A Holographic Model For Quantum Critical Responses
We analyze the dynamical response functions of strongly interacting quantum critical states described by conformal field theories (CFTs). We construct a self-consistent holographic model that incorporates the relevant scalar operator driving the quantum critical phase transition. Focusing on the finite temperature dynamical conductivity , we study its dependence on our model parameters, notably the scaling dimension of the relevant operator. It is found that the conductivity is well-approximated by a simple ansatz proposed in katz () for a wide range of parameters. We further dissect the conductivity at large frequencies using the operator product expansion, and show how it reveals the spectrum of our model CFT. Our results provide a physically-constrained framework to study the analytic continuation of quantum Monte Carlo data, as we illustrate using the O(2) Wilson-Fisher CFT. Finally, we comment on the variation of the conductivity as we tune away from the quantum critical point, setting the stage for a comprehensive analysis of the phase diagram near the transition.
A quantum critical (QC) system can be broadly defined as a quantum many-body system with a gapless energy spectrum, and generically taken to be interacting. Some of the best understood instances are described by conformal field theories (CFTs). A canonical example of a CFT is the QC phase transition at zero temperature in the quantum Ising model in 1+1 or 2+1 spacetime dimensions book (), which results from tuning the transverse magnetic field across a critical value. QC systems essentially come in two flavors: QC phase transitions or QC phases. The former fundamentally necessitate tuning, such as the QC point in the quantum Ising model which results from tuning the transverse magnetic field across a critical value. In contrast, a QC phase exists without fine-tuning. A simple example is a two-component Dirac fermion in 2+1 dimensions. A mass term breaks time-reversal symmetry and is thus forbidden if we demand that the symmetry be preserved. (One could turn on a chemical potential to obtain a metal but this is not the type of tuning we are referring to, as we shall see). In contrast, the mass term of the scalar -theory, describing the QC Ising transition, is invariant under all the symmetries of the theory and thus needs to be fine-tuned to reach the quantum phase transition point.
An important challenge in the study of CFTs/QC systems is to understand their real-time dynamics fisher1990 (), especially at finite temperature damle (). In the linear response regime, important examples are the frequency-dependent conductivity and dynamical shear viscosity . Because the corresponding theories are strongly interacting, perturbative QFT methods are of limited use in analyzing the dynamics. At the same time, nonperturbative quantum Monte Carlo simulations suffer from the perennial problem of analytically continuing Euclidean data to real time. In contrast, holography yields real-time results for strongly interacting systems lacking quasiparticles. However, in the context where the duality is best understood, these CFTs correspond to large- gauge theories Maldacena (). It is thus important to identify which of their dynamical properties are generic, and which are special to the holographic regime.
Progress in applying holography and general non-perturbative CFT methods to these questions was recently made in selfdual (); Myers:2010pk (); sum-rules (); ws (); ws2 (); natphys (); katz (); chen (); will-hd (). For instance, new sum rules for the dynamical conductivity of (conformal) QC systems were first discovered using holography sum-rules (); ws (); ws2 () (see also justin2 () in the context of doped holographic SCFTs), and subsequently proved for a large class of CFTs katz (), including the Wilson-Fisher CFTs. Further, references natphys (); katz () constructed holographic models which allowed comparison with quantum Monte Carlo (QMC) results for the dynamical conductivity in the O(2) Wilson-Fisher fixed-point theory. In particular, ref. katz () recognized that the relevant scalar operator that needs to be tuned to reach the QC phase transition plays an important role in determining the dynamics. Hence the holographic studies in katz () incorporated this operator in an essential way. However, a shortcoming of their construction was that the dual of the relevant operator in the boundary theory was not incorporated as a dynamical field in the bulk gravity theory. Our primary goal in this paper then is to construct a new holographic model where the relevant boundary operator is incorporated in a self-consistent way. The key feature, which distinguishes our holographic model from previous models, is that it incorporates a natural bulk interaction which ensures that the relevant operator acquires a thermal expectation value. Further, as shown in figure 2, it allows us to easily study the dynamical conductivity for a wide range of conformal dimensions and of the two holographic parameters, and (which are proportional to the OPE coefficients, and , respectively – see further explanation in section 3). Our model also provides a holographic framework where we can examine the response functions as we tune away from the quantum critical point. Although we focus on the dynamical conductivity in 2+1 dimensions, our analysis can be extended to treat other response functions, such as the shear viscosity , in arbitrary dimensions.
The paper is organized as follows: in section 2, based on general CFT considerations, we present the key ingredients that a holographic model will need to describe QC response functions. In section 3, we present our holographic model and focus on evaluating the dynamical conductivity. We then analyze in detail the large-frequency asymptotics of the conductivity in section 4, and compare the results with those predicted by the operator product expansion (OPE). We conclude in section 5 with a brief discussion of our results and we also make some preliminary comments on the behaviour of the boundary theory when we detuned away from the QCP. This paves the way for the holographic study of observables in the entire phase diagram surrounding a QCP. We have four appendices to discuss certain technical details: Appendix A provides the details of calculating various vacuum correlators in the boundary CFT, which are used in section 4. Appendix B describes some of the details for the calculation of the dynamical conductivity made in section 3.1. In appendix C, we consider the bulk scalar profile and conductivity for special cases of the conformal dimension of the relevant operator. Appendix D extends the high frequency expansion of the conductivity in section 4.1 to second order in the expansion.
2 Required ingredients: CFT analysis
In our holographic study, we will be mainly concerned with canonical QC phase transitions described by CFTs. These are realized by tuning a (single) coupling to a specific value, which will be zero here:
where the local scalar operator is relevant, i.e., its scaling dimension satisfies . Unitarity also requires that . At this point, it may be useful to recall the action of the QFT in :
where is a real -component vector. For all , the RG fixed point at finite interaction corresponds to a non-trivial CFT, often called the O() Wilson-Fisher (quantum critical) fixed point. For the case of a single real scalar, , this critical point corresponds to the Ising CFT. The relevant scalar here is the mass operator, and the corresponding coupling that needs to be tuned to zero to reach the QCP. In general, is an important operator in the spectrum, and it is not surprising that it plays a key role in determining the quantum dynamics of various observables. In our holographic model, we must include a scalar field in the bulk gravity theory to be dual to in the boundary theory.
At finite temperature, typically acquires an expectation value:
where is a pure number determined by CFT data (scaling dimensions and OPE coefficients). Of course, the expectation value (3) vanishes at zero temperature since, by definition, is not sourced at the QCP. That is, at , the vacuum of the corresponding CFT contains no scales and so the expectation value of all operators must vanish. The Wilson-Fisher CFT described above provides a simple example with this behaviour, with the mass operator acquiring an expectation value as shown in eq. (3) at finite katz (). However, not all CFTs describe QCPs (by the present definition), since in some cases there is no relevant scalar that is invariant under the full symmetry group of the CFT. An elementary example is the free Dirac fermion CFT, where the mass operator breaks time-reversal symmetry. As a consequence, it does not acquire a thermal expectation value, i.e., . Symmetry requirements alone are sufficient to set the mass to zero, so that the Dirac CFT does not need to be fine-tuned, unlike (2), and the theory describes a quantum critical phase not a point. Typical holographic theories that have been studied up to this point do not exhibit the behaviour shown in eq. (3). Rather, at finite temperature, only the stress tensor acquires a nonvanishing expectation value in these models. Hence, a key ingredient of our holographic model will be a natural mechanism which ensures that eq. (3) holds.
Finally, the large-frequency/momentum structure of two-point correlation functions is determined by the OPE of the corresponding operators willprl (). For example, the conductivity is determined by the current-current correlator and hence the large-frequency structure is given by the OPE. In this context, the first non-trivial operator in the OPE is the relevant scalar katz (). Hence to study the conductivity, we first need to introduce a bulk gauge field in our holographic model to match the current in the boundary theory. Further, we will need include appropriate bulk interactions to realize the property that the OPE coefficient corresponding to the fusion is non-zero in the boundary theory. Alternatively, the vacuum three-point function must be non-zero, as will be illustrated in section 4.
3 Holographic model
Here we describe an explicit holographic model with all of the ingredients described in the previous section. We will be focusing our attention on three-dimensional CFTs and so in the bulk, we begin with four-dimensional Einstein gravity coupled to a negative cosmological constant,
Here, is the Planck length, which is related to Newton’s gravitational constant by . The vacuum solution is then simply the anti-de Sitter (AdS) geometry with the curvature scale . The ratio of these two scales determines the central charge of the boundary CFT, e.g., see airport (): . Another useful solution, which will set the background geometry for our calculations, is the planar black hole:
with . The position of the event horizon is and taking yields the familiar Poincaré patch of AdS space. According to the usual AdS/CFT correspondence, this solution (5) is dual to the CFT at finite temperature (and zero chemical potential), where the temperature is given by
It will simplify our calculations to change to a dimensionless radial coordinate , with which the metric becomes
where . In these coordinates, corresponds to the asymptotic AdS boundary and is the black hole horizon.
|Bulk coupling||Bulk operator||CFT correlator ()||Observable|
To ensure that the boundary CFT also contains a (conserved) current and a scalar operator with conformal dimension , we introduce the following bulk actions for a (massless) gauge field and a scalar field with mass :222Latin (Greek) indices are used to indicate Lorentz vector or tensor quantities in the bulk (boundary).
where is the field strength of , and is the Weyl curvature tensor. The scalar action (8) is normalized with a factor of to ensure that the scalar field is dimensionless, which will be convenient in the following calculations. The gauge field has the usual dimension of inverse length and so the Maxwell coupling is dimensionless. The scaling dimension is taken to be above the unitary bound for dimensional CFTs, . We further note that in the range , the theory will contain at least one other relevant scalar, which can be thought of as . In this regime, the CFT dual thus describes a multicritical point instead of a simple critical point; we refer the reader to section 4.3 for further details. Further, although our motivation in the previous section considered relevant operators with , the following holographic analysis easily extends to irrelevant operators with as well. However, certain technical issues arise for — see further comments in footnotes 4 and appendix C.
If we supplement eq. (4) with the free actions in eqs. (8) and (9), i.e., with , a thermal state (with vanishing chemical potential) in the boundary CFT is still described by the above black hole solution (7). In particular, and would both vanish in the bulk solution.333The gauge field vanishes because we have assumed that the black hole is not charged, i.e., the chemical potential vanishes in the boundary theory. If bulk scalar has a positive mass-squared, i.e., , there are no hair theorems which ensure that vanishes, e.g., Torii:2001pg (). However, with a negative mass-squared, i.e., , stable black hole solutions can be found with nontrivial scalar hair, e.g., Torii:2001pg (); Winstanley:2002jt (); Buchel:2007vy (); Buchel:2013lla (). However, from a holographic perspective, the latter solutions involve turning on the (dimensionful) coupling constant for the corresponding operator in the boundary theory e.g., Buchel:2007vy (); Buchel:2013lla (). However, as explained below, we wish to focus on the critical theory in which this coupling vanishes and so we impose boundary conditions where the only black hole solutions have vanishing for the free theory. However, a key ingredient, which we wanted to include in our holographic model, is that the scalar operator should acquire a nonvanishing thermal expectation value. Therefore the dual scalar must be sourced to have a nontrivial profile in the black hole background. The latter is engineered by adding the new interaction in eq. (8) which couples the scalar field to the Weyl curvature. The Weyl curvature vanishes in the vacuum AdS geometry since the latter is conformally flat and hence the vacuum of the boundary CFT remains stable. However, provides a nontrivial source for the scalar in the black hole background (7) and as desired then, in the CFT. We show in appendix A that the (dimensionless) coupling is related to the CFT parameter controlling the vacuum three-point function .
Lastly, as described above, the three-point function must be nonvanishing in the vacuum of the boundary theory. The simplest way to accomplish the latter is to add the interaction in eq. (9). The (dimensionless) coupling is then dual to the CFT parameter which controls the desired three-point function. The four dimensionless couplings which characterize the bulk gravitational theory and their role in the dual boundary CFT are summarized in table 1.
Now in principle, one would want to solve the full nonlinear equations of the total action to solve for a new black hole solution in which the scalar field has a nontrivial profile. However, in the present paper, we only approach this problem to leading order in a perturbative approach. In particular, we will construct the background perturbatively in the amplitude of the scalar field and in fact, we only perform the present calculations to leading order in this expansion. Alternatively, since the bulk scalar is sourced by the interaction in eq. (8), one can think that we are working to leading order in a small expansion.
Because the black hole background (7) is translation invariant in the boundary directions, only depends on . Hence we can solve eq. (10) with a simple ansatz , in which case the above equation reduces to
This equation has an exact solution:444This representation of the solution is only valid for . In particular, the integral defining in eq. (13) diverges for — see further comments in appendix C. Further, the two independent solutions presented in eq. (12) are actually identical for . Of course, the coefficients of and also diverge for this particular value of . The correct solution for is presented in appendix C. However, we note that the conductivity is still a smooth function of at this special value and so where results are presented for in the following, we have actually evaluated our expressions with a nearby value of the conformal dimension, i.e., .
where and are integration constants and denotes the standard hypergeometric function. Further, and are given by
Given the definitions in eq. (13), we have at the AdS boundary.
The above solution has the expected asymptotic behaviour for with
Note that since we are using the dimensionless radial coordinate here, both of the coefficients, and , are also dimensionless. Recall the first term is the non-normalizable mode, and the coefficient corresponds to the coupling which deforms the boundary theory as in eq. (1). Hence, as is standard in the AdS/CFT correspondence, tuning this boundary condition for the bulk scalar field corresponds to tuning the dual coupling constant in the boundary field theory. In particular, we set since we want to study the behaviour of the critical theory (to compare to katz ()).555This choice corresponds to the tuning needed to reach a QC phase transition discussed in section 2. In section 5, we provide some preliminary remarks on tuning away from the critical point by choosing instead a nonvanishing value of , but leave this situation for detailed study in new (). Of course, setting is also what allows us to consider irrelevant operators in the following. As is evident from eq. (14), the scalar would diverge near the boundary with and and hence its gravitational back-reaction would destroy the asymptotic AdS geometry. Further the second term in eq. (14) corresponds to the normalizable mode, and the corresponding coefficient is dual to the expectation value . To fix this integration constant , we demand that the scalar field be regular at the horizon. As , the solution (12) has a (potential) logarithmic divergence which is eliminated by setting
Note that and are both finite and can be determined by numerically evaluating the integrals in eq. (13).
Figure 3 shows the resulting scalar profiles for and 4, in comparison to a simple power law , as used in katz (). The value of the coefficient in the power-law profile was chosen to match that in the holographic solution so that the two profiles exactly agree as . Then we find that for relevant operators (i.e., ), the scalar profile produced by eq. (12) is larger than the power-law profile in the vicinity of the horizon (i.e., ). Further, the relative separation of the two profiles is increased as is decreased (below 3). In contrast, for irrelevant operators (i.e., ), the solution (12) is smaller than the power-law profile near the horizon. When the scalar operator is marginal (i.e., ), in fact, the exact solution and the simple power-law are identical, so that as shown in appendix C.1.
Hence to leading order in our perturbative expansion, our background is the black hole metric (7) with scalar field solution (12) with set as in eq. (15) and . As mentioned above, is dual to the expectation value of the operator and using the usual holographic dictionary, we find
where is given in eq. (15). We note that for a fixed dimension, the expectation value above can be positive or negative depending on the sign of . Our holographic calculation recovers the expected form given in eq. (3). Recall our perturbative framework assumes that the amplitude of the bulk scalar is small (), which is equivalent to or .
The above expression may appear to vanish when , however, as noted in footnote 4, our scalar field solution eq. 12 breaks down at this point. Hence the scalar profile and any subsequent calculations must be reconsidered for this particular value of the conformal dimension, as discussed in appendix C — the resulting expectation value is given in eq. 96.
3.1 Holographic conductivity
Next we examine the charge response, in particular the frequency-dependent conductivity, of the boundary theory in our holographic model. Note that in our perturbative approach, the scalar profile is directly proportional to the coupling and further the scalar modifies the charge response through the interaction in eq. (9), which in turn is controlled by . Therefore we will find that the charge response only depends on the product , not on their separate values. Thus, for example, the normalized dynamical conductivity is only a function of two parameters, and , as illustrated in figure 2.
where is the outward-pointing radial unit vector. The charge density then satisfies the diffusion equation Kovtun:2003wp ()
The value of the scalar field at the horizon is given by
where is the digamma function.
Figure 4 shows the diffusion constant as a function of the scaling dimension of the scalar operator (while holding the combination fixed). For relevant operators (i.e., ), the diffusion constant calculated from the exact solution is larger than for the pure power-law , while for irrelevant scalars (i.e., ), the ratio of the two results is reversed. As expected, the two curves cross at where the two scalar profiles are identical.
The conductivity at zero frequency is given by Ritz:2008kh (); Myers:2010pk ()666Implicitly, we have set here, where is the charge of the quantum charge carriers — see katz (). Recall that in our holographic model.
Of course, the results shown in figure 4 are readily understood in terms of the behaviour of the scalar profiles illustrated in figure 3. That is, we found that the profile produced by our holographic model is smaller (larger) than the simple power-law profile near the horizon for (). Note that is finite in our holographic model, even in the absence of momentum dissipation. This phenomenon is possible for systems where momentum and current are distinct, like in CFTs damle (). However, in general “small-N” CFTs like the Wilson-Fisher QCPs with a finite symmetry group (2), it is expected that will show a weak logarithmic divergence that arises from the phenomenon of long-time tails of hydrodynamics kovtun-rev (); natphys (). This is tantamount to saying that current-current correlations decay more slowly at long-times because of current conservation. It was shown simon () that these long-time tails can be recovered in holography by including quantum corrections in the bulk, i.e., they are suppressed by a factor of .
The frequency-dependent conductivity is given by Myers:2010pk ()
where the temperature is given in eq. (6) and is the Fourier transform of (the -component of) the gauge field. The latter profile is determined by numerically solving the gauge field equations of motion resulting from eq. (9), with appropriate boundary conditions at the event horizon — see details in appendix B.
We plot the resulting as a function of real and Euclidean frequency in figures 5 and 6 for various values of the scaling dimension . In each case, we compare the conductivity calculated with our holographic model to that calculated with a simple power-law profile for the bulk scalar , as in katz (). The two results are nearly in agreement. In particular, in figure 5, we adjust the amplitude of the scalar profile with to fit to conductivity for Euclidean frequencies to the quantum Monte Carlo data of katz (); natphys () and we see that the two results agree almost exactly for Euclidean frequencies — see further discussion in section 5. The largest discrepancies in all of these comparisons appear at the origin , where the conductivity probes the holographic background near the event horizon. As noted above, the conductivity in our holographic model is higher (lower) than for the power-law profile when ().
4 Asymptotic expansion of conductivity & OPEs
The asymptotic expansion of the conductivity for frequencies which are large compared to the temperature is useful for many reasons. First, it reveals important properties about the operators with low scaling dimensions. It also allows us to establish non-trivial sum rules, e.g., Son09 (); sum-rules (); justin2 (); ws (); katz (). Further, it plays a role in the comparison of holographic response functions with Euclidean data for the conductivity, as the latter is available from Monte Carlo simulations for frequencies exceeding , e.g., natphys (); katz (). In this section, we first obtain the expansion in our holographic model directly from the equation of motion for the gauge field dual to the current. Then we re-derive the expansion by using the operator product expansion (OPE) of the boundary CFT. This analysis reveals fundamental properties of our model, and the corresponding dynamical charge response. Let us also note that similar analyses of the modifications of the high frequency behaviour of the conductivity and viscosity due to scalar expectation values was made for a variety of other holographic backgrounds in justin1 (); justin2 (). In those studies, the scalars were chiral primaries that acquired an expectation value as a result of turning on a chemical potential.
4.1 High frequency expansion
We now compute the conductivity at frequencies much greater than the temperature. Working in Euclidean frequencies, this corresponds to evaluating with .777Our notation alludes to Matsubara frequencies that arise in finite temperature quantum field theory. In this case, these frequencies would be discrete multiples of , however, can be thought of as a continuous variable in the following.
In the following, we will calculate the high frequency asymptotics perturbatively in the dimensionless coupling . Recall that this coupling controls the strength of the interaction in eq. (9), which determines how the scalar operator in the boundary modifies the conductivity. In this approach, it is convenient to first change coordinates from to , where . The boundary, , corresponds to , however, the horizon is stretched to in these new coordinates. With this coordinate choice, the equation determining the gauge field profile — see eq. (74) — becomes
where we have introduced the rescaled (dimensionless) Euclidean frequency
Now in our perturbative approach, we expand the gauge profile as . Similarly, expanding the gauge equation (23), the zeroth order component satisfies , and the solution (which is regular or “in-falling” at the horizon) is
Next at first order in , eq. (23) yields
This equation can be solved with the use of the following Green’s function
where and vanishes at and at . The solution to eq. (26) is then given by
To calculate the conductivity, we must evaluate
Substituting the power series for into eq. (29) yields
The first few terms in the near boundary expansion (i.e., ) of the scalar field profile (12) are
Given this result,888As well as using . we obtain the first few terms for the conductivity at :
If we recall that in eq. (15), we see explicitly here that in this expansion, the normalized conductivity is only a function of the two model parameters, and , as well as the frequency .
One can easily extend the above analysis to second order in the coupling — see appendix D. Here we note that at that expansion order, the leading correction to the high-frequency expansion (32) is proportional to and therefore the leading term above remains unchanged. The fact that the leading term above is exact can be anticipated by the arguments in the next section which determine the coefficient of this contribution from the OPE.
4.2 OPE analysis
To gain a deeper physical insight into the asymptotic expansion (32), we now reconstruct it using the OPE and the CFT data corresponding to our holographic model. Here, we focus on the first two terms. The leading term is simply the ground state conductivity, which obtains from the vacuum current-current correlator (appendix A.1). The second term is nontrivial as it arises because the relevant operator , the CFT operator dual to , appears in the OPE and acquires an expectation value at katz ().
First, let us recall the OPE of two (conserved) currents in the CFT written in momentum space katz ()
where are Euclidean 3-momenta. is the tensorial structure satisfying the conformal symmetries and the Ward identity arising from current conservation, i.e., . Above, we have only included the contributions from the identity and from the scalar with dimension . The ellipsis denotes the appearance of higher dimension operators in the OPE, e.g., the stress tensor katz ().999Implicitly, we are assuming that is a relevant operator with for the stress tensor to appear as a higher dimension operator. To obtain the asymptotic expansion of the finite temperature conductivity, we take the thermal expectation value of eq. (33) setting and with :
where we have used . Further, to connect this result to the expansion (32), we recall that the conductivity can be evaluated with the Kubo formula
Now, recall that for our holographic model, and , where is given in eq. (16). We can use the results in appendix A to derive the value of the OPE coefficient for the boundary CFT. In particular, inserting (the component of) eq. (33) in a vacuum correlator with yields
which is understood to be in the limit . Our notation above emphasizes that this is the singular part, as the full three-point function also contains terms regular in as , but these are not relevant for the OPE. Now comparing this expression with the holographic result in eq. (63), we find that the OPE coefficient in our model is
which is proportional to , as advertised previously.101010Again, the pole at in eq. (37) signals that our calculations have to be reconsidered for this special value of the conformal dimension — see appendix C. Substituting this expression into eq. (34) then yields
Using the expression for in (16), as well as , we find
which matches precisely with the first two terms of eq. (32), if we recall the definition the rescaled frequency in eq. (24). We note that eq. (39) can be analytically continued to real frequencies, , so that for generic both the real and imaginary parts of will contain a term at large frequencies Caron-Huot (); willprl ().
At this point, let us observe that generically we expect the stress tensor will appear in the OPE (33) and so there would be additional contributions to the asymptotic expansion (30), beginning at the order . Of course, the latter would in fact be the dominant frequency-dependent contribution when . It is an ‘exceptional’ feature of our holographic model that the vacuum correlator vanishes and such contributions are not present in the asymptotic expansion above. In fact, if the same holographic model was studied for , we would find that is nonvanishing and additional terms appear in the analog of eq. (33). Alternatively, the holographic model could be extended to include a new bulk interaction , as in Myers:2010pk (); natphys ().
4.3 Fingerprints of large- factorization
We first consider the higher order terms in the high frequency expansion of the conductivity given in eq. (32), which is valid to linear order in our expansion. As shown in eq. (30), the expansion of the scalar field controls the high frequency expansion of conductivity and the powers in the high frequency expansion matches the powers of in the expansion of . Hence, examining eq. (31) and the translation between the and coordinates — see footnote 8 — we conclude that that beyond , the only powers of which will appear in the expansion of conductivity (32) will be and with .
First, let us consider the sequence of terms with where . These contributions should arise from the thermal expectation value of a local operator with conformal dimension , which appears in the OPE in eq. (33). If is a primary operator, one might naively think that these higher dimension operators are descendants of . For example, the operator would have dimension . However, it cannot contribute the term proportional to in the asymptotic expansion because its thermal expectation value vanishes by symmetry. Indeed, is space- and time-independent. The natural interpretation is that this asymptotic term arises from the composite operator , obtained by “composing” and the stress tensor. In a general CFT, such a “composition” (reminiscent of free theories) is not well-defined and thus one cannot interpret the result as a well-defined local operator. However, in the large- limit (or alternatively, the limit of large central charge ) implicit in our holographic model, such a composition is natural because of the large- factorization arising in such theories el-showk (). Similarly, one can attach a string of stress tensors to to obtain an operator with scaling dimension for higher values of . We note that these operators have non-zero thermal expectation values and that in our model, their OPE coefficients with two currents are determined by . By the same token, the same composition explains the presence of terms with powers , as these will correspond to strings of stress tensors.
In appendix D, we find that at second order in the coupling , the asymptotic expansion of the conductivity acquires a new term proportional to . Following the above discussion, it is natural to interpret this contribution as arising from the composite operator . Usually these composite operators are irrelevant, however, we observe then that when the original conformal dimension lies in the range , then the conformal dimension of this new operator is . That is, in this regime, our holographic model has at least two relevant scalar operators, and hence it describes a quantum multicritical point, rather than a simple critical point. It would be interesting to further study the interplay of these two operators in the dynamics of the multicritical point using the holographic techniques established for so-called “multi-trace” operators, e.g., multi1 (); multi2 (); multi3 (); newmulti ()
To recap, ref. katz () recognized the important role of the relevant operator at a quantum critical phase transition in determining the dynamics of the corresponding QCP. They also took some steps to investigating this question in a holographic framework. A shortcoming of their construction was that the dual of the relevant operator in the boundary theory was not incorporated as a dynamical field in the bulk gravity theory. Of course, it is well understood that including a bulk scalar field with the appropriate mass, i.e., will introduce a scalar operator with conformal dimension in the boundary theory, e.g., see revue (). However, for the present purposes, a weakness of holographic theories studied up to this point is that the corresponding operator will not acquire a nonvanishing expectation value at finite temperature. Hence the key innovation of our holographic model was to include a natural mechanism which ensures that , as in eq. (3). That is, the bulk scalar is sourced to have a nontrivial profile in the dual black hole background, which then allows us to study the dynamical conductivity in a self-consistent holographic model. However, let us add the nontrivial observation that the conductivity obtained using our model is well-approximated by the simple Ansatz of katz () for a wide range of parameters, as illustrated in figures 5 and 6. We examine this point in more detail below. Further, we will also discuss below (section 5.4) how our holographic model provides a starting point to examine the response functions as a function of the relevant coupling — see eq. (1) — as we tune away from the QCP.
In section 2, we motivated the construction of our holographic model with a discussion of QC phase transitions which involve a relevant operator, with . However, our holographic analysis easily extends to considering irrelevant boundary operators, with , as well. In the latter case, the results may be interesting to better understand the dynamical response of certain QC phases (where there is no relevant scalar operator whose coupling needs to be fine-tuned). In this case, we could consider to be the leading irrelevant operator controlling RG flows down to this critical phase. The stress tensor would be the minimal dimension operator which acquires a thermal expectation value and hence one would also want to include the bulk interaction considered in Myers:2010pk (). This would ensure, e.g., that the stress tensor produces the leading contribution in the high-frequency expansion (32) proportional to ,111111We note that in certain CFTs, supersymmetry will forbid this contribution coming from the stress tensor william9 (). whereas that coming from the irrelevant operator is higher order being proportional to . However, this contribution could still be significant when is nearly marginal, i.e., when is only slightly larger than 3.
5.1 Minimality and related models
Again, the key new feature of our holographic model is that the scalar operator in the boundary theory acquires a nonvanishing thermal expectation value, as in eq. (3). This feature was engineered by adding the new interaction in eq. (8) which couples the dual scalar field to the Weyl curvature of the bulk geometry. This choice was motivated by the observation that the Weyl curvature vanishes in the vacuum AdS geometry but is nonvanishing in the black hole geometry (7). Hence the resulting equation (10) for the bulk scalar has no source in the AdS vacuum and the relevant solution is just . However, the equation has a nonvanishing source in the black hole geometry and acquires a nontrivial profile in this background. As desired then, in the boundary theory.
As noted before, previous holographic models did not reproduce this simple physical behaviour in the boundary theory. Certainly, one could imagine more complex approaches to produce the same physics and so one might think of our approach as providing the minimal holographic model with this feature. One simple modification would be to introduce an interaction with higher powers of the Weyl curvature, however, the behaviour found in our model would not be modified in an essential way. For example, with a interaction (with ), the leading term in the high-frequency expansion would still be proportional to and in fact, it would still be given by exactly the same expression as in eq. (32) if there are no other changes to the holographic action. The effect of this new interaction would only appear at higher orders. In particular, the term in eq. (32) would be replaced by a new contribution proportional to . One defining feature of the boundary CFT which would be modified is that the three-point correlator would vanish with this new bulk interaction. However, this then indicates that in general there is no direct connection between the CFT parameter controlling this three-point function and the thermal expectation value .
5.2 Perturbative bulk expansion
Next we discuss the perturbative nature of our calculations, however, let us first comment on the fact that we are using a higher curvature interaction in the scalar action (8) to generate . Similar higher curvature interactions will generically appear in string theoretic models, e.g., as corrections in the low-energy effective action gross (). However, rather than constructing explicit top-down holographic models, our approach here is to examine simple toy holographic models involving higher curvature interactions in the bulk gravity theory (see Refs. Ritz:2008kh (); Myers:2010pk (); will-hd (); Bai2013 () for different such models without scalar operators). Our perspective is that if there are interesting universal properties which hold for all CFTs, then they should also appear in the holographic CFTs defined by these toy models as well. This approach has been successfully applied before, e.g., in the discovery of the F-theorem Myers:2010xs (); Myers:2010tj () and more recently, in uncovering universal behaviour in the corner entanglement entropy for CFTs corner1 (); corner2 ().
We also stress that we are only working perturbatively in the dimensionless coupling for our new interaction. Higher curvature actions are typically regarded as problematic because generically they lead to “unstable” higher derivative equations of motion. However, these issues are essentially overcome when treating the higher curvature (or more generally, higher derivative) interactions as providing “small” perturbative corrections to a second-order theory JZ (). Hence our perturbative approach evades this problem.
At the outset, we said that our construction of the holographic background was perturbative in the amplitude of the bulk scalar. As indicated by eq. (15), this is equivalent to a perturbative expansion in terms of the dimensionless coupling , which controls the strength of the source in the scalar wave equation (10). In terms of the boundary theory, we can characterize this approach as considering the regime where the thermal expectation value of is much smaller than the thermal energy density, i.e., , where .
In fact, we only carried out our analysis to linear order in and so the holographic background consisted of the unmodified black hole geometry along with the scalar field profile given in eqs. (7) and (12), respectively. The next step in extending our perturbative construction would be to include the contributions of the scalar action (8) in the gravitational equations of motion. The back-reaction of the scalar would then produce perturbations in the black hole metric. Evaluating the conductivity would then extend the analysis in appendix B by considering the gauge field equation of motion in this modified metric. As a result, one would then find contributions in the conductivity proportional to . Hence we may conclude that the full conductivity in our holographic model depends independently on the three parameters, , and . That is, finding that the charge response in section 3.1 was a function of only and the product was an artifact of only carrying out our perturbative construction to first order. Working beyond first order also suggests the possibility of obtaining bounds on the holographic couplings and from the boundary theory, in analogy to the bounds found in, e.g., Myers:2010pk (); Brigante:2008gz (). However, we leave all of these interesting research directions for future work.
5.3 Monte Carlo data and analytic continuation
We are building on the holographic studies in natphys (); katz () and our construction is a next step in developing holography as a useful tool in studying the real-time dynamics of QCPs. One of the successes of these previous works was using quantum Monte Carlo (QMC) to study the dynamical conductivity of the O(2) Wilson-Fisher fixed-point theory and fitting the numerical results for imaginary frequencies with a holographic model. Further the holographic results are easily analytically continued to real frequencies, which is not possible for the QMC data, which only provides for the discrete Matsubara frequencies with . For this fixed-point theory, the conformal dimension of the relevant operator is very close to Campostrini01 (); bootstrap (). Figure 5 show the results of fitting the QMC data with our holographic model with and compares it to the results in katz (), which used a simple power-law profile for the bulk scalar. Both the conductivity fit for imaginary frequencies and the analytic continuation to real frequencies are almost identical for the two holographic models. Hence in this case, the two approaches do not differ in any essential way.
However, the power-law profile considered in katz () is a more or less ad hoc choice and we would like to emphasize the importance of developing a self-consistent holographic model for potential future studies. To illustrate this point, we show the result of fitting the QMC data with holographic models constructed in the same spirit as katz () with a new simple scalar profile:
As shown figure 7, the model with this new profile fits the QMC data for imaginary frequencies essentially as well as that with the profile or our holographic model. However, as the figure also shows, evaluating the conductivity for real frequencies with the new profile yields rather different behaviour for . In particular, the scalar profile in eq. (40) was designed to yield .
Let us consider the fit for the imaginary-frequency conductivity in more detail. As noted above, the QMC studies only yield for the discrete Matsubara frequencies with . In particular, the first data point appears at or at , in terms of the dimensionless frequency introduced in eq. (24). Now examining eq. (29), we see that the contribution of the scalar profile to is suppressed near the horizon by the exponential factor in the integral. Roughly, we can say that only probes to holographic background up to . Hence we might conclude that the fit to all of the QMC data points is only probing the bulk geometry up to or in our holographic model.121212Note that . On the other hand, the analytic continuation of the conductivity to real frequencies clearly relies much more on the detailed structure of the holographic model, including the near horizon region. Hence it is not difficult to engineer scalar field profiles which provide a good fit to the QMC data but yield disparate (and even peculiar) results for the real-frequency conductivity. For example, beyond the example given in eq. (40), one can easily construct examples where the conductivity seems to be vortex-like rather than particle-like, in the sense discussed in Myers:2010pk (), i.e., with . However, this simply illustrates the hazards of applying holography in an unprincipled manner, and we conclude that the most constrained and most reliable approach is focus on constructing self-consistent holographic models.
It might be interesting to extend this comparison to the QMC data by including the contribution of the coupling, i.e., one would extend the gauge field action (9) to include an additional interaction proportional to , as in Myers:2010pk (). As noted above, this new coupling would modify the high-frequency expansion (32) of the conductivity by introducing a new contribution proportional to . Including these contributions may improve the fit to the QMC data. However, a priori, it is not clear if extending the calculations to higher orders in the expansion will produce equally important modifications of the conductivity. Of course, our model can be easily adapted with other conformal QCPs, such as the Ising CFT in . It is likely that the stress tensor contributions will become more important as the conformal dimension of moves closer to 3.
5.4 Tuning away from criticality
Throughout the main text, we were considering a critical boundary theory which required setting the coefficient of the non-normalizable mode in eq. (14) to zero. As was commented above, this coefficient is dual to the coupling to the scalar operator in the boundary theory, as in eq. (1). More precisely, we have
Hence setting corresponds to the tuning needed to reach a QC phase transition as discussed in section 2. However, our holographic model then also provides a starting point to examine the response functions as a function of the relevant coupling as we tune away from the QCP. To study the off-critical behaviour of the boundary theory, we simply need to extend our analysis to scalar profiles (12) having nonvanishing .
As in the main text, we would still calculate perturbatively in the amplitude of the scalar field and so our analysis would be limited to the regime where . We must also assume that is a relevant operator, i.e., . For , the non-normalizable mode of the bulk scalar diverges asymptotically, e.g., see eq. (14), and as a result, the back-reaction of the scalar field cannot be controlled for . In order for to be regular at the black hole horizon, the coefficient must be chosen as
where is the value given in eq. (15). Hence as might be expected, the boundary theory responds linearly to the introduction of a small coupling . For example, the shift in the expectation value of the scalar operator becomes131313We also expect that away from the QCP, however, our perturbative analysis does not capture this contribution which would be nonanalytic in the coupling .
where is given by eq. (16) and is a numerical coefficient depending only on the conformal dimension.
Given the new scalar profile, it is straightforward to again evaluate the dynamical conductivity, as described in appendix B. Figure 8 shows the response of the conductivity to variations of . One might note the similarity of the plot for imaginary frequencies to the QMC results, shown in figure 6(a) of natphys () and also in gazit14 (). The extension of the analysis of the high-frequency expansion given in section 4.1 is also straightforward. In particular, turning on both coefficients in the near-boundary expansion (14) of the bulk scalar, the leading terms in the asymptotic expansion of the conductivity take the form
With , the second term proportional to is precisely the term in eq. (32). Hence we see that tuning away from criticality introduces a small shift in the contribution but it also generates a new term proportional to which is completely independent of the temperature. Let us emphasize that the above off-critical behaviour applies for . In terms of the phase diagram illustrated in figure 1, we are studying the theory deep in the “fan” where the physics is still dominated by the QCP. We plan to investigate the off-critical response further in new (), with the goal of shedding light on the response functions in the entire phase diagram near a quantum critical point.
We would like to thank A. Buchel, S. Hartnoll, C. Herzog, E. Katz, A. L. Fitzpatrick, A. Lucas, P. McFadden, and S. Sachdev for useful discussions. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research & Innovation. WWK was supported in part by a postdoctoral fellowship from NSERC. TS is supported in part by the Ontario Graduate Scholarship. RCM and TS are also supported in part by an NSERC Discovery grant. RCM is also supported by research funding from the Canadian Institute for Advanced Research and from the Simons Foundation through the “It from Qubit” Collaboration.
Appendix A Vacuum correlation functions
In this appendix, we provide some of the details of calculating various vacuum correlators in our holographic model, which are used in section 4. In order to calculate correlation functions, we will be working with Euclidean time, i.e., the time coordinate for Euclidean spacetime is given by the Wick rotation .
a.1 Two-point functions
To evaluate the two-point correlation functions, we begin with the ‘free part’ of the Euclidean bulk action