1008/1008.1581.md

Holographic quantum criticality from multi-trace deformations

Thomas Faulkner

Kavli Institute for Theoretical Physics
University of California
Santa Barbara, CA 93106-4030

Gary T. Horowitz

and

Matthew M. Roberts

Department of Physics
University of California
Santa Barbara, CA 93106-4030

Abstract

We explore the consequences of multi-trace deformations in applications of gauge-gravity duality to condensed matter physics. We find that they introduce a powerful new “knob” that can implement spontaneous symmetry breaking, and can be used to construct a new type of holographic superconductor. This knob can be tuned to drive the critical temperature to zero, leading to a new quantum critical point. We calculate nontrivial critical exponents, and show that fluctuations of the order parameter are ‘locally’ quantum critical in the disordered phase. Most notably the dynamical critical exponent is determined by the dimension of an operator at the critical point. We argue that the results are robust against quantum corrections and discuss various generalizations.# Contents

1	Introduction	2
2	Double trace deformations	5
2.1	Gravity setup and boundary conditions . . . . .	5
2.2	Symmetry breaking from double trace deformations . . . . .	7
2.3	A novel holographic superconductor . . . . .	9
2.4	Non-zero density and stability conditions . . . . .	11
3	Quantum critical point	13
3.1	The flow of double trace couplings and the 2 point function . . . . .	13
3.2	Properties of the critical point . . . . .	17
3.3	Renormalization group interpretation . . . . .	20
3.4	Parametric dependence on bulk couplings . . . . .	26
4	Constructing the ordered phase	26
4.1	Ansatz for the background . . . . .	28
4.2	Asymptotic $AdS\_4$ data and the free energy . . . . .	28
4.3	IR fixed point and shooting . . . . .	31
4.4	Confirming the scaling relations close to the critical point . . . . .	33
4.5	Critical solution . . . . .	36
5	Discussion	37
5.1	Summary of critical exponents . . . . .	37
5.2	Applications of our results . . . . .	38
5.3	Magnetic fields . . . . .	39
5.4	Quantum corrections . . . . .	41
5.5	Lifshitz normal phase . . . . .	42
A	Equations of motion	45
A.1	Field equations . . . . .	45
A.2	Our ansatz . . . . .	46
A.3	Calculating conductivity . . . . .	46

B	Critical temperature at zero chemical potential	48
C	More on the 2 point function	49
D	Complete $AdS\_4$ expansion and boundary terms	51

1 Introduction

Over the past couple of years, gauge/gravity duality has been applied to a number of problems in condensed matter physics (for reviews see [1, 2, 3]). An important feature of some condensed matter systems is the existence of quantum critical points, marking continuous phase transitions at zero temperature. One goal of the present work is to introduce and study a new mechanism for generating quantum critical points in the context of gauge/gravity duality. We will see that the behavior near the critical points is described by nontrivial critical exponents and goes beyond the usual Landau-Ginzburg-Wilson description of phase transitions at zero temperature.

A second goal is to introduce a new type of holographic superconductor. The key ingredient in constructing a gravitational dual of a superconductor is to find an instability which breaks a $U(1)$ symmetry at low temperature and causes a condensate to form. Previous constructions have started with a charged anti de Sitter (AdS) black hole which has such an instability when coupled, e.g., to a charged scalar field [4, 5, 6]. We will show that there is another source of instability which applies even for Schwarzschild AdS black holes. So these new superconductors can exist even at zero chemical potential and no net charge density.

Both of these goals are achieved by adding a multi-trace operator to the dual field theory action¹. For example, given a (single trace) scalar operator $\mathcal{O}$ of dimension $\Delta_- < 3/2$ in a $2 + 1$ dimensional field theory (which will be our main focus) one can modify the action

$$S\to S-\int d^{3}x\bar{\kappa }{\mathcal{O}}^{†}\mathcal{O}(1.1)S\; \backslash rightarrow\; S\; -\; \backslash int\; d^3x\; \backslash bar\{\backslash kappa\}\; \backslash mathcal\{O\}^\backslash dagger\; \backslash mathcal\{O\}\; \backslash quad\; (1.1)$$

where for convenience later the coupling will be rescaled as $\bar{\kappa} = 2(3 - 2\Delta_-)\kappa$ . Since this is a relevant deformation, it is unnatural to exclude such a term, and it has important consequences. If $\mathcal{O}$ is the operator dual to the bulk charged scalar field in conventional holographic superconductors, then adding this term (with $\kappa > 0$ ) makes it harder to form the condensate and lowers the critical temperature. We will see that in some cases $T_c$ vanishes at a finite value of $\kappa = \kappa_c$ . This defines a new quantum critical point which we will study in

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¹For another recent discussion of multi-trace operators in gauge/gravity duality, see [7].detail. Since $T_c$ can be quite large at $\kappa = 0$ , adding this double trace perturbation introduces a sensitive new knob for adjusting the critical temperature.

For $\kappa > \kappa_c$ (and nonzero chemical potential $\mu$ ), the ground state is described by the extremal Reissner-Nördstrom (RN) AdS black hole which has an emergent $AdS_2$ geometry in the IR. For $\kappa < \kappa_c$ there are various possible IR geometries depending on details of the bulk potential. However, as $\kappa$ approaches $\kappa_c$ from below, the bulk solution develops an intermediate $AdS_2$ geometry. It is this intermediate region which controls the behavior near the critical point. For example, we will show that the critical exponents do not take mean field values, but are determined by the scaling dimension of certain operators in the $0 + 1$ dimensional CFT dual to this region. This is closely analogous to the way properties of holographic non-Fermi liquids [8, 9, 10] were described in terms of a dual $0 + 1$ dimensional CFT [11, 12]. In addition, the instability for $\kappa < \kappa_c$ can be interpreted as turning on a double trace term with negative coefficient in the $0 + 1$ dimensional CFT dual to this region.

Let us contrast this with the usual argument for why the RN AdS black hole becomes unstable at low temperature in the presence of a scalar field [13]. In the $AdS_2$ near horizon geometry of the $T = 0$ solution, the scalar field has an effective mass $m_{\text{eff}}$ which depends on the original mass $m$ and charge $q$ of the scalar field. When this effective mass squared is below the Breitenlohner-Freedman (BF) bound [14] for $AdS_2$ , this near horizon region becomes unstable. Since $m^2$ is above the BF bound of the asymptotic $AdS_4$ geometry, the asymptotic region is stable, and the solution settles down to a black hole with scalar hair. It is now clear that this argument is sufficient but not necessary. It overlooks the possibility of instabilities with $m_{\text{eff}}^2$ above the BF bound which are allowed due to modified boundary conditions for the scalar field. The boundary conditions may be modified due to the addition of a multi-trace deformation in the dual field theory [15, 16], or simply due to alternative quantization of the bulk theory [17].

It was widely believed that if one added a double trace term with $\kappa < 0$ , then the theory would not have a stable ground state. However, we have recently shown that this is not necessarily the case [18]. For a large class of dual gravity theories, there is still a stable ground state with $\langle \mathcal{O} \rangle \neq 0$ when the boundary conditions correspond to $\kappa < 0$ . As one increases the temperature, there is a second order phase transition to a state with $\langle \mathcal{O} \rangle = 0$ . This provides a new mechanism for spontaneously breaking a $U(1)$ symmetry and constructing novel holographic superconductors. This mechanism does not require a charged black hole and works for Schwarzschild AdS as well. In other words, one can set $\mu = 0$ and still break the $U(1)$ symmetry at low temperature. The critical temperature is now set by $\kappa$ . We will discuss some properties of these novel holographic superconductors in section 2.It is worth pointing out that the coupling $\kappa$ , as the coefficient of the square of the order parameter, is the usual tuning parameter in the context of Landau-Ginzburg theory. Also if $\mathcal{O}$ is a gauge invariant trace of a fermion bilinear then the double trace is a 4 fermion interaction, a natural interaction to consider.

Although our discussion so far has focussed on the case where the operator $\mathcal{O}$ is charged, our results apply equally well when $\mathcal{O}$ is neutral. In this case, the ordered phase breaks a $Z_2$ symmetry. More generally, one can imagine different symmetry breaking scenarios where for example $\mathcal{O}$ could be part of a triplet of operators forming a representation of $SU(2)$ which is spontaneously broken at low temperature. This is a particular attractive possibility as outlined in [19, 20], since in many condensed matter systems $SU(2)$ spin is a global symmetry (ignoring spin orbit effects.) Including an exact global $SU(2)$ symmetry then allows us to model magnetism in a holographic setup. The triplet in which one embeds $\mathcal{O}$ can be interpreted as the staggered order parameter associated with anti-ferromagnetic transitions. Since the boundary theory has a global $SU(2)$ symmetry the bulk will have an $SU(2)$ gauge symmetry distinct from the $U(1)$ electromagnetic charge. For the rest of this paper the $SU(2)$ gauge fields and triplet structure of the order parameter will not play a roll.

It is useful to bear this possibility in mind, in particular so we can compare our results to quantum phase transitions in metallic systems, where anti-ferromagnetic order plays an important role. The “standard” theory of which was given in [21, 22] is based on the renormalization group Landau-Ginzburg paradigm. However experimental measurements (see for example [23, 24], and references therein) of heavy fermion systems with quantum critical points show that the “standard” theory can break down, as a result new theoretical methods are required. Subsequently several different methods were developed (see for example [25, 26, 27, 28].) One such method [25, 26] which is formally justified in a large $d$ expansion [29] shows local quantum critical behavior similar to the new quantum critical point that we find. It will be useful to compare and contrast our results to those of the standard theory and the other theoretical methods used in the study of quantum criticality in heavy fermion systems.

Since the finite density normal phase we consider is governed by $AdS_2 \times R^2$ in the IR, at zero temperature the theory has a finite entropy density. This has lead many people to suggest that this state must not be the true ground state, since otherwise one finds unnatural violations of the third law of thermodynamics. Of course this may be natural in the context of applications to heavy fermion system where superconductivity instabilities can be observed close to criticality. So it may be that the state we work in is the correct one for a large range of temperatures, but ultimately at low temperatures something else takes over.Indeed as is discussed and extended in this paper $AdS_2$ has many possible forms of instability. This however motivates us to attempt to extend our results in various directions to directly address this problem. One extension we consider is adding a magnetic field. We show that while a magnetic field suppresses the superconducting instability, it can enhance the neutral (anti-ferromagnetic) instability. Another extension we consider is replacing $AdS_2 \times R^2$ with other possible IR geometries, such as a Lifshitz geometry which does not have finite entropy density at zero temperature. We find our results are rather robust here.

The organization of the paper is as follows: In the next section we show how double trace deformations can induce spontaneous symmetry breaking and use this to construct a new type of holographic superconductor with zero net charge density. We then extend this to the finite density case and show that the coefficient of the double trace deformation provides a sensitive knob by which one can tune the critical temperature $T_c$ to zero. In section 3 we study the new quantum critical point that arises and analytically compute the nontrivial critical exponents. In section 4 we numerically construct the backreacted geometries that correspond to the ordered (condensed) phase away from the phase transition. We confirm the critical exponents near the critical point. In the discussion section, we summarize our results and discuss generalizations to magnetic fields and Lifshitz normal phases. The Appendices contain additional details.

2 Double trace deformations

In this section we begin by emphasizing the simple under appreciated fact that double trace deformations are useful for studying symmetry breaking in gauge/gravity duality. We then note that this system provides a simple holographic model for superconductivity with zero total charge density. With nonzero charge density, we show that double trace deformations introduce a new parameter by which one can tune the critical temperature of the superconductor.

2.1 Gravity setup and boundary conditions

The theory we study is gravity in $3 + 1$ dimensions with a negative cosmological constant, a $U(1)$ gauge field, and a scalar field $\Psi$ which may or may not be charged under the $U(1)$ symmetry. By general arguments of gauge/gravity duality this theory is dual to a $CFT_{2+1}$ with a conserved current operator $J^\mu$ and a scalar operator $\mathcal{O}$ .

The action is that of the Einstein-Abelian Higgs model with a negative cosmological constant, where we parameterize the phase and modulus of the charged scalar as $\Psi = \psi e^{i\theta}$ ,following e.g. [30]

$$S=\int d^{4}x\sqrt{-g}(R-\frac{1}{4}G(\psi )F^2-(\mathrm{\nabla }\psi {)}^2-J(\psi )(\mathrm{\nabla }\theta -qA{)}^2-V(\psi ))(2.1)S\; =\; \backslash int\; d^4x\; \backslash sqrt\{-g\}\; \backslash left(\; R\; -\; \backslash frac\{1\}\{4\}\; G(\backslash psi)\; F^2\; -\; (\backslash nabla\backslash psi)^2\; -\; J(\backslash psi)\; (\backslash nabla\backslash theta\; -\; qA)^2\; -\; V(\backslash psi)\; \backslash right)\; \backslash quad\; (2.1)$$

We require that the coupling functions $G$ and $J$ and the potential $V$ be even functions of $\psi$ , since we need to preserve our $U(1)$ symmetry. We will assume an expansion of the form

$$V=-6+m^2{\psi }^2+\mathcal{O}({\psi }^4),G=1+g{\psi }^2+\mathcal{O}({\psi }^4),J={\psi }^2+\mathcal{O}({\psi }^4)(2.2)V\; =\; -6\; +\; m^2\backslash psi^2\; +\; \backslash mathcal\{O\}(\backslash psi^4),\; \backslash quad\; G\; =\; 1\; +\; g\backslash psi^2\; +\; \backslash mathcal\{O\}(\backslash psi^4),\; \backslash quad\; J\; =\; \backslash psi^2\; +\; \backslash mathcal\{O\}(\backslash psi^4)\; \backslash quad\; (2.2)$$

where we have set the $AdS_4$ radius to one. The coefficient of $\psi^2$ in $J$ is fixed by regularity at $\Psi = 0$ . This is all that we will need to determine the behavior near the critical point in section 3. We will specify the potential and coupling functions more fully later when they are needed to construct the ordered phase away from the critical point.

The mass $m$ of the field around the symmetric point $\psi = 0$ determines the conformal dimension of the dual operator $\mathcal{O}$ in the $CFT_{2+1}$

$${\mathrm{\Delta }}_{\pm }=3\mathrm{/}2\pm \sqrt{9\mathrm{/}4+m^2}(2.3)\backslash Delta\_\{\backslash pm\}\; =\; 3/2\; \backslash pm\; \backslash sqrt\{9/4\; +\; m^2\}\; \backslash quad\; (2.3)$$

This also controls the asymptotic behavior of the scalar field

$$\psi (r)=\frac{\alpha }{r^{{\mathrm{\Delta }}_{-}}}+\frac{\beta }{r^{{\mathrm{\Delta }}_{+}}}+\dots \text{as }r\to \mathrm{\infty }(2.4)\backslash psi(r)\; =\; \backslash frac\{\backslash alpha\}\{r^\{\backslash Delta\_-\}\}\; +\; \backslash frac\{\backslash beta\}\{r^\{\backslash Delta\_+\}\}\; +\; \backslash dots\; \backslash quad\; \backslash text\{as\; \}\; r\; \backslash rightarrow\; \backslash infty\; \backslash quad\; (2.4)$$

with the metric asymptotically approaching

$$d{s}^2=r^2(-d{t}^2+d{x}_{i}d{x}^i)+\frac{d{r}^2}{r^2}(2.5)ds^2\; =\; r^2(-dt^2\; +\; dx\_i\; dx^i)\; +\; \backslash frac\{dr^2\}\{r^2\}\; \backslash quad\; (2.5)$$

In order for us to be able to add a double trace operator as in (1.1) in a controlled fashion (without destroying the $AdS_4$ asymptotics) we require that the mass be in the range:

In this range, both sets of modes are normalizable and one has a choice of boundary conditions for quantizing the bulk theory. Standard quantization corresponds to setting $\alpha = 0$ , and $\mathcal{O}$ has dimension $\Delta_+$ . Alternative quantization corresponds to $\beta = 0$ , and $\mathcal{O}$ has dimension $\Delta_-$ [17]. We will refer to these two theories as $AdS_4^{(\text{std.})}$ and $AdS_4^{(\text{alt.})}$ . We want to start with alternative quantization, so the dimension of $\mathcal{O}$ is

and add a double trace deformation. In this range, adding the double trace operator (1.1) amounts to studying the gravitational theory in asymptotically $AdS_4$ space with new boundary conditions for the scalar [15, 16]

$$\beta =\kappa \alpha (2.8)\backslash beta\; =\; \backslash kappa\backslash alpha\; \backslash quad\; (2.8)$$

Up to an overall normalization, $\alpha = \langle \mathcal{O} \rangle$ . Note that $\kappa$ has dimension $\Delta_+ - \Delta_- = 3 - 2\Delta_-$ and is thus a relevant coupling in the range of interest (2.7). Hence adding the perturbation will induce an RG flow from the original CFT to a new theory in the IR which can be understood by sending $\kappa \rightarrow \infty$ . Dividing (2.8) by $\kappa$ as we take this limit we see that we arrive at $\alpha = 0$ , the theory with standard boundary conditions. If we had started with $\kappa = 0$ we would stay at the unstable fixed point in alternative quantization. Many details of this flow have been studied (see, e.g., [31]).

2.2 Symmetry breaking from double trace deformations

We will first study a simple model of spontaneous symmetry breaking. We will work with zero chemical potential, and hence can set the bulk Maxwell field to zero. This is natural in a theory which preserves charge conjugation. Adding a double trace term (1.1) with negative coefficient is expected to destabilize the vacuum. This is easily seen as follows. Consider the two-point function for the operator $\mathcal{O}$ in the vacuum (at zero chemical potential, temperature, and double-trace coefficient.) The retarded Green's function is (up to overall normalization)

$$G_{\pm }^R(p)=p^{2{\mathrm{\Delta }}_{\pm }-3}{p}^2=-(\omega +iϵ{)}^2+{\vec{p}}^2(2.9)G\_\{\backslash pm\}^R(p)\; =\; p^\{2\backslash Delta\_\{\backslash pm\}-3\}\; \backslash quad\; p^2\; =\; -(\backslash omega\; +\; i\backslash epsilon)^2\; +\; \backslash vec\{p\}^2\; \backslash quad\; (2.9)$$

where the sign $+$ ( $-$ ) indicates the Green's function for the standard (alternative) quantized theory. Also $\omega$ and $\vec{p}$ are the energy and momentum respectively. If we start with the alternative quantized theory and add a double-trace term of the form (1.1), one finds²

$$G^R(p)=\frac{1}{\frac{1}{G_{-}^R(p)}+\kappa }+\mathcal{O}(1\mathrm{/}{N}^2)\mathrm{.}(2.10)G^R(p)\; =\; \backslash frac\{1\}\{\backslash frac\{1\}\{G\_-^R(p)\}\; +\; \backslash kappa\}\; +\; \backslash mathcal\{O\}(1/N^2).\; \backslash quad\; (2.10)$$

For $\kappa > 0$ , this just introduces a new massive pole with a width at $p_{pole}^2 = (-\kappa)^{1/(3-2\Delta_-)}$ . However, for $\kappa < 0$ , we have a tachyonic instability, with a pole at real positive $p_{pole}^2$ . All of this is directly analogous to a massless free scalar field getting a massive deformation of either sign. In [32] an exponentially growing tachyonic mode was explicitly found in the bulk

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²This can be calculated either in the field theory at large $N$ by summing up a geometric series of diagrams, or on the gravity side with proper treatment of boundary conditions.precisely when $\kappa$ had the wrong sign³. It was widely believed that theories with $\kappa < 0$ would not have a stable vacuum, but in [18] it was proven that for many scalar gravity theories, there was a stable ground state with nonzero $\alpha = \langle \mathcal{O} \rangle$ .

The stability of the dual gravitational system depends on the global existence of a superpotential $P_c(\psi)$ , and the zero temperature broken symmetry ground state is given entirely by $P_c(\psi)$ . The details can be found in [18]. The key result is the behavior of the off-shell potential $\mathcal{V}(\alpha)$ . It turns out that⁴

$$\mathcal{V}=2({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})(W+W_0),(2.11)\backslash mathcal\{V\}\; =\; 2(\backslash Delta\_+\; -\; \backslash Delta\_-)(W\; +\; W\_0),\; \backslash quad\; (2.11)$$

$W(\alpha)$ is given by our boundary conditions $\beta = W'(\alpha)$ , and for our double trace deformation is simply $W(\alpha) = \kappa\alpha^2/2$ . $W_0(\alpha)$ is found from a scaling limit of smooth horizonless static solitons in global $AdS$ , again see [18] for details. At $T = \mu = 0$ , scale invariance implies

$$W_0(\alpha )=\frac{s_c{\mathrm{\Delta }}_{-}}{3}\mathrm{\mid }\alpha {\mathrm{\mid }}^{3\mathrm{/}{\mathrm{\Delta }}_{-}}\mathrm{.}(2.12)W\_0(\backslash alpha)\; =\; \backslash frac\{s\_c\; \backslash Delta\_-\}\{3\}\; |\backslash alpha|^\{3/\backslash Delta\_-\}.\; \backslash quad\; (2.12)$$

The coefficient $s_c$ depends on the full bulk potential $V(\psi)$ , and is generally positive. If $s_c$ is negative the theory is somewhat sick since $W_0$ is unbounded, and the alternative quantized theory is unstable, as it has states with arbitrarily negative energy. We do not consider this case any further. The full off shell potential is thus

$$\mathcal{V}(\alpha )=({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})(\kappa {\alpha }^2+\frac{2s_c{\mathrm{\Delta }}_{-}}{3}\mathrm{\mid }\alpha {\mathrm{\mid }}^{3\mathrm{/}{\mathrm{\Delta }}_{-}})\mathrm{.}(2.13)\backslash mathcal\{V\}(\backslash alpha)\; =\; (\backslash Delta\_+\; -\; \backslash Delta\_-)\; \backslash left(\; \backslash kappa\backslash alpha^2\; +\; \backslash frac\{2s\_c\; \backslash Delta\_-\}\{3\}\; |\backslash alpha|^\{3/\backslash Delta\_-\}\; \backslash right).\; \backslash quad\; (2.13)$$

For our range of interest (2.7) the second term dominates at large $\alpha$ , and we have a classic example of spontaneous symmetry breaking with a saddle-shaped potential for negative $\kappa$ (see Fig. 1). The ground state which minimizes (2.13) has

$$\alpha =⟨\mathcal{O}⟩={(-\frac{\kappa }{s_c})}^{{\mathrm{\Delta }}_{-}\mathrm{/}(3-2{\mathrm{\Delta }}_{-})},{\mathcal{V}}_{min}=-\frac{1}{3}{\left(\frac{3-2{\mathrm{\Delta }}_{-}}{\frac{{\mathrm{\Delta }}_{-}}{s_c\mathrm{/}(3-2{\mathrm{\Delta }}_{-})}}\right)}^2(-\kappa {)}^{3\mathrm{/}(3-2{\mathrm{\Delta }}_{-})}\mathrm{.}(2.14)\backslash alpha\; =\; \backslash langle\; \backslash mathcal\{O\}\; \backslash rangle\; =\; \backslash left(\; -\backslash frac\{\backslash kappa\}\{s\_c\}\; \backslash right)^\{\backslash Delta\_-/(3-2\backslash Delta\_-)\},\; \backslash quad\; \backslash mathcal\{V\}\_\{min\}\; =\; -\backslash frac\{1\}\{3\}\; \backslash left(\; \backslash frac\{3\; -\; 2\backslash Delta\_-\}\{\backslash frac\{\backslash Delta\_-\}\{s\_c/(3-2\backslash Delta\_-)\}\}\; \backslash right)^2\; (-\backslash kappa)^\{3/(3-2\backslash Delta\_-)\}.\; \backslash quad\; (2.14)$$

As mentioned above, the gravitational description of this ground state is uniquely determined by the superpotential $P_c(\psi)$ .

Putting the theory at finite temperature can lift this instability. The detailed calculation is in appendix B, but the result is that as we heat the system up to

$$T_c=k_1(-\kappa {)}^{1\mathrm{/}(3-2{\mathrm{\Delta }}_{-})},\text{with }{k}_1=\frac{3}{4\pi }{(-\frac{\mathrm{\Gamma }(({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})\mathrm{/}3)\mathrm{\Gamma }({\mathrm{\Delta }}_{-}\mathrm{/}3{)}^2}{\mathrm{\Gamma }(({\mathrm{\Delta }}_{-}-{\mathrm{\Delta }}_{+})\mathrm{/}3)\mathrm{\Gamma }({\mathrm{\Delta }}_{+}\mathrm{/}3{)}^2})}^{1\mathrm{/}(3-2{\mathrm{\Delta }}_{-})}(2.15)T\_c\; =\; k\_1\; (-\backslash kappa)^\{1/(3-2\backslash Delta\_-)\},\; \backslash quad\; \backslash text\{with\; \}\; k\_1\; =\; \backslash frac\{3\}\{4\backslash pi\}\; \backslash left(\; -\backslash frac\{\backslash Gamma((\backslash Delta\_+\; -\; \backslash Delta\_-)/3)\backslash Gamma(\backslash Delta\_-/3)^2\}\{\backslash Gamma((\backslash Delta\_-\; -\; \backslash Delta\_+)/3)\backslash Gamma(\backslash Delta\_+/3)^2\}\; \backslash right)^\{1/(3-2\backslash Delta\_-)\}\; \backslash quad\; (2.15)$$

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³Strictly speaking, [32] studied the theory on a sphere, where coupling to background curvature induces a positive double trace term for scalars. In that case, $\kappa$ needed to be sufficiently negative to find the instability. Since we are studying the theory on Minkowski space, the critical point is simply when $\kappa$ changes sign.

⁴The overall factor of 2 (which was not present in [18]) arises since we do not have a 1/2 in front of our action (2.1).Figure 1: The off-shell potential as we tune $\kappa$ . The black dashed curve is the fine-tuned theory with $\kappa = 0$ , the blue curve is $\kappa < 0$ , and the red curve is $\kappa > 0$ . This is a strongly coupled version of the standard Landau-Ginzburg symmetry breaking mechanism.

the system returns to the symmetry preserving state. Note that everything scales as a power of $\kappa$ , since this is the only scale in the problem. Another way of studying this system at $T > 0$ is to construct the finite temperature generalization of $\mathcal{V}$ . This can be obtained from the family of hairy black holes in the bulk, as described in [33].

2.3 A novel holographic superconductor

We saw above that adding a renormalizable double trace coupling can break a $U(1)$ symmetry at low temperature. This provides a new mechanism for constructing holographic superconductors. Unlike the previous approach, which required a nonzero charge density to generate a low temperature condensate, we can now work at zero net charge. In this case, the Maxwell field remains strictly zero, even when the charged scalar hair is present in the bulk.

We computed the critical temperature above. To study the system away from $T_c$ , we need to specify the full nonlinear bulk potential. Working in units of $\kappa$ (which is analogous to working in units of $\mu$ or $\rho$ in cases of finite density), we find that the order parameter behaves just as it does in the case where we find an instability by lowering $T/\mu$ in the standard holographic superconductor setup. We can also calculate the difference between the free energy of the hairy black hole and the normal black hole, which is simply AdS-Schwarzschild, and find generically that below the critical temperature the hairy black hole is always preferred. An example which comes from a consistent string theory truncation is shown in Fig. 2.

As usual, to compute the conductivity, one starts by perturbing the Maxwell field andFigure 2: The order parameter $\alpha = \langle \mathcal{O} \rangle$ and free energy density $f$ across the second order phase transition down to zero temperature. In the figure on the right, the dashed red line is the free energy of the normal phase (Schwarzschild AdS solution) and the black line is the free energy of the condensed phase. We used a case with $\Delta_- = 1$ , and bulk potential $V(\psi) = \sinh^2(\psi/\sqrt{2})(\cosh(\sqrt{2}\psi) - 5) = -6 - 2\psi^2 + \mathcal{O}(\psi^4)$ [34, 35]. In [18] it was found that $s_c = 0.56$ for this potential. From (2.15), we have $T_c/(-\kappa) \approx 0.62$ .

metric. As shown in Appendix A, the conductivity can be simply related to the reflection coefficient in a one dimensional Schrodinger problem:

$$-b^{\mathrm{\prime }\mathrm{\prime }}(z)+V_{Sch}(z)b(z)={\omega }^{2}b,(2.16)-b\text{'}\text{'}(z)\; +\; V\_\{Sch\}(z)b(z)\; =\; \backslash omega^2\; b,\; \backslash quad\; (2.16)$$

where $\delta A_x = a_x e^{-i\omega t}$ and $b = \sqrt{G(\psi)} a_x$ . The Schrodinger potential is bounded (and given explicitly in (A.24)). $z$ is a new radial coordinate that vanishes at infinity and goes to minus infinity at the horizon. To obtain the required ingoing wave boundary condition at the horizon, we assume $b = e^{-i\omega z} + \mathcal{R}e^{i\omega z}$ near $z = 0$ so that $b = \mathcal{T}e^{-i\omega z}$ near the horizon. The conductivity is simply given by

$$\sigma =\frac{1-\mathcal{R}}{1+\mathcal{R}}\mathrm{.}(2.17)\backslash sigma\; =\; \backslash frac\{1\; -\; \backslash mathcal\{R\}\}\{1\; +\; \backslash mathcal\{R\}\}.\; \backslash quad\; (2.17)$$

Since the potential is bounded, this will produce the usual behavior of the optical conductivity. At low temperature there will typically be a gap at frequencies below the height of the potential⁵, and at higher frequencies the conductivity will approach its normal state value. There will be a delta function at $\omega = 0$ in the condensed phase, which can be seen from a pole in the imaginary part of the conductivity.

The key difference from the holographic superconductors at nonzero charge density, is that there is no delta function in $\text{Re}[\sigma]$ at $\omega = 0$ in the normal phase. This awkward feature of the previous construction arose since a state with net charge can be boosted, yielding a

⁵Since the Schrodinger potential is no longer positive definite, one can sometimes get peaks at low frequency in the conductivity [36].nonzero current with zero applied electric field. This implies infinite DC conductivity. Since we can now start in a state with zero charge, we no longer have this problem. Mathematically, the delta function arose since the Schrodinger potential was nonzero in the normal phase due to a contribution from the background electric field. Here, the Schrodinger potential vanishes in the normal phase. The background is just the Schwarzschild AdS black hole, and $\sigma = 1$ with no delta-function contributions.

2.4 Non-zero density and stability conditions

In addition to providing another way to construct holographic superconductors, the addition of a double trace perturbation provides a new knob for adjusting the critical temperature of traditional holographic superconductors. As discussed in the introduction, adding a term with $\kappa > 0$ makes it harder to condense the operator $\mathcal{O}$ and lowers the critical temperature. We will see below that in some cases, $T_c \rightarrow 0$ as $\kappa$ approaches a finite value, $\kappa_c$ . This is a new quantum critical point which will be studied in detail in the next section.

With a nonzero charge density, the normal phase is described by the Reissner-Nördstrom-AdS (RN-AdS) black hole. The critical temperature is determined by looking for a static normalizable mode of the scalar field in this background [13]. This marks the onset of the instability to form scalar hair. This problem only requires the leading terms in the functions $V(\psi)$ , $G(\psi)$ , $J(\psi)$ given in (2.2). The addition of the double trace term changes the critical temperature since it changes the boundary condition on the normalizable mode.

For reference, the RN AdS black hole is described by the metric and gauge potential,

$$d{s}^2=-fd{t}^2+r^{2}d{\vec{x}}^2+\frac{d{r}^2}{f},\varphi =\mu -\frac{\rho }{r},f=r^2-\frac{m_0}{2r}+\frac{{\rho }^2}{4r^2}(2.18)ds^2\; =\; -f\; dt^2\; +\; r^2\; d\backslash vec\{x\}^2\; +\; \backslash frac\{dr^2\}\{f\},\; \backslash quad\; \backslash phi\; =\; \backslash mu\; -\; \backslash frac\{\backslash rho\}\{r\},\; \backslash quad\; f\; =\; r^2\; -\; \backslash frac\{m\_0\}\{2r\}\; +\; \backslash frac\{\backslash rho^2\}\{4r^2\}\; \backslash quad\; (2.18)$$

$$m_0=\frac{2{\rho }^3}{{\mu }^3}+\frac{\rho \mu }{2},T=\frac{\mu }{4\pi }(\frac{3\rho }{{\mu }^2}-\frac{{\mu }^2}{4\rho })(2.19)m\_0\; =\; \backslash frac\{2\backslash rho^3\}\{\backslash mu^3\}\; +\; \backslash frac\{\backslash rho\backslash mu\}\{2\},\; \backslash quad\; T\; =\; \backslash frac\{\backslash mu\}\{4\backslash pi\}\; \backslash left(\; \backslash frac\{3\backslash rho\}\{\backslash mu^2\}\; -\; \backslash frac\{\backslash mu^2\}\{4\backslash rho\}\; \backslash right)\; \backslash quad\; (2.19)$$

The horizon is located at $r_0 = \rho/\mu$ , where $\rho$ and $\mu$ are the charge density and chemical potential respectively.

On this background the linear fluctuations are given by,

$$(r^{2}f{\psi }^{\mathrm{\prime }}{)}^{\mathrm{\prime }}=(m^2{r}^2+{\vec{p}}^2-\frac{g{\rho }^2}{2r^2}-\frac{r^2(\omega +q\varphi {)}^2}{f})\psi (2.20)(r^2\; f\; \backslash psi\text{'})\text{'}\; =\; \backslash left(\; m^2\; r^2\; +\; \backslash vec\{p\}^2\; -\; \backslash frac\{g\backslash rho^2\}\{2r^2\}\; -\; \backslash frac\{r^2(\backslash omega\; +\; q\backslash phi)^2\}\{f\}\; \backslash right)\; \backslash psi\; \backslash quad\; (2.20)$$

where we have included momentum dependence $\vec{p}$ and frequency dependence $\omega$ and $\Psi = \psi e^{-i\omega t + i\vec{x} \cdot \vec{p}}$ . We will need these in Section 3 but for now they can be set to zero. Before going onto the case of double trace boundary conditions, we first recall some known results on stability conditions. One natural question to ask is what is the condition for the absence ofan instability at any $T$ . In other words, when is the normal state stable at zero temperature? A necessary condition was identified in [13]: the extremal RN black hole in the near horizon limit becomes $AdS_2 \times R^2$ , so if the effective mass of $\psi$ derived from (2.20) is below the $AdS_2$ BF bound, the RN BH will be unstable below some critical temperature. So one condition for stability is demanding $m_{\text{eff}}^2 > -1/4$ where

$$m_{\text{eff}}^2=\frac{m^2}{6}-\frac{q^2}{3}-g(2.21)m\_\{\backslash text\{eff\}\}^2\; =\; \backslash frac\{m^2\}\{6\}\; -\; \backslash frac\{q^2\}\{3\}\; -\; g\; \backslash quad\; (2.21)$$

This is equivalent to demanding the conformal dimension of the operator dual to $\psi$ in the $AdS_2$ region, $\delta_{\pm}$ , are real, where

$${\delta }_{\pm }=\frac{1}{2}\pm \sqrt{\frac{1}{4}+m_{\text{eff}}^2}\mathrm{.}(2.22)\backslash delta\_\{\backslash pm\}\; =\; \backslash frac\{1\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash frac\{1\}\{4\}\; +\; m\_\{\backslash text\{eff\}\}^2\}.\; \backslash quad\; (2.22)$$

This condition is too weak however, and we would like to refine it. As we will see below, the stability condition $m_{\text{eff}}^2 > -1/4$ is sufficient for standard boundary conditions for the scalar $\alpha = 0$ ( $\kappa = \infty$ ). However since alternative boundary conditions $\beta = 0$ ( $\kappa = 0$ ) are weaker, the scalar field can still be unstable to forming hair despite the BF bound in $AdS_2$ being satisfied. This is because for any effective mass, there are always unstable modes in $AdS_2$ . It is just that they are usually thrown out by the boundary conditions. With alternative boundary conditions in the asymptotic $AdS_4$ region, some of these unstable modes are allowed.

As further indication of the fact that alternative boundary conditions are more unstable, it was noticed in [13] that $T_c$ diverges as one approaches the unitarity bound $\Delta_- = 1/2$ . As shown in Fig. 3, this divergence actually takes the form

$$T_c\sim \frac{\mu q}{({\mathrm{\Delta }}_{-}-1\mathrm{/}2{)}^{1\mathrm{/}2}}(2.23)T\_c\; \backslash sim\; \backslash frac\{\backslash mu\; q\}\{(\backslash Delta\_-\; -\; 1/2)^\{1/2\}\}\; \backslash quad\; (2.23)$$

Interestingly, for neutral scalar fields there is no divergence and $T_c$ approaches a finite limit as $\Delta_- \rightarrow 1/2$ .

We now want to include the effect of nonzero $\kappa$ . Working at fixed $\mu$ the relevant scale invariant quantity that we will vary is $\kappa/\mu^{\Delta_+ - \Delta_-}$ (as well as $T/\mu$ ). It turns out to be easy to study $T_c$ as a function of $\kappa$ simply by changing the definition of “normalizable”.⁶ Increasing $\kappa$ always decreases $T_c$ . If the mass and charge of the scalar field is such that extreme RN AdS is unstable with standard boundary conditions, then $T_c$ remains nonzero for all $\kappa$ . However, if extreme RN AdS is stable with standard boundary conditions, then $T_c$ must vanish at a

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⁶Amusingly this is a much simpler problem than the usual shooting problem. Rather than adjusting $T$ to find a static normalizable mode we can simply fix $T = T_c$ , shoot to the boundary and read off $\kappa(T_c)$ .Figure 3: The critical temperature as a function of $\Delta_-$ for $q = .1, .25, .5$

finite value $\kappa = \kappa_c$ . Both cases are illustrated in Fig. 4. Since $T_c$ can be arbitrarily large at $\kappa = 0$ and vanish at $\kappa_c$ , we see that $\kappa$ is a very sensitive knob to adjust the critical temperature of the superconductor. The point $\kappa = \kappa_c$ is the quantum critical point that we will study in the next section. Notice that as $\kappa$ becomes large and negative in Fig. 4, $\mu$ becomes less important, and both curves approach the scaling (2.15).

3 Quantum critical point

We now turn to a more precise discussion of the point $\kappa = \kappa_c$ where the critical temperature $T_c/\mu \rightarrow 0$ . Below we will outline the various requirements on bulk parameters to achieve this critical point. Note in particular we need not fine tune these bulk parameters. The coupling that we do tune $\kappa/\mu^{\Delta_+ - \Delta_-}$ is a well defined boundary theory coupling.

3.1 The flow of double trace couplings and the 2 point function

It is useful to think of the extremal RN black hole as representing a flow from a $CFT_{2+1}$ in the UV to a $CFT_{0+1}$ in the IR. This flow is induced in the UV by turning on a source ( $\mu$ ) for the charge density operator $J^t$ . The IR CFT can be seen by taking a scaling limit of (2.18)Figure 4: The critical temperature, in units of chemical potential, as a function of the UV double trace coupling $\kappa$ for fixed $\Delta_- = 1$ and $q = 1/2$ . The top curve has $g = 0.2$ and has nonzero critical temperature for all $\kappa$ . The lower curve has $g = -0.2$ and ends at a quantum critical point.

towards $r \rightarrow r_0$ at extremality.⁷ Rescale coordinates as:

$$\widehat{r}=ϵ(r-r_{\star })\widehat{t}=ϵt\widehat{x}=r_{\star }x(3.1)\backslash hat\{r\}\; =\; \backslash epsilon(r\; -\; r\_\backslash star)\; \backslash quad\; \backslash hat\{t\}\; =\; \backslash epsilon\; t\; \backslash quad\; \backslash hat\{x\}\; =\; r\_\backslash star\; x\; \backslash quad\; (3.1)$$

where $r_\star$ is the location of the horizon at extremality $r_\star \equiv r_0|{T=0} = \rho_\star/\mu$ with $\rho_\star \equiv \rho|{T=0} = \mu^2/\sqrt{12}$ . Formally we can scale towards $r \rightarrow r_\star$ by expanding in $\epsilon$ then setting $\epsilon = 1$ . This yields

$$d{s}_0^2=(-6{\widehat{r}}^{2}d{\widehat{t}}^2+\frac{d{\widehat{r}}^2}{6{\widehat{r}}^2})+(d{\widehat{x}}^2+d{\widehat{y}}^2),A_0=\sqrt{12}\widehat{r}d\widehat{t}(3.2)ds\_0^2\; =\; \backslash left(\; -6\backslash hat\{r\}^2\; d\backslash hat\{t\}^2\; +\; \backslash frac\{d\backslash hat\{r\}^2\}\{6\backslash hat\{r\}^2\}\; \backslash right)\; +\; (d\backslash hat\{x\}^2\; +\; d\backslash hat\{y\}^2),\; \backslash quad\; A\_0\; =\; \backslash sqrt\{12\}\backslash hat\{r\}d\backslash hat\{t\}\; \backslash quad\; (3.2)$$

This is the classic $AdS_2 \times R^2$ geometry found in the IR of many extremal black hole solutions. Notice in particular that the scale $\mu$ has dropped out. Since this geometry is supposed to be dual to a scale invariant theory, this had to be the case. The only knowledge that this theory has of the scale $\mu$ is encoded in higher order irrelevant terms which we have dropped. For example keeping the next order terms in the expansion in $\epsilon$ one finds:

$$d{s}^2=d{s}_0^2+{\delta }_{h}d{s}_1^2+\dots ,A=A_0+{\delta }_h{A}_1+\dots (3.3)ds^2\; =\; ds\_0^2\; +\; \backslash delta\_h\; ds\_1^2\; +\; \backslash dots,\; \backslash quad\; A\; =\; A\_0\; +\; \backslash delta\_h\; A\_1\; +\; \backslash dots\; \backslash quad\; (3.3)$$

where $ds_1^2$ and $A_1$ have energy scaling dimension 1 under the $AdS_2$ scaling. The chemical potential now appears through $\delta_h = 1/\mu$ . Note that since $\delta_h$ has dimensions of $-1$ this

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⁷This limit can be taken more carefully, keeping a finite but small $T$ . See Appendix C. The scaling limit was discussed in [11] and the discussion here follows that paper closely.represents an irrelevant coupling, which when turned on induces a flow in the UV to $AdS_4$ . It is useful to think of $\delta_h$ as opening up the $R^2$ directions of the metric.

We now return to linearized fluctuations of $\psi$ in the extremal RN background. Our goal will be to compute the two point function of the order parameter at small frequencies and momenta $\omega, p \ll \mu$ . We proceed heuristically, leaving details to Appendix C. Following [11] we do a matched asymptotic expansion where we split the geometry into two regions. In both regions we do a perturbative expansion in $\epsilon$ where we redefine

$$\omega \to ϵ\omega ,\vec{p}\to ϵ\vec{p}(3.4)\backslash omega\; \backslash rightarrow\; \backslash epsilon\backslash omega,\; \backslash quad\; \backslash vec\{p\}\; \backslash rightarrow\; \backslash epsilon\backslash vec\{p\}\; \backslash quad\; (3.4)$$

so as to access small frequencies and momenta. In the inner region we rescale coordinates as in (3.1). In the outer region we leave the coordinates unscaled. A systematic expansion in both regions is defined in this way, matching occurs in an intermediate region connecting the two.

At zeroth order the outer region simply follows from setting $\omega = 0, T = 0, \vec{p} = 0$ in the full RN background. A general solution to the resulting equation can be characterized by the behavior at the $AdS_4$ boundary and at the extremal horizon,

$$\psi (r\to \mathrm{\infty })\to {\alpha }_0{r}^{-{\mathrm{\Delta }}_{-}}+{\beta }_0{r}^{-{\mathrm{\Delta }}_{+}},\psi (r\to r_{\star })\to {\widehat{\alpha }}_0(r-r_{\star }{)}^{-{\delta }_{-}}+{\widehat{\beta }}_0(r-r_{\star }{)}^{-{\delta }_{+}}(3.5)\backslash psi(r\; \backslash rightarrow\; \backslash infty)\; \backslash rightarrow\; \backslash alpha\_0\; r^\{-\backslash Delta\_-\}\; +\; \backslash beta\_0\; r^\{-\backslash Delta\_+\},\; \backslash quad\; \backslash psi(r\; \backslash rightarrow\; r\_\backslash star)\; \backslash rightarrow\; \backslash hat\{\backslash alpha\}\_0\; (r\; -\; r\_\backslash star)^\{-\backslash delta\_-\}\; +\; \backslash hat\{\backslash beta\}\_0\; (r\; -\; r\_\backslash star)^\{-\backslash delta\_+\}\; \backslash quad\; (3.5)$$

where $a^\pm$ and $b^\pm$ are constants (in units of $\mu$ ) and can only be computed numerically.⁸ If we impose linear boundary conditions (2.8) on the allowed fluctuations, then this maps into the following condition near $r_\star$ ,

$$\frac{{\widehat{\beta }}_0}{{\widehat{\alpha }}_0}=\frac{(a^{+}{)}^2(\kappa -{\kappa }_c)}{\det L-a^{-}{a}^{+}(\kappa -{\kappa }_c)}\text{where}{\kappa }_c\equiv b^{+}\mathrm{/}{a}^{+}\text{and}\det L={\mu }^2\frac{{\delta }_{+}-{\delta }_{-}}{{\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-}}(3.7)\backslash frac\{\backslash hat\{\backslash beta\}\_0\}\{\backslash hat\{\backslash alpha\}\_0\}\; =\; \backslash frac\{(a^+)^2(\backslash kappa\; -\; \backslash kappa\_c)\}\{\backslash det\; L\; -\; a^-a^+(\backslash kappa\; -\; \backslash kappa\_c)\}\; \backslash quad\; \backslash text\{where\}\; \backslash quad\; \backslash kappa\_c\; \backslash equiv\; b^+/a^+\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash det\; L\; =\; \backslash mu^2\; \backslash frac\{\backslash delta\_+\; -\; \backslash delta\_-\}\{\backslash Delta\_+\; -\; \backslash Delta\_-\}\; \backslash quad\; (3.7)$$

We would like to make the identification of $\hat{\beta}_0/\hat{\alpha}0$ above with the value of a double trace coupling in the IR $AdS_2$ CFT, $\kappa{\text{IR}}$ . We will only be interested in $\kappa$ close to $\kappa_c$ such that,

$${\kappa }_{\text{IR}}=\frac{(a^{+}{)}^2}{\det L}(\kappa -{\kappa }_c)(3.8)\backslash kappa\_\{\backslash text\{IR\}\}\; =\; \backslash frac\{(a^+)^2\}\{\backslash det\; L\}(\backslash kappa\; -\; \backslash kappa\_c)\; \backslash quad\; (3.8)$$

It is then natural to identify $\kappa = \kappa_c$ as the critical point that we observed in the previous section. One main reason for this identification is the fact that for $\kappa < \kappa_c$ the double trace coupling in the IR is negative, and thus analogous to the discussion in Section 2, there will be a new state with lower free energy and scalar hair.

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⁸ In [11] these same constants were called $a_\pm^{(0)}, b_\pm^{(0)}$ .We are now in a position to complete the computation of the two point function of the order parameter. The retarded Green's function follows from imposing incoming boundary conditions at the extremal horizon in the inner region. Then to zeroth order in the $\epsilon$ expansion one finds that $\hat{\beta}_0/\hat{\alpha}_0 = \Sigma_R(\omega)$ where $\Sigma_R$ is the retarded $AdS_2$ Green's function for fluctuations on the background (3.2). The Green's function in the full CFT can be computed using any of the usual prescription [37, 38] generalized to include nonstandard boundary conditions. The result is up to overall normalization,

$${\chi }_R(\omega ,\vec{p})=\frac{\alpha }{-\beta +\kappa \alpha }=\frac{Z+\dots }{{\kappa }_{\text{IR}}-{\mathrm{\Sigma }}_R(\omega )+\mathbb{X}(\omega ,\vec{p})+\dots }Z=(a^{+}{)}^2\mathrm{/}\det L(3.9)\backslash chi\_R(\backslash omega,\; \backslash vec\{p\})\; =\; \backslash frac\{\backslash alpha\}\{-\backslash beta\; +\; \backslash kappa\backslash alpha\}\; =\; \backslash frac\{Z\; +\; \backslash dots\}\{\backslash kappa\_\{\backslash text\{IR\}\}\; -\; \backslash Sigma\_R(\backslash omega)\; +\; \backslash mathbb\{X\}(\backslash omega,\; \backslash vec\{p\})\; +\; \backslash dots\}\; \backslash quad\; Z\; =\; (a^+)^2\; /\; \backslash det\; L\; \backslash quad\; (3.9)$$

where we have included higher order terms that can be important in $\mathbb{X}$ . These higher order terms always come from perturbative corrections in the outer region and are thus real. In contrast, $\Sigma_R$ is in general complex. The elipses above represent even higher order terms that we have dropped. We compute $\mathbb{X}$ in Appendix C. The result can be written as,

$$\mathbb{X}(\omega ,\vec{p})=c_p{\vec{p}}^2-c_{\omega }{\omega }^2-c_T{\kappa }_{c}T+c_{q}q(-\omega +\frac{2\pi }{\sqrt{3}}Tq)(3.10)\backslash mathbb\{X\}(\backslash omega,\; \backslash vec\{p\})\; =\; c\_p\; \backslash vec\{p\}^2\; -\; c\_\backslash omega\; \backslash omega^2\; -\; c\_T\; \backslash kappa\_c\; T\; +\; c\_q\; q\; \backslash left(\; -\backslash omega\; +\; \backslash frac\{2\backslash pi\}\{\backslash sqrt\{3\}\}\; T\; q\; \backslash right)\; \backslash quad\; (3.10)$$

where $c_i$ are constants in units of $\mu$ . They have the following positivity constraints depending on the value of $\delta_-$ : $c_p > 0, c_T > 0$ always, $c_\omega > 0$ for $\delta_- < -1/2$ and $c_q > 0$ for $\delta_- < 0$ .

The $AdS_2$ Green's function plays the role of the self energy in (3.9) and is given by

$${\mathrm{\Sigma }}_R(\omega ,T=0)=h{e}^{i\varphi }(-i\omega {)}^{1-2{\delta }_{-}}(3.11)\backslash Sigma\_R(\backslash omega,\; T\; =\; 0)\; =\; h\; e^\{i\backslash phi\}\; (-i\backslash omega)^\{1-2\backslash delta\_-\}\; \backslash quad\; (3.11)$$

where $h$ is a real positive number, and $e^{i\phi}$ is a phase, the precise form of which does not matter. We can generalize the above discussion to finite but small $T$ . At finite temperature $\Sigma_R$ takes the form of a nontrivial scaling function,

$${\mathrm{\Sigma }}_R(\omega ,T)={\left(\frac{2\pi T}{3}\right)}^{{\delta }_{+}-{\delta }_{-}}\frac{\mathrm{\Gamma }({\delta }_{+}-{\delta }_{-})\mathrm{\Gamma }({\delta }_{+}-iq\mathrm{/}\sqrt{3})\mathrm{\Gamma }({\delta }_{+}+iq\mathrm{/}\sqrt{3}-i\omega \mathrm{/}(2\pi T))}{\mathrm{\Gamma }({\delta }_{-}-{\delta }_{+})\mathrm{\Gamma }({\delta }_{-}-iq\mathrm{/}\sqrt{3})\mathrm{\Gamma }({\delta }_{-}+iq\mathrm{/}\sqrt{3}-i\omega \mathrm{/}(2\pi T))}(3.12)\backslash Sigma\_R(\backslash omega,\; T)\; =\; \backslash left(\; \backslash frac\{2\backslash pi\; T\}\{3\}\; \backslash right)^\{\backslash delta\_+\; -\; \backslash delta\_-\}\; \backslash frac\{\backslash Gamma(\backslash delta\_+\; -\; \backslash delta\_-)\; \backslash Gamma(\backslash delta\_+\; -\; iq/\backslash sqrt\{3\})\; \backslash Gamma(\backslash delta\_+\; +\; iq/\backslash sqrt\{3\}\; -\; i\backslash omega/(2\backslash pi\; T))\}\{\backslash Gamma(\backslash delta\_-\; -\; \backslash delta\_+)\; \backslash Gamma(\backslash delta\_-\; -\; iq/\backslash sqrt\{3\})\; \backslash Gamma(\backslash delta\_-\; +\; iq/\backslash sqrt\{3\}\; -\; i\backslash omega/(2\backslash pi\; T))\}\; \backslash quad\; (3.12)$$

Importantly the $AdS_2$ Green's function always satisfies the constraint,

$$\omega \text{Im}{\mathrm{\Sigma }}_R(\omega ,T)>0(3.13)\backslash omega\; \backslash text\{Im\}\; \backslash Sigma\_R(\backslash omega,\; T)\; >\; 0\; \backslash quad\; (3.13)$$

which is necessarily true for any bosonic spectral density.

Generally speaking since the quantities computed in the IR $AdS_2$ geometry depend simply on two numbers $q$ and $\delta_-$ we will call these quantities “universal”. Since they are associated with a CFT this language seems appropriate. Other quantities that come from the outer region such as $a^\pm, b^\pm$ and the $c_i$ are “nonuniversal”, they can be computed only numerically.### 3.2 Properties of the critical point

Given the two point function (3.9) we can now understand the physics close to the critical point. First we study the phase boundary in the $(\kappa, T)$ plane where the order parameter condenses, or equivalently, where the correlation length diverges.

We again look for a static normalizable mode at $T = T_c$ which manifests itself as a zero frequency pole in (3.9). Since the instability kicks in first for the homogenous mode we can take $\vec{p} = 0$ . There are two cases depending on the IR conformal dimension $\delta_-$ . For $0 < \delta_- < 1/2$ we can ignore $\mathbb{X}$ altogether, however for $\delta_- < 0$ the analytic correction $\propto T$ is larger than $\Sigma_R \propto T^{1-2\delta_-}$ . Thus we find for small $\kappa_{\text{IR}}$ ,

where,

$$k_2=\frac{3}{2\pi }{(-\frac{\mathrm{\Gamma }({\delta }_{-}-{\delta }_{+})}{\mathrm{\Gamma }({\delta }_{+}-{\delta }_{-})}{\mid \frac{\mathrm{\Gamma }({\delta }_{-}-iq\mathrm{/}\sqrt{3})}{\mathrm{\Gamma }({\delta }_{+}-iq\mathrm{/}\sqrt{3})}\mid }^2)}^{1\mathrm{/}(1-2{\delta }_{-})}{k}_3=\frac{1}{-c_T{\kappa }_c+c_q{q}^{2}2\pi \mathrm{/}\sqrt{3}}(3.16)k\_2\; =\; \backslash frac\{3\}\{2\backslash pi\}\; \backslash left(\; -\backslash frac\{\backslash Gamma(\backslash delta\_-\; -\; \backslash delta\_+)\}\{\backslash Gamma(\backslash delta\_+\; -\; \backslash delta\_-)\}\; \backslash left|\; \backslash frac\{\backslash Gamma(\backslash delta\_-\; -\; iq/\backslash sqrt\{3\})\}\{\backslash Gamma(\backslash delta\_+\; -\; iq/\backslash sqrt\{3\})\}\; \backslash right|^2\; \backslash right)^\{1/(1-2\backslash delta\_-)\}\; \backslash quad\; k\_3\; =\; \backslash frac\{1\}\{-c\_T\; \backslash kappa\_c\; +\; c\_q\; q^2\; 2\backslash pi/\backslash sqrt\{3\}\}\; \backslash quad\; (3.16)$$

Note that while $k_2$ is a universal number (one to be compared with (2.15)) $k_3$ is nonuniversal, depending on quantities $c_T$ and $c_q$ defined in the outer region. Indeed it is not clear the sign of $k_3$ is fixed. Although $c_q > 0$ and $c_T > 0$ for the range of dimensions of interest, $\kappa_c$ does not have a fixed sign. Generically it seems that for a critical point occurring with $\delta_- < 0$ and $q = 0$ then $\kappa_c < 0$ so that $k_3$ is fixed to be positive in such a case. However we do not know a proof of the positivity of $k_3$ .

To check the scaling relations (3.14) and (3.15), we have numerically computed the critical temperature. In Fig. 5 we plot $T_c$ as a function of $\kappa$ for various $\delta_-$ . The results are perfectly consistent with our scaling relations.

Now we move away from the phase boundary and study the disordered phase at zero temperature. Here we can examine the structure of the retarded Green's function in the complex $\omega$ plane. We will be particularly interested in the dispersion of the mode that becomes tachyonic on the ordered side. Again depending on the value of $\delta_-$ the dispersion will differ, also now there will be a difference if the order parameter is charged or not. Examining (3.9) at $T = 0$ for the charged case the dispersion of the pole in the complexFigure 5: The critical temperature close to $\kappa_c$ for different values of $\delta_-$ . These are for theories with $q = 0$ , $\Delta_- = 1$ , and from left to right: $\delta_- = 0.45, 0.30, 0.26, 0.15, 0, -0.15$ . Note that when $\delta_- > 0$ the critical temperature vanishes with a power law, but for $\delta_- \leq 0$ it vanishes linearly.

plane is,

where in the last case the width of the quasiparticle scales like $\text{Im}\Sigma_R(\omega_R) \propto |\omega_R|^{1-2\delta_-}$ , which is smaller than the energy $|\omega_R|$ . In this case we have a genuine quasiparticle with a mass $\propto \kappa_{\text{IR}}$ . In the first case (3.17) the width scales like the energy and thus the pole does not represent a genuine quasiparticle.

For the neutral case we again get the same behavior as in (3.17) however now for the range of conformal dimensions $-1/2 < \delta_- < 1/2$ . For the remaining neutral case,

where there are now two quasiparticles at positive and negative energies $\pm\omega_R$ . The width of these quasiparticles scales as $|\omega_R|^{-2\delta_-}$ .

As expected, $\kappa_{\text{IR}}$ is playing the role of a mass (or an energy gap.) For all cases one can show via application of the constraint (3.13) that for $\kappa_{\text{IR}} > 0$ the quasi particle pole alwayslies in the lower half plane and there is no instability. There is a “mass gap”⁹, $E_g$ , for all cases which is roughly given by the closest approach of the pole to $\omega = 0$ . For the less universal cases (3.18) and (3.19) one finds the expected results, $E_g \propto \kappa_{\text{IR}}$ and $\kappa_{\text{IR}}^{1/2}$ for $q \neq 0$ and $q = 0$ respectively. However for the more interesting “critical” case the gap scales as,

$$E_g\sim ({\kappa }_{\text{IR}}{)}^{\frac{1}{1-2{\delta }_{-}}}(3.20)E\_g\; \backslash sim\; (\backslash kappa\_\{\backslash text\{IR\}\})^\{\backslash frac\{1\}\{1-2\backslash delta\_-\}\}\; \backslash quad\; (3.20)$$

The correlation length for all cases scales as $\xi \propto \kappa_{\text{IR}}^{-1/2}$ .

When $\kappa_{\text{IR}} < 0$ , application of the constraint (3.13) shows that the pole always lies in the upper half plane for momenta $p < \sqrt{-\kappa_{\text{IR}}/c_p}$ , representing an instability.

Exactly at the critical point $\kappa_{\text{IR}} = 0$ we find a free gapless mode which disperses in the complex plane as

$$\omega \sim \mathrm{\mid }\vec{p}{\mathrm{\mid }}^z\text{with}z=\frac{2}{1-2{\delta }_{-}}(3.21)\backslash omega\; \backslash sim\; |\backslash vec\{p\}|^z\; \backslash quad\; \backslash text\{with\}\; \backslash quad\; z\; =\; \backslash frac\{2\}\{1-2\backslash delta\_-\}\; \backslash quad\; (3.21)$$

for the “critical” case, and $z = 2$ for $q \neq 0$ and $\delta_- < 0$ and $z = 1$ for $q = 0$ and $\delta_- < -1/2$ . We thus conclude that the physics of the critical point has a nontrivial dynamical critical exponent determined by the dimension of an operator $\delta_-$ in the IR $AdS_2$ CFT. This identification is consistent with the relationship $E_g \propto \xi^{-z}$ in all cases. Interestingly $z$ has a lower bound in this model, with $z > 2$ for the charged case and $z > 1$ for the neutral case.

To summarize, in the most interesting “critical case” where $0 < \delta_- < 1/2$ for $q \neq 0$ and $-1/2 < \delta_- < 1/2$ for $q = 0$ the two point function close to the critical point has the universal scaling form,

$${\chi }_R=\frac{Z}{{\kappa }_{\text{IR}}+c_p{\vec{p}}^2+T^{2\mathrm{/}z}g(\omega \mathrm{/}T)}(3.22)\backslash chi\_R\; =\; \backslash frac\{Z\}\{\backslash kappa\_\{\backslash text\{IR\}\}\; +\; c\_p\; \backslash vec\{p\}^2\; +\; T^\{2/z\}\; g(\backslash omega/T)\}\; \backslash quad\; (3.22)$$

where $z = \frac{2}{1-2\delta_-}$ and $g(\omega/T)$ is a universal scaling function that follows from (3.12). Note that since the correlation function is analytic in $\vec{p}$ (the self energy is momentum independent) the critical point is ‘locally’ quantum critical [25]. This type of criticality goes beyond the usual Landau Ginzburg paradigm often applied to quantum critical points, due to the existence of the locally critical modes associated with $AdS_2$ . These results are compatible with experimental measurements of the spin susceptibility in a heavy fermion compound $CeCu_{6-x}Au_x$ at criticality [39]. In order to compare the spin susceptibility to the two point function of the triplet staggered order parameter (which is effectively $\chi_R$ ), one must shift the momentum in (3.22) by the ordering vector associated to the anti-ferromagnetic order $\vec{p} \rightarrow \vec{p} - \vec{K}$ . Very similar results were also found theoretically for the spin susceptibility in

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⁹ This is not a genuine gap, since there will always be gapless incoherent junk coming from the $AdS_2$ Green’s function. However it does represent a gap to the coherent part of the 2 point function, which is represented by the dispersing pole.[25, 26]. The most important feature for comparing to experiments was $\omega/T$ scaling of the susceptibility at the ordering vector $\vec{p} = \vec{K}$ and a nontrivial exponent $z \approx 2.7$ or $\delta_- \approx .13$ . Indeed we capture both features here, although since our $\delta_-$ does not take a universal value, it is hard to make a prediction for this exponent without a real string embedding where $\delta_-$ will be fixed.

3.3 Renormalization group interpretation

We would like to now give an RG interpretation of the above results and in so doing try to understand what to expect of the zero temperature ordered phase when $\kappa_{\text{IR}} < 0$ .

We have already discussed a major aspect of the RG flow: the extreme RN black hole represents a flow from one $2 + 1$ CFT in the UV to a $0 + 1$ CFT in the IR. We would like to now understand how this picture changes in the presence of the scalar $\psi$ . For this purpose there is clearly a set of distinct cases depending on the value of $\delta_-$ the conformal dimension of the operator dual to $\psi$ in the IR CFT. Firstly for $\delta_-$ complex the IR CFT will never be realized and there will be no critical point. We do not consider this case further here. There are two remaining cases $0 < \delta_- < 1/2$ and $\delta_- < 0$ which we turn to now.

This range of dimensions is in the “critical” range identified above where the 2 point function takes the more universal form (3.22) for both neutral and charged cases. Here we argue that this result is universally controlled by the $AdS_2$ theory supplemented by double trace deformations. The bulk field $\psi$ in the $AdS_2 \times R^2$ geometry is dual to a set of operators $\Psi_{\vec{p}}$ where $\vec{p}$ labels the momentum in the $R^2$ direction, which one should think of as a charge under the KK reduction down to $AdS_2$ . $\Psi_{\vec{p}}$ can be viewed as the image of $\mathcal{O}(t, \vec{p})$ under RG flow.

The operator dimension of $\Psi_{\vec{p}}$ in the $0 + 1$ dimensional CFT is given by $\delta_{\pm} + \mathcal{O}(p^2)$ , where in this range of dimensions we can take either value. The two point function of $\Psi_{\vec{p}}$ is then,

$${\mathrm{\Sigma }}_R^{\pm }\propto {\omega }^{2{\delta }_{\pm }-1}(3.23)\backslash Sigma\_R^\{\backslash pm\}\; \backslash propto\; \backslash omega^\{2\backslash delta\_\{\backslash pm\}-1\}\; \backslash quad\; (3.23)$$

where the previously defined $AdS_2$ Green’s function (3.11) is $\Sigma_R^+ = \Sigma_R$ . If we take the dimension of $\Psi_{\vec{p}}$ to be $\delta_-$ such that we are working in alternative quantization then we can reproduce the result of (3.22) at $T = 0$ simply by including the following deformation of the $0 + 1$ CFT [20],

$$S_{0+1}\to S_{0+1}-\int dt\frac{d^2\vec{p}}{(2\pi {)}^2}({\kappa }_{\text{IR}}+c_p{\vec{p}}^2){\mathrm{\Psi }}_{\vec{p}}^{†}{\mathrm{\Psi }}_{\vec{p}}(3.24)S\_\{0+1\}\; \backslash rightarrow\; S\_\{0+1\}\; -\; \backslash int\; dt\; \backslash frac\{d^2\; \backslash vec\{p\}\}\{(2\backslash pi)^2\}\; (\backslash kappa\_\{\backslash text\{IR\}\}\; +\; c\_p\; \backslash vec\{p\}^2)\; \backslash Psi\_\{\backslash vec\{p\}\}^\backslash dagger\; \backslash Psi\_\{\backslash vec\{p\}\}\; \backslash quad\; (3.24)$$Note for example when the deformation is zero, $\kappa_{\text{IR}} + c_p \vec{p}^2 = 0$ , the two point function (3.22) scales as $\omega^{2\delta_- - 1}$ consistent with $\Psi_{\vec{p}}$ having dimension $\delta_-$ .

At zero momentum there is a single relevant double trace coupling $\kappa_{\text{IR}}$ at the critical point, that does not explicitly break the symmetry. Just as in the $AdS_4$ case discussed in Section 2, a positive $\kappa_{\text{IR}}$ will induce a flow to standard quantization where the operators $\Psi_{\vec{p}}$ now have dimension $1 - \delta_- = \delta_+$ . On the other hand, a negative $\kappa_{\text{IR}}$ will induce an instability which will lead to a symmetry broken state. Since $c_p > 0$ the zero momentum mode will go unstable first, so in the ordered phase the homogenous mode $\Psi_{\vec{p}=0}$ develops a vev.

The ordered state can then be studied using gravity with the bulk field $\psi$ turned on. For $\delta_- > 0$ , $m_{\text{eff}}^2$ defined in (2.21) is negative which means that for the disordered phase the scalar $\psi$ is sitting at a maximum of an effective potential; in the ordered phase it will roll away from this maximum. The resulting geometry will tell us where the theory ends up in the deep IR. In fact there are many possibilities which will depend on the bulk functions $V(\psi)$ , $G(\psi)$ and $J(\psi)$ . One possibility which we highlight in the next section is the theory flows to a different $\widetilde{AdS_2} \times R^2$ geometry, where the field $\psi$ sits at the minimum of a particular effective potential to be discussed later. In Section 4 we will outline some other possible examples of deep IR geometries. At this stage to be appropriately noncommittal we call this geometry $X$ .

The above discussion can be understood within the context of the UV completion of $AdS_2$ (a fancy name for the RN black hole) where we identify the coefficient of the double trace operator $\kappa$ as the microscopic control parameter which allows us to probe the critical point. The critical value $\kappa_c$ is thus not special from the perspective of the UV theory. The blue region of Fig. 6 is a pictorial description of the RG flows represented by the (extremal) RN black hole phase of the theory. In the large- $N$ limit, this flow is trivial, only effecting the fluctuations of fields in the bulk through boundary conditions. When $\kappa < \kappa_c$ , the double trace coupling runs negative in the IR theory, and the instability ensues. The full $AdS_4$ theory then allows us to view the end point of this instability - the geometry in the extreme IR flows to a new attractive fixed point $X$ . These flows are represented by the white region in Fig. 6.

Finally we make an argument as to how the order parameter behaves in the condensed phase. The argument¹⁰ is based on an RG analysis close to the critical fixed point and relies on the picture given in Fig. 6. We will confirm the result with a numerical calculation in the next section. Consider shooting from the theory $X$ in the extreme IR up to $AdS_4$ . As suggested by Fig. 6 there will be two tuning parameters, of which only an appropriate scale

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¹⁰A similar argument appeared in [40] in the context of a BKT type transition, and related arguments appear in [41].Figure 6: Renormalization group flow interpretation of the quantum critical point (denoted by the red dot) for $0 < \delta_- < 1/2$ . This picture is heuristic, since the couplings are only well defined close to each fixed point. The blue region is the disordered phase which we have studied in this section. A distinction is made between the theory in alternative (alt) and standard (std) quantizations. The white region denotes the ordered phase which we will study more carefully in the next section. The theories denoted by $X$ and $Y$ depend on bulk couplings. $Y$ is the end point of the flow induced by turning on a negative double trace coupling in the $AdS_4$ theory and setting $\mu = 0$ . These flows were considered in Section 2. $X$ is discussed in the text above.

invariant ratio of the two will matter. As we tune this ratio we can shoot closer and closer to the $AdS_2$ critical theory. The dashed line in Fig. 6 is an example of such a flow. In the $AdS_2$ region, the metric and Maxwell field are given by (3.2) and the scalar field takes the form

$$\psi (\widehat{r})\sim v{\widehat{r}}^{-{\delta }_{-}}+w{\widehat{r}}^{-{\delta }_{+}}(3.25)\backslash psi(\backslash hat\{r\})\; \backslash sim\; v\; \backslash hat\{r\}^\{-\backslash delta\_-\}\; +\; w\; \backslash hat\{r\}^\{-\backslash delta\_+\}\; \backslash quad\; (3.25)$$

Here $v$ and $w$ are constrained by scale invariance:

$$w=-s_c^{\text{IR}}{v}^{{\delta }_{+}\mathrm{/}{\delta }_{-}}(3.26)w\; =\; -s\_c^\{\backslash text\{IR\}\}\; v^\{\backslash delta\_+/\backslash delta\_-\}\; \backslash quad\; (3.26)$$

where $s_c^{\text{IR}}$ is a number which can be computed numerically. It is analogous to the $s_c$ in (2.12).

We can now match (3.25) onto linear fluctuations in the extreme RN background which we have considered above (what we called the outer region above, see for example (3.5)).

$$\widehat{r}=\mathrm{\Lambda }(r-r_{\ast }),\widehat{t}=t\mathrm{/}\mathrm{\Lambda },{\widehat{\alpha }}_0=v{\mathrm{\Lambda }}^{-{\delta }_{-}},{\widehat{\beta }}_0=w{\mathrm{\Lambda }}^{-{\delta }_{+}}(3.27)\backslash hat\{r\}\; =\; \backslash Lambda(r\; -\; r\_*),\; \backslash quad\; \backslash hat\{t\}\; =\; t/\backslash Lambda,\; \backslash quad\; \backslash hat\{\backslash alpha\}\_0\; =\; v\backslash Lambda^\{-\backslash delta\_-\},\; \backslash quad\; \backslash hat\{\backslash beta\}\_0\; =\; w\backslash Lambda^\{-\backslash delta\_+\}\; \backslash quad\; (3.27)$$where we have allowed for an arbitrary rescaling $\Lambda$ which will drop out in the end. Additionally we have for $\kappa$ close to $\kappa_c$ ,

$${\alpha }_0=a^{+}{\widehat{\alpha }}_0,{\widehat{\beta }}_0={\kappa }_{\text{IR}}{\widehat{\alpha }}_0(3.28)\backslash alpha\_0\; =\; a^+\; \backslash hat\{\backslash alpha\}\_0,\; \backslash quad\; \backslash hat\{\backslash beta\}\_0\; =\; \backslash kappa\_\{\backslash text\{IR\}\}\; \backslash hat\{\backslash alpha\}\_0\; \backslash quad\; (3.28)$$

Putting these results together we find,

$$⟨\mathcal{O}⟩=\alpha \approx a^{+}{(-\frac{{\kappa }_{\text{IR}}}{s_c^{\text{IR}}})}^{\frac{{\delta }_{-}}{1-2{\delta }_{-}}},(3.29)\backslash langle\; \backslash mathcal\{O\}\; \backslash rangle\; =\; \backslash alpha\; \backslash approx\; a^+\; \backslash left(\; -\backslash frac\{\backslash kappa\_\{\backslash text\{IR\}\}\}\{s\_c^\{\backslash text\{IR\}\}\}\; \backslash right)^\{\backslash frac\{\backslash delta\_-\}\{1-2\backslash delta\_-\}\},\; \backslash quad\; (3.29)$$

This is precisely an IR analog of the scaling we derived in section 2 for a negative double trace perturbation (2.14). Note we have made an assumption that $s_c^{\text{IR}}$ is positive. This is a nontrivial assumption and will depend on the bulk couplings, as for example was the case for the analogous $AdS_4$ parameter $s_c$ [18]. However whereas in the $AdS_4$ case, a negative $s_c$ meant a somewhat sick theory, we suspect a negative $s_c^{\text{IR}}$ will simply result in a first order transition. Since we are working with the assumption of a continuous transition we cannot say much about the case with $s_c^{\text{IR}}$ negative.

More generally we can consider a black hole solution with a non-zero temperature which has $\psi$ nonzero and comes close to $AdS_2 \times R^2$ . Again this is a nontrivial shooting problem. Now there is a one parameter family of solutions labeled by the temperature $\hat{T}$ and asymptotic to (3.25). Scale invariance imposes the following relationship between $v$ and $w$ ,

$$w=-S\left(\frac{\widehat{T}}{v^{1\mathrm{/}{\delta }_{-}}}\right)v^{{\delta }_{+}\mathrm{/}{\delta }_{-}}(3.30)w\; =\; -S\; \backslash left(\; \backslash frac\{\backslash hat\{T\}\}\{v^\{1/\backslash delta\_-\}\}\; \backslash right)\; v^\{\backslash delta\_+/\backslash delta\_-\}\; \backslash quad\; (3.30)$$

where $S$ is a scaling function with $S(0) = s_c^{\text{IR}}$ . It can only be computed numerically. Going through the same matching procedure and rescaling $\hat{T} = \Lambda T$ one finds the following scaling relation,

$$-{\kappa }_{\text{IR}}=(⟨\mathcal{O}⟩\mathrm{/}{a}^{+}{)}^{\frac{1-2{\delta }_{-}}{{\delta }_{-}}}S\left(\frac{T}{(⟨\mathcal{O}⟩\mathrm{/}{a}^{+}{)}^{1\mathrm{/}{\delta }_{-}}}\right)(3.31)-\backslash kappa\_\{\backslash text\{IR\}\}\; =\; (\backslash langle\; \backslash mathcal\{O\}\; \backslash rangle\; /\; a^+)^\{\backslash frac\{1-2\backslash delta\_-\}\{\backslash delta\_-\}\}\; S\; \backslash left(\; \backslash frac\{T\}\{(\backslash langle\; \backslash mathcal\{O\}\; \backslash rangle\; /\; a^+)^\{1/\backslash delta\_-\}\}\; \backslash right)\; \backslash quad\; (3.31)$$

This is of course consistent with the dimension of the IR operator which gets a vev

$$⟨{\mathrm{\Psi }}_{\vec{p}}⟩={\widehat{\alpha }}_0{\delta }^2(\vec{p})(3.32)\backslash langle\; \backslash Psi\_\{\backslash vec\{p\}\}\; \backslash rangle\; =\; \backslash hat\{\backslash alpha\}\_0\; \backslash delta^2(\backslash vec\{p\})\; \backslash quad\; (3.32)$$

being $\delta_-$ in alternative quantization and $\kappa_{\text{IR}}$ having dimensions $1 - 2\delta_-$ . In the next section we will construct these flows on the condensed side. We will confirm amongst other things the result (3.29).

Actually this story is incomplete, for $0 < \delta_- < 1/4$ the $AdS_2$ CFT has higher order multi-trace operators that are relevant (first $\hat{\alpha}^4$ then $\hat{\alpha}^6$ etc.). In this situation the scalingarguments above fail, since the critical point is now multi-critical. Away from the multi-critical point, on the continuous side we now expect a mean field¹¹ relationship $\langle \mathcal{O} \rangle \sim (-\kappa_{\text{IR}})^{1/2}$ . The argument for this goes as follows. Nonlinear corrections $\mathcal{O}(\psi^3)$ to the linear equation for $\psi$ in the outer region produce additional terms in the matching (3.27). Most importantly

$$\widehat{\beta }={\widehat{\beta }}_0+{\widehat{\beta }}_{NL}+\cdots \approx {\kappa }_{\text{IR}}{\widehat{\alpha }}_0+u_{\text{IR}}{\widehat{\alpha }}_0^3=w{\mathrm{\Lambda }}^{-{\delta }_{+}},\widehat{\alpha }={\widehat{\alpha }}_0+{\widehat{\alpha }}_{NL}+\cdots \approx {\widehat{\alpha }}_0=v{\mathrm{\Lambda }}^{-{\delta }_{-}}(3.33)\backslash hat\{\backslash beta\}\; =\; \backslash hat\{\backslash beta\}\_0\; +\; \backslash hat\{\backslash beta\}\_\{NL\}\; +\; \backslash dots\; \backslash approx\; \backslash kappa\_\{\backslash text\{IR\}\}\; \backslash hat\{\backslash alpha\}\_0\; +\; u\_\{\backslash text\{IR\}\}\; \backslash hat\{\backslash alpha\}\_0^3\; =\; w\; \backslash Lambda^\{-\backslash delta\_+\},\; \backslash quad\; \backslash hat\{\backslash alpha\}\; =\; \backslash hat\{\backslash alpha\}\_0\; +\; \backslash hat\{\backslash alpha\}\_\{NL\}\; +\; \backslash dots\; \backslash approx\; \backslash hat\{\backslash alpha\}\_0\; =\; v\; \backslash Lambda^\{-\backslash delta\_-\}\; \backslash quad\; (3.33)$$

where $u_{\text{IR}}$ can be computed along the lines of Appendix C. It is obvious that we should interpret $u_{\text{IR}}$ as the RG flow of the quadruple trace operator from $AdS_4$ to $AdS_2$ analogous to $\kappa \rightarrow \kappa_{\text{IR}}$ . Thus at zero temperature we find,

$${\kappa }_{\text{IR}}{\widehat{\alpha }}_0+u_{\text{IR}}{\widehat{\alpha }}_0^3=-s_c^{\text{IR}}({\widehat{\alpha }}_0{)}^{(1-{\delta }_{-})\mathrm{/}{\delta }_{-}}(3.34)\backslash kappa\_\{\backslash text\{IR\}\}\; \backslash hat\{\backslash alpha\}\_0\; +\; u\_\{\backslash text\{IR\}\}\; \backslash hat\{\backslash alpha\}\_0^3\; =\; -s\_c^\{\backslash text\{IR\}\}(\backslash hat\{\backslash alpha\}\_0)^\{(1-\backslash delta\_-)/\backslash delta\_-\}\; \backslash quad\; (3.34)$$

For $\delta_- > 1/4$ , the $u_{\text{IR}}$ term is not important for small $\hat{\alpha}0$ and we reproduce the result (3.29). However for $\delta- < 1/4$ the non-analytic term in $\hat{\alpha}_0$ is less important and we get the mean field answer assuming that $u_{\text{IR}} > 0$ ,

$$⟨\mathcal{O}⟩=a^{+}(-{\kappa }_{\text{IR}}\mathrm{/}{u}_{\text{IR}}{)}^{1\mathrm{/}2}(3.35)\backslash langle\; \backslash mathcal\{O\}\; \backslash rangle\; =\; a^+\; (-\backslash kappa\_\{\backslash text\{IR\}\}/u\_\{\backslash text\{IR\}\})^\{1/2\}\; \backslash quad\; (3.35)$$

for $u_{\text{IR}} < 0$ we get a first order transition that we cannot say much about. These nonlinear corrections to the linear $\psi$ equation come from potential terms, as well as from back-reaction on gravity. They are rather complicated, however they are most likely computable in the probe approximation introduced in [19] for the neutral case. We leave their explicit computation to future work.

$\delta_- < 0$

In this case, $m_{\text{eff}}^2$ defined in (2.21) is positive, so the IR $AdS_2$ with $\psi = 0$ is stable. However, one can still have phase transitions which turn on $\psi$ at larger radius. In this case, we will see the critical exponents are not governed by the $0+1$ CFT but take mean field values.

We can reproduce the more general 2 point function (3.9) for $\mathbb{X}$ given in (3.10) using the following semi-holographic action [20],

$$S=S_{0+1}+\int d^{3}x(\mathrm{\mid }{\mathrm{\partial }}_t\mathrm{\Phi }{\mathrm{\mid }}^2+q_{\mathrm{\Phi }}(i{\mathrm{\Phi }}^{†}{\mathrm{\partial }}_t\mathrm{\Phi }+\text{h.c.})-c^2\mathrm{\mid }\vec{\mathrm{\partial }}\mathrm{\Phi }{\mathrm{\mid }}^2-{\kappa }_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Phi }{\mathrm{\mid }}^2-\frac{1}{2}{u}_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Phi }{\mathrm{\mid }}^4)(3.36)S\; =\; S\_\{0+1\}\; +\; \backslash int\; d^3x\; \backslash left(\; |\backslash partial\_t\; \backslash Phi|^2\; +\; q\_\backslash Phi\; (i\backslash Phi^\backslash dagger\; \backslash partial\_t\; \backslash Phi\; +\; \backslash text\{h.c.\})\; -\; c^2\; |\backslash vec\{\backslash partial\}\; \backslash Phi|^2\; -\; \backslash kappa\_\backslash Phi\; |\backslash Phi|^2\; -\; \backslash frac\{1\}\{2\}\; u\_\backslash Phi\; |\backslash Phi|^4\; \backslash right)\; \backslash quad\; (3.36)$$

$$+\eta \int dt\frac{d^2\vec{p}}{(2\pi {)}^2}({\mathrm{\Phi }}_{\vec{p}}^{†}{\mathrm{\Psi }}_{\vec{p}}+\text{h.c.})(3.37)+\; \backslash eta\; \backslash int\; dt\; \backslash frac\{d^2\; \backslash vec\{p\}\}\{(2\backslash pi)^2\}\; \backslash left(\; \backslash Phi\_\{\backslash vec\{p\}\}^\backslash dagger\; \backslash Psi\_\{\backslash vec\{p\}\}\; +\; \backslash text\{h.c.\}\; \backslash right)\; \backslash quad\; (3.37)$$

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¹¹We thank Kristan Jensen for drawing our attention to this possibility.where $\Phi$ is a boundary field which we have coupled to the $AdS_2$ CFT operator $\Psi_{\vec{p}}$ and $\Phi_{\vec{p}}$ is the spatial fourier transform of $\Phi$ . To reproduce (3.9) the dimension of $\Psi$ must be $\delta_+ + \mathcal{O}(\vec{p}^2)$ . The two point function for $\Phi$ will then agree with $\chi_R$ with the following identifications,

$$\mathrm{\mid }\eta {\mathrm{\mid }}^2=1\mathrm{/}{c}_{\omega },{\kappa }_{\mathrm{\Phi }}={\kappa }_{\text{IR}}\mathrm{/}{c}_{\omega },q_{\mathrm{\Phi }}=q{c}_q\mathrm{/}{c}_{\omega }{c}^2=c_p\mathrm{/}{c}_{\omega },\mathrm{\Phi }\sim \mathcal{O}(\sqrt{c_{\omega }\mathrm{/}Z})(3.38)|\backslash eta|^2\; =\; 1/c\_\backslash omega,\; \backslash quad\; \backslash kappa\_\backslash Phi\; =\; \backslash kappa\_\{\backslash text\{IR\}\}/c\_\backslash omega,\; \backslash quad\; q\_\backslash Phi\; =\; qc\_q/c\_\backslash omega\; \backslash quad\; c^2\; =\; c\_p/c\_\backslash omega,\; \backslash quad\; \backslash Phi\; \backslash sim\; \backslash mathcal\{O\}(\backslash sqrt\{c\_\backslash omega/Z\})\; \backslash quad\; (3.38)$$

Note that $\mathcal{O}$ will also have an overlap with $\Psi$ but this will lead to a subdominant correction to the two point function. We have also included a nonlinear interaction term $u_\Phi$ in (3.36) which should be generated in the flow from $AdS_4$ to $AdS_2$ . We will assume that $u_\Phi > 0$ , but this need not be the case.

In order to construct the ordered phase we first assume that we can treat $\eta$ perturbatively. We will show this is a consistent assumption. So for now we set $\eta = 0$ and work in the mean field approximation for $\Phi$ . We do this because we are working in the classical gravity approximation. (The whole action above should be multiplied by $1/G_N$ , and for small $G_N$ mean field applies.) Then for $\kappa_\Phi$ negative, $\Phi$ develops a vev:

$$⟨\mathrm{\Phi }⟩=\sqrt{-{\kappa }_{\mathrm{\Phi }}\mathrm{/}{u}_{\mathrm{\Phi }}}(3.39)\backslash langle\; \backslash Phi\; \backslash rangle\; =\; \backslash sqrt\{-\backslash kappa\_\backslash Phi/u\_\backslash Phi\}\; \backslash quad\; (3.39)$$

Turning on $\eta$ , this will now act as a source for the homogenous mode of the operator $\Psi_{\vec{p}}$ ,

$$S=S_{0+1}+\eta \sqrt{-\frac{{\kappa }_{\mathrm{\Phi }}}{u_{\mathrm{\Phi }}}}\int dt\frac{d^2\vec{p}}{(2\pi {)}^2}{\delta }^2(\vec{p})({\mathrm{\Psi }}_{\vec{p}}^{†}+\text{h.c.})(3.40)S\; =\; S\_\{0+1\}\; +\; \backslash eta\; \backslash sqrt\{-\backslash frac\{\backslash kappa\_\backslash Phi\}\{u\_\backslash Phi\}\}\; \backslash int\; dt\; \backslash frac\{d^2\; \backslash vec\{p\}\}\{(2\backslash pi)^2\}\; \backslash delta^2(\backslash vec\{p\})\; \backslash left(\; \backslash Psi\_\{\backslash vec\{p\}\}^\backslash dagger\; +\; \backslash text\{h.c.\}\; \backslash right)\; \backslash quad\; (3.40)$$

Since the dimension of $\Psi$ is irrelevant ( $\delta_+ > 1$ ) this source will scale away in the IR. Thus we do not expect the vev of $\Phi$ to back-react on the $AdS_2$ CFT which is now a stable fixed point. This is consistent with the fact that for $\delta_- < 0$ , the effective mass square $m_{\text{eff}}^2 > 0$ , and the bulk field $\psi$ sits at a minimum in the disordered phase and has nowhere to go in the ordered phase. Thus the ordering is controlled by the boundary field $\Phi$ .

It is now clear how to construct the ordered phase from gravity, we simply shoot from $AdS_2$ with a non-zero source term for the irrelevant operator $\Psi$ . (One also needs to turn on the irrelevant coupling $\delta_h$ discussed around (3.3).) This situation is depicted in Fig. 7. Again we can match this onto the perturbations of the extreme RN black hole with

$$\widehat{\alpha }\approx {\widehat{\alpha }}_0=\eta \sqrt{-{\kappa }_{\mathrm{\Phi }}\mathrm{/}{u}_{\mathrm{\Phi }}},\widehat{\beta }\approx {\widehat{\beta }}_0+u_{\text{IR}}{\widehat{\alpha }}_0^3=0(3.41)\backslash hat\{\backslash alpha\}\; \backslash approx\; \backslash hat\{\backslash alpha\}\_0\; =\; \backslash eta\; \backslash sqrt\{-\backslash kappa\_\backslash Phi/u\_\backslash Phi\},\; \backslash quad\; \backslash hat\{\backslash beta\}\; \backslash approx\; \backslash hat\{\backslash beta\}\_0\; +\; u\_\{\backslash text\{IR\}\}\; \backslash hat\{\backslash alpha\}\_0^3\; =\; 0\; \backslash quad\; (3.41)$$

where we have included an important nonlinear correction as in (3.33). The condition $\hat{\beta} = 0$ (which is the requirement that $\psi$ not blow up in the IR) and $\hat{\beta}0 = \kappa{\text{IR}} \hat{\alpha}_0$ then allow us to match $u_\Phi$ ,

$$u_{\mathrm{\Phi }}=u_{\text{IR}}\mathrm{/}{c}_{\omega }^2(3.42)u\_\backslash Phi\; =\; u\_\{\backslash text\{IR\}\}/c\_\backslash omega^2\; \backslash quad\; (3.42)$$The vev then follows from the value of the source term $\hat{\alpha}_0$ ,

$$⟨\mathcal{O}⟩=\alpha =(a^{+}\mathrm{/}{c}_{\omega })(-{\kappa }_{\text{IR}}\mathrm{/}{u}_{\mathrm{\Phi }}{)}^{1\mathrm{/}2}=a^{+}(-{\kappa }_{\text{IR}}\mathrm{/}{u}_{\text{IR}}{)}^{1\mathrm{/}2}(3.43)\backslash langle\; \backslash mathcal\{O\}\; \backslash rangle\; =\; \backslash alpha\; =\; (a^+/c\_\backslash omega)(-\backslash kappa\_\{\backslash text\{IR\}\}/u\_\backslash Phi)^\{1/2\}\; =\; a^+(-\backslash kappa\_\{\backslash text\{IR\}\}/u\_\{\backslash text\{IR\}\})^\{1/2\}\; \backslash quad\; (3.43)$$

which is same as (3.35).

Figure 7: RG interpretation for $\delta_- < 0$ . In this case $m_{\text{eff}}^2 > 0$ , so from the gravity perspective $AdS_2$ is stable and the three fixed points on the right side are all the same. The condensation is controlled by an $AdS_2$ boundary field $\Phi$ . The three cases are distinguished by whether $\Phi$ has a nonzero vev (bottom), is part of the IR dynamics but does not condense (middle), or is massive so it drops out of the IR theory (top).

3.4 Parametric dependence on bulk couplings

In this section we compile some numerical results relating to the critical point. The universal features discussed above depend on two parameters $q$ and $\delta_-$ . However the location of the critical point itself $\kappa_c$ and other nonuniversal constants appearing in the dynamic susceptibility depend on three bulk parameters, $m^2, g, q$ . In Fig. 8 and Fig. 9 we plot $\kappa_c$ through two different slices of this three parameter space, one with $q = 0$ and the other with $\Delta = 1$ .

4 Constructing the ordered phase

We have left many of the details of constructing the ordered phase to this section. We will consider the full back-reaction of the condensing field $\psi$ on the metric so the problem is highlyFigure 8: Contour plots of $\kappa_c$ for $q = 0$ . The lines $\delta_- = 0$ and $\kappa_c = 0$ are shown. The solid (blue) region represents the excluded BF bound in $AdS_2$ and $AdS_4$ . Note that positive $\kappa_c$ tends to occur close to this region since then the theory is more unstable. For theories with $\kappa_c$ negative, introducing a chemical potential fails to destabilize the theory with alternative boundary conditions. So these theories are more stable.

Figure 9: Contour plots of $\kappa_c$ for $\Delta_- = 1$ . The lines $\delta_- = 0$ and $\kappa_c = 0$ are shown. Clearly increasing $q$ tends to destabilize the theory since $\kappa_c$ increases. Decreasing the bulk coupling $g$ also tends to increase stability.

nonlinear and we must proceed numerically. We will focus on the zero temperature case where the basic problem will be to find the appropriate IR solution and to integrate outwards from there. Perturbing around the IR solution one finds various shooting parameters that take the form of irrelevant couplings. These couplings generate flows in the UV to an asymptotically $AdS_4$ solution, representing the UV fixed point of the theory. From here we can read off various data such as the vev of the order parameter, the double trace couplings $\kappa$ and the free energy.As one tunes the irrelevant couplings in the IR we find that one can shoot closer and closer to the critical point that we identified in the previous section. As we will see this involves a flow whose geometry is described, for a large chunk of radial proper distance, by the $AdS_2 \times R^2$ critical solution.

4.1 Ansatz for the background

We will use the following metric and field ansatz,

$$d{s}^2=-f(r)d{t}^2+\frac{d{r}^2}{f(r)}+h^2(r)d{\vec{x}}^2,A=\varphi (r)dt,\psi =\psi (r)(4.1)ds^2\; =\; -f(r)dt^2\; +\; \backslash frac\{dr^2\}\{f(r)\}\; +\; h^2(r)d\backslash vec\{x\}^2,\; \backslash quad\; A\; =\; \backslash phi(r)dt,\; \backslash quad\; \backslash psi\; =\; \backslash psi(r)\; \backslash quad\; (4.1)$$

This metric differs from the one used in [6] in which the radial coordinate was chosen to be $\sqrt{g_{xx}}$ . The form (4.1) is more convenient since it allows $AdS_2 \times R^2$ as a solution. Also numerically it is more convenient to work with this metric as we will demonstrate later.

The equations of motion which follow from the action (2.1) are given in Appendix A (A.5-A.8). For our ansatz (4.1) this reduces to the system of ODEs (A.10-A.13). There are three important reparameterizations that leave the metric form invariant, thus allowing us to generate new solutions from old solutions. They are:

$$\text{Shift : }r\to r+a\mathrm{.}(4.2)\backslash text\{Shift\; :\; \}\; r\; \backslash rightarrow\; r\; +\; a.\; \backslash quad\; (4.2)$$

$$\text{Spatial rescaling : }\vec{x}\to \vec{x}\mathrm{/}s,h\to sh\mathrm{.}(4.4)\backslash text\{Spatial\; rescaling\; :\; \}\; \backslash vec\{x\}\; \backslash rightarrow\; \backslash vec\{x\}/s,\; h\; \backslash rightarrow\; sh.\; \backslash quad\; (4.4)$$

We will fix these reparameterizations by demanding certain asymptotic boundary conditions, which we specify in the next subsection.

4.2 Asymptotic $AdS_4$ data and the free energy

The asymptotic UV fixed point will always be $AdS_4$ . For $\Delta_+ < 3$ there are no irrelevant (single trace) operators within our truncation so $AdS_4$ is always an attractive fixed point in the UV. A complete expansion about $AdS_4$ can be systematically derived as discussed in Appendix D. The leading terms include

Additional terms in these expansions will be needed for some of the manipulations which follow, however due to their cumbersome form we leave these corrections to the appendix.A general solution is parameterized by 5 constants $\alpha, \beta, \mu, \rho, m_0$ . We have used the shift (4.2) to fix the sub-leading $r^{-1}$ terms in the metric to zero, and used the spatial rescaling in (4.4) to fix the normalization of $h$ and thus the spatial components of the boundary metric. The remaining conformal symmetry (4.3) can be used to fix one of these five constants. We will work with fixed chemical potential $\mu$ , and often set $\mu = 1$ . Numerical results of dimensionful quantities will be quoted in units of $\mu$ .

We will need to compute the thermodynamic potential, which in our ensemble will be the grand potential $G$ . We will go through this in some detail by computing the Euclidean on-shell action. While the results for general scalar boundary conditions are known using other methods the inclusion of a gauge field is new.

The grand potential is $G/T = S_E + S_{ct}$ where $S_E$ is the Euclidean action and $S_{ct}$ are boundary counter terms. These counter terms are required for a good variational problem, as well as to regulate divergences in $S_E$ . We want to keep the metric fixed on the boundary (necessitating the Gibbons Hawking term), and the chemical potential fixed. For the scalar field $\psi$ we would like to require $\alpha, \beta$ to be constrained by $\beta = W'(\alpha)$ . We will work initially with a fixed source $\beta$ in alternative quantization where $W(\alpha) = \beta\alpha$ , and generalize later. The counter terms in the case of fixed $\beta$ , $S_{ct}^\beta$ , and $S_E$ are given in Appendix D, with the free energy density evaluating to:

$$g_{\beta }=-\frac{1}{2}{m}_0+\frac{2}{3}\alpha \beta {\mathrm{\Delta }}_{+}({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})(4.6)g\_\backslash beta\; =\; -\backslash frac\{1\}\{2\}m\_0\; +\; \backslash frac\{2\}\{3\}\backslash alpha\backslash beta\backslash Delta\_+(\backslash Delta\_+\; -\; \backslash Delta\_-)\; \backslash quad\; (4.6)$$

where we have defined $g = G/V$ and $V$ is the field theory volume.

We can also compute how the free energy varies as we move in solution space parameterized by changes in the five integration constants $\delta\alpha, \delta\beta, \delta\mu, \delta\rho, \delta m_0$ . Varying the on shell action we derive an analog of the first law of thermodynamics,

$$\delta g_{\beta }=-s\delta T-\rho \delta \mu +2({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})\alpha \delta \beta (4.7)\backslash delta\; g\_\backslash beta\; =\; -s\backslash delta\; T\; -\; \backslash rho\backslash delta\backslash mu\; +\; 2(\backslash Delta\_+\; -\; \backslash Delta\_-)\backslash alpha\backslash delta\backslash beta\; \backslash quad\; (4.7)$$

where $s$ is the entropy density. From (4.7) it is clear that $g_\beta$ is stationary at fixed $T$ , $\mu$ , and $\beta$ . To generalize the boundary conditions for $\psi$ we simply define

$$g_W=g_{\beta }-2({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})(\alpha \beta -W)(4.8)g\_W\; =\; g\_\backslash beta\; -\; 2(\backslash Delta\_+\; -\; \backslash Delta\_-)(\backslash alpha\backslash beta\; -\; W)\; \backslash quad\; (4.8)$$

who's variation is given by,

$$\delta g_W=-s\delta T-\rho \delta \mu -2({\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-})\delta \alpha (\beta -W^{\mathrm{\prime }}(\alpha ))(4.9)\backslash delta\; g\_W\; =\; -s\backslash delta\; T\; -\; \backslash rho\backslash delta\backslash mu\; -\; 2(\backslash Delta\_+\; -\; \backslash Delta\_-)\backslash delta\backslash alpha(\backslash beta\; -\; W\text{'}(\backslash alpha))\; \backslash quad\; (4.9)$$

Notice that $g_W$ is stationary if $\beta = W'(\alpha)$ (and temperature and chemical potential are fixed). The free energy $g_W$ is what we will be concerned with in this section, the preferred state of the system must have lowest $g_W$ .