Imsc/2009/09/12 Properties of CFTs dual to Charged BTZ black-hole
We study properties of strongly coupled CFT’s with non-zero background electric charge in 1+1 dimensions by studying the dual gravity theory - which is a charged BTZ black hole. Correlators of operators dual to scalars, gauge fields and fermions are studied at both and . In the case we are also able to compare with analytical results based on and find reasonable agreement. In particular the correlation between log periodicity and the presence of finite spectral density of gapless modes is seen. The real part of the conductivity (given by the current-current correlator) also vanishes as as expected. The fermion Green’s function shows quasiparticle peaks with approximately linear dispersion but the detailed structure is neither Fermi liquid nor Luttinger liquid and bears some similarity to a ”Fermi-Luttinger” liquid. This is expected since there is a background charge and the theory is not Lorentz or scale invariant. A boundary action that produces the observed non-Luttinger-liquid like behavior (-independent non-analyticity at ) in the Greens function is discussed.
A variety of ill-understood strongly coupled field theories are expected to describe phenomena
such as High- superconductivity   . A technique that
seems tailor made for this is the AdS/CFT
correspondence  , where a string theory in AdS background is dual to a conformal field theory
on the boundary.
If the CFT is strongly coupled, then the curvature of the dual gravity background is small
and the massive string excitations do not play much of a role, and so pure gravity is a good approximation. This is the reason
this technique is ideal for strongly coupled theories.
If one wants to study the finite temperature behavior of the CFT, one can study dual backgrounds that are asymptotically AdS but have a finite temperature. This could be just pure AdS with a thermal gas of photons or other massless particles, but also includes backgrounds that contain a black hole in the interior. There is typically a critical temperature above which the black hole is the favored background . The critical temperature is in fact zero if the boundary is taken to be flat rather than . This is what happens when we represent the AdS by a Poincare patch. Equivalently there is a scaling limit that gives starting from the boundary of global AdS. Recently such backgrounds with a charged Reisner Nordstrom black hole in 3+1 dimensions have been studied to understand 2+1 dimensional CFT’s with charge. In particular fermionic fields have been studied and interesting ”non-Fermi liquid behavior”  has been discovered    . The same system was studied with magnetic field turned on in  .
In this paper we study analogous situations in 2+1 dimensions
We should note that the correspondence works in full string theory, so ideally we should consider embeddings of dimensional black hole in full string theory in dimensions. In our present study we assume the existence of such an embedding. By analogy with the best studied example of AdS/CFT correspondence, where Type IIB string theory on corresponds to super Yang Mills, one expects that tree level calculation in the bulk corresponds to the planar limit of the boundary theory. In our case there is no identifiable parameter . We can take it generically as a measure of the number of species of particles in the boundary theory. Thus when we refer to Luttinger liquid we are actually referring to a conformal field theory with a large (infinite) number of scalar fields. Similarly the gravity approximation (neglecting stringy modes) corresponds to large ’t Hooft coupling in Yang-Mills. In our case we do not have an action for the boundary theory and hence no well defined notion of coupling constant. However we can assume that the theory describes some non trivial ”strong coupling” fixed point with consequent large anomalous dimensions. This can be taken as the operational meaning of ”strong coupling”.
In the above approximation (planar, strong coupling) used here the full string theory embedding is not required. However the full embedding would in principle determine the charges and dimensions of the operators of the boundary theory, which for us are free parameters. Also to go beyond this approximation would require a knowledge of the embedding.
We study Green’s functions corresponding to scalars, fermions and gauge fields at both zero and non zero temperature. The gauge field calculation gives the conductivity. This is studied at both zero and non zero temperatures. At zero temperature the small behavior is universal (as shown in ) and we see that our numerical results are consistent with this expectation. The same analysis can also be done for fermions and scalars and again there is consistency with the analytic results at zero temperature and low frequencies. This gives some confidence in the numerical calculations. The fermion Green’s function does show quasiparticle peaks at specific momenta and these can be identified as Fermi surfaces. The dispersion relation is approximately linear. The log periodicity in the Green’s function are also observed and is consistent with the analytical expectations. Qualitatively the curves are also similar to the Fermi-Luttinger liquid curves.
We also attempt to reproduce in the boundary theory the intriguing non analyticity in the fermion Green’s function at for any . We show that it can be explained if one assumes that there are modes that have their velocity renormalized to zero, interacting with fermions. These could be thus some localized modes representing impurities.
This paper is organized as follows. In Section 2 we very briefly describe the charged BTZ background. In Section 3 we describe what to expect from the boundary physics point of view. The paradigm of the Fermi-Luttinger liquid is very useful and is briefly outlined. The numerical results for the Fermion Green’s function is also presented. Section 4 contains a discussion of the gauge field equation and conductivities are extracted again for zero and non zero temperature. Section 5 contains a discussion of the scalar. Section 6 contains a discussion of the possible boundary theory that could give rise to these non-analyticities at . The Appendices contain some background material in the context of the charged BTZ black-hole, that are useful for some of these calculations and also some details of the calculation.
2 The Background Geometry: Charged dimensional Black Hole
In our analysis we will consider background of a charged black hole in dimension. The Einstein-Maxwell action is given by,
Where is Newton constant, is the cosmological constant ( is the AdS-length). One of the solutions to the above action is given by the following metric and vector potential after some rescaling( , ) (See Appendix (A))
As , the metric asymptotes to metric and is called boundary. The metric is also singular at called horizon. The black hole temperature is given by ( ).
3 Green’s function: Fermion
A bulk Dirac spinor field with charge is mapped to a fermionic operator in CFT of the same charge and conformal dimension . In dimension is a chiral spinor. By studying the Dirac equation in dimension asymptotically AdS space, we can find the retarded Green’s Function of the dimensional boundary CFT.    . We will study behavior for simple case where bulk fermion is massless (i.e. ). Following the calculation of , the boundary Green’s function is given by
where, and . defined in the above equation follows a first order non-linear differential equation which follows directly from bulk Dirac equation,
where . In order get the retarded correlation function for the dual fermionic operator , we need to impose ingoing boundary condition for at the horizon which is equivalent to . The boundary condition has to be modified for and and is given by
A detailed analysis for Fermions is given in Appendix (B).
3.1 What to expect
As a consistency check of our numerics we can use the following symmetry properties of the Green’s function obtained by direct inspection of the equation of motion (B.0.8) with , , , , .
Since our background geometry (2) asymptotes to , in the ultra violet () we expect the effects of finite density and temperature become negligible and it will recover conformal invariance. If we choose our background geometry as pure , the Green’s function (massless bulk fermion) can be easily obtained as (),
where . This is Green’s function for a dimension chiral operator in dimensional CFT. or the spectral function has a symmetry under (“Particle-hole symmetry’) and has an edge-singularity along . is zero in the range . Also , and .
In the ultra violet we expect same scaling behavior () and linear dispersion with velocity unity. Note that the scaling dimension of the fermionic operator in the boundary is compared to usual dimension fermionic operators (viz. electron operator) in dimension. The scaling dimension of the operator in the IR of the boundary theory may be very different from as the boundary theory may flow to a different fixed point in IR as described in next section.
As shown in (,) theories dual to charged extremal black holes (in dimensions, ) have a universal IR behavior controlled by the region in the bulk. A similar analysis goes through in case, a brief sketch is given in Appendix (D). At zero temperature, the the background geometry (Appendix (D)) is described by in the near horizon limit. From the boundary field theory point of view, although even at the conformal invariance was broken by , the theory will have an scale invariance in the IR limit () and will be controlled by an IR CFT dual to . The IR behavior suggests that the spectral function where . For real, the spectral function vanishes as and is “log-periodic” in if is imaginary ( where with ). There may also exist particular values of where we can have a pole as . The spectral function at those values of is given by,
Condensed Matter Systems
For dimension boundary theory one typically expects to get Luttinger Liquid behavior. “Particle-hole symmetry’(, is the deviation of momentum from the Fermi momentum ), absence of quasi-particle peak (i.e. Lorentzian peak known as quasi-particle peak in usual Landau Fermi liquid theory is replaced by power law edge singularity) , linear dispersion, spin-charge separation are the hallmarks of Luttinger liquid in dimension. In spinless luttinger liquid the spectral function for electron (dimension operator) has a behavior(,) (for ),
is Luttinger liquid exponent, gives the IR scaling dimension which is related to “anomalous scaling dimension” of the fermion operator (See Fig.(1)). Deviation from these behavior can be seen in modified Luttinger models e.g. Fermi-Luttinger liquid (). In Fermi-Luttinger liquid the dispersion is modified by a non-linear term, and as a consequence the edge singularity for particle (Fig.(1)) is replaced by a Lorentzian peak like Fermi-liquid, but corresponding behavior for hole remains same (“Particle-hole asymmetry”). The scaling exponent also becomes function of . Very close to the singularity the behavior of the particle spectral function become similar to (3.1.2), which resembles a Fermi-liquid. We should note that in our model, both scale invariance and Lorentz invariance is broken for the boundary theory, so we can expect deviations from Luttinger Liquid. In a Luttinger liquid one could construct correlators like,
where are defined as deviations from Fermi point. At a fixed point one expects , and has a edge singularity like (Fig.(1)). But at a generic point on the RG between fixed points one expects some finite imaginary part which smooths out the singularity to a peak (viz. Fermi-Luttinger liquid ). The Green’s function have branch cut singularity is along and ; the peak is along the second line of singularity. The Green’s function along the peak becomes,
where . clearly increases with () as long as does not have an dependence - as at a fixed point Luttinger liquid. In general if is sub linear then one expects . If with then .
On the other hand for a Fermi liquid one expects a behavior like (3.1.2) but with . But the IR dictates that the scaling exponent of is generically different from usual Fermi liquid. In analyzing the example we must keep these points in mind. We expect on general grounds that at least for weak coupling it should behave as a Luttinger liquid. More generally it could go into a massive phase. But we do not find evidence of a mass gap in the numerics below. Any deviation from a Luttinger liquid therefore is something noteworthy.
3.2 Numerical results for Fermion
Since charge of the bulk field fixes the charge of boundary operator under global , we must keep fixed to study a particular boundary operator. For (), , we will study calculate the Fermi-momentum and velocity for fixed . We will also vary or to study the “log-periodicity” discussed in section (3.1.3). For finite temperature case, we will fix and study the Green’s function for various values of temperature (equivalently ).
We will mainly study the qualitative features of zero temperature Green’s function as the numerics is not very precise. Quantitative studies are made at finite temperature where we have very good numerical handle.
UV behavior: For , the Green’s function matches over a large range with pure AdS Green’s Function (3.1.1). For finite , the as for fixed which implies the UV scaling dimension of the operator is (Fig.(2)). For fixed , the Green’s function always matches near which is consistent with symmetry properties (Section (3.1.1)). The qualitative nature of the peak () matches with Luttinger Liquid Green’s function considered in eq.(3.1.4) (Compare Fig.(1) & (3)). As expected for from pure AdS behavior (eqn.3.1.1), the spectral function should be zero in the range and a peak near (), is approximately seen in Fig.3.
IR behavior: For a fixed density plot of the spectral function (fig.(4)) shows a sharp quasi-particle like peak at and some value of momentum , called Fermi momentum. At Fermi point the peak height goes to infinity and the width goes to zero. The behavior is as described in section (3.1.3), this can be seen in modified Luttinger liquids as describe in the paragraph below the equation (3.1.5). We expect that the theory flow to a different fixed point in IR. But at small , along the dispersion curve we see the spectral function has a minima at compared to the quasi-particle like peak, as expected in Luttinger liquid (Fig.(7)). Fermi momentum changes from to for due to the symmetry (section 3.1.1). For , and obtained from the dispersion plot which is linear.(fig.5). If similar analysis was done for , where and . Note changes sign if we change . Now, as we can see from the expression of scaling dimension of IR operator at Fermi momentum () turns out to be . In this regime according to analysis  the dispersion relation should be linear. Also as described in section (3.1.3), the spectral function should go to zero if real (Fig.(6)) and is “log-periodic” where is imaginary (fig.6). We found a very good match of the numerical and analytical results for periodicity. Non-analyticity of Green’s function at independent of is not typical of Luttinger liquids and in Section (6) we have discussed a possible resolution of this.
Particle-Hole (A)symmetry: We find “particle-hole asymmetry” for , i.e. the spectral function behavior is different under reflection at Fermi point (fig.(4)). This is again very different from Luttinger liquid behavior, but it can be observed in Fermi-Luttinger Liquids as discussed in (). The particle hole symmetry is restored for (fig.(7)), and the asymmetry slowly increases with .
Gapless phase Also a closer look at density plot (fig.(4)) shows the system is in gapless phase which is as expected because in the IR the theory is expected to have conformal invariance.
The behavior of the spectral function at finite temperature can be summarized as follows:
IR behavior: At sufficiently small temperatures, the quasi-particle like peak still survives. But the width gets broadened with temperature and also the Fermi frequency shifts to some non-zero value. Table (1) shows various peak properties at various temperature with . Data shows that is a finite temperature effect and goes to zero as . Also (FWHM or “Full Width at Half Maximum” at Fermi point) goes to with . Fermi velocity is independent of temperature. The FWHM along the dispersion varies as where at (fig.(8) ,fig.(9)). varies linearly with as . Which implies at , which matches with the zero temperature result.
Dispersion: The dispersion curve (fig.(10),fig.(8) ) is linear near the Fermi point, but deviates from linearity away from the Fermi point. approaches as as expected from UV behavior, but becomes large . We expect would go to for , but we were unable to explore that region as the peak gets very broad in that range and numerical error increases.
Particle-Hole (A)symmetry: Again in finite temperature case also the spectral function shows “particle-hole asymmetry” at low temperature, but the symmetry gets restored for (large temperature) with fixed . We can conclude that the particle-hole asymmetry is controlled by the parameter .
Large Temperature: For large temperature, , the Green’s function approaches to that for uncharged non-rotating BTZ (UBTZ) as given in () (with , ). The quasi-particle peak is completely lost at large temperature and has a minima along the line of dispersion at .
4 Gauge Field: Conductivity
In this section we consider aspects of the vector field and its perturbation. This gives the current-current correlators of the boundary theory. The vector field is given by the solution ( is the usual radial coordinate and often we will use . is the location of the horizon).
If we consider the effective scalar given by we get
is an arbitrary normalization scale - denoting violation of conformal invariance. This is a situation where the eigenvalues are degenerate so and are the two independent solution. There is some ambiguity regarding which solution corresponds to the source and which one to the expectation value of the conjugate operator, which in this case is the charge density. Thus if following  we take as the value of the source then the charge density is . The analysis of the degenerate case for the scalar does seem to give this - see Appendix A. On the other hand reversing the roles gives as the charge density.
where . Also . The equation of motion also gives:
Thus we get for the charge dependent part of the action :
The chemical potential at the boundary is . Thus the free energy is . And the expectation value of the charge density is . There is no dependence on . This suggests that we should treat the coefficient of as the expectation value and not the source. But we will follow the argument given in appendix and consider coefficient of the log term as source.
We now turn to the calculation of the Green’s function as a function of the frequency, keeping no dependence (i.e.zero wave vector). This is obtained as the ratio of the two solutions in time dependent perturbations . The perturbations in and , which we call and respectively obeys coupled differential equation. We can combine the two equations (Appendix) to get an equation for - which is (we are assuming a time dependence of for both and ):
Analyzing the leading behavior at we see that the solution has degenerate eigenvalues and is given by . The Green’s function for the conjugate operator is given by the ratio . ( We have to put in-going boundary condition at the horizon, described in detail in the Appendix (C)) The ratio between and is the conductivity . So the Green’s function divided by gives the conductivity (See fig. (11),(12)). In the next section we will discuss about the low energy and zero temperature limit of the conductivity.
4.1 limit of Conductivity at
As we have discussed for the fermion case, for this case also we do the same co-ordinate transformation for the near horizon limit of the extremal black hole. So, in the near horizon the equation of motion for the gauge field, The leading leading order behavior of the IR Green’s function is given by (see Appendix(D)),
. So, as mentioned earlier the real part of the conductivity is
So, we see that a dimension 2 IR CFT operator determines the limiting behavior of the optical conductivity in low frequency limit of boundary dimensional field theory. This behavior should be contrasted with behavior of doped Mott insulator (), where the real part of conductivity goes as . In addition, the imaginary part of has a pole at which is also clear from the plot (Fig.12),
Now, from the well known Kramers-Kronig relation
at . From the numerical analysis (fig.12) we were not able to see the delta function peak. But this is a natural expectation that for any translational invariant theory the DC conductivity (for ), should be infinity.
5 Complex Scalar
In this section we consider the complex scalar field in the background geometry (2) and derive the Green’s function numerically for the dual operators in the boundary. The equation of motion of the scalar is given by,
The solution to the equation (5.0.1) near the boundary, , for non-zero mass of the scalar field
behaves as, (we have dropped the log term coming from the potential term, in (5.0.1) assuming a finite mass for the scalar field
Here is the boundary space-time dimension and . As per  the boundary Green’s function is given by, . The solution for is obtained numerically and the boundary Green’s function can be written as,
5.1 limit of Scalar Green’s Function at Zero Temperature
Following a calculation similar to Fermion (see Appendix (D)), the Green’s functions in the goes as where . This is just the same as the analysis of , who use this to further conclude that the imaginary part of the full Green’s function has this behavior. If is not real one has also the interesting periodicity in .
We should note that the effective mass of the charged scalar field in in the presence of the gauge field is . The condition for log-periodicity implies that this effective mass is lower than the BF bound in which is . This means that for the choice of the parameters which exhibit log-periodicity of the Green’s function the scalar field is unstable.
Figure 13 shows that the log-periodic behavior (in ) of the real and the imaginary parts of the Green’s function. We have plotted these for , and for . Note that this value of mass is well above the BF bound for the scalar field in (which is ), although the effective mass in is unstable. The scalar field though asymptotically stable is unstable in the near-horizon region. We expect that this instability will lead to the condensation of the scalar field. In the case of charged scalars in this instability leads to a transition to a hairy black hole phase which in the dual theory has been identified as the transition to a superconductor phase (see  for nice review). For our case we shall analyse this aspect in our upcoming work.
The imaginary part of the Green’s function becomes negative for smaller values of . We presume that this is due to instabilities occurring from the fact that the effective mass for the scalar is tachyonic in the region of parameter values where the Green’s function shows log-periodicity. However the log-period that appears in the plot matches with the analytical value which in this case is .
5.2 Numerical results
Let us now note some general properties of the Green’s functions before discussing the numerical results. From the differential equations it can be seen the Green’s function is even in , . For , complex conjugation of the Green’s function has the same effect as . This implies that Im Im , and Re Re . For non-zero values of , this symmetry is lost due to the presence of the covariant derivative, however, . The asymmetric behavior is visible in all the plots for nonzero . We now turn to our numerical results.
Zero temperature: Figure 14 shows the real and imaginary parts of the boundary Green’s function for the extremal
case, or , and for various values of . At zero temperature, the Green’s function has peaks lying between and .
These peaks shift towards as is increased which
disappear for larger values of . The plot (Figure 14) shows a range of for which the peaks exist. We would like to
also point out that the values of the parameters considered here lie in the region where the Green’s function is periodic in for small . This is discussed in more detail in the next section.
Finite temperature: For a fixed value of , the peaks lying in the positive region smoothen out beyond a particular temperature. Figure 15 shows the plot for the real and the imaginary parts of the Green’s functions for various values of or and and for . The Green’s function is very sensitive to variations in temperature near the zero temperature region. The number of peaks reduce to one in the finite temperature regions for a particular value of . Eventually this peak smoothens out as the temperature is increased further. At the highest temperature , the curves (not shown in the figure) have the symmetries mentioned at the beginning of this section. For large values of , the functions are monotonically increasing or decreasing. Like the zero temperature case the peaks shift towards the origin of and then disappear for larger values of . In fact, like at zero temperature there exists a range of for which there are peaks. This window of for which the peaks exist shrinks as the temperature is increased.
Large : In the large limit for fixed the Green’s function should behave as, , With . This scaling behavior can be verified numerically for our Green’s function. We get the following results from our numerical computation at finite temperature: , , . These may be compared to the analytical values: , , . We have verified that results correct to the first decimal place can be obtained for the real and imaginary parts of the Green’s function both for finite and zero temperatures. Similarly the same scaling can be checked for large with fixed finite . We get the numerical answers correct to the first decimal place as above.
6 Non Fermi Liquid Behavior in the Boundary Field Theory
Since our boundary theory lies on an RG trajectory connecting a UV fixed point CFT dual to and a CFT in the IR (because it is gapless) we can expect it to be described by the action of a Luttinger liquid like theory modified by the addition of some irrelevant perturbations. These terms are allowed because the background charge density (implied by the chemical potential ) breaks scale invariance. It also breaks Lorentz invariance, which means that one cannot assume that all propagating modes propagate with the same velocity. The effect of these irrelevant terms have been investigated in  in perturbation theory. One expected effect is that the linear dispersion relation is corrected by non linear terms. This has the consequence that the interaction terms induce a finite imaginary part to the self energy correction and in some approximation it introduces a Lorentzian peak modifying the edge singularity of the Luttinger liquid. This is called a Fermi-Lutttinger liquid in  and was discussed in Sec. 3.
The holographic analysis also showed an interesting non Luttinger behavior in that the zero frequency behavior is peculiar. It has a singularity at that is -independent. In this section we attempt to reproduce this in the boundary theory.
Let us consider the following action:
is the quasiparticle fermion field whose Green’s function we are interested in determining. It has dimension half unlike the fermion operator that was studied in the last section, which had dimension one. Nevertheless this is a matter of detail and will not affect the main point of this section. is the bosonized version of some other charged fermion, . They are related by .
The above action assumes Lorentz invariance and we have set . But because Lorentz invariance is absent one can expect the two quasiparticles to have different velocities:
Both and can get renormalized by interactions. For instance adding a charge-charge interaction term for adds to the action. This modifies . In fact we will assume that is renormalized to zero. This will be crucial as we will see below.
We now add an interaction term
We can perform a Wick rotation () to Euclidean space for calculations. Using the fact that we see that and has dimension . The interaction term that has been added is thus irrelevant when . One also sees that
The propagator in momentum space can be obtained by dimensional analysis and is (in Minkowski space) . If we now assume is renormalized to zero, we get
Thus a one loop correction (Fig.(16)) to the two point function (inverse propagator) is:
We can do the integral to get and then do the integral to get . This is a independent non-analyticity that one is seeing in the AdS computation. The crucial element that made this happen was the fact that the velocity of the was zero. This could correspond to some effective non propagating or localized particle.
We have attempted to understand the behavior of Green’s functions of a 1+1 field theory with some background charge density using the AdS/CFT correspondence. The forms of Green’s functions of scalars, fermions and currents have been obtained at zero as well as non-zero temperatures. The fermion Green’s function shows interesting behavior - not typical of a Landau Fermi liquid and can be termed non-Fermi liquid. In 1+1 dimensions one may expect Luttinger liquid behavior under some circumstances. At high frequencies this is seen. However at low frequencies the behavior is quite different and seem to have similarities with Fermi-Luttinger Liquid. We have also made detailed comparison for with expectations from and found reasonable agreement. In particular the intriguing log periodicity found in [20, 23] is also seen. We have also suggested a possible explanation for the -independent non-analyticity at in the fermion (or scalar) Green’s function from the point of view of the boundary theory. The non-analyticity can be explained if there are modes with almost zero velocity (i.e. non propagating) that interact with these fermions. These could be impurities for instance.
Finally there is the obvious question of understanding what experimental setup would correspond to these theories. Also because the boundary is 1+1 dimensional, more detailed theoretical calculations may be tractable. This would be a good way to test the AdS/CFT correspondence in more general situations such as the one discussed in this paper.
We would like to thank T. Senthil, T Vojta and especially G. Baskaran for extremely useful discussions.
Appendix A The Background Geometry
where, and . Also let be the horizon radius determined by the largest real root of . The coordinates have following range: , and . We scale the coordinates and redefine parameters in the following way such that can be replaced by , where ,
We can also define . Then the metric is given by ,
with . Note that now, and the vector potential is given by,
We have chosen a gauge such that potential at horizon is zero. We now introduce a further
redefinition of coordinates to make all the parameters and coordinates dimensionless
Thus consider the redefinitions:
After which the metric becomes (dropping primes for simplicity),
We now choose a coordinate and set , we get the metric (2)
Appendix B Spinor Green’s Function
The action for bulk spinor field is given by,
and is the spin connection
where are complex functions. Now writing
the Dirac equation becomes,
where . Let us consider the following transformation
where is some constant. The equations for becomes,
In order get the retarded correlation function for the dual fermionic operator , we need to impose ingoing boundary condition for at the horizon.
We substitute in (B.0.8) to obtain the leading behavior as ,
Assume that the dominant term is the one with the -derivative acting on . This gives