The phase structure of QCD with imaginary chemical potential provides information on the phase diagram of QCD with real chemical potential. With imaginary chemical potential $iμ_I=iπT$, previous studies show that the Roberge-Weiss (RW) transition end points are triple points at both large and small quark masses, and second order transition points at intermediate quark masses. The triple and second order end points are separated by two tricritical ones. We present simulations with $ N_f=2 $ Wilson fermions to investigate the nature of RW transition end points. The simulations are carried out at 8 values of the hopping parameter $κ$ ranging from 0.020 to 0.140 on different lattice volumes. The Binder cumulant, susceptibility and reweighted distribution of the imaginary part of Polyakov loop are employed to determine the nature of RW transition end points. The simulations show that the two tricritical points are within the range $0.070-0.080$ and $0.120-0.140$, respectively.
We make simulations with 2 flavor Wilson fermions to investigate the nature of the end points of Roberge-Weiss (RW) first order phase transition lines. The simulations are carried out at 9 values of the hopping parameter $κ$ ranging from 0.155 to 0.198 on different lattice spatial volume. The Binder cumulants, susceptibilities and reweighted distributions of the imaginary part of Polyakov loop are employed to determine the nature of the end points of RW transition lines. The simulations show that the RW end points are of first order at the values of $κ$ in our simulations.
We study the Roberge-Weiss phase transition numerically. The phase transition is associated with the discontinuities in the quark-number density at specific values of imaginary quark chemical potential. We parameterize the quark number density $ρ_q$ by the polynomial fit function to compute the canonical partition functions. We demonstrate that this approach provides a good framework for analyzing lattice QCD data at finite density and a high temperature. We show numerically that at high temperature, the Lee-Yang zeros lie on the negative real semi-axis provided that the high-quark-number cont…
We study lattice QCD with $N_f=2$ Wilson fermions at nonzero imaginary chemical potential and nonzero temperature. We relate the Roberge - Weiss phase transition to the properties of dyons which are constituents of the KvBLL calorons. We present numerical evidence that the characteristic features of the spectral gap of the overlap Dirac operator as function of an angle modifying the boundary condition are determined by the $Z_3$ sector of the respective imaginary chemical potential. We then demonstrate that dyon excitations in thermal configurations could be responsible (in line with perturbat…
We discuss results for the Roberge Weiss (RW) phase transition at nonzero imaginary baryon and isospin chemical potentials, in the plane of temperature and quark masses. Our study focuses on the light tricritical endpoint which has already been used as a starting point for extrapolations aiming at the chiral limit at vanishing chemical potentials. In particular, we are interested in determining how imaginary isospin chemical potential shifts the tricritical mass with respect to earlier studies at zero imaginary isospin chemical potential. A positive shift might allow one to perform the chiral …
The finite temperature chiral and deconfinement phase transitions at zero density for light and heavy quarks, respectively, have analytic continuations to imaginary chemical potential. At some critical imaginary chemical potential, they meet the Roberge-Weiss transition between adjacent $Z(3)$ sectors. For light and heavy quarks, where the chiral and deconfinement transitions are first order, the transition lines meet in a triple point. For intermediate masses chiral or deconfinement transitions are crossover and the Roberge-Weiss transition ends in a second order point. At the boundary betwee…
We investigate N=4 SYM coupled to fundamental flavours at nonzero imaginary quark chemical potential in the strong coupling and large N limit, using gauge/gravity duality applied to the D3-D7 system, treating flavours in the probe approximation. The interplay between Z(N) symmetry and the imaginary chemical potential yields a series of first-order Roberge-Weiss transitions. An additional thermal transition separates phases where quarks are bound/unbound into mesons. This results in a set of Roberge-Weiss endpoints: we establish that these are triple points, determine the Roberge-Weiss temperat…
At imaginary values of the quark chemical potential $μ$, Quantum Chromodynamics shows an interesting phase structure due to an exact center, or Roberge-Weiss (RW), symmetry. This can be used to constrain QCD at real $μ$, where the sign problem prevents Monte Carlo simulations of the lattice theory. In previous studies of this region with staggered fermions it was found that the RW endpoint, where the center transition changes from first-order to a crossover, depends non-trivially on the quark mass: for high and low masses, it is a triple point connecting to the deconfinement and chiral transit…
Result on the locations of the tricritical points of $N_f=2$ lattice QCD with imaginary chemical potential is presented. Simulations are carried out with Symanzik improved gauge action and Asqtad fermion action. With imaginary chemical potential $iμ_I=iπT$, previous studies show that the Roberge-Weiss (RW) transition endpoints are triple points at both large and small quark masses, and second order transition points at intermediate quark masses. The triple and second order endpoints are separated by two tricritical ones. Our simulations are carried out at 7 values of quark mass $am$ ranging fr…
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…
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…
QCD with imaginary chemical potential is free of the sign problem and exhibits a rich phase structure constraining the phase diagram at real chemical potential. We simulate the critical endpoint of the Roberge-Weiss (RW) transition at imaginary chemical potential for $N_\text{f}=2$ QCD on $N_τ=6$ lattices with standard Wilson fermions. As found on coarser lattices, the RW endpoint is a triple point connecting the deconfinement/chiral transitions in the heavy/light quark mass regions and changes to a second-order endpoint for intermediate masses. These regimes are separated by two tricritical v…
In the absence of a genuine solution to the sign problem, lattice studies at imaginary quark chemical potential are an important tool to constrain the QCD phase diagram. We calculate the values of the tricritical quark masses in the Roberge-Weiss plane, $μ=\imathπT/3$, which separate mass regions with chiral and deconfinement phase transitions from the intermediate region, for QCD with $N_\text{f}=2$ unimproved staggered quarks on $N_τ=6$ lattices. A quantitative measure for the quality of finite size scaling plots is developed, which significantly reduces the subjective judgement required for…
We present the results of extensive simulations regarding the critical behavior at the endpoint of the Roberge-Weiss transition for N_f = 2 QCD. We confirm early evidence, presented in arXiv:0909.0254, according to which the Roberge-Weiss endpoint is first order in the limit of large or small quark masses, and second order for intermediate masses. A systematic study of the transition strength as a function of the quark mass in the first order regions, permits us to estimate the tricritical values of the quark mass separating the second order region from the first order ones.
The QCD phase diagram at imaginary chemical potential exhibits a rich structure and studying it can constrain the phase diagram at real values of the chemical potential. Moreover, at imaginary chemical potential standard numerical techniques based on importance sampling can be applied, since no sign problem is present. In the last decade, a first understanding of the QCD phase diagram at purely imaginary chemical potential has been developed, but most of it is so far based on investigations on coarse lattices ($N_τ=4$, $a=0.3\:$fm). Considering the $N_f=2$ case, at the Roberge-Weiss critical v…
We extend the work of Aarts et al., including an imaginary chemical potential for quark number into the Sakai-Sugimoto model and codimension k defect theories. The phase diagram of these models are a function of three parameters, the temperature, chemical potential and the asymptotic separation of the flavour branes, related to a mass for the quarks in the boundary theories. We compute the phase diagrams and the pressure due to the flavours of the theories as a function of these parameters and show that there are Roberge-Weiss transitions in the high temperature phases, chiral symmetry restore…
We consider the endpoint of the Roberge-Weiss (RW) first order transition line present for imaginary baryon chemical potentials. We remark that it coincides with the finite size transition relevant in the context of large $N_c$ QCD and study its order in the theory with two degenerate flavors. The RW endpoint is first order in the limit of large and small quark masses, while it weakens for intermediate masses where it is likely in the Ising 3d universality class. Phenomenological implications and further speculations about the QCD phase diagram are discussed.
We study the localization properties of the low-lying Dirac eigenmodes in QCD at imaginary chemical potential $\hatμ_I=π$ at temperatures above the Roberge-Weiss transition temperature $T_{\rm RW}$. We find that modes are localized up to a temperature-dependent "mobility edge" and delocalized above it, and that the mobility edge extrapolates to zero at a temperature compatible with $T_{\rm RW}$. This supports the existence of a strong connection between localization of the low Dirac modes and deconfinement, studied here for the first time in a model with a genuine deconfinement transition in t…
Roberge and Weiss showed that for SU(N) gauge theories, phase transitions occur in the presence of an imaginary quark chemical potential. We show that at asymptotically high temperature, where the phase transition is of first order, that even with dynamical quarks 't Hooft loops of arbitrary Z(N) charge are well defined at the phase boundary. To leading order in weak coupling, the 't Hooft loop satisfies Casimir scaling in the pure glue theory, but not with quarks. Because the chemical potential is imaginary, typically the interaction measure is negative on one side of the phase transition. Us…
The effect of an external electric field on the quark matter is an important question due to the presence of strong electric fields in heavy ion collisions. In the lattice QCD approach, the case of a real electric field suffers from the `sign problem', and a classical electric field is often used similar as the case of chemical potential. Interestingly, in axial gauge a uniform classical electric field actually can correspond to an inhomogeneous imaginary chemical potential that varies with coordinate. On the other hand, with imaginary chemical potential, Roberge-Weiss~(R-W) phase transition o…
The Roberge-Weiss (RW) phase transition in the imaginary chemical potential region is analyzed by the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. In the RW phase transition, the charge-conjugation symmetry is spontaneously broken, while the extended Z3 symmetry (the RW periodicity) is preserved. The RW transition is of second order at the endpoint. At the zero chemical potential, a crossover deconfinement transition appears as a remnant of the second-order RW phase transition at the endpoint, while the charge-conjugation symmetry is always preserved.
Ample numerical evidence from lattice calculations shows a strong connection between the confining properties of gauge theories at finite temperature and the localisation properties of the low-lying Dirac eigenmodes. In this contribution we discuss recent progress on this topic, focussing on results for QCD at imaginary chemical potential $μ_I/T = π$ at temperatures above the Roberge-Weiss transition temperature. These confirm the general picture of low modes turning from delocalised to localised at the deconfinement transition, in a previously unexplored setup with a genuine, physical transit…
To obtain deeper understanding of QCD properties at finite temperature, we consider the Fourier decomposition of the grand-canonical partition function based on the canonical ensemble method via the imaginary chemical potential. Expectation values are, then, represented by summation over each canonical sector. We point out that the modified Polyakov-loop can play an important role in the canonical ensemble; for example, the Polyakov-loop paradox which is known in the canonical ensemble method can be evaded by considering the quantity. In addition, based on the periodicity issue of the modified…
We study the phase diagram of $N_f = 2+1$ QCD in the $T - μ_B$ plane and investigate the critical point corresponding to the onset of the Roberge-Weiss transition, which is found for imaginary values of $μ_B$. We make use of stout improved staggered fermions and of the tree level Symanzik gauge action, and explore four different sets of lattice spacings, corresponding to $N_t = 4,6,8,10$, and different spatial sizes, in order to assess the universality class of the critical point. The continuum extrapolated value of the endpoint temperature is found to be $T_{\rm RW} = 208(5)$ MeV, i.e. $T_{\r…
We investigate the fate of the Roberge-Weiss endpoint transition and its connection with the restoration of chiral symmetry as the chiral limit of $N_f = 2+1$ QCD is approached. We adopt a stout staggered discretization on lattices with $N_t = 4$ sites in the temporal direction; the chiral limit is approached maintaining a constant physical value of the strange-to-light mass ratio and exploring three different light quark masses, corresponding to pseudo-Goldstone pion masses $m_π\simeq 100, 70$ and 50 MeV around the transition. A finite size scaling analysis provides evidence that the transiti…
We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the $\mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a $\mathbb{Z}_2$ 't Hooft anomaly of this system, which involves the $\mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrai…
Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations have confirmed that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudo-critical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the f…
We discuss the interplay between chiral and center sector phase transitions that occur in QCD with an imaginary quark chemical potential $μ=i(2n+1) πT/3$. Based on a finite size scaling analysis in (2+1)-flavor QCD using HISQ fermions with a physical strange quark mass and a range of light quark masses, we show that the endpoint of the line of first-order Roberge-Weiss (RW) transitions between center sectors is second order for light quark masses $m_l\ge m_s/320$, and that it belongs to the $3$-d, $Z(2)$ universality class. The operator for the chiral condensate behaves like an energy-like ope…
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…
The pseudo-critical temperature of the confinement-deconfinement transition and the phase transition surface are investigated by using the complex chemical potential. We can interpret the imaginary chemical potential as the Aharonov-Bohm phase, then the analogy of the topological order suggests that the Roberge-Weiss endpoint would define the pseudo-critical temperature. The behavior of the Roberge-Weiss endpoint at small real quark chemical potential is investigated with the perturbative expansion. The expected QCD phase diagram at complex chemical potential is presented.
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…
The order of the thermal transition in the chiral limit of QCD with two dynamical flavours of quarks is a long-standing issue. Still, it is not definitely known whether the transition is of first or second order in the continuum limit. Which of the two scenarios is realized has important implications for the QCD phase diagram and the existence of a critical endpoint at finite densities. Settling this issue by simulating at successively decreased pion mass was not conclusive yet. Recently, an alternative approach was proposed, extrapolating the first order phase transition found at imaginary ch…
We investigate the nature of the deconfinement and Roberge-Weiss transition in the heavy quark regime for finite real and imaginary chemical potential within the functional approach to continuum QCD. We extract the critical phase boundary between the first order and cross-over regions, and also explore tricritical scaling. Our results confirm previous ones from finite volume lattice studies.
We present a calculation of the net baryon number density as a function of imaginary baryon number chemical potential, obtained with highly improved staggered quarks (HISQ) at temporal lattice extent of $N_τ=4,6$. We construct various rational function approximations of the lattice data and discuss how poles in the complex plane can be determined from them. We compare our results of the singularities in the chemical potential plane to the theoretically expected positions of the Lee-Yang edge singularity in the vicinity of the Roberge-Weiss and chiral phase transitions. We find a temperature sc…
We calculate Fourier coefficients of the net-baryon number as a function of a purely imaginary chemical potential. The asymptotic behavior of these coefficients is governed by the singularity structure of the QCD partition function and thus encodes information on phase transitions. For the calculation of the Fourier coefficients from lattice data of the Bielefeld-Parma collaboration we use a novel Filon-type quadrature, designed for highly oscillatory integrals. We find sensitivity to chiral scaling in a narrow temperature interval below the Roberge-Weiss transition temperature. Scaling fits y…
We report Lee-Yang zeros behavior at finite temperature and density. The quark number densities, <n>, are calculated at the pure imaginary chemical potential, where no sign problem occurs. Then, the canonical partition functions, Z_C(n,T,V), up to some maximal values of n are estimated through fitting theoretically motivated functions to <n>, which are used to compute the Lee-Yang zeros. We study the temperature dependence of the distributions of the Lee-Yang zeros around the pseudo-critical temperature region T/T_c = 0.84 - 1.35. In the distributions of the Lee-Yang zeros, we observe the Robe…
We investigate the quark number density at finite imaginary chemical potential by using the Dirac-mode expansion. In the large quark mass region, it is found that the quark number density can be expressed by the Polyakov loop and its conjugate in all order of the large quark mass expansion. Then, there are no specific Dirac-modes which dominantly contribute to the quark number density. In comparison, the small quark mass region is explored by using the quenched lattice QCD simulation. We found that the absolute value of the quark number density strongly depends on the low-lying Dirac-modes, bu…
A new approach is presented to explore the singularity structure of lattice QCD at imaginary chemical potential. Our method can be seen as a combination of the Taylor expansion and analytic continuation approaches. Its novelty lies in using rational (Padé) approximants for studying Lee Yang edge singularities. The motivation for using rational approximants will be exhibited. We will provide some confidence in our approach based on numerical experiments performed on well-motivated "toy models". Our focus lies in identifying singularities of the net-baryon number density in the complex $μ_B$ pla…
A new approach is presented to explore the singularity structure of lattice QCD in the complex chemical potential plane. Our method can be seen as a combination of the Taylor expansion and analytic continuation approaches. Its novelty lies in using rational (Padé) approximants for studying Lee-Yang edge singularities. We present a calculation of the cumulants of the net-baryon number as a function of a purely imaginary baryon number chemical potential, obtained with highly improved staggered quarks at temporal lattice extent of $N_τ=4,6$. We construct various rational function approximations o…
We present results from an ongoing study of the phase diagram of (2+1)-flavor QCD using the HISQ action with smaller than physical light quark masses in the Roberge-Weiss (RW) plane on lattices with temporal extent $N_τ=4$. We find that the endpoint of the $1^{st}$ order RW transition line remains $2^{nd}$ order at least down to light quark masses corresponding to Goldstone pion masses of $55$ MeV. Furthermore, we show that the chiral condensate is sensitive to the RW transition and may serve as a energy-like operator characterizing universal behavior close to the RW endpoint.
Properties of QCD at finite imaginary chemical potential are revisited to utilize for the model building of QCD in low energy regimes. For example, the electric holonomy which is closely related to the Polyakov-loop drastically affects thermodynamic quantities beside the Roberge-Weiss transition line. To incorporate several properties at finite imaginary chemical potential, it is important to introduce the holonomy effects to the coupling constant of effective models. This extension is possible by considering the entanglement vertex. We show justifications of the entanglement vertex based on t…
We extend the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model by introducing an effective four-quark vertex depending on Polyakov loop. The effective vertex generates entanglement interactions between Polyakov loop and chiral condensate. The new model is consistent with lattice QCD data at imaginary quark-number chemical potential and real and imaginary isospin chemical potentials, particularly on strong correlation between the chiral and deconfinement transitions and also on the quark-mass dependence of the order of the Roberge-Weiss endpoint predicted by lattice QCD very lately. We in…
The chemical-potential dependence of pi and sigma meson masses is analyzed at both real and imaginary chemical potentials, $μ_\mathrm{R}$ and $μ_\mathrm{I}$, by using the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model that possesses both the extended ${\mathbb Z}_3$ symmetry and the chiral symmetry. In the $μ_\mathrm{I}$ region, the meson masses have the Roberge-Weiss periodicity. Assuming that the meson masses will be measured at finite $μ_\mathrm{I}$ by lattice QCD in future, we simulate how meson masses at finite $μ_\mathrm{R}$ are extracted from those at finite $μ_\mathrm{I}$, and…
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 …
We present results of the location of the closest singularities in the complex chemical potential plane using a novel method. These results are obtained with (2+1)-flavor of highly improved staggered quarks (HISQ) on lattices with temporal extent of Nt=4,6. We show that the scaling is consistent with the expected scaling of the Lee-Yang edge singularities in the vicinity of the Roberge-Weiss (RW) transition. We determine various non-universal parameters using 3D Ising model scaling functions that map QCD in the scaling region of the RW transition. Furthermore, as a preliminary result we discus…
