Metric and Spin-3 Field with Chemical Potentials
13.4 Flat Space Einstein Gravity with Chemical Poten- Poten-tials
13.5.2 Grand Canonical Free Energy and Phase Transitions
There are three branches of solutions of all the holonomy conditions as shown in the previous section. The proposal for the correct branch to choose was the branch which connects continuously to the spin-2 results in the limit of vanishing spin-3 charges. However, it is not guaranteed that this branch is actually the correct one from a thermodynamical perspective in the whole parameter space. A way to check whether or not the chosen branch is thermodynamically sensible is to compare the free energies of all branches for given values of the chemical potentials and check which of the branches leads to the lowest free energy.
The first step will be to write the general result for the (grand canonical) free energy, regardless of the specific branch6. Since the entropy has been determined previously, which is a thermodynamic potential in terms of extensive quantities (charges), the only thing to do is to perform a Legendre transformation with respect to all pairs of charges and chemical potentials.7
F(T, Ω,ΩV,ΩU) =−QM−T S−ΩQL−ΩVQV−ΩUQU. (13.54) The zero mode charges are given by (13.6) and the intensive quantities by the chemical potentials.
T−1 =β =− ∂S
∂QM
L,V,U =−βL(1 +µM), (13.55) βΩ =− ∂S
∂QL
M,V,U =−βLµL, (13.56)
βΩV=− ∂S
∂QV
M,L,U =−βLµV, (13.57)
βΩU =− ∂S
∂QU
M,L,V =−βLµU. (13.58)
In order to express free energy in terms of intensive variables one has to invert the holonomy conditions (13.28),(13.29),(13.32) and (13.33) and solve for the charges in terms of chemical potentials.
Before doing so, however, it is instructive to consider the free energy expressed in terms of charges in certain limits. In the large Rlimit (weak contribution from spin-3 charges) one recovers the spin-2 result
Fweak =−M
2 +O(P/√
R) +O(1/R). (13.59) In theR →3limit (strong contribution from spin-3 charges) one obtains
Fstrong=−M
6 +O(R −3)2. (13.60)
6I setk= 1in this subsection.
7Alternatively, one could use the on-shell action method by Bañados, Canto and Theisen [168].
Thus, one finds a universal ratio
Fweak
Fstrong
= 3. (13.61)
The results (13.59)-(13.61) are valid on all branches and show that the free energy approaches the correct spin-2 value.
Performing the Legendre transformation (13.54) with the entropy (13.34) yields F =−QM+T βLµVQV =−M
2 −4ΩVV. (13.62)
In order to obtain the free energy as function of intensive variables one has to solve the nonlinear holonomy conditions (13.28), (13.29) for the charges in terms of the chemical potentials. Solving (13.28) forV allows one to express free energy in terms of the massMand chemical potentials.
F =−M
2 +MΩΩV
6ΩU +2M2ΩUΩV
9Ω −2π2T2ΩV
3ΩΩU . (13.63)
Plugging the solution for the spin-3 chargeV in terms of the massMinto the other holonomy condition (13.29) establishes a quartic equation for the massM, which leads to four branches of solutions for the free energy. The discriminant of that equation is positive, provided the spin-3 chemical potential obeys the bound
Ω2U< 9(2√
Another way to read the inequality (13.64) is that it provides an upper bound on the temperature for given spin-3 chemical potentialΩU. The maximal temperature is given by
In the limit of small ΩU it turns out that only one of the branches has finite free energy. This is the branch that continuously connects with spin-2 results, on which free energy yields The term before the parentheses reproduces the spin-2 result for free energy. The term in the parentheses depends only on two linear combinations of the chemical potentials8. As in the spin-2 case [63] there will be a phase transition between flat space cosmologies and hot flat space at some critical temperature.
A novel feature of the spin-3 case is that there are additional phase transitions
8To be more precise, ontandvintroduced in (13.68).
between the various flat space cosmology branches. To see this, I consider the difference between the free energies of two branches
∆F12= 2ΩUΩV
9Ω M1− M2M1+M2+3Ω(ΩΩV−3ΩU) 4Ω2UΩV
. (13.67)
There are two zeros in the difference (13.67), an obvious one when the masses of the two branches coincide,M1 =M2, and a non-obvious one when the expression in the last parentheses in (13.67) vanishes. In the following the focus will be on the difference between the branch that continuously connects to spin-2 results (branch 1) and the other branch that ceases to exist if the bound (13.64) is violated (branch 2). The other two branches are then branch 3 and 4, which will only play minor roles.
To reduce clutter it will be assumed from now on that temperature and the chemical potentials are non-negative. Moreover, it is convenient to introduce dimensionless combinations of chemical potentials as
t= 2πT ΩU
Ω2, v= ΩV Ω
ΩU . (13.68)
The quantitytis a dimensionless temperature, whilevis essentially a ratio of odd over even spin-3 chemical potential. Expressing the difference of free energies (13.67) between branches 1 and 2 as function of these two combinations, up to a non-negative overall constant, yields
∆F12∝15v−18−v s
64t2+ 9 +8t(64t2+ 27)
N(t) + 8tN(t). (13.69) with
N(t) = 512t3+ 648t+ 9p4096t4+ 3456t2−2431/3. (13.70) The positive real zero of the term under the square-root in (13.70) corresponds precisely to the critical temperature (13.65). For each value of dimensionless temperature t there is a simple zero in∆F12 since it depends linearly onv. The corresponding value ofvwill be called “critical” and denoted by a subscript “c”. For vanishing temperature one finds from setting (13.69) to zero
vc|t=0= 3
2, (13.71)
while at the critical temperature (13.65) one finds similarly vc|
t=tc=3 8
√
2√
3−3 = 2. (13.72)
The corresponding free energy differences near these temperatures read, respec-tively
∆F12∝12v−18−12tv+83t2v+O(t3), (13.73)
∆F12∝9v−18−8q1 +√2
3(t−tc)v−1627(t−tc)2v+O(t−tc)3. (13.74) Hence one arrives at the following picture, depending on the value of the parameter v:9
0<v< 32: Branch 1 is thermodynamically unstable for all temperatures.
v= 32: Branch 1 degenerates with branch 2 at vanishing temperature and is thermodynamically unstable for all positive temperatures.
32 <v<2: Branch 1 degenerates with branch 2 at some positive temperature.
Below that temperature branch 1 is thermodynamically unstable. At that temperature there is a phase transition from branch 2 to branch 1. Above that temperature branch 1 is stable (modulo the phase transition to hot flat space [63]).
v = 2: Branch 1 degenerates with branch 2 at the maximal temperature (13.65) and is thermodynamically stable for all temperatures (again modulo
the phase transition to hot flat space).
v>2: Branch 1 is thermodynamically stable for all temperatures (with the same caveat as above).
To illustrate the results above I show an example in Figure13.1. In all six graphs the thick line depicts free energy for branch 1 and the dashed line for branch 210. The three upper plots show explicitly the phase transition between branches 1 and 2, depending on the choice ofv. The three lower plots show that there are further phase transitions involving the branches 3 and 4 if branch 1 is unstable for all values of temperature. In addition to all these new phase transitions there is the ‘usual’
phase transition to hot flat space [63], which in the present case can be of zeroth, first or second order. Since there are several phase transitions possible, there exist also multi-critical points where three or four phases co-exist.
The most striking difference between the AdS results by David, Ferlaino and Kumar [169] and this flat space results is that one observes the possibility of first order phase transitions between various branches (see the right upper and middle lower plot in figure 13.1). In contrast, for AdS the only phase transitions (other than
9Positivity of entropy imposes additional constraints on the existence of branches. The existence of the first order phase transition between branches 1 and 2 described below is not influenced by such constraints.
10The other two branches are not essential for this discussion. If visible they are plotted as dotted lines.
0.1 0.2 0.3 0.4 T
Fig. 13.1.: Plots of free energy as function of temperature. In all plotsΩ = 1,ΩU = 0.1.
Upper Left: ΩV= 0.4. Upper Middle:ΩV= 0.2. Upper Right:ΩV= 0.18. Lower Left: ΩV= 0.15. Lower Middle: ΩV= 0.12. Lower Right:ΩV= 0.01. The branch with smooth spin-2 limit is displayed as thick line, the second branch as dashed line, the other two branches as dotted lines (in the upper plots these lines are at positiveF).
Hawking–Page like) arise because two of the branches end, at which point the free energy jumps (These zeroth order phase transitions are also recovered in flat space, see e.g. the left lower plot in figure13.1).
13.6 Conclusions
This PartIIIwas mainly focused on the gravity aspects of a flat space (higher-spin) holographic correspondence. In Chapter10I have shown how suitable limits from known results related to AdS3 holography can be used to determine an efficient Chern-Simons description of flat space, the asymptotic symmetries of flat space (higher-spin) gravity theories and a flat space (higher-spin) Cardy formula. This shows that for certain instances the limit of vanishing cosmological constant of the AdS3results can yield physically sensible results which can be used for a holographic correspondence involving asymptotically flat spacetimes.
Chapter11discussed unitarity of linear and nonlinearF W-algebras alike. Similar to the results obtained forW-algebras, the requirement of unitary representations yields restrictions for bothcLandcM under certain assumptions. In order to have unitary representations one has to set cM = 0. This has severe consequences for the presence of higher-spin excitations of nonlinearF W-algebras. I showed that cM = 0also renders all higher-spin excitations for general nonlinearF W-algebras unphysical and thus concluded that this is a NO-GO result for having flat space, unitarity and higher-spins at the same time. Since I assumed that theF W-algebras are realized by highest-weight representations, I also argued that there could be possible loopholes circumventing my NO-GO result. I then showed how to exploit a
specific loophole, namely the nonlinearity of theF W-algebras, by considering the linearF W∞algebra. Using this algebra I showed that for a linearF W-algebra it is indeed possible to have flat space, unitarity and higher-spins at the same time.
Following up on this discussion I reviewed in Chapter12how to describe higher-spin gravity in flat space using a Chern-Simons formulation. Building up on this I showed how to add chemical potentials to flat space (higher-spin) gravity in Chapter13. I also performed consistency checks and closely examined the special case of flat space Einstein gravity with chemical potentials. Furthermore, I determined the entropy of flat space cosmological solutions with chemical (higher-spin) potentials turned on. Following up on this I also showed how to determine the grand canonical free energy which then led to the discovery of new first order phase transitions between various flat space cosmological solutions.
Building up on these results from the gravity side I will provide explicit checks of a holographic correspondence in asymptotically flat spacetimes in the following PartIV of this thesis.