QGP universality in a magnetic field?

We use top-down holographic models to study the thermal equation of state of strongly coupled quark-gluon plasma in external magnetic field. We identify different conformal and non-conformal theories within consistent truncations of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 8 gauged supergravity in five dimensions (including STU models, gauged N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2* theory) and show that the ratio of the transverse to the longitudinal pressure PT /PL as a function of T /B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{B} $$\end{document} can be collapsed to a ‘universal’ curve for a wide range of the adjoint hypermultiplet masses m. We stress that this does not imply any hidden universality in magnetoresponse, as other observables do not exhibit any universality. Instead, the observed collapse in PT /PL is simply due to a strong dependence of the equation of state on the (freely adjustable) renormalization scale: in other words, it is simply a fitting artifact. Remarkably, we do uncover a different universality in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2* gauge theory in the external magnetic field: we show that magnetized N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2* plasma has a critical point at Tcrit/B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{B} $$\end{document} which value varies by 2% (or less) as m/B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{B} $$\end{document} ∈ [0, ∞). At criticality, and for large values of m/B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{B} $$\end{document}, the effective central charge of the theory scales as ∝ B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{B} $$\end{document}/m.


Introduction and summary
In [1] the authors used the recent lattice QCD equation of state (EOS) data in the presence of a background magnetic field [2,3], and the holographic EOS results 1 for the strongly coupled N = 4 SU(N ) maximally supersymmetric Yang-Mills (SYM) to argue for the universal magnetoresponse. While N = 4 SYM is conformal, the scale invariance is explicitly broken by the background magnetic field B and its thermal equilibrium stress-energy tensor is logarithmically sensitive to the choice of the renormalization scale. It was shown in [1] that both the QCD and the N = 4 data (with optimally adjusted renormalization scale) for the pressure anisotropy R, i.e., defined as a ratio of the transverse P T to the longitudinal P L pressure, 2 collapse onto a single universal curve as a function of T / √ B, at least for T / √ B 0.2 or correspondingly for R 0.5, see figure 6 of [1]. The authors do mention that the 'universality' is somewhat fragile: besides the obvious fact that large-N N = 4 SYM is not QCD (leading to inherent JHEP06(2020)149 ambiguities as to how precisely one would match the renormalization schemes in both theories -hence the authors opted for the freely-adjustable renormalization scale in SYM), one observes the universality in R, but not in other thermodynamic quantities (e.g., P T /E -the ratio of the transverse pressure to the energy density).
So, is there a universal magnetoresponse? In this paper we address this question in a controlled setting: specifically, we consider holographic models of gauge theory/string theory correspondence [5,6] where all the four-dimensional strongly coupled gauge theories discussed have the same ultraviolet fixed point -N = 4 SYM. We discuss two classes of theories: • conformal gauge theories corresponding to different consistent truncations of N = 8 gauged supergravity in five dimensions 3 [7]; • non-conformal N = 2 * gauge theory (N = 4 SYM with a mass term for the N = 2 hypermultiplet) [7][8][9] (PW).
In the former case, the anisotropic thermal equilibrium states are characterized by the temperature T , the background magnetic field B and the renormalization scale µ; in the latter case, we have additionally a hypermultiplet mass scale m.
Before we present results, we characterize more precisely the models studied.
CFT diag : N = 4 SYM has a global SU(4) R-symmetry. In this model magnetic field is turned on for the diagonal U(1) of the R-symmetry. This is the model of [1], see also [4]. See section 2.1 for the technical details.
CFT STU : Holographic duals of N = 4 SYM with U(1) 3 ⊂ SU(4) global symmetry are known as STU-models [10,11]. In this conformal theory the background magnetic field is turned on for one of the U(1)'s. This model is a consistent truncation of N = 8 five-dimensional gauged supergravity with two scalar fields dual to two dimension ∆ = 2 operators. As we show in section 2.2, in the presence of the background magnetic field these operators will develop thermal expectation values.
nCFT m : As we show in section 2.3, within consistent truncation of N = 8 five-dimensional gauged supergravity presented in [7], it is possible to identify a holographic dual to N = 2 * gauge theory with a single U(1) global symmetry. In this model the background magnetic field is turned on in this U (1). The label m ∈ (0, +∞) denotes the hypermultiplet mass of the N = 2 * gauge theory.
CFT PW,m=0 : This conformal gauge theory is a limiting case of the nonconformal nCFT m model: Its bulk gravitational dual contains two scalar fields dual to dimension ∆ = 2 and ∆ = 3 operators of the N = 2 * gauge theory. As we show in section 2.3.1, in the presence of the background magnetic field these operators will develop thermal expectation values.

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CFT PW,m=∞ : This conformal gauge theory is a limiting case of the nonconformal nCFT m model: Its holographic dual can be obtained from the N = 8 five dimensional gauged supergravity of [7] using the "near horizon limit" of [12], 4 followed by the uplift to six dimensions -the resulting holographic dual is Romans F (4) gauged supergravity in six dimensions [15,16]. 5 The six dimensional gravitational bulk contains a single scalar, dual to dimension ∆ = 3 operator of the effective CFT 5 . There is no conformal anomaly in odd dimensions. Furthermore, there is no invariant dimension-five operator that can be constructed only with the magnetic field strength -as a result, the anisotropic stressenergy tensor of CFT PW,m=∞ plasma is traceless, and is free from renormalization scheme ambiguities. Details on the CFT PW,m=∞ model are presented in section 2.3.2. The renormalization scheme-independence of CFT PW,m=∞ is a welcome feature: we will use the pressure anisotropy (1.1) of the theory as a benchmark to compare with the other conformal and non-conformal models.
And now the results. There is no universal magnetoresponse. Qualitatively, among conformal/non-conformal models we observe three different IR regimes (i.e., when T / √ B is small): In CFT diag it is possible to reach deep IR, i.e., the T / √ B → 0 limit. For T / √ B 0.1 the thermodynamics is BTZ-like with the entropy density 6 [4] Both in CFT PW,m=0 and CFT PW,m=∞ (and in fact in all nCFT m models) there is a terminal critical temperature T crit which separates thermodynamically stable and unstable phases of the anisotropic plasma. Remarkably, this T crit is universally determined by the magnetic field B, (almost) independently 7 of the mass parameter m of nCFT m : See appendix D of [13] for details of the isotropic (no magnetic field) thermal states of N = 2 * plasma in the limit m/T → ∞. The first hint that N = 2 * plasma in the infinite mass limit is an effective five dimensional CFT appeared in [14]. 5 See [17] for a recent discussion. 6 We independently reproduce this result. 7 A very weak dependence on the mass parameter has been also observed for the equilibration rates in N = 2 * isotropic plasma in [18].

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i.e., the variation of T crit / √ B with mass about its mean value is 2% or less, see figure 7 (left panel). We leave the extensive study of this critical point to future work, and only point out that the specific heat at constant B at criticality has a critical exponent 8 α = 1 2 : where F is the free energy density, see figure 6.
The CFT STU model in the IR is different from the other ones. We obtained reliable numerical results in this model for T / √ B 0.06: we neither observe the critical point as in the CFT PW,m=0 and CFT PW,m=∞ models, nor the BTZ-like behavior (1.2) as in the CFT diag model, see figure 3 (left panel).
In figure 1 we present the pressure anisotropy parameter R (1.1) for the conformal theories: CFT diag (black curves), CFT STU (blue curves), CFT PW,m=0 (green curves) and CFT PW,m=∞ (red curves) as a function of 9 T / √ B. R is renormalization scheme independent in the CFT PW,m=∞ model, while in the former three conformal models it is sensitive to where µ is the renormalization scale. We performed high-temperature perturbative analysis, i.e., as T / √ B 1, to ensure that the definition of δ is consistent across all the conformal models sensitive to it, see appendix B. In the { top left, top right, bottom left, bottom right } panel of figure 1 we set {δ = 4 , δ = 2.5 , δ = 3.5 , δ = 7} (correspondingly) for R CFT diag , R CFT STU and R CFT PW,m=0 -notice that while all the curves exhibit the same high-temperature asymptotics, the anisotropy parameter R is quite sensitive to δ; in fact, R CFT diag diverges for δ = 2.5 (because P L crosses zero with P T remaining finite). Varying δ, it is easy to achieve R CFT diag , R CFT STU and R CFT PW,m=0 in the IR to be "to the left" of the scheme-independent (red) curve R CFT PW,m=∞ (top panels and the bottom left panel); or "to the right" of the scheme-independent (red) curve R CFT PW,m=∞ (the bottom right panel).
In figure 1 we kept δ the same for the conformal models CFT diag , CFT STU and CFT PW,m=0 . This is very reasonable given that one can match δ across all the models by comparing the UV, i.e., T / √ B 1 thermodynamics (see appendix B) -there are no other scales besides T and B, and thus by dimensional analysis, 10 If we give up on maintaining the same renormalization scale for all the conformal models, it is easy to 'collapse' all the curves for the pressure anisotropy, see figure 2. We will not perform sophisticated fits as in [1], and instead, adjusting δ independently for each model, 8 The critical point with the same mean-field exponent α has been observed in isotropic thermodynamics of N = 2 * plasma with different masses for the bosonic and fermionic components of the hypermultiplet [19]. 9 We use the same normalization of the magnetic field in holographic models as in [1]. 10 The asymptotic AdS5 radius L always scales out from the final formulas.  Figure 1. Anisotropy parameter R = P T /P L for conformal models CFT diag (black curves), CFT STU (blue curves), CFT PW,m=0 (green curves) and CFT PW,m=∞ (red curves) as a function of T / √ B. R CFTPW,m=∞ is renormalization scheme independent; for the other models there is a strong dependence on the renormalization scale δ = ln B µ 2 : different panels represent different choices for δ; all the models in the same panel have the same value of δ, leading to identical high-temperature asymptotics, T / √ B 1.
we require that in all models the pressure anisotropy R = 0.5 is attained at the same value of T / √ B (represented by the dashed brown lines): Specifically, we find that (1.6) is true, provided In a nutshell, this is what was done in [1] to claim a universal magnetoresponse for R 0.5. Rather, we interpret the collapse in figure 2 as nothing but a fitting artifact, possible due to a strong dependence of the anisotropy parameter R on the renormalization scale.
To further see that there is no universal physics, we can compare renormalization scheme-independent anisotropic thermodynamic quantities of the models: the entropy densities, see figure 3. The color coding is as before: CFT diag (black curves), CFT STU (blue curves), CFT PW,m=0 (green curves) and CFT PW,m=∞ (red curves). We plot the entropy densities relative to the entropy density of the UV fixed point at the corresponding tem- perature (see eq. (D.13) for the CFT PW,m=∞ model in [13]): (1.8) The dashed vertical lines in the left panel indicate the terminal (critical temperature) T crit / √ B for CFT PW,m=0 (green) and CFT PW,m=∞ (red) models which separates thermodynamically stable (top) and unstable (bottom) branches. Notice that s/s UV diverges for the CFT diag model as T / √ B → 0 -this is reflection of the IR BTZ-like thermodynamics (1.2); the dashed black line is the IR asymptote In nCFT m models it is equally easy to 'collapse' the data for the pressure anisotropy. In these models we have an additional scale m -the mass of the N = 2 hypermultiplet. In the absence of the magnetic field, i.e., for isotropic N = 2 * plasma, the thermodynamics is renormalization scheme-independent 11 [20]. Once we turn on the magnetic field, there is a scheme-dependence. In figure 4 we show the pressure anisotropy for N = 2 * gauge 11 Scheme-dependence arises once we split the masses of the fermionic and bosonic components of the N = 2 * hypermultiplet [20].  The dashed red curve represents the anisotropy parameter of the conformal CFT PW,m=∞ model, which is renormalization scheme-independent. In the left panel the renormalization scale δ = 4 for all the nCFT m models. In the right panel, we adjusted δ = δ m for each nCFT m model independently, so that the pressure anisotropy R nCFTm = 0.5 at the same temperature as in the CFT PW,m=∞ model, see (1.6). This matching point is denoted by dashed brown lines. In the right panel we show this for the model with m/ √ 2B = 1: the brown lines identify the critical temperature T crit / √ B and the relative entropy at the criticality s crit /s UV (these quantities are presented in figure 7). "Top" solid black curve denotes the thermodynamically stable branch and "bottom" dashed black curve denotes the thermodynamically unstable branch (see figure 6 for further details).
As in conformal models, the entropy densities (which are renormalization scheme independent thermodynamic quantities) are rather distinct, see left panel of figure 5. The color coding is as in figure 4, except that we collected more data 12 in addition to (1.10): these are the dashed and dotted curves. The entropy density of the UV fixed point is defined as in (1.8). All the nCFT m models studied, as well as the CFT PW,m=0 and CFT PW,m=∞ conformal models, have a terminal critical point T crit that separates the thermodynamically stable (top solid) and unstable (bottom dashed) branches, which we presented for the m √ 2B = 1 nCFT m model in the right panel. The dashed brown lines identify the critical temperature T crit and the entropy density s crit at criticality. In figure 6 we present results for the specific heat c B in this model defined as in (1.3). Indeed, the (lower) thermodynamically unstable branch has a negative specific heat (left panel); approaching the critical temperature from above we observe the divergence in the specific heat, both for the stable and the unstable branches. To extract a critical exponent α, defined as we plot (right panel) the dimensionless quantity c 2 B /s 2 as a function of T / √ B. Both the stable (solid) and the unstable (dashed) curves approach zero, signaling the divergence of the specific heat at the critical temperature (vertical dashed brown line), with a finite slope -this implies that the critical exponent is (1.12) 12 To have a better characterization of the critical points.  There is a remarkable universality of the critical points in nCFT m and conformal CFT PW,m=0 and CFT PW,m=∞ models. In figure 7 (left panel) we present the results for the critical temperature as a function of m/ √ 2B in nCFT m models (points). The horizontal dashed lines indicate the location of the critical points for the CFT PW,m=0 (green) and CFT PW,m=∞ (red) conformal models. In the right panel the dots represent the relative entropy, (1.14) One can understand the origin of the asymptote (1.13) from the fact that nCFT m models in the large m limit should resemble the conformal model CFT PW,m=∞ ; thus, we expect that γ ∞ ≈ γ CFT PW,m=∞ . Indeed, where we extracted numerically the value of s crit s UV for the CFT PW,m=∞ conformal model, used (1.8) to analytically compute the second factor in the first line, and substituted the numerical value for T crit / √ B of the CFT PW,m=∞ model in the second line. We now outline the rest of the paper, containing technical details necessary to obtain the results reported above. In section 2 we introduce the holographic theory of [7] and explain how the various models discussed here arise as consistent truncations of the latter: CFT diag in section 2.1, CFT STU in section 2.2, and nCFT m in section 2.3. The conformal models CFT PW,m=0 and CFT PW,m=∞ are special limits of the nCFT m model and are discussed in sections 2.3.1 and 2.3.2 correspondingly. Holographic renormalization is by now a standard technique [21], and we only present the results for the boundary gauge theory observables. Our work is heavily numerical. It is thus important to validate the numerical results in the limits where perturbative computations (analytical or numerical) are available. We have performed such validations in appendix B, i.e., when T √ B 1. We did not want to overburden the reader with details, and so we did not present the checks of the agreement of the numerical parameters (e.g., as in (2.23)) with the corresponding perturbative counterparts -but we have performed such checks in all models. There are further important constraints on the numerically obtained energy density, pressure, entropy, etc., of the anisotropic plasma: the first law of the thermodynamics dE = T ds (at constant magnetic field and the mass parameter, if available), and the thermodynamic relation between the free energy density and the longitudinal pressure F = −P L . The latter relation can be proved (see appendix A) at the level of the equations of motion, borrowing the holographic arguments of [22] used to establish the universality of the shear viscosity to the entropy density in the holographic plasma models. Still, as the first law of thermodynamics, it provides an important consistency check on the numerical data -we verified these constraints in all the models, both perturbatively in the high-temperature limit, to O B 4 T 8 inclusive, see appendix B, and for finite values of B/ √ T , see appendix C -once again, we present only partial results of the full checks.

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Our paper is a step in broadening the class of strongly coupled magnetized gauge theory plasmas (both conformal and massive) amenable to controlled holographic analysis. We focused on the equation of state, extending the work of [1]. The next step is to analyze the magneto-transport in these models, in particular the magneto-transport at criticality.

Technical details
The starting point for the holographic analysis is the effective action of [7]: where the F (J) are the field strengths of the U(1) gauge fields, A (J) , and P is the scalar potential. We introduced The scalar potential, P, is given in terms of a superpotential In what follows we set gauged supergravity coupling g = 1, this corresponds to setting the asymptotic AdS 5 radius to L = 2. The five dimensional gravitational constant G 5 is related to the rank of the supersymmetric N = 4 SU(N ) UV fixed point as The models discussed below, i.e., CFT diag , CFT STU and nCFT m , have holographic duals which are consistent truncations of (2.1). It would be interesting to study the stability of these truncations following [23].
The holographic dual to the CFT diag conformal model is a consistent truncation of (2.1) with leading to where we used the normalization of the bulk U(1) to be consistent with [1]. This model has been extensively studied in [1,4] and we do not review it here.

CFT STU
The holographic dual to the CFT STU is a special case of the STU model [10,11,24], a consistent truncation of the effective action (2.1) with leading to 9) and the scalar potential We would like to keep a single bulk gauge field, so we can set two of them to zero and work with the remaining one. The symmetries of the action allow us to choose whichever gauge field we want. To see this, notice that the action (2.9) is invariant under F (2) µν = 0 for the gauge fields and with the scalar field redefinitions ρ → ν 1/2 ρ −1/2 and ν → ν 1/2 ρ 1/2 . Thus, we arrive to the holographic dual of CFT STU as where once again we used the normalization of the remaining gauge field as in [1]. Solutions to the gravitational theory (2.11) representing magnetic black branes dual to anisotropic magnetized CFT STU plasma correspond to the following background ansatz: 13

12)
13 Note that we fixed the radial coordinate r with the choice of the metric warp factor in front of dz 2 .

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As shown in appendix B.1, the renormalization scheme choice (2.28) implies that in the high-temperature limit T 2 B, We can not solve the equations (2.16)-(2.20) analytically; adapting numerical techniques developed in [25], we solve these equations (subject to the asymptotics (2.21) and (2.22)) numerically. The results of numerical analysis are data files assembled of parameters (2.23), labeled by b. It is important to validate the numerical data (in addition to the standard error analysis). There are two important constraints that we verified for CFT STU (and in fact all the other models): • The first law of thermodynamics (FL), dE/(T ds) − 1 (with B kept fixed), leads to the differential constrain on data sets (2.23) (here = d db ): (2.30) • Anisotropy introduced by the external magnetic field results in P T = P L . From the elementary anisotropic thermodynamics (see [1] for a recent review), the free energy density of the system F is given by We emphasize that holographic renormalization (even anisotropic one) naturally enforces (2.27) (see [26] for one of the first demonstrations), but not (2.31). In appendix A we present a holographic proof 14 15 Technical details presented here are enough to generate the CFT STU model plots reported in section 1.
14 The proof follows the same steps as in the first proof of the universality of the shear viscosity to the entropy density in holography [22]. 15 Additionally, as in the nCFTm model with m/ √ 2B = 1 (see appendix C), we checked both relations for finite b.

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As in the CFT STU model, r 0 is completely scaled out from all the equations of motion. Eqs. (2.36)-(2.40) have to be solved subject to the following asymptotics: in the UV, i.e., as x → 0 + , (2.41) in the IR, i.e., as y ≡ 1 − x → 0 + , (2.42) The non-normalizable coefficients α 1,1 (of the dimension ∆ = 2 operator) and χ 0 (of the dimension ∆ = 3 operator) are related to the masses of the bosonic and the fermionic components of the hypermultiplet of N = 2 * gauge theory. When both masses are the same (see [20]) Furthermore, carefully matching to the extremal PW solution [8,9] (following the same procedure as in [20]) we find B m 2 = 2b where m is the hypermultiplet mass. We find it convenient to use to label different mass parameters in nCFT m models, see (1.10). In total, given η and b, the asymptotics expansions are specified by 8 parameters: {a 2,2,0 , a 4,2,0 , α 1,0 , χ 1,0 , a 1,h,0 , a 2,h 0 , r h,0 , c h,0 } , (2.46) which is the correct number of parameters necessary to provide a solution to a system of three second order and two first order equations, 3 ×2+2×1 = 8. Parameters α 1,0 and χ 1,0

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correspond to the expectation values of dimensions ∆ = 2 (O 2 ) and ∆ = 3 (O 3 ) operators (correspondingly) of the boundary nCFT m ; the other two parameters, a 2,2,0 and a 4,2,0 , determine the expectation value of its stress-energy tensor. Using the standard holographic renormalization [27] we find: for the components of the boundary stress-energy tensor, for the entropy density and the temperature. Note that, as expected [27],  [25], we solve these equations (subject to the asymptotics (2.41) and (2.42)) numerically. The results of numerical analysis are data files assembled of parameters (2.46), labeled by b and η. As for the CFT STU model, we validate the numerical JHEP06(2020)149 data verifying the differential constraint from the first law of the thermodynamics dE = T ds (FL) and the algebraic constraint from the thermodynamic relation F = −P L (TR): In appendix C we have verified FT and TR in the nCFT m model with m/ √ 2B = 1 numerically.
Technical details presented here are enough to generate nCFT m model plots reported in section 1.

CFT PW,m=0
The CFT PW,m=0 model is a special case of the nCFT m model when the hypermultiplet mass m is set to zero. This necessitates setting the non-normalizable coefficients α 1,1 and χ 0 to zero =⇒ η = 0 in (2.45). From (2.40) it is clear that this m = 0 limit is consistent with implying that the Z 2 symmetry of the holographic dual, i.e., the symmetry associated with χ ↔ −χ, is unbroken. In what follows, we study the Z 2 -symmetric phase of the CFT PW,m=0 anisotropic thermodynamics, 16 In appendix B.2 we verified FT and TR in CFT PW,m=0 to order O(b 4 ) inclusive; we also present O(B 4 /T 8 ) results for R CFT PW,m=0 and confirm that the renormalization scheme choice of κ as in (2.28) leads to (2.55) JHEP06(2020)149

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As before, r 0 is completely scaled out of all the equations of motion. Eqs. (2.68)-(2.71) have to be solved subject to the following asymptotics: in the UV, i.e., as x → 0 + , (2.72) in the IR, i.e., as y ≡ 1 − x → 0 + , In total, givenb, the asymptotic expansions are specified by 6 parameters: which is the correct number of parameters necessary to provide a solution to a system of two second order and two first order equations, 2 × 2 + 2 × 1 = 6. The parameter p 3 corresponds to the expectation value of a dimension ∆ = 3 operator of the boundary theory; the other two parameters, a 1,5 and a 2,5 , determine the expectation value of its stress-energy tensor. Using the standard holographic renormalization we find: for the components of the boundary stress-energy tensor, and for the entropy density and the temperature. Note that, There is no renormalization scheme dependence in (2.75), and the trace of the stress-energy tensor vanishes -there is no invariant dimension-five operator that can be constructed only with the magnetic field strength. The (holographic) free energy density is given by the standard relation (2.27). In (2.75)-(2.76) we used the subscript [5] to indicate that JHEP06(2020)149 the thermodynamic quantities are measured from the perspective of the effective fivedimensional boundary conformal theory; to convert to the four-dimensional perspective, we need to account for (2.65), see also [13], E, P T , P L = E [5] , P [5]T , P [5]L × 2k 3 As for the other models discussed in this paper, the first law of thermodynamics dE = T ds (at fixed magnetic field) and the thermodynamic relation F = −P L lead to constraints on the numerically obtained parameter set (2.74) (here = d db ): The proof follows the argument for the universality of the shear viscosity to the entropy density in holographic plasma [22]. Consider a holographic dual to a four dimensional 19 gauge theory in an external magnetic field. We are going to assume that the magnetic field is along the z-direction, as in (2.12). We take the (dimensionally reduced -again, this can be relaxed) holographic background geometry to be At extremality (whether or not the extremal solution is singular or not within the truncation is irrelevant), the Poincare symmetry of the background geometry guarantees that 19 Generalization to other dimensions is straightforward.

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where R µν is the Ricci tensor in the orthonormal frame. Clearly, an analogous condition must be satisfied for the full gravitational stress tensor of the matter supporting the geometry Because turning on the nonextremality will not modify (A.3), we see that (A.2) is valid away from extremality as well. Computing the Ricci tensor for (A.1) reduces (A.2) to Explicitly evaluating the ratio of the const in (A.4) in the UV (r → ∞) and IR (r → r horizon ) we recover for each of the models we study. We should emphasize that the condition (A.2) can be explicitly verified using the equations of motion in each model studied. The point of the argument above (as the related one in [22]) is that this relation is true based on the symmetries of the problem alone. B Conformal models in the limit T / √ B 1 In holographic models, supersymmetry at extremality typically guarantees that equilibrium isotropic thermodynamics is renormalization scheme independent (compare the N = 2 * model with the same masses for the bosonic and the fermionic components m 2 b = m 2 f , versus the same model with m 2 b = m 2 f [20]). This is not the case for the holographic magnetized gauge theory plasma in four space-time dimensions, e.g., see [1] for N = 4 SYM. In this appendix we discuss the high temperature anisotropic equilibrium thermodynamics of the conformal (supersymmetric in vacuum) models. For the (locally) four dimensional models (CFT diag , CFT STU and CFT PW,m=0 ) matching high-temperature equations of state is a natural way to relate renormalization schemes in various theories. In the CFT PW,m=∞ model, which is locally five dimensional, magnetized thermodynamics is scheme independent.

B.1 CFT STU
The high temperature expansion corresponds to the perturbative expansion in b. In what follows we study anisotropic thermodynamics to order O(b 4 ) inclusive. Introducing

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Using the results (B.12) and (B.13) (rather, we use more precise values of the parameters reported -obtained from numerics with 40 digit precision) we find at order n = 1: It is important to keep in mind that the value a 2,2,(2) is sensitive to the matter content of the gravitational dual -set of relevant operators in CFT STU that develop expectation values in anisotropic thermal equilibrium.