Inverse magnetic catalysis in bottom-up holographic QCD

We explore the effect of magnetic field on chiral condensation in QCD via a simple bottom-up holographic model which inputs QCD dynamics through the running of the anomalous dimension of the quark bilinear. Bottom-up holography is a form of effective field theory and we use it to explore the dependence on the coefficients of the two lowest order terms linking the magnetic field and the quark condensate. In the massless theory, we identify a region of parameter space where magnetic catalysis occurs at zero temperature but inverse magnetic catalysis at temperatures of order the thermal phase transition. The model shows similar nonmonotonic behavior in the condensate with $B$ at intermediate $T$ as the lattice data. This behavior is due to the separation of the transition at which a thermal width develops for the mesons and the chiral transition in the holographic framework. The introduction of quark mass raises the scale of $B$ where inverse catalysis takes over from catalysis until the inverse catalysis lies outside the regime of validity of the effective description leaving just catalysis.

Figure 1

Lattice results for the change in the quark condensate as a function of magnetic field strength over a range of temperatures—figure taken from [7].

Figure 2

The holographic model’s results for the change in the condensate of light quarks as a function of magnetic field strength over a range of temperatures for $N_c=N_f=3$. $T_{C0}$ is the thermal transition temperature at $B=0$ which is used to set our holographic energy scale (and is approximately 160 MeV according to the lattice simulations). For this plot the model parameters are taken as $\kappa =0.05$ and $b=0.037$.

Figure 3

Thermal phase transitions in the holographic model with $N_c=N_f=3$. For values of the parameter $\kappa$ close to $\kappa =1$, a first-order chiral transition is present. As the value of $\kappa$ is reduced and the black hole is deformed along the $L$ axis, the phase transition switches to becoming second order. The introduction of a background magnetic field can be seen to affect the value of the transition order parameter, $\sigma$. Here we show an example with magnetic catalysis at low temperature and inverse magnetic catalysis at higher temperatures, a phenomenon which reduces the critical temperature, $T_c(B)<T_c(0)$.

Figure 4

Plot showing the chiral embeddings (at $B=0$) for a range of temperatures. Each embedding is colored to match the black-hole horizon pertaining to the relevant temperature. The second-order nature of the transition is evident; the embedding smoothly transforms into the flat $L=0$ embedding at the critical temperature, $T_c$.

Figure 5

The phase structure of the holographic model in terms of the phenomenological parameters $a$ and $b$. The $a–b$ plane can be dissected into four sectors wherein the condensate is affected differently with temperature and an external magnetic field.

Figure 6

Plots of the critical temperature against $eB$, $T_{C0}=160\mathrm{MeV}$. We show the best fit lattice data taken from [8] and the holographic model’s best fit (in the chiral limit) to those data ($\kappa =0.05$, $a=0$, $b=0.037$). We also show the holographic model’s prediction for another value of $b=0.33$—the model depends on the quantity $b{B}^2$ so the $eB$ axis is simply rescaled by this change.

Figure 7

Plot showing the chiral embeddings at $T=0.75T_{C0}$ for a range of magnetic field. Increasing $B$ moves the embedding towards and onto the horizon but initially also increases the condensate value.

Figure 8

Plot showing the change in the chiral condensate as a function of $B$ for different quark masses. The parameters are those used to make Fig. 5.