FIGURE 5.2: The normal-ized baryon diffusion coeffi-cient as a function of normal-ized inverse temperature. At densities belowd˜= 0.00315 we observe a multivalued de-pendence on m, signaling a phase transition. For the behav-ior of the coefficient in a larger range ofmsee figure 4.1.
1.29 1.30 1.31 1.32 1.33 1.34
0.0 0.1 0.2 0.3 0.4
m
2πTD
d˜= 0 d˜= 0.00105 d˜= 0.00210 d˜= 0.00315 d˜= 0.00420 d˜= 0.00525 d˜= 0.00630
At finite density, we know that black hole embeddings capture the physics at all temperatures,i. e.the entire parameter regime ofm. The fundamental phase transition in this case is a transition between two different black hole embeddings[58]. As discussed in ref. 59, the baryon density affects the location and the presence of the fundamental phase transition. The transition is of first order only very close to the separation line between the regions of zero and non-zero baryon density shown in figure 3.1. Note that as discussed in refs. 58–60 there exists a region in the( ˜d, T)phase diagram at smalld˜andT where the embeddings are unstable. This instability disappears for larged.˜
We study the baryon diffusion coefficient at different baryon densities.
Figure 5.2 shows the Diffusion coefficientDas a function of the ratio of quark mass to temperaturem. By fixing the quark mass we may think ofmas the inverse of the temperature.
We find that the phase transition is slightly shifted towards smaller temper-atures when we increase the density. At a critical density ofd˜∗ = 0.00315the phase transition temperature is given bym= 1.31. Beyond the critical density the transition vanishes, in agreement with the critical densityd˜∗ for the phase transition in the quark condensate, discussed in ref. 58.
0 1 2 3 4 0
50 100 150 200 250 300 350
0 1 2 3 4
0 100 200 300 400 500 600 700
w w
R RX
m= 3 m= 3
d˜= 1 d˜= 2 d˜= 20 d˜= 200 d˜= 2000
d˜I= 2 d˜I= 8 d˜I= 14
FIGURE5.3: Spectral functions for various baryon densities (left) and isospin densities (right), again normalized toNfNcT2/4. At increasing densitiesd˜the peaks are smeared out, as we saw in the discussion of the spectral functions in chapter 3. At very high densities a new structure forms at smallw.
figure 3.11. The magnitude of this splitting of the spectral lines is determined by the chemical potential and the undetermined couplingcA.
In the limit of zero frequencyw→0, equations(3.82)and(3.83)coincide and will result in identical solutions X and Y. In this limit the solution E3, though, differs fromX andY, by means of the last term. So for small frequenciesw, we expect differences between the solutions E3 and X, Y. All three equations of motion depend on the particle densityd˜parametrically, since the density has influence on the background fields.
Spectral functions and quasi normal modes at high densities
We work in the canonical ensemble and will now investigate the effects of variations in d. Spectral functions for various finite baryonic and isospin˜ densitiesd˜are shown in figure 5.3. As in section 3.3, the peaks in these spectral functions indicate that quarks form bound states. At low baryon densities we recognized the positions of the peaks to agree with the supersymmetric result (3.67). Increasing the quark density leads to a broadening of the peaks, which indicates decreasing stability of mesons at increasing baryon density[81,160]. At the same time the positions of the peaks change, which indicates a dependence of the meson mass on the baryon density. Now, further increasing the quark density leads to the formation of a new structure atw<1. We will discuss this structure below together with the results at finite isospin density.
We now turn to the effects of finite isospin density on the spectrum. The peaks in the spectral functions again correspond to mesons. An interesting feature at finite isospin chemical potential is the formation of a new peak in the spectral function in the regime of smallwat high density/high chemical potential, see figure 5.3. Notice that compared to the baryonic case, the density at which the new peak forms is about two orders of magnitude smaller. As
-0.02 -0.01 0.00 0.01 0.02 -4000
-2000 0 2000 4000
RY
Rew
m= 3 d˜I= 15.00 d˜I= 15.35 d˜I= 15.70
FIGURE5.4: Plot of the spectral function for the modeY aroundw= 0. At a value ofd˜I= 15.35a pole appears at the origin.
This behavior is due to the movement of poles in the complexw-plane, illustrated in figure 5.6.
Imw
Rew
¨
⌣
¨
⌢
FIGURE5.5: A sketch of the positions and movements of the quasinormal fre-quencies under changes ofd˜I. Color in-dicates the function:red=Y,green=X, blue = E3. The symbols indicate the range ofd˜I:◦<d˜crit,•= ˜dcrit,>d˜crit. Poles in the gray region introduce instabili-ties.
in the baryonic case, the excitations related to the supersymmetric spectrum broaden, the corresponding mesons become unstable.
We pointed out that the structure of the spectral function is determined by the pole structure of the retarded correlator, see section 3.1. The poles of this function are located in the complexω-plane at positionsΩn ∈ C. The spectral functions show the imaginary part of the correlator at real valuedω.
Any pole in the vicinity of the real axis will therefore introduce narrow peaks in the spectral function, while poles far from the real axis have less influence and merely introduce small and broad structures.
In section 3.1 we outlined how the imaginary part of the quasinormal modes describes damping, as long asIm Ωn<0. The short note on the pole structure demonstrated the dependence of the position of the quasinormal modes on the chemical potential/particle density. From figure 5.3 we deduce that at higher densities than studied so far, a quasinormal mode approaches the origin of the complexωplane as the particle density is increased. We observe a pole atw= 0for a certain particle densityd˜crit, the value depends onm. An impression of the variation in the spectral function is given in figure 5.4.
In figure 5.5 we qualitatively sketch the result from the investigation of the behavior of the quasinormal modes closest to the origin of the complex w-plane. These modes donotproduce the peaks corresponding to the spectrum (3.67). At low densities all quasinormal modes are located in the lower half plane. When increasing the isospin density, the lowest frequency modes of the solutionsX andY to(3.82)and(3.83)move towards the origin of the frequency plane. At the same time two quasinormal modes ofE3move towards each other and merge on the negative imaginary axis, then travel along the
-160 160
0 -80 80
m
d I
d I
:
d I
:
-0.05 0.00 0.05 -0.05
0.00 0.05
Rew
Imw
m= 3 d˜I= 10
-0.05 0.00 0.05 -0.05
0.00 0.05
Rew
Imw
m= 3 d˜I= 15.35
-0.05 0.00 0.05 -0.05
0.00 0.05
Rew
Imw
m= 3 d˜I= 20.70
FIGURE5.6: Contour plots of the spectral function for the modeY aroundw= 0in the complexw-plane. The density increases from the left plot at sub-critical density to the right one at super-critical density. Here, the pole in the upper half plane introduces an instability.
axis towards the origin as one single pole. At the critical value ofd˜= ˜dcritthe modes fromXandY meet at the origin, the quasinormal modes fromE3still reside in the lower half plane. This observation matches the discussion at the beginning of this section, where we expectedXandY to behave similarly at smallw, whileE3should differ from this behavior. Upon further increasing the isospin density, the modesΩfrom XandY enter the upper half plane, maintaining their distinct directions. The sign change inIm ΩfromIm Ω<0 toIm Ω>0indicates that a damped resonance changes into a self-enhancing one, and thus introduces an instability to the system. Figure 5.6 illustrates the transition of a quasinormal mode ofY from the lower half plane to the upper half plane. TheE3-mode does not enter the upper half plane at any value of d˜we considered. Compare this to the values ofd˜in figure 5.3 at which the pole induces visible structures at smallw. A comparable movement of poles in a different but related setup was found in ref. 161. There the quasinormal modes of correlation functions of electromagnetic currents were investigated as a function of temperature.
In the following we interpret the observation of decaying mesons and the emergence of a new peak in the spectral function in terms of field theory quantities. In particular we speculate on a new phase in the phase diagram for fundamental matter in the D3/D7setup.
In the far UV, the field theory dual to our setup is supersymmetric, thus containing scalars as well as fermions, both of which contribute to the bound states we identified with mesons, even when supersymmetry is eventually broken. The meson decay at non-vanishing particle densities may be explained by the change of the shape of the potential for the scalars in the field theory upon the introduction of a non-vanishing density. As outlined in appendix E, a chemical potential may lead to an instability of the theory, since it induces a runaway potential for the scalar fields at small field values[162]. Nevertheless, interactions ofφ4-type lead to a Mexican hat style potential for larger field values. In this way the theory is stabilized at finite densityd˜while the scalar fields condensate. This squark condensate presumably contributes to the vev
FIGURE 5.7: the (µI, T )-plane. In the blue shaded re-gion D7-branes have the topol-ogy of Minkowski embeddings, the white and brown regions are modeled by black hole em-beddings. These become unsta-ble in the brown region. The boundary of the unstable re-gion asymptotically seems to agree with the thin gray line of
constant densityd˜I= 20.5. 0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.2 0.4 0.6 0.8
µI/mq
1/m
Minkowski unstable
black hole
of the scalar flavor current, d˜∝
J0
∝ψ γ¯ 0ψ +
φ ∂0φ
. (5.1)
In theAdS/CFT context, the presence of an upside-down potential for the squark vev has been shown in ref. 54 using an instanton configuration in the dual supergravity background.
The occurrence of a pole in the upper half plane of complex frequencies at finited˜critindicates an instability of the theory. A comparable observation was made in ref. 81, where in fact the vector meson becomes unstable by means of negative values for its mass beyond some critical chemical potential. The difference between this work and ref. 81 is that our model includes scalar modes in addition to the fundamental fermions. Nevertheless, in both models an instability occurs at a critical value of the chemical potential. The theory may still be stabilized dynamically by vector condensation[163]. In this case the system would enter a new phase of condensed vectors at densities larger than d˜crit, in accordance with the expectation from QCD calculations[88, 164, 165].
We perform the analysis of the pole structure atw= 0for variousm, and interpret the phenomenon of the transition of poles into the upper half plane at finite critical particle density as a sign of the transition to an unstable phase.
We relate the critical particle densityd˜critto the according chemical potential
˜
µIbyµI = limρ→∞A30(ρ)and use the pairs ofmand critical dimensionful µIto trace the line of the phase transition in the phase diagram of fundamental matter in the D3/D7setup. The result is drawn in figure 5.7. The picture shows the(µI, T)-plane of the phase diagram and contains three regions, drawn as blue shaded, white, and brown shaded, as well as solid lines, separating the different regions.
The blue shaded region marks the range of parameters, in which fundamen-tal matter is described by D7-branes with Minkowski embeddings. The line, delimiting the blue region, marks the line of phase transitions to the black hole phase, where fundamental matter is described by D7-branes which have black hole embeddings. Using the symmetry of the DBI action, this phase transition line can be mapped to the line of phase transitions between Minkowski and
black hole embeddings, present at finite baryon chemical potential[41, 59, 60]. The brown shaded region in the phase diagram in figure 5.7 marks the observation made in this section. The line delimiting the brown region marks the values ofd˜critat which the pole in the spectral function appears atw= 0.
Beyond this line we enter the brown shaded unstable region.
We observe that the separation line of the unstable phase asymptotes to a straight line at high temperatures. Within the values computed by us, this line agrees with the asymptotic behavior of the contour of particle density withd˜≈ 20.5, drawn as a thin gray line in the phase diagram. We thus speculate on a finite critical particle density beyond which the black hole phase is unstable. This interpretation is supported by analogous studies of the phase diagram ofN = 4super-Yang-Mills theory with R-symmetry chemical potentials, where a similar line in the phase diagram was discovered[166, 167]. The remaining question is whether the brown shaded phase in figure 5.7 indeed is unstable in the sense that it inaccessible for any physical setup, or if there is a way to stabilize the system in the parameter range of question. Recent publications revealed that the introduction of a further vev for a different gauge field component on the stack of probe branes leads to a stabilization of the system[168, 169]. The resulting setup exhibits a second order phase transition to the new phase, which bears analogies to the theories of superfluidity and superconductivity[168–171].
Note that the location of the transition line to the unstable phase in fig-ure 5.7 as well as the results shown in figfig-ure 5.6 and figfig-ure 5.5 are obtained from the analysis of poles in the spectral functions. These functions in turn are obtained as solutions to equations(3.82)to(3.84), which do depend on the so far unknown factorcA in determining the self coupling of the gauge field on the brane. The computation of this factor is left to future work. It will determine the exact position of the boundary of the brown shaded region in figure 3.1. This will answer the question whether there is a triple point in the phase diagram and if the color shaded regions meet at a common border.
Moreover, other poles than the ones investigated here may have influence on the stability of this system.