Heng-Tong Ding
In this talk I review the current status of lattice QCD calculations at nonzero temperature and density. I focus on the QCD phase structure and bulk QCD thermodynamics at zero and nonzero chemical potentials.
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We summarise recent progress in simulating QCD at nonzero baryon density using complex Langevin dynamics. After a brief outline of the main idea, we discuss gauge cooling as a means to control the evolution. Subsequently we present a status report for heavy dense QCD and its phase structure, full QCD with staggered quarks, and full QCD with Wilson quarks, both directly and using the hopping parameter expansion to all orders.
These lecture notes contain an elementary introduction to lattice QCD at nonzero chemical potential. Topics discussed include chemical potential in the continuum and on the lattice; the sign, overlap and Silver Blaze problems; the phase boundary at small chemical potential; imaginary chemical potential; and complex Langevin dynamics. An incomplete overview of other approaches is presented as well. These lectures are meant for postgraduate students and postdocs with an interest in extreme QCD. A basic knowledge of lattice QCD is assumed but not essential. Some exercises are included at the end.
We present preliminary results for meson spectral functions at nonzero momentum, obtained from quenched lattice QCD simulations at finite temperature using the Maximal Entropy Method. Twisted boundary conditions are used to have access to many momenta p~T. For light quarks, we observe a drastic modification when heating the system from below to above Tc. In particular, for the vector spectral density we find a nonzero spectral weight at all energies.
In this letter we give the equation of state of QCD at finite temperatures and densities. The recently proposed overlap improving multi-parameter reweighting technique is used to determine observables at nonvanishing chemical potentials. Our results are obtained by studying n_f=2+1 dynamical staggered quarks with semi-realistic masses on N_t=4 lattices.
We extend our work on QCD thermodynamics with 2+1 quark flavors at nonzero chemical potential to finer lattices with $N_t=6$. We study the equation of state and other thermodynamic quantities, such as quark number densities and susceptibilities, and compare them with our previous results at $N_t=4$. We also calculate the effects of the addition of the charm and bottom quarks on the equation of state at zero and nonzero chemical potential. These effects are important for cosmological studies of the early Universe.
We study the phase diagram and the thermodynamic properties of QCD at nonzero isospin asymmetry at physical quark masses with staggered quarks. In particular, continuum results for the phase boundary between the normal and the pion condensation phases and the chiral/deconfinement transition are presented. Our findings indicate that the pion condensation phase is restricted to $T\lesssim170$~MeV for isospin chemical potentials up to 325~MeV. We also use the data to test the range of validity of the Taylor expansion method and show first results for the equation of state.
We discuss some selected recent developments in the field of lattice QCD at nonzero temperature and density, describing in particular the transition from the hadronic gas to the quark-gluon plasma, as seen in simulations using Wilson fermions.
For large isospin asymmetries, perturbation theory predicts the QCD ground state to be a superfluid phase of $u$ and $\bar{d}$ Cooper pairs. This phase, which is denoted as the BCS phase, is expected to be smoothly connected to the standard phase with Bose-Einstein condensation (BEC) of charged pions at $μ_I\ge m_π/2$ by an analytic crossover. A first hint for the existence of the BCS phase, which is likely characterised by the presence of both, deconfinement and charged pion condensation, is coming from the lattice observation that the deconfinement crossover smoothly penetrates into the BEC …
We study QCD at non-zero quark density, zero temperature, infinite coupling using the Glasgow algorithm. An improved complex zero analysis gives a critical point μ_c in agreement with that of chiral symmetry restoration computed with strong coupling expansions, and monomer-dimer simulations. We observe, however, two unphysical critical points: the onset for the number density μ_o, and μ_s the saturation threshold, coincident with pathological onsets observed in past quenched QCD calculations. An analysis of the probability distributions for particle number supports our physical interpretation …
Recently, a non-Hermitian chiral random matrix model was proposed to describe the eigenvalues of the QCD Dirac operator at nonzero chemical potential. This matrix model can be constructed from QCD by mapping it to an equivalent matrix model which has the same symmetries as QCD with chemical potential. Its microscopic spectral correlations are conjectured to be identical to those of the QCD Dirac operator. We investigate this conjecture by comparing large ensembles of Dirac eigenvalues in quenched SU(3) lattice QCD at nonzero chemical potential to the analytical predictions of the matrix model.…
We show that QCD Dirac spectra well below Lambda_{QCD}, both at zero and at nonzero chemical potential, can be obtained from a chiral Lagrangian. At nonzero chemical potential Goldstone bosons with nonzero baryon number condense beyond a critical value. Such superfluid phase transition is likely to occur in any system with a chemical potential with the quantum numbers of the Goldstone bosons. We discuss the phase diagram for one such system, QCD with two colors, and show the existence of a tricritical point in an effective potential approach.
The QCD phase diagram is studied in the presence of an isospin asymmetry using continuum extrapolated staggered quarks with physical masses. In particular, we investigate the phase boundary between the normal and the pion condensation phases and the chiral/deconfinement transition. The simulations are performed with a small explicit breaking parameter in order to avoid the accumulation of zero modes and thereby stabilize the algorithm. The limit of vanishing explicit breaking is obtained by means of an extrapolation, which is facilitated by a novel improvement program employing the singular va…
We study the effective action for strong-coupling lattice QCD with one-component staggered fermions in the case of nonzero chemical potential and zero temperature. The structure of this action suggests that at large chemical potentials its ground state is a crystalline `chiral density wave' that spontaneously breaks chiral symmetry and translation invariance. In mean-field theory, on the other hand, we find that this state is unstable. We show that lattice artifacts are partly responsible for this, and suggest that if this phase exists in QCD, then finding it in Monte-Carlo simulations would r…
We investigate the phase diagram of QCD with heavy quarks at finite temperature and chemical potential in the context of background field methods. In particular, we use a massive extension of the Landau-DeWitt gauge which is motivated by previous studies of the deconfinement phase transition in pure Yang-Mills theories. We show that a simple one-loop calculation is able to capture the richness of the phase diagram in the heavy quark region, both at real and imaginary chemical potential. Moreover, dimensionless ratios of quantities directly measurable in numerical simulations are in good agreem…
Two approaches are pursued to determine the magnetization of the QCD vacuum at zero and nonzero temperatures using lattice simulations. The first method builds on pressure anisotropies which are induced by the magnetic field on the lattice. The second approach is a novel, generalized version of the integral method, which exploits the independence of the theory on B for asymptotically large quark masses. Both approaches give consistent results and confirm that the QCD medium is paramagnetic. Finally, an interesting relation between QCD paramagnetism, magnetic catalysis of the QCD condensate and…
Progress in simulating QCD at nonzero baryon density requires, amongst others, substantial numerical effort. Here we propose two different expansions to all orders in the hopping parameter, preserving the full Yang-Mills action, which are much cheaper to simulate. We carry out simulations using complex Langevin dynamics, both in the hopping expansions and in the full theory, for two flavours of Wilson fermions, and agreement is seen at sufficiently high order in the expansion. These results provide support for the use of complex Langevin dynamics to study QCD at nonzero density, both in the fu…
In a Euclidean space functional integral treatment of the free energy of QCD, a chemical potential enters only through the functional determinant of the Dirac operator which for any flavor is $\dslash + m - μ_f γ_0$ (where $μ_f$ is the chemical potential for the given flavor). Any nonzero $μ$ alters all of the eigenvalues of the Dirac operator relative to the $μ=0$ value, leading to a naive expectation that the determinant is altered and which thereby alters the free energy. Phenomenologically, this does not occur at T=0 for sufficiently small $μ$, in contradiction to this naive expectation. T…
Progress in the application of the complex Langevin method to full QCD at non-zero chemical potential is reported. The method evades the sign problem which makes naive simulations at nonzero density impossible. The procedure 'gauge cooling' is used to stabilize the simulations at small enough lattice spacings. The method allows simulations also at high densities, all the way up to saturation. Simulations in a systematic hopping parameter expansion are also performed and good convergence is observed, validating the full as well as the expanded simulations.
The properties of hot hadronic matter are of great importance to the studies of heavy-ion collisions, cosmology and compact star formation. I briefly outline the current methods in use in the lattice simulations of QCD thermodynamics at zero and nonzero density. I discuss the most recent results for the QCD phase transition, critical behavior and the equation of state.
We study lattice QCD at non-vanishing chemical potential using the complex Langevin equation. We compare the results with multi-parameter reweighting both from $μ=0$ and phase quenched ensembles. We find a good agreement for lattice spacings below $\approx$0.15 fm. On coarser lattices the complex Langevin approach breaks down. Four flavors of staggered fermions are used on $N_t=4, 6$ and 8 lattices. For one ensemble we also use two flavors to investigate the effects of rooting.
Lattice QCD at finite chemical potential is difficult due to the sign problem. We use stochastic quantization and complex Langevin dynamics to study this issue. First results for QCD in the hopping expansion are encouraging. U(1) and SU(3) one link models are used to gain further insight into why the method appears to be successful.
We use the Nambu--Jona-Lasinio model to study the effects of the isospin chemical potential on the QCD phase diagram at nonzero temperature and baryon chemical potential. We find that the phase diagram is qualitatively altered by a small isospin chemical potential. There are two first order phase transitions that end in two critical endpoints, and there are two crossovers at low baryon chemical potential. These results have important consequences for systems where both baryon and isospin chemical potentials are nonzero, such as heavy ion collision experiments. Our results are in complete agree…
We determine the phase diagram of QCD on the μ-T plane for small to moderate chemical potentials. Two transition lines are defined with two quantities, the chiral condensate and the strange quark number susceptibility. The calculations are carried out on N_t =6,8 and 10 lattices generated with a Symanzik improved gauge and stout-link improved 2+1 flavor staggered fermion action using physical quark masses. After carrying out the continuum extrapolation we find that both quantities result in a similar curvature of the transition line. Furthermore, our results indicate that in leading order the …
The complex Langevin method is extended to full QCD at non-zero chemical potential. The use of gauge cooling stabilizes the simulations at small enough lattice spacings. At large fermion mass the results are compared to the HQCD approach, in which the spatial hoppings of fermionic variables are neglected, and good agreement is found. The method allows simulations also at high densities, all the way up to saturation.
The basic form of the quark condensate for arbitrary values of external magnetic field and temperature, is derived using the field equations with account of confinement. The resulting expression of the chiral condensate is shown to be proportional to square of the singlet $q\bar q$ ground state wave function at origin, $|ϕ_0(0)|^2$. For light quarks without magnetic field the condensates are proportional to $σ^{3/2}$ ($σ$ is the string tension). Numerical results are presented in 5 Tables and shown to be in good agreement with the lattice data, both for nonzero magnetic field $eB$ and temperat…
We analyse discretization effects in the calculation of high-temperature meson spectral functions at nonzero momentum and fermion mass on the lattice. We do so by comparing continuum and lattice spectral functions in the infinite temperature limit. Complete analytical results for the spectral densities in the continuum are presented, along with simple expressions for spectral functions obtained with Wilson and staggered fermions on anisotropic lattices. We comment on the use of local and point split currents.
Lattice simulations for (2+1)-flavor QCD with external magnetic field demonstrated that the quark mass is one of the important parameters responsible for the (inverse) magnetic catalysis. We discuss the dependences of chiral condensates and susceptibilities, the Polyakov loop on the magnetic field and quark mass in three degenerate flavor QCD. The lattice simulations are performed using standard staggered fermions and the plaquette action with spatial sizes Ns = 16 and 24 and a fixed temporal size Nt = 4. The value of the quark masses are chosen such that the system undergoes a first order chi…
We study the thermodynamic properties of QCD at nonzero isospin chemical potential using improved staggered quarks at physical quark masses. In particular, we discuss the determination of the equation of state at zero and nonzero temperatures and show results. Using the results for the isospin density $n_I$, we also determine the phase diagram in the $(n_I,T)$-plane.
The Hadron Resonance Gas (HRG) model is considered to study the QCD equation of state for the case of nonzero external magnetic fields. Thermodynamic observables including the pressure, energy density, entropy density, magnetization and the speed of sound are presented as functions of the temperature and the magnetic field. The magnetization is determined to be positive, indicating that the hadronic phase of QCD is paramagnetic. The behavior of the speed of sound suggests that the deconfinement transition temperature is lowered as the magnetic field grows. Moreover, a simple correspondence is …
Simulations of full QCD at nonzero baryon density using light quark masses are presented. The sign problem is evaded by the usage of the complex Langevin equation. The simulations are stabilized by the gauge cooling procedure for small lattice spacings. The method allows simulations at high densities, up to the saturation. The sign average is measured in the full as well as the phasequenched theory. Results are compared to the HQCD approach, in which the spatial hopping terms of fermionic variables are dropped, and good agreement is found at large masses.
We determine the pseudo-critical line in the temperature - chemical potential plane of 2-color QCD by direct Monte Carlo simulations and by analytic continuation from imaginary chemical potential.
In nonzero magnetic field, the magnetic properties and thermodynamics of the quantum-chromodynamic (QCD) matter is studied in the hadron resonance gas and the Polyakov linear-sigma models and compared with recent lattice calculations. Both models are fairly suited to describe the degrees of freedom in the hadronic phase. The partonic ones are only accessible by the second model. It is found that the QCD matter has paramagnetic properties, which monotonically depend on the temperature and are not affected by the hadron-quark phase-transition. Furthermore, raising the magnetic field strength inc…
We study the QCD phase diagram at nonzero baryon and isospin chemical potentials using the 1/Nc expansion. We find that there are two phase transitions between the hadronic phase and the quark gluon plasma phase. We discuss the consequences of this result for the universality class of the critical endpoint at nonzero baryon and zero isospin chemical potential.
QCD Equation of State at nonzero baryon density in external magnetic field
本文通过格点模拟在虚重子化学势下研究外部磁场对QCD状态方程的影响，发现磁场显著增强了压力展开的三个主要系数。
We determine the equation of state of QCD for nonzero chemical potentials via a Taylor expansion of the pressure. The results are obtained for N_f=2+1 flavors of quarks with physical masses, on various lattice spacings. We present results for the pressure, interaction measure, energy density, entropy density, and the speed of sound for small chemical potentials. At low temperatures we compare our results with the Hadron Resonance Gas model. We also express our observables along trajectories of constant entropy over particle number. A simple parameterization is given (the Matlab/Octave script p…
We discuss the analytic continuation beyond $μ/T \simeq 1$ in QCD at nonzero T and $μ$ by use of the Pade' approximants. The slope of the critical line obtained in this way increases at large $μ$ with respect to the second order Taylor result.In the hot phase Pade' and Taylor approximants coincide, suggesting a very large, and possibly infinite, radius of convergence of the Taylor series in this thermodynamic region.
We present results for the QCD equation of state, quark densities and susceptibilities at nonzero chemical potential, using 2+1 flavor asqtad ensembles with $N_t=4$. The ensembles lie on a trajectory of constant physics for which $m_{ud}\approx0.1m_s$. The calculation is performed using the Taylor expansion method with terms up to sixth order in $μ/T$.
We compute the equation of state of isospin asymmetric QCD at zero and non-zero temperatures using direct simulations of lattice QCD with three dynamical flavors at physical quark masses. In addition to the pressure and the trace anomaly and their behavior towards the continuum limit, we will particularly discuss the extraction of the speed of sound. Furthermore, we discuss first steps towards the extension of the EoS to small non-zero baryon chemical potentials via Taylor expansion.
We study scalar QCD at nonzero density in the strong coupling limit. It has a sign problem which looks structurally similar to the one in QCD. We show first data for the reweighting factor. After introducing dual variables by integrating out the SU(3) gauge links, we find that at least 3 flavors are needed for a nontrivial dependence on the chemical potential. In this dual representation there is no sign problem remaining. The dual variables are partially constrained, thus we propose to use a hybrid approach for the updates: For unconstrained variables local updates can be used, while for cons…
We use a hadron resonance gas model to study the QCD phase diagram at nonzero temperature, baryon, isospin and strangeness chemical potentials. We determine the temperature of the transition from the hadronic phase to the quark gluon plasma phase using two different methods. We find that the critical temperatures derived in both methods are in very good agreement. We find that the critical surface has a small curvature. We also find that the critical temperature's dependence on the baryon chemical potential at zero isospin chemical potential is almost identical to its dependence on the isospin…
We derive exact constraints relating QCD at nonzero baryon chemical potential and temperature to QCD at nonzero isospin chemical potential and temperature, a theory which can be simulated by conventional methods. These results challenge the consistency of dynamical models of superconductivity and superfluidity in QCD.
We discuss a few aspects of the phase diagram of two color lattice QCD: we investigate the long distance screening analyzing the behavior of the interquark potential at large distances; we present a first set of results for vector mesons and diquarks; we note similarities and differences between features at high temperature and high baryon density.
We present results for the topological susceptibility at nonzero temperature obtained from lattice QCD with four dynamical quark flavours. We apply different smoothing methods, including gradient Wilson flow and over--improved cooling, before calculating the susceptibility. It is shown that the considered smoothing techniques basically agree among each other, and that there are simple scaling relations between flow time and the number of cooling/smearing steps. The topological susceptibility exhibits a surprisingly slow decrease at high temperature.
We investigate the eigenvalue spectrum of the staggered Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis.
The static longitudinal and transverse gluon propagators in the Landau gauge are studied in $2+1$ lattice QCD with nonzero isospin chemical potential $μ_I$. Parameterization of the momentum dependence of the propagators is provided for all values of the chemical potential under study. We find that the longitudinal propagator is infrared suppressed at nonzero $μ_I$ with suppression increasing with increasing $μ_I$. It is found, respectively, that the electric screening mass is increasing with increasing $μ_I$. Additionally, we analyze the difference between two propagators as a function of the …
We study what happens to D and D_s mesons as the temperature increases, using lattice QCD simulations with N_f=2+1 dynamical flavours on anistropic lattices. We have access to five temperatures in the hadronic phase. Using the determined groundstate mass at the lowest temperature, we investigate the effect of rising temperature by analysing ratios of mesonic correlators, without the need for further fitting or spectral reconstruction. In the pseudoscalar and vector channels, we demonstrate that temperature effects are at the percent level and can be captured by a reduction of the groundstate m…
The maximum entropy method is used to compute the quark spectral function at nonzero temperature. We solve the gap equation of quantum chromodynamics (QCD) self-consistently, employing a rainbow kernel which phenomenologically models results from Dyson-Schwinger equations (DSE) and lattice QCD. We use the criterion of positivity restoration of the spectral function as a signal for deconfinement. Our calculation indicates that the critical temperature of deconfinement $T_d$ is slightly smaller than the one of chiral symmetry restoration $T_c$: $T_d\sim 94% T_c$ in the chiral limit, and $T_d\sim…
We determine the phase diagram and chiral condensate for lattice QCD with two flavors of twisted-mass fermions in the presence of nondegenerate up and down quarks, discretization errors and a nonzero value of $Θ_{\rm QCD}$. Although such a theory has a complex action and cannot, at present, be simulated, the results are needed to understand how to tune to maximal twist in the presence of electromagnetism, a topic discussed in a companion paper. We find that, in general, the only phase structure is a first-order transition of finite length. Pion masses are nonvanishing throughout the phase plan…
QCD Equation of State with Strong Magnetic Fields and Nonzero Baryon Density
本文通过格点QCD模拟和泰勒展开方法，研究了强磁场与非零重子化学势下$(2+1)$味QCD的状态方程，给出了主要热力学量领头阶系数的连续极限结果，并揭示了强磁场的显著影响。
We identify the Goldstone modes appropriate to the low and high density phases of SU(2) lattice gauge theory with staggered fermions. We present hybrid Monte Carlo simulation results for susceptibilities on a 6^4 lattice at beta=1.5, m=0.05. We specify how to implement a diquark source term in a lattice simulation and present first measurements of the lattice diquark condensate as a function of the diquark source.
In the high-temperature phase of QCD meson spectral functions at nonzero momentum are expected to have a nontrivial and interesting structure. In order to provide a reference point for lattice studies employing e.g. the Maximal Entropy Method, we discuss several characteristics of meson spectral functions in the infinite-temperature limit. We report on ongoing work in quenched QCD with staggered fermions.
In the $ε$-domain of QCD we have obtained exact analytical expressions for the eigenvalue density of the Dirac operator at fixed $θ\ne 0$ for both one and two flavors. These results made it possible to explain how the different contributions to the spectral density conspire to give a chiral condensate at fixed $θ$ that does not change sign when the quark mass (or one of the quark masses for two flavors) crosses the imaginary axis, while the chiral condensate at fixed topological charge does change sign. From QCD at nonzero density we have learnt that the discontinuity of the chiral condensate …
The behavior of quenched QCD at nonzero chemical potential $μ$ has been a long-standing puzzle. An explicit solution is found using the random matrix approach to chiral symmetry breaking. At nonzero $μ$ the quenched QCD is not a simple $n\to0$ limit of a theory with $n$ quarks: a naive `replica trick' fails. A limit that leads to the quenched QCD is that of a theory with $2n$ quarks: $n$ quarks with original action and $n$ quarks with conjugate action.
According to perturbation theory predictions, QCD matter in the zero-temperature, high-density limits of QCD at nonzero isospin chemical potential is expected to be in a superfluid Bardeen-Cooper-Schrieffer (BCS) phase of $u$ and $\bar{d}$ Cooper pairs. It is also expected, on symmetry grounds, that such phase connects via an analytical crossover to the phase with Bose-Einstein condensation (BEC) of charged pions at $μ_\text{I}\geq m_π/2$. With lattice results, showing some indications that the deconfinement crossover also smoothly penetrates the BEC phase, the conjecture was made that the for…
Numerical simulations of quantum chromodynamics at nonzero temperature provide information from first principles about the physical properties of the quark gluon plasma. Because the lattice approximation can be refined indefinitely, results of lattice simulations now provide the most reliable basis for our understanding of the nonperturbative characteristics of the plasma and of the high temperature phase transition. Following a brief overview of the methodology of lattice gauge theory at nonzero temperature, recent results and insights from lattice simulations are discussed. These include our…
We introduce a random matrix model with the symmetries of QCD at finite temperature and chemical potentials for baryon number and isospin. We analyze the phase diagram of this model in the chemical potential plane for different temperatures and quark masses. We find a rich phase structure with five different phases separated by both first and second order lines. The phases are characterized by the pion condensate and the chiral condensate for each of the flavors. In agreement with lattice simulations, we find that in the phase with zero pion condensate the critical temperature depends in the s…
Generalized Parton Distributions from Lattice QCD with Asymmetric Momentum Transfer: Unpolarized Quarks at Nonzero Skewness
将非对称参考系形式推广到非零斜率，通过洛伦兹不变振幅从晶格数据提取广义部分子分布H和E，并讨论主要挑战。
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) and U(1) gauge theory as well as in full QCD with two colors and finite chemical potential. Along the strong-coupling axis up to the phase transition, the low-lying Dirac spectrum of these quantum field theories is well described by random matrix theory and exhibits universal behavior. Related results for gauge theories with minimal coupling are discussed in the chirally symmetric phase and no universality is seen for the microscopic spectral densities.
The simulation of dense fermionic matters is a long-standing problem in lattice gauge theory. One hopeful solution would be the use of quantum computers. In this paper, digital quantum simulation is designed for lattice gauge theory at nonzero density. The quantum variational algorithm is adopted to obtain the ground state at nonzero density. A benchmark test is performed in the lattice Schwinger model.
We extend a previous investigation of the QCD phase diagram with heavy quarks in the context of background field methods by including the two-loop corrections to the background field effective potential. The nonperturbative dynamics in the pure-gauge sector is modeled by a phenomenological gluon mass term in the Landau-DeWitt gauge-fixed action, which results in an improved perturbative expansion. We investigate the phase diagram at nonzero temperature and (real or imaginary) chemical potential. Two-loop corrections yield an improved agreement with lattice data as compared to the leading-order…
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