Physical Response Functions of Strongly Coupled Massive Quantum Liquids
We study physical properties of strongly coupled massive quantum liquids from their spectral functions using the AdS/CFT correspondence.
The generic model that we consider is dense, heavy fundamental matter coupled to super Yang-Mills theory at finite temperature above the deconfinement phase transition but below the scale set by the baryon number density. In this setup, we study the current-current correlators of the baryon number density using new techniques that employ a scaling behavior in the dual geometry.
Our results, the AC conductivity, the quasi-particle spectrum and the Drude-limit parameters like the relaxation time are simple temperature-independent expressions that depend only on the mass-squared to density ratio and display a crossover between a baryon- and meson-dominated regime. We concentrated on the (2+1)-dimensional defect case, but in principle our results can also be generalized straightforwardly to other cases.
In the last decades, the AdS/CFT correspondence [1, 2, 3] has become a powerful tool to study various properties of the strong coupling limit of conformal field theories, with applications to QCD and also more recently to some aspects of condensed matter physics. In principle, one has to distinguish between top-down setups that are constructed within string theory and imply consistency and bottom-up setups in which the gravitational duals are constructed from a phenomenological point of view. In this paper, we use the former approach as we would like to explore what happens to a particular consistent theory.
A common, and in particular top-down, approach how to introduce fundamental matter in AdS/CFT is the probe brane approach, where one considers a small number of “probe branes” in an black hole background which are dual to the fundamental matter coupled to an adjoint gauge theory. For example one considers the well-known (black hole) solution (above the deconfinement phase transition) that is a solution to a stack D3-branes in the decoupling limit and is dual to a (thermal) supersymmetric Yang-Mills theory . Then one inserts intersecting Dp branes, giving families of (charged) fields in the fundamental representation of the , living along the directions of the intersection [4, 5] - such as the above-mentioned D3-Dp intersections.
In condensed matter applications, there has been particular interest in dimensional systems that can be typically constructed using M2 branes , or as a defect in a dimensional background using D3-D5 [7, 8] and D3-D7 [7, 8, 9] intersections - and also as bottom-up setups in various contexts such as superconductivity. As our world is dimensional, the defect setup may be more realistic, even though there are some problems with the consistency of the D3-D7 setup [7, 8]. In recent years, there has also arisen significant interest in quantum-liquid-like aspects that arise when the temperature is small compared to the density, for example in a dimensional D3-D7 setup [10, 11], a dimensional D3-D3 configuration [12, 11] or in a dimensional D3-D5 setup . In these systems several physical aspects have been studied, such as the zero-sound mode [10, 13], a fermionic instability , the quantum Hall effect  and the response functions of fermionic matter at finite magnetic fields . This low-temperature limit, in particular the construction of  will be in the focus of this paper
One interesting property of these configurations is a scaling behavior that occurs at small temperatures in setups that are dual to a finite density of fundamental matter with finite mass. Then one finds that solutions at different temperatures but fixed mass-squared to density ratios are equivalent through a simple scaling and display thermodynamic features reminiscent of quantum liquids . In this paper, we will exploit this scaling to further explore the properties of this state of matter through their two-point functions. This limit in parameter space corresponds to large quark number density with finite temperature or finite quark number density with small temperature but still above the deconfinement temperature, and any choice of quark mass in each case.
The paper is organized as follows: In section 2, we review the setup and the formalism developed in . Then we will develop a method to solve the appropriate equations of motion in a temperature-independent parametrization in section 3 and obtain the physical results in section 4. There we first give an overview of the AC conductivity in section 4.1, then obtain the small-frequency (Drude) limit in section 4.2 and the spectrum of quasiparticle excitations in section 4.3. Finally we conclude in section 5.
The supergravity background of a planar black hole in is
This corresponds to the decoupling limit of black D3-branes dual to super-Yang-Mills theory at finite temperature , living along the flat directions of the AdS . The temperature is given by the Hawking temperature and the Yang-Mills coupling by . Since the curvature is given in terms of the string coupling and string length as , the ’t Hooft coupling can be written as . Hence the “supergravity limit” in which the type IIB supergravity action and the solution (1) are valid corresponds to the strong coupling .
In practice, it is convenient to define dimensionless coordinates
in which the metric becomes
In this setup all the fields transform in the adjoint representation of the . In QCD or condensed matter physics, however, one would like to consider also matter that is charged under this symmetry, i.e. that transforms in the fundamental representation. Hence, the use of this solution for such applications is very much limited. To introduce the fundamental matter one may create an intersection of “probe” Dp branes with the D3 branes, such that at the string theory side there are fields at the massless level of field theory at the intersection. From the point of view of the probe branes, they correspond to endpoints of D3-Dp strings, and in the field theory dual they correspond to fundamental fields of a (defect) field theory.
The dual field theory is now the SYM gauge theory coupled to fundamental hypermultiplets, which are confined to a (2+1)-dimensional defect. This construction is still supersymmetric, but the supersymmetry has been reduced from to by the introduction of the defect. In the limit , the D5-branes may be treated as probes in the supergravity background, i.e. we may ignore their gravitational back-reaction.
We assume translational invariance along the flat directions and rotational invariance on the sphere. Hence, the pullback on D5-brane gives us one scalar field corresponding to the position in the direction, which was extensively studied in [7, 8], and another scalar which describes the size of the compact sphere and corresponds to turning on the mass of the fundamental matter, studied in [20, 8] and more extensively in the similar D3-D7 system in [21, 22, 23, 24, 25]. Parametrizing the as and putting the branes on the first of the , the induced metric on the probe branes is given by
where we defined the scalar as .
We would like to turn on only the overall factor of the world-volume gauge field, so the the probe branes are governed by the DBI action
where . This gauge field is dual to the current operator of the that gives rise to the “flavor symmetry”. In particular, the flux of the electric field
corresponds in the field theory side to the baryon or quark number density (see e.g. ), where we explicitly keep the quark number density
From the action (6), it is straightforward to obtain the solution for the gauge field
where . The equation of motion for is
On the horizon , this equation reduces to
The asymptotic solution at is
where and are dimensionless parameters that are determined by the value of on the horizon. Following arguments of the T-dual dimensional D3-D7 setup [21, 22, 23, 24, 25], the quark mass and dual condensate are given by
One can understand this identification of the mass from the separation between the D3 and D5 branes in flat space, such that is the mass of a stretched D3-D5 string and the condensate is just the thermodynamic dual of the mass.
At vanishing density, there is a critical temperature-mass ratio below which the probe branes do not extend down to the horizon [27, 21]. At finite densities, unless one turns on the scalar in the direction considered in [8, 20], the branes always extend down to the horizon even though a phase transition may still be observed at small densities . In the limit considered in this paper, however, this phase transition is of no concern.
To obtain the low-temperature scaling solutions, we make a transformation for (10) and expand it in the large- (i.e. ) limit. The leading order gives
which doesn’t depend on explicitly and there remains only implicit dependence from the scaling of . In the large limit, we have a series expansion solution
which serves as the boundary condition to solve (2) numerically. In the small limit, we obtain
The parameters and are now related to the quark mass, condensate and density by
In general, equation (2) has no analytic solution and (15) implies that in practice one has to start integrating the equation from the large- region – that replaced the horizon boundary condition – to obtain the mass and the related condensate at the asymptotic boundary, rather than setting either of them in the beginning.
3 Solving the Fluctuation Field
To obtain the conductivity though linear response theory and also the spectrum of excitations, we need to know the two-point functions of current operators. In AdS/CFT, they are obtained from the variation of the on-shell action – since the baryon number current is dual to the gauge field of the subgroup of the , the variation with respect to :
and is the boundary value of the gauge field at the asymptotic boundary . Hence we perform a pertubation of the gauge field and expand the action up to quadratic order,
where . The variation of eq. (18) then becomes
To obtain temperature-independent expression in the spirit of , we do this however in a smart manner that represents the temperature independence. To do so, we obtain a solution at large radius or and fix it in an overlap in the region or , with an analytic solution in terms of the original radial coordinate that is valid at relatively small radius or , and reflects the horizon boundary condition.
We fix the gauge to , and we work in the case of vanishing momentum where and and the x and y component equations are the same. As discussed in [7, 8] in the context of the duality, the -component is also related to the spatial components, such that we only need to obtain the solution for . After Fourier transformation, the x-component equation of motion is
where . In the near horizon region, (3) is reduced to
where . The solution is
where are integration constants. The in-falling boundary condition only picks up the second term. We set to be 1 for convenience in the later discussion, which will not affect the conductivity and other physical results.
First let us obtain the solution for . In the limit , (3) reduces to
where we made use of the scaling in eq. (15). To solve this equation, we take the Ansatz
with some arbitrary number , the E.O.M. in this limit becomes
This equation can be solved order by order in the limit of large (i.e. ) and we obtain the leading terms
Hence the solution for is
Next, we consider the E.O.M. in the outer region . After transforming (3) into -coordinate, in the large expansion the leading order gives the simplified E.O.M.
and . Note that the explicit dependence has dropped out, which is the advantage of the -coordinate. In the large limit, this equation is further simplified to
and the solution is
where and are fixed by matching this solution with (30) in the region. We can take (35) as the boundary condition to solve (31) numerically, replacing what would be usually the boundary condition on the horizon.
4 Physical results
In this section, we give an overview of some interesting properties that we can extract from the correlator eq. (18). For the transport properties, we obtain the conductivity from linear response theory, i.e. using the Kubo formula
4.1 AC Conductivity
We should note that the conductivity doesn’t explicitly depend on , since it appears only in the pre-factor of and gets canceled out in the conductivity. The AC conductivity for various and is shown in the fig. 2.
We see at large frequencies the “plasma” resonances that were discussed in , and at small frequencies the absence of the Drude-peak. The latter is due to the fact that we took the limit to obtain (30), and we will reconstruct it in the following section. We will also discuss the spectrum of “plasmons” that gives rise to the resonances in section 4.3.
An interesting observation is that the temperature-scaled AC conductivity goes to some finite value (i.e. the conductivity scales as ) as the frequency approaches 0 (but still above the Drude peak), which we plot in fig. 3 as a function of .
This constant, , approaches the constant 0.29 in the small region, and vanishes as when is large. Physically, it reflects an accumulated contribution from the Drude peak and all the plasmon modes, and the scaling implies that they contribute less as the quark mass becomes much larger than the scale set by the density – which is in line with the fact that the resonances become more stable, i.e. narrower with increasing mass as we will find out in section 4.3.
4.2 Relaxation Time in the Drude limit
The Drude model describes usually the DC and small-frequency conductivity of weakly coupled systems, such as electrons in simple metals. In  it was found, however, that the Drude conductivity given by (see e.g. )
also applies to strongly coupled systems and describes the conductivity very well in a consistent manner, up to a “conformal” contribution that is subleading at large densities. While the relaxation time was given only numerically, the DC conductivity was found to be
For convenience, we define the relaxation time such that .
To obtain a closed-form expression for , we look for an approximate solution to (31) by considering only the first term. The solution for is then
This is purely imaginary and has a pole at with the residue
Due to the Kramers-Kronig relations, the real part of conductivity becomes a -function in with a pre-factor which we will compare to the Drude conductivity (38) by only considering the area under the peak, as our methods do not “resolve” its detailed frequency dependence. To do so, we integrate the real part of the conductivity eq. (38) over the frequency
and identify this with the integral over the frequency in the low-frequency regime of our low-temperature-limit result, i.e. with the weight of the delta function. This is justified by the fact that our scaling and approximation has the effect of “shrinking” the Drude-region to zero size in . Using expression for the DC conductivity from  (39) and the expression for the residue, (42), we can obtain an expression for the relaxation time
At , this result can be solved analytically as such that
In fig. 4, we compare the Drude conductivity with our numerically-obtained relaxation time to the result obtained from the full equations of motion and the result from our low-temperature limit for an illustrative large value and finite value . We see reasonably close agreement. Already for , there would be no difference visible between the result for the full conductivity and the one from the Drude conductivity.
In fig. 5, we show the relaxation time as a function of the quark mass. At small , i.e. when the system is dominated by its density, we find . From a weak-coupling “mean free path” point of view, this is somewhat counter intuitive, but it can be understood as the system is strongly coupled and the conductivity is due to collective excitations that become more stable as the density is increased. At large quark mass, we have which is independent of the density as we are in the limit were the shortest length scale is the Compton wavelength of the quark mass (or more precisely of the energy scale of mesons  that dominates the type of system at large quark mass [21, 24, 25, 8]). While this impression implies that the relaxation time increases with increasing mass and decreasing temperature, it is not of the form of the weak-coupling geometric scattering. Also the factor of the meson mass might be misleading, as they cannot carry the baryon number current.
4.3 Quasiparticle Spectrum
The spectrum of quasiparticles corresponds to the locations of the poles of the correlator in the complex frequency plane, and hence to quasinormal modes in the gravity side. Thus, it can be obtained by tuning the complex frequency such that the fluctuation fields vanish at the asymptotic boundary . It is straightforward to search the exact locations of poles for large by directly solving (31) for complex numerically, but the numerics start to break down when we approach to the small region, where we can only obtain few first order poles. Hence, to study for higher order excitations and small values of we developed a method based on analytic continuation.
4.3.1 Analytic Continuation
Our general strategy is to find a closed-form expression that is exact for large . As this expression numerically still does not converge for sufficiently large imaginary parts of , we then do a parametric fit for real frequencies and then perform an analytic continuation to obtain the location of the poles in the complex frequency plane. First, we take the Ansatz
to re-cast (31) into
Then, we let , where is the leading term in and satisfies the algebraic equation
with the solutions
The subleading term should thus satisfy
This is essentially a transformation of a linear second order equation into a non-linear first order one.
The boundary condition at large can be obtained from the large- expansion of beyond the one of eq. (35),
which gives us
By matching this with the large expansion of ,
we know that we should choose the positive root for and the resulting large- expansion of is given by
where the integration constant is fixed by the boundary condition to .
Given a solution for , the solution for is then from the Ansatz (46)
where is some large value of . Our approximation holds for when is small for some finite . When approaches 0, the incoming and outgoing modes decouple such that (4.3.1) is still valid. This can be easily seen in eqs. (31) or (47) as the coupling is proportional to which actually vanishes in both (large and small) limits of . Thus, we could find the locations of poles by demanding .
In practice, however, we only solve (50) for real , then we fit the real part of into the real part of a generic complex model
where and are complex parameters, is a non-periodic function and asymptotically proportional to . Then, we can solve this model for in the complex frequency plane to find the locations of poles in the complex frequency plane. One may obtain same result by fitting imaginary part accordingly and the result should also be independent of the particulars of the model. In particular, at large , the first term is the most generic one and as we are only interested in the parameter to give us the tower of excitations.
For validation of our procedure, we compare our results for each the first pole at various values of the mass in the range at which we can find the exact location to the exact result in fig. 6. Even though the first poles are expected to be the least reliable ones and most dependent on the model, they match with the exact locations very well.
Now that we have obtained the locations of the poles of the two-point function – exactly for large and with a good approximation also for small – we can use them to analyze the quasiparticle spectrum of our theory in the field theory side. To do so, we parametrize the poles corresponding to a fixed value of asymptotically (using the highest few excitations) as
for some complex parameters and that represent the spacing of excitation levels (and their inverse lifetimes) and some “ground state energy” which will obviously not be exactly the value of the excited state. The other way round, one can interpret the real part of as some induced length scale that depends on , i.e. on the ratio .
That this parametrization is well-motivated can be seen in fig. 7, where we show the first few excitations for selected values of .
We see that parametrization is actually a very good fit, i.e. we have an almost linear tower of excitations, or an induced length scale that persists at high energies. We also find already now that both the excited states and the ground states get heavier and more stable with increasing quark mass.
To look at the data from a different perspective, we also show the energy of the quasiparticle states, i.e. the real part of , and their inverse lifetime to mass ratio as a function of the mass in fig. 8.
We see that except for a small deviation at small mass (or large density), the energy of the excitations depends approximately linearly on the the quark mass. The inverse lifetime to mass ratio decreases inversely with increasing mass and, for the higher orders of the excitations, does not depend on the excitation number, such that the asymptotic behavior is approximately .