Finite-Temperature QCD on the Lattice

Research ObjectivesThe MILC Collaboration is engaged in a broad research program in Quantum Chromodynamics (QCD).This research addresses fundamental questions in high energy and nuclear physics,and is directly related to major experimental programs in thesefields.It includes studies of the mass spectrum of strongly interacting particles,the weak interactions of these particles,and the behavior of strongly interacting matter under extreme conditions.The Standard Model of High Energy Physics encompasses our current knowledge of the funda-mental interactions of subatomic physics.It consists of two quantumfield theories:the Weinberg-Salaam theory of electromagnetic and weak interactions,and QCD,the theory of the strong interac-tions.The Standard Model has been enormously successful in explaining a wealth of data produced in accelerator and cosmic ray experiments over the past thirty years;however,our knowledge of it is incomplete because it has been difficult to extract many of the most interesting predictions of QCD,those that depend on the strong coupling regime of the theory,and therefore require non-perturbative calculations.At present,the only means of carrying out non-perturbative QCD calculations fromfirst principles and with controlled errors,is through large scale numerical sim-ulations within the framework of lattice gauge theory.These simulations are needed to obtain a quantitative understanding of the physical phenomena controlled by the strong interactions,to de-termine a number of the fundamental parameters of the Standard Model,and to make precise tests of the Standard Model’s range of validity.Despite the many successes of the Standard Model,it is believed by high energy physicists that to understand physics at the shortest distances,a more general theory,which unifies all four of the fundamental forces of nature,will be required.The Standard Model is expected to be a limiting case of this more general theory,just as classical mechanics is a limiting case of the more general quantum mechanics.A central objective of the experimental program in high energy physics,and of lattice QCD simulations,is to determine the range of validity of the Standard Model,and to search for new physics beyond it.Thus,QCD simulations play an important role in efforts to obtain a deeper understanding of the fundamental laws of physics.QCD is formulated in the four-dimensional space-time continuum;however,in order to carry out numerical calculations one must reformulate it on a lattice or grid.It should be emphasized that the lattice formulation of QCD is not merely a numerical approximation to the continuum formu-lation.The lattice regularization of QCD is every bit as valid as continuum regularizations.The lattice spacing a establishes a momentum cutoffπ/a that removes ultraviolet divergences.Stan-dard renormalization methods apply,and in the perturbative regime they allow a straightforward conversion of lattice results to any of the standard continuum regularization schemes.Lattice QCD calculations proceed in two steps.In thefirst,one uses importance sampling tech-niques to generate gauge configurations,which are representative samples from the Feynman path integrals that define QCD.These configurations are saved,and in the second step they are used to calculate a wide variety of physical quantities.It is necessary to generate configurations with a range of lattice spacings,and then perform extrapolations to the zero lattice spacing limit.Fur-thermore,the computational cost of calculations rises as the masses of the quarks,the fundamental constituents of strongly interacting matter,decrease.Until recently,it has been too expensive to carry out calculations with the masses of the two lightest quarks,the up and the down,set to their physical values.Instead,one has performed calculations for a range of up and down quark masses, and extrapolated to their physical values guided by chiral perturbation theory,an effectivefield theory that determines how physical quantities depend on the masses of the lightest quarks.The extrapolations in lattice spacing(continuum extrapolation)and quark mass(chiral extrapolation) are the major sources of systematic errors in QCD calculations,and both must be under control in order to obtain trustworthy results.In our current simulations,we are,for thefirst time,working at or near the physical masses of the up and down quarks.The gauge configurations produced in these simulations greatly reduce,and will eventually eliminate,the systematic errors associatedwith the chiral extrapolation.A number of different formulations of QCD on the lattice are currently in use by lattice gauge theorists,all of which are expected to give the same results in the continuum limit.In recent years, major progress has been made in thefield through the development of improved formulations(im-proved actions)which reducefinite lattice spacing artifacts.Approximately twelve years ago,we developed one such improved action called asqtad[1],which significantly increased the accuracy of our simulations for a given amount of computing resources.We have used the asqtad action to generate an extensive library of gauge configurations with small enough lattice spacings and light enough quark masses to perform controlled calculations of a number of physical quantities. Computational resources provided by the DOE and NSF have enabled us to complete our program of generating asqtad gauge configurations.These configurations are publicly available,and have been used by us and by other groups to study a wide range of physical phenomena of importance in high energy and nuclear physics.Ours was thefirst set of full QCD ensembles that enabled control over both the continuum and chiral extrapolations.We have published a review paper describing the asqtad ensembles and the many calculations that were performed with them up to2009[2]. Over the last decade,a major component of our work has been to use our asqtad gauge config-urations to calculate quantities of importance to experimental programs in high energy physics. Particular emphasis was placed on the study of the weak decays and mixings of strongly interact-ing particles in order to determine some of the least well known parameters of the standard model and to provide precise tests of the standard model.The asqtad ensembles have enabled the calcu-lation of a number of physical quantities to a precision of1%–5%,and will enable many more quantities to be determined to this precision in the coming years.These results are already having an impact on experiments in high energy physics;however,in some important calculations,partic-ularly those related to tests of the standard model,higher precision is needed than can be provided by the existing asqtad ensembles.In order to obtain the required precision,we are now working with the Highly Improved Staggered Quark(HISQ)action developed by the HPQCD Collabora-tion[3].We have performed tests of scaling in the lattice spacing using HISQ valence quarks with gauge configurations generated with HISQ sea quarks[4].We found that lattice artifacts for the HISQ action are reduced by approximately a factor of2.5from those of the asqtad action for the same lattice spacing,and taste splittings in the pion masses are reduced by approximately a factor of three,which is sufficient to enable us to undertake simulations with the mass of the Goldstone pion at or near the physical pion mass.(“Taste”refers to the different ways one can construct the same physical particle in the staggered quark formalism.Although particles with different tastes become identical in the continuum limit,their masses can differ atfinite lattice spacing).More-over,the improvement in the quark dispersion relation enables us to include charm sea quarks in the simulations.The properties of the HISQ ensembles are described in detail in Ref.[5],and the first physics calculations using the physical quark mass ensembles in Refs.[6,7,8].The current status of the HISQ ensemble generation project is described at the link HISQ Lattice Generation and some initial calculations with them at Recent Results.The HISQ action also has major advan-tages for the study of QCD at high temperatures,so we have started to use it in our studies of this subject.Projects using the HISQ action will be a major component of our research for the next several years.Our research is currently focused on three major areas:1)the properties of light pseudoscalar mesons,2)the decays and mixings of heavy-light mesons,3)the properties of strongly interacting matter at high temperatures.We briefly discuss our research in each of these areas at the link Recent Results.References[1]The MILC Collaboration:C.Bernard et al.,Nucl.Phys.(Proc.Suppl.),60A,297(1998);Phys.Rev.D58,014503(1998);G.P.Lepage,Nucl.Phys.(Proc.Suppl.),60A,267(1998);Phys.Rev.D59,074501(1999);Kostas Orginos and Doug Toussaint(MILC),Nucl.Phys.(Proc.Suppl.),73,909(1999);Phys.Rev.D59,014501(1999);Kostas Orginos,Doug Tou-ssaint and R.L.Sugar(MILC),Phys.Rev.D60,054503(1999);The MILC Collaboration:C.Bernard et al.,Phys.Rev.D61,111502(2000).[2]The MILC Collaboration: A.Bazavov et al.,Rev.Mod.Phys.82,1349-1417(2010)[arXiv:0903.3598[hep-lat]].[3]The HPQCD/UKQCD Collaboration: E.Follana et al.,Phys.Rev.D73,054502(2007)[arXiv:hep-lat/0610092].[4]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D82,074501(2010)[arXiv:1004.0342].[5]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D87,054505(2013)[arXiv:1212.4768].[6]The MILC Collaboration: A.Bazavov et al.,Phys.Rev.Lett.110,172003(2013)[arXiv:1301.5855].[7]The Fermilab Lattice and MILC Collaborations:A.Bazavov,et al.,Phys.Rev.Lett.112,112001(2014)[arXiv:1312.1228].[8]The MILC Collaboration:A.Bazavov et al.,Proceedings of Science(Lattice2013)405(2013)[arXiv:1312.0149].。

a r X i v :h e p -l a t /9709021v 1 9 S e p 1997Dual variables for the SU (2)lattice gauge theory at finite temperatureSrinath CheluvarajaTheoretical Physics GroupTata Institute of Fundamental ResearchHomi Bhabha Road,Mumbai 400005,IndiaWe study the three-dimensional SU (2)lattice gauge theory at finite temperature using an observable which is dual to the Wilson line.This observable displays a behaviour which is the reverse of that seen for the Wilson line.It is non-zero in the confined phase and becomes zero in the deconfined phase.At large distances,it’s correlation function falls offexponentially in the deconfined phase and remains non-zero in the confined phase.The dual variable is non-local and has a string attached to it which creates a Z (2)interface in the system.It’s correlation function measures the string tension between oppositely oriented Z (2)domains.The construction of this variable can also be made in the four-dimensional theory where it measures the surface tension between oppositely oriented Z (2)domains.e-mail:srinath@theory.tifr.res.in1Dual variables have played an important role in statistical mechanical systems[1].These variables display a behaviour which is the opposite of that seen for the order parameters.They are non-zero in the disordered phase and remain zero in the ordered phase.Hence they are commonly referred to as disorder variables.Unlike the order parameters which are local observables and measure long range order in a statistical mechanical system,the dual variables are non-local and are sensitive to disordering effects which often arise as a consequence of topological excitations supported by a system-like vortices,magnetic monopoles etc.Disorder variables for the U(1)LGT have been studied recently[2].In this paper we study thefinite temperature properties of the three-dimensional SU(2)lattice gauge theory using an observable which is dual to the Wilson line.We explain the sense in which this is dual to the Wilson line and show that it’s behaviour is the reverse of that observed for the Wilson line.Unlike the Wilson line which creates a static quark propagating in a heat bath,the dual variable creates a Z(2)interface in the system.The definition of this variable can also be extended to the four-dimensional theory.Before we consider the three-dimensional SU(2)lattice gauge theory let us briefly recall the construction of the dual variable for the two-dimensional Ising model[3].The variable dual to the spin variableσ( n) is denoted byµ(⋆ n)and is defined on the dual lattice.This variable which is shown in Fig.1has a string attached to it which pierces the bonds connecting the spin variables.The position of the string is notfixed and it can be varied using a Z(2)(σ( n)→−σ( n))transformation.The average value of the dual variable is defined asZ(˜K)<µ(⋆ n)>=The dual variableµ(⋆ n)thus creates an interface beginning from⋆ n.It has the following behaviour at high and low temperatures[3]<µ(⋆ n)>≈1for K small<µ(⋆ n)>≈0for K large.It is in this sense that the variableµ(⋆ n)is dual to the variableσ( n)which behaves as<σ( n)>≈0for K small<σ( n)>≈1for K large.The spin and dual correlation functions satisfy the relation<µ(⋆ n)µ(⋆ n′)>K>>1=<σ( n)σ( n′)>K<<1.(3) Using theσ→−σtransformation it can be shown that the correlation function of theµ’s is independent of the shape of the string joining⋆ n and⋆ n′.The variablesσ( n)andµ(⋆ n)satisfy the algebraσ( n)µ(⋆ n)=µ(⋆ n)σ( n)exp(iω),(4) whereω=0if the variableσdoes not lie on a bond pierced by the string attached toµ(⋆ n)andω=πotherwise.The above considerations generalize easily to the three-dimensional Z(2)gauge theory.The dual variables are again defined on the sites of the dual lattice and the string attached to them will now pierce plaquettes instead of bonds.Whenever a plaquette is pierced by a string the coupling constant changes sign just as in the case of the Ising model.One can similarly define correlation functions of these variables.Since the three-dimensional Z(2)gauge theory is dual to the the three-dimensional Ising model,the correlation functions of these variables will have a behaviour which is the reverse of the spin-spin correlation function in the three-dimensional Ising model.For the case of the SU(2)lattice gauge theory which is our interest here,the definition of these variables is more involved.However,since Z(2)is a subgroup of SU(2)one can define variables which are dual to the Z(2)degrees of freedom by following the same prescription as in the three-dimensional Z(2)gauge theory.The relevance and effectiveness of these variables will depend3on the role played by the Z(2)degrees of freedom in the SU(2)lattice gauge theory.The role of the center degrees of freedom in the SU(2)lattice gauge theory was also examined in[4].Since thefinite temperature transition in SU(N)lattice gauge theories is governed by the center(Z(N) for SU(N))degrees of freedom[5],we expect these variables to be useful in studying this transition.The usual analysis offinite temperature lattice gauge theories is carried out by studying the behaviour of the Wilson line which becomes non-zero across thefinite temperature transition[5].The non-zero value of the Wilson line indicates deconfinement of static quarks.The spatial degrees of freedom undergo no dramatic change across the transition and only serve to produce short-range interactions between the Wilson lines. Thus one gets an effective theory of Wilson lines in one lower dimension[6].The deconfinement transition can be monitored by either measuring the expectation value of the Wilson line or by looking at the behaviour of the Wilson line correlation function[7].In the confining phase,the correlation function is(for| n− n′| large)<L( n)L( n′)>≈exp(−σT| n− n′|)(5) while in the deconfining phase<L( n)L( n′)>≈constant.(6) We define the variableµ(⋆ n)on the dual lattice site⋆ n asµ(⋆ n)=Z(˜β)2ptr U(p).(8)The variablesµ(⋆ n)and L( n)satisfy the algebra4L( n)µ(⋆ n)=µ(⋆ n)L( n)exp(iω)(9) whereω=0if the plaquette pierced by the string attached toµ(⋆ n)is not touching any of the links belonging to L( n)andω=πif the plaquette makes contact with any of the links of L( n).The variables µ(⋆ n)and L( n)satisfy the same algebra as theσandµvariables in the Ising model.This is the same as the algebra of the order and disorder variables in[8].Note that this algebra is only satisfied if the string is taken to be in the spatial direction.The location of the string can again be changed by local Z(2) transformations.The correlation function of the dual variables is defined to beZ(˜β)<µ(⋆ x)µ(⋆ y)>=)Nτ n n′J( n− n′)trL( n)trL( n′).(12)2The term which gives this contribution is shown in Fig.2.When we calculate the correlation function in Eq.10(where x and y are only separated in space)using this approximation,one plaquette occurring in this diagram will contribute with the opposite sign(shown shaded in Figure.2)and will cause the bond between n and n′to have a coupling with the opposite sign.In Eq.12J( n− n′)contains the sign induced5on the bond.This feature will persist for every diagram contributing to the effective two-dimensional Ising model and it’s effect will be to create a disorder line from x to y.Thus this correlation function will behave exactly like the disorder variable in the two-dimensional Ising model and at large distances will fall offexponentially in the ordered phase and will approach a constant value in the disordered phase.We expect it to behave(for large| x− y|)as<µ( x)µ( y)>≈exp(−| x− y|/ξ)β>βcr<µ( x)µ( y)>≈µ2β<βcrWriting the above correlation function as<µ( x)µ( y)>=exp(−βτ(F( x− y))(13) we can interpret F as the free energy of an interface of length| x− y|.The inverse temperature is denoted byβτto distinguish it from the gauge theory couplingβ.In the ordered phase the interface energy increases linearly with the length of the interface while in the disordered phase it is independent of the length.In thefinite temperature system high temperature results in the ordering of the Wilson lines and low temperature results in the disordering of the Wilson lines.Therefore the dual variables will display ordering at low temperatures and disordering at high temperatures.A direct measurement of the dual variable results in large errors because the dual variable is the expo-nential of a sum of plaquettes andfluctuates greatly.We have directly measured the dual variable and the correlation function and found that they fall to zero at high temperatures and remain non-zero at low temperatures.Since the measurement had large errors we prefer to use the method in[11]where a similar problem was encountered in the measurement of the disorder variable in the U(1)LGT.Instead of directly measuring the correlation function we measure∂ln<µ>ρ( x, y)=−where p′denotes the plaquettes which are dual to the string joining x and y.In our case this quantity directly measures the free energy of the Z(2)interface between x and y.Hence we expect it to increase linearly with the interface length in the deconfining phase and approach a constant value in the confining phase.Also this variable is like any other statistical variable and is easier to measure numerically.The variableρcan be used to directly measure the interface string tension between oppositely oriented Z(2) domains.The behaviour of the quantityρis shown in Fig.3and Fig.4.In the confined phaseρapproaches a constant value at large distances while it increases linearly with distance in the deconfined phase.The slope of the straight line in Fig.3gives the interface string tension.The calculation ofρwas made on a 12∗∗23lattice with200000iterations.The values ofβused were2.5in the confined phase and5.5in the deconfined phase.The deconfinement transition on the Nτ=3lattice occurs atβ=4.1[10].The errors were estimated by blocking the data.We would now like to point out a few applications of these dual variables.The mass gap in the high temperature phase is determined by studying the large distance behaviour of the Wilson line correlation function.Since the Wilson line correlation function remains non-zero in the deconfined phase the long distance part is subtracted out to get the leading exponential.The dual variable correlation function already displays an exponential fall offin the high temperature phase and provides us with another method of estimating the mass gap.Also,since dual variables reverse the roles of strong and weak coupling,they provide an alternate way of looking at the system which may be convenient to address certain questions. In this case they can be used to determine the string tension between oppositely oriented Z(2)domains in the SU(2)gauge theory.The surface tension between oppositely oriented Z(2)domains in the four-dimensional theory has been calculated semi-classically in[12].The above construction of the dual variable can also be made in four dimensions.The only difference is that in four dimensions the dual variables are defined on loops in the dual lattice.The spatial string in three-dimensions is replaced by a spatial surface which has the loops as it’s the boundary.The dual variables are functionals of the surface bounding the loops.The correlation function of the dual variables is defined to be<µ(C,C′)>=<exp(−β p′tr U(p))>(16)7where the summation is over all plaquettes which are dual to the surface joining C and C′.Since the surface is purely spatial the plaquettes contributing to the summation are all space-time plaquettes.This correlation function will fall of exponentially as the area of the surface joining C and C′in the deconfined phase and will approach a constant value in the confined phase.A similar measurement ofρcan be used to determine the surface tension between oppositely oriented Z(2)domains in the four-dimensional gauge theory.8........................X FIG.1.Dual variable in the Ising model.10n n′333333 FIG.3.ρin the deconfining phase.12333333 FIG.4.ρin the confining phase.13。

a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。

当前LatticeQCD的国内外研究现状Z 4430 Belle 在道发现一个共振结构如果实验上确认其存在，则最小的夸克组分应该是其质量与和的阈十分接近量子数还没有确定。

一些理论家把它解释为可能的 S-wave分子态或者四夸克态候选者。

对此，我们进行了两方面的研究：计算的散射；用四夸克场算符计算可能的质量谱，用单粒子态和两粒子态有限个点上对体积的依赖关系来判断所得到的态是散射态还是束缚态。

这两方面的都得到了一些初步结果。

散射根据Luscher公式，在有限的格点上，两粒子散射系统的能量、两粒子自由能量以及低能散射的散射长度之间有如下关系散射能量两粒子自由能量散射长度约化质量空间体积 a 0.204fm 400confs. a0.144fm 400confs. Beta 2.5的结果 Beta 2.8的结果表示 A1 A2 B1 B2 E 2.5 0.370 33 0.494 60 0.397 64 0.432 63 0.267 33 2.8 0.624 56 0.507 76 0.523 77 0.517 78 0.667 60 手征外推后散射长度（单位：fm 的结果：初步结论：在所有的情况下，的S-wave散射的散射长度为正，说明之间的相互作用是吸引性的。

至于这种吸引性的相互作用的强弱、以及能否构成束缚态，还需要进一步的理论分析。

小结：井冈山长征抗战战略防御战略相持战略进攻解放战争建国萌芽生存成长成功 2.我们的方针制胜三大法宝：群众路线：CLQCD努力工作，合作攻关。

理论联系实际：密切结合BES试验，关注国际实验新结果。

统一战线：希望大家多多支持，多多建议。

1.我们所处的阶段谢谢！基态是单粒子态还是两粒子态？可能的判据：Weight的体积效应。

单粒子态：两粒子态：当前Lattice QCD的国内外研究现状陈莹中国科学院高能物理研究所报告内容一、Lattice QCD简介二、国际研究现状三、国内研究现状四、关于X 3872 和Z 4430 五、小结一、Lattice QCD 简介 Wick 转动Euclidean 空间QCD作用量路径积分时空离散化连续时空四维超立方格点体系无限自由度有限自由度路径积分量子化生成泛函: 物理观测量: 算符的真空平均值淬火近似 quenched approximation : 物理含义: 不考虑夸克真空极化图,即忽略海夸克效应 Monte Carlo 模拟重点抽样：根据Boltzmann 分布产生由有限数量的位形构成的统计系综，计算可观测量的系综平均值，样本越大，统计误差越小。

量子色动力学相变的临界行为量子色动力学（Quantum Chromodynamics，简称QCD）是描述强相互作用的理论，它在粒子物理中起着重要的作用。

在高温和高能量密度条件下，QCD相变会导致强子系统中的自由夸克和胶子相互作用的改变，这种相变被称为量子色动力学相变（Quantum Chromodynamics Phase Transition）。

本文将讨论量子色动力学相变的临界行为。

1. 引言量子色动力学是标准模型的一部分，它描述了夸克和胶子之间的相互作用。

在冷却高温夸克胶子等离子体时，会发生从强子相到夸克-胶子等离子体的相变。

在相变过程中，系统的热力学性质发生了显著变化，这种变化被称为临界现象。

量子色动力学相变的临界行为一直是研究者关注的焦点。

2. 临界行为的表征量子色动力学相变的临界行为可以通过临界指数来表征。

临界指数是指在临界点附近各种物理量的行为方式。

其中，最常用的是比热容、磁化率和相关长度的临界指数。

3. 临界指数（1）比热容的临界指数在量子色动力学相变的临界点附近，比热容的行为可以用下式描述：C_v \sim |T - T_c|^{-\alpha}$$式中，$C_v$为比热容，$T$为温度，$T_c$为临界温度，$\alpha$为比热容的临界指数。

临界指数$\alpha$的数值决定了比热容在临界点附近的行为。

（2）磁化率的临界指数磁化率是描述系统磁现象的物理量，它在临界点附近的行为可以用下式表示：$$\chi \sim |T - T_c|^{-\gamma}$$式中，$\chi$为磁化率，$\gamma$为磁化率的临界指数。

磁化率的临界指数$\gamma$决定了磁化率在临界点附近的变化行为。

（3）相关长度的临界指数临界点附近的相关长度也能够描述临界行为。

相关长度可以用下式表示：$$\xi \sim |T - T_c|^{-\nu}式中，$\xi$为相关长度，$\nu$为相关长度的临界指数。