Many Polyakov loop models can be written in a dual formulation which is free of sign problem even when a non-vanishing baryon chemical potential is introduced in the action. Here, results of numerical simulations of a dual representation of one such effective Polyakov loop model at finite baryon density are presented. We compute various local observables such as energy density, baryon density, quark condensate and describe in details the phase diagram of the model. The regions of the first order phase transition and the crossover, as well as the line of the second order phase transition, are established. We also compute several correlation functions of the Polyakov loops.
We compare the grand canonical partition function at fixed chemical potential mu with the canonical partition function at fixed baryon number B, formally and by numerical simulations at mu=0 and B=0 with four flavours of staggered quarks. We verify that the free energy densities are equal in the thermodynamic limit, and show that they can be well described by the hadron resonance gas at T < T_c and by the free fermion gas at T>T_c. Small differences between the two ensembles, for thermodynamic observables characterising the deconfinement phase transition, vanish with increasing lattice size. T…
The 't Hooft-Veneziano limit of various U(N) and SU(N) Polyakov loop models is constructed at finite temperature and non-zero baryon chemical potential. In this paper we calculate the free energy, its derivatives, the Polyakov loop expectation values and the baryon density. The phase diagram is described in details.
We compare SU(2) Polyakov loop models with different effective actions with data from full two-color QCD simulations around and above the critical temperature. We then apply the effective theories at finite temperature and density to extract quantities like Polyakov loop correlators, effective Polyakov loop potentials and baryon density.
We present a study of the 1+1 flavor system of strongly interacting matter in terms of the Polyakov$-$Nambu$-$Jona-Lasinio model. We find that though the small isospin symmetry breaking brought in through unequal light quark masses is too small to affect the thermodynamics of the system in general, it may have significant effect in baryon-isospin correlations and have a measurable impact in heavy-ion collision experiments.
We discuss the quenched limit of lattice QCD at non-zero baryon number density. We find evidence for a mixed phase that becomes broader with increasing baryon number. Although the action is explicitly Z(3) symmetric the Polyakov loop expectation value becomes non-zero already in the low temperature phase. It indicates that the heavy quark potential stays finite at large distances, i.e. the string between static quarks breaks at non-zero baryon number density already in the hadronic phase. This behaviour is validated by calculating the heavy quark potential using Polyakov loop correlations.
We construct a novel observable for finite temperature QCD that relates confinement and chiral symmetry. It uses phases as boundary conditions for the fermions. We discuss numerical and analytical aspects of this observable, like its spectral behavior below and above the critical temperature, as well as the connection to chiral condensate, center symmetry and the canonical ensemble.
We show how all the contributions to the determinant of the Dirac-Kogut-Susskind operator of QCD at finite temperature containing a net number of Polyakov loops become irrelevant in the infinite volume limit. We discuss also on two of the most interesting physical implications of this result: i) the restoration of the Polyakov symmetry in the full theory with dynamical fermions and ii) the total suppression of baryonic thermal fluctuations in QCD at finite temperature.
We report our recent numerical studies on two-color QCD with Wilson fermions. First the phase structure of the system in (kappa,mu)-plane is analyzed by measuring Polyakov line, baryon number density, gluon energy density and Polyakov line susceptibility. Then meson and baryon (di-quark) propagators are investigated. We find that the vector meson mass decreases near mu_c. This phenomena can be related to the low-mass lepton-pair enhancement observed in CERES experiment. Since our lattice is small, in order to confirm the results, we calculate both periodic and anti-periodic boundary condition …
Motivated by the sign problem, we calculate the effective Polyakov line action corresponding to certain SU(3) lattice gauge theories on a ${16^3 \times 6}$ lattice via the "relative weights" method introduced in our previous articles. The calculation is carried out at $β=5.6,5.7$ for the pure gauge theory, and at $β=5.6$ for the gauge field coupled to a relatively light scalar particle. In the latter example we determine the effective theory also at finite chemical potential, and show how observables relevant to phase structure can be computed in the effective theory via mean field methods. In…
While the Polyakov loop is an order parameter of the deconfinement transition in the heavy quark mass regime of QCD, its sensitivity to the deconfinement of light, dynamical quarks in QCD is not apparent. On the other hand, the quark mass dependence of the Polyakov loop is sensitive to the appearance of a chiral phase transition. Using lattice QCD calculations in the staggered fermion discretization scheme at finite values of the lattice spacing, $aT = 1/8$, we show here, for the first time, that the Polyakov loop expectation value, and the heavy quark free energy extracted from it, behave lik…
The broad class of U(N) and SU(N) Polyakov loop models on the lattice are solved exactly in the combined large N, Nf limit, where N is a number of colors and Nf is a number of quark flavors, and in any dimension. In this 't Hooft-Veneziano limit the ratio N/Nf is kept fixed. We calculate both the free energy and various correlation functions. The critical behavior of the models is described in details at finite temperatures and non-zero baryon chemical potential. Furthermore, we prove that the calculation of the N-point (baryon) correlation function reduces to the geometric median problem in t…
A three-dimensional effective theory of Polyakov loops has recently been derived from Wilson's Yang-Mills lattice action by means of a strong coupling expansion. It is valid in the confined phase up to the deconfinement phase transition, for which it predicts the correct order and gives quantitative estimates for the critical coupling. In this work we study its predictive power for further observables like correlation functions and the equation of state. We find that the effective theory correctly reproduces qualitative features and symmetries of the full theory as the continuum is approached.…
We consider a dual representation of an effective three-dimensional Polyakov loop model for the SU(3) theory at nonzero real chemical potential. This representation is free of the sign problem and can be used for numeric Monte-Carlo simulations. These simulations allow us to locate the line of second order phase transitions, that separates the region of first order phase transition from the crossover one. The behavior of local observables in different phases of the model is studied numerically and compared with predictions of the mean-field analysis. Our dual formulation allows us to study als…
The lattice simulations of $QC_2D$ with two flavors of staggered fermions and non-zero quark chemical potential $μ_q$ have been performed. Dependencies of the Polyakov loop, chiral condensate and baryon number density on $μ_q$ were studied. We found that an increase of the baryon chemical potential leads to chiral symmetry restoration. At small values of $μ_q$, our results for the baryon number density agree with ChPT predictions.
We construct a novel observable for finite temperature QCD that relates confinement and chiral symmetry. It uses phases as boundary conditions for the fermions. We discuss numerical and analytical aspects of this observable, like its spectral behavior below and above the critical temperature, as well as the connection to chiral condensate and center symmetry.
We study genuine finite density effects in QCD-like theories with three-dimensional Polyakov-loop effective theories for heavy quarks. These are derived from the full QCD-like theories by combined strong-coupling and hopping expansions. In particular, we investigate the cold and dense regimes of phase diagrams where we expect to find Bose-Einstein-condensation of diquark baryons or a fermionic first-order liquid-gas transition, depending on the gauge group of the theory. In two-color QCD, for example, we observe evidence of a continuous zero-temperature transition to finite diquark density whe…
Effective Polyakov loop theories are a useful tool for an investigation of pure Yang-Mills theory and full QCD. A systematic derivation of the effective action can be done in a spatial strong coupling expansion. Quite accurate predictions for the deconfinement phase transition of Yang-Mills theory have been obtained in this approach. Besides the critical couplings, further observables can be measured in the effective theory. These provide additional tests for the reliability of the strong coupling approach and the truncation of the effective action. In this contribution we will present recent …
We take a first step towards the solution of QCD in 1+1 dimensions at nonzero density. We regularize the theory in the UV by using a lattice and in the IR by putting the theory in a box of spatial size L. After fixing to axial gauge we use the coherent states approach to obtain the large-N classical Hamiltonian H that describes color neutral quark-antiquark pairs interacting with spatial Polyakov loops in the background of baryons. Minimizing H we get a regularized form of the `t Hooft equation that depends on the expectation values of the Polyakov loops. Analyzing the L-dependence of this equ…
A popular observable in finite-temperature lattice QCD is the so-called singlet quark-antiquark free energy, conventionally defined in Coulomb gauge. In an effort to interpret the existing numerical data on this observable, we compute it at order O(alpha_s^2) in continuum, and analyze the result at various distance scales. At short distances (r << 1/pi T) the behaviour matches that of the gauge-independent zero-temperature potential; on the other hand at large distances (r >> 1/pi T) the singlet free energy appears to have a gauge-fixing related power-law tail. At infinite distance the result …
We have developed a method for extracting equilibrium observables from non-equilibrium simulations by rapidly changing the temperature and recording the subsequent evolution of the Polyakov loop. Both nucleation and spinodal decomposition are observed to occur. In the latter case the Polyakov loop correlation function shows exponential growth for wavenumbers less than or equal to the critical wavenumber k_c. We have constructed the bare as wee as the effective potential for the Polyakov loop, from which k_c and m_D/k_c can be extracted as a function of temperature. The shift from spinodal deco…
In the heavy, static quark mass regime of QCD, the Polyakov loop is well known to be an order parameter of the deconfinement phase transition; however, the sensitivity of the Polyakov loop to the deconfinement of light, dynamical quarks is less clear. On the other hand, from the perspective of an effective Lagrangian written in the vicinity of the chiral transition, the Polyakov loop is an energy-like operator and should hence scale as any energy-like operator would. We show here that the Polyakov loop and heavy-quark free energy are sensitive to the chiral transition, i.e. their scaling is co…
The thermodynamics of QCD with sufficiently heavy dynamical quarks can be described by a three-dimensional Polyakov loop effective theory, obtained after a truncated character and hopping expansion. We investigate the resulting phase diagram for low temperatures by mean field methods. Taking into account chemical potentials for both baryon number and isospin, we obtain clear signals for a liquid-gas type transition to baryon matter at $μ_I=0$ and a Bose-Einstein condensation transition at $μ_B=0$, as well as for their connection when both chemical potentials are non-zero.
Properties of local Polyakov loops are studied in finite temperature lattice QCD and SU(3) lattice gauge theory. We evaluate local Polyakov loops, identify the closest center element for each loop and investigate cluster properties of these center phases. For a suitable definition of the clusters we find that the deconfinement transition of pure SU(3) gauge theory may be characterized by percolation of the center clusters. For the case of full QCD cluster observables show a behavior which seems to be compatible with a smooth crossover type of transition.
Using the example of the SU(2) gauge theory in 3+1 dimensions we consider the construction of a 3-dimensional effective model in terms of Polyakov loops. We demonstrate the application of an equilibrium self-consistency condition to the systematic analysis of the contribution of various (global Z(2) symmetric) terms in the effective model action. We apply this analysis to the construction of a simple effective action with the minimum necessary number of operators. Such an action is shown to be capable of reproducing relevant observables, e.g. the Polyakov loop ensemble average, within the desi…
Imaginary chemical potentials serve as a useful tool to constrain the QCD phase diagram and to gain insight into the thermodynamics of strongly interacting matter. In this study, we report on the first determination of the phase diagram for arbitrary imaginary baryon and isospin chemical potentials at high temperature using one-loop perturbation theory, revealing a nontrivial structure of Roberge-Weiss (RW) phase transitions in this plane. Subsequently, this system is simulated numerically with $N_{\rm f}=2$ unimproved staggered quarks on $N_τ=4$ lattices at a range of temperatures at one of t…
We investigate high density state of SU(2) QCD in the case of the Wilson fermions with Iwasaki improved action. Based on the Woodbery formula, the ratio of fermion determinants is evaluated at each step of the Metropolis link update. At $β=0.7$, for $κ= 0.150$, and $κ= 0.175$, we calculate thermodynamical quantities, such as, baryon number density, Polyakov line, and the energy density of gluon sector with chemical potential $μ$=0 to 0.9 on the $4^{3} \times 8$ lattice. The susceptibility of the Polyakov line to the chemical potential shows peak at around $μ= 0.7$ which indicates the deconfine…
We report on the first steps of an ongoing project to add gauge observables and gauge corrections to the well-studied strong coupling limit of staggered lattice QCD, which has been shown earlier to be amenable to numerical simulations by the worm algorithm in the chiral limit and at finite density. Here we show how to evaluate the expectation value of the Polyakov loop in the framework of the strong coupling limit at finite temperature, allowing to study confinement properties along with those of chiral symmetry breaking. We find the Polyakov loop to rise smoothly, thus signalling deconfinemen…
It is argued that the adjoint torelon loop, i.e. a Polyakov loop in the adjoint representation running in a spatial, rather than temporal, direction, is an observable which is sensitive to the presence of long color electric flux tubes at high temperatures. We show via lattice Monte Carlo simulations that this observable has a sharp peak at the deconfinement transition, remains much larger than the vacuum value for some range of $T>T_c$, and falls below the vacuum value for $T > 2T_c$. This result suggests that long electric flux tubes may persist for a finite range of temperatures past the de…
We explore the phase diagram of two-flavor QCD at imaginary values of baryon and isospin chemical potentials analyzing the thermodynamic potential of QCD analytically and that of the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model numerically. QCD has no pion condensation at imaginary baryon and isospin chemical potentials, and therefore has discrete symmetries that are not present at real baryon and isospin chemical potentials. The PNJL model possesses all the discrete symmetries. The PNJL model can reproduce qualitatively lattice QCD data presented very lately.
Properties of the quark-gluon plasma(QGP) in the presence of the baryon chemical potential $μ_B$ are studied using the Field Correlator Method(FCM). The nonperturbative FCM dynamics includes the Polyakov line, computed via colorelectric string tension $σ^E(T)$ and the quark and gluon Debye masses, defined by the colormagnetic string tension $σ^H(T)$. The resulting QGP thermodynamics at $μ_B \le 400$ MeV is in a good agreement with the available lattice data,both pressure and the sound velocity do not show any sign of a critical behavior in this region.
We consider the spectrum of transfer matrix eigenvalues associated with Polyakov loops in lattice QCD at strong coupling. The transfer matrix at finite density is non-Hermitian, and its eigenvalues become complex as a manifestation of the sign problem. We show that the symmetry under charge conjugation and complex conjugation ensures that the eigenvalues are either real or part of a complex conjugate pair, and the complex pairs lead to damped oscillatory behavior in Polyakov loop correlation functions, which also appeared in our previous phenomenological models using complex saddle points. We …
The Polyakov loop has been used repeatedly as an order parameter in the deconfinement phase transition in QCD. We argue that, in the confined phase, its expectation value can be represented in terms of hadronic states, similarly to the hadron resonance gas model for the pressure. Specifically, L(T) \approx 1/2\sum_αg_α\,e^(-Δ_α/T), where g_αare the degeneracies and Δ_αare the masses of hadrons with exactly one heavy quark (the mass of the heavy quark itself being subtracted). We show that this approximate sum rule gives a fair description of available lattice data with N_f=2+1 for temperatures…
We discuss the thermodynamics of gluons in the background of static quark sources. In order to do so we formulate the quenched limit of QCD at non-zero baryon number. A first numerical analysis of this system shows that it undergoes a smooth deconfining transition. We find evidence for a region of coexisting phases that becomes broader with increasing baryon number density. Although the action is in our formulation explicitly Z(3) symmetric the Polyakov loop expectation value becomes non-zero already in the low temperature phase. It indicates that the heavy quark potential stays finite at larg…
Using two-color lattice QCD with Wilson fermions, we report a study of the finite baryon number density system with two-flavors. First we investigate the Polyakov line and thermodynamical uantities in the $(κ,μ)$ plane, where $κ$ and $μ$ are the hopping parameter and chemical potential in the fermion action, respectively. Then we calculate propagators of meson ($\bar{q}Γq$) and baryon ($qΓq$) states. We find that the vector meson propagators are strongly modified in large $μ$ regions, indicating the reduction of the mass. This anomalous behavior of the vector meson is observed for the first ti…
We test the finite density algorithm in the canonical ensemble which combines the HMC update with the accept/reject step according to the ratio of the fermion number projected determinant to the unprojected one as a way of avoiding the determinant fluctuation problem. We report our preliminary results on the Polyakov loop in different baryon number sectors which exhibit deconfinement transitions on small lattices. The largest density we obtain around $T_c$ is an order of magnitude larger than that of nuclear matter. From the conserved vector current, we calculate the quark number and verify th…
We investigate the temperature and magnetic field dependence of the Polyakov loop and heavy quark entropy in a bottom-up Einstein-Maxwell-dilaton (EMD) holographic model for the strongly coupled quark-gluon plasma (QGP) that quantitatively matches lattice data for the $(2+1)$-flavor QCD equation of state at finite magnetic field and physical quark masses. We compare the holographic EMD model results for the Polyakov loop at zero and nonzero magnetic fields and the heavy quark entropy at vanishing magnetic field with the latest lattice data available for these observables and find good agreemen…
At finite imaginary values of the chemical potential, QCD is free of the sign problem. Moreover, at high temperatures the partition function exhibits a new symmetry (the Roberge-Weiss symmetry) connecting phases with different orientations of the Polyakov loop, and the corresponding phase transitions between these. In this contribution we investigate the perturbative one-loop effective potential for the Polyakov loop in the presence of imaginary isospin as well as baryon chemical potentials. This leads to a novel phase diagram, which reveals an interesting insight about the rich phase structur…
Deconfinement phase transition due to disappearance of confining colorelectric field correlators is described using nonperturbative equation of state. The resulting transition temperature $T_c(μ)$ at any chemical potential $μ$ is expressed in terms of the change of gluonic condensate $ΔG_2$ and absolute value of Polyakov loop $L_{fund} (T_c)$, known from lattice and analytic data, and is in good agreement with lattice data for $ΔG_2 \approx 0.0035 $ GeV$^4$. E.g. $T_c(0) =0.27; 0.19; 0.17$ GeV for $n_f=0,2,3$ respectively.
We compare different Polyakov loop actions yielding effective descriptions of finite-temperature SU(2) Yang-Mills theory on the lattice. The actions are motivated by a simultaneous strong-coupling and character expansion obeying center symmetry and include both Ising and Ginzburg-Landau type models. To keep things simple we limit ourselves to nearest-neighbor interactions. Some truncations involving the most relevant characters are studied within a novel mean-field approximation. Using inverse Monte-Carlo techniques based on exact geometrical Schwinger-Dyson equations we determine the effectiv…
We give an analytical derivation of the confinement/deconfinement phase transition at finite temperature in the $SU(N)$ Yang-Mills theory in the $D$-dimensional space time for $D>2$. For this purpose, we use a novel reformulation of the Yang-Mills theory which allows the gauge-invariant gluonic mass term, and calculate analytically the effective potential of the Polyakov loop average concretely for the $SU(2)$ and $SU(3)$ Yang-Mills theories by including the gauge-invariant dynamical gluonic mass $M$. For $D=4$, we give an estimate on the transition temperature $T_d$ as the ratio $T_d/M$ to th…
By using the dressed Polyakov loop or dual chiral condensate as an equivalent order parameter of the deconfinement phase transition, we investigate the relation between the chiral and deconfinement phase transitions at finite temperature and density in the framework of three-flavor Nambu--Jona-Lasinio (NJL) model. It is found that in the chiral limit, the critical temperature for chiral phase transition coincides with that of the dressed Polyakov loop in the whole $(T,μ)$ plane. In the case of explicit chiral symmetry breaking, it is found that the phase transitions are flavor dependent. For e…
Renormalized Polyakov loop in accelerated gluodynamics
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research（鲍里索夫理论物理实验室，联合核研究所）；Pacific Quantum Center, Far Eastern Federal University（太平洋量子中心，远东联邦大学）；Institute of Automation and Control Processes, Far Eastern Branch, Russian Academy of Science（自动化与控制过程研究所，俄罗斯科学院远东分院）；Dubna State University（杜布纳国立大学）
The first principle lattice QCD methods allow to calculate the thermodynamic observables at finite temperature and imaginary chemical potential. These can be compared to the predictions of various phenomenological models. We argue that Fourier coefficients with respect to imaginary baryochemical potential are sensitive to modeling of baryonic interactions. As a first application of this sensitivity, we consider the hadron resonance gas (HRG) model with repulsive baryonic interactions, which are modeled by means of the excluded volume correction. The Fourier coefficients of the imaginary part o…
The phase structure of two-flavor QCD is investigated at real isospin and imaginary quark chemical potentials by using the Polyakov-loop extended Nambu--Jona-Lasinio model. In the region, parity symmetry is spontaneously broken by the pion superfluidity phase transition, whereas charge-conjugation symmetry is spontaneously violated by the Roberge-Weiss transition. The chiral (deconfinement) crossover at zero isospin and quark chemical potentials is a remnant of the parity (charge-conjugation) violation. The interplay between the parity and charge-conjugation violations are analyzed, and it is …
Fluctuations of conserved charges such as baryon number, electric charge and strangeness may provide a test for completeness of states in lattice QCD for three light flavors. We elaborate on the idea that the corresponding susceptibilities can be saturated with excited baryonic states with an underlying quark-diquark structure with a linearly confining interaction. Using Polyakov-loop correlators we show that in the static limit, the quark-diquark potential coincides with the quark-antiquark potential in marked agreement with recent lattice studies. We thus study in a quark-diquark model the b…
Dual representations are constructed for non-abelian lattice spin models with U(N) and SU(N) symmetry groups, for all N and in any dimension. These models are usually related to the effective models describing the interaction between Polyakov loops in the strong coupled QCD. The original spin degrees of freedom are explicitly integrated out and a dual theory appears to be a local theory for the dual integer-valued variables. The construction is performed for the partition function and for the most general correlation function. The latter include the two-point function corresponding to quark-an…
We present results on the nucleon scalar, axial and tensor charges, as well as, on the first moments of the unpolarized, polarized and transversity parton distributions using $N_f=2$ and $N_f=2+1+1$ twisted mass fermions. These include an ensemble that yields the physical value of the ratio of the nucleon to the pion mass. Results on the axial charges of hyperons and charmed baryons are also presented for a range of pion masses including the physical one.
We discuss the extension of gauge-invariant electric and magnetic screening masses in the Quark-Gluon Plasma to the case of a finite baryon density, defining them in terms of a matrix of Polyakov loop correlators. We present lattice results for $N_f=2+1$ QCD with physical quark masses, obtained using the imaginary chemical potential approach, which indicate that the screening masses increase as a function of $μ_B$. A separate analysis is carried out for the theoretically interesting case $μ_B/T=3 i π$, where charge conjugation is not explicitly broken and the usual definition of the screening …
The phase structure of two-flavor QCD is explored for thermal systems with finite baryon- and isospin-chemical potentials, mu_B and mu_{iso}, by using the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. The PNJL model with the scalar-type eight-quark interaction can reproduce lattice QCD data at not only mu_{iso}=mu_B=0 but also mu_{iso}>0 and mu_B=0. In the mu_{iso}-mu_{B}-T space, where T is temperature, the critical endpoint of the chiral phase transition in the mu_B-T plane at mu_{iso}=0 moves to the tricritical point of the pion-superfluidity phase transition in the mu_{iso}-T pl…
We study the Z_N symmetry in SU(N)+Higgs theories with the Higgs field in the fundamental representation. The distributions of the Polyakov loop show that the Z_N symmetry is explicitly broken in the Higgs phase. On the other hand, inside the Higgs symmetric phase the Polyakov loop distributions and other physical observables exhibit the Z_N symmetry. This effective restoration of the Z_N symmetry changes the nature of the confinement-deconfinenement transition. We argue that the Z_N symmetry will lead to time independent topological defect solutions in the Higgs symmetric deconfined phase whi…
We determine the baryon spectrum of 1 + 1 + 1-flavor QCD in the presence of strong background magnetic fields using lattice simulations at physical quark masses for the first time. Our results show a splitting within multiplets according to the electric charge of the baryons and reveal, in particular, a reduction of the nucleon masses for strong magnetic fields. This first-principles input is used to define constituent quark masses and is employed to set the free parameters of the Polyakov loop-extended Nambu-Jona-Lasinio (PNJL) model in a magnetic field-dependent manner. The so constructed mo…
