YITP-20-57
Numerical and analytical studies of a matrix model with non-pairwise contracted indices

Quantization of gravity is one of the most challenging problems in theoretical physics. There have been various proposals to solve the problem. Discretized setups are used by some of them, including tensor models [1, 2, 3, 4]. In discretized approaches, the main question is whether the dynamics generate global spacetimes with the continuum description of GR. In the tensor models in Euclidean forms, however, the produced objects are singular, such as branched polymers [5], or unnatural fine-tunings of the models are required to generate global spaces [6, 7]. Currently, it is an open question whether there exist discretized theories with sensible spacetime emergence.

Stimulated by the success of CDT [8], a Lorentzian formulation of the tensor model was proposed by the current author [9, 10]. Rather than introducing an explicit time direction, the tensor model was formulated as a first-class constrained system in the Hamilton formalism. Namely, Hamiltonian is given by

$\displaystyle H=N_{a}{\cal H}_{a}+N_{[ab]}{\cal L}_{[ab]},$

(1)

where $N_{a}$ and $N_{[ab]}$ are arbitrary multipliers, and ${\cal H}_{a}$ and ${\cal L}_{[ab]}$ are the first-class constraints satisfying the following structure of a closed Poisson algebra [9, 10],

$\displaystyle\begin{split}\{{\cal H},{\cal H}\}\sim{\cal L},\\ \{{\cal H},{\cal L}\}\sim{\cal H},\\ \{{\cal L},{\cal L}\}\sim{\cal L}.\end{split}$

(2)

Here the indices take the values, $1,2,\ldots,N$, and the repeated indices are assumed to be summed over. Intriguingly, this constraint Poisson algebra has the similar form as that of the ADM formalism of GR with the correspondence of ${\cal H}_{a}$ and ${\cal L}_{[ab]}$ to the Hamiltonian and momentum constraints, respectively. In particular, the similarity includes the variable dependence of the structure function in the first line of (2), which makes the dynamics non-trivial. Under some reasonable assumptions such as time-reversal symmetry and locality for the tensor model, the formulation was shown to be unique [10]. We call the tensor model the canonical tensor model for short, since this is based on the canonical formalism.

The quantization of the canonical tensor model is straightforward with the canonical quantization [11]. After quantization, the first class constraint algebra basically keeps the form (2) with the replacement of the Poisson brackets to commutators. The physical state conditions are given by

$\displaystyle\hat{\cal H}_{a}|\Psi\rangle=\hat{\cal L}_{[ab]}|\Psi\rangle=0,$

(3)

where $\hat{\cal H}_{a}$ and $\hat{\cal L}_{[ab]}$ are the quantized constraints. These conditions give a system of partial differential equations with non-linear coefficients for wave functions. Surprisingly, these equations have an exact solution [12],

$\displaystyle\Psi(P)=\left(\int_{\mathbb{R}^{N}}\prod_{a=1}^{N}d\phi_{a}\,\exp\left(I\,P_{abc}\phi_{a}\phi_{b}\phi_{c}\right){\rm Ai}\left(\kappa\,\phi_{a}\phi_{a}\right)\right)^{\frac{\lambda_{H}}{2}},$

(4)

where $\lambda_{H}=(N+2)(N+3)/2$, $P$ designates a real symmetric tensor $P_{abc}\ (a,b,c=1,2,\ldots,N)$, $I$ is the imaginary unit ($I^{2}=-1$), and Ai$(\cdot)$ is the Airy Ai function. $\kappa$ is a real parameter, which has the opposite sign to the cosmological constant, which has been argued by comparing the $N=1$ case with the mini-superspace approximation of GR [13]. An important thing is that the relation $\lambda_{H}=(N+2)(N+3)/2$ is determined by the hermiticity of the quantized constraint $\hat{\cal H}_{a}$, and should therefore have this fixed relation with $N$.

The properties of the wave function (4) are largely unknown. In particular it is unknown whether this has ridges which can be understood as spacetime trajectories, as in the ideal situation shown in the left figure of Figure 1. On the other hand, the wave function is known to have peaks (or ridges) on the locations where $P_{abc}$ satisfies Lie group symmetries (namely, $P_{abc}=h_{a}^{a^{\prime}}h_{b}^{b^{\prime}}h_{c}^{c^{\prime}}P_{a^{\prime}b^{\prime}c^{\prime}}\ (h\in H)$ with a Lie group representation $H$), as shown for the $N=3$ case in the right figure of Figure 1 [14, 15]. A striking fact is that the Lie-groups associated to such ridges of the wave function (4) have Lorentz signatures. Comparing with the left figure, the similarity encourages the study of the canonical tensor model as a model for emergent spacetimes. It would also be noteworthy that Lie group symmetries, such as Lorentz, de Sitter, and gauge, are ubiquitous in the universe, as on the peaks (or ridges) of the wave function.

To understand more of the properties of the wave function, we take a different strategy in this section. The simplest observable would be

$\displaystyle\langle\Psi|e^{-\alpha\hat{P}_{abc}\hat{P}_{abc}}|\Psi\rangle,$

(5)

where $\alpha$ is positive real. For example, one of the interests in this wave function is the limit $\alpha\rightarrow+0$. If the wave function contains a ridge which has a spacetime interpretation with an infinite extension of a time direction, the norm will linearly diverge in this time direction in the $\alpha\rightarrow+0$ limit. Therefore the behavior of (5) in the limit is related to whether the tensor model wave function contains an infinite extension of time direction or not.

Currently, it is not easy to study (5). Therefore our strategy is to carry out some simplifications to the wave function (4). When $\kappa>0$, which corresponds to the negative cosmological constant, the Airy Ai function is a fast decaying function of $\phi_{a}\phi_{a}$. Therefore, as an initial study, it would be interesting to simplify the Airy function to $e^{-k\phi_{a}\phi_{a}}$ with positive $k$, and study the properties of (5). With this simplification we obtain

$\displaystyle\langle\Psi_{simple}|e^{-\alpha\hat{P}_{abc}\hat{P}_{abc}}|\Psi_{simple}\rangle={\cal N}\alpha^{-\#P/2}Z_{N,\lambda_{H}}((4\alpha)^{-1},k),$

(6)

where ${\cal N}$ is an unimportant normalization factor, $\#P=N(N+1)(N+2)/6$, namely, the number of independent components of $P_{abc}$, and $Z_{N,R}(\lambda,k)$ is the partition function of the matrix model defined by [16]

$\displaystyle Z_{N,R}(\lambda,k)=\int\prod_{i=1}^{R}\prod_{a=1}^{N}d\phi_{a}^{i}\ e^{-\lambda\sum_{i,j=1}^{R}(\phi_{a}^{i}\phi_{a}^{j})^{3}-k\sum_{i=1}^{R}\phi_{a}^{i}\phi_{a}^{i}}.$

(7)

Here the repeated lower indices are assumed to be summed over, but the summation over the upper indices are written explicitly. This is because the upper indices are not always contracted pairwise: the first term of the exponent contains triple contractions of the upper indices. We also assume $\lambda,k>0$ for the obvious convergence of the integration.

It is particularly important to note that, while the matrix model (7) itself has both $N$ and $R$ as free parameters, they are restricted by $R=\lambda_{H}=(N+2)(N+3)/2$, if we consider the relation (6) to the tensor model. Surprisingly, we will see later that this relation agrees with the location of the continuous phase transition (or crossover) point in the leading order of $N$.

Lastly, let us comment on the relation between our matrix model and the one that appears in the replica trick applied to the the spherical $p$-spin model ($p=3$) for spin glasses [17, 18]. The partition function of the spherical $p$-spin model for $p=3$ is given by

$\displaystyle Z_{p\hbox{-}{\rm spin}}=\int_{\phi_{a}\phi_{a}=1}d\phi\ e^{-P_{abc}\phi_{a}\phi_{b}\phi_{c}}$

(8)

with a random coupling $P_{abc}$ to simulate spin glasses. Simulating the random coupling by Gaussian distribution, $e^{-\alpha P^{2}}$, and applying the replica trick, one obtains

$\displaystyle\int dP\,e^{-\alpha P^{2}}(Z_{p\hbox{-}{\rm spin}})^{R}=\int_{\phi_{a}^{i}\phi_{a}^{i}=1}d\phi\ e^{\frac{1}{4\alpha}\sum_{i,j=1}^{R}(\phi_{a}^{i}\phi_{a}^{j})^{3}}.$

(9)

This is very similar to our matrix model (7), but there are a few major differences. One is that the coupling parameter has the opposite sign compared to our case. This means that the dominant configurations are largely different between the spin glass and our cases. Another difference is that, in the replica trick, the replica number $R$ is finally taken to the limit $R\rightarrow 0$, while we are rather interested in the case of the tensor model $R=\lambda_{H}=(N+2)(N+3)/2$. In particular, this means that, in the thermodynamic limit $N\rightarrow\infty$, $R$ should also be taken infinite for the correspondence to the tensor model.

A perturbative computation of the matrix model (7) was carried out in [16]. Regarding the exponent of (7) as the action of the matrix model, Feynman rules can be written down for the perturbative computation in $\lambda$. The most dominant diagrams in large $R$ for each order in $\lambda$ are the necklace diagrams depicted in Figure 2, where the examples of the order $n=3$ are given. Summing over all $n$, it has been obtained that

$\displaystyle Z_{N,R}(\lambda,k)={\cal N^{\prime}}\int dr\,r^{NR-1}f_{N,R}(\lambda r^{6})\,e^{-kr^{2}},$

(10)

where

$\displaystyle f_{N,R}(t)=\left(1+\frac{12t}{N^{3}R^{2}}\right)^{-\frac{N(N-1)(N+4)}{12}}\left(1+\frac{6(N+4)t}{N^{3}R^{2}}\right)^{-\frac{N}{2}}.$

(11)

A generalization of the perturbative computation above was performed in [19] by employing a slightly different strategy. In the generalization, the coupling term in the exponent was generalized to

$\displaystyle\sum_{i,j=1}^{R}\Lambda_{ij}(\phi_{a}^{i}\phi_{a}^{j})^{3}$

(12)

with a symmetric matrix $\Lambda$. In this case, the above $f_{N,R}(t)$ is generalized to

$\displaystyle f_{N,R}(t)=\prod_{e_{\Lambda}}h_{N,R}(e_{\Lambda}t),$

(13)

where the product is over the eigenvalues of the matrix $\Lambda$ with the degeneracies taken into account, and

$\displaystyle h_{N,R}(t)=(1+12\gamma_{3}t)^{-\frac{N(N+4)(N-1)}{12}}\,(1+6(N+4)\gamma_{3}t)^{-\frac{N}{2}}$

(14)

with

$\displaystyle\gamma_{3}=\frac{\Gamma\left(\frac{NR}{2}\right)}{8\,\Gamma\left(\frac{NR}{2}+3\right)}.$

(15)

This indeed agrees with (11) for the special case $\Lambda_{ij}=\lambda$ in the leading order of $R$, since the eigenvalues of $\Lambda$ are one $R\lambda$ and $R-1$ zeros. This generalization makes it possible to compute the expectation values of the products of $(\phi_{a}^{i}\phi_{a}^{j})^{3}$ by taking the derivatives of the partition function with respect to $\Lambda_{ij}$.

The Monte Carlo simulations were performed by the Metropolis update method in [19]. We observed a phase transition or a crossover around $R\sim N^{2}/2$. This location curiously agrees with $R=\lambda_{H}=(N+2)(N+3)/2$ in the leading order of $N$, which is required by the consistency of the tensor model, as explained in Section 2.

Since the overall coefficient of the exponent of (7) can be absorbed by the rescaling of $\phi_{a}^{i}$, we set $\lambda=1$ in the simulations, leaving $N,R,k$ as variables.

We have studied a matrix model which represents the simplest observable for an exactly solved physical state of the canonical tensor model. This matrix model has the unusual property that it contains non-pairwise index contractions, and cannot be exactly solved by the well-known methods applied to the usual matrix models. This matrix model has the similar form as the one that appears in the replica trick of the spherical $p$-spin model ($p=3$) for spin glasses. However, our case has different ranges of parameters and variables from the spin glass case, and new analysis is needed. We have performed some analytical and numerical computations for the matrix model. The most striking result is the presence of the continuous phase transition point or a crossover at the location which is required by the consistency of the canonical tensor model. This means that the canonical tensor model exists exactly on or in the vicinity of the continuous phase transition point. This means that the canonical tensor model automatically exists at the location where its continuum limit could be taken, as continuum theories are often obtained by taking continuum limits around continuous phase transition points. The non-trivial behavior of the dimensions of the configurations, represented by point clouds, would also be encouraging for the possibility of spacetime emergence in the canonical tensor model.

Further improvement of the results after the conference has been presented in our recent paper [21].

The Monte Carlo simulations in this study were performed using KEKCC, the cluster system of KEK. N.S. would like to thank L. Lionni and S. Takeuchi for the collaborations, on which this presentation is based. The work of N.S. is supported in part by JSPS KAKENHI Grant No.19K03825.