Contents

CQUeST–2012-0550

TAUP-2955/12




Aging Logarithmic Conformal Field Theory

: a holographic view



Seungjoon Hyun, Jaehoon Jeong and Bom Soo Kim

Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea

Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea

Raymond and Beverly Sackler School of Physics and Astronomy,

Tel Aviv University, 69978, Tel Aviv, Israel


sjhyun@yonsei.ac.kr,   jhjeong@sogang.ac.kr,   bskim@post.tau.ac.il



We consider logarithmic extensions of the correlation and response functions of scalar operators for the systems with aging as well as Schrödinger symmetry. Aging is known to be the simplest nonequilibrium phenomena, and its physical significances can be understood by the two-time correlation and response functions. Their logarithmic part is completely fixed by the bulk geometry in terms of the conformal weight of the dual operator and the dual particle number.

Motivated by recent experimental realizations of Kardar-Parisi-Zhang universality class in growth phenomena and its subsequent theoretical extension to aging, we investigate our two-time correlation functions out of equilibrium, which show several qualitatively different behaviors depending on the parameters in our theory. They exhibit either growing or aging, i.e. power-law decaying, behaviors for the entire range of our scaling time. Surprisingly, for some parameter ranges, they exhibit growing at early times as well as aging at later times.

1 Introduction

Most of the physical systems are out of equilibrium [2][3]. Thus it is very interesting to understand various physical properties of those systems. In general, it has been known to be difficult to understand the nonequilibrium phenomena despite active studies. Still, progresses have been made in the context of the simple nonequilibrium critical phenomena, where one can use their underlying local scaling invariance, universality classes and critical exponents to capture universal properties. One simple example is Kardar-Parisi-Zhang (KPZ) universality class [4][5]. This universality class is known to describe many interesting systems in nature, such as paper wetting, epidemics and fluid flow of porous media and so on [2][6]. In the last few years, this class is experimentally realized in the context of surface growth in turbulent liquid crystals [7][8], which provides the experimental evidence of the dynamical scaling of the KPZ theory in 1+1 dimensions.

Inspired by these developments, the authors of [9] investigate the possibility of aging phenomena, which can arise in the same KPZ universality class. It is shown that the systems undergo simple aging in the correlation and response functions, whose dynamical scaling is characterized by some aging exponents. Moreover, it is claimed that the form of the autoresponse scaling function is well described by the logarithmic extension of the scaling function with local scale invariance [10]. This provides us the main physical motivation to pursue the logarithmic extension of the aging correlation functions in the context of holography, following [11][12]. In the rest of the introduction, we briefly explain aging and logarithmic conformal field theories (LCFT).

Aging is one of the simplest nonequilibrium phenomena of many body systems, which is realized when the system is rapidly brought out of equilibrium. The basic properties of aging can be characterized by two-time correlation functions with two-time scales, “waiting time,” when the system is perturbed after brought out of the equilibrium, and “response time,” when the perturbation is measured. The correlation functions show the typical behaviors: older systems with a bigger waiting time relax in a slower manner than younger systems with smaller waiting time, see e.g. [3][6][13] for reviews.

In the context of Anti-de Sitter space / Conformal field theory correspondence (AdS/CFT) [14], the geometric realization of aging is put forward in [11] by generalizing Schrödinger background [15][16] with explicit time dependent terms.111See also [17] for some general time dependent deformations of Schrödinger backgrounds. The authors of [11] observe that aging algebra can be realized as a subset of Schrödinger algebra using a singular time dependent coordinate transformation. Thus it is established that the time translation symmetry is broken globally, and the aging symmetry is realized as conformal Schrödinger symmetry modulo time translation symmetry. Yet, it is claimed that the time itself necessarily has to be complex to show a relaxation process, which is one of the central properties of the aging [11]. This complexification of the time is certainly not preferred.

This drawback is overcome in [12], by introducing some decay modes of the bulk scalar field along the ‘internal’ spectator direction , which is not explicitly visible from the dual field theory with Schödinger and aging symmetry. The resulting two-time correlation functions show a dissipative behavior and exhibit the characteristic features of the aging system: power law decay, broken time translation invariance and dynamical scaling between the time and spatial directions [12], see also e.g. [3][6][13] from the field theory point of view. These properties are shown in the geometric realizations of the aging background in the context of [15][16] as well as of the aging in light-cone in the context of [18][19]. The corresponding finite temperature generalizations for these two different backgrounds with asymptotic aging invariance are also studied in [12].

LCFT is a CFT, in which some correlation functions have logarithmic singularities. For the review on LCFT, see, for example, [20][21]. Recently, there have been interesting developments in LCFT as a dual CFT of higher derivative gravity model. It started in three dimensional gravity models [22]-[26] and then in their higher dimensional analogues, namely, critical gravity [27]-[31]. Generically, these theories include higher derivative terms as well as Einstein-Hilbert and cosmological constant terms and admit the AdS spacetime as a vacuum solution. In four and higher dimensional spacetime, the higher derivative terms are given by some combination of and terms. When the coupling of the higher derivative terms is tuned, the linearized perturbations of the metric around the AdS vacuum solution include the massless and logarithmic modes [27].222In the theory without these fine-tuned coupling of the higher derivative terms, the fluctuations around AdS spacetime include the massive and massless graviton modes, one of which is ghost-like. Since AdS spacetime has timelike boundary, one may impose a boundary condition and truncate the ghost-like massive mode if it falls off more slowly than the massless one[32][31]. It was shown in [33][34] that the theory with the truncation becomes effectively identical to the usual Einstein gravity at the classical but full nonlinear level and thus the correlation functions of the dual CFT can be readily read off from those in the CFT dual to Einstein gravity. If we choose special values on the coupling of higher derivative terms, the massive and massless modes become degenerate and turn into the massless and the logarithmic modes [27]. This is called critical gravity and its dual CFT becomes LCFT which includes stress-energy tensor and its logarithmic counterpart.

The holographic descriptions of LCFT have been attempted since early days of the AdS/CFT correspondence using a pair of scalar fields [35][36][37]. Recently, the correlation functions of a pair of scalar operators in non-relativistic LCFT have been considered in the context of AdS/CFT correspondence using the Lifshitz background [38]. In this paper, we study the correlation functions of a pair of scalar operators, which would play the role of order parameters, in the non-relativistic LCFT with Schrödinger and aging invariance.

We introduce a new type of action with Schrödinger invariance and generalize the correlation function to obtain the logarithmic corrections in detail in section §2. This is generalized for the system with aging invariance in section §3. In section §4, we investigate various physical properties of the logarithmic aging correlation functions and also seek possibilities to connect to the physical systems with surface growth [7][8] and their extension with logarithmic corrections [9]. We conclude in section §5.

2 Schrödinger LCFT

Let us consider some strongly coupled dimensional dual field theory with Schrödinger symmetry, living at the boundary of the Schrödinger background

(1)

where indicates the fact that we consider two different cases on the same footing, the Schrödinger background with [15][16] and the AdS in light-cone [18][19] at zero temperature. It turns out that these two geometric realizations give the same physical properties for the correlation functions at zero temperature for the scalar operators of given conformal dimensions. Their finite temperature generalizations have been realized in [39]-[43] for and in [44][45] for . It is also observed that there exist clear distinctions in metric fluctuations between these two backgrounds [12].

Motivated by the recent interests on LCFT from the holographic point of view [38], we consider two scalar fields and in this background, whose action has the following form

(2)

where represents a cutoff near the boundary, , and . Upon integrating out the field , one can obtain an equivalent action with second order time derivatives and fourth order spatial derivatives. In the context of AdS/CFT, the logarithmic extension on Lifshitz backgrounds can be found in [38], where one can find some general comments for this action.

Let us comment the general solution space of the theory described by the action (2). In [38], a bigger solution space for the Lifshitz theory with two spin fields and with different parameters and is suggested. When specialized with , the theory describes the Lifshitz LCFT. It is straight forward to follow this features and generalize the solution space of the theory by introducing two different parameters and for the fields and .

Contrasted to Lifshitz theories, Schrödinger theories have additional features due to the isometry associated with the coordinate , called “dual particle mass.” This can be shown explicitly as and considering the fields and with two different values of and , respectively.333As mentioned in the introduction, we are concerned with the aging properties which shows dissipative properties in the correlation functions. Thus we consider only the decaying modes of the scalar fields and . For the detailed discussion for the differences between the decaying modes and oscillating modes, see [12]. Thus the solution space of our fields and can be described by the parameters and . In this paper, we consider the case and , which gives the Schrödinger LCFT.

The linearized field equations for and of the action (2) become

(3)
(4)

where

(5)

which is a differential operator for the Schrödinger case.

Following [12][38], we construct the bulk to boundary Green’s function as

(6)

where we use , which give . We choose , which is in accord with the structure of the equations of motion given in (3). The Green’s functions satisfy

(7)

where is defined in (5) and . The Green’s functions and are given by

(8)

where . The normalization constants can be determined by requiring that [46].

There exists another Green’s function due to the action we choose (2), which satisfies

(9)

To evaluate , we use the same methods used in [36]. Using

(10)

and the fact that , we get

(11)

Thus

(12)

We will not write down the explicit expression, as it is lengthy and not so illuminating.

After plugging the bulk equation of motion into the action (2), the boundary action becomes of the form

(13)

For the Schrödinger case, the system has local time translation invariance, thus the time integral is trivially evaluated to give delta function. The ’s are given by

(14)
(15)

can be expressed, using

(16)

as

(17)

Note that we keep only the first non-trivial contribution in the small expansion and use the normalization for the wave function to be normalized to unity at the boundary [46]. This result is already noticed in several different places including [16][18][12]. The momentum space correlation function is given by

(18)

where we use as the dual operator of the bulk field .

The coordinate space correlation function can be computed using the inverse Fourier transformation

(19)

This is the result we are looking for. Note that the final result has the same form as that of the Schrödinger case which is given in [12].

Now let us turn to evaluate the using (12)

(20)

This is our main result in this section. The correlation function of the dual field theory acquires the logarithmic time dependence, which is very interesting. We give comments on it after we get more general results in the aging case below.

3 Aging LCFT

In the previous section, we evaluate the two point correlation function of scalar order parameters of the dual field theory with Schrödinger symmetry. Similarly, we would like to compute the correlation functions with aging symmetry, which is intrinsically out of equilibrium and time dependent.

Let us consider the following aging background

(21)

whose construction is explained in [11][12]. The parameter signifies the fact that the system has an explicit time dependence, while the parameter represents the difference between the aging in Schrödinger background with and the aging in light-cone with . The system is one of the simplest time dependent background, with which we can explicitly explore important physical properties related to out of equilibrium. The reader can find more detailed information in [11][12].

Consider two scalar fields and whose action is given by (2). The linearized field equations for and become

(22)
(23)

where we treat coordinate special by assigning and and use

(24)

which is a modified differential operator for the aging case.

The bulk to boundary Green’s function may be defined similar to the Schrödinger case given in (2).

(25)
(26)

Once again, we choose and the Green’s functions satisfy

(27)

where is defined in (24). We find that , where satisfies equations (7) - (12) of the Schrödinger case.

After plugging the bulk equation of motion, the boundary action becomes of the form

(28)

where ’s are given by the equations (14) and (15).

Note that represents our restriction on the region of time to be , which is very natural for our setup from the metric point of view. The physical significances and effects of the time boundary and associated boundary condition are explained in detail in [12].

To evaluate the momentum space correlation functions, we first evaluate the time integral for the range and define . Then

(29)

where . These results are the same as those evaluated by Laplace transform.

Thus the momentum space correlation function is given by

(30)

where is given by (17). We use the spatial translational invariance to get the delta function for , but not for due to the lack of the time translation symmetry.

Let us calculate the coordinate space correlation function as

(31)

After carrying out the integral for the first

(32)

the expression reduces to the Schrödinger case in the time independent part, whose calculation is done similarly. Finally we obtain

(33)

This result has exactly the same form as that of the aging case computed in [12][11].444 We have a slightly different expression compared to [12] using the identity of the function. This form is slightly more convenient for the evaluation of .

We can similarly evaluate correlation function as

(34)

This expression is our main result, which has all the essential properties related to the logarithmic extension of the aging holography. Let us comment about this coordinate space correlation function.

  • There are clear differences between the two functions and . The latter can be obtained simply by taking a derivative of the former with respect to the parameter . This is clear from the structure of the differential operator. The same result can be obtained by a direct, yet lengthy, calculation outlined in the appendix §A.

  • The correlation function reveals an additional logarithmic dependence only of the time difference , besides some irrelevant constant factors. This is the main feature of the aging LCFT from our bulk construction. Interestingly, the spatial dependence only shows up in the exponential part, which is responsible for the well defined decaying properties of the Schrödinger and aging correlation functions for both the large distance and small time regimes.

  • We confirm the relation between the aging correlation functions and those of the Schrödinger

    (35)
    (36)

    which is one of the important properties of aging observed in [11][12].

  • We stress that the final form of the correlation function with logarithmic extension is fixed by the backgrounds motivated by string theory solutions. Schrödinger and aging backgrounds can be constructed directly from the AdS solution using the null Melvin twist [47][48] for and the light-cone reduction [40][44] for . Attempts to construct the corresponding dual field theories have been put forward, e.g. in [41][12].

4 Physics of two-time correlation and response functions

We obtained the correlation functions and their logarithmic extensions of the Schrödinger case in §2 and aging case in §3. In this section, we describe the physical properties of them in detail and seek their applications to the KPZ universality class, motivated by its recent theoretical extension to the aging regime [9]. Aging cases offer more general class of correlation functions with explicit breaking of time translational invariance compared to the Schrödinger case. Thus we concentrate on the aging phenomena and comment on the Schrödinger case briefly.

Let us first comment about the physical identifications of and . From the point of view of our detailed construction outlined in §2, these are naturally identified as correlation functions of the logarithmic pair and . Yet, it is compelling to identify them as response functions by coupling the logarithmic pair with some external fields as . Starting from one point function , the variations with respect to and will give the response functions and , respectively. Similar identification in the field theory side is already adapted in the context of aging and its logarithmic extension [9][10]. Thus we would like to identify and as our response functions. In this section, we freely take the view them as correlation functions or response functions. We investigate the properties of and in this section and try to seek possible connections to KPZ class.

4.1 Correlation function

The two-time correlation function of the geometric realizations with aging isometry is identified and analyzed in the aging regime [12], reproducing the characteristic features of its time dependence: slow power-law decay, broken time translation invariance and dynamical scaling [13]. Here we would like to analyze the full profile of the correlation function without the logarithmic extensions, motivated by the recent interest in the field theory side [9][10]. As already noted above, the response function has the same form as the correlation function obtained in [12]. Thus we analyze its physical properties here together.

Let us consider the two-time correlation function obtained by identifying from (3)

(37)

where . This depends on two different time scales, and . is called a waiting time, which is the time passed after the system is quenched or prepared. Now the system is disturbed with small perturbations at this waiting time , and subsequently measured at which is called a response time. Explicit dependence on the waiting and response times in the two-time correlation function is a crucial difference of the aging system. The correlation functions of the Schrödinger invariant system or of the systems with time translational invariance, in general, only depends on , the time difference between the response time and the waiting time. These time dependent features manifest themselves in the term with dependence. The correlation function without the logarithmic corrections is explicitly depicted in [12] for the aging regime, which is and . There, it is demonstrated that the plots show the slow power law dynamics with explicit waiting time dependence, in addition to the excellent collapse in the aging regime revealing the universal features.

Here we would like to explore the entire range of the waiting and response times motivated by recent interesting applications [9][10]. It is more transparent to adopt the parametrization, , under which

(38)

where is the waiting time, and . Note that it is only a function of the scaling variable apart from the dependence of the waiting time .

4.2 Correlation function, roughness and exponents

Growth phenomena have attracted much attention due to their wide applications in physics and beyond. Their examples include paper wetting, epidemics, flux lines in a superconductor and fluid flow in porous media, see e.g. [5][6][2]. We would like to point out two characteristic properties of these phenomena, which are relevant for the following discussions. First, this is intrinsically nonequilibrium process and breaks the detailed balance [2], while the time independent treatments still provide much insight [5]. Simply the system is out of equilibrium due to the fact that noise or randomness is an essential part of its dynamics. Second, the concept of the local scale invariance and associated universal exponents are useful descriptions, which are similar to the equilibrium critical phenomena. The second property is particularly important for our purpose. We consider the aging as well as the growth properties both for the equilibrium and for the out of equilibrium systems, using the Schrödinger and aging holography, respectively.

Recently, Kardar-Parisi-Zhang type universality class in 1+1 dimensions is experimentally realized in the growing interfaces of the turbulent liquid crystals [7][8]. The roughness of interfaces is quantified by their width defined as the standard deviation of the interface height over a length scale at time , or equivalently the radius along the circle due to the fact that the universality class is independent of the shapes of the interface [8]. The authors measure the interfacial width and the height-difference correlation function where and denote the average over a segment of length and all over the interface and ensembles, respectively [7]. Both and are common quantities for characterizing the roughness,555We concentrate on the correlation function due to the fact that both the and show similar scaling properties, and the latter is readily available for the holographic theories. for which the so-called Family-Vicsek scaling [49] is expected to hold. The dynamical scaling property is

(39)

with two characteristic exponents: the roughness exponent and the growth exponent . The dynamical exponent is given by , and the cross over length scale is . For an infinite system, the correlation function behaves as at large times.

In the holographic approach, the dynamical exponent and the number of spatial directions are built-in. In our case, and . Thus we can fix the roughness exponent by determining the growth exponent from the limiting behavior of the correlation function, , at large times for our infinite system. Once we take , the correlation functions (2) and (3) behave as

(40)
(41)

where we assume in the latter. Thus

(42)
(43)

where the ratio between and is fixed, .666Surely, these exponents are not belong to the KPZ universality class ( and in + dimensions), which is realized in growing interfaces of the turbulent liquid crystals [7][8]. It is necessary to study the holographic theory with to directly contact with the KPZ class. Our non-relativistic conformal case with is more close to the Edwards-Wilkinson class[50] with the same dynamical exponent. These relations show that the growth exponent in the Schrödinger case is given by the parameter , which measures the conformal weight of the dual scalar operator, while the dynamical exponent is fixed from the construction as . Similar story holds for the aging case, where the growth exponent is a function of in the metric, as well as . Note that there are interesting qualitative changes for or , which become clear below.

Figure 1: Left: Log-log plot of the correlation function vs. given in (38), which shows a typical behavior of aging and is compared with the typical aging plot from numerical study, e.g. fig. 1 of [9]. Right: Log-log plot of vs. .

With these discussions in mind, let us investigate our correlation functions in detail. First of all, it is natural to have in our construction. For this range, we depict the log-log plot of the correlation functions for the Schrödinger and aging cases together in figure 1. We remind that the roughness shows the similar behaviors as the square root of the correlation function, . In the left plot, we observe the typical aging behavior, which can be compared to fig. 1 of [9]. In the right plot, we put both the Schrödinger and aging correlation functions to see clear differences. Note that our aging correlation function (38) changes its behavior around , while that of the Schrödinger does not. The latter represents the characteristic features of the system with translational invariance.

Figure 2: Log-log plot of the correlation function . We clearly observe the growing or decaying patterns. The top left panel (A) shows only decaying behaviors, valid for , while the bottom right panel (D) shows only growing ones valid for . The bottom left panel (C) shows an interesting behavior, changing from growing to decaying, valid for . Qualitative change is observed near as shown in panel B.

It is interesting to extend our analysis to the range from the phenomenological point of view. As mentioned above, there are qualitative changes at and . Let us concentrate on aging case, whose correlation function is fixed by our bulk geometry. It is interesting to see the qualitative different behaviors of the correlation functions by changing the values of parameters, and . The typical behaviors are depicted in figure 2. Their behavior for , given in panel A of figure 2, is similar to that of the depicted in figure 1. As approaches the critical value , it becomes almost flat for small, positive . On the other hand, it shows the growing behavior for .

Surprisingly, the bottom left panel (C), which is valid for the range , shows a typical growing behavior for , while decaying (aging) behavior for . We can see these changes because we choose the horizontal axis as . If we choose the horizontal axis as instead of , we only see the aging behaviors because the entire region shrinks to very small part around the origin of .

4.3 Response function

Let us consider the two-time response function , which includes the logarithmic contribution obtained from (3)

(44)

where . This is again the function of the waiting time and response time. See some relevant explanations on them near the equation (38).

Figure 3: Log-linear plot of the two-time response function with logarithmic extension (45) for with fixed . This parameter range of naturally arises from our holographic model.

To analyze , it is convenient to introduce a scaling time variable