N$_f$ = 1 QCD in External Magnetic Fields: Staggered Fermions

We investigate N$_f$ = 1 QCD in external magnetic fields on the lattice. The background field is introduced by means of the so-called Schrodinger functional. We adopt standard staggered fermions with constant bare mass $am = 0.025$ and magnetic fields with constant magnetic flux up to $a^2 e H \simeq 2.3562$. We find that the the deconfinement and chiral symmetry restoration temperatures do not depend on the strength of the applied magnetic field. Our method allow us to easily study the effects of the external magnetic fields on the QCD thermodynamics. We determine the influences of applied magnetic fields to the free energy, pressure, and equation of state of strongly interacting matter.


Introduction
Strong interactions are described by quantum chromodynamics (QCD), a local relativistic non-abelian quantum field theory which is not amenable to perturbation theory in the lowenergy, large-distance regimes. However, many fundamental questions are linked to the large scale behavior of QCD. In particular, non-perturbative approaches to QCD can be used to account for the different phases of hadronic matter under extreme conditions. Recently, the study of the effects of strong magnetic fields on the QCD phase diagram has become a topic of increasing interest (for a recent review, see Refs. [1,2]). In the nonperturbative regimes this problem can be efficiently approached by lattice QCD simulations with dynamical quarks. The study of lattice gauge theories with external background fields has been pioneered in Refs. [3,4] for the U(1) Higgs model in an external electromagnetic field. In the continuum a background field can be introduced by writing: In the lattice approach one deals with link variables U µ (x). Accordingly, on the lattice Eq. (1.1) becomes: where U ext µ (x) is the lattice version of the background field A ext µ (x). As a consequence the lattice action gets modified as: so that the action gets modified as: (1.5) where: The main disadvantage of these approaches resides on the lack of gauge invariance for nonabelian gauge theories. The issue of gauge invariance, however, does not pose if one is interested in QCD in external magnetic fields. In fact, let us consider the lattice partition function of QCD with f flavors of dynamical staggered quarks: where S G is the gauge field action and M is the fermion matrix for a staggered quark with bare mass am f : Since magnetic fields couple only to quarks, external magnetic fields can be introduced in the lattice action by replacing in the fermion mass matrix Eq. (1.8) the gauge field links according to Eq. (1.2), where the U ext µ (x)'s are U(1) elements corresponding to the external magnetic fields with continuum gauge potential A ext µ (x). For instance, if we consider constant magnetic fields directed along the x 3 direction, then the continuum gauge potential in the Landau gauge reads: A ext k ( x) = δ k,2 x 1 H .
An alterative approach to put background fields on the lattice has been proposed since long time [24]. Indeed, that proposal allows to overcome the problem of gauge invariance by implementing background fields on the lattice by means of the manifestly gauge-invariant lattice Schrödinger functional. In this paper we present an exploratory study of lattice QCD immersed in a uniform external magnetic field. The background field is introduced by using the Schrödinger functional. Moreover, for simplicity, we restrict ourself to one flavor of staggered dynamical quark. The plan of the paper is as follows. In Sect. 2, for completeness, we briefly discuss our method to introduce background fields on the lattice. In Sect. 3 we present the results of our numerical simulations for several local observables. We also address the problem of the possible dependence of the pseudoritical couplings on the magnetic field strengths. Sect. 4 is devoted to the discussion of the effects of magnetic fields on QCD thermodynamics. Finally, our conclusions are relegated in Sect. 5.

Magnetic Fields within the Schrödinger Functional
For reader's convenience, let us briefly review background fields in lattice gauge theories within the Schrödinger functional. Firstly, we illustrate the method in pure gauge theories. In Ref. [24], to overcome the gauge invariance problem in presence of background fields, it was proposed that background fields on the lattice could be implemented by means of the gauge invariant lattice Schrödinger functional: where the functional integration is extended over links on a lattice with the hypertorus geometry and satisfying the constraints (x t ≡ x 4 is the temporal coordinate) One also imposes that links at the spatial boundaries are fixed according to Eq. (2.2). In fact, in the continuum this last condition amounts to the requirement that fluctuations over the background field vanish at infinity. This approach has been applied for both abelian and non-abelian gauge theories with different background fields [25][26][27][28][29][30][31][32][33][34]. The effects of dynamical fermions can be accounted for quite easily. Indeed, when including dynamical fermions, the lattice Schrödinger functional in presence of a static external background gauge field becomes: where S F is the fermion action and M indicates the generic fermion matrix. Notice that the fermion fields are not constrained and the integration constraint is only relative to the gauge fields. This leads to the appearance of the gauge invariant fermion determinant after integration on the fermion fields. As usual we impose on fermion fields periodic boundary conditions in the spatial directions and anti-periodic boundary conditions in the temporal direction. In fact, Eq. (2.3) has been employed to study the dynamics of QCD with two degenerate staggered quarks [35,36], as well as the quantum Hall effect in graphene [37].
In the case of QCD in constant magnetic fields the constraints in the lattice Schrödinger functional need to be slightly modified to take into account that the magnetic field is coupled only to quarks. To this end, we impose that during the upgrade of the gauge links U ext k ( x) = I, while for the upgrade of the fermion fields U ext k ( x) = I × e iθ ext k ( x) where: Since our Schrödinger functional Z[U ext k ] is defined on a lattice with periodic boundary conditions, usually we impose that: As a consequence the magnetic field H turns out to be quantized: with n ext integer. However, it should be kept in mind that we are dealing with a periodic lattice with fixed boundary conditions, so that is is not strictly necessary to impose the quantization Eq. (2.6) and the "integer" n ext can be an arbitrary real number.

Numerical Results
We perform simulations of lattice QCD with one-flavor of rooted staggered quark. Our numerical results were obtained by choosing as gauge action the Wilson action: where U µν (x) are the plaquettes in the (µ, ν)-plane and β = 6 g 2 . Therefore we are led to consider the following lattice Schrödinger functional: where the staggered fermion matrix is given by Eq. (1.8). To perform the functional integration over the SU(3) links we have made use of the publicly available MILC code [38] which has been suitably modified by us in order to introduce the boundary constraints Eq. (2.2). All simulations make use of the rational hybrid Monte Carlo (RHMC) algorithm. The functional integration is performed over the lattice links, but the links at the spatial boundaries are fixed according to Eq. (2.2). Accordingly, the links which are frozen are not evolved during the molecular dynamics trajectory and the corresponding conjugate momenta are set to zero. The length of each RHMC trajectory has been set to 1.0 in molecular dynamics time units. For each value of the gauge coupling β and the magnetic field eH we collected 4000 -5000 trajectories, and about 10000 trajectories around the critical coupling. To allow thermalization we typically discarded 1000 trajectories. The statistical errors were estimated by means of boostrap combined with binning.
In the present exploratory study we consider lattices of size L s = 24 and L t = 4 and fixed bare fermion mass m 0 ≡ am = 0.025. At fixed L t the temperature of the gauge system T = 1 aLt is changed by varying the coupling constant β. Since the smallest quark electric charge is |q| = 1/3, from Eq. (2.6) we get: Different strengths of the external magnetic field are labelled by the parameter n ext according to Eq. (3.3). We performed simulations for n ext = 0 , 1 , 3, corresponding to field strength a 2 eH = 0 , 0.7854 , 2.3562 in lattice units, and assumed q f = 2 3 (up quark). Note that, the case of down quark q f = − 1 3 can be recovered with n ext = − 1 2 (a 2 eH = − 0.3927). In fact, to check the dependence of the free energy on the magnetic field we have also performed numerical simulations for n ext = − 1 2 . For the sake of completeness let us discuss, briefly, how the background magnetic field influences the dynamics of the gauge system. We said that to update the gauge system we used the rational hybrid Monte Carlo algorithm. As it is well known (see for instance Ref. [39]), to simulate the fermion determinant one introduces color-triplet scalar pseudofermion fields. The pseudofermion action depends on the inverse of the staggered fermion matrix Eq. (1.8). In the molecular dynamics one solves the equations of motion of the momenta conjugated to the gauge links. The derivative of the gauge momentum is called the force term, which is the formal derivative of the effective action with respect to the gauge potential. Thus, the force term consists of two contributions, namely the gauge force term and the fermion force term. Our boundary conditions correspond to set U ext k = I on the x t = 0 hypersurface and at the spatial boundaries of the lattice in the gauge force term, while U ext k ( x) = I × e iθ ext k ( x) in the fermion force term. To maintain the above constraints during the molecular dynamics, the momenta conjugated to the frozen gauge links are set to zero.

Local Observables
In this Section we are interested in the effects of the external magnetic field on several local observables. First, we consider the gauge action which, following the MILC convention, we define as: In Fig. 1 we display the gauge action as a function on the gauge coupling β for three different values of the magnetic field. Since G action is a pure gauge quantity, it couples to the magnetic field only through quark loops. Therefore we expect that this quantity should manifest a very weak dependence on the magnetic field. Indeed, Fig. 1 shows that the effects of the magnetic field on the gauge action are at most of order 10 −2 (see Fig. 5).
Interestingly enough, we see that the gauge action increases as a function of eH in the strong coupling region whereas it decreases in the weak coupling region. In fact the three different curves displayed in Fig. 1 cross near the critical coupling β c ≃ 5.0. A more interesting quantity is the quark chiral condensate: which should display a pronunciate dependence on the magnetic field. In Fig. 2 we display ψ ψ versus the gauge coupling β for three different values of the magnetic field eH.
The chiral condensate was computed by noise estimators with 4 random vectors. It is evident that the chiral condensate increases as a function of eH for all temperatures. For comparison, in Fig. 2 we also display the real part of the Polyakov loop expectation value: The Polyakov loop L, likewise the gauge action, is a pure gauge observable. Nevertheless, Fig. 2 shows that the Polyakov loop displays a sizable dependence on the magnetic field. In particular, we see that L increases with eH for all temperatures as for the chiral condensate. This behavior can be qualitatively understood if the quark free energy decreases with the applied magnetic field. In fact, later on we will show that the strongly interacting system behaves like a paramagnetic medium, i.e. positive magnetic susceptibility. Another interesting feature of Fig. 2 is the crossing of the chiral condensate and the Polyakov loop near the critical temperature. In fact, Fig. 2 seems to suggest that the pseudocritical gauge coupling β c does not manifest a strong dependence on the magnetic field eH.

Pseudocritical couplings
In this section we address the problem of the possible dependence of the pseudocritical coupling on the magnetic field. In general, the (pseudo)critical coupling is determined as the value for which some relevant susceptibilities exhibit a peak. In the present paper, to precisely localize the peak in the relevant susceptivity we parametrize the peak region with a Lorentzian function: Our estimate of the critical coupling β c is obtained by fitting the susceptivity to Eq. (3.7) in the peak region. We use the Polyakov loop susceptibility as well as the disconnected part of the chiral susceptibility to locate the transition temperature to the high temperature phase of QCD. First, we consider the disconnected chiral susceptibility: In Fig. 3 we show the disconnected chiral susceptibility as a function of the gauge coupling for three different values of eH. As usual the disconnected chiral susceptibility displays a sharp peak near the chiral critical coupling. Interestingly enough, the dependence of the chiral susceptibility on the magnetic field is almost relegated to the peak region. Moreover, the peak values increase with eH signaling that the chiral transition sharpens in presence of a non-zero magnetic field. Notwithstanding, we find that, within our statistical uncertainties, the chiral critical coupling does not depend on the magnetic field. We have also considered the Polyakov loop susceptibility: (3.9) Results for the Polyakov loop susceptibility are shown in Fig. 4. In this case we see that the dependence of the Polyakov loop susceptibility on the magnetic field is less pronounced with respect to the disconnected chiral susceptibility. Moreover, Fig. 4 shows that the peak of χ L decreases with increasing eH. This means that the applied magnetic field tends to smooth out the deconfinement transition. However, even in this case the deconfinement critical couplings does not display any appreciable dependence on eH. Moreover, we find that the chiral and deconfinemnet critical couplings agree for any values of the magnetic field strengths considered in this work. Finally, as further check, we have considered the variation of the gauge action: which is known to display a peak in the critical region. Of course, by using ∆G action we may estimate the pseudocritical couplings only for non-zero magnetic field strengths.
In   on eH is clearly displayed in Fig. 5. In any case, we see that ∆G action does display a well developed peak in the critical region. We find that the peaks in ∆G action are located at a systematically slightly larger values of the gauge coupling with respect to the chiral and Polyakov loop susceptibilities. In a finite volume this is, of course, not unexpected. Indeed, we recall that we are using lattices with fixed boundary conditions and, as previous studies showed, the gauge action turns out to be more susceptible to finite volume effects. Nonetheless, what it is relevant is that the pseudocritical couplings do not depend on the magnetic field eH. For reader convenience, in Table 1 we summarize our estimates of the critical couplings β c as function of the magnetic field eH. From Table 1 we may safely conclude that the critical temperature does not depend on the external magnetic field.

Thermodynamics in external Magnetic Fields
The partition function Z[U ext k ] allows us to define observables that can be used to establish the equation of state of the theory. Such observables play an important role in describing the thermodynamic properties of the system. The free energy density is related to the logarithm of the partition function as: The pressure is given by the derivative of T log Z[U ext k ] with respect to the volume. Assuming that we have a large, homogeneous system, differentiation with respect to V is equivalent to dividing by the volume. Therefore in the thermodynamic limit the pressure can be written as minus the free energy density: Using the well-known relation between the trace anomaly (also called interaction measure) and the derivative of the pressure: As usual, we need to renormalize the free energy density by subtracting the divergent zero-point energy. To do this it is enough to subtract the zero temperature contribution. Thus we define: (4.7) The zero temperature contributions are conventionally obtained by performing simulations on lattices with L t = L s . Moreover, since we are interested in the thermal magnetic properties of our system, we will focus on: In a Monte Carlo simulation, one cannot compute the partition function directly. The most frequently used method in practice is the integral method, in which a derivative of the free energy with respect to some parameter serves as observable, which then gets integrated again to yield the free energy density. Since we are doing simulations at fixed L t , it is convenient to take derivatives with respect to the bare gauge coupling β. The expectation values of the derivatives with respect to β of our partition function correspond to the average Wilson action. Thus, we have: In Fig. 6 we report our results for the β-derivative of ∆ p r (T, H) (normalized to T 4 ) versus the ratio T /T c for three different values of the magnetic field. We recall that the temperature corresponding to a given value of the gauge coupling is given by the relation T = 1 Lta(β) . For the dependence of the lattice spacing on the gauge coupling we used the two-loop β-function. Accordingly, we have: where f QCD (β) is the asymptotic scaling function of QCD with one dynamical fermion N f = 1: For definitiveness, the critical temperatures have been obtained by using the pseudocritical gauge coupling estimated by means of the chiral susceptibilities. Obviously, a direct determination of the physical scale should be preferable. However, it is known that the lattice violations to the asymptotic scaling law Eq. (4.10) are within a few percent. So that, the adopted approximation is adequate to the purpose of the present exploratory study. From the derivative of the pressure, after numerical integration, we may easily obtain ∆ p r (T, H) = −∆ f r (T, H). In Fig. 7 we display the normalized pressure ∆ p r (T, H) versus the temperature for three different values of the magnetic field. Fig. 7 shows that the magnetic contributions to the renormalized pressure is clearly different from zero even for T < T c , and it seems to vanish rapidly for low temperatures. This behavior can be naturally accounted for within the Hadron Resonance Gas model (see, for instance, Ref. [40,41]). On the other hand, for temperatures above the critical temperature the pressure increases rapidly in qualitative agreements with perturbative calculations in the high-temperature regime [42]. As concern the energy density, using Eq. (4.3) we may write: where R β (β) is the lattice β QCD -function. According to our approximation, we have: . (4.14) Using Eqs. (4.12), (4.13), and (4.14) we determined the so-called interaction measure displayed in Fig. 8 for three different values of the magnetic field. After that, the magnetic contributions to the renormalized energy density can be straightforwardly obtained as: In Fig. 9 we show the renormalized energy density. Even for the magnetic contribution to the energy density we find two different regimes for T < T c (confined phase) and T > T c (deconfined phase). Having determined the magnetic contribution to the pressure and energy density, in principle one can construct the equation of state and obtain the entropy density and the speed of sound by means of Eqs. (4.5) and (4.6). For the purposes of the present paper we do not discuss any further this matter. We, merely, observe that in the deconfined phase T > T c the magnetic contribute to the energy density increases slower with respect to the pressure by increasing the temperature. This behavior leads to a stiffening of the equation of state. Let us, finally, address the problem of the magnetic susceptibility. As is well known, for small magnetic field strengths we may write for the free energy density (see, for instance, Ref. [43]): where χ mag is the magnetic susceptibility. Therefore, to determine the magnetic susceptibility we need to check if ∆ f r (T, H) = −∆ p r (T, H) scales with H 2 at least for small enough magnetic field strengths. To this end, in Fig 10 we display ∆ fr(T,H) (q f eH) 2 for different values of the magnetic field strength. In fact, we see that the free energy density seems to scale with H 2 for (q f = 2 3 , n ext = 1), and (q f = − 1 3 , n ext = 1), and for temperatures not too far from the critical temperature. We note, however, that in the high-temperature region physical observables are more affected by finite volume and cutoff effects. On the other hand, we see clearly that for the strongest magnetic field used in this paper (q f = 2 3 , n ext = 3) the scaling with H 2 is badly violated. It is useful to give the corresponding values of the magnetic field in physical units. To this end, we use the known flavor dependence of the QCD critical temperature reported in Ref. [44] to infer that for N f = 1 the critical temperature is T c ∼ 190 MeV. This corresponds to a lattice spacing a ≃ 0.26 fm. So that, in the critical region, we estimate |eH| ≃ 0.48, 0.67, 1.17 GeV corresponding to n ext = −0.5 , 1 , 3 respectively. Thus, we see that for magnetic field strengths not exceeding 1.0 GeV the free energy density seems to display an approximate scaling with H 2 within our statistical uncertainties. This is in qualitative agreement with the results in Ref. [7] where the strongest magnetic field used was |eH| ≃ 0.85 GeV. In any case, for physical applications, we recall that the magnetic fields relevant for heavy-ion collision experiments are of order |eH| ∼ 0.1 GeV. To determine the magnetic susceptibility we are lead to consider the free energy density for |eH| ≤ 1.0 GeV where we can safely apply Eq. (4.16). Indeed, in Fig. 11 we report our determination of the magnetic susceptibility as function of the temperature. For comparison, we also display our determination of the magnetic susceptibility for |eH| ≃ 1.17 GeV. From Fig. 11 we see that the magnetic susceptibility is positive in the whole temperature range explored in the present study. Moreover, the magnetic susceptibility increases monotonically with the temperature. Therefore, the strongly interacting medium behaves as a paramagnetic substance both below and above the critical temperature T c . It is remarkable that our results for the magnetic susceptibility is in fair qualitative and quantitative agreement with Ref. [19] where it has been considered N f = 2 + 1 QCD with physical quark masses, discretized on a lattice by stout improved staggered fermions and a tree level improved Symanzik pure gauge action.

Conclusions and Discussion
In conclusion, let us summarize briefly the main results of the present paper. We investigated QCD with one flavor of staggered quark in an external magnetic field on the lattice. The external magnetic field has been introduced by means of the so-called Schrödinger functional. We have investigated the magnetic properties of one-flavour quarks and gluons in thermal equilibrium for magnetic field strengths up to |eH| ≤ 1.17 GeV. In particu-  Figure 11. The magnetic susceptibility χ q 2 f e 2 as a function of the temperature for different magnetic field strengths. lar, we focused on the effects of the magnetic field on several local observables and found results in qualitative agreement with recent results in the literature obtained with a different method, as described in Sect. 1, to implement external magnetic fields in QCD on the lattice. We have, also, looked for the effects of the magnetic field on the critical temperature. Surprisingly, we found that the critical temperature does not change even for the strongest magnetic field used in the present work. This is in striking contrast with the results in the literature. However, since we used one flavor rooted staggered quark which is known to be strongly affected by taste symmetry violation effects, one could suspect that our results on the critical temperature is merely due to lattice artifacts. Indeed, presently we are simulating the same physical system by adopting highly improved staggered quarks (HISQ) where the taste symmetry violations are dramatically reduced. Nevertheless, our preliminary simulations adopting HISQ quarks to do not yet display a clear dependence of the pseudocritical temperature on the background magnetic field. In any event, we plan to report progress on this subject in a future paper. We evaluated the magnetic contributions to the pressure, energy density, and free energy. Our results are in qualitative agreement with previous investigations. In particular, we confirm that the free energy density scales with H 2 for small enough magnetic field strengths. Moreover, we determined the magnetic susceptibility and found that the strongly interacting medium behaves like a paramagnetic substance both below and above the critical temperature in agreement with previous results in the literature.