Critical Exponents of Nonequilibrium Phase Transitions in AdS/CFT Correspondence

We study critical phenomena of nonequilibrium phase transitions by using the AdS/CFT correspondence. Our system consists of charged particles interacting with a heat bath of neutral gauge particles. The system is in current-driven nonequilibrium steady state, and the nonequilibrium phase transition is associated with nonlinear electric conductivity. We define a susceptibility as a response of the system to the current variation. We further define a critical exponent from the power-law divergence of the susceptibility. We find that the critical exponent and the critical amplitude ratio of the susceptibility agree with those of the Landau theory of equilibrium phase transitions, if we identify the current as the external field in the Landau theory.


I. INTRODUCTION
Nonequilibrium phenomena have wider variety compared to equilibrium phenomena. The number of parameters that control the nonequilibrium systems is larger than that of the equilibrium systems, in general. For example, a current density J, under the presence of an external electric field E along the current, can be an additional parameter for a current-driven non-equilibrium steady state (NESS). In this case, the equilibrium state is realized when J = 0, and the magnitude of J (under a constant E) measures the "distance" from the equilibrium state. It is important to study how the system depends on this additional parameter. When the system is far from equilibrium, we need a non-perturbative method of analysis with respect to J (or E) that is applicable in the nonlinear regime.
Recently, the anti-de Sitter/conformal field theory (AdS/CFT) correspondence has been paid attention as a tool of analysis beyond the linear response regime. 1 The AdS/CFT correspondence is a duality between a classical gravity theory and a strongly coupled quantum gauge field theory [2,3]. This correspondence provides a computational method for the gauge field theory beyond the linear response regime in terms of general relativity.
For example, a NESS of strongly interacting gauge theory plasma was studied in Ref. [4]. It was shown in Ref. [4] that the system exhibits not only positive differential conductivity (PDC) but also negative differential conductivity (NDC) in the NESS driven by a constant current. Furthermore, a first-order and a secondorder nonequilibrium phase transitions associated with the nonlinear conductivity were discovered in the same system [5]. 2 In Ref. [5], critical exponents β and δ for the nonequilibrium phase transition were defined. 3 In the Landau theory of equilibrium phase transitions, critical exponent δ is defined from the power-law dependence 1 A review on application of the AdS/CFT correspondence to nonequilibrium physics is Ref. [1]. 2 A same type of phase transition was also observed later in a different setup of holography in Ref. [6]. 3 The critical exponent δ in this paper isδ in Ref. [5].
of the order parameter with respect to an external field (e.g. a magnetic field in ferromagnets). In this case, the external field does not drive the system into nonequilibrium states. However, the exponent δ in the nonequilibrium phase transition was defined as the exponent of the power behavior of the order parameter in the variation of the current. Interestingly, the obtained values of the β and the δ in the nonequilibrium phase transition agreed with those of β and δ in the Landau theory of equilibrium phase transitions, respectively, within the numerical error [5].
In this paper, we further proceed calculations of critical exponents in the nonequilibrium phase transition. We define a susceptibility as a response of the system to the current variation. We also define a critical exponent γ from the power-law divergence of the susceptibility. We find that the value of the critical exponent γ agrees with that in the Landau theory of equilibrium phase transitions. The critical amplitude ratio of the susceptibility also agrees with that in the Landau theory where the susceptibility is defined with respect to the external field. Together with the results for β and δ, our results state that the critical phenomena in the nonequilibrium phase transition in question have remarkable similarity with those in the Landau theory of equilibrium phase transitions, if we identify the current as the external field. 4 The organization of the paper is as follows. In Section II, we review the Landau theory of equilibrium phase transitions. In Section III, we explain the setup of our model. We focus on the so-called D3-D7 system. In Section IV, we propose the definitions of the susceptibility and the critical exponent γ mentioned above. We compute them and the results will be exhibited. We conclude in Section V.

II. LANDAU THEORY AND CRITICAL EXPONENT
We begin with a brief review of the Landau theory of equilibrium phase transitions. The new definitions of the susceptibility and the critical exponents in Section IV will be given by using an analogy with the Landau theory.
Here we consider a phase transition of ferromagnets. In the Landau theory, we assume the free energy is given as an even function of the magnetization because of the symmetry under the spin flip. Since we are interested in the critical point, we assume the magnetization is sufficiently small. Then we can expand the free energy as a power series in the magnetization and ignore the higher order terms: where M is the magnetization as an order parameter. F 0 and b are constants, whereas a = k(T − T c )/T c with a constant k. T c is the critical temperature of the secondorder phase transition. Since a thermal equilibrium state is realized as a minimum of the free energy, we consider to obtain the thermal equilibrium magnetization M 0 : for T < T c . If there is an external magnetic field in this system, the free energy is written as where H is the external magnetic field. Then, the thermal equilibrium state is determined by the following relation: Although the solution of Eq. (5) is complicated, M 0 becomes simple for T = T c : .
The magnetic susceptibility is defined as χ = ∂M/∂H. We obtain this from Eq. (5) as For T > T c with M = 0, we have whereas for T < T c . From Eqs. (8) and (9), the ratio of the coefficients of |T −T c |, which is called as critical amplitude ratio, becomes two: χ T >Tc /χ T <Tc = 2. Note that this value is independent of k and T c we have introduced. Let us derive the specific heat from the free energy. The specific heat is defined by Thus we find that the specific heat is constant. The divergent behaviors of various quantities at the critical point are characterized by the critical exponents. The definitions of the critical exponents in ferromagnets are given by In the Landau theory, the critical exponents are determined by Eqs. (3), (6), (8), (9), and (10) as: and these values are the same as those in the mean-field theory. Note that there are further two critical exponents η and ν, which are related to the correlation functions. However, these critical exponents cannot be determined within the above discussion. We are not going to deal with η and ν in the present paper.

III. SETUP
To realize a system in NESS, we employ (3+1)dimensional SU (N c ) N = 4 super-symmetric Yang-Mills (SYM) theory with a fundamental N = 2 hypermultiplet as the microscopic theory. The theory contains the gauge particles in the adjoint representation (which we call gluon sector) and the charged particles (quark sector) in the fundamental and antifundamental representation.
Here the charge is that of the global U (1) B symmetry, and not that of the color. In this sense, the gluon sector is neutral. We apply an constant external electric field acting on this charge. We take the large-N c limit in order to realize a NESS. This is because the degree of freedom of the gluon sector, which is O(N 2 c ), becomes sufficiently larger than that of the quark sector, which is O(N c ). Then we can ignore the back reaction to the gluon sector in this limit. As a result, the gluon sector acts as a heat bath for the quark sector. Then the system realizes a NESS with a constant current of the charge. The D3-D7 system is the gravity dual of our microscopic theory [7]. The D7-brane is embedded in the background geometry which is a direct product of a 5-dimensional AdS-Schwarzschild black hole (AdS-BH) and S 5 . The gluon sector and the quark sector correspond to AdS-BH and the D7-brane, respectively.
The metric of the AdS-BH part is given by is the radial coordinate of the geometry. The boundary is located at z = 0, whereas the horizon is located at z = z H . The Hawking temperature is given by T = √ 2/(πz H ). t and x denote the (3+1)dimensional spacetime coordinates of the gauge theory. The metric of the S 5 part is given by where 0 ≤ θ ≤ π/2 and dΩ d is the volume element of a d-dimensional unit sphere. For simplicity, the radius of the S 5 part has been taken to be 1. This is equivalent to choosing the 't Hooft coupling λ of the gauge theory at λ = (2π) 2 /2. In our D3-D7 system, the D7-brane is wrapped on the S 3 part of the S 5 . Since the radius of the S 3 part is cos θ, the configuration of the D7-brane is determined by the function θ(z). The asymptotic form of θ(z) is given by where qq denotes the chiral condensate and m q is the current quark mass [9,10]. (See also Ref. [8]. The D7-brane action is given by the Dirac-Born-Infeld (DBI) action: Here T D7 is the D7-brane tension, ξ are the worldvolume coordinates, g ab is the induced metric and F ab is the U (1) field strength on the D7-brane. The Wess-Zumino term does not contribute in our setup. Assuming the external electric field E is applied along the x direction, the asymptotic form of the gauge field A x on the D7-brane is related to E as Here we have employed the gauge ∂ x A t = 0. Thus, the Lagrangian density in the D7-brane action (20) is explicitly written as where the prime and the dot denote the differentiation with respect to z and t, respectively. The induced metric agrees with the metric of AdS-BH (17) except for g zz = 1/z 2 + θ (z) 2 . According to the AdS/CFT dictionary, the current density J (in the x direction) is given by J = ∂L D7 /∂A x . Let us perform a Legendre transformatioñ so that J becomes a control parameter.
The Euler-Lagrange equation for θ is In addition, requiring the on-shell D7-brane action (23) to be real, we can determine the relationship between J and E as where z * is the point at whichL D7 equals zero [8].
We need to solve the equation of motion (EOM) (24) numerically in order to obtain θ(z * ) explicitly. We employ two boundary conditions: θ(z)/z| z=0 = m q and θ | z=z * = [B − √ B 2 + C 2 ]/(Cz * ). Here B = 3z 8 H + 2z 4 H z 4 * + 3z 8 * and C = 3(z 8 * − z 8 H ) tan θ(z * ). The second boundary condition is derived from the EOM at z = z * [11]. (See also Ref. [4].) After solving the EOM numerically under these boundary conditions, we pick out the corresponding values of J and E so that m q agrees with the designed value. Since the numerical analysis becomes unstable at z = 0, z = z H and z = z * , we avoid these points by introducing cutoffs.
We choose m q = 1 and N = 1 for simplicity. In other words, our J is understood as J/N when we assign general value to N . The J-E characteristics at various temperatures are shown in Fig. 1.
For T < T c the NDC region, where the slope of the J-E curve is negative, is smoothly connected to the PDC region, where the slope is positive. On the other hand, for T > T c there is an intermediate region between the NDC region and the PDC region, where E has three different possible values at a given J. Since the value of E has to be selected to one of them, E jumps to another value at some point in this region. It was proposed that the transition point is determined by a thermodynamic potential defined by using the Hamiltonian of the D7-brane [5]. The Hamiltonian density is given bỹ Then the thermodynamic potential is defined as where L count denotes the counterterms that renormalize the divergence at the boundary z = 0. L count is given by where each term of (28) is given in Ref. [8] as Here γ ij is the induced metric on the z = slice and κ is a factor in order to make the argument of the logarithm dimensionless. The value of κ is scheme dependent, and we have chosen this value as one of the possible choices so that ∂ 2 L D7 /∂E 2 = 0 for vacuum (T = 0, E = 0, m q = 0). It has been found that the stable state has the lowest E at a given J [5]. As a result, the transition point between the NDC phase and the PDC phase is the point indicated by the arrow between A and B in Fig. 1. We call the transition for T > T c the first-order transition because E changes discontinuously. We call the transition for T = T c the second-order transition because the differential resistivity ∂E/∂J diverges there while the σ = J/E changes continuously [5].

IV. RESULTS
In this section, we consider the critical phenomena of the nonequlibrium phase transition given in the previous section.

A. β and δ
In our nonequilibrium phase transition, the critical exponents β and δ are defined in Ref. [5] as where T is the heat-bath temperature and ∆σ is the difference of the conductivity between the PDC phase and the NDC phase at a transition point. σ c and J c are the conductivity and the current density at the critical point, respectively. ∆σ is evaluated along the line of the first-order phase transition. The value of δ is evaluated along the line of T = T c . These definitions correspond to Eqs. (11) and (12) in the Landau theory: the definitions (33) were proposed by using an assumption that σ − σ c and J − J c play a role of the order parameter and that of the external field, respectively. Our numerical data are shown in Fig. 2. We obtain β = 0.505 ± 0.008 and δ = 3.008 ± 0.032. It has been proposed in Ref. [5] that the chiral condensate qq is another candidate for the order parameter. Then we have another definition of the critical exponents: where qq c is qq at the critical point. We show the numerical results for chiral condensate in Fig. 3. We find that these critical exponents are β chiral = 0.515 ± 0.029 and δ chiral = 2.999 ± 0.061. We have reconfirmed the results found in Ref. [5]. Note that all of these values agree with those of the Landau theory given in (16) within the numerical error.

B. γ
This section is the main part of the present paper which is about the definition and calculation of the critical exponent γ. First we define the critical exponent γ for our nonequilibrium phase transition. In Section II, we have reviewed that the critical exponent γ in the Landau theory is defined by using the magnetic susceptibility χ = ∂M/∂H, where M is the magnetization and H is the external magnetic field. Near the critical point, the magnetic susceptibility behaves as χ ∝ |T − T c | −γ . In our nonequilibrium phase transition, since we use either the conductivity or the chiral condensate as the order parameter, it is natural to generalize the definition of χ asχ where J is again assumed to act as the external field. We can rewriteχ by using the definition of conductivity σ = J/E,χ so that it can be calculated from the J-E characteristics. We propose to define the critical exponent γ as in our nonequilibrium phase transition. 5 There are two possible definitions ofχ for T > T c : that in the NDC phase and that in the PDC phasẽ As shown in Fig. 4, the behaviors of the susceptibilities in these phases are similar to each other and it is found that each value of γ is γ NDC = 1.018 ± 0.043 and γ PDC = 1.014±0.042. We find that they agree with that from the Landau theory (16), γ = 1, within the numerical error.
In addition, we evaluate the γ for T < T c . In the liquid-vapor phase transition, the susceptibility should be calculated along the critical isochore in the crossover region. Therefore, it is necessary to determine the line that corresponds to the critical isochore for our nonequilibrium phase transition. In analogy with the ferromagnet phase transition or the liquid-vapor transition, we choose this point as the inflection point in the J-σ curve. The phase diagram is shown in Fig. 5 and it is found that the inflection point for T < T c is nearly constant with σ = σ c = 0.0156.
We show the relationship between the values ofχ and the temperature along the σ = σ c line in Fig. 6. The numerical data gives γ crossover = 1.022 ± 0.025. Furthermore, if we assume that γ NDC = γ PDC = γ crossover , we find χ NDC /χ crossover = 2.2 ± 0.4 andχ PDC /χ crossover = 2.0 ± 0.4. These results agree with the fact that the critical amplitude ratio in the Landau theory is 2. The critical phenomena are exhibited in Fig. 6. All of the above arguments go along with the chiral condensate instead of the conductivity.

V. CONCLUSION AND DISCUSSION
We found that the critical exponents of our nonequilibrium phase transition agree with those in the Landau theory: β = 1/2, δ = 3, and γ = 1. The critical amplitude ratio ofχ also agreed with that of the Landau theory. Our results satisfy the scaling laws such as the Widom scaling, γ = β(δ − 1), within the numerical error.
We have two remarks. There are models of equilibrium phase transitions in which a deviation of the law of rectilinear diameter gives the critical exponent α [12][13][14]. Let us see how it goes for our case. If we assume that the foregoing method is valid in our system, we may define α as where A is a constant, σ ave = (σ NDC + σ PDC )/2 and σ c is the critical conductivity at T = T c . In Fig. 8, we show the conductivities in the PDC phase, those in the NDC phase, and their averages. We obtain the value of the exponent α = 0.048 ± 0.111, which agrees with (16) of the Landau theory. We may define α chiral as where B is a constant, qq ave = ( qq NDC + qq PDC ) /2 and qq c is the critical value of the chiral condensate.
However, our numerical data shows that B 0: the values of the chiral condensates in each phase are arranged symmetrically with respect to the critical value, as is the case with the ferromagnet phase transition. For this reason, we cannot determine the value of α chiral accurately in this manner. We leave more concrete definition of the critical exponent α for our phase transition to future work. The second remark is on the relationship with the Landau theory. In our definitions of the critical exponents, we assumed that J − J c plays a role of the magnetic field H in (4). We find that the critical exponent and the critical amplitude ratio of the susceptibility we defined agree with those of the Landau theory of equilibrium phase transitions. Together with the results for β and δ, our results state that the critical phenomena in the nonequilibrium phase transition in question have remarkable similarity with those in the Landau theory of equilibrium phase transition. On the other hand, J − J c (under the presence of the external electric field along the current) is a parameter that appears only in the non-equilibrium systems. It is important to clarify in detail the similarity and the difference between the equilibrium phase transitions of the Landau theory and that studied in the present paper. This is an issue for future research to explore.