PHYSICAL REVIEW D 103, 066022 (2021) Finite density effects on chiral symmetry breaking in a magnetic field in 2 + 1 dimensions from holography † ‡ Diego M. Rodrigues ,1,2,* Danning Li ,3, Eduardo Folco Capossoli ,4,5, and Henrique Boschi-Filho 4,§ 1Centro de Matemática, Universidade Federal do ABC, Santo Andr´e 09580-210, Brazil 2Centro de Física, Universidade Federal do ABC, Santo Andr´e 09580-210, Brazil 3Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China 4Instituto de Física, Universidade Federal do Rio de Janeiro, 21.941-972 Rio de Janeiro, RJ, Brazil 5Departamento de Física and Mestrado Profissional em Práticas de Educação Básica (MPPEB), Col´egio Pedro II, 20.921-903 Rio de Janeiro, RJ, Brazil (Received 5 January 2021; accepted 2 March 2021; published 26 March 2021) In this work, we study finite density effects in spontaneous chiral symmetry breaking as well as chiral phase transition under the influence of a background magnetic field in 2 þ 1 dimensions. For this purpose, we use an improved holographic soft wall model based on an interpolated dilaton profile. We find inverse magnetic catalysis at finite density. We observe that the chiral condensate decreases as the density increases, and the two effects (addition of magnetic field and chemical potential) sum up, decreasing even more the chiral condensate. DOI: 10.1103/PhysRevD.103.066022 I. INTRODUCTION Ref. [11], the authors proposed a solution to this problem and provided extremely accurate results for the QCD Systems with finite chemical potential for fermions are a transition, extrapolating from an imaginary chemical poten- challenging and actual subject. The main reason for this tial up to real baryonic potential μ ¼ 300 MeV. interest is that in many physical systems we have to take B There is an alternative approach to nonperturbative into account a fermionic density, such as in heavy-ion QCD, or even LQCD, based on the AdS=CFT correspon- collisions, neutron stars, and condensed matter theories, dence [12,13]. Presented in 1997, this correspondence, among others. For a review, see, e.g., Refs. [1,2]. generically referred to as holography, relates a strong In particular, nuclear matter can be treated within a very coupling theory, without gravity, in a four-dimensional well-established theory, which is based on first principles and a nonperturbative approach, called lattice QCD space to a weak coupling theory, including gravity, in a (LQCD). The calculations in LQCD are numerical and curved higher-dimensional space. The theoretical frame- usually via Monte Carlo simulations. However, at finite work to deal with nuclear matter in the presence of a finite chemical potential, LQCD seems to crash due to the sign chemical potential within the AdS=CFT correspondence problem, meaning that action of the theory becomes was put forward in many important works. See, for instance, Refs. [14–17] and, more recently, Refs. [18–28]. complex [3,4]. 3 þ 1 Some proposals have been made to overcome this Despite real QCD being a ( )-dimensional gauge difficulty. For instance, in Refs. [5,6], a complex chemical theory, the numerical calculations in such a background (in potential was used. In Refs. [7,8], the authors have used the presence of a magnetic field) are extremely hard and reweighting approaches; in Ref. [9], nonrelativistic expan- reliable only for low values of the magnetic field, as can be – sions were used; and in Ref. [10], the authors have dealt with seen in Refs. [29 35] in the holographic context as well as the reconstruction of the partition function. Very recently, in in the nonholographic approach [36,37]. Here, in this work, our focus is to study the finite density effects on chiral symmetry breaking in the presence of a *[email protected] † background constant magnetic field B in 2 þ 1 dimensions [email protected] ‡ [email protected] based on holographic studies done at zero density in §[email protected] Refs. [38–41]. Our choice for a dimensional reduction comes from the fact that, in 2 þ 1 dimensions, our model Published by the American Physical Society under the terms of has a computational task easier than in real QCD even when the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to considering both nonzero chemical potentials and magnetic the author(s) and the published article’s title, journal citation, fields. This approach is very useful, since we can learn from and DOI. Funded by SCOAP3. this model and try to extrapolate it to real QCD. Some 2470-0010=2021=103(6)=066022(8) 066022-1 Published by the American Physical Society RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 103, 066022 (2021) 2 þ 1 previous nonholographic works in dimensions with z B magnetic fields can be seen, for instance, in Refs. [42–47]. A ¼ μ 1 − dt þ ðxdy − ydxÞ: ð6Þ z 2 This work is organized as follows. In Sec. II, we describe h our holographic model. In Sec. II A, we detail the back- The temperature of the black hole solution can be ground geometry, which is an AdS4-Reissner-Nordstrom obtained through the Hawking formula black hole in the presence of a background magnetic field, and present all relevant quantities for our further calcula- 0ð Þ ¼ − f zh ð Þ tions. In Sec. II B, we show the holographic description of T 4π 7 our effective action for a complex scalar field interacting with a non-Abelian gauge field. Such a complex scalar field and is given by will be related to the chiral condensate at the boundary theory. In Sec. III, we present our numerical results where we 1 ð μ Þ¼ ð3 − 2 4 − μ2 2Þ ð Þ observe inverse magnetic catalysis (IMC) at finite density as T zh; ;B B zh zh : 8 4πzh well as the decreasing of the chiral condensate when the density increases. In Sec. IV, we make our conclusions. Note that this equation for zh has four roots for fixed T, μ, and B, but only one is real and positive, which means II. HOLOGRAPHIC MODEL physically acceptable [note that, by taking Eq. (7),wehave chosen the outer horizon]. So, from now on, we just A. Background geometry consider the branch zh > 0. In this section, we will establish the holographic description of our background geometry. Considering the 2 1. IR near-horizon geometry: Emergence of AdS2 × R Einstein-Maxwell action on AdS4 (for more details, see [38,39], and references therein): Here, we briefly discuss an important feature of the near- Z horizon (z → zh) geometry of extremal (T ¼ 0) charged 1 pffiffiffiffiffiffi 6 ¼ 4 − − − 2 MN ð Þ black holes in asymptotically AdS spacetime, which is the S 2 d x g 2 L FMNF ; 1 2 2κ4 L emergence of the AdS2 × R space [50–52]. An extremal black hole is characterized by the fact that its temperature T κ2 where 4 is the four-dimensional coupling constant (with [Eq. (8)] is zero. In this case the horizon function [Eq. (5)] 2κ2 ≡ 16π ’ the relation 4 G4 with 4D Newton s constant G4), becomes M ¼ð Þ x t; z; x; y , with z being the holographic coordinate, and F ¼ ∂ A − ∂ A is the field strength for the 3 4 MN M N N M ð Þj ¼ 1 − 4 z þ 3 z ð Þ ð1Þ f z T¼0 ; 9 U gauge field AM. Throughout the text, we will work in z z 2 h h units such that 2κ4 ¼ L ¼ 1. The field equations from Eq. (1) are where it has a double zero at the horizon and can be Taylor 1 expanded as R ¼ 2 FP F − g F2 − 3g ; ð2Þ MN M NP 4 MN MN 6 ð Þ ≈ ð − Þ2 ð Þ f z 2 z zh : 10 MN zh ∇MF ¼ 0; ð3Þ Now, defining a dimensionless coordinate w through the where R is the Ricci tensor and g is the metric tensor. MN MN rescaling w ≔ z=z and changing variables according to Since we want to include a nonzero chemical potential μ h w ¼ 1 þ z η, we find that the geometry in the near-horizon and a constant magnetic field B, from now on in this work, h region (z → z ) becomes it is worthwhile to mention that the magnetic field B that we h are considering is always a background field. The ansatz we 2 2 η L 1 are going to consider is the dyonic AdS/Reissner- ds2 ≈ − dt2 þ eff dη2 þ ðdx2 þ dy2Þ; ð Þ 2 η2 2 11 Nordstrom black hole solution [48,49], with both electric Leff zh and magnetic charge, given by 2 which is the AdS2 × R , with AdS2 curvature radius pffiffiffi 1 2 L ≡ 1= 6, in units of L ¼ 1. Thus, in the near-horizon 2 ¼ − ð Þ 2 þ dz þ 2 þ 2 ð Þ eff ds 2 f z dt dx dy ; 4 R2 z fðzÞ regime (IR), the AdS2 × space controls the low-energy physics of the dual gauge theory on the boundary. 3 4 Therefore, it seems that supergravity on AdS4 flows in ð Þ¼1−ð1þμ2 2 þ 2 4Þ z þðμ2 2 þ 2 4Þ z f z zh B zh zh B zh ; the IR to a gravity theory on AdS2 which, in turn, is zh zh dual to a (0 þ 1)-dimensional effective conformal quantum ð Þ 5 theory, which is referred to as an IR CFT1. For a more 066022-2 FINITE DENSITY EFFECTS ON CHIRAL SYMMETRY … PHYS.