PPrepared for submission to JHEP
Quenched free energy in random matrix model
Kazumi Okuyama
Department of Physics, Shinshu University,3-1-1 Asahi, Matsumoto 390-8621, Japan
E-mail: [email protected]
Abstract:
We compute the quenched free energy in the Gaussian random matrix modelby directly evaluating the matrix integral without using the replica trick. We find thatthe quenched free energy is a monotonic function of the temperature and the entropyapproaches log N at high temperature and vanishes at zero temperature. a r X i v : . [ h e p - t h ] N ov ontents N = 2 , Recently, it is shown that the path integral of Jackiw-Teitelboim (JT) gravity [1, 2] isequivalent to a certain double-scaled matrix model [3]. From the viewpoint of holography,this implies that the holographic dual of JT gravity is not a single quantum mechanicalsystem but an ensemble of systems with random Hamiltonians. Moreover, it is realizedin [4] that the Euclidean wormhole connecting different boundaries of spacetime plays animportant role in explaining the so-called “ramp” of the spectral form factor [5, 6] in theSachdev-Ye-Kitaev (SYK) model [7, 8]. The importance of the wormhole in quantumgravity is also emphasized in the recent computation of the Page cure using the replicamethod [9, 10], where the inclusion of the so-called replica wormhole is essential for theresolution of the apparent paradox in the original Hawking’s calculation. In recent papers [11, 12] the replica method is applied to the computation of the freeenergy in JT gravity. As emphasized in [11], this problem is very interesting to reveal therole of replica wormholes and explore the possibility of replica symmetry breaking and theputative spin glass phase of quantum gravity. However, it is reported in [11] that a naiveapplication of the replica method leads to a pathological behavior of the free energy. In arecent paper [12] it is emphasized that the non-perturbative effect is important to resolvethis problem.In this paper, we will consider a simple toy model for the computation of free energyin JT gravity. Instead of the matrix model of JT gravity in [3], we consider the free energyin the Gaussian matrix model where the Hamiltonian is regarded as a random hermitian However, the recovering of a unitary Page curve is far from sufficient to completely resolve the paradox.We still lack an understanding of by what mechanism information is able to exit an evaporating black hole. The replica symmetry breaking in the SYK model is discussed in [13, 14]. The idea of replica symmetrybreaking is developed by Parisi [15, 16] to solve the sping glass model of Sherrington and Kirkpatrick [17].See e.g. [18–20] for reviews of spin glasses. – 1 –atrix with Gaussian distribution. We are interested in the the so-called quenched freeenergy (cid:104) log Z ( β ) (cid:105) in the matrix model, where Z ( β ) = Tr e − βH is the partition functionwith the inverse temperature β = T − and the expectation value is defined by the integralover the N × N hermitian matrix H (cid:104) f ( H ) (cid:105) = (cid:82) dHe − N Tr H f ( H ) (cid:82) dHe − N Tr H . (1.1)One can compute the quenched free energy by the replica method (cid:104) log Z ( β ) (cid:105) = lim n → (cid:104) Z ( β ) n (cid:105) − n . (1.2)In the high temperature regime the n -point correlator (cid:104) Z ( β ) n (cid:105) is approximated by thedisconnected correlator (cid:104) Z ( β ) n (cid:105) ≈ (cid:104) Z ( β ) (cid:105) n , (1.3)and the n → (cid:104) log Z ( β ) (cid:105) ≈ lim n → (cid:104) Z ( β ) (cid:105) n − n = log (cid:104) Z ( β ) (cid:105) . (1.4)The right hand side of this equation is known as the annealed free energy. On the otherhand, in the low temperature regime it is not clear how to define the analytic continuationof (cid:104) Z ( β ) n (cid:105) to n <
1. This is the origin of the difficulty found in [11].It turns out that we can avoid this difficulty of analytic continuation by directly eval-uating the quenched free energy by the matrix integral (cid:104) log Z ( β ) (cid:105) = (cid:82) dHe − N Tr H log Tr e − βH (cid:82) dHe − N Tr H . (1.5)We can rewrite this integral (1.5) as an integral over the N eigenvalues of the matrix H and study the physical quantities like the free energy F and the entropy SF = − T (cid:104) log Z ( β ) (cid:105) , S = − ∂F∂T . (1.6)In order for the entropy to be positive, the free energy F should be a monotonically decreas-ing function of T . In the replica computation of the quenched free energy of JT gravity[11], a pathological non-monotonic behavior of F is found under a certain prescription of This problem is suggested in the discussion section in [11]. Strictly speaking the quenched free energy is defined by including the factor of temperature F = − T (cid:104) log Z ( β ) (cid:105) as in (1.6), but we will loosely use the name “quenched free energy” to indicate either (cid:104) log Z ( β ) (cid:105) or F = − T (cid:104) log Z ( β ) (cid:105) depending on the context. We believe that which one we are referring tois clear from the context and this will not cause a confusion to the readers. The correlators of the resolvent Tr( E − H ) − in the Gaussian matrix model are analyzed by the replicamethod in [21]. It is found that the replica symmetry breaking is important to reproduce the known resultsof the correlators of resolvents. In this paper we are dealing with the different quantity (cid:104) log Z ( β ) (cid:105) and thecomputation in [21] cannot simply be generalized to our case. – 2 –he analytic continuation in n . We find that the direct computation of the quenched freeenergy in the Gaussian matrix model (1.5) gives rise to a well-defined monotonic behaviorof the free energy F .This paper is organized as follows. In section 2, we find the explicit integral repre-sentation of the quenched free energy (1.5) and study its behavior in the high and lowtemperature regimes. In section 3, we study the exact free energy and entropy for N = 2 , T . In section 4, we comment on the computation using the replica method. Wepropose a necessary condition for the analytic continuation of (cid:104) Z ( β ) n (cid:105) to satisfy. Finally,we conclude in section 5 with some discussions on the interesting future problems. In this paper we will analyze the quenched free energy in Gaussian matrix model (1.5)directly without using the replica trick. From the standard argument, the matrix integralin (1.5) is written as an integral over the N eigenvalues { E , · · · , E N } of H (cid:104) log Z ( β ) (cid:105) = 1 Z N ! (cid:90) ∞−∞ N (cid:89) i =1 dE i e − N E i (cid:89) i 1) isrestricted to E i > E N . With this remark in mind, the quenched free energy is written as (cid:104) log Z ( β ) (cid:105) = − βE + 1 Z NN ! (cid:90) ∞−∞ dE N e − N E N (cid:90) ∞ E N N − (cid:89) i =1 dE i e − N E i × N − (cid:89) i =1 ( E i − E N ) (cid:89) ≤ i 1) and integrating out E N (cid:104) log Z ( β ) (cid:105) = − βE + 1 Z √ πN ! (cid:90) ∞ N − (cid:89) i =1 dE i e − N (cid:80) i E i + ( (cid:80) i E i ) × N − (cid:89) i =1 E i (cid:89) ≤ i 3. For N = 2 the integral (2.14) becomes (cid:104) log Z ( β ) (cid:105) = (cid:114) π β + (cid:114) π (cid:90) ∞ dEe − E E log(1 + e − βE ) , (3.1)– 5 –nd for N = 3 we find (cid:104) log Z ( β ) (cid:105) = 9 √ √ π β + 27 √ π (cid:90) ∞ dE (cid:90) ∞ dE e − ( E + E )+ ( E + E ) E E ( E − E ) log(1 + e − βE + e − βE ) . (3.2)One can easily evaluate these integrals numerically. In Fig. 1 we show the plot of free energyas a function of temperature. At high temperature, the quenched free energy approachesthe annealed free energy F ann = − T log (cid:104) Z ( β ) (cid:105) (orange dashed curve) as expected. In Fig. 2we show the plot of entropy S . One can see that S approaches log N at high temperatureand vanishes at zero temperature. T - - - - - - - F (a) Free energy at N = 2 T - - - - - F (b) Free energy at N = 3 Figure 1 : Plot of free energy for (a) N = 2 and (b) N = 3 as a function of temperature T . The solid curves are the quenched free energy while the orange dashed curves representthe annealed free energy F ann = − T log (cid:104) Z ( β ) (cid:105) with the exact one-point function in (2.3). T S (a) Entropy at N = 2 T S (b) Entropy at N = 3 Figure 2 : Plot of entropy for (a) N = 2 and (b) N = 3 as a function of temperature T .The solid curves are the exact result while the horizontal gray lines represent the maximumvalue of entropy S = log N .Let us take a closer look at the low temperature regime for N = 2. The small T expansion of the integral (3.1) is obtained by rescaling the integration variable E → T E – 6 – .05 0.10 0.15 0.20 0.25 0.30 T - - - - - - F Figure 3 : Plot of free energy for N = 2 at low temperature. The solid curve is the exactresult while the orange dashed curve represents the small T expansion (3.3).and expanding the Gaussian factor in (3.1) F = − (cid:114) π (cid:34) T (cid:90) ∞ dEE ∞ (cid:88) n =0 n ! (cid:18) − T E (cid:19) n log(1 + e − E ) (cid:35) = − (cid:114) π (cid:34) π T − π T + O ( T ) (cid:35) . (3.3)Note that the first small T correction is of order T which is consistent with the generalresult in (2.15). In Fig. 3 we show the plot of quenched free energy for N = 2 and itssmall T expansion up to T in (3.3). One can see that the exact quenched free energy isa monotonic function of T even in the low temperature regime and F becomes E at zerotemperature. A pathological non-monotonic behavior found in [11] using the replica trickdoes not occur in the exact result of quenched free energy. Let us compare our direct calculation of quenched free energy with the replica method(1.2). As we mentioned in section 1, one can easily apply the replica method in the hightemperature regime and obtain the result (1.4). In particular, in the high temperature limit β → 0, the partition function Z ( β ) = Tr e − βH reduces to the dimension of the Hilbert spacelim β → Z ( β ) = Tr 1 = N. (4.1)Thus the quenched free energy approaches the maximal entropy of the system in the limit T → ∞ lim β → (cid:104) log Z ( β ) (cid:105) = lim n → N n − n = log N. (4.2)– 7 –n the other hand, the application of the replica trick in the low temperature regimeis rather subtle. Under a certain prescription of the analytic continuation in the number ofreplicas n , it is found that the free energy exhibits a non-monotonic behavior as a functionof temperature [11].Our direct computation of the quenched free energy puts a certain constraint on thepossible form of the analytic continuation in n . At low temperature, the smallest eigenvalue E of H becomes dominant and thus we expectlim β →∞ (cid:104) Z ( β ) n (cid:105) = e − nβE . (4.3)We can regard (4.3) as a condition for the possible analytic continuation of (cid:104) Z ( β ) n (cid:105) tosatisfy. Then we can apply the replica method in the low temperature regimelim β →∞ (cid:104) log Z ( β ) (cid:105) = lim n → e − nβE − n = − βE , (4.4)which reproduces the correct behavior of the quenched free energy (2.9).Note that there is no log N entropy term in (4.4) since only a single eigenvalue (thelowest energy state) contributes to (cid:104) Z ( β ) n (cid:105) in the low temperature limit. This explainsthe vanishing of entropy at zero temperature (2.17).We would like to understand the role of replica symmetry breaking in a possible large N phase transition. When n is a positive integer, the n -replica correlator (cid:104) Z ( β ) n (cid:105) is expandedin terms of the connected correlators (cid:104) Z ( β ) n (cid:105) = n ! (cid:88) (cid:80) pν p = n n (cid:89) p =1 ν p !( p !) ν p (cid:16) (cid:104) Z ( β ) p (cid:105) conn (cid:17) ν p . (4.5)Here [ p ν p ] = [1 ν ν · · · n ν n ] denotes a partition of n . In the high temperature regime thedisconnected part (cid:104) Z ( β ) (cid:105) n corresponding to the partition [1 n ] is dominant, while at lowtemperature the totally connected part (cid:104) Z ( β ) n (cid:105) conn corresponding to the partition [ n ] isdominant [26]. Then one might naively think that the quenched free energy in the lowtemperature regime is given by the totally connected correlator (cid:104) Z ( β ) n (cid:105) conn (cid:104) log Z ( β ) (cid:105) = lim n → (cid:104) Z ( β ) n (cid:105) conn − n . (4.6)One can try to compute (cid:104) Z ( β ) n (cid:105) conn for integer n and analytically continue it to n = 0.However, this analytic continuation is very subtle since (cid:104) Z ( β ) n (cid:105) conn scales as N − n in thelarge N limit and the naive n → (cid:104) Z ( β ) n (cid:105) conn is not 1 and the limit (4.6) doesnot exist. It is not clear how to define the analytic continuation of (cid:104) Z ( β ) n (cid:105) conn whichsatisfies the condition (4.3). We believe that (4.6) is not the correct way to compute thelow temperature regime of quenched free energy. In other words, the two limits β → ∞ and n → In [27]this problem is circumvented by promoting ( p, ν p ) in (4.5) as a continuous variable and the The random energy model is defined as a model with N randomly distributed energy eigenvalues withGaussian distribution but the correlation among eigenvalues is ignored. It is known that the random energymodel is equivalent to the p → ∞ limit of a p -spin generalization of the Sherrington-Kirkpatrick model[27, 28]. – 8 –orrect low temperature behavior is obtained by plugging ν p = np and extremizing the term( (cid:104) Z ( β ) p (cid:105) conn ) n/p in (4.5) with respect to p . The n -point function (cid:104) Z ( β ) n (cid:105) obtained withthis prescription indeed satisfies the necessary condition (4.3) and we can safely take the n → F exhibits a phase transition in thelarge N limit at a certain critical temperature T c : for T > T c , F agrees with the annealedfree energy which takes the form F ann = aT + bT − with some coefficients a, b , while for T < T c , F is constant [27]. This F is a monotonic function of T as expected. It wouldbe interesting to see if the same prescription works in the present case of random matrixmodel. We leave this as an interesting future problem. In this paper we have analyzed the quenched free energy in Gaussian matrix model directlywithout using the replica method. We find an integral representation of the exact quenchedfree energy (2.14). The exact quenched free energy is a monotonic function of temperatureas expected, and the entropy computed from this free energy approaches log N at hightemperature and vanishes at zero temperature.There are many interesting open questions. It is very interesting to see if there is aphase transition in the large N limit. In the case of random energy model, it is knownthat there is a phase transition associated with the replica symmetry breaking and thelow temperature phase corresponds to a spin glass [29]. Since the random matrix modelconsidered in this paper can be thought of as a generalization of the random energy model,it is tempting to speculate that the quenched free energy of the random matrix model alsoexhibits a phase transition. To settle this issue it is important to understand the analyticcontinuation of (cid:104) Z ( β ) n (cid:105) to n < 1. We proposed a simple condition (4.3) for the analyticcontinuation of (cid:104) Z ( β ) n (cid:105) to satisfy.It would be very interesting to generalize our analysis to the JT gravity matrix modeland see if the spin glass phase is realized at low temperature [11]. In [11] the quenchedfree energy is computed by a certain prescription of the analytic continuation of (cid:104) Z ( β ) n (cid:105) and it leads to a pathological behavior at low temperature. It is argued in [12] that thisproblem is resolved by including the non-perturbative effect. It would be very interestingto complete the program of replica computation of the free energy in JT gravity.Our analysis suggests that at low temperature the smallest eigenvalue (or the lowestenergy state) gives a dominant contribution to the quenched free energy. This reminds us ofthe “eigenbrane” introduced in [30]. Perhaps the spacetime picture of the low temperaturephase is described by an eigenbrane with one of the eigenvalues pinned to the edge of thespectral density. It would be interesting to investigate this picture further. As discussed in [26], all contributions in the decomposition (4.5) become comparable around T ∼ N − / and the dominance of disconnected part (cid:104) Z ( β ) (cid:105) n is lost below this temperature. In the strict N → ∞ limitthis temperature T ∼ N − / vanishes. To keep this scale finite, we can take a scaling limit N → ∞ , T → T N / fixed. As discussed in [11], this amounts to focusing on the edge of the Wigner distribution(2.4) and this scaling limit corresponds to the so-called Airy limit. – 9 – cknowledgments This work was supported in part by JSPS KAKENHI Grant No. 19K03845. References [1] R. Jackiw, “Lower Dimensional Gravity,” Nucl. Phys. B252 (1985) 343–356.[2] C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,” Phys. Lett. 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