We study the expectation value of the phase of the fermion determinant for Wilson lattice fermions with chemical potential. We use quenched SU(3) ensembles and implement a recently proposed exact dimensional reduction of the fermion determinant. Ensembles at several temperatures below and above the phase transition are studied and we analyze the role of the quark mass, the temperature, the volume and the topological sectors. We compare our numerical results to predictions from chiral perturbation theory.
We report our recent results on the QCD phase diagram obtained from the lattice QCD simulation. The location of the phase boundary between hadronic and QGP phases in the two-flavor QCD phase diagram is investigated. The imaginary chemical potential approach is employed, which is based on Monte Carlo simulations of the QCD with imaginary chemical potential and analytic continuation to the real chemical potential region.
An algorithm to simulate full QCD with 3 colours at nonzero chemical potential on the lattice is proposed. The algorithm works for small values of the chemical potential and can be used to extract expectation values of CPT invariant operators.
A connection between the operator fermionic currents $\hatJ$ and corresponding `Grassmannian' currents $J$ in the functional integral formalism is studied. The QCD action with non--zero chemical potential $μ$ is derived. A connection between the fermionic Fock space and boundary conditions along the forth direction is discussed.
Chiral chemical potential does not cause the sign problem in the Monte Carlo simulation of lattice QCD. Using the chiral chemical potential, we study the chiral magnetic effect in two-flavor full QCD. We show that a strong external magnetic field induces an electric current in a chirally imbalanced QCD matter. The qualitative feature of the induced current is consistent with an analytical prediction.
In the presence of a non-vanishing chemical potential the eigenvalues of the Dirac operator become complex. We use a random matrix model approach to calculate analytically all correlation functions at weak and strong non-Hermiticity for three-dimensional QCD with broken flavor symmetry and four-dimensional QCD in the bulk.
It is shown that starting from one and the same transfer matrix formulation of QCD on a lattice, it is possible to obtain both the action of Hasenfratz and Karsch as well as an action where the chemical potential is not coupled to the temporal links.
We investigate the real and imaginary chemical-potential dependence of pion and $ρ$-meson screening masses in both the confinement and the deconfinement region by using two-flavor lattice QCD. The spatial meson correlators are calculated in the imaginary chemical potential region with lattice QCD simulations. We extract pion and $ρ$-meson screening masses from the correlators. The obtained meson screening masses are extrapolated to the real chemical potential region by assuming some analytic function. In the real chemical potential region, the resulting pion and $ρ$-meson screening masses mono…
The regime of small isospin chemical potential in QCD is investigated. Using the phase quenched partition function in the $ε$-regime an expression for the chiral condensate is given, which is studied in the temperature isospin chemical potential plane. Lines of constant values of the condensate are shown and it is estimated how the critical temperature varies as a function of the isospin chemical potential. Finally, the dependency of the fermion sign problem on the chemical potential and temperature is examined.
We use a model based on the hopping parameter expansion to study QCD at large $μ$. We find interesting behavior in the region expected to show flavor-color locking.
In this paper we report on lattice simulations of SU(3)-QCD with non-zero chiral chemical potential. We focus on the influence of the chiral chemical potential on the confinement/deconfinement phase transition and the breaking/restoration of chiral symmetry. The simulation is carried out with dynamical Wilson fermions. We find that the critical temperature rises as the chiral chemical potential grows.
Using a chiral random matrix theory we can now derive the low energy partition functions and Dirac eigenvalue correlations of QCD with different chemical potentials for the dynamical and valence quarks. The results can also be extended to complex (and purely imaginary) chemical potential. We also discuss possible applications such as fitting to low energy constants and understanding the phase diagrams of the full and partially quenched theories.
In this work we study the finite temperature SU(2) gauge theory with staggered fermions for real and imaginary chemical potential. We test the method of analytical continuation of Monte Carlo results obtained for imaginary chemical potential, by comparison with those obtained {\em directly} for real chemical potential.
We propose a new method to investigate the thermal properties of QCD with a small chemical potential $(μ)$. The derivatives of the phase transition point with respect to $μ$ are computed for 2 flavors of p4-improved staggered fermions. We moreover comment on the complex phase of the fermion determinant in finite-density QCD.
The method of analytic continuation from imaginary to real chemical potential is one of the most powerful tools to circumvent the sign problem in lattice QCD. Here we test this method in a theory, 2-color QCD, which is free from the sign problem. We find that the method gives reliable results, within appropriate ranges of the chemical potential, and that a considerable improvement can be achieved if suitable functions are used to interpolate data with imaginary chemical potential.
We present an N_t=4 lattice study for the equation of state of 2+1 flavour staggered, dynamical QCD at finite temperature and chemical potential. We use the overlap improving multi-parameter reweighting technique to extend the equation of state for non-vanishing chemical potentials. The results are obtained on the line of constant physics and our physical parameters extend in temperature and baryon chemical potential upto \approx 500-600 MeV.
The method of analytical continuation from imaginary to real chemical potential is tested in 2-color QCD. In comparison to previous studies in the same theory, an exact updating algorithm is used and simulations are performed closer to the thermodynamic limit. It is shown that the method considerably improves if suitable functions are used to interpolate data with imaginary chemical potential.
The phase diagram of two-color QCD with non-zero chiral chemical potential is studied by means of lattice simulation. We focus on the influence of a chiral chemical potential on the confinement/deconfinement phase transition and the breaking/restoration of chiral symmetry. The simulation is carried out with dynamical staggered fermions without rooting. The dependences of the Polyakov loop, the chiral condensate and the corresponding susceptibilities on the chiral chemical potential and the temperature are presented. The critical temperature is observed to increase with increasing chiral chemic…
We define the chemical potential as the Lagrange multiplier of the baryon charge operator in the transfer matrix formalism of QCD on a lattice. Transforming the partition function into an euclidean path integral we get the Hasenfratz-Karsh action both for Wilson and Kogut-Susskind fermions. In the latter case the chemical potential in the spin-diagonal basis is half that in the flavour basis. Some open problems in the spin-diagonal basis are pointed out.
We study the thermodynamic singularities of QCD in the complex chemical potential plane by a numerical simulation of lattice QCD, and discuss a method to understand the nature of the QCD phase transition at finite density from the information of the singularities. The existence of singular points at which the partition function (Z) vanishes is expected in the complex plane. These are called Lee-Yang zeros or Fisher zeros. We investigate the distribution of these singular points using the data obtained by a simulation of two-flavor QCD with p4-improved staggered quarks. The convergence radius o…
We discuss a new Random Matrix Model for QCD with a chemical potential that is based on the symmetries of the Dirac operator and can be solved exactly for all eigenvalue correlations for any number of flavors. In the microscopic limit of small energy levels the results should be an accurate description of QCD. This new model can also be scaled so that all physical observables remain at their $μ=0$ values until a first order chiral restoration transition is reached. This gives a more realistic model for the QCD phase diagram than previous RMM. We also mention how the model might aid in determin…
In this paper we study the finite temperature SU(2) gauge theory with staggered fermions for non-zero imaginary and real chemical potential. The method of analytical continuation of Monte Carlo results from imaginary to real chemical potential is tested by comparison with simulations performed {\em directly} for real chemical potential. We discuss the applicability of the method in the different regions of the phase diagram in the temperature -- imaginary chemical potential plane.
The first results on lattice at finite temperature $T$ and chemical potential $μ$ with 2+1 flavors of Wilson quarks are presented. We have studied the dependence of chiral phase transition and deconfinement phase transition on quark mass. Finite volume size analysis and Binder cumulants are used to determine the properties of phase transition. Phase diagram of QCD with 2+1 flavors of Wilson quarks are presented.
We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.
In this lecture we discuss various properties of the phase factor of the fermion determinant for QCD at nonzero chemical potential. Its effect on physical observables is elucidated by comparing the phase diagram of QCD and phase quenched QCD and by illustrating the failure of the Banks-Casher formula with the example of one-dimensional QCD. The average phase factor and the distribution of the phase are calculated to one-loop order in chiral perturbation theory. In quantitative agreement with lattice QCD results, we find that the distribution is Gaussian with a width $\sim μT \sqrt V$ (for $m_π…
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 phase diagram of two-color QCD with a chiral chemical potential is studied on the lattice. The focus is on the confinement/deconfinement phase transition and the breaking/restoration of chiral symmetry. The simulations are carried out with dynamical staggered fermions without rooting. The dependence of the Polyakov loop, the chiral condensate and the corresponding susceptibilities on the chiral chemical potential and the temperature are presented.
One suggestion for determining the properties of QCD at finite temperatures and densities is to carry out lattice simulations with an imaginary chemical potential whereby no sign problem arises, and to convert the results to real physical observables only afterwards. We test the practical feasibility of such an approach for a particular class of physical observables, spatial correlation lengths in the quark-gluon plasma phase. Simulations with imaginary chemical potential followed by analytic continuation are compared with simulations with real chemical potential, which are possible by using a…
Within the reweighting approach, one has the freedom to choose the Monte Carlo action so that it provides a good overlap with the finite-μmeasure but remains simple to simulate. We explore several choices of action in the regime of small μ. Simulating with a finite isospin chemical potential μ_I=μgives a better overlap than the standard choice μ=0, with no computational overhead.
We review our knowledge of the phase diagram of QCD as a function of temperature, chemical potential and quark masses. The presence of tricritical lines at imaginary chemical potential mu=i pi/3 T, with known scaling behaviour in their vicinity, puts constraints on this phase diagram, especially in the case of two light flavors. We show first results in our project to determine the finite-temperature behaviour in the two-flavour chiral limit.
We consider thermodynamic singularities appearing in the complex chemical potential plane in the vicinity of QCD critical point. In order to investigate what the singularities are like in a concrete form, we resort to an effective theory based on a mean field approach. We study the behavior of extrema of the real part of the complex effective potential in the complex order parameter plane.
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 $μ/T$.
I describe the results for the critical line and the thermodynamics of different phases of QCD which have been obtained by lattice simulations with an imaginary chemical potential, and review motivations and merits of the different strategies -- Taylor expansion, Fourier analysis and Pade' approximants -- for analytic continuation from imaginary to real chemical potential. I consider phenomenological models which can be easily extended to the complex chemical potential plane, thus affording a direct comparison with lattice data at imaginary $μ$: the hadronic phase and the high temperature limi…
We present a quantitative analysis of the microscopic Dirac spectrum which is complex in the presence of a non-vanishing quark chemical potential. Data from quenched SU(3) lattice simulations for different volumes V and small values of the chemical potential are compared to analytical predictions from matrix models. We confirm the existence of two distinct limits for weakly and strongly nonhermitian Dirac operators. Good agreement is found in both limits, confirming the different scaling of chemical potential and eigenvalues with the volume.
We perform the first study of lattice QCD with overlap fermions at finite temperature $T$ and chemical potential $μ$. We start from the Taylor expanded overlap fermion action, and derive in the strong coupling limit the effective free energy by mean field approximation. On the ($μ,T$) plane and in the chiral limit, there is a tricritical point, separating the second order chiral phase transition line at small $μ$ and large $T$, and first order chiral phase transition line at large $μ$ and small $T$.
We discuss the distribution of the quark number over the gauge fields for QCD at nonzero quark chemical potential. As the quark number operator is non-hermitian, the distribution is over the complex plane. Moreover, because of the fermion determinant, the distribution is not real and positive. The computation is carried out within leading order chiral perturbation theory and gives direct insight into the delicate cancellations that take place in contributions to the total baryon number.
We compute derivatives of thermodynamic quantities with respect to $μ$, at $μ=0$ for 2 and 3 flavors of degenerate quark masses. This allows us to estimate the phase transition line in the $T,μ$ plane and quantify the influence of a non vanishing chemical potential on the equation of state by computing lines of constant energy, pressure and density. Moreover we evaluate the order of the QCD phase transition by measuring the Binder Cumulant of the chiral condensate. This gives access to the chiral critical point on the phase transition line.
New approach to computation of canonical partition functions in $N_f=2$ lattice QCD is presented. We compare results obtained by new method with results obtained by known method of hopping parameter expansion. We observe agreement between two methods indicating validity of the new method. We use results for the number density obtained in the confining and deconfining phases at imaginary chemical potential to determine the phase transition line at real chemical potential.
We consider antisymmetric perturbations of real symmetric matrices in the context of random matrix theory and two-color quantum chromodynamics. We investigate the level spacing distributions of eigenvalues that remain real or become complex conjugate pairs under the perturbation. We work out analytical surmises from small matrices and show that they describe the level spacings of large random matrices. As expected from symmetry arguments, these level spacings also apply to the overlap Dirac operator for two-color QCD with chemical potential.
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.
We study QCD with 2+1 flavors at nonzero temperature and nonzero chemical potential. We present preliminary results obtained from lattice calculations performed with an improved staggered fermions action (p4-action) on lattice with temporal extent N_t = 4 on a line of constant physics with the strange quark mass adjusted to its physical value and a pion mass of about 220 MeV. We compute at imaginary chemical potential and compare with Taylor expansion results. We focus our study on a range of temperatures 0.94 < T/T_c < 1.08.
The QCD thermodynamics on the lattice provides fundamental theoretical grounds to analyze the various experimental data in relativistic heavy ion collisions. So far, most of the numerical simulations on the lattice have been performed by using the staggered-type fermion actions. Therefore it is important to carry out studies using different fermion formulations to test the uncertainties of the lattice QCD results. For this purpose, we perform systematic simulations of two-flavor QCD with an improved Wilson quark action to investigate the equation of state. We report the current status of our p…
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 …
The QCD phase diagram is studied in the lattice QCD simulation with the imaginary chemical potential approach. We employ a clover-improved Wilson fermion action of two-flavors and a renormalization-group improved gauge action, and perform the simulation at an intermediate quark mass on a $8^3\times 4$ lattice. The QCD phase diagram in the imaginary chemical potential $μ_I$ region is investigated by performing the simulation for more than 150 points on the $(β,μ_I)$ plane. We find that the Roberge-Weiss phase transition at $μ_I/T=π/3$ is first order and its endpoint is second order, which are i…
We investigate the phase diagram in the temperature, imaginary chemical potential plane for QCD with three degenerate quark flavors using Wilson type fermions. While more expensive than the staggered fermions used in past studies in this area, Wilson fermions can be used safely to simulate systems with three quark flavors. In this talk, we focus on the (pseudo)critical line that extends from $μ=0$ in the imaginary chemical potential plane, trace it to the Roberge-Weiss line, and determine its location relative to the Roberge-Weiss transition point. In order to smoothly follow the (pseudo)criti…
We exploit analytic continuation to prolongate to the region of real chemical potentials the (pseudo)critical lines of QCD with two degenerate staggered fermions at nonzero temperature and quark or isospin density obtained in the region of imaginary chemical potentials. We determine the curvatures at zero chemical potential and quantify the deviation between the cases of finite quark and of finite isospin chemical potential. In both circumstances deviations from a quadratic dependence of the pseudocritical lines on the chemical potential are clearly seen. We try different extrapolations and, f…
After giving an overview of recently invented methods for simulating lattice QCD at small mu/T, we discuss some results for bulk thermodynamic quantities of QCD matter coming from those methods. We focus on the transition line and the critical endpoint in the QCD phase diagram.
We present preliminary results about the critical line of QCD with two degenerate staggered quarks at nonzero temperature and chemical potential, obtained by the method of analytic continuation. As in our previous studies with different numbers of colors and flavors, we find deviations from a simple quadratic dependence on the chemical potential. We comment on the shape of the critical line at real chemical potential and give an estimate of the curvature of the critical line, both for quark chemical potential and isospin chemical potential.
We present our new results for the QCD equation of state at nonzero chemical potential at N_t=6 and compare them with N_t=4. We use the Taylor expansion method with terms up to sixth order in simulations with 2+1 flavors of improved asqtad quarks along a line of constant physics with m_l=0.1 m_s and approximately physical strange quark mass m_s.
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…
In heavy ion collision experiments as well as in neutron stars, both baryon and isospin chemical potentials are different from zero. In particular, the regime of small isospin chemical potential is phenomenologically important. Using a random matrix model, we find that the phase diagram at non-zero temperature and baryon chemical potential is greatly altered by an arbitrarily small isospin chemical potential: There are two first order phase transitions at low temperature, two critical endpoints, and two crossovers at high temperature. As a consequence, in the region of the phase diagram explor…
We compare higher moments of baryon numbers measured at the RHIC heavy ion collision experiments with those by the lattice QCD calculations. We employ the canonical approach, in which we can access the real chemical potential regions avoiding the sign problem. In the lattice QCD simulations, we study several fits of the number density in the pure imaginary chemical potential, and analyze how these fits affects behaviors at the real chemical potential. In the energy regions between $\sqrt{s}_{NN}$=19.6 and 200 GeV, the susceptibility calculated at $T/T_c=0.93$ is consistent with experimental da…
We study the high density region of QCD within an effective model obtained in the frame of the hopping parameter expansion. The model still acknowledges the sign problem peculiar to non-zero chemical potential, but it permits the development of refined algorithms which ensure a good overlap of the Monte Carlo ensemble with the true one. We review the main features of the model, including the most explicit form of the resumed expansion, and present calculations concerning the dependence of various observables on the chemical potential and on the temperature, in particular of the charge density …
We have developed a method to derive the (approximate) quark contribution to the fermion free energy of QCD on a lattice, at finite temperature and chemical potential, with Kogut-Susskind fermions in the flavor basis. We show here the expression at zero temperature. This result has been obtained at the lowest order of the nilpotency expansion. At this order the well known "sign problem" does not arise and the quark contribution to the action can be used as a statistical weight in the Monte Carlo simulations.
We present results from lattice calculations on the thermodynamics of QCD at non-zero temperature and baryon chemical potential and discuss the role of resonances for the occurrence of the transition to the quark-gluon plasma in hot and dense matter. Properties of a hadronic resonance gas are compared to lattice results on the equation of state at zero as well as non-zero baryon chemical potential. Furthermore, it is shown that the quark mass dependence of the transition temperature can be understood in terms of lines of constant energy density in a resonance gas.
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.
We simulate lattice QCD at a finite chemical potential $μ_I$ for isospin ($I_3$) at zero and finite temperatures. At some $μ_I=μ_c$ QCD has a second order transition with mean-field critical exponents to a state where ($I_3$) is broken spontaneously by a charged pion condensate. Heating the system with $μ_I > μ_c$ we find there is some temperature at which this condensate evaporates. This transition appears to be second order and mean-field at lower $μ_I$ values, and first order for $μ_I$ sufficiently large. We are determining the dependence of the finite temperature crossover $T_c$ on $μ_I$ f…
We explore the phase diagram spanned by the temperature and the chemical potential for 4-flavor QCD by the phase-reweighting approach. In order to determine the order of phase transition, we perform finite size scaling studies for various quantities, for example, susceptibility, kurtosis and Challa-Landau-Binder cumulant. At the parameter (beta=1.60, kappa=0.1371, c_sw=1.9655 and N_T=4), where the Kentucky group reported a first-order phase transition in their canonical simulation, we observe that the transition is consistent with being of first order.
The response of the QCD vacuum to quark chemical potentials on the order of the pion mass is studied within the context of chiral perturbation theory. For two colour QCD diquark condensation sets in when the baryon chemical potential becomes larger than the pion mass. This leading order result is confirmed at next to leading order. Next a set of massive spin-one vectors is included in the effective Lagrangian and the vector masses are calculated at leading order. Massless vector modes appear when the baryon chemical potential is equal to the mass of the lightest vector at zero chemical potenti…
We analyze the eigenvalue spectrum of the staggered Dirac matrix in two-color QCD at nonzero baryon density when the eigenvalues become complex. The quasi-zero modes and their role for chiral symmetry breaking and the deconfinement transition are examined. The bulk of the spectrum and its relation to quantum chaos is considered. A comparison with predictions from random matrix theory is presented. An analogous analysis is performed for the spectrum of the Gor'kov representation of the fermionic action.
After presenting a brief review of how simulations of QCD with imaginary chemical potential can be used to extract physical results, we analyse the phase structure of QCD with four flavours of dynamical fermions in the finite temperature - imaginary chemical potential plane, and discuss perspectives for realistic calculations.
