The effect of an external (electro)magnetic field on the finite temperature transition of QCD is studied. We generate configurations at various values of the quantized magnetic flux with N f = 2 + 1 flavors of stout smeared staggered quarks, with physical masses. Thermodynamic observables including the chiral condensate and susceptibility, and the strange quark number susceptibility are measured as functions of the field strength. We perform the renormalization of the studied observables and extrapolate the results to the continuum limit using N t = 6, 8 and 10 lattices. We also check for finite volume effects using various lattice volumes. We find from all of our observables that the transition temperature T c significantly decreases with increasing magnetic field. This is in conflict with various model calculations that predict an increasing T c (B). From a finite volume scaling analysis we find that the analytic crossover that is present at B = 0 persists up to our largest magnetic fields eB ≈ 1 GeV 2 , and that the transition strength increases mildly up to this eB ≈ 1 GeV 2 .

We present a comprehensive analysis of the light condensates in QCD with 1+1+1 sea quark flavors (with mass-degenerate light quarks of different electric charges) at zero and nonzero temperatures of up to 190 MeV and external magnetic fields B < 1 GeV 2 /e. We employ stout smeared staggered fermions with physical quark masses and extrapolate the results to the continuum limit. At low temperatures we confirm the magnetic catalysis scenario predicted by many model calculations while around the crossover the condensate develops a complex dependence on the external magnetic field, resulting in a decrease of the transition temperature.

We study the physical mechanism of how an external magnetic field influences the QCD quark condensate. Two competing mechanisms are identified, both relying on the interaction between the magnetic field and the low quark modes. While the coupling to valence quarks enhances the condensate, the interaction with sea quarks suppresses it in the transition region. The latter `sea effect' acts by ordering the Polyakov loop and, thereby, reduces the number of small Dirac eigenmodes and the condensate. It is most effective around the transition temperature, where the Polyakov loop effective potential is flat and a small correction to it by the magnetic field can have a significant effect. Around the critical temperature, the sea suppression overwhelms the valence enhancement, resulting in a net suppression of the condensate, named inverse magnetic catalysis. We support this physical picture by lattice simulations including continuum extrapolated results on the Polyakov loop as a function of temperature and magnetic field. We argue that taking into account the increase in the Polyakov loop and its interaction with the low-lying modes is essential to obtain the full physical picture, and should be incorporated in effective models for the description of QCD in magnetic fields in the transition region.Comment: 19 pages, 8 figures, new reference added, version accepted for publicatio

Abstract:We determine the equation of state of 2+1-flavor QCD with physical quark masses, in the presence of a constant (electro)magnetic background field on the lattice. To determine the free energy at nonzero magnetic fields we develop a new method, which is based on an integral over the quark masses up to asymptotically large values where the effect of the magnetic field can be neglected. The method is compared to other approaches in the literature and found to be advantageous for the determination of the equation of state up to large magnetic fields. Thermodynamic observables including the longitudinal and transverse pressure, magnetization, energy density, entropy density and interaction measure are presented for a wide range of temperatures and magnetic fields, and provided in ancillary files. The behavior of these observables confirms our previous result that the transition temperature is reduced by the magnetic field. We calculate the magnetic susceptibility and permeability, verifying that the thermal QCD medium is paramagnetic around and above the transition temperature, while we also find evidence for weak diamagnetism at low temperatures.

We construct a new order parameter for finite temperature QCD by considering the quark condensate for U(1)-valued temporal boundary conditions for the fermions. Fourier transformation with respect to the boundary condition defines the dual condensate. This quantity corresponds to an equivalence class of Polyakov loops, thereby being an order parameter for the center symmetry. We explore the duality relation between the quark condensate and these dressed Polyakov loops numerically, using quenched lattice QCD configurations below and above the QCD phase transition. It is demonstrated that the Dirac spectrum responds differently to changing the boundary condition, in a manner that reproduces the expected Polyakov loop pattern. We find the dressed Polyakov loops to be dominated by the lowest Dirac modes, in contrast to thin Polyakov loops investigated earlier.PACS numbers: 12.38. Aw, 11.15.Ha, 11.10.Wx Introductory remarksUnderstanding the nature of confinement has been a challenging task for many years. Several scenarios with different candidates for the relevant gluonic excitations were proposed, but no closed picture has emerged yet (it is even still debated whether confinement is predominantly an infrared or an ultraviolet phenomenon). Also a connection of confinement to chiral symmetry and its breaking has been conjectured, but not been shown either.In recent work [1, 2, 3] we have explored the idea of connecting quantities sensitive to confinement to spectral sums for Dirac and covariant Laplace operators. These ideas were developed further in [4,5,6] where it was shown that also other quantities such as quark propagators and heat kernels may be turned into order parameters for the breaking of center symmetry. Spectral sums provide a natural decomposition into infrared (IR) and ultraviolet (UV) parts and allow one to analyze their respective role in confinement, as studied numerically using quenched [2,3,4,5,6] and dynamical [7] lattice configurations.In this letter we build on those results and develop a new order parameter for center symmetry. In particular we Fourier transform the quark condensate (that turns into the chiral condensate in the massless limit) with respect to a U(1)-valued temporal boundary condition for the fermions, which we parameterize with a phase ϕ ∈ [0, 2π). We show that this duality transformation turns the quark condensate into the expectation value of an equivalence class of Polyakov loops which all have the same winding number n ∈ Z. The winding number n is the conjugate variable to the phase ϕ. To the equivalence class of loops with winding number n = 1, which transforms under center transformations in the same way as the conventional thin Polyakov loop, we refer to as the "dressed Polyakov loop".Since for pure gauge theory the deconfinement transition can be understood as spontaneous breaking of the center symmetry [8], the dressed Polyakov loop is an order parameter for confinement in pure gauge theory. The center transformation property of the dressed Polyakov loop is indep...