Quenched free energy in random matrix model

Recently, it is shown that the path integral of Jackiw-Teitelboim (JT) gravity Jackiw:1984je ; Teitelboim:1983ux is equivalent to a certain double-scaled matrix model Saad:2019lba . From the viewpoint of holography, this implies that the holographic dual of JT gravity is not a single quantum mechanical system but an ensemble of systems with random Hamiltonians. Moreover, it is realized in Saad:2018bqo that the Euclidean wormhole connecting different boundaries of spacetime plays an important role in explaining the so-called “ramp” of the spectral form factor Garcia-Garcia:2016mno ; Cotler:2016fpe in the Sachdev-Ye-Kitaev (SYK) model Sachdev ; kitaev2015simple . The importance of the wormhole in quantum gravity is also emphasized in the recent computation of the Page cure using the replica method Penington:2019kki ; Almheiri:2019qdq , where the inclusion of the so-called replica wormhole is essential for the resolution of the apparent paradox in the original Hawking’s calculation.¹¹1However, 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.

In recent papers Engelhardt:2020qpv ; Johnson:2020mwi the replica method is applied to the computation of the free energy in JT gravity. As emphasized in Engelhardt:2020qpv , this problem is very interesting to reveal the role of replica wormholes and explore the possibility of replica symmetry breaking and the putative spin glass phase of quantum gravity.²²2The replica symmetry breaking in the SYK model is discussed in Gur-Ari:2018okm ; Arefeva:2018vfp . The idea of replica symmetry breaking is developed by Parisi parisi1979infinite ; parisi1980sequence to solve the sping glass model of Sherrington and Kirkpatrick sherrington1975solvable . See e.g. Denef:2011ee ; castellani2005spin ; sherrington for reviews of spin glasses. However, it is reported in Engelhardt:2020qpv that a naive application of the replica method leads to a pathological behavior of the free energy. In a recent paper Johnson:2020mwi it is emphasized that the non-perturbative effect is important to resolve this problem.

In this paper, we will consider a simple toy model for the computation of free energy in JT gravity. Instead of the matrix model of JT gravity in Saad:2019lba , we consider the free energy in the Gaussian matrix model where the Hamiltonian is regarded as a random hermitian matrix with Gaussian distribution.³³3This problem is suggested in the discussion section in Engelhardt:2020qpv . We are interested in the the so-called quenched free energy $\langle\log Z(\beta)\rangle$⁴⁴4 Strictly speaking the quenched free energy is defined by including the factor of temperature $F=-T\langle\log Z(\beta)\rangle$ as in (6), but we will loosely use the name “quenched free energy” to indicate either $\langle\log Z(\beta)\rangle$ or $F=-T\langle\log Z(\beta)\rangle$ depending on the context. We believe that which one we are referring to is clear from the context and this will not cause a confusion to the readers. in the matrix model, where $Z(\beta)=\operatorname{Tr}e^{-\beta H}$ is the partition function with the inverse temperature $\beta=T^{-1}$ and the expectation value is defined by the integral over the $N\times N$ hermitian matrix $H$

One can compute the quenched free energy by the replica method⁵⁵5 The correlators of the resolvent $\operatorname{Tr}(E-H)^{-1}$ in the Gaussian matrix model are analyzed by the replica method in kamenev1999wigner . It is found that the replica symmetry breaking is important to reproduce the known results of the correlators of resolvents. In this paper we are dealing with the different quantity $\langle\log Z(\beta)\rangle$ and the computation in kamenev1999wigner cannot simply be generalized to our case.

In the high temperature regime the $n$-point correlator $\langle Z(\beta)^{n}\rangle$ is approximated by the disconnected correlator

and the $n\to 0$ limit in (2) gives rise to

The right hand side of this equation is known as the annealed free energy. On the other hand, in the low temperature regime it is not clear how to define the analytic continuation of $\langle Z(\beta)^{n}\rangle$ to $n<1$. This is the origin of the difficulty found in Engelhardt:2020qpv .

It turns out that we can avoid this difficulty of analytic continuation by directly evaluating the quenched free energy by the matrix integral

We can rewrite this integral (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 $S$

In order for the entropy to be positive, the free energy $F$ should be a monotonically decreasing function of $T$. In the replica computation of the quenched free energy of JT gravity Engelhardt:2020qpv , a pathological non-monotonic behavior of $F$ is found under a certain prescription of the analytic continuation in $n$.⁶⁶6As emphasized in Engelhardt:2020qpv the analytic continuation of $\langle Z(\beta)^{n}\rangle$ from positive integer $n$ to $n<1$ is not unique. The non-monotonic behavior of the free energy is a consequence of the particular choice of the analytic continuation used in Engelhardt:2020qpv . However, as explained in Engelhardt:2020qpv their choice of analytic continuation is not meant to be the correct one but it is just an illustrative example to demonstrate the importance of the replica wormholes in the computation of free energy. We find that the direct computation of the quenched free energy in the Gaussian matrix model (5) gives rise to a well-defined monotonic behavior of the free energy $F$.

This paper is organized as follows. In section 2, we find the explicit integral representation of the quenched free energy (5) and study its behavior in the high and low temperature regimes. In section 3, we study the exact free energy and entropy for $N=2,3$ as examples. We find that the free energy exhibits a well-defined monotonic behavior as a function of $T$. In section 4, we comment on the computation using the replica method. We propose a necessary condition for the analytic continuation of $\langle Z(\beta)^{n}\rangle$ 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 (5) directly without using the replica trick. From the standard argument, the matrix integral in (5) is written as an integral over the $N$ eigenvalues $\{E_{1},\cdots,E_{N}\}$ of $H$

Here the normalization factor $\mathcal{Z}$ is given by

where $G_{2}(N+1)$ denotes the Barnes $G$-function. Using this expression (7), in subsection 2.1 and 2.2 we will study the behavior of quenched free energy in the high temperature and the low temperature regimes, respectively.

In this section we study numerically the integral representation of the quenched free energy (20) for $N=2,3$. For $N=2$ the integral (20) becomes

and for $N=3$ we find

One can easily evaluate these integrals numerically. In Fig. 1 we show the plot of free energy as a function of temperature. At high temperature, the quenched free energy approaches the annealed free energy $F_{\text{ann}}=-T\log\langle Z(\beta)\rangle$ (orange dashed curve) as expected. In Fig. 2 we show the plot of entropy $S$. One can see that $S$ approaches $\log N$ at high temperature and vanishes at zero temperature.

Let us take a closer look at the low temperature regime for $N=2$. The small $T$ expansion of the integral (24) is obtained by rescaling the integration variable $E\to TE$ and expanding the Gaussian factor in (24)

Note that the first small $T$ correction is of order $T^{4}$ which is consistent with the general result in (21). In Fig. 3 we show the plot of quenched free energy for $N=2$ and its small $T$ expansion up to $T^{6}$ in (26). One can see that the exact quenched free energy is a monotonic function of $T$ even in the low temperature regime and $F$ becomes $E_{0}$ at zero temperature. A pathological non-monotonic behavior found in Engelhardt:2020qpv using the replica trick does not occur in the exact result of quenched free energy.

Let us compare our direct calculation of quenched free energy with the replica method (2). As we mentioned in section 1, one can easily apply the replica method in the high temperature regime and obtain the result (4). In particular, in the high temperature limit $\beta\to 0$, the partition function $Z(\beta)=\operatorname{Tr}e^{-\beta H}$ reduces to the dimension of the Hilbert space

Thus the quenched free energy approaches the maximal entropy of the system in the limit $T\to\infty$

On the other hand, the application of the replica trick in the low temperature regime is rather subtle. Under a certain prescription of the analytic continuation in the number of replicas $n$, it is found that the free energy exhibits a non-monotonic behavior as a function of temperature Engelhardt:2020qpv .

Our direct computation of the quenched free energy puts a certain constraint on the possible form of the analytic continuation in $n$. At low temperature, the smallest eigenvalue $E_{0}$ of $H$ becomes dominant and thus we expect

We can regard (29) as a condition for the possible analytic continuation of $\langle Z(\beta)^{n}\rangle$ to satisfy. Then we can apply the replica method in the low temperature regime

which reproduces the correct behavior of the quenched free energy (15).

Note that there is no $\log N$ entropy term in (30) since only a single eigenvalue (the lowest energy state) contributes to $\langle Z(\beta)^{n}\rangle$ in the low temperature limit. This explains the vanishing of entropy at zero temperature (23).

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 $\langle Z(\beta)^{n}\rangle$ is expanded in terms of the connected correlators

Here $[p^{\nu_{p}}]=[1^{\nu_{1}}2^{\nu_{2}}\cdots n^{\nu_{n}}]$ denotes a partition of $n$. In the high temperature regime the disconnected part $\langle Z(\beta)\rangle^{n}$ corresponding to the partition $[1^{n}]$ is dominant, while at low temperature the totally connected part $\langle Z(\beta)^{n}\rangle_{\text{conn}}$ corresponding to the partition $[n^{1}]$ is dominant Okuyama:2019xvg . Then one might naively think that the quenched free energy in the low temperature regime is given by the totally connected correlator $\langle Z(\beta)^{n}\rangle_{\text{conn}}$

One can try to compute $\langle Z(\beta)^{n}\rangle_{\text{conn}}$ for integer $n$ and analytically continue it to $n=0$. However, this analytic continuation is very subtle since $\langle Z(\beta)^{n}\rangle_{\text{conn}}$ scales as $N^{2-n}$ in the large $N$ limit and the naive $n\to 0$ limit of $\langle Z(\beta)^{n}\rangle_{\text{conn}}$ is not $1$ and the limit (32) does not exist. It is not clear how to define the analytic continuation of $\langle Z(\beta)^{n}\rangle_{\text{conn}}$ which satisfies the condition (29). We believe that (32) is not the correct way to compute the low temperature regime of quenched free energy. In other words, the two limits $\beta\to\infty$ and $n\to 0$ do not commute.

A similar problem has appeared in the so-called random energy model derrida1981random .⁷⁷7 The random energy model is defined as a model with $N$ randomly distributed energy eigenvalues with Gaussian distribution but the correlation among eigenvalues is ignored. It is known that the random energy model is equivalent to the $p\to\infty$ limit of a $p$-spin generalization of the Sherrington-Kirkpatrick model derrida1980random ; derrida1981random . In derrida1981random this problem is circumvented by promoting $(p,\nu_{p})$ in (31) as a continuous variable and the correct low temperature behavior is obtained by plugging $\nu_{p}=\frac{n}{p}$ and extremizing the term $(\langle Z(\beta)^{p}\rangle_{\text{conn}})^{n/p}$ in (31) with respect to $p$. The $n$-point function $\langle Z(\beta)^{n}\rangle$ obtained with this prescription indeed satisfies the necessary condition (29) and we can safely take the $n\to 0$ limit derrida1981random . The resulting quenched free energy $F$ exhibits a phase transition in the large $N$ limit at a certain critical temperature $T_{c}$: for $T>T_{c}$, $F$ agrees with the annealed free energy which takes the form $F_{\text{ann}}=aT+bT^{-1}$ with some coefficients $a,b$, while for $T<T_{c}$, $F$ is constant derrida1981random . This $F$ is a monotonic function of $T$ as expected. It would be interesting to see if the same prescription works in the present case of random matrix model. We leave this as an interesting future problem.

In this paper we have analyzed the quenched free energy in Gaussian matrix model directly without using the replica method. We find an integral representation of the exact quenched free energy (20). The exact quenched free energy is a monotonic function of temperature as expected, and the entropy computed from this free energy approaches $\log N$ at high temperature and vanishes at zero temperature.

There are many interesting open questions. It is very interesting to see if there is a phase transition in the large $N$ limit. In the case of random energy model, it is known that there is a phase transition associated with the replica symmetry breaking and the low temperature phase corresponds to a spin glass gross1984simplest . Since the random matrix model considered 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 also exhibits a phase transition.⁸⁸8As discussed in Okuyama:2019xvg , all contributions in the decomposition (31) become comparable around $T\sim N^{-2/3}$ and the dominance of disconnected part $\langle Z(\beta)\rangle^{n}$ is lost below this temperature. In the strict $N\to\infty$ limit this temperature $T\sim N^{-2/3}$ vanishes. To keep this scale finite, we can take a scaling limit $N\to\infty,T\to 0$ with $TN^{2/3}$ fixed. As discussed in Engelhardt:2020qpv , this amounts to focusing on the edge of the Wigner distribution (10) and this scaling limit corresponds to the so-called Airy limit. To settle this issue it is important to understand the analytic continuation of $\langle Z(\beta)^{n}\rangle$ to $n<1$. We proposed a simple condition (29) for the analytic continuation of $\langle Z(\beta)^{n}\rangle$ to satisfy.

It would be very interesting to generalize our analysis to the JT gravity matrix model and see if the spin glass phase is realized at low temperature Engelhardt:2020qpv . In Engelhardt:2020qpv the quenched free energy is computed by a certain prescription of the analytic continuation of $\langle Z(\beta)^{n}\rangle$ and it leads to a pathological behavior at low temperature. It is argued in Johnson:2020mwi that this problem is resolved by including the non-perturbative effect. It would be very interesting to 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 lowest energy state) gives a dominant contribution to the quenched free energy. This reminds us of the “eigenbrane” introduced in Blommaert:2019wfy . Perhaps the spacetime picture of the low temperature phase is described by an eigenbrane with one of the eigenvalues pinned to the edge of the spectral density. It would be interesting to investigate this picture further.