iσσ′ˆn i1σˆn i2σ′+V<ij>ˆn iˆn j.(5)In the above U,U′and V are the intra-and inter-orbital and the nearest neighbour Coulomb interaction strengths respectively.We treat the t2g spin subsystem quasi-classically as in BK(this is the usual practice in many of the treatments of the double exchange model [8,9,68]),but we choose to work with the more realistic limit offinite values of the Hund’s coupling.In an uncanted homogeneous ground state we choose S=S0exp(i Q.r)where the choice of Q determines the different spin arrangements for the core(t2g)spins.For example,Q=(0,0,0)would be the pure ferromagnetic phase,Q=(π,π,π)gives the G-type antiferromagnetic phase,Q=(π,π,0)is for C-type antiferromagnetic phase and finally Q=(0,0,π)reproduces A-type antiferromagnetic phase.In the infinite J H limit, the e g electron spins are forced to follow the t2g spins leading to the freezing of their spin degrees of freedom.Atfinite J H,however,the quantum nature of the transport allows for fluctuations and the e g spin degrees of freedom,along with anisotropic hopping across the two orbitals,play a central role.For canted magnetic structures where the angle between two nearest-neighbour t2g spins is different from that of the pure phases,S i is given by S i=S0(sinθi,0,cosθi)withθi taking all values between0andπ.The t2g spins are allowed to cant only in the xz−plane(this does not cause any loss of generality in the treatments that follow).We will discuss the canted structures at length in the foregoing.We begin our discussion by considering the model without the interaction terms U,U′and V.The interactions and their effects will be dealt with in detail later.。

NO Abbr aa1ABM Activity-based Management2AO Application Outsourcing3APICS American Production and Inventory4APICS Applied Manufacturing Education S5APO Advanced Planning and Optimizatio6APS Advanced Planning and Scheduling7ASP Application Service/Software Prov8ATO Assemble To Order9ATP Available To Promise10B2B Business to Business11B2C Business to Consumer12B2G Business to Government13B2R Business to Retailer14BIS Business Intelligence System15BOM Bill Of Materials16BOR Bill Of Resource17BPR Business Process Reengineering18BPM Business Process Management19BPS Business Process Standard20C/S Client/Server(C/S)\Browser/Server21CAD Computer-Aided Design22CAID Computer-Aided Industrial Design23CAM Computer-Aided Manufacturing24CAPP Computer-Aided Process Planning25CASE Computer-Aided Software Engineeri26CC Collaborative Commerce27CIMS Computer Integrated Manufacturing28CMM Capability Maturity Model29COMMS Customer Oriented Manufacturing M30CORBA Common Object Request Broker Arch31CPC Collaborative Product Commerce32CPIM Certified Production and Inventor33CPM Critical Path Method34CRM Customer Relationship Management35CRP capacity requirements planning36CTI Computer Telephony Integration37CTP Capable to Promise38DCOM Distributed Component Object Mode39DCS Distributed Control System40DMRP Distributed MRP41DRP Distribution Resource Planning42DSS Decision Support System43DTF Demand Time Fence44DTP Delivery to Promise45EAI Enterprise Application Integratio46EAM Enterprise Assets Management47ECM Enterprise Commerce Management48ECO Engineering Change Order49EDI Electronic Data Interchange50EDP Electronic Data Processing51EEA Extended Enterprise Applications 52EIP Enterprise Information Portal53EIS Executive Information System54EOI Economic Order Interval55EOQ Economic Order Quantity56EPA Enterprise Proficiency Analysis 57ERP Enterprise Resource Planning58ERM Enterprise Resource Management59ETO Engineer To Order60FAS Final Assembly Schedule61FCS Finite Capacity Scheduling62FMS Flexible Manufacturing System63FOQ Fixed Order Quantity64GL General Ledger65GUI Graphical User Interface66HRM Human Resource Management67HRP Human Resource Planning68IE Industry Engineering/Internet Exp 69ISO International Standard Organizati 70ISP Internet Service Provider71ISPE International Society for Product 72IT/GT Information/Group Technology73JIT Just In Time74KPA Key Process Areas75KPI Key Performance Indicators76LP Lean Production77MES Manufacturing Executive System78MIS Management Information System79MPS Master Production Schedule80MRP Material Requirements Planning81MRPII Manufacturing Resource Planning 82MTO Make To Order83MTS Make To Stock84OA Office Automation85OEM Original Equipment Manufacturing 86OPT Optimized Production Technology 87OPT Optimized Production Timetable88PADIS Production And Decision Informati 89PDM Product Data Management90PERT Program Evaluation Research Techn 91PLM Production Lifecycle Management 92PM Project Management93POQ Period Order Quantity94PRM Partner Relationship Management95PTF Planned Time Fence96PTX Private Trade Exchange97RCCP Rough-Cut Capacity Planning98RDBM Relational Data Base Management99RPM Rapid Prototype Manufacturing100RRP Resource Requirements Planning101SCM Supply Chain Management102SCP Supply Chain Partnership103SFA Sales Force Automation104SMED Single-Minute Exchange Of Dies105SOP Sales And Operation Planning106SQL Structure Query Language107TCO Total Cost Ownership108TEI Total Enterprise Integration109TOC Theory Of Constraints/Constraints110TPM Total Productive Maintenance111TQC Total Quality Control112TQM Total Quality Management113WBS Work Breakdown System114XML eXtensible Markup Language115ABC Classification(Activity Based Classification) 116ABC costing117ABC inventory control118abnormal demand119acquisition cost ,ordering cost120action message121action report flag122activity cost pool123activity-based costing(ABC)124actual capacity125adjust on hand126advanced manufacturing technology127advanced pricing128AM Agile Manufacturing129alternative routing130Anticipated Delay Report131anticipation inventory132apportionment code133assembly parts list134automated storage/retrieval syste135Automatic Rescheduling136available inventory137available material138available stock139available work140average inventory141back order142back scheduling143base currency144batch number145batch process146batch production147benchmarking148bill of labor149bill of lading150branch warehouse151bucketless system152business framework153business plan154capacity level155capacity load156capacity management157carrying cost158carrying cost rate159cellular manufacturing160change route161change structure162check point163closed loop MRP164Common Route Code(ID)165component-based development 166concurrent engineering167conference room pilot168configuration code169continuous improvement170continuous process171cost driver172cost driver rate173cost of stockout174cost roll-up175crew size176critical part177critical ratio178critical work center179CLT Cumulative Lead Time180current run hour181current run quantity182customer care183customer deliver lead time 184customer loyalty185customer order number186customer satisfaction187customer status188cycle counting189DM Data Mining190Data Warehouse191days offset192dead load193demand cycle194demand forecasting195demand management196Deming circle197demonstrated capacity198discrete manufacturing199dispatch to200DRP Distribution Requirements Plannin 201drop shipment202dunning letter203ECO workbench204employee enrolled205employee tax id206end item207engineering change mode flag208engineering change notice209equipment distribution210equipment management211exception control212excess material analysis213expedite code214external integration215fabrication order216factory order217fast path method218fill backorder219final assembly lead time220final goods221finite forward scheduling222finite loading223firm planned order224firm planned time fence225FPR Fixed Period Requirements226fixed quantity227fixed time228floor stock229flow shop230focus forecasting231forward scheduling232freeze code233freeze space234frozen order235gross requirements236hedge inventory237in process inventory238in stock239incrementing240indirect cost241indirect labor242infinite loading243input/output control244inspection ID245integrity246inter companies247interplant demands248inventory carry rate249inventory cycle time250inventory issue251inventory location type 252inventory scrap253inventory transfers254inventory turns/turnover 255invoice address256invoice amount gross257invoice schedule258issue cycle259issue order260issue parts261issue policy262item availability263item description264item number265item record266item remark267item status268job shop269job step270kit item271labor hour272late days273lead time274lead time level275lead time offset days276least slack per operation 277line item278live pilot279load leveling280load report281location code282location remarks283location status284lot for lot285lot ID286lot number287lot number traceability288lot size289lot size inventory290lot sizing291low level code292machine capacity293machine hours294machine loading295maintenance ,repair,and operating 296make or buy decision297management by exception298manufacturing cycle time299manufacturing lead time300manufacturing standards301master scheduler302material303material available304material cost305material issues and receipts306material management307material manager308material master,item master309material review board310measure of velocity311memory-based processing speed312minimum balance313Modern Materials Handling314month to date315move time , transit time316MSP book flag317multi-currency318multi-facility319multi-level320multi-plant management321multiple location322net change323net change MRP324net requirements325new location326new parent327new warehouse328next code329next number330No action report331non-nettable332on demand333on-hand balance334on hold335on time336open amount337open order338order activity rules339order address340order entry341order point342order point system343order policy344order promising345order remarks346ordered by347overflow location348overhead apportionment/allocation 349overhead rate,burden factor,absor 350owner's equity351parent item352part bills353part lot354part number355people involvement356performance measurement357physical inventory358picking359planned capacity360planned order361planned order receipts362planned order releases363planning horizon364point of use365Policy and procedure366price adjustments367price invoice368price level369price purchase order370priority planning371processing manufacturing372product control373product family374product mix375production activity control376production cycle377production line378production rate379production tree380PAB Projected Available Balance 381purchase order tracking382quantity allocation383quantity at location384quantity backorder385quantity completion386quantity demand387quantity gross388quantity in389quantity on hand390quantity scrapped391quantity shipped392queue time393rated capacity394receipt document395reference number396regenerated MRP397released order398reorder point399repetitive manufacturing 400replacement parts401required capacity402requisition orders403rescheduling assumption404resupply order405rework bill406roll up407rough cut resource planning 408rounding amount409run time410safety lead time411safety stock412safety time413sales order414scheduled receipts415seasonal stock416send part417service and support418service parts419set up time420ship address421ship contact422ship order423shop calendar424shop floor control425shop order , work order426shrink factor427single level where used428standard cost system429standard hours430standard product cost431standard set up hour432standard unit run hour433standard wage rate434status code435stores control436suggested work order437supply chain438synchronous manufacturing439time bucket440time fence441time zone442top management commitment443total lead time444transportation inventory445unfavorable variance, adverse446unit cost447unit of measure448value chain449value-added chain450variance in quantity451vendor scheduler,supplier schedul 452vendor scheduling453Virtual Enterprise(VE)/ Organizat 454volume variance455wait time456where-used list457work center capacity458workflow459work order460work order tracking461work scheduling462world class manufacturing excelle 463zero inventories464465Call/Contact/Work/Cost center 466Co/By-product467E-Commerce/E-Business/E-Marketing 468E-sales/E-procuement/E-partner 469independent/dependent demand470informal/formal system471Internet/Intranet/Extranet472middle/hard/soft/share/firm/group ware 473pegging/kitting/netting/nettable474picking/dispatch/disbursement lis475preflush/backflush/super backflus476yield/scrap/shrinkage (rate)477scrap/shrinkage factor478479costed BOM480engineering BOM481indented BOM482manufacturing BOM483modular BOM484planning BOM485single level BOM486summarized BOM487488account balance489account code490account ledger491account period492accounts payable493accounts receivable494actual cost495aging496balance due497balance in hand498balance sheet499beginning balance500cash basis501cash on bank502cash on hand503cash out to504catalog505category code506check out507collection508cost simulation509costing510current assets511current liabilities512current standard cost513detail514draft remittance515end of year516ending availables517ending balance518exchange rate519expense520financial accounting521financial entity522financial reports523financial statements524fiscal period525fiscal year526fixed assets527foreign amount528gains and loss529in balance530income statement531intangible assets532journal entry533management accounting534manual reconciliation535notes payable536notes receivable537other receivables538pay aging539pay check540pay in541pay item542pay point543pay status544payment instrument545payment reminder546payment status547payment terms548period549post550proposed cost551simulated cost552spending variance,expenditure var 553subsidiary554summary555tax code556tax rate557value added tax558559as of date , stop date560change lot date561clear date562date adjust563date available564date changed565date closed566date due567date in produced568date inventory adjust569date obsolete570date received571date released572date required573date to pull574earliest due date575effective date576engineering change effect date 577engineering stop date578expired date579from date580last shipment date581need date582new date583pay through date584receipt date585ship date586587allocation588alphanumeric589approver590assembly591backlog592billing593bill-to594bottleneck595bulk596buyer597component598customer599delivery600demand601description602discrete603ergonomics604facility605feature606forecast607freight608holidays609implement610ingredient611inquire612inventory613item614job615Kanban616level617load618locate619logistics620lot621option622outstanding623overhead624override625overtime626parent627part628phantom629plant630preference631priority632procurement633prototyping634queue635quota636receipt637regeneration638remittance639requisition640returned641roll642routing643schedule644shipment645ship-to646shortage647shrink648spread649statement650subassembly651supplier652transaction653what-if654655post-deduct inventory transaction 656pre-deduct inventory transaction 657generally accepted manufacturing658direct-deduct inventory transacti 659Pareto Principle660Drum-buffer-rope661663Open Database Connectivity664Production Planning665Work in Process666accelerated cost recovery system 667accounting information system668acceptable quality kevel669constant purchasing power account 670break-even analysis671book value672cost-benefit analysis673chief financial office674degree of financial leverage675degree of operating leverage676first-in , first-out677economic lot size678first-in ,still-here679full pegging680linear programming681management by objective682value engineering683zero based budgeting684CAQ computer aided quality assurance 685DBMS database management system686IP Internet Protocol687TCP T ransmission Control Protocol 689690API Advanced Process Industry691A2A Application to Application692article693article reserves694assembly order695balance-on-hand-inventory696bar code697boned warehouse698CPA Capacity Requirements Planning 699change management700chill space701combined transport702commodity inspection703competitive edge704container705container transport706CRP Continuous Replenishment Program707core competence708cross docking709CLV Customer Lifetime Value710CReM Customer Relationship Marketing 711CSS Customer Service and Support712Customer Service Representative 713customized logistics714customs declaration715cycle stock716data cleansing717Data Knowledge and Decision Suppo 718data level integration719data transformation720desktop conferencing721distribution722distribution and logistics723distribution center724distribution logistics725distribution processing726distribution requirements727DRP distribution resource planning 728door-to-door729drop and pull transport730DEM Dynamic Enterprise Module731ECR Efficient Consumer Response732e-Government Affairs733EC Electronic Commerce734Electronic Display Boards735EOS Electronic order system736ESD Electronic Software Distribution 737embedding738employee category739empowerment740engineering change effect work or 741environmental logistics742experiential marketing743export supervised warehouse744ERP Extended Resource Planning745field sales/cross sale/cross sell 746franchising747FCL Full Container Load748Global Logistics Management749goods collection750goods shed751goods shelf752goods stack753goods yard754handing/carrying755high performance organization756inland container depot757inside sales758inspection759intangible loss760internal logistics761international freight forwarding 762international logistics763invasive integration764joint distribution765just-in-time logistics766KM Knowledge Management767lead (customer) management768learning organization769LCL less than container load770load balancing771loading and unloading772logistics activity773logistics alliance774logistics center775logistics cost776logistics cost control777logistics documents778logistics enterprise779logistics information780logistics management781logistics modulus782logistics network783logistics operation784LRP Logistics Resource Planning785logistics strategy786logistics strategy management787logistics technology788MES Manufacture Execute System789mass customization790NPV Net Present Value791neutral packing792OLAP On-line Analysis Processing793OAG Open Application Group794order picking795outsourcing796package/packaging797packing of nominated brand798palletizing799PDA Personal Digital Assistant800personalization801PTF Planning time fence802POS Point Of Sells803priority queuing804PBX Private Branch Exchange805production logistics806publish/subscribe807quality of working life808Quick Response809receiving space810REPs Representatives811return logistics812ROI Return On Investment813RM Risk Management814sales package815scalability816shipping space817situational leadership818six sigma819sorting/stacking820stereoscopic warehouse821storage822stored procedure823storehouse824storing825SRM Supplier Relationship Management 826tangible loss827team building828TEM Technology-enabled Marketing829TES Technology-enabled Selling830TSR TeleSales Service Representative 831TPL Third-Part Logistics832through transport833unit loading and unloading834Value Management835value-added logistics service 836Value-chain integration837VMI Vender Managed Inventory838virtual logistics839virtual warehouse840vision841volume pricing model842warehouse843waste material logistics844workflow management845zero latency846ZLE Zero Latency Enterprise847ZLP Zero Latency Process848zero-inventory technologyCC S F NUM基于作业活动管理F10应用程序外包E21美国生产与库存管理协会ext L651实用制造管理系列培训教材ext C652先进计划及优化技术F14高级计划与排程技术F15应用服务/软件供应商L22定货组装L24可供销售量(可签约量)L31企业对企业(电子商务)F51企业对消费者(电子商务)F52企业对政府(电子商务)F53企业对经销商(电子商务)F54商业智能系统E47物料清单bom L471资源清单L43业务/企业流程重组E49业务/企业流程管理E49业务/企业流程标准E50客户机/服务器\浏览器/服务器abr L457计算机辅助设计L75计算机辅助工艺设计L76计算机辅助制造L77计算机辅助工艺设计L78计算机辅助软件工程L79协同商务E68计算机集成制造系统L73能力成熟度模型L55面向客户制造管理系统ext L653通用对象请求代理结构F70协同产品商务E69生产与库存管理认证资格ext F654关键线路法L92客户关系管理L102能力需求计划L60电脑电话集成(呼叫中心)L74可承诺的能力F56分布式组件对象模型F121分布式控制系统L122分布式MRP L123分销资源计划L125决策支持系统L110需求时界L115可承诺的交货时间F111企业应用集成E140企业资源管理E141企业商务管理F142工程变更订单D139电子数据交换L131电子数据处理F132扩展企业应用系统F152企业信息门户E143高层领导信息系统F150经济定货周期L129经济订货批量(经济批量法)L130企业绩效分析144企业资源计划L145企业资源管理L145专项设计，按订单设计L136最终装配计划L160有限能力计划L162柔性制造系统L171固定定货批量法L167总账cid D522图形用户界面F178人力资源管理L181人力资源计划L182工业工程/浏览器188国际标准化组织F194互联网服务提供商F195国际生产力促进会ext F655信息/成组技术abr F458准时制造/准时制生产L218关键过程域L220关键业绩指标F219精益生产L227制造执行系统L254管理信息系统L252主生产计划L259物料需求计划L268制造资源计划D256定货(订货)生产L249现货(备货)生产L250办公自动化L292原始设备制造商E311最优生产技术E300最优生产时刻表E301生产和决策管理信息系统L346产品数据管理L342计划评审技术L352产品生命周期管理E348项目管理353周期定量法L323合作伙伴关系管理F320计划时界L330自用交易网站F339粗能力计划L385关系数据库管理F372快速原形制造F367资源需求计划D380供应链管理L420供应链合作伙伴关系L421销售自动化L392快速换模法L408销售与运作规划L391结构化查询语言F417总体运营成本F428全面企业集成F429约束理论/约束管理L423全员生产力维护F431全面质量控制L432全面质量管理L433工作分解系统F448可扩展标记语言F153 ABC分类法T1作业成本法F2 ABC 库存控制D3反常需求D4定货费L5行为/活动(措施)信息D6活动报告标志D7作业成本集L8作业基准成本法/业务成本法L9实际能力D11调整现有库存量D12先进制造技术L13高级定价系统D16敏捷制造L17替代工序(工艺路线)D18拖期预报T19预期储备L20分摊码D23装配零件表D25自动仓储/检索系统C26计划自动重排T27可达到库存D28可用物料D29达到库存T30可利用工时T32平均库存D33欠交(脱期)订单L34倒排(序)计划/倒序排产?L35本位币D36批号D37批流程L38批量生产D39标杆瞄准(管理)sim F586工时清单D41提货单D42分库D44无时段系统L45业务框架D46经营规划L48能力利用水平L57能力负荷D58能力管理L59保管费L61保管费率D62单元式制造T63修改工序D64修改产品结构D65检查点sim D66闭环MRP L67通用工序标识T71组件(构件)开发技术F72并行(同步)工程L80会议室模拟L81配置代码D82进取不懈C84连续流程L85作业成本发生因素L86作业成本发生因素单位费用L87短缺损失L88成本滚动计算法L89班组规模D90急需零件D91紧迫系数L93关键工作中心L94累计提前期L95现有运转工时D96现有运转数量D97客户关怀D98客户交货提前期L99客户忠诚度F100客户订单号D101客户满意度F103客户状况D104周期盘点L105数据挖掘F106数据仓库F107偏置天数L108空负荷T109需求周期L112需求预测D113需求管理L114戴明环ext L116实际能力C117离散型生产L119调度D120分销需求计划L124直运C126催款信D127 ECO工作台D128在册员工D133员工税号D134最终产品D135工程变更方式标志D137工程变更通知D138设备分配D146设备管理D147例外控制D148呆滞物料分析D149急送代码T151外部集成F154加工订单T155工厂订单D156快速路径法D157补足欠交D158总装提前期D159成品D161有限顺排计划L163有限排负荷L164确认的计划订单L165确认计划需求时界L166定期用量法L168固定数量法D169固定时间法D170作业现场库存L172流水车间T173调焦预测T174顺排计划L175冻结码D176冷冻区D176冻结订单D177毛需求L179囤积库存L180在制品库存D183在库D184增值D185间接成本D186间接人工D187无限排负荷L189投入/产出控制L190检验标识D191完整性D192公司内部间D193厂际需求量T196库存周转率D197库存周期D197库存发放D198仓库库位类型D199库存报废量D200库存转移D201库存(资金)周转次数L202发票地址D203发票金额D204发票清单D205发放周期D206发送订单T207发放零件D208发放策略D209项目可供量D210项目说明D211项目编号D212项目记录T213项目备注D214项目状态D215加工车间L216作业步骤D217配套件项目D221人工工时D222延迟天数D223提前期L224提前期水平D225提前期偏置(补偿)天数D226最小单个工序平均时差C228单项产品T229应用模拟L230负荷量T231负荷报告T232仓位代码D233仓位备注T234仓位状况T235按需定货(因需定量法/缺补法)L236批量标识T237批量编号T238批号跟踪D239批量D240批量库存L241批量规划L242低层(位)码L243机器能力D244机时D245机器加载T246维护修理操作物料C247外购或自制决策D248例外管理法L251制造周期时间T253制造提前期D255制造标准D257主生产计划员L260物料L261物料可用量L262物料成本D263物料发放和接收D264物料管理L265物料经理L266物料主文件L267物料核定机构L269生产速率水平C270基于存储的处理速度F271最小库存余量L272现代物料搬运C273月累计D274传递时间L275 MPS登录标志T276多币制D277多场所D278多级D279多工厂管理F280多重仓位T281净改变法L282净改变式MRP T283净需求L284新仓位D285新组件D286新仓库D287后续编码D288后续编号D289不活动报告D290不可动用量C291急需的D293现有库存量D294挂起D295准时D296未清金额D298未结订单/开放订单L299订单活动规则D302订单地址D303订单输入T304定货点T305定货点法L306定货策略L307定货承诺T308定货备注T309定货者D310超量库位D312间接费分配L313间接费率L314所有者权益L315母件L316零件清单D317零件批次D318零件编号D319全员参治C321业绩评价L322实际库存D324领料/提货D325计划能力L326计划订单L327计划产出量L328计划投入量L329计划期/计划展望期L331使用点C332工作准则与工作规程L333价格调整D334发票价格D335物价水平D336采购订单价格D337优先计划D338流程制造D340产品控制D341产品系列D343产品搭配组合C344生产作业控制L345生产周期L347产品线D349产品率D350产品结构树T351预计可用库存(量)L354采购订单跟踪D355已分配量D356仓位数量T357欠交数量D358完成数量D359需求量D360毛需求量D361进货数量T362现有数量D363废品数量D364发货数量D365排队时间L366额定能力L368收款单据D369参考号D370重生成式MRP T371下达订单L373再订购点D374重复式生产(制造)L375替换零件D376需求能力L377请购单D378重排假设T379补库单L381返工单D382上滚D383粗资源计划D384舍入金额D386加工(运行)时间L387安全提前期L388安全库存L389保险期T390销售订单D393计划接收量(预计入库量/预期到货量)L394季节储备L395发送零件T396服务和支持D397维修件T398准备时间L399发运地址D400发运单联系人D401发货单D402工厂日历(车间日历)L403车间作业管理(控制)L404车间订单L405损耗因子(系数)D406单层物料反查表D407标准成本体系L409标准工时D410标准产品成本D411标准机器设置工时T412标准单位运转工时T413标准工资率T414状态代码D415库存控制T416建议工作单D418供应链L419同步制造/同期生产C422时段(时间段)L424时界L425时区L426领导承诺C427总提前期L430在途库存L434不利差异L435单位成本T436计量单位D437价值链L438增值链C439量差D440采购计划员/供方计划员L442采购计划法T443虚拟企业/公司L444产量差异L445等待时间L446反查用物料单L447工作中心能力L449工作流L450工作令T451工作令跟踪T452工作进度安排T453国际优秀制造业C454零库存T455456呼叫/联络/工作/成本中心abr X459联/副产品abr X460电子商务/电子商务/电子集市abr X461电子销售/电子采购/电子伙伴abr独立需求/相关需求件abr X462非/规范化管理系统abr X463互联网/企业内部网/企业外联网abr X464中间/硬/软/共享/固/群件abr X465追溯(反查)/配套出售件/净需求计算abr X466领料单(或提货单)/派工单/发料单abr X467预冲/倒冲法/完全反冲abr X468成品率/废品率/缩减率abr X469残料率（废品系数）/损耗系数abr fromchen470成本物料清单bom D472设计物料清单bom L473缩排式物料清单bom L474制造物料清单bom L475模块化物料清单bom L476计划物料清单bom L477单层物料清单bom D478汇总物料清单bom L479480账户余额cid D481账户代码cid D482分类账cid D483会计期间cid D484应付账款cid L485应收账款cid L486实际成本cid D487账龄cid D488到期余额cid D489现有余额cid D490资产负债表cid D491期初余额cid D492现金收付制cid D493银行存款cid L494现金cid L495支付给cid D496目录cid D497分类码cid D498结帐cid D499催款cid D500成本模拟cid D501成本核算cid D502流动资产cid L503流动负债cid L504现行标准成本cid C505明细cid D506汇票汇出cid D507年末cid D508期末可供量cid D509期末余额cid D510汇率cid D511费用cid D512财务会计cid L513财务实体cid L514财务报告cid D515财务报表cid D516财务期间cid D517财政年度cid D518固定资产cid L519外币金额cid D520损益cid D521平衡cid D523损益表cid D524无形资产cid L525分录cid D526管理会计cid L527手工调账cid D528应付票据cid L529应收票据cid L530其他应收款cid L531付款账龄cid D532工资支票cid D533缴款cid D534付款项目cid D535支付点cid D536支付状态cid D537付款方式cid D538催款单cid D539付款状态cid D540付款期限cid D541期间cid D542过账cid D543建议成本cid L544模拟成本cid L545开支差异cid L546明细账cid D547汇总cid D548税码cid D549税率cid D550增值税cid D551552截止日期dat D553修改批量日期dat D554结清日期dat D555调整日期dat D556有效日期dat D557修改日期dat D558结束日期dat D559截止日期dat560生产日期dat D561库存调整日期dat D562作废日期dat D563收到日期dat D564交付日期dat D565需求日期dat D566发货日期dat D567最早订单完成日期dat L568生效日期dat D569工程变更生效日期dat D570工程停止日期dat D571失效日期，报废日期dat D572起始日期dat D573最后运输日期dat T574需求日期dat D575新日期dat D576付款截止日期dat D577收到日期dat D578发运日期dat D579580已分配量sim D581字母数字sim C582批准者sim D583装配(件)sim D584未结订单/拖欠订单sim L585开单sim D587发票寄往地sim C588瓶颈资源sim L589散装sim D590采购员sim T591子件/组件sim L592客户sim D593交货sim D594需求sim D595说明sim D596离散sim D597工效学(人类工程学)sim L598设备、功能sim D599基本组件/特征件sim L600预测sim D601运费sim D602例假日sim D603实施sim D604配料、成分sim D605查询sim D606库存sim L607物料项目sim D608作业sim D609看板sim T610层次(级)sim D611负荷sim D612定位sim D613后勤保障体系；物流管理sim L614批次sim D615可选件sim L616逾期未付sim D617制造费用sim D618覆盖sim C619加班sim D620双亲(文件)sim D621零件sim D622虚拟件sim L623工厂，场所sim D624优先权sim D626优先权(级)sim D627采购sim628原形测试sim L629队列sim T630任务额，报价sim D631收款、收据sim D632全重排法sim C633汇款sim D634请购单sim L635退货sim D636滚动sim D637工艺线路sim L638计划表sim D639发运量sim D640交货地sim C641短缺sim D642损耗sim D643分摊sim D644报表sim D645子装配件sim D646供应商sim D647事务处理sim F648如果怎样-将会怎样sim C649650后减库存处理法ext T656前减库存处理法ext T657通用生产管理原则ext 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笔译测试一、解释术语1、Translation consists in?________________________________________________________________________ 2Translation as a science or an art?________________________________________________________________________ 3 Criteria of Translation________________________________________________________________________ 4What is dynamic equivalence?________________________________________________________________________ 5The methods of Translation?________________________________________________________________________ ________________________________________________________________________ 6What is method?________________________________________________________________________ 7What is diction?________________________________________________________________________ 8What is Amplification?________________________________________________________________________ 9What are purposes and obligatory?________________________________________________________________________ 10Note to omission?________________________________________________________________________二、句子翻译(2做3没做 )My holiday afternoons were spent in ramble about thesurrounding country.2）The sight of his native town called back his childhood.、I love my love with an E, because she's enticing; I hate her with anE, because she's engaged. I took her to the sign of the exquisite,and treated her with an elopement, her name's Emily, and she livesin the East.一朝春尽红颜老，花落人忘两不知。

a r X i v :h e p -p h /0605010v 1 30 A p r 2006TKYNT-06-5,IMSc/2005/4/11Is the 2-Flavor Chiral Transition of First Order?Sanatan DigalDepartment of Physics,University of Tokyo,Tokyo 113-0033,JapanRadiation Lab.,RIKEN,2-1Hirosawa,Wako,Saitama,351-0198,JapanandInstitute of Mathematical Sciences,C.I.T.Campus,Taramani,Chennai 600113,IndiaWe study the effects of the interaction between the Chiral condensate and the Polyakov loop on the chiral transition within an effective Lagrangian.We find that the effects of the interaction change the order of the phase transition when the explicit breaking of the Z N symmetry of the Polyakov loop is large.Our results suggest that the chiral transition in 2-flavor QCD may be first order.PACS numbers:PACS numbers:12.38.MhI.INTRODUCTION QCD matter with N f flavors,N colors and zero baryon chemical potential undergoes two finite temperature phase transitions,the chiral transition and the deconfinement.The chiral transition is associated with the spontaneous breaking of the chiral symmetry,SU (N f )L ×SU (N f )R →SU (N f )V ,below the critical temperature (T χ)for massless quarks.The order parameter for this transition is the chiral condensate.On the other hand the deconfinement transition is associated with the spontaneous symmetry breaking,Z N ⊂SU (N )→1,above the critical temperature T d for infinitely heavy quarks.The order parameter for this transition is the Polyakov loop expectation value.For finite non-zero quark masses both the chiral symmetry and the Z N symmetry are explicitly broken.Nevertheless these two transitions show up as crossover,second or first order transitions depending on the values of the quark masses.The chiral transition depends on the number of quark flavors and the deconfinement transition depends on the number of colors.Since the chiral symmetry and the Z N symmetry seem nothing to do with each other one would expect that these two transitions occur independently.However lattice QCD calculations have shown that both these transitions occursimultaneously,T χ=T d [1-5].Furthermore strong correlation between the chiral condensate and the Polyakov loop is observed around the phase transition point[6].This is clear evidence that there is interaction between these two order parameters.So studying the interaction between the chiral condensate and the Polyakov loop is fundamental to understanding the interplay between the chiral transition and deconfinement.There are several studies on the possible causes of the simultaneous occurrence of the chiral transition and the deconfinement transition.Mixing between the gauge and matter field operators has been suggested to explain the simultaneous chiral and deconfinement transitions [7,8].Some other studies consider that for small quark masses the chiral transition drives the deconfinement transition [6,9].Though most of the studies are concerned with why T χ=T d only a few have considered the effect of the interaction on the phase transition itself [10].It seems natural to expect that if the interaction between the two order parameters can result in the simultaneous occurrence of the two transitions then the interaction may also have important effect on the phase transition.One of the most interesting cases to study for the possible effects of this interaction is the 2-flavor chiral ttice QCD calculations have not yet been able to settle on the order of this phase transition.2 Lattice calculations by different groups do not agree on the order of this phase transition.Some lattice groupsfind the transition is second order[3]and other groupsfind the transition isfirst order[11].Conventionally this transition is believed to be second order and in the universality class of O(4)Heisenberg magnet[12].But the effect of interaction between the chiral order parameter (Φ)and the Polyakov loop(L)may change the behavior of this transition.So in the present work we investigate the effect of interaction betweenΦand L on the2-flavor chiral transition within an effective Lagrangian.Previously,the effect of interaction between the chiral order parameter and the Polyakov loop has been studied in the renormalization group approach[10].The main difference between the present work and previous studies is that we consider the explicit breaking of the Z N symmetry. Explicit breaking of the Z N symmetry can be introduced in the effective Lagrangian by terms such as∼(L+L†),∼(L+L†)Φ†Φ.Previous studies[10]have considered interaction term,such as (LL†)(Φ†Φ),which respects both the chiral and the Z N symmetry.However,in the chiral limit,the interaction terms need not respect the Z N symmetry.So terms such as∼(L+L†)Φ†Φshould be considered.As we discuss later such a interaction term is always present if the explicit Z N symmetry breaking is large,for example in the chiral limit.For simplicity we consider N=2color QCD.We expect that different N will not qualitatively change the physics we are discussing here.For N f=2 the chiral order parameterΦis a four component vectorfield whereas for N=2the Polyakov loop L is a real scalarfield.In this work we basically study the effect of the three terms,L,LΦ†Φand L2Φ†Φin the effective Lagrangian.Our main result is that the strong explicit breaking of the Z N symmetry can make the chiral transitionfirst order.We show that the chiral transition can befirst order even at the meanfield level.We also carry out numerical Monte Carlo simulations which confirm thefirst order phase transition for large enough explicit Z N symmetry breaking.We mention here that the N f=2chiral transition can befirst order from the interaction term L2Φ†Φwithout Z N symmetry breaking[13].However lattice QCD results indicate that effect of the Z N symmetric interaction term is small.A possiblefirst order chiral transition can result more likely from the explicit Z N symmetry breaking as we will argue later.Conventionally explicit symmetry breaking weakens a phase transition.But our results suggest that for a system with two order parameterfields explicit symmetry breaking can make the transition stronger.It is interesting to note that the chiral order parameterΦdoes not couple to gaugefields directly.The gaugefields seem to affect the chiral phase transition indirectly,through the Polyakov loop.We mention here that the effect of explicit symmetry breaking discussed here should not be restricted to the Z N symmetry.We expect that the explicit breaking of chiral symmetry may have some effect on the deconfinement transition in the large quark mass region.We mention here that interaction between the chiral order parameter and the diquarkfields is considered to study the chiral/color-superconducting transition at low temperature and high density[14].This paper is organized as follows.In section-II we describe the effective Lagrangian and discuss the effect of the interaction betweenΦand L on the chiral transtition.We describe our numerical Monte Carlo calculations and results in section-III.The discussions and conclusions are presented in section-IV.II.THE EFFECTIVE LAGRANGIAN AND THE EFFECT OF THE BROKEN Z NSYMMETRYWe consider the following Lagrangian[9]in3dimension for theΦand the Lfields,3 L=12(∇L)2+V(Φ,L),V(Φ,L)=m2Φ4|Φ|4+m2L4L4−gL2|Φ|2−cL|Φ|2−eL(1)Some of the parameters of this reduced3D Lagrangian depend explicitly on temperature.This Lagrangian is invariant under O(4)rotation of theΦfield.The last two terms of V(Φ,L)break the Z2symmetry(L→−L)explicitly.The interactions between the chiral order parameter and the Polyakov loop are taken care by5th and6th terms in V(Φ,L).The signs of the couplings g and c decides the correlation between thefluctuations of theΦand the Lfields.For example when c>0,the thermal average of the correlation between theΦand L fluctuations, (δL)(δ|Φ|) ,is positive.When c<0, (δL)(δ|Φ|) is negative.Such”anti-correlation”is seen between thefluctuations of the the chiral condensate and the Polyakov loop in the results of lattice QCD calculations[6].The correlation between the variations of the two order parameters with respect to temperature is of the same sign as the correlation between thefluctuations.Note that in the above Lagrangian when c=0theΦfield acts like an orderingfield for the Lfield.In the chiral symmetric phase the chiral order parameter is small and the Polyakov loop expectation value is large.The large expectation value of L,in the chiral symmetric phase,can result only from the last term in V(Φ,L)(with e>0)because|Φ|is small.As we have mentioned before previous studies[10]have considered only the interaction term gL2|Φ|2.When the coupling parameters c=0=e,the chiral transition and the deconfinement transition do not always occur simultaneously.For large value of the coupling g the transitions can occur simultaneously and are offirst order when coefficients of|Φ|2and L2are negative in Eq.1 [13,14].A large positive g would increase the critical temperature for the deconfinement transition as the coefficent of L2term is negative for high ttice results on the other hand show that inclusion of dynamical quarks decrease the deconfinement transition temperature.So in QCD the coupling g should be small.For small g both the transitions are second order and do not occur simultaneously.The coupling parameters c and e represent the strength of the Z N explicit breaking.So they should increase with decreasing quark masses as the Z N breaking becomes severe.This can be seen explicitly in the large quark mass region[15].To see the effect of the explicit Z N breaking let us consider g=0=e and c=0.At the meanfield level one can consider the temperature variation of the parameters m2Φand m2L.For simplicity wefixλφ,L>0,m2L>0and vary the m2Φparameter. Tofind the m2Φdependence of L andΦexpectation values one has to solve the following coupled equations,λΦ|Φ|3+m2Φ|Φ|−2c|Φ|L=0λL L3+m2L L−c|Φ|2=0.We have checked numerically that for large enough c these equations give two solutions which correspond to a degenerate minima of the effective potential V(Φ,L)at some particular value of m2Φ.It may seem surprising that the potential V(Φ,L)has degenerate minima even though there is no cubic term forΦand L in it.However because of the coupling c these twofields are mixed. Though the mixing anle varies as m2Φis varied.With variation of m2Φthe minimum of V(Φ,L), in the|Φ|−L plane,moves in directions other than the|Φ|or L axes.To understand how the minimum of V(Φ,L)behaves one should express V(Φ,L)in terms of variables which are the linear combinations of|Φ|and L.This would invariably lead to cubic terms of the newfields in the effective potential.For some choice of parameters the cubic term may then be important to cause degenerate4 minima of V(Φ,L).Even if the explicit symmetry breaking is not strong enough at meanfield level fluctuations at higher order can make the transitionfirst order.At one loop thefluctuations of the Φfield will contribute to a non zero3-point function of the Lfield.This3-point function can be calculated perturbatively.In the high temperature approximation the3-point function is given by,∼T c3κΦκ2Φ,m2Φ→2−4λΦ−6κΦκLκ2L,m2L→2−4λL−6κLκΦκL ,c→cκLκΦ,e→eκL.(3)a is the lattice spacing andκΦandκL are the hopping parameters for theΦand the Lfields respectively.After the rescaling of thefields and the coupling parameters the discretized lattice5 action becomes,S= x−κΦ µΦxΦx+µ+|Φx|2+λΦ(|Φx|2−1)2−κL µL x L x+µ+L2x+λL(L2x−1)2−gL2x|Φx|2−cL x|Φx|2−cL x(4) HereΦx(L x)represents the value of theΦ(L)field at the lattice site x.x+µrepresents the six nearest neighbor lattice sites to x.We adopted the pseudo heat-bath method used for the Higgs updating in SU(2)+Higgs studies[16].To updateΦx at a lattice site x we write the probability distribution P(Φx)ofΦx as,P(Φx)∼Exp[−S1(Φx)−S2(Φx)],withS1(Φx)=α Φx−A2λΦ+α2λΦ(5)The coefficientαis a parameter chosen so that we get a reasonable acceptance rate for the new Φx.Onceαis chosen,a Gaussian random number is generated according to the distribution Exp[−S1(Φx)].Then this random number is accepted as the new value ofΦx with the proba-bility Exp[−S2(Φx)].Using this procedureΦx is updated at all the lattice sites.Then we do the updating of L x along the same steps.The process of updatingΦx and L x on the entire lattice is repeated about20times between successive measurements.We measure the magnetizationsMΦ=1V x L x,(6)where V=N3s is the number of lattice sites.The expectation values of theΦand the Lfields are given by the thermal averages(average over the measurements), Φ = |MΦ| and L = M L .We take the absolute value of MΦfor Φ because a normal average of MΦis usually not a well behaved observable.The numerical simulations were carried out on a N s=16lattice.In this work we do not intent to explore the phase diagram of the model(Eq.1)but to show that for suitable choice of parameters the phase transition can change from second order tofirst order.Here we present results for two sets of parameters.For one set wefix the couplings g=e=0and for the second set wefix the coupling c=0.For thefirst set of parameters we choose,λΦ=0.004,λL=0.0020,κL=0.01and c=0.1.We observe the hysteresis of Φ and L by varying the parameterκΦ.In FIG.1we show the hysteresis loop of Φ and in FIG.2we show the hysteresis loop of L .Since we take c to be positive we see Φ and L increase or decrease simultaneously.The choice of the values for the parameters is such that the variation of Φ and L are similar in magnitude. For c<0increase in Φ should lead to decrease in L .So in this case the hysteresis loop for Φ will look somewhat similar to Fig.1while the hysteresis loop for L will be more or less inverted about the y-axis.For the second set of parameters we chooseλΦ=0.0055,λL=0.0010,κL=0.14,g=−0.02and e=0.9.The hysteresis loop of Φ and L are observed by again varying the parameterκΦ.The choice ofκL and e is such that the expectation value of L is non-zero and positive.The sign of g60 1234 5 670.138 0.14 0.142 0.144<φ>κφFIG.1:The hysteresis of Φ vs κΦ 0.51 1.52 2.53 3.544.5 5<L >κφFIG.2:The hysteresis of L vs κΦassures that increase in Φ leads to decrease in L and vice-versa as the parameter κΦis varied.The hysteresis curves of the two order parameters are shown in FIG.3and FIG.4.The values of λΦ,L in our calculations are chosen so that we can see first order phase transition clearly and the variation of Φ and L are of order O(1)across the transition point.Note that with suitable choice of κL and λL one change the average of the Polyakov loop across the transition point.The choice g was such that the chiral transition turned out to be second order when the coupling e was set to zero.We also did simulations on a 4D lattice.The results in this case are very similar to the 3D simulations.0 0.5 1 1.52 2.50.2163 0.2166 0.2169 0.2172 0.2175<φ>κφFIG.3:The hysteresis of Φ vs κΦ1.7 1.81.922.12.22.32.42.52.6 2.7 0.2163 0.2166 0.2169 0.2172 0.2175<L >κφFIG.4:The hysteresis of L vs κΦThe results in FIG.1-4show strong first order phase transition for the Φand L fields.By fixing the coupling parameters g ,c and e we observed that the strength of the transition depends on the values of the quartic couplings λΦand λL .The transition becomes weaker with increase in any of the quartic couplings λΦ,L .However even for larger quartic couplings a suitable choice of the parameters g ,c and e made the transition strong first order.7 We also did calculations with small explicit symmetry breaking for theΦfield by considering a linearΦterm in the Lagrangian.In this case we found that the hysteresis loops of both Φ and L shrinking in size with increase in the coefficient of linearΦterm in the Lagrangian.These results suggest that the transition becomes weaker when the chiral symmetry is explicitly broken.IV.DISCUSSION AND CONCLUSIONSUsing a simple effective Lagrangian,which captures the chiral symmetry and Z N symmetry of QCD,we have investigated the effect of the explicit symmetry breaking on the chiral phase transition. Since we consider the chiral limit the Z2symmetry of the Polyakov loop is explicitly broken.As we incorporate the explicit Z2breaking interaction terms in the effective Lagrangian wefind the chiral transition becomesfirst order.We observed that thefirst order transition becomes weaker when a small explicit symmetry breaking is considered for the chiral order parameterΦ.The results of our calculations show that twoflavor chiral transition is offirst order for large enough Z N explicit breaking.As we have mentioned before at present some lattice studies suggest that the transition isfirst order[11]and some other studies show the transition is second order[3].These conflicting results may be because the quark masses studied are not small enough or the lattices used are not big enough.In the previous study,for N f=2and N=3[10],the second order chiral transition becomesfirst order when coupled to the Polyakov loop.This is because the deconfinement transition isfirst order for N=3with a cubic term∼(L3+L∗3)in the effective potential with exact Z3symmetry.However when the quark masses arefinite and decrease the deconfinement transition becomes weaker.This can be understood due to explicit breaking of Z3symmetry.Already in the large quark mass region the deconfinement transition becomes crossover which implies the the explicit symmetry breaking dominates over the effects of the above Z3symmetric cubic term.For smaller quark masses the explicit Z3symmetry breaking likely grows and expected to be maximal in the chiral limit.So for smaller quark masses,i.e in the chiral limit one should rather consider the effect of the explicit breaking of the Z3symmetry.The effects of explicit symmetry breaking discussed in this work should not be restricted to the explicit breaking of the Z N symmetry.For2-flavor and2-color QCD the deconfinement transition in the large quark mass region may have the effects coming from the explicit breaking of the chiral symmetry.It may be possible that this effect change the phase transition behavior of the deconfinement transition in the heavy quark mass region. 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