1007/1007.4854.md

Gravity/Spin-model correspondence and holographic superfluids

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Umut Gürsoy

Institute for Theoretical Physics, Utrecht University; Leuvenlaan 4, 3584 CE Utrecht, The Netherlands.

ABSTRACT: We propose a general correspondence between gravity and spin models, inspired by the well-known IR equivalence between lattice gauge theories and the spin models. This suggests a connection between continuous type Hawking-phase transitions in gravity and the continuous order-disorder transitions in ferromagnets. The black-hole phase corresponds to the ordered and the graviton gas corresponds to the disordered phases respectively. A simple set-up based on Einstein-dilaton gravity indicates that the vicinity of the phase transition is governed by a linear-dilaton CFT. Employing this CFT we calculate scaling of observables near $T_c$ , and obtain mean-field scaling in a semi-classical approximation. In case of the XY model the Goldstone mode is identified with the zero mode of the NS-NS two-form. We show that the second speed of sound vanishes at the transition also with the mean field exponent.

KEYWORDS: AdS/CFT, gauge theories, black-holes, thermodynamics super-fluids, spin-models.---

Contents

1. Introduction	2
2. Gravity - spin model duality	6
2.1 Correspondence between gauge theories and spin systems	6
2.2 Holographic superfluidity	9
2.3 Spontaneous breaking of $U(1)\_B$, the Goldstone mode and the second speed of sound	10
3. A model based on Einstein-scalar gravity	14
3.1 The model	14
3.2 Scaling of the free energy	16
3.3 The large N limit and string perturbation theory	18
3.4 Parameters of the model	21
4. Non-critical string theory and the IR CFT	22
4.1 Linear-dilaton in the deep interior	22
4.2 The CFT in the IR	23
5. Spin-model observables from strings	26
5.1 What can we learn from the Gravity-Spin model duality?	26
5.2 Identification of observables	29
5.3 One-point function	30
5.3.1 Warm-up: classical computation	31
5.3.2 Semi-classical computation	33
5.4 The two-point function	43
5.4.1 Warm-up: classical computation	44
5.4.2 Semi-classical computation	48
5.5 D-strings and vortices	55
5.5.1 One-point function	56
5.5.2 Two-point function	57
5.6 Vanishing of the second sound	59
6. A proposal for gravity-spin model correspondence in the general case	62
7. Discussion	65
7.1 Summary	65
7.2 Outlook	67

Acknowledgments	70
Appendices	71
A. Simplest example of the LGT-spin equivalence	71
B. Relation between non-critical strings and the linear-dilaton theory	72
C. Some background in statistical mechanics	74
C.1 Landau action	74
C.2 Landau approximation	75
C.3 Mean-field approximation	75
C.4 Gaussian fluctuations	76
C.5 Vanishing of the second sound	79
C.6 BKT theory	79
D. Fundamental string action	80
D.1 The Polyakov loop	81
D.2 The Polyakov loop correlator	83
D.3 The ‘t Hooft loop	87
E. Spectrum of bulk fluctuations	91
E.1 Graviton and dilaton	91
E.2 B-field	92
E.3 Tachyon	92

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1. Introduction

There has been great progress recently in applications of holography [1, 2, 3] to condensed matter systems such as superconductors following the pioneering works of [4] and [5]. These authors managed to find a simple gravitational background in Einstein-Maxwell gravity coupled to a complex scalar field where a second order normal-to-superfluid type transition occurs at finite temperature. The basic interest behind application of holographic ideas to condensed matter theory (CMT) lies in the hope that the strongly correlated condensed matter systems may secretly possess a gravitational description. Indeed, computations of certain observables in the gravity picture, such as conductivity provides supporting evidence, see [6][7][8] for reviews. It is a considerable possibility that [9] the underlying dynamics behind the phase transition in high $T_c$ superconducting materials is a strongly coupled quantum phase transition at zero $T$ . Then the hope is that, a dual gravity description of the stronglycoupled field theory around this critical point may also shed light over the finite temperature transition in the quantum critical region.

On the other hand, there are several issues of fundamental importance in the proposed gravity-CMT models, such as the role of the large $N$ limit and the notion of weak-strong duality, that are not entirely clarified. We have a much better understanding in the holographic constructions of gauge theories, thanks to the basic example [1, 2, 3] of the $\mathcal{N} = 4$ super Yang-Mills theory where the D3 brane picture provide the link between the gauge side and the gravity side. Such a “top-bottom” approach is missing in the gravity-CMT models.

In this work, we entertain the possibility that such a link may be established under certain assumptions, at least for certain simple condensed matter systems, i.e. spin models, by analogy with the better understood gauge-gravity case.

The building blocks of such a connection are already present in the well-known literature. First of all, we recall the famous equivalence between lattice gauge theories (LGT) and spin-models (SpM)[10, 11]: Integrating out the gauge invariant degrees of freedom in the partition function of a LGT with gauge group $G$ , one arrives at an effective action for the lowest lying mode, namely the Polyakov loop $P$ . This effective action is invariant under the leftover center symmetry $\mathcal{C} = \text{Center}(G)$ of the original gauge invariance. Identifying the Polyakov loop $P$ with a spin field $\vec{s}$ , one then obtains the partition function of a spin-model with the global spin invariance $\mathcal{C}$ . Using this equivalence between lattice gauge theories and spin-models Polyakov and Susskind were able to show the existence of confinement-deconfinement phase transition on the lattice, long time ago. Based on these works, than Svetitsky and Yaffe [12] further proposed that, if continuous critical phenomena prevails in the continuum limit of a certain lattice gauge theory, then it should fall in the same universality class as the corresponding spin-model.

It is interesting to employ the same idea in the opposite direction in order to study a spin-model that is strongly coupled at criticality. In particular, one would like to compute the critical exponents, the transition temperature $T_c$ , certain thermodynamic functions etc., by analytic methods. If one is lucky enough to find a gauge-theory that corresponds to the spin-model under the aforementioned equivalence, then one may be able to study the strongly coupled phenomena by the gauge-gravity correspondence.

One purpose of this paper is to emphasize that this chain of dualities may provide a well-defined setting in understanding fundamental issues in the gravity-CMT correspondence. In particular, if one can figure out the relevant D-brane configuration that describes the gauge theory which arises in the continuum limit of the LGT under question, then one may be able to take the decoupling limit and obtain a gravity description of the LGT—and of the equivalent spin model—around criticality. Despite being abstract, in principle this provides a top-bottom approach to the problem. In particular, such an approach would hopefully provide a microscopicdescription that is long sought for in holographic applications to CMT.

Another purpose of this paper is to provide a concrete realization of these ideas in a simple setting. For this purpose we consider $SU(N)$ gauge theory in $d$ -dimensions (with possible adjoint matter) in the strict $N \rightarrow \infty$ limit. In this limit the center $\mathcal{C}$ becomes $U(1)$ ¹. We imagine that the adjoint matter is arranged such that the deconfinement transition of the gauge theory is of continuous type. This transition is then in the same universality class with the order-disorder transition in the corresponding $U(1)$ rotor model in $d - 1$ dimensions—that is sometimes called the XY model. The XY-models—and their $O(n)$ generalizations—provide canonical examples of superfluidity that arises as spontaneous breaking of the global $U(1)$ symmetry in a continuous phase transition.

To realize this phenomenon in the dual gravity setting we consider the NS-NS sector of non-critical string theory in $d + 1$ dimensions with Euclidean time direction $x^0$ compactified. It was shown in [15] that this theory in the two-derivative sector exhibits a continuous Hawking-Page transition at some finite temperature $T_c$ . The background is of the type $AdS^{d+1}$ near the boundary and linear-dilaton in the deep-interior. Building upon the ideas in [13], we argue that the $U(1)$ symmetry (in the strict $N \rightarrow \infty$ limit) corresponds to the shift symmetry $\int_M B \rightarrow \int_M B + \text{const}$ where $B$ is the NS-NS two-form field and $M$ is the $(r, x^0)$ subspace of the background geometry. The only objects that are charged under this symmetry are string states winding the time circle. In the thermal gas phase these states have infinite energy and cannot be excited, hence the symmetry is unbroken and this phase corresponds to the normal phase of the spin system. In the black-hole phase on the other-hand they have finite energy (with an appropriate regularization) and the black-hole corresponds to the superfluid phase.

It was further observed in [15] that the geometry becomes exactly linear-dilaton in the transition region. Therefore, we argue that the transition region of the XY model is governed by the linear-dilaton CFT on the string side. Although in general the $\alpha'$ corrections can not be ignored in the type of backgrounds that we will consider in this paper², one can still perform calculations in the critical regime, precisely because the linear-dilaton background is known to be an $\alpha'$ -exact background in non-critical string theory[16]. In particular the calculations that involve probe strings can be performed by employing the exact CFT description of the linear-dilaton background, (in the limit $g_s \rightarrow 0$ ).

The spin operator $\vec{s}(x)$ is related to a fundamental string that wraps the time-

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¹This idea in the AdS/CFT context was considered before[13], see also [14] and [12] for earlier discussions

²We recall that in the case of $\mathcal{N} = 4$ sYM theory the $\alpha'$ corrections can be ignored both for the bulk and the string computations at strong 't Hooft coupling $\lambda$ . In the theories we consider here we do not have a similar modulus that serves as a parameter to suppress the $\alpha'$ corrections. Generally, the string scale and the scale of the background geometry may be of the same order.circle and connected to the boundary at point $x$ . Consequently one can compute correlation functions of the operator $\vec{s}$ by studying the string propagation in the linear-dilaton CFT in the (single) winding sector. We perform such calculations in a semi-classical limit where we only take into account the contribution of the lowest-lying string states. It is shown that in this approximation, one obtains mean-field scaling near $T_c$ . We find that the “magnetization” behaves as

$$M\sim (T-T_c{)}^{\frac{1}{2}},\text{as, }T\to T_c\mathrm{.}M\; \backslash sim\; (T\; -\; T\_c)^\{\backslash frac\{1\}\{2\}\},\; \backslash quad\; \backslash text\{as,\; \}\; T\; \backslash rightarrow\; T\_c.$$

A similar calculation with string propagation connecting the points $x$ and $y$ on the (spatial) boundary corresponds to the spin two-point function. We show that the expected behavior of the spin-system arises in the large $|x - y|$ limit near $T_c$ indeed arises from this calculation in a non-trivial manner. In particular, in order to show that the correlation length $\xi$ diverges at $T_c$ , one has to identify the transition with the Hagedorn temperature where the lowest lying single-winding mode becomes massless [17]. With such an identification one indeed finds the expected behavior

$$\xi \sim (T-T_c{)}^{-\frac{1}{2}},\text{as, }T\to T_c,\backslash xi\; \backslash sim\; (T\; -\; T\_c)^\{-\backslash frac\{1\}\{2\}\},\; \backslash quad\; \backslash text\{as,\; \}\; T\; \backslash rightarrow\; T\_c,$$

again in a semi-classical approximation.

One can also study scaling of the speed of second sound that is associated with the Goldstone mode in the superfluid phase. This mode is identified with fluctuations of the zero-mode of the NS two-form field $B$ . We find that the speed of sound indeed vanishes at $T_c$ precisely with the expected mean-field scaling,

$$c_s^2\sim (T-T_c),\text{as }T\to T_{c}c\_s^2\; \backslash sim\; (T\; -\; T\_c),\; \backslash quad\; \backslash text\{as\; \}\; T\; \backslash rightarrow\; T\_c$$

in a second order Hawking-Page transition. We also argue that this finding is not altered by possible $\alpha'$ corrections.

The identification of spin operators with the F-strings suggest a similar identification between the vortex configurations—that play an important role in the 2D XY model—with D-strings in the gravity dual. We study correlation functions of such D-string configurations and find that they exhibit the expected behavior in the spin-model.

The paper is organized as follows. In the next section, we review basic ideas in the past literature which indicate a general duality between spin-systems and gravity. We first focus on the case of $SU(N)$ in the $N \rightarrow \infty$ limit and postpone the general discussion to section 6. Section 3 reviews the Einstein-dilaton system that was studied in [15]. In section 4 we argue that the IR limit of the model is described by a linear-dilaton CFT and review basic features of such CFTs. Section 5 contains main technical results of this paper. We first review the basic statistical mechanics results that are relevant in what follows. Then we propose the precise identification betweenthe F-string configurations and the spin correlation functions. We calculate the one-point and two-point functions near criticality in the semi-classical approximation making use of the linear-dilaton CFT. Finally we present calculations related to vortex configurations. In section 6 we take a first step in formulating a gravity spin-model duality in general. In the last section we discuss various issues and possible future directions of research.

Several appendices detail our presentation. In appendix A, we review the simplest example of the equivalence between lattice gauge theories and the spin-systems. In appendix B, we review the connection between non-critical string theory and the linear-dilaton background. Appendix C provides some basic background material in statistical mechanics of the XY models for the unfamiliar reader. Finally, Appendices D and E contain details of our calculations in section 5.

2. Gravity - spin model duality

Our goal in this section is to propose a particular approach to the gravity-CMT correspondence that relates the spin-models in CMT to gravity by a two step procedure: The first step is to employ a well-known equivalence between spin-models and lattice gauge theories [10, 11] followed by a second step that is to utilize the gauge-gravity duality to relate the (continuum limit) of the lattice gauge theory to a dual gravitational background.

2.1 Correspondence between gauge theories and spin systems

Existence of the confinement-deconfinement phase transition in lattice gauge theories at strong coupling is rigorously proved [10, 11] long time ago. The proof is based on an equivalence between lattice gauge theories (LGT) and spin systems with nearest-neighbor ferromagnetic interactions, [10, 11, 12]. In the original papers of Polyakov and Susskind, this equivalence was established for the cases of $U(1)$ and $SU(2)$ gauge theories. Subsequently it was generalized to general Lie groups³. We shall refer to this equivalence as the LGT-spin model equivalence. We review how the spin systems arise from the lattice gauge theories in the Hamiltonian formalism, and in the simplest case of $U(1)$ gauge group in Appendix A.

This equivalence has profound implications in the continuum limit: As argued and verified with various examples by Svetitsky and Yaffe [12], the critical phenomena—if exists—in the continuum limit of the LGT, should be in the same universality class with the corresponding spin model. Therefore, a continuous order-disorder type transition in a $d-1$ dimensional spin-model with global symmetry group

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³See [18] for a recent presentation of how the map works in a general case.$\mathcal{C}$ is directly related to a continuous type confinement-deconfinement transition of the gauge theory with gauge group $G$ where $\mathcal{C} = \text{Center}(G)$ .⁴

Let us briefly review the argument of [12]. The basic observation is that the magnetic fluctuations are always gapped both in the high and the low $T$ limit of the lattice gauge theory. Therefore, they are expected to be gapped for any $T$ on a trajectory crossing the phase boundary in figure 1. This means that the magnetic fluctuations should not play an essential role at criticality in the vicinity of a continuous confinement-deconfinement transition. Integrating these short-range fluctuations, one indeed obtains an effective theory that only involves the Polyakov loops, which in turn can be mapped on a spin model. Therefore the critical phenomena, e.g. the critical exponents etc. of the lattice gauge theory around a continuous transition should be governed by the corresponding spin model.

Figure 1: Typical phase diagram of a lattice gauge theory with non-trivial center. Low $T$ phase is confining with vanishing expectation value for the Polyakov loop $P$ and high $T$ phase is de-confined. We assume that (at least a portion) of the phase boundary that separates these phases is of second or higher order. Then the critical phenomena around the phase boundary is determined by the corresponding spin-model.

The magnetic sector is gapped at low $T$ by assumption. We assume that the (bare) coupling constant is large enough (see figure 1) so that the low $T$ theory is confined. The argument at high $T$ is as follows. In the Lagrangian formulation of the LGT one can take the action to be,

$${\mathcal{A}}_{lgt}=\sum _{\vec{r}}\text{Re}\{{\beta }_t\sum _i\text{Tr}{U}_{\vec{r},0i}+{\beta }_s\sum _{ij}\text{Tr}{U}_{\vec{r},ij}\};U_{\vec{r},\mu \nu }=U_{\vec{r},\mu }{U}_{\vec{r}+\widehat{\mu },\nu }{U}_{\vec{r}+\widehat{\nu },\mu }^{†}{U}_{\vec{r},\nu }^{†}(2.1)\backslash mathcal\{A\}\_\{lgt\}\; =\; \backslash sum\_\{\backslash vec\{r\}\}\; \backslash text\{Re\}\; \backslash left\backslash \{\; \backslash beta\_t\; \backslash sum\_i\; \backslash text\{Tr\}\; U\_\{\backslash vec\{r\},0i\}\; +\; \backslash beta\_s\; \backslash sum\_\{ij\}\; \backslash text\{Tr\}\; U\_\{\backslash vec\{r\},ij\}\; \backslash right\backslash \};\; \backslash quad\; U\_\{\backslash vec\{r\},\backslash mu\backslash nu\}\; =\; U\_\{\backslash vec\{r\},\backslash mu\}\; U\_\{\backslash vec\{r\}+\backslash hat\{\backslash mu\},\backslash nu\}\; U\_\{\backslash vec\{r\}+\backslash hat\{\backslash nu\},\backslash mu\}^\backslash dagger\; U\_\{\backslash vec\{r\},\backslash nu\}^\backslash dagger\; \backslash quad\; (2.1)$$

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⁴Of course, not all of the spin-models exhibit continuous transitions. See [12] for a list of examples.where $\vec{r}$ labels sites on the square lattice, $U_{\vec{r},\mu\nu}$ are the product of link variables on a plaquette with corner $\vec{r}$ . The first term in the action above corresponds to the electric contribution and the second to the magnetic. The electric and magnetic coupling constants are related to the bare coupling constant of the LGT and the temperature as follows:

$$\frac{2}{g^2}=a^{4-d}\sqrt{{\beta }_t{\beta }_s},T=\sqrt{\frac{{\beta }_t}{{\beta }_s}\frac{1}{N_{t}a}}(2.2)\backslash frac\{2\}\{g^2\}\; =\; a^\{4-d\}\; \backslash sqrt\{\backslash beta\_t\; \backslash beta\_s\},\; \backslash quad\; T\; =\; \backslash sqrt\{\backslash frac\{\backslash beta\_t\}\{\backslash beta\_s\}\; \backslash frac\{1\}\{N\_t\; a\}\}\; \backslash quad\; (2.2)$$

where $a$ is the lattice spacing and $N_t$ is the number of the lattice sites in the Euclidean time direction. One observes that, at fixed coupling, $\beta_t \sim T$ . Then, for sufficiently high $T$ , only configurations with vanishing electric flux contributes in the partition function. This reduces the system to static configurations at high $T$ , hence the theory can be thought of a $d - 1$ dimensional LGT at zero $T$ , with a coupling constant $g_{d-1}^2 = g_d^2 T$ . Any such LGT with a non-trivial center is confined and exhibits magnetic screening at strong coupling [12]. As mentioned before, the equivalence to the spin-models can be shown exactly at strong coupling, [10, 11].

Svetitsky and Yaffe [12] were able to make reliable predictions concerning the critical phenomena of a wide range of lattice gauge theories making use of this connection and the well-known results on the critical phenomena of the corresponding spin-models.

First of all, they correctly predicted that the 2nd order transition in the pure $SU(2)$ theory in 4D is in the same universality class with the 3D Ising model (see e.g. [19] and references therein). As another check of these arguments [12] presents the example of $SU(N)$ theory for $d - 1 = 2$ , $N > 4$ where the dual spin model is again $Z_N$ symmetric and exhibit a BKT type continuous transition. In this case, it was argued that for large $N$ , the theory approximates that of a $U(1)$ LGT in 2+1 and the corresponding spin-model should be the XY-model in 2D. It was explicitly checked in [12] that, for the $U(1)$ LGT the critical phenomena is in the same universality class as that of the 2D XY model. More generally, if the $d = 2 + 1$ $SU(N)$ gauge theory—or a suitable deformation with additional adjoint matter—involves continuous critical phenomena than it should be the BKT type.

A particularly interesting case concerns $SU(N)$ gauge theory in $d - 1 > 2$ spatial dimensions with $N > 4$ (that includes the large $N$ ) where the dual spin model is $Z_N$ symmetric. We then consider the large $N$ limit that is most relevant for the gauge-gravity duality. It is reasonable to believe that in the strict $N \rightarrow \infty$ limit (with or without adjoint matter), the center $Z_N$ is promoted to $U(1)$ . See [13] for an argument in favor of this, in the case of $\mathcal{N} = 4$ SYM at strong-coupling⁵. Another indication that this happens in $Z_N$ invariant LGT at $d = 2$ is explained in [12]. Therefore,

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⁵See however [20] which shows that the $U(1)$ symmetry is expected to arise only in the strict $N \rightarrow \infty$ limit.by the universality arguments above, if there exist a continuous phase transition it should be governed by a $U(1)$ invariant spin model ⁶.

We review how the equivalence of LGTs and spin-systems work at strong coupling in Appendix A for the unfamiliar reader. Here we shall mention two salient features.

• • The temperature of the spin-system is inversely related to temperature in the original gauge theory:

$$T_s\sim T_l^{-1}\mathrm{.}(2.3)T\_s\; \backslash sim\; T\_l^\{-1\}.\; \backslash quad\; (2.3)$$

Consequently, the low temperature (confined) phase of the gauge theory corresponds to the high temperature (disordered) phase of the ferromagnet, whereas the high temperature (de-confined) phase of the gauge theory corresponds to the low temperature (ordered) phase of the ferromagnet. ⁷

• • Quite generally, the LGT-spin model equivalence can be generalized to incorporate (adjoint) matter. This is mainly because the basic ingredient in the calculation i.e. the center symmetry of the LGT remains intact upon addition of adjoint matter. See [14] for a related recent discussion.

2.2 Holographic superfluidity

Here and until section 6, we specify to the particular case of $U(1)$ invariant spin-models. Continuous critical phenomena in such models include the interesting case of superfluidity, that requires spontaneous breaking of the global $U(1)$ symmetry. As reviewed above this transition is directly connected to the confinement-deconfinement transition in the gauge theory. In the original derivation of [10][11], the de-confined phase of the $U(1)$ invariant LGT was understood as an ordered phase of the $U(1)$ spin model. This is clear from the discussion of section 2, as the center of $U(1)$ is $U(1)$ itself.

Instead, here we shall adopt an alternative approach where the $U(1)$ factor arises from the large $N$ limit of an $SU(N)$ gauge theory (pure or with adjoint matter). In

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⁶One can ask whether there is any evidence, for or against criticality at $N = \infty$ . There are two independent arguments that argue for a second order transition [21][22] in the case of pure YM in 3+1 dimensions. On the other hand, there is the usual argument against a continuous transition at large $N$ that claims, since the number of degrees of freedom in the system changes from $\mathcal{O}(1)$ to $\mathcal{O}(N^2)$ in a confinement-deconfinement transition, latent heat should be finite. In [15] we presented a counter-example to this reasoning, albeit in a gravitational setting: although the degrees of freedom change abruptly as the graviton gas deconfines in the black-hole phase, the entropy difference may vanish at the transition. See [14] for other examples of second order transitions at large $N$ . Finally, even if the transition is first-order for pure YM, the situation may change when one adds adjoint matter.

⁷In $d = 2$ , IR divergence of the spin waves prevent ordinary long range order. Instead, a topological long-range order in terms of the vortex-anti-vortex pairs arises [23][24]. The gauge theory partition function is capable of describing the vortex configurations [12].this case the deconfinement transition can be understood in the gravity dual as a Hawking-Page transition by a generalization of the arguments in [13]. Assuming that the following assumptions hold,

• • There exists a suitable $SU(N)$ lattice gauge theory with coupling to adjoint matter chosen such that, at large $N$ it flows to an IR fixed point with a continuous confinement-deconfinement transition,
• • Gauge-gravity correspondence holds and maps this to a Hawking-Page type transition,

then one should be able to map the normal-to-superfluid transition in the XY model to a continuous Hawking-Page type transition on the gravity side.

The $U(1)$ symmetry of the spin-model follows on the GR side from the shift symmetry $\psi \rightarrow \psi + \text{const.}$ where $\psi$ is the flux of the B-field

$$\psi ={\int }_{M}B=\text{const.}(2.4)\backslash psi\; =\; \backslash int\_M\; B\; =\; \backslash text\{const.\}\; \backslash quad\; (2.4)$$

on the subspace $M$ of the BH geometry that is spanned by the coordinates $r$ and $x_0$ . In this paper we consider gravitational set-ups where the B-field is either constant or pure gauge $B = d\xi$ so that it does not back-react on the solution with $H = dB = 0$ . Of course such a B-field has no visible effect on the gravitational solution and in the second case it can be removed by a gauge transformation. This ceases to be the case in presence of objects that are charged under this shift symmetry.

In the classical approximation where one keeps only the low-lying gravity fields, there are no bulk fields that carry the extra $B$ charge. However, strings that wind around the time-circle couple to the B-field through the term $i\psi$ , thus they are charged under the shift symmetry with the identification $\psi \sim \psi + 2\pi$ . We shall denote this topological $U(1)$ symmetry as $U(1)_B$ ⁸ Therefore a non-vanishing string one-point function signals a breakdown of the $U(1)_B$ symmetry. On the spin-model side this corresponds to an order-disorder transition upon identification of the $U(1)_B$ symmetry with the $U(1)$ spin symmetry of the spin-model. Below, we would like to review these ideas in more detail.

2.3 Spontaneous breaking of $U(1)_B$ , the Goldstone mode and the second speed of sound

For simplicity, let us consider the (critical or non-critical) bosonic string theory on a background with $U(1) \times E(d-1)$ isometry where the $U(1)$ corresponds to the

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⁸This symmetry should be broken down to $Z_N$ for finite $N$ by quantum effects, see [20]. However we only consider the $N \rightarrow \infty$ limit in this paper.temporal $S^1$ , and $E(d-1)$ to translations and rotations on the spatial part. The general background with these symmetries is of the form⁹

$$d{s}^2=A(r,\mathrm{\Omega })d{x}_0^2+B(r,\mathrm{\Omega })d{K}^2+C(r,\mathrm{\Omega })d{r}^2+D(r,\mathrm{\Omega })d\mathrm{\Omega };\bar{\mathrm{\Phi }}=\bar{\mathrm{\Phi }}(r,\mathrm{\Omega })H=dB=0,(2.5)ds^2\; =\; A(r,\; \backslash Omega)dx\_0^2\; +\; B(r,\; \backslash Omega)dK^2\; +\; C(r,\; \backslash Omega)dr^2\; +\; D(r,\; \backslash Omega)d\backslash Omega;\; \backslash quad\; \backslash bar\{\backslash Phi\}\; =\; \backslash bar\{\backslash Phi\}(r,\; \backslash Omega)\; \backslash quad\; H\; =\; dB\; =\; 0,\; \backslash quad\; (2.5)$$

where $x_0 \sim x_0 + 1/T$ , $K$ is the $d-1$ dimensional transverse part, and $\Omega$ is some internal compact manifold. There can be additional bulk fields but we are only interested in the NS-NS sector.

Most of the following traces the arguments in [13]. The order parameter for the transition is the vev of the Polyakov loop, $\langle P[C] \rangle$ , where $C$ is a loop isomorphic to the time-circle. This maps to the expectation value of the F-string path integral,

$$⟨P[C]⟩\propto ⟨{\mathcal{W}}_F{⟩}_{SG}(2.6)\backslash langle\; P[C]\; \backslash rangle\; \backslash propto\; \backslash langle\; \backslash mathcal\{W\}\_F\; \backslash rangle\_\{SG\}\; \backslash quad\; (2.6)$$

where $\mathcal{W}_F$ denotes the F-string path integral over all of the string configurations with the boundary ending on $C$ , and the final averaging is path integral over the super-gravity fields that couple to the string. The string path integral is

$${\mathcal{W}}_F=\int \mathcal{D}{X}_{\mu }\mathcal{D}{h}_{ab}{e}^{-\int (G+iB+\bar{\mathrm{\Phi }}{R}^{(2)})},(2.7)\backslash mathcal\{W\}\_F\; =\; \backslash int\; \backslash mathcal\{D\}X\_\backslash mu\; \backslash mathcal\{D\}h\_\{ab\}\; e^\{-\backslash int\; (G\; +\; iB\; +\; \backslash bar\{\backslash Phi\}R^\{(2)\})\},\; \backslash quad\; (2.7)$$

where $R^{(2)}$ is the Ricci scalar on the sub-manifold $M$ that the F-string wraps and $X_\mu$ denotes the matter fields. We also use the short-hand notation

$$G\equiv \sqrt{\det h_{ab}}{h}^{ab}{\mathrm{\partial }}_a{X}^{\mu }{\mathrm{\partial }}_b{X}^{\nu }{G}_{\mu \nu },B\equiv \sqrt{\det h_{ab}}{ϵ}^{ab}{\mathrm{\partial }}_a{X}^{\mu }{\mathrm{\partial }}_b{X}^{\nu }{B}_{\mu \nu }\mathrm{.}(2.8)G\; \backslash equiv\; \backslash sqrt\{\backslash det\; h\_\{ab\}\}\; h^\{ab\}\; \backslash partial\_a\; X^\backslash mu\; \backslash partial\_b\; X^\backslash nu\; G\_\{\backslash mu\backslash nu\},\; \backslash quad\; B\; \backslash equiv\; \backslash sqrt\{\backslash det\; h\_\{ab\}\}\; \backslash epsilon^\{ab\}\; \backslash partial\_a\; X^\backslash mu\; \backslash partial\_b\; X^\backslash nu\; B\_\{\backslash mu\backslash nu\}.\; \backslash quad\; (2.8)$$

One has to make sure that $\mathcal{W}$ is finite by an appropriate regularization of infinite volume of the space time¹⁰ and factoring out diffeo-Weyl gauge volume a la Faddeev-Popov.

In the original discussion of [13] $\mathcal{W}$ is dominated by the classical saddles that minimize the action in (2.7). The boundary condition for these classical strings is such that at $\tau = 0$ , $X^\mu(\sigma, \tau)$ ends on the temporal circle $x_0$ , some point $x$ in $K$ and at the cut-off of the radial coordinate $r = \epsilon$ .

The string path integral is dominated by classical saddles when $\ell/\ell_s \gg 1$ where $\ell$ is the typical curvature of the target space and $\ell_s$ is the string length. In the original AdS/CFT correspondence this ratio is proportional to the 't Hooft coupling of the dual $\mathcal{N} = 4$ SYM theory, $\ell/\ell_s \propto \lambda^{\frac{1}{4}}$ and indeed the classical strings dominate in the limit of strong interactions. In the general case here one has to consider the full path integral.

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⁹In order to distinguish the dilaton and the scalar field that appears in the Einstein-frame potential, which is related to the dilaton by some rescaling, we denote the former (dilaton itself) by $\bar{\Phi}$ and the latter (rescaled dilaton) by $\Phi$ .

¹⁰A cut-off in $r$ that we call $\epsilon$ close to the boundary would suffice for the sake of the discussion here. We elaborate on this regularization in appendix D.1.The vev of $P[C]$ is given by the path integral of $\mathcal{W}F$ over the super-gravity fields that couple the F-string, weighted by the SG action. The non-trivial SG fields are the space-time metric $G{\mu\nu}$ , the B-field $B_{\mu\nu}$ and the dilaton $\bar{\Phi}$ . Thus one has,

$$⟨P[C]⟩\propto \int \mathcal{D}{G}_{\mu \nu }\mathcal{D}{B}_{\mu \nu }\mathcal{D}\bar{\mathrm{\Phi }}{e}^{-{\mathcal{A}}_{sg}}{\mathcal{W}}_F,(2.9)\backslash langle\; P[C]\; \backslash rangle\; \backslash propto\; \backslash int\; \backslash mathcal\{D\}G\_\{\backslash mu\backslash nu\}\; \backslash mathcal\{D\}B\_\{\backslash mu\backslash nu\}\; \backslash mathcal\{D\}\backslash bar\{\backslash Phi\}\; e^\{-\backslash mathcal\{A\}\_\{sg\}\}\; \backslash mathcal\{W\}\_F,\; \backslash quad\; (2.9)$$

where $\mathcal{A}{sg}$ is the gravity action. As we are interested in the large $N$ limit of the dual field theory, we can send the string coupling $g_s \rightarrow 0$ and the SG path integral is dominated by the classical saddles of $\mathcal{A}{sg}$ . For given asymptotic boundary conditions of $G$ , $\bar{\Phi}$ and $B$ , the saddles of interest involve only two type of solutions, the thermal gas (TG) and the black-hole (BH). At an arbitrary temperature $T$ (that partially determines the asymptotic boundary condition for $G$ ), only one of these saddles will dominate the SG path integral as a result of the classical limit $g_s \rightarrow 0$ .

Let us assume that TG dominates at $T < T_c$ and BH dominates at $T > T_c$ . Let us also assume that the BH solution only exists above a certain temperature $T_{min}$ . Backgrounds that exhibit confinement generically satisfy $T_c \geq T_{min}$ [25, 26]. As explained in detail in [15] and reviewed in the next section, only in the case $T_{min} = T_c$ the transition is second or higher order.

On the TG phase, the classical world-sheet $M$ has infinite area. Therefore the string path-integral $\mathcal{W}_F$ , hence $\langle P[C] \rangle$ in (2.9) vanishes. One concludes that the TG solution is $U(1)_B$ symmetric and the center in the dual gauge theory is unbroken. This means that the dual spin-model is in the normal (disordered) phase. This is precisely as one expects from the behavior of the dual spin model in the high temperature phase, recalling that the temperature of the spin model is inversely proportional to the temperature on the gravity side $T_s \propto T^{-1}$ .

On the BH solution $T > T_{min}$ however, the classical string saddle $M$ has finite area and one has to evaluate (2.9) carefully. One has to include all of the configurations over the classical fields $G, B$ and $\bar{\Phi}$ with the same on-shell value of the SG action.

The path integral over $G$ and $\bar{\Phi}$ in (2.9) is replaced by the classical solution (2.5) that is a BH in this case. Sum over these saddles include the following important contribution from the B-field. In the black-hole case the sub-manifold $M$ has finite area¹¹ and the B-field has a flux $\psi = \int_M B$ . $\psi$ in (2.7) has angular nature because it appears with a factor of $i$ and it can attain any value in the range $\psi \sim \psi + 2\pi$ . This identification yields the $U(1)_B$ invariance¹². The sum over classical saddles then should include various different values of $\psi$ . As $dB = 0$ all different values of $\psi$ yield the same on-shell gravity action.

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¹¹The divergence near boundary is regularized in the familiar way, cf. appendix D.1.

¹²In the critical IIB theory this identification arises as a result of discrete gauge transformations that shift the value of $\psi$ by a multiple of $2\pi$ [13].We can thus write,

$$P[C]\propto \int \mathcal{D}\psi e^{-S_{sg}[\psi ]}\int \mathcal{D}{X}_{\mu }\mathcal{D}{h}_{ab}{e}^{i\psi }{e}^{-{\int }_M(G+\bar{\mathrm{\Phi }}{R}^{(2)})},(2.10)P[C]\; \backslash propto\; \backslash int\; \backslash mathcal\{D\}\backslash psi\; e^\{-S\_\{sg\}[\backslash psi]\}\; \backslash int\; \backslash mathcal\{D\}X\_\backslash mu\; \backslash mathcal\{D\}h\_\{ab\}\; e^\{i\backslash psi\}\; e^\{-\backslash int\_M\; (G\; +\; \backslash bar\{\backslash Phi\}\; R^\{(2)\})\},\; \backslash quad\; (2.10)$$

where $S_{sg}$ now is evaluated on the saddle solution and is only a functional of $\psi$ . On the other hand the $\psi$ path integral includes the classical saddle $\psi = \text{const}$ and the fluctuations $\delta\psi(K)$ around it.

When $K$ is non-compact and $\dim(K) > 2$ , then the fluctuations $\delta\psi(K)$ viewed as a massless bosonic field on $K$ has long-range order, hence $\psi$ should condense¹³. Thus, on the black-hole solution the $U(1)B$ symmetry breaks down¹⁴. This happens exactly at the point where the black-hole forms, right above $T{min}$ . As a result, the fluctuation $\delta\psi$ in (2.10) becomes a Goldstone mode on the transverse space $K$ .

Considering the wave equation for $\delta\psi$ one expects to find,

$${\omega }^2=c_{\psi }^2(T){\mathbf{q}}^2+\mathcal{O}({\mathbf{q}}^4),(2.11)\backslash omega^2\; =\; c\_\backslash psi^2(T)\; \backslash mathbf\{q\}^2\; +\; \backslash mathcal\{O\}(\backslash mathbf\{q\}^4),\; \backslash quad\; (2.11)$$

where $c_\psi$ is the speed of sound of $\psi$ and there is no mass term for $\psi$ for $T > T_c$ . It is well-known that (see appendix C for a review, and section 5.6 for a holographic derivation in gravity), the speed of sound $c_\psi$ of the Goldstone mode vanishes continuously as one approaches the transition temperature $T_c$ from above, only if the transition is of continuous type. This is exactly what happens in super-fluidity, where the “second speed of sound”, i.e. the speed of sound associated with the entropy waves vanish as one approaches $T_c$ of the XY model from below (recall that temperature in gravity and in the XY model are inversely related). In order to mimic this property of the spin model, we should require that the Hawking-Page transition in gravity is of continuous type, hence $T_c = T_{min}$ [15]. In section 5.6 we show by an explicit gravity calculation that indeed the second sound vanishes with the expected mean-field exponents.

Our conclusion is: whenever a second order (or higher order) Hawking phase transition occurs in the gravitational background, it is natural to associate it with super-fluidity. Here the thermal gas phase is dual to the normal phase of the system, and the black-hole phase is dual to the super-fluid. The “first speed of sound” i.e. the sound of the density waves is associated with the graviton fluctuations (that we are considered in [15]), and the “second speed of sound” is associated with the fluctuations of the B-field that we consider in section 5.6.

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¹³The situation at $\dim(K) = 2$ exactly parallels the analogous situation in the 2D dual field theory, where IR divergences kill long-range order.

¹⁴As a technical aside, in the computation above, one should check that the dilaton term in the action does not spoil the arguments. In the particular case of the geometries considered in this paper, $\bar{\Phi}$ diverges in the deep interior, hence this check especially becomes important. We check in appendix D.1 that this term indeed remains finite in our case.

	Lattice gauge theory	Gravity	Spin model
$T\; \backslash uparrow$	Deconfined, $U(1)\_c$	Black-hole, $U(1)\_B$	Superfluid, $U(1)\_S$	$T\; \backslash downarrow$
$T\_c$	Confined, $U(1)\_c$	Thermal gas, $U(1)\_B$	Normal phase, $U(1)\_S$	$T\_c^\{-1\}$

One can summarize the various phases of the theories by the table above. The various $U(1)$ factors in this table are as follows: The $U(1)_B$ is the dual symmetry that arises from compactifying the B field on the temporal circle. The $U(1)_c$ is the center symmetry of the corresponding lattice gauge theory that is proposed to arise in the large N limit of $SU(N)$ (with or without) adjoint matter. Finally the $U(1)_S$ is the spin symmetry of the corresponding XY model. The arrow of increasing T is the same for the LGT and gravity picture and opposite in the spin model picture.

3. A model based on Einstein-scalar gravity

The arguments put forward in favor of a gravity-spin model correspondence above are general. In this section we would like to introduce a simple set-up which allows for computations of quantities such as the scaling of magnetization and spin-spin correlation function on the gravity side. The model is inspired by non-critical string theory and it becomes precisely non-critical string theory in the interesting regime near the continuous phase transition.

3.1 The model

The action in the Einstein frame reads,

$$\mathcal{A}=\frac{1}{16\pi G_N}\int d^{d+1}x\sqrt{-g}(R-\frac{4}{d-1}(\mathrm{\partial }\mathrm{\Phi }{)}^2+V(\mathrm{\Phi })-\frac{1}{12}{e}^{-\frac{8}{d-1}\mathrm{\Phi }}{H}^2+\dots )+G\mathrm{.}H\mathrm{.}(3.1)\backslash mathcal\{A\}\; =\; \backslash frac\{1\}\{16\backslash pi\; G\_N\}\; \backslash int\; d^\{d+1\}x\; \backslash sqrt\{-g\}\; \backslash left(\; R\; -\; \backslash frac\{4\}\{d-1\}\; (\backslash partial\backslash Phi)^2\; +\; V(\backslash Phi)\; -\; \backslash frac\{1\}\{12\}\; e^\{-\backslash frac\{8\}\{d-1\}\backslash Phi\}\; H^2\; +\; \backslash dots\; \backslash right)\; +\; G.H.\; \backslash quad\; (3.1)$$

where the kinetic terms of the dilaton¹⁵ and the B-field $H = dB$ are inspired by non-critical string theory in $d+1$ dimensions. The ellipsis denote higher derivative corrections. The last term in (3.1), that we shall not need to specify here, is the Gibbons-Hawking term on the boundary.

We allow for a non-trivial dilaton potential $V(\Phi)$ that should be specified by matching the thermodynamics of the dual field theory. In the case of non-critical string theory in $d+1$ dimensions the potential is given by,

$$V_{nc}(\mathrm{\Phi })=\frac{\delta c}{{\mathrm{\ell }}_s^2}{e}^{\frac{4}{d-1}\mathrm{\Phi }},(3.2)V\_\{nc\}(\backslash Phi)\; =\; \backslash frac\{\backslash delta\; c\}\{\backslash ell\_s^2\}\; e^\{\backslash frac\{4\}\{d-1\}\backslash Phi\},\; \backslash quad\; (3.2)$$

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¹⁵The scalar field $\Phi$ here is related to the original dilaton of the non-critical string $\bar{\Phi}$ by some rescaling that is defined in section 3.3. By $\Phi$ we will always mean the “rescaled dilaton” throughout the paper.where $\ell_s$ is the string length and $\delta c$ is the central deficit, see section 3.3 for more detail. $G_N$ in (3.1) is the Newton's constant in $D = d + 1$ dimensions. It is related to $N$ of the dual field theory¹⁶ by,

$$\frac{1}{16\pi G_N}=M_p^{d-1}{N}^2,(3.3)\backslash frac\{1\}\{16\backslash pi\; G\_N\}\; =\; M\_p^\{d-1\}\; N^2,\; \backslash quad\; (3.3)$$

where $M_p$ is a “normalized” Planck scale, that is generally of the same order as the typical curvature of the background $\ell$ . The limit of large $N$ corresponds to classical gravity as usual. One should be careful in attaining this classical limit: The correct way of achieving this is described in section 3.3. On the gravity side the parameter $N$ arises from the RR-sector, where it is the integration constant of a space-filling $F_{(d+1)}$ form, $F_{(d+1)} \propto N$ . Then the large $N$ limit is defined as sending this value to infinity and sending the boundary value of the dilaton $\Phi_0$ to $-\infty$ such that $N \exp(\Phi_0)$ remains constant and yields $M_p$ in (3.3). We refer to section 3.3 for details.

In what follows we shall only consider solutions with either constant or pure-gauge $B$ -field whose legs are taken to lie along $r$ and $x_0$ directions:

$$B_{\mu \nu }=B_{r0},(3.4)B\_\{\backslash mu\backslash nu\}\; =\; B\_\{r0\},\; \backslash quad\; (3.4)$$

In this case $H = 0$ in (3.1) and the $B$ -field contributes to neither the equations of motion nor the on-shell value of the action. However, it contributes the F-string and D-string solutions as we study in section 5.

There are only two types of backgrounds at finite $T$ (with Euclidean time compactified), with Poincaré symmetries in $d - 1$ spatial dimensions, and an additional $U(1)$ symmetry in the Euclidean time direction. These are the thermal graviton gas,

$$d{s}^2=e^{2A_0(r)}(d{r}^2+d{x}_{d-1}^2+d{x}_0^2),\mathrm{\Phi }={\mathrm{\Phi }}_0(r),(3.5)ds^2\; =\; e^\{2A\_0(r)\}\; (dr^2\; +\; dx\_\{d-1\}^2\; +\; dx\_0^2),\; \backslash quad\; \backslash Phi\; =\; \backslash Phi\_0(r),\; \backslash quad\; (3.5)$$

and the black-hole,

$$d{s}^2=e^{2A(r)}(f^{-1}(r)d{r}^2+d{x}_{d-1}^2+d{x}_0^{2}f(r)),\mathrm{\Phi }=\mathrm{\Phi }(r)\mathrm{.}(3.6)ds^2\; =\; e^\{2A(r)\}\; (f^\{-1\}(r)dr^2\; +\; dx\_\{d-1\}^2\; +\; dx\_0^2\; f(r)),\; \backslash quad\; \backslash Phi\; =\; \backslash Phi(r).\; \backslash quad\; (3.6)$$

We define the coordinate system such that the boundary is located at $r = 0$ . For the potentials $V$ that we consider in this paper, there is a curvature singularity in the deep interior, at $r = r_s$ . In (3.5), $r$ runs up to singularity $r_s$ . In (3.6) there is a horizon that cloaks this singularity at $r_h < r_s$ where $f(r_h) = 0$ . $x_0$ is the Euclidean time that is identified as $x_0 \sim x_0 + 1/T$ . This defines the temperature $T$ of the associated thermodynamics. In the black-hole solution, the relation between the temperature and $r_h$ is obtained in the standard way, by demanding absence of a conical singularity at the horizon:

$$4\pi T=-f^{\mathrm{\prime }}(r_h)\mathrm{.}(3.7)4\backslash pi\; T\; =\; -f\text{'}(r\_h).\; \backslash quad\; (3.7)$$

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¹⁶As explained above, $N$ may either be the number of colors in $SU(N)$ gauge theory or the number of spin states at each site in a $Z_N$ spin-model.This identifies $T$ and the surface gravity in the BH solution.

In the $r$ -frame defined by (3.5) and (3.6) one derives the following Einstein and scalar equations of motion from (3.1):

$$A^{\mathrm{\prime }\mathrm{\prime }}-A^{\mathrm{\prime }2}+\frac{\xi }{d-1}{\mathrm{\Phi }}^{\mathrm{\prime }2}=0,(3.8)A\text{'}\text{'}\; -\; A\text{'}^2\; +\; \backslash frac\{\backslash xi\}\{d-1\}\; \backslash Phi\text{'}^2\; =\; 0,\; \backslash quad\; (3.8)$$

$$f^{\mathrm{\prime }\mathrm{\prime }}+(d-1)A^{\mathrm{\prime }}{f}^{\mathrm{\prime }}=0,(3.9)f\text{'}\text{'}\; +\; (d-1)A\text{'}f\text{'}\; =\; 0,\; \backslash quad\; (3.9)$$

$$(d-1)A^{\mathrm{\prime }2}+A^{\mathrm{\prime }}{f}^{\mathrm{\prime }}+A^{\mathrm{\prime }\mathrm{\prime }}f-\frac{V}{d-1}{e}^{2A}=0.(3.10)(d-1)A\text{'}^2\; +\; A\text{'}f\text{'}\; +\; A\text{'}\text{'}f\; -\; \backslash frac\{V\}\{d-1\}e^\{2A\}\; =\; 0.\; \backslash quad\; (3.10)$$

One easily solves (3.9) to obtain the “blackness function” $f(r)$ in terms of the scale factor as,

$$f(r)=1-\frac{{\int }_0^r{e}^{-(d-1)A}dr}{{\int }_0^{r_h}{e}^{-(d-1)A}dr}\mathrm{.}(3.11)f(r)\; =\; 1\; -\; \backslash frac\{\backslash int\_0^r\; e^\{-(d-1)A\}\; dr\}\{\backslash int\_0^\{r\_h\}\; e^\{-(d-1)A\}\; dr\}.\; \backslash quad\; (3.11)$$

Then the temperature of the BH is given by eq. (3.7):

$$T^{-1}=4\pi e^{(d-1)A(r_h)}{\int }_0^{r_h}{e}^{-(d-1)A(r)}dr\mathrm{.}(3.12)T^\{-1\}\; =\; 4\backslash pi\; e^\{(d-1)A(r\_h)\}\; \backslash int\_0^\{r\_h\}\; e^\{-(d-1)A(r)\}\; dr.\; \backslash quad\; (3.12)$$

The difference between the entropy densities of the BH and the TG solutions is given by the BH entropy density up to $1/N^2$ corrections¹⁷ that we ignore from now on [15]:

$$\mathrm{\Delta }S=\frac{1}{4G_N{N}^2}{e}^{(d-1)A(r_h)}\mathrm{.}(3.13)\backslash Delta\; S\; =\; \backslash frac\{1\}\{4G\_N\; N^2\}\; e^\{(d-1)A(r\_h)\}.\; \backslash quad\; (3.13)$$

The difference in the free energy densities can be evaluated by integrating the first law of thermodynamics, [26]:

$$\mathrm{\Delta }F(r_h)=-\frac{1}{4G_N{N}^2}{\int }_{r_c}^{r_h}{e}^{(d-1)A({\tilde{r}}_h)}\frac{dT}{d{\tilde{r}}_h}d{\tilde{r}}_h,(3.14)\backslash Delta\; F(r\_h)\; =\; -\backslash frac\{1\}\{4G\_N\; N^2\}\; \backslash int\_\{r\_c\}^\{r\_h\}\; e^\{(d-1)A(\backslash tilde\{r\}\_h)\}\; \backslash frac\{dT\}\{d\backslash tilde\{r\}\_h\}\; d\backslash tilde\{r\}\_h,\; \backslash quad\; (3.14)$$

where $r_c$ is the value of the horizon size that corresponds to the phase transition temperature $T(r_c) = T_c$ , at which the difference in free energies should vanish.

3.2 Scaling of the free energy

In [15] we showed that there exists a continuous type Hawking-Page transition between the TG and the BH solutions when the black-hole horizon marginally traps a curvature singularity: $r_h = r_c \rightarrow \infty$ . This happens only when the IR asymptotics of the dilaton potential is chosen such that,

$$V(\mathrm{\Phi })\to V_{\mathrm{\infty }}{e}^{\frac{4}{d-1}\mathrm{\Phi }}(1+V_{sub}(\mathrm{\Phi })),\mathrm{\Phi }\to \mathrm{\infty }(3.15)V(\backslash Phi)\; \backslash rightarrow\; V\_\backslash infty\; e^\{\backslash frac\{4\}\{d-1\}\backslash Phi\}\; (1\; +\; V\_\{sub\}(\backslash Phi)),\; \backslash quad\; \backslash Phi\; \backslash rightarrow\; \backslash infty\; \backslash quad\; (3.15)$$

where $V_\infty$ is a constant and $V_{sub}$ denote subleading corrections that vanish as $\Phi \rightarrow \infty$ . It is also shown in [15] that the transition temperature $T_c$ that follows from (3.12) with $r_h \rightarrow \infty$ stays finite.

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¹⁷We choose to normalize the thermodynamic quantities by an extra factor of $1/N^2$ so that the entropy on the BH becomes $\mathcal{O}(1)$ and on the TG it becomes $\mathcal{O}(1/N^2)$ .Given the asymptotics in (3.15) one solves the equations of motion (3.8) and (3.8) to obtain the IR behavior, as $r \rightarrow \infty$ ,

$$\mathrm{\Phi }(r)\to \frac{\sqrt{V_{\mathrm{\infty }}}}{2}r+\dots (3.16)\backslash Phi(r)\; \backslash rightarrow\; \backslash frac\{\backslash sqrt\{V\_\backslash infty\}\}\{2\}\; r\; +\; \backslash dots\; \backslash quad\; (3.16)$$

$$A(r)\to -\frac{\sqrt{V_{\mathrm{\infty }}}}{d-1}r+\dots (3.17)A(r)\; \backslash rightarrow\; -\backslash frac\{\backslash sqrt\{V\_\backslash infty\}\}\{d-1\}\; r\; +\; \backslash dots\; \backslash quad\; (3.17)$$

where the subleading terms vanish in the limit.

Depending on $V_{sub}$ there are various different possibilities for types of transitions. We consider only two classes of potentials with:

$$\text{Case i : }{V}_{sub}=C{e}^{-\kappa \mathrm{\Phi }},\kappa >0,\mathrm{\Phi }\to \mathrm{\infty }(3.18)\backslash text\{Case\; i\; :\; \}\; V\_\{sub\}\; =\; C\; e^\{-\backslash kappa\backslash Phi\},\; \backslash quad\; \backslash kappa\; >\; 0,\; \backslash quad\; \backslash Phi\; \backslash rightarrow\; \backslash infty\; \backslash quad\; (3.18)$$

$$\text{Case ii : }{V}_{sub}=C{\mathrm{\Phi }}^{-\alpha },\alpha >0,\mathrm{\Phi }\to \mathrm{\infty }(3.19)\backslash text\{Case\; ii\; :\; \}\; V\_\{sub\}\; =\; C\; \backslash Phi^\{-\backslash alpha\},\; \backslash quad\; \backslash alpha\; >\; 0,\; \backslash quad\; \backslash Phi\; \backslash rightarrow\; \backslash infty\; \backslash quad\; (3.19)$$

Defining the normalized temperature,

$$t=\frac{T-T_c}{T_c},(3.20)t\; =\; \backslash frac\{T\; -\; T\_c\}\{T\_c\},\; \backslash quad\; (3.20)$$

the scaling of thermodynamic functions with $t$ can be found from the following set of formulae: The reduced temperature directly follows from the subleading term in the potential,

$$t=V_{sub}({\mathrm{\Phi }}_h),(3.21)t\; =\; V\_\{sub\}(\backslash Phi\_h),\; \backslash quad\; (3.21)$$

where $\Phi_h$ is the value of the dilaton at the horizon. Then the free-energy as a function of $t$ follows from by (3.14) as,

$$\mathrm{\Delta }F(t)\propto {\int }_0^{t}d\tilde{t}{e}^{(d-1)A(\tilde{t})}\mathrm{.}(3.22)\backslash Delta\; F(t)\; \backslash propto\; \backslash int\_0^t\; d\backslash tilde\{t\}\; e^\{(d-1)A(\backslash tilde\{t\})\}.\; \backslash quad\; (3.22)$$

Here, the dependence of the scale factor on $t$ should be found by inverting (3.21), and comparing the (leading term) asymptotics of the scale factor $A(r)$ with the dilaton $\Phi(r)$ [15]. In the cases (3.18) and (3.19) one finds that,

$$\text{Case i : }A(t)=\frac{2}{\kappa (d-1)}\log (t\mathrm{/}C)+\dots ,t\to 0^{+}(3.23)\backslash text\{Case\; i\; :\; \}\; A(t)\; =\; \backslash frac\{2\}\{\backslash kappa(d-1)\}\; \backslash log(t/C)\; +\; \backslash dots,\; \backslash quad\; t\; \backslash rightarrow\; 0^+\; \backslash quad\; (3.23)$$

$$\text{Case ii : }A(t)=-\frac{2}{\kappa (d-1)}(t\mathrm{/}C{)}^{-\frac{1}{\alpha }}+\dots ,t\to 0^{+}\mathrm{.}(3.24)\backslash text\{Case\; ii\; :\; \}\; A(t)\; =\; -\backslash frac\{2\}\{\backslash kappa(d-1)\}\; (t/C)^\{-\backslash frac\{1\}\{\backslash alpha\}\}\; +\; \backslash dots,\; \backslash quad\; t\; \backslash rightarrow\; 0^+.\; \backslash quad\; (3.24)$$

The free energy then follows from (3.22) as :

$$\text{Case i : }\mathrm{\Delta }F(t)\propto t^{\frac{2}{\kappa }+1},t\to 0^{+}(3.25)\backslash text\{Case\; i\; :\; \}\; \backslash Delta\; F(t)\; \backslash propto\; t^\{\backslash frac\{2\}\{\backslash kappa\}+1\},\; \backslash quad\; t\; \backslash rightarrow\; 0^+\; \backslash quad\; (3.25)$$

$$\text{Case ii : }\mathrm{\Delta }F(t)\propto e^{C^{\mathrm{\prime }}{t}^{-\frac{1}{\alpha }}}{t}^{1+\frac{1}{\alpha }},t\to 0^{+},(3.26)\backslash text\{Case\; ii\; :\; \}\; \backslash Delta\; F(t)\; \backslash propto\; e^\{C\text{'}t^\{-\backslash frac\{1\}\{\backslash alpha\}\}\}\; t^\{1+\backslash frac\{1\}\{\backslash alpha\}\},\; \backslash quad\; t\; \backslash rightarrow\; 0^+,\; \backslash quad\; (3.26)$$

where $C' = 2C^{\frac{1}{\alpha}}$ in the second equation. We see that $F$ vanishes, as it should, for arbitrary but positive constants $\xi$ , $\kappa$ and $\alpha$ . Other thermodynamic quantities suchas the entropy, specific heat, speed of sound etc, all follow from the free energy above [15].

In the special case of

$$\kappa =\frac{2}{n-1},(3.27)\backslash kappa\; =\; \backslash frac\{2\}\{n-1\},\; \backslash quad\; (3.27)$$

in (3.25) one finds an $n$ th order phase transition. On the other hand, the special case of $\alpha = 2$ in (3.19) corresponds to the BKT type scaling¹⁸.

One can also obtain the value of the transition temperature $T_c$ in terms of the coefficient of the dilaton potential in the IR as [15]:

$$T_c=\frac{\sqrt{V_{\mathrm{\infty }}}}{4\pi }\mathrm{.}(3.28)T\_c\; =\; \backslash frac\{\backslash sqrt\{V\_\backslash infty\}\}\{4\backslash pi\}.\; \backslash quad\; (3.28)$$

Finally, we should note the following issue. As mentioned above, the transition region $t \approx 0$ generically coincides with the singular region $\Phi_h \gg 1$ in this setting. We do not need to worry about the $\alpha'$ corrections because they vanish in the interesting region $r \gg 1$ in the interesting limit $r_h \gg 1$ [15]. However, one should worry about the string loops. In a generic situation the higher string loops cannot be ignored near the transition region. We are however interested in the situation with $g_s \rightarrow 0$ ( $N \rightarrow \infty$ ) that corresponds to the $U(1)$ invariant spin-model. This can be achieved by sending the boundary value of the dilaton to $-\infty$ . We will now dwell on this point in more detail.

3.3 The large $N$ limit and string perturbation theory

The effective Einstein frame action in (3.1) is supposed to arise from a (fermionic) non-critical string theory which also involves an RR-sector. The string frame action is,

$${\mathcal{A}}_s=\frac{1}{g_s^2{\mathrm{\ell }}_s^{d-1}}\int d^{d+1}x\sqrt{-g_s}{e}^{-2\bar{\mathrm{\Phi }}}(R_s+4(\mathrm{\partial }\bar{\mathrm{\Phi }}{)}^2+\frac{\delta c}{{\mathrm{\ell }}_s^2}-\frac{1}{12}{H}_{(3)}^2)-\frac{1}{2(d+1)!}{F}_{(d+1)}^2+\dots (3.29)\backslash mathcal\{A\}\_s\; =\; \backslash frac\{1\}\{g\_s^2\; \backslash ell\_s^\{d-1\}\}\; \backslash int\; d^\{d+1\}x\; \backslash sqrt\{-g\_s\}\; e^\{-2\backslash bar\{\backslash Phi\}\}\; \backslash left(\; R\_s\; +\; 4(\backslash partial\backslash bar\{\backslash Phi\})^2\; +\; \backslash frac\{\backslash delta\; c\}\{\backslash ell\_s^2\}\; -\; \backslash frac\{1\}\{12\}\; H\_\{(3)\}^2\; \backslash right)\; -\; \backslash frac\{1\}\{2(d+1)!\}\; F\_\{(d+1)\}^2\; +\; \backslash dots\; \backslash quad\; (3.29)$$

The ellipsis denote higher derivative ( $\alpha'$ ) corrections, subscript $s$ denote string-frame objects and $\delta c$ is the central deficit that—depending on the fermionic or the bosonic string theory—reads¹⁹,

$$\delta c=c_f(9-d),\text{fermionic};\delta c=c_b(25-d),\text{bosonic}\mathrm{.}(3.30)\backslash delta\; c\; =\; c\_f\; (9\; -\; d),\; \backslash quad\; \backslash text\{fermionic\};\; \backslash quad\; \backslash delta\; c\; =\; c\_b\; (25\; -\; d),\; \backslash quad\; \backslash text\{bosonic\}.\; \backslash quad\; (3.30)$$

---

¹⁸Very recently holographic realizations of (quantum) BKT scalings were obtained in [27] and [28].

¹⁹The constants $c_f, c_b$ depend on the particular CFT on the world-sheet as there are various possibilities for the boundary conditions and GSO projections on the world-sheet fermions possible twisted or shifted boundary conditions for the scalar matter $X^\mu$ [29]. In the case of bosonic world-sheet with periodic scalars, one has $c_b = 2/3$ which is indeed what one obtains from solving (3.10) with the asymptotics (3.17) and (3.16). See the next section for details of the IR CFT.$F_{(d+1)}$ is a space filling RR-form whose presence is motivated by holography: it should couple to the $D_{d-1}$ branes that are responsible for producing the $SU(N)$ gauge group. As it is space-filling, its effect in the theory can be obtained by replacing it in the action by its on-shell solution [30]. This solution in general will be very complicated as the higher derivative corrections will also depend on $F_{(d+1)}$ . Let us ignore these higher derivative solutions for the moment in order to be definite—the following discussion will not qualitatively depend on the higher derivative corrections.

The equation of motion for $F_{(d+1)}$ is $d * F_{(d+1)} = 0$ . The solution is

$$F_{(d+1)}=\frac{c_{F}N}{{\mathrm{\ell }}_s^2}\frac{{ϵ}_{(d+1)}}{\sqrt{-g_s}},(3.31)F\_\{(d+1)\}\; =\; \backslash frac\{c\_F\; N\}\{\backslash ell\_s^2\}\; \backslash frac\{\backslash epsilon\_\{(d+1)\}\}\{\backslash sqrt\{-g\_s\}\},\; \backslash quad\; (3.31)$$

where $\epsilon_{(d+1)}$ is the Levi-Civita symbol in $d + 1$ dimensions and $c_F$ is some $\mathcal{O}(1)$ constant. We chose the integration constant to be proportional to $N$ motivated by the fact that $F$ should couple to $N$ $D_{d-1}$ branes before the decoupling limit. Inserting the solution in the action gives (we ignore the NS-NS two-form in the following discussion),

$${\mathcal{A}}_s=\frac{1}{g_s^2{\mathrm{\ell }}_s^{d-1}}\int d^{d+1}x\sqrt{-g_s}{e}^{-2\bar{\mathrm{\Phi }}}(R_s+4(\mathrm{\partial }\bar{\mathrm{\Phi }}{)}^2+\frac{\delta c}{{\mathrm{\ell }}_s^2})+\frac{c_F^2}{2{\mathrm{\ell }}_s^2}{N}^2+\dots (3.32)\backslash mathcal\{A\}\_s\; =\; \backslash frac\{1\}\{g\_s^2\; \backslash ell\_s^\{d-1\}\}\; \backslash int\; d^\{d+1\}x\; \backslash sqrt\{-g\_s\}\; e^\{-2\backslash bar\{\backslash Phi\}\}\; \backslash left(\; R\_s\; +\; 4(\backslash partial\backslash bar\{\backslash Phi\})^2\; +\; \backslash frac\{\backslash delta\; c\}\{\backslash ell\_s^2\}\; \backslash right)\; +\; \backslash frac\{c\_F^2\}\{2\backslash ell\_s^2\}\; N^2\; +\; \backslash dots\; \backslash quad\; (3.32)$$

Now we define a shifted dilaton field

$$\mathrm{\Phi }=\bar{\mathrm{\Phi }}+\log N,(3.33)\backslash Phi\; =\; \backslash bar\{\backslash Phi\}\; +\; \backslash log\; N,\; \backslash quad\; (3.33)$$

and go to the Einstein frame by

$$g_{s,\mu \nu }=e^{\frac{4}{d-1}\mathrm{\Phi }}{g}_{\mu \nu }\mathrm{.}(3.34)g\_\{s,\backslash mu\backslash nu\}\; =\; e^\{\backslash frac\{4\}\{d-1\}\backslash Phi\}\; g\_\{\backslash mu\backslash nu\}.\; \backslash quad\; (3.34)$$

We obtain,

$$\mathcal{A}=\frac{N^2}{g_s^2{\mathrm{\ell }}_s^{d-1}}\int d^{d+1}x\sqrt{-g}(R-\frac{4}{d-1}(\mathrm{\partial }\mathrm{\Phi }{)}^2+V(\mathrm{\Phi }))+\dots (3.35)\backslash mathcal\{A\}\; =\; \backslash frac\{N^2\}\{g\_s^2\; \backslash ell\_s^\{d-1\}\}\; \backslash int\; d^\{d+1\}x\; \backslash sqrt\{-g\}\; \backslash left(\; R\; -\; \backslash frac\{4\}\{d-1\}(\backslash partial\backslash Phi)^2\; +\; V(\backslash Phi)\; \backslash right)\; +\; \backslash dots\; \backslash quad\; (3.35)$$

where the dilaton potential becomes,

$$V(\mathrm{\Phi })=\frac{1}{{\mathrm{\ell }}_s^2}(\delta c{e}^{\frac{4}{d-1}\mathrm{\Phi }}+\frac{c_F^2}{2}{e}^{\frac{2(d+1)}{d-1}\mathrm{\Phi }}+\dots )(3.36)V(\backslash Phi)\; =\; \backslash frac\{1\}\{\backslash ell\_s^2\}\; \backslash left(\; \backslash delta\; c\; e^\{\backslash frac\{4\}\{d-1\}\backslash Phi\}\; +\; \backslash frac\{c\_F^2\}\{2\}\; e^\{\backslash frac\{2(d+1)\}\{d-1\}\backslash Phi\}\; +\; \backslash dots\; \backslash right)\; \backslash quad\; (3.36)$$

We denote the corrections coming from the higher-derivative terms by the ellipsis. This is what one would obtain by ignoring the higher derivative terms²⁰.

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²⁰In the phenomenological approach that we adopted in the previous section, one assumes that there exist a string theory that would produce a potential of the form (3.15) instead of (3.36). In particular the leading term with exponent $2(d+1)/(d-1)$ should either be absent or renormalized to $4/(d-1)$ .On the other hand the solution of the dilaton equation of motion follows from (3.29) generically involves an integration constant that we shall denote as $\overline{\Phi}_0$ . For example in the kink solutions of [15] this corresponds to the boundary value of the dilaton on the AdS boundary. One can write

$$\bar{\mathrm{\Phi }}={\bar{\mathrm{\Phi }}}_0+\delta \bar{\mathrm{\Phi }}(r)(3.37)\backslash overline\{\backslash Phi\}\; =\; \backslash overline\{\backslash Phi\}\_0\; +\; \backslash delta\backslash overline\{\backslash Phi\}(r)\; \backslash quad\; (3.37)$$

to make explicit the integration constant. Now, we are ready to define the large-N limit. We send $N \rightarrow \infty$ , $\overline{\Phi}_0 \rightarrow -\infty$ such that the shifted dilaton in (3.33) remains constant

$$e^{{\bar{\mathrm{\Phi }}}_0}N\to \lambda ⟹e^{\mathrm{\Phi }}=\lambda e^{\delta \bar{\mathrm{\Phi }}}(3.38)e^\{\backslash overline\{\backslash Phi\}\_0\}\; N\; \backslash rightarrow\; \backslash lambda\; \backslash implies\; e^\backslash Phi\; =\; \backslash lambda\; e^\{\backslash delta\backslash overline\{\backslash Phi\}\}\; \backslash quad\; (3.38)$$

where $\lambda$ is some $\mathcal{O}(1)$ constant.

The shifted dilaton $\Phi$ is the one that we used in the previous section to discuss thermodynamics and it is what we will refer in the next sections to study the observables of the spin-system from the gravity point of view. Whether $\Phi$ is large or small does not matter neither for the loop-counting of strings nor for the strength of gravitational interactions: The latter is determined by the coefficient in the action (3.35). Identification with (3.1) yields the Newton's constant

$$G_N=\frac{g_s^2{\mathrm{\ell }}_s^{d-1}}{16\pi N^2},(3.39)G\_N\; =\; \backslash frac\{g\_s^2\; \backslash ell\_s^\{d-1\}\}\{16\backslash pi\; N^2\},\; \backslash quad\; (3.39)$$

which shows that the gravitational interactions among the bulk fields can be safely ignored in the large N limit. This equation also defines the “rescaled” Planck energy that was introduced in (3.3) in terms of $g_s$ and $\ell_s$ as,

$$M_p={\mathrm{\ell }}_s^{-1}{g}_s^{-\frac{2}{d-1}}\mathrm{.}(3.40)M\_p\; =\; \backslash ell\_s^\{-1\}\; g\_s^\{-\backslash frac\{2\}\{d-1\}\}.\; \backslash quad\; (3.40)$$

The string loops on the other hand are counted by the coupling of the original dilaton $\overline{\Phi}$ to a world-sheet $M$ with genus $g$ as,

$$e^{-\frac{1}{4\pi }{\int }_M\sqrt{h}{R}^{(2)}\bar{\mathrm{\Phi }}}=e^{-{\bar{\mathrm{\Phi }}}_0\chi (M)}{e}^{-\frac{1}{4\pi }{\int }_M\sqrt{h}{R}^{(2)}\delta \bar{\mathrm{\Phi }}}=N^{\chi (M)}{e}^{-\frac{1}{4\pi }{\int }_M\sqrt{h}{R}^{(2)}\mathrm{\Phi }},(3.41)e^\{-\backslash frac\{1\}\{4\backslash pi\}\; \backslash int\_M\; \backslash sqrt\{h\}\; R^\{(2)\}\; \backslash overline\{\backslash Phi\}\}\; =\; e^\{-\backslash overline\{\backslash Phi\}\_0\; \backslash chi(M)\}\; e^\{-\backslash frac\{1\}\{4\backslash pi\}\; \backslash int\_M\; \backslash sqrt\{h\}\; R^\{(2)\}\; \backslash delta\backslash overline\{\backslash Phi\}\}\; =\; N^\{\backslash chi(M)\}\; e^\{-\backslash frac\{1\}\{4\backslash pi\}\; \backslash int\_M\; \backslash sqrt\{h\}\; R^\{(2)\}\; \backslash Phi\},\; \backslash quad\; (3.41)$$

where $\chi(M) = 2(1 - g)$ is the Euler characteristic. We observe that the above definition of the large N limit does the job and suppresses the strings with higher genus.

One might still worry about the viability of the string perturbation expansion if the additional term proportional to $\int_M \sqrt{h} R^{(2)} \Phi$ in (3.41) becomes very large in some limit. Indeed, as we argued above the interesting physics concerns the region $\Phi \gg 1$ which corresponds to the vicinity of the phase transition. We check in appendix D.1 and D.2 that for all of the string paths that we consider in this paper the world-sheet Ricci scalar suppresses the linear divergence in $\Phi$ . For example in case of (3.18) one finds that in the transition region $\sqrt{h} R^{(2)} \sim \exp(-\kappa\Phi)$ .All of the discussion we presented above can be understood in the following equivalent way. To be definite let us consider the simplest effective (rescaled) dilaton potential that corresponds to case 3.18:

$$V(\mathrm{\Phi })=V_{\mathrm{\infty }}{e}^{\frac{4}{d-1}\mathrm{\Phi }}(1+C{e}^{-\kappa \mathrm{\Phi }})\mathrm{.}(3.42)V(\backslash Phi)\; =\; V\_\backslash infty\; e^\{\backslash frac\{4\}\{d-1\}\backslash Phi\}\; (1\; +\; Ce^\{-\backslash kappa\backslash Phi\}).\; \backslash quad\; (3.42)$$

It was shown in [15] that this potential has a kink solution that flows from the AdS extremum at

$$e^{{\mathrm{\Phi }}_0}=C^{\frac{1}{\kappa }}a(3.43)e^\{\backslash Phi\_0\}\; =\; C^\{\backslash frac\{1\}\{\backslash kappa\}\}\; a\; \backslash quad\; (3.43)$$

—where $a$ is some number independent of $C$ —to the linear dilaton geometry in the IR $\Phi \rightarrow \infty$ . Then the subleading term in the potential can be written as,

$$V_{sub}(\mathrm{\Phi })=a^{-\kappa }{e}^{\kappa {\mathrm{\Phi }}_0-\kappa \mathrm{\Phi }}=a^{-\kappa }{e}^{\kappa {\bar{\mathrm{\Phi }}}_0}{e}^{-\kappa \bar{\mathrm{\Phi }}}(3.44)V\_\{sub\}(\backslash Phi)\; =\; a^\{-\backslash kappa\}\; e^\{\backslash kappa\backslash Phi\_0\; -\; \backslash kappa\backslash Phi\}\; =\; a^\{-\backslash kappa\}\; e^\{\backslash kappa\backslash bar\{\backslash Phi\}\_0\}\; e^\{-\backslash kappa\backslash bar\{\backslash Phi\}\}\; \backslash quad\; (3.44)$$

where we used (3.33). The statement that “the transition region corresponds to large dilaton” now can be quantified. What we really mean by this is that the reduced temperature $t$ (3.20) is small enough, so that the scaling behavior of observables set in. Now, from (3.21) we see that this is given as,

$$t=a^{-\kappa }{e}^{\kappa {\bar{\mathrm{\Phi }}}_0}{e}^{-\kappa \bar{\mathrm{\Phi }}}\mathrm{.}(3.45)t\; =\; a^\{-\backslash kappa\}\; e^\{\backslash kappa\backslash bar\{\backslash Phi\}\_0\}\; e^\{-\backslash kappa\backslash bar\{\backslash Phi\}\}.\; \backslash quad\; (3.45)$$

On the other hand the large $N$ limit (3.38) involves $\bar{\Phi}_0 \rightarrow -\infty$ , therefore we see that in order for $t$ to be small, one needs not the actual dilaton $\bar{\Phi}$ but the difference $\delta\Phi = \bar{\Phi} - \bar{\Phi}_0$ to be large. The same reasoning can be generalized to general potentials that involve an AdS extremum.

To conclude, we can safely ignore higher string loops in the computations below.

3.4 Parameters of the model

In the model constructed above there are various parameters. Here we shall list the parameters without derivation and refer to [26] for a detailed discussion.

• • Parameters of the action: In the weak gravity limit, $G_N \rightarrow 0$ , $N \rightarrow \infty$ and $M_p^{1-d} = 16\pi G_N N^2 = \text{fixed.}$ , there are two parameters in the action: $M_p$ and the overall size of the potential $\ell$ . The latter fixes the units in the theory. One can construct a single dimensionless parameter from the two: $M_p \ell$ which determines the overall size of thermodynamics functions in the dual field theory and it can be fixed e.g. by comparison with the value of the free energy at high temperatures, see [26]. In the present paper we are only interested in scaling of functions near $T_c$ , thus this parameter will play no role in what follows.

• • Parameters of the potential: We have not specified the potential apart from its IR asymptotics. The IR piece will be enough to determine the scaling behaviors and also the transition temperature through equation (3.28). Therefore wehave only three (dimensionless) parameters: $V_\infty \ell^2$ , $C$ and $\kappa$ or $\alpha$ that appear in (3.15), (3.18) and (3.19). The first determines the (dimensionless) transition temperature $T_c \ell$ through (3.28), the second one is related to the boundary value of the dilaton (cf. the discussion above), and the third one determines the type of the transition. For example $\kappa = 2$ for a second order transition, equation (3.27).

• • Integration constants: In [15] we solve the Einstein-dilaton system and work out the thermodynamics in the reduced system of “scalar variables” that is a coupled system of two first order differential equations. One boundary condition can be interpreted as the value of $T$ , and the other is just regularity of the solution at the horizon. Therefore the only dimensionless parameter that arise among the integration constants is $T\ell$ .²¹

4. Non-critical string theory and the IR CFT

4.1 Linear-dilaton in the deep interior

The leading asymptotics (3.15) of the dilaton potential which follows from the requirement of a continuous Hawking-Page transition is precisely the same as the potential that follows from $d + 1$ dimensional non-critical string theory. This is easily seen by transforming (3.1) with the potential (3.15) to string frame with $g_{s,\mu\nu} = \exp(2\Phi/(d-1))g_{\mu\nu}$ . Not only that but we also have the asymptotics (3.16), which imply that the asymptotic solution in the IR corresponds to a linear-dilaton background that is—very conveniently—an $\alpha'$ exact solution to (3.29) and corresponds to an exact world-sheet CFT. Indeed, in [15] it is shown that, with the subleading terms of the form (3.15), the string-frame curvature invariants both on the TG and the BH backgrounds vanish in the deep interior region near criticality i.e. for $r_h \rightarrow \infty$ , ( $T \rightarrow T_c$ ). Hence the higher derivative terms denoted by ellipsis in (3.29) become unimportant in the IR theory.

This implies that the dynamics in the transition region should be governed by the linear-dilaton CFT. More precisely, we expect that quantities that receive dominant contributions from the deep interior region near criticality should be determined by the linear-dilaton CFT.

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²¹In the fifth order system of (3.8-3.9) it is a little harder to work out the non-trivial integration constants. There it works as follows [26]: In (3.9), one requires $f \rightarrow 1$ as $r \rightarrow 0$ . This fix one constant, and the other is gives $T$ . In the rest, one is fixed by requirement of regularity at the horizon, one is just a reparametrization in $r$ , and the last is fixed either by the asymptotic value of the dilaton in the case $\Phi(r) = \Phi_0$ is constant at the boundary, or the integration constant $\Lambda$ that determines the running of the dilaton near the boundary $\Phi \sim \log(-\log(\Lambda r))$ near $r \rightarrow 0$ . In either case the thermodynamic functions can be shown to be independent of this constant [26].In the next section we shall make use of this observation to argue that the various observables in the corresponding spin-model scale precisely with the expected critical exponents near $T_c$ .

Another implication of this is that an asymptotically linear dilaton geometry (with corrections governed by the subleading terms in (3.15)) develops an instability at a finite temperature $T_c$ into formation of black-holes. It is quite reasonable to expect that in the limit of weak $g_s$ this point coincides with the Hagedorn temperature of strings on the linear-dilaton background [17]. We have more to say on this in section 5.4.2.

Finally, we note that in the case when the model is embedded in non-critical string theory, all of the parameters in the model are entirely fixed. To illustrate this let us assume that the entire potential is given by the leading term, ignoring the subleading terms etc. Then the coefficient $V_\infty$ in (3.15) and the transition temperature would be given as,

$$V_{\mathrm{\infty },nc}=\frac{c_b(25-d)}{{\mathrm{\ell }}_s^2},T_{c,nc}=\frac{1}{4\pi {\mathrm{\ell }}_s}\sqrt{c_b(25-d)},(4.1)V\_\{\backslash infty,nc\}\; =\; \backslash frac\{c\_b(25-d)\}\{\backslash ell\_s^2\},\; \backslash quad\; T\_\{c,nc\}\; =\; \backslash frac\{1\}\{4\backslash pi\backslash ell\_s\}\; \backslash sqrt\{c\_b(25-d)\},\; \backslash quad\; (4.1)$$

in the case of bosonic world-sheet CFT and

$$V_{\mathrm{\infty },nc}=\frac{c_f(9-d)}{{\mathrm{\ell }}_s^2},T_{c,nc}=\frac{1}{4\pi {\mathrm{\ell }}_s}\sqrt{c_f(9-d)},(4.2)V\_\{\backslash infty,nc\}\; =\; \backslash frac\{c\_f(9-d)\}\{\backslash ell\_s^2\},\; \backslash quad\; T\_\{c,nc\}\; =\; \backslash frac\{1\}\{4\backslash pi\backslash ell\_s\}\; \backslash sqrt\{c\_f(9-d)\},\; \backslash quad\; (4.2)$$

in the case of fermionic world-sheet CFT. These results follow from (3.30) and (3.28). Of course, in reality these numbers should be renormalized because the theory is not just given by the leading piece: a potential with only the leading exponential behavior do not possess any phase transition. The corrections will depend on the UV physics where the $\alpha'$ corrections kick in and renormalize these coefficients. We shall argue for another way to fix these numbers in section 5. We will also show in that section that the scaling exponents are also determined completely, once the CFT is fixed.

4.2 The CFT in the IR

The arguments presented above point towards the conclusion that, on the string side the criticality of the dual spin-system should be governed by a linear-dilaton CFT. Here we want to spell out some of the salient features of this IR CFT. We start with the bosonic case and then mention generalization to fermionic CFT in the end.

We reviewed the intimate connection between non-critical string theory and the linear-dilaton background in appendix B. Utilizing this relation one can obtain the stress-tensor of the (bosonic) linear-dilaton CFT as [29],

$$T(z)=-\frac{1}{{\alpha }^{\mathrm{\prime }}}:\mathrm{\partial }{X}^{\mu }\mathrm{\partial }{X}_{\mu }:+v_{\mu }{\mathrm{\partial }}^2{X}^{\mu }(4.3)T(z)\; =\; -\backslash frac\{1\}\{\backslash alpha\text{'}\}\; :\; \backslash partial\; X^\backslash mu\; \backslash partial\; X\_\backslash mu\; :\; +\; v\_\backslash mu\; \backslash partial^2\; X^\backslash mu\; \backslash quad\; (4.3)$$

for the left-movers, with an analogous expression for the right movers. $v_\mu$ are the proportionality constants in the dilaton solution $\bar{\Phi} = v_\mu X^\mu$ . The indices are raisedand lowered by the flat metric. The total central charge of the theory (including the ghost sector) vanishes for $v_\mu$ satisfying (B.5). In our case we have,

$$v_{\mu }=\frac{\sqrt{V_{\mathrm{\infty }}}}{2}{\delta }_{\mu ,r}\equiv m_0{\delta }_{\mu ,r}\mathrm{.}(4.4)v\_\backslash mu\; =\; \backslash frac\{\backslash sqrt\{V\_\backslash infty\}\}\{2\}\; \backslash delta\_\{\backslash mu,r\}\; \backslash equiv\; m\_0\; \backslash delta\_\{\backslash mu,r\}.\; \backslash quad\; (4.4)$$

The reason for denoting this constant $m_0$ will be clear when we analyze the spectrum of fluctuations in this geometry, see appendix E. The total central charge of the theory (including ghosts) vanishes only for,

for the bosonic and fermionic CFT's, [31].

Now we discuss the spectrum in the case we are interested in: The Euclidean $d+1$ dimensional world sheet with (4.4) and the Euclidean $X^0$ dimension compactified on a radius $R = 1/2\pi T$ . There are various ways to obtain the spectrum. Both the light-cone and the covariant quantization is discussed in [31, 29]. Here we trivially extend these results in our case.

The Virasoro generators are now

$$L_m=\frac{1}{2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}:{\alpha }_{m-n}^{\mu }{\alpha }_{n,\mu }:+i\frac{\sqrt{{\alpha }^{\mathrm{\prime }}}}{\sqrt{2}}(m+1)m_0{\alpha }_m^r\mathrm{.}(4.6)L\_m\; =\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}\; :\; \backslash alpha\_\{m-n\}^\backslash mu\; \backslash alpha\_\{n,\backslash mu\}\; :\; +\; i\; \backslash frac\{\backslash sqrt\{\backslash alpha\text{'}\}\}\{\backslash sqrt\{2\}\}\; (m+1)\; m\_0\; \backslash alpha\_m^r.\; \backslash quad\; (4.6)$$

The center-of-mass momenta are related to the zero mode oscillators as usual, $p_L^\mu = \sqrt{\frac{2}{\alpha'}} \alpha_0^\mu$ and $p_R^\mu = \sqrt{\frac{2}{\alpha'}} \tilde{\alpha}_0^\mu$ . Decomposing into components one has,

$$p_{0,L}=2\pi Tk+\frac{w}{2\pi T{\alpha }^{\mathrm{\prime }}},p_{0,R}=2\pi Tk-\frac{w}{2\pi T{\alpha }^{\mathrm{\prime }}},(4.7)p\_\{0,L\}\; =\; 2\backslash pi\; T\; k\; +\; \backslash frac\{w\}\{2\backslash pi\; T\; \backslash alpha\text{'}\},\; \backslash quad\; p\_\{0,R\}\; =\; 2\backslash pi\; T\; k\; -\; \backslash frac\{w\}\{2\backslash pi\; T\; \backslash alpha\text{'}\},\; \backslash quad\; (4.7)$$

$$p_{i,L}=p_{i,R}=p_i,p_{r,L}=p_{r,R}=p_r\mathrm{.}(4.8)p\_\{i,L\}\; =\; p\_\{i,R\}\; =\; p\_i,\; \backslash quad\; p\_\{r,L\}\; =\; p\_\{r,R\}\; =\; p\_r.\; \backslash quad\; (4.8)$$

In the first line the integer $k$ denotes the Matsubara frequency and the integer $w$ denotes the winding number on the time-circle. As a result of the linear piece in the zeroth level Virasoro generator (4.6) one obtains the following mass-shell conditions (we adopt the definition of mass in [29]) in the light-cone gauge:

$$-m_{d+1}^2=p_{\perp }^2+p_r^2+2i{m}_0{p}_r+(2\pi kT{)}^2+{\left(\frac{w}{2\pi T{\alpha }^{\mathrm{\prime }}}\right)}^2=-\frac{2}{{\alpha }^{\mathrm{\prime }}}(N+\tilde{N}-2),(4.9)-m\_\{d+1\}^2\; =\; p\_\backslash perp^2\; +\; p\_r^2\; +\; 2im\_0\; p\_r\; +\; (2\backslash pi\; k\; T)^2\; +\; \backslash left(\; \backslash frac\{w\}\{2\backslash pi\; T\; \backslash alpha\text{'}\}\; \backslash right)^2\; =\; -\backslash frac\{2\}\{\backslash alpha\text{'}\}\; (N\; +\; \backslash tilde\{N\}\; -\; 2),\; \backslash quad\; (4.9)$$

$$0=kw+N-\tilde{N},(4.10)0\; =\; kw\; +\; N\; -\; \backslash tilde\{N\},\; \backslash quad\; (4.10)$$

where $p_\perp$ , $N$ and $\tilde{N}$ denote the center-of-mass momentum, the left (right) number of oscillations in the space transverse to motion, respectively,

$$N=\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{\perp ,-n}\cdot {\alpha }_{\perp ,n},\tilde{N}=\sum _{n=1}^{\mathrm{\infty }}{\tilde{\alpha }}_{\perp ,-n}\cdot {\tilde{\alpha }}_{\perp ,n}\mathrm{.}(4.11)N\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash alpha\_\{\backslash perp,-n\}\; \backslash cdot\; \backslash alpha\_\{\backslash perp,n\},\; \backslash quad\; \backslash tilde\{N\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash tilde\{\backslash alpha\}\_\{\backslash perp,-n\}\; \backslash cdot\; \backslash tilde\{\backslash alpha\}\_\{\backslash perp,n\}.\; \backslash quad\; (4.11)$$In (4.9) $m_{d+1}^2$ denote the $d + 1$ dimensional mass. One important difference between the linear-dilaton and the flat case is that the definition of the mass of the string excitations in terms of their momentum gets modified[29] due to the linear oscillator piece in (4.3). The flat case follows by setting $m_0 = 0$ , hence sending dilaton to constant.

Once the modified definition of mass is attained, the physical spectrum of the linear dilaton is exactly the same as the critical string: the lowest level $N = \tilde{N} = 0$ is a tachyon with mass $-4/\alpha'$ , the next level is massless and corresponds to the fluctuations of the metric, the B-field and the dilaton, etc. [29].

All of these results are readily extended to the fermionic case with $N = 1$ world-sheet supersymmetry [29]. In the light-cone gauge, one obtains the following spectra for the NS and Ramond sectors,

$$m_{d+1}^2=N+\sum _{q>0}q{b}_{-q}^{\perp }{b}_q^{\perp }-\frac{1}{2},(NS)(4.12)m\_\{d+1\}^2\; =\; N\; +\; \backslash sum\_\{q>0\}\; q\; b\_\{-q\}^\backslash perp\; b\_q^\backslash perp\; -\; \backslash frac\{1\}\{2\},\; \backslash quad\; (NS)\; \backslash quad\; (4.12)$$

$$m_{d+1}^2=N+\sum _{q>0}q{b}_{-q}^{\perp }{b}_q^{\perp },(R),(4.13)m\_\{d+1\}^2\; =\; N\; +\; \backslash sum\_\{q>0\}\; q\; b\_\{-q\}^\backslash perp\; b\_q^\backslash perp,\; \backslash quad\; (R),\; \backslash quad\; (4.13)$$

where $N$ denotes the number of bosonic oscillations in the transverse space (4.11) and $q \in \mathbb{Z}$ for the R-fermions and $q \in \mathbb{Z} + \frac{1}{2}$ for the NS-fermions. This is again the same spectra that one finds in the critical super-string.

However one finds crucial differences at the one-loop level: Modular invariance does not allow for NS-R fermions except in particular dimensions given by multiples of 8. This is quite convenient for our holographic purposes, because we do not want any fermionic operators in the dual spin-model. Thus in a generic dimension $d+1 < 8$ one has only two sectors R-R and NS-NS. Furthermore, in the generic case, there is no analog of the GSO projection of the superstring. Therefore the tachyon in the NS-NS sector survives.

Existence of tachyon in the physical spectrum is a very generic feature of the linear-dilaton CFT in any dimensions. The mass of the tachyon changes depending on which particular CFT chosen. With the definition of mass adopted above it is given by $m_T^2 = -4/\alpha'$ for the bosonic case, $m_T^2 = -2/\alpha'$ for the NS-NS fermions, $m^2 = -15/4\alpha'$ for an orbifold in the r-direction, etc., but we stress that the ground state for $k = w = 0$ in linear-dilaton CFT in arbitrary dimensions is always a tachyon.²²

This fact renders the linear-dilaton theory unattractive from many perspectives. In our case however, it is a desired feature of the IR CFT. We recall that the background geometry becomes asymptotically linear-dilaton only in the transition region

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²²With a more conventional definition of mass [31], one finds a tachyon only for $d > 1$ in a $d + 1$ dimensional theory. In our case, the equivalent statement is that if we consider propagation of the tachyonic mode, we find a smooth propagation for $d \leq 1$ but oscillatory behavior for $d > 1$ .$T \sim T_c$ and only in the large $r$ region. We do not expect that the complete sigma-model which corresponds to the black-hole for an arbitrary $T$ have tachyon as a ground state. This would imply that the black-hole geometry is unstable at any temperature. Instead the linear dilaton CFT describes the physics near the transition and we do expect instability in this region. In fact, as we show in sections (5.3.2) and (5.4.2), it is the presence of the tachyon which guarantees vanishing of magnetization as $M \sim (T - T_c)^\beta$ and divergence of the correlation length as $\xi \sim |T - T_c|^{-\nu}$ at the transition!

5. Spin-model observables from strings

F-strings and D-branes are important probes in the standard examples of the gauge-gravity correspondence. In case of the holographic models for QCD-like theories, the phase of the field theory at finite temperature, the quark-anti-quark potential, the force between the magnetic quarks etc, can all be read off from classical F-string and D-string solutions in the dual gravitational background. In this section we argue that the probe strings constitute indispensable tools also in the spin-model-gravity correspondence. In particular, the Landau potential, the correlation length, the various critical exponents, the scaling of order parameters near the transition, the phase of the system, spin-spin correlation function, etc. can all be computed from the probe strings in the dual background. In this section we discuss how to obtain the various observables of the spin-model from the probe string solutions.

5.1 What can we learn from the Gravity-Spin model duality?

In order to answer this question, one has to identify the Landau and the mean-field approximations on the gravity side. The Landau approach is based on integrating out the “fast” degrees of freedom in the spin-model in order to obtain a free-energy functional for the “slow” degrees of freedom, i.e. the order parameter $\vec{M}$ . We refer to appendix C for a review of the statistical mechanics background and in particular a description of the Landau approach. This is exactly analogous to integrating out the gauge invariant states to obtain an effective action for the Polyakov loop on the LGT side, as illustrated in appendix A. In the context of the gauge-gravity correspondence, this is, in essence, very similar to keeping only the lowest lying degrees of freedom in string theory, i.e. the supergravity multiplet. It is tempting to think that the complicated step of integrating over the spin configurations in (C.4) to obtain (C.5) can be side-stepped by use of the gravity-spin model duality²³.

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²³We note a very interesting paper [32] that dwells on these issues. In this paper Headrick argues that one can generate the Landau functional at strong coupling in terms of classical string solutions.In the most general case, the correspondence between the spin model and gravity should relate the Landau functional (C.5) with the string path integral²⁴:

$$Z_L=Z_{st}\mathrm{.}(5.1)Z\_L\; =\; Z\_\{st\}.\; \backslash quad\; (5.1)$$

As in the original gauge-gravity duality we expect that there is a simple corner of the correspondence where both sides of (5.1) become classical and one approximates the path integrals by the classical saddles.

This is the large N limit: On the LHS this is given by the Landau approximation (C.6). On the RHS, it is given by the saddle-point approximation to the string theory where one can ignore string interactions $g_s \rightarrow 0$ . Then (5.1) reduces to,

$$e^{-\beta F_L}=e^{-{\mathcal{A}}_{st}},(5.2)e^\{-\backslash beta\; F\_L\}\; =\; e^\{-\backslash mathcal\{A\}\_\{st\}\},\; \backslash quad\; (5.2)$$

where the action on the RHS is the full target-space action including all the $\alpha'$ corrections, evaluated on the classical saddle. It is the effective action for all excitations of a single string²⁵ and in principle it can be obtained from the sigma model on the world-sheet.

At this point, it is clear that scaling of any quantity on both side of (5.2) near $T_c$ should be characterized by the mean-field scaling. This is just a consequence of the saddle point approximation. Therefore, any operator in the spin-model that is given by a fluctuation of $\mathcal{A}{st}$ should obey the standard mean-field scaling. We shall refer to these operators as local operators. The only possible exceptions to this—within the classical approximation of (5.2)—are operators that can not be obtained as fluctuations of $\mathcal{A}{st}$ . These correspond to non-local operators on the gauge theory, they are governed by probe F-strings or D-branes on the string side. Yet, as we will show in the next section, they can correspond to quite ordinary quantities such as the magnetization on the spin-model side. Thus, magnetization is an example of a non-local operator. Even for the “non-local observables” though the mean-field scaling is expected to hold in a semi-classical approximation, where one only keeps the lowest-lying string excitations in string path integrals. These excitations correspond to bulk gravity modes (levels $N = 0$ and $N = 1$ of the string spectrum). We confirm this expectation in the sections (5.3.2) and (5.4.2) below.

In practice, it is usually very hard to reckon with (5.2), and one further considers the weak-curvature limit where one can replace the RHS with the (super)gravity action:

$$\frac{1}{T}{F}_L\approx T{V}_{d-1}{\mathcal{L}}_{gr}\mathrm{.}(5.3)\backslash frac\{1\}\{T\}\; F\_L\; \backslash approx\; T\; V\_\{d-1\}\; \backslash mathcal\{L\}\_\{gr\}.\; \backslash quad\; (5.3)$$

Here, $\mathcal{L}_{gr}$ is the (super)gravity action evaluated on-shell, on the classical saddle. We also assumed a trivial dependence on the spatial volume and made use of the fact

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²⁴We shall be schematic in what follows.

²⁵It is important to note that this is not a string field theory action, the excitations governed by $\mathcal{A}_{st}$ are particles, rather than strings.that the temperatures on the spin-model and the gravity sides are inversely related, cf. appendix A.

Influenced by the standard lore of the gauge-gravity correspondence, we expect that the weak curvature limit corresponds to strong correlations on the spin-model side. On the other hand, one quantifies “strong correlations” by the Ginzburg criterion in the spin-model, as reviewed in section C. Quite generally, the system will be in a regime of strong correlations around the phase transition where the mean-field approximation usually breaks down. Therefore, one may hope that the gravity side provides a better description in (5.3) precisely within this interesting region. This can be checked explicitly by computing curvature invariants in the string frame. Even though one shows that the Ricci scalar (and the various contractions of Ricci two-form and the Riemann tensor) vanish in the limit (see [15]) there exists invariants such as $d\Phi^2$ that asymptote to a constant that is generically the same order as the string length scale $\ell_s^{-2}$ . Therefore, in a generic case one is forced to include the higher derivative corrections. Luckily this can be done precisely in the interesting critical regime, because the background asymptotes to a linear-dilaton theory.

What observables can we actually calculate on the gravity side? Because going beyond the large N limit is very hard, one can (at present) only hope to obtain results in the Landau approximation. The main observables then include the Landau coefficients²⁶ $\alpha_0(T)$ , $\alpha_1(T)$ , $\alpha_2(T)$ , the basic scaling exponents $\beta$ , $\nu$ , $\eta$ , $\gamma$ etc., and the spin correlation functions. Moreover, the scaling exponents of operators that are dual to fluctuations of the bulk fields in $\mathcal{L}_{gr}$ in (5.3) are necessarily given by the mean-field scaling. Therefore one can only hope to obtain results beyond mean-field in the scaling exponents of operators dual to stringy objects, such as magnetization or the spin-spin correlator.

Once again, we would like to emphasize the distinction between “mean-field scaling” and the “mean-field approximation”. The former is unavoidable for local operators in the Landau approximation (large N). On the other hand, gravity description is expected to go beyond the latter. Therefore for quantities such as $T_c$ , the Landau coefficients at $T_c$ , etc., and correlation functions of the non-local observables we expect gravity to provide better answers than the mean-field approximation.

One may still ask the question, what is the use gravity-spin-model duality if one can compute all of these quantities by employing Monte-Carlo simulations, or RG techniques? First of all, the RG techniques are limited in the case of strong correlations. Secondly, the calculations on the gravity side are much easier to perform, much easier than the Monte-Carlo simulations, and one can usually obtain analytic results. However a more fundamental reason is that, there are situations where applicability of the Monte-Carlo simulations are limited. The well-known examples are the computation of real-time correlators or spin-models with fermionic degrees of

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²⁶We refer to appendix C for a definition of these coefficients.freedom. By the gravity-spin-model correspondence, one expects to overcome such fundamental difficulties.

5.2 Identification of observables

The duality between the lattice gauge theories and spin-models [10], [11] relate the magnetization directly to the Polyakov loop. On the other hand, the Polyakov loop is related to the classical F-string solution as discussed in section 2.3. Therefore we propose the following chain of relations:

$$⟨P(x)⟩\leftrightarrow ⟨\vec{m}(x)⟩\leftrightarrow e^{-S_{NG}[C_x]}\mathrm{.}(5.4)\backslash langle\; P(x)\; \backslash rangle\; \backslash leftrightarrow\; \backslash langle\; \backslash vec\{m\}(x)\; \backslash rangle\; \backslash leftrightarrow\; e^\{-S\_\{NG\}[C\_x]\}.\; \backslash quad\; (5.4)$$

Here the boundary condition $C_x$ for the string is just a point $x$ in the spatial part and a loop on the temporal circle.

The spin field is valued under $U(1)_S$ . Similarly the Polyakov loop is valued under the center $\mathcal{C} = Z_N$ that becomes a $U(1)$ in the large $N$ limit. One should think of this as the exponents becoming angles in the transformation,

$$P\to e^{2\pi i\frac{k}{N}}P,k=1,2\dots NP\; \backslash rightarrow\; e^\{2\backslash pi\; i\; \backslash frac\{k\}\{N\}\}\; P,\; \backslash quad\; k\; =\; 1,\; 2\; \backslash dots\; N$$

at large $N$ . We shall denote this $U(1)$ as $U(1)_C$ . Similarly, as discussed in section 2.2 at length, the F-string that winds the time-circle is charged under the $U(1)_B^{27}$ , because it couples to the B-field. Thus one should identify

$$U(1{)}_S=U(1{)}_C=U(1{)}_B\mathrm{.}(5.5)U(1)\_S\; =\; U(1)\_C\; =\; U(1)\_B.\; \backslash quad\; (5.5)$$

as in table in section 2.3.

One should work out the identification in (5.4) carefully. In particular the first entry is a complex number and the second entry is a vector in 2D spin space. The precise identification of the two is provided with the standard isomorphism between $U(1)$ and $O(2)$ representations. We imagine the vector $\vec{m}$ on the XY plane represented by the magnitude $|\vec{m}|$ and the phase $\psi$ . Then the simplest option is to set $m_x = \text{Re}(P)$ and $m_y = \text{Im}(P)$ . There is a little complication though, because in fact the identification should depend on the value of $\psi$ . This is because the physically preferred reference frame is set by the direction of the magnetization vector $v_i$ in (C.16). All of the correlation functions should be decomposed into components parallel and perpendicular to $v_i$ . Represented by the phase, the direction of magnetization reads

$$\vec{v}=(\cos (\psi ),\sin (\psi ))\mathrm{.}(5.6)\backslash vec\{v\}\; =\; (\backslash cos(\backslash psi),\; \backslash sin(\backslash psi)).\; \backslash quad\; (5.6)$$

Thus, the naive identification mentioned above is correct only for $\psi = 0$ . For a different value of $\psi$ one should obtain the correct identification by a $U(1)$ rotation: $P = \exp(i\psi)(m_x + im_y)$ . Thus, in general we have,

$$\text{Re}(P)=m_{\parallel }=\cos (\psi )m_x-\sin (\psi )m_y,\text{Im}(P)=m_{\perp }=\sin (\psi )m_x+\cos (\psi )m_y,(5.7)\backslash text\{Re\}(P)\; =\; m\_\{\backslash parallel\}\; =\; \backslash cos(\backslash psi)m\_x\; -\; \backslash sin(\backslash psi)m\_y,\; \backslash quad\; \backslash text\{Im\}(P)\; =\; m\_\{\backslash perp\}\; =\; \backslash sin(\backslash psi)m\_x\; +\; \backslash cos(\backslash psi)m\_y,\; \backslash quad\; (5.7)$$

²⁷The charge is determined by the winding number. Here we are only interested in strings that wind the time circle once.