Holographic real-time non-relativistic correlators at zero and finite temperature
Edwin Barnes111Present address: CMTC, University of Maryland, College Park, MD 20742, firstname.lastname@example.org Diana Vaman222dv3h@virginia.edu, Chaolun Wu333cw2an@virginia.edu
Department of Physics, The University of Virginia
McCormick Rd, Charlottesville, VA 22904
We compute a variety of two and three-point real-time correlation functions for a strongly-coupled non-relativistic field theory. We focus on the theory conjectured to be dual to the Schrödinger-invariant gravitational spacetime introduced by Balasubramanian, McGreevy, and Son, but our methods apply to a large class of non-relativistic theories. At zero temperature, we obtain time-ordered, retarded, and Wightman non-relativistic correlators for scalar operators of arbitrary conformal dimension directly in field theory by applying a certain lightlike Fourier transform to relativistic conformal correlators, and we verify that non-relativistic AdS/CFT reproduces the results. We compute thermal two and three-point real-time correlators for scalar operators dual to scalar fields in the black hole background which is the finite temperature generalization of the Schrödinger spacetime. This is done by first identifying thermal real-time bulk-to-boundary propagators which, when combined with Veltman’s circling rules, yield two and three-point correlators. The two-point correlators we obtain satisfy the Kallen-Lehmann relations. We also give retarded and time-ordered three-point correlators.
- 1 Introduction and Summary
2 Zero-temperature two-point functions from CFT correlators
- 2.1 Time-ordered and reverse-time-ordered two-point functions in position space
- 2.2 Time-ordered and reverse-time-ordered two-point functions in momentum space
- 2.3 Wightman two-point functions in position space
- 2.4 Wightman two-point functions in momentum space
- 2.5 Retarded and advanced two-point functions in position space
- 2.6 Retarded and advanced two-point functions in momentum space
- 3 Zero-temperature three-point functions from CFT correlators
- 4 Zero-temperature bulk-to-boundary propagators in momentum space
- 5 Finite-temperature correlators
- A Zero-temperature time-ordered three-point functions in position space
- B Zero-temperature time-ordered three-point functions in momentum space
- C Zero-temperature Wightman three-point functions in position space
- D Zero-temperature retarded and advanced three-point functions in position space
- E metric fluctuation spectrum
1 Introduction and Summary
The AdS/CFT correspondence [1, 2, 3] has enjoyed much success in elucidating aspects of strongly-interacting relativistic field theories over the past twelve years. However, there exists a broad array of strongly-coupled systems which admit a field-theoretic description but do not exhibit Lorentz invariance. Cold atomic gases in the so-called unitarity limit provide a prime example of this. The unitarity limit refers to a regime in which the system resides precisely at the crossover point between a gas of Cooper pairs and a Bose-Einstein condensate. This regime arises when the interatomic potential is tuned such that the range of the potential vanishes while the scattering length diverges, thus removing all scales in the problem. The ability to realize this scaling limit experimentally by exploiting the phenomena of Feshbach resonances  has drawn wide attention to this system and emphasized the need for more progress on the theory front to balance the successes in the laboratory. Recent approaches to this problem that employ field theory techniques directly can be found in [5, 6, 7, 8].
The unitary Fermi gas is a natural candidate for testing the waters of non-relativistic AdS/CFT as it possesses not just scale invariance but a larger group of symmetries known as the Schrödinger group, which can be thought of as the non-relativistic analog of the conformal group. The fact that the Schrödinger group is not only a close cousin of conformal symmetry but that it can also be embedded in the conformal group was a key observation in formulating the first proposal for a non-relativistic AdS/CFT correspondence [9, 10]. This proposal posited that the non-relativistic (Schrödinger) correspondence works in much the same way as its relativistic counterpart, but with a particular plane-wave-type geometry (sourced by additional non-trivial background fields) replacing AdS as the dual gravitational spacetime at zero temperature.
The form of the Schrödinger spacetime suggests that we should think of the dual Schrödinger theory as being embedded within a parent theory living in one higher dimension. If we denote the time coordinate in the parent theory by , then the Schrödinger theory is obtained by switching to light cone coordinates , where is one of the spatial directions, and then projecting the spectrum onto a sector of fixed momentum. The coordinate plays the role of time in the Schrödinger theory. If is also compactified, then the momentum assumes a discrete set of values that are naturally interpreted as particle number for the non-relativistic Schrödinger theory, and the procedure is referred to as discrete light cone quantization (DLCQ). At zero temperature, there are subtleties involved with this compactification since is lightlike, and one needs to consider large particle number in order to trust the gravity dual [11, 12, 13]. This idea of embedding the Schrödinger theory within a parent theory is closely related to the notion of Bargmann spacetimes [14, 15, 16], and it greatly facilitates the formulation of a non-relativistic version of AdS/CFT.
The original conjecture of Refs. [9, 10] has since been placed on firmer ground by subsequent work which uplifted the dual gravitational background to a solution of IIB supergravity and extended it to a finite temperature black hole solution [11, 12, 13]. Furthermore, it was shown that at finite temperature, is no longer a lightlike direction, so that subtleties pertaining to its compactification no longer arise. The work of [11, 12, 13] also clarified that the parent theory related to the Schrödinger metric constructed in [9, 10] is an example of a well known class of non-commutative theories known as dipole theories [17, 18, 19]. The dipole theories are obtained by starting from an ordinary commutative field theory and replacing regular multiplication of fields with a non-commutative star product. The particular parent theory described by the Schrödinger metric is the dipole theory that results when super Yang-Mills is deformed in this way, and the corresponding Schrödinger theory is just the DLCQ of this. In what follows, we will always refer to these theories simply as the parent and Schrödinger theories111Although Schrödinger invariance is broken at finite temperature, we will continue to use the term “Schrödinger theory” even in this case., and we will also use these terms to distinguish between their gravity duals, which are also related by DLCQ. These discoveries constitute an important milestone toward extending AdS/CFT to the realm of non-relativistic theories and applying it to improve our understanding of systems like the unitary Fermi gas.
Some of the most important information that AdS/CFT provides about a field theory are its correlation functions. As for theories exhibiting conformal invariance, the two and three-point correlation functions for Schrödinger-invariant theories also provide crucial checks of the conjectured correspondence since these correlators are largely determined by the symmetry group alone [20, 21]. In particular, the two-point functions are completely fixed by the symmetry up to an overall constant. Unlike the conformally-invariant case, however, it is not true that the functional form of the three-point functions are fully determined. It was verified in [22, 23, 24] that AdS/CFT reproduces both two and three-point scalar correlation functions.222Intriguingly, the authors of  also computed the three-point function using a certain field theory model for a unitary Fermi gas and showed that, in addition, AdS/CFT correctly computes the piece of the three-point function which is not determined by Schrödinger symmetry.
In this paper, we will focus on computing real-time correlation functions for scalar fields in Schrödinger field theories. Real-time correlation functions are of particular interest since these in turn yield quantities like conductivity and viscosity. However, real-time correlation functions in AdS/CFT pose more technical challenges relative to Euclidean ones since it is less clear how to formulate basic AdS/CFT recipes in Minkowski signature, where the bulk spacetime tends to be more complicated. Early attempts to deal with this problem yielded a case-by-case treatment [25, 26], and only in the last two years have more systematic approaches appeared in the literature [27, 28, 29]. In the context of Schrödinger-invariant theories, the methods of [27, 28] were employed by Hoang and Leigh  to compute time-ordered and Wightman two-point functions at zero temperature. We will show that the approach given in  reproduces their findings in a simple way, and we further use it to compute a wide variety of real-time two and three-point correlators at both zero and finite temperature. Our results for the zero-temperature two-point functions are also consistent with the bulk-to-bulk correlators computed in .
The complete functional form of the zero-temperature two and three-point correlation functions can in fact be computed in real time without having to invoke AdS/CFT or any other method for solving the Schrödinger field theory due to the intimate relation between the Schrödinger and conformal groups. More specifically, we may obtain correlation functions of the Schrödinger theory directly from the correlation functions of a CFT by switching to light cone coordinates and Fourier-transforming with respect to , as was originally noticed some time ago in Ref.  and more recently exploited in the context of AdS/CFT in Refs. [22, 23]. This procedure is essentially equivalent to performing DLCQ, with the role of the Fourier transform being to project onto a sector of fixed momentum. However, as stressed in , it is important to note that the Fourier-transform trick must be performed keeping noncompact so as to avoid having to construct CFT correlators on a lightlike circle. We will still refer to this procedure as DLCQ since we are free to consider fixed particle number in the resulting expressions, keeping in mind the zero-temperature subtleties mentioned earlier.
We will apply the Fourier-transform technique to compute various zero-temperature correlators in real time. Although the authors of [22, 23] computed real-time correlators, they did not keep track of the type of correlator (e.g. time-ordered, retarded, etc.), as their primary focus was on testing whether non-relativistic AdS/CFT reproduces Schrödinger-invariant expressions. Since the prescriptions which distinguish between the different types of real-time correlators are well known in the case of a relativistic CFT (see  for a review), it is a straightforward task to perform DLCQ on these correlators to produce real-time Schrödinger correlators, and we will show that the type of correlator is preserved under DLCQ. In addition, we verify that the standard Kallen-Lehmann relations are satisfied by the various real-time Schrödinger correlators.
We stress that the CFT correlators that we Fourier transform are not the correlators of the parent theory. In particular, note that the parent dipole theory is not Lorentz-invariant [17, 18, 19]. Therefore, the resulting Schrödinger-invariant correlators will not contain the same overall constants as those of the DLCQ of the dipole theory. Our aim in applying the Fourier transform to CFT correlators is to ascertain the form of the various real-time Schrödinger correlators at zero temperature; this provides us with an important check of the exact Schrödinger theory correlators we will compute from non-relativistic AdS/CFT.
We compute the zero-temperature correlators in momentum space as well as position space. DLCQ applied to momentum space correlators simply amounts to performing a rotation in the plane spanned by and , the energy and momentum associated with and . In the case of three-point functions, we express the time-ordered and retarded correlators in a way that is very reminiscent of AdS/CFT, namely in terms of an integral over what are readily interpreted as three bulk-to-boundary propagators in the Schrödinger spacetime. This allows us to define in a natural way non-relativistic Feynman and retarded bulk-to-boundary propagators which in turn are easily verified to reproduce the corresponding two-point correlators that we compute directly in field theory. Wightman propagators are then defined in such a way that they reproduce Wightman two-point functions. The exercise of constructing the different real-time bulk-to-boundary propagators and establishing their interrelations not only provides additional consistency checks of non-relativistic AdS/CFT, but also helps set the stage for the computations of thermal correlation functions, which rely heavily on correctly identifying these propagators.
One of the main objectives of this work is to compute real-time two-point and three-point non-relativistic correlation functions for scalar operators dual to minimally coupled scalars at finite temperature. The Schrödinger theory at finite temperature also possesses a nonzero chemical potential which breaks Schrödinger invariance, meaning that we can no longer receive guidance by applying the Fourier-transform trick to CFT correlators as in the zero-temperature case. With the confidence gained from cross-checking AdS/CFT with field theory at zero temperature, we therefore focus solely on AdS/CFT in the finite temperature case.
The first step is to compute real-time bulk-to-boundary propagators. This is done by starting with the Euclidean version of the metric which describes a black hole in the Schrödinger spacetime [11, 12, 13] and solving the scalar wave equation to obtain the Euclidean bulk-to-boundary propagator in momentum space. This is the Euclidean propagator in the gravity dual of the parent dipole theory. We analytically continue this to a retarded propagator and then perform DLCQ to get the retarded Schrödinger bulk-to-boundary propagator. In , it was shown that in the relativistic case, the other real-time thermal propagators can be derived from the retarded one using certain identities. These identities naturally extend to the parent dipole theory, where we can then apply DLCQ to show that similar relations between Schrödinger propagators also hold.
Once we have the real-time Schrödinger bulk-to-boundary propagators, we can assemble them into real-time two and three-point correlators. This problem was solved in the case of thermal SYM in  by extending Veltman’s circling rules [31, 32, 33] to the gravitational theory on the black hole spacetime, and similar arguments should apply in the present context as well. We compute two-point correlators in this way and confirm that the results satisfy standard Kallen-Lehmann relations. The retarded three-point was worked out explicitly in , and we borrow the result to immediately write down the three-point function for the parent dipole theory. A simple application of DLCQ to the result in turn gives the retarded Schrödinger three-point correlator. In addition, we apply the circling rules to construct the time-ordered Schrödinger three-point function.
Before proceeding with the calculations, we pause for a moment to briefly discuss the dimensionality of the theories we consider. We keep the number of spatial dimensions (denoted ) arbitrary as much as possible. There is certainly no obstacle to doing so in our discussion of DLCQ applied directly to CFT correlators at zero temperature. Furthermore, we may still consider the dimensional Schrödinger spacetime as a “bottom-up” holographic model even though this spacetime can only be uplifted to a solution of IIB supergravity in the case . Leaving the dimension arbitrary here also facilitates comparison with our field theory results. We could adopt a similar philosophy toward the Schrödinger black hole spacetime, however it is considerably more difficult to solve the wave equation for values of not equal to 2, so we restrict ourselves to in this case.
We will consider various types of correlators and bulk-to-boundary propagators throughout the paper, and we distinguish between these with different fonts of the letter , as we will now clarify. Relativistic CFT correlators are denoted by , while non-relativistic Schrödinger correlators are denoted by . The symbol specifies bulk-to-boundary propagators in the gravity dual of the parent theory. We also define bulk-to-boundary propagators for the gravity dual of the Schrödinger theory, and these we denote by . In addition, we use tildes to denote momentum space functions, the superscript to denote causal three-point correlators, and additional decorations to distinguish between the different types of real-time functions. In particular, the subscripts , , and denote time-ordered, retarded, and advanced functions respectively. Reverse-time-ordered functions are distinguished from time-ordered functions by an additional bar (e.g. ) which should not be confused with complex conjugation, for which we use the symbol . Wightman two-point functions have superscripts , while Wightman three-point functions are specified by a subscript which shows the order of operator insertions. For example, is a Schrödinger Wightman three-point function in momentum space proportional to , where the are arbitrary scalar operators in the Schrödinger theory.
The paper is organized as follows. In section 2, we compute all types of scalar real-time Schrödinger two-point correlators at zero temperature in both position and momentum space by applying the Fourier-transform trick to relativistic CFT correlators. We also verify that the results satisfy the Kallen-Lehmann relations. In section 3, we perform similar computations for real-time zero-temperature three-point correlators. We confirm that standard relations among the three-point functions hold, and we also express the results in a way that is very suggestive of an AdS/CFT calculation, allowing us to identify Schrödinger “bulk-to-boundary propagators”. In section 4, we construct real-time bulk-to-boundary propagators directly in the zero-temperature gravity dual and show that the results are consistent with the answers obtained using the Fourier-transform trick in sections 2 and 3. Finally in section 5, we apply the intuition developed in the zero-temperature case to construct thermal real-time bulk-to-boundary propagators for scalar fields. These are then employed to compute two and three-point scalar correlators for the Schrödinger theory at finite temperature. The first four appendices each contain details about the application of the Fourier-transform trick to particular zero-temperature three-point correlators considered in section 3. In appendix E, we construct an explicit example of a massive scalar in the effective 5d gravity theory which arises when a fluctuation in a particular off-diagonal metric component has non-trivial charges on the 5d internal space.
2 Zero-temperature two-point functions from CFT correlators
In this section, we will obtain Schrödinger-invariant non-relativistic zero-temperature two-point functions in position space by starting with real-time relativistic conformal two-point functions in light cone coordinates and Fourier-transforming with respect to one of these coordinates. We also obtain momentum space two-point functions by translating the Fourier-transform trick to momentum space, where it becomes a simple redefinition of momenta. We will pay particular attention to how the different types of relativistic real-time functions (e.g. time-ordered, retarded, etc.) map to the different types of non-relativistic functions. We will see that the type of real-time correlator is preserved under the special Fourier transform. We also verify that standard Kallen-Lehmann relations between the various real-time functions are satisfied by the non-relativistic correlators we obtain.
2.1 Time-ordered and reverse-time-ordered two-point functions in position space
Consider first the time-ordered relativistic conformal function:
Here, , where is the relativistic time coordinate, and is one of the spatial coordinates. We use the mostly plus signature for Minkowski space. The coordinate vector represents the remaining spatial coordinates which coincide with the spatial coordinates of the non-relativistic theory: . Following [21, 23], we can obtain a non-relativistic two-point function by performing a Fourier-transform with respect to the direction:
This integral can be done by introducing a Schwinger parameter:
The integral on yields a -function:
The -function will evaluate to zero unless . Therefore, we find
This has the form of a non-relativistic time-ordered two-point function where the non-relativistic time coordinate is . Hoang and Leigh  found the same result employing a very different approach [27, 28]. We have retained the in the final expression because it removes a potential singularity at . It is clear from (2.2) that this singularity should not be present.
If we instead start with the reverse-time-ordered relativistic two-point function:
the same calculation now gives
This has the form of a non-relativistic reverse-time-ordered function.
2.2 Time-ordered and reverse-time-ordered two-point functions in momentum space
To make it clear how the Fourier-transform trick translates to momentum space, we first consider a simple example. The Fourier transform of the conformal time-ordered two-point function with and is
where and . Introducing the light cone coordinates , we can write this as
We now define the non-relativistic energy and mass in terms of the relativistic energy and momentum component :
Applying the inverse of this map,
This is the non-relativistic time-ordered propagator in momentum space with non-relativistic energy
It is easy to check that (2.12) is the Fourier transform of (2.5) with and :333Our conventions for both relativistic and non-relativistic Fourier transforms are established in this subsection. In particular, note that we define our non-relativistic Fourier transform with an additional overall factor of because the non-relativistic time coordinate is .
Obviously the same approach can be applied to any other momentum-space function since, from the point of view of the Fourier transform in (2.8) and (2.9), we are simply redefining momentum-space variables. Starting from the relativistic time-ordered two-point function with arbitrary and ,
the map (2.11) gives the non-relativistic time-ordered function444The second line of (LABEL:nrfeynmanmomentumtwopt) can be obtained with the help of the identity , .:
The reverse-time-ordered two-point function is just minus the complex conjugate of :
2.3 Wightman two-point functions in position space
Next consider the following relativistic Wightman function:
We can facilitate taking the Fourier transform by rewriting this as
Before taking the Fourier transform, it is useful to first factor out a factor of from the denominator:
One can then use the positivity of to introduce a Schwinger parameter as we did in the case of time-ordered functions in the previous subsection. The integral then gives a -function:
Since the argument of the -function does not contain , a step function involving does not arise in this case. The result is then
This agrees with the result found in  using the methods of [27, 28]. The role of the in the exponent is to ensure that vanishes at , while the in the denominator controls the location of the branch cut for non-integer :555Throughout this paper, we adopt the convention that if no prescription is given explicitly.
If we instead consider starting with the other relativistic Wightman function,
the same steps lead to an -integration as above, but with instead of appearing in the integrand, yielding
are satisfied.666When , the in the exponents can be discarded relative to . When , , , and each vanish identically, and (2.26) is satisfied trivially.
2.4 Wightman two-point functions in momentum space
Now consider the relativistic Wightman functions with and in momentum space:
Under (2.11), these transform to
For (), the -function has support at positive (negative) values of , so that
which is the Fourier transform of (2.22) for , , and
To obtain the non-relativistic Wightman functions for general and , we start with the Fourier transform of (2.18):
The map (2.11) then gives
We have used that
to rewrite the argument of the step function before applying the map. The second relativistic Wightman function has the form
Now observing that
we see that the map (2.11) leads to
So far, we have seen that the relativistic (reverse-)time-ordered two-point function transforms into a non-relativistic (reverse-)time-ordered two-point function, and the relativistic Wightman functions map to non-relativistic Wightman functions. It remains for us to check what happens to relativistic retarded and advanced functions under this mapping.
2.5 Retarded and advanced two-point functions in position space
The relativistic conformal retarded two-point correlator can be written in terms of Wightman functions:
In terms of light cone coordinates, we therefore have
Factoring out the in the denominators, assuming , and Fourier-transforming leads to
Notice that we have chosen to write instead of in the denominators. Since we are assuming , we may rewrite , and the term can be eliminated by rescaling the in the denominators of the integrands. The in the factors appearing outside the integrals cannot be neglected even though because the in these factors control branch cuts, as we discussed in the previous subsection on Wightman functions.
Changing the integration variable to , we have
The integrals can be evaluated using
and we find
This has the form of a non-relativistic retarded two-point function. This expression is valid for ; when , it is easy to check that . (This can be done by setting in (2.38) and Fourier-transforming.) An analogous calculation reveals that if we start with the relativistic advanced function, we obtain the non-relativistic advanced function:
A shortcut to obtaining this result is to write down the analog of (2.39) for and to notice that can be obtained from by sending and in the final answer. It is easy to check using (2.22) and (2.25) that standard relations between retarded/advanced and Wightman functions:
hold for the non-relativistic two-point functions we have obtained.
We conclude that the type of real-time two-point functions is preserved under the mapping which takes us from relativistic to non-relativistic functions.
2.6 Retarded and advanced two-point functions in momentum space
The Fourier transform of the relativistic retarded two-point function (2.38) with and is given by
Applying (2.11), we obtain
This function has a pole at
Since this pole lies in the lower half -plane regardless of the value of or , we see that this function has the analytic properties expected of a retarded Green’s function. We may therefore redefine to absorb the positive coefficient which multiplies it in :
The advanced function is the conjugate of this:
Comparing these results with (2.12), we see that , and obey the following relation:
On the other hand, we can obtain what appears to be a slightly different relation directly from (2.46):
At first glance, it may seem surprising that these two relations hold simultaneously. However, their compatibility has a simple origin in the form of the imaginary part of the retarded two-point correlator. Using (2.46), we can write this as . Note that only the imaginary part is affected by the presence of the step functions in (2.50) and (2.51). Since the -function requires that and have the same sign, this is equivalent to , which agrees with (2.48). Also notice that we can write the relation in a third way:
The Fourier transform of (2.38) for general and is
where was given in (2.15). This yields
Similar reasoning leads to the following advanced two-point function:
It is not difficult to check that the relations (2.50), (2.51), and (2.52) hold for general and . Furthermore, the relation (2.50) provides a useful check of the consistency of our results since it must also hold in position space, and one can readily verify that (2.5), (2.42), and (2.43) satisfy (2.50). We will see later on in section 5 that the second relation, (2.51), generalizes to a relation that is also valid at finite temperature.
3 Zero-temperature three-point functions from CFT correlators
In this section, we apply the Fourier-transform trick to relativistic three-point functions to obtain Schrödinger-invariant non-relativistic three-point functions in position space. We do this for time-ordered and Wightman functions and show that the usual identity relating these types of correlators is satisfied. In position space, retarded and advanced functions can be expressed in terms of the Wightman functions .
We compute non-relativistic time-ordered and retarded/advanced three-point functions in momentum space as well. This can be done either by computing the Fourier transforms of the non-relativistic position space three-point functions, or by starting with relativistic conformal three-point functions in momentum space and performing an appropriate redefinition of momenta as we did for two-point functions in section 2. We will apply the former approach to obtain time-ordered functions and the latter approach for retarded/advanced functions. Wightman three-point functions are quite unwieldy in momentum space, so we do not include them here.
The final answers we obtain for the momentum space three-point functions are of the form one would expect from an AdS/CFT calculation. In particular, we express the results as integrals of products of three functions which are naturally interpreted as bulk-to-boundary propagators for scalar fields in non-relativistic AdS/CFT [9, 10]. In section 4, we reproduce these bulk-to-boundary propagators by applying DLCQ to propagators in the gravity dual of the dipole theory and show that they give rise to two-point functions which are consistent with the non-relativistic correlators we computed in section 2.
3.1 Time-ordered three-point functions in position space
The relativistic time-ordered three-point function in position space has the form
We have introduced the notation . The constants are related to the conformal dimensions of the scalar fields:
To obtain the non-relativistic correlator , we must Fourier-transform with respect to the . The integrals can be computed with the help of Schwinger parameters, and the details can be found in appendix A. The result is