YITP-22-123
Exact analytic expressions of real tensor eigenvalue
distributions of Gaussian tensor model for small $N$

Eigenvalue distributions are important dynamical quantities in studies of matrix models. They model energy eigenvalue distributions of complex dynamical systems [1]. They provide a major technique in solving matrix models [2]. Their topological properties differentiate phases of matrix models²²2For more details, see for instance [3]., providing insights into the dynamics of gauge theories [4, 5].

Recently, tensor models [6, 7, 8, 9] attract much attention in various contexts [10]. While it is important to develop efficient techniques of computing eigenvalues/vectors for certain tensors in various practical applications of tensors [11], it is also interesting to study their distributions for ensembles of tensors, because tensors are dynamical in tensor models.

While there are already some interesting results [12, 13, 14, 15, 16, 17], eigenvalue distributions in tensor models still remain largely unexplored. In our previous studies [16, 17], the problem was rewritten as computations of partition functions of zero-dimensional fermion systems. In [16], an exact formula of signed distributions of real eigenvalue/vector distributions for real symmetric order-three tensors with Gaussian distributions was obtained, where each eigenvalue/vector contributed to the distribution by $\pm 1$, depending on the sign of a Hessian matrix associated to each eigenvalue/vector. In [17], real eigenvalue/vector distributions for the same ensemble of tensors was studied. In this study, however, only an approximate large-$N$ expression³³3$N$ denotes the dimension of the index vector space of the tensor. was obtained by truncating a Schwinger-Dyson equation, though some closely related exact results were also obtained. While the functional form of the large-$N$ expression, which was a Gaussian, agreed well with Monte Carlo simulations, the overall factor did not and an improvement was expected.

In this paper, we rewrite the real eigenvalue/vector problem as a computation of a partition function of a boson-fermion system with four-interactions, instead of the fermion systems in the previous studies [16, 17]. We exactly compute the real eigenvalue/vector distributions for real symmetric order-three tensors with Gaussian distributions for $N\leq 8$, and extrapolate the expressions to guess a large-$N$ expression, including the overall factor, which was not correctly obtained by the approximation of the previous paper [17]. We compare the exact expressions for small $N$ with Monte Carlo simulations, and obtain precise agreement. We also compare the large-$N$ expression, and obtain good agreement.

Computing the real eigenvalue/vector distributions is essentially the same as the computation of complexity in the $p$-spin spherical model of spin glasses [18, 19] (see Appendix D). By using a rotational symmetry, the problem can be mapped to a random matrix model, which can subsequently be solved by the standard matrix model methods [20, 12, 13]. In fact the results of Section 6 can be obtained from those in [20], as explained in Appendix D. However we expect that the field theoretical method employed in this paper may provide new insights and new applications in future analysis of the tensor eigenvalue/vector problems, which are still largely unexplored.

This paper is organized as follows. In Section 2, we define what we compute, the real eigenvalue/vector distributions for real symmetric order-three tensors with Gaussian distributions. In Section 3, we rewrite the problem as computation of a partition function of a zero-dimensional boson-fermion system with four-interactions. In Section 4, we determine the general form which is obtained by integrating out the fermions in the partition function. In Section 5, we perform the remaining bosonic integrals but one. In Section 6, we explicitly perform the last bosonic integral for each case of $N\leq 8$, and obtain the explicit exact expressions of the distributions. Interestingly, the apparently complicated bosonic integrals are feasible and the final results have simple expressions. In Section 7, we perform some Monte Carlo simulations, and precise agreement with the exact expressions is obtained. In Section 8, we perform an extrapolation of the exact results for small $N$ to guess a large-$N$ expression. We compare it with Monte Carlo simulations, and good agreement is obtained. The last section is devoted to a summary and future prospects.

In this paper, we restrict ourselves to the real symmetric order-three tensors, i.e., $C_{abc}\in\mathbb{R}\ (a,b,c=1,2,\ldots,N)$, satisfying $C_{abc}=C_{bac}=C_{bca}$, as the simplest case. As for eigenvalues/vectors of tensors, there are a few similar but slightly different definitions in the literature [21, 22, 23]. In this paper we employ the definition that the real eigenvectors $v$ of a given $C$ are the non-zero solutions to

where the repeated indices are assumed to be summed over, as will be assumed in the rest of this paper. Then the distribution of the eigenvectors for a given $C$ is given by

for $v\neq 0$ under the volume measure $d^{N}v=\prod_{a=1}^{N}dv_{a}$, where $v^{i}\ (i=1,2,\ldots,n_{C})$ are all the solutions to (1), $|\cdot|$ denotes the absolute value, and

Here the absolute value of the determinant, $|{\rm det}M|$, is the Jacobian associated to the change of the arguments of the $\delta$-functions performed in (2).

When the tensor $C$ has a Gaussian distribution, the real eigenvector distribution is given by

where $dC=\prod_{a\leq b\leq c=1}^{N}dC_{abc}$, $A=\int_{\mathbb{R}^{\#C}}dC\,e^{-\alpha C^{2}}$, $C^{2}=C_{abc}C_{abc}$, $\alpha>0$, and $\#C=N(N+1)(N+2)/6$, which is the total number of the independent components of $C$.

It is worth commenting on the eigenvalue distribution corresponding to the real eigenvector distribution above. An eigenvalue $\zeta$ accompanied with a real eigenvector $v$ (a Z-eigenvalue in the terminology of [21]) is defined by

where $|w|=\sqrt{w_{a}w_{a}}$. Comparing with (1), we obtain the relation,

Therefore, by using (6) and the fact that $\rho(v)$ is actually a function of $|v|$ because of the rotational symmetry of the distribution, we obtain the real eigenvalue distribution,

where $1/\zeta$ in the argument abusively represents an arbitrary vector of size $1/\zeta$, and $S_{N-1}=2\pi^{N/2}/\Gamma[N/2]$, the surface volume of a unit sphere in an $N$-dimensional space.

Here we would like to stress that we are specifically considering real eigenvalues/vectors only. Unlike real symmetric matrices, real symmetric tensors can also have complex eigenvalues/vectors by allowing complex solutions to the equation in (1). To include such complex solutions in our analysis, the expressions in this section need to appropriately be modified. Note also that the connection to the $p$-spin spherical model explained in Appendix D is lost in this case. Therefore the end results for the complex case may have non-trivial differences from the real case.

In this section, we will rewrite (4) with (2) as a partition function of a zero-dimensional boson-fermion system with four-interactions. An immediate obstacle in doing this is the presence of an absolute value in (2), which is not an analytic function. To rewrite it in an analytic form, we take

where $I_{ab}=\delta_{ab}$ is an identity matrix, and the parameter $\epsilon$ is a positive small regularization parameter, which assures the convergence of the integrals below.

Determinant factors like (8) can be managed inside partition functions of zero-dimensional field theoretical systems. This technique is common in supersymmetric approaches to disorder averaging in statistical physics (see for instance [24] and references therein). The numerator $\det(M^{2}+\epsilon I)$ can be rewritten as $\det(M^{2}+\epsilon I)=\int d{\bar{\psi}}d\psi\,e^{{\bar{\psi}}_{a}(M^{2})_{ab}\psi_{b}+\epsilon{\bar{\psi}}_{a}\psi_{a}}$ by introducing a fermion pair, ${\bar{\psi}}$ and $\psi$ [25]. However, the exponent of this expression contains $C$ in a quadratic manner (see (3)) and is difficult to handle when we perform an integration over $C$ as will be done below. Therefore, as was done previously in [17], we introduce another fermion pair, $\bar{\varphi},{\varphi}$, to rewrite the exponent in a form linear in $C$:

where the contracted indices are suppressed for brevity: $\bar{\varphi}{\varphi}=\bar{\varphi}_{a}{\varphi}_{a}$, ${\bar{\psi}}M{\varphi}={\bar{\psi}}_{a}M_{ab}{\varphi}_{b}$, and so on. The equality can be shown by noting that $\bar{\varphi}{\varphi}+{\bar{\psi}}M{\varphi}+\bar{\varphi}M\psi=(\bar{\varphi}+{\bar{\psi}}M)({\varphi}+M\psi)-{\bar{\psi}}M^{2}\psi$. In a similar manner we obtain

where $\sigma$ is a new bosonic variable, and $\sigma^{2}=\sigma_{a}\sigma_{a}$, and so on.

By using (8), (9), (10) and a well-known formula, $\int_{\mathbb{R}}dx\,e^{ipx}=2\pi\delta(p)$, (4) with (2) can be rewritten as

where

with a new bosonic variable $\lambda_{a}\ (a=1,2,\ldots,N)$ to rewrite the $\delta$-functions.

Next, let us perform the integrations over $C$ and $\lambda$ in the expression (11). Let us first consider $C$. The terms containing $C$ in (12) are

Then we obtain

where

where the summation is over all the permutations of $a,b,c$, which is necessary because $C$ is a symmetric tensor. By explicitly expanding (15), we obtain

where

Here $\hat{v}=v/|v|$, and ${\bar{\psi}}_{\parallel}=\bar{\psi}_{a}\hat{v}_{a}$, etc., and $I_{\parallel}$ and $I_{\perp}$ are respectively the projection matrices to the parallel and transverse subspaces to $\hat{v}$.

Next let us perform the integration over $\lambda$. Picking up the terms containing $\lambda$ (with no $C$) in (12) and (16), we obtain

Considering that $B$ is a sum of the projection matrices as in (17), we obtain

where $\det B=3$ from (17), and

Here we have used $B^{-1}=I_{\perp}+\frac{1}{3}I_{\parallel}$, and $D_{\perp}\cdot\tilde{D}_{\perp}=D_{\perp\,a}\tilde{D}_{\perp\,a}$, and so on, where $X_{\perp}\ (X=D,\tilde{D})$ denotes vector $X$ projected to the transverse subspace to $\hat{v}$. From (17) the projections of $D$ and $\tilde{D}$ are more explicitly given by

The first term in the second line of (20) is a constant contribution to the potential. The second and the third terms in the same line are corrections to the kinetic terms of the parallel components of the fermions and the bosons, respectively. The other terms describe the four-interaction terms among the bosons and the fermions. Collecting the results above, we obtain the following intermediate expression,

where

Here $\tilde{K}_{F},\tilde{V}_{F}$ contain only the fermions, $\tilde{K}_{B},\tilde{V}_{B}$ only the bosons, and $\tilde{V}_{BF}$ is a mixture. Note that the quadratic terms of the parallel components in $\tilde{K}_{F}$ and $\tilde{K}_{B}$ have been corrected by the aforementioned terms from the second line of (20).

Now we want to compute $\tilde{V}_{B},\tilde{V}_{F},\tilde{V}_{BF}$ more explicitly. The computation of $\tilde{V}_{F}$ in (23) is essentially the same as the derivation of the four-fermi interaction terms in [17] for $R=1$. By noting that two of the terms in [17] do not appear, because $\bar{\psi}_{\perp}\cdot\bar{\psi}_{\perp}=\bar{\varphi}_{\perp}\cdot\bar{\varphi}_{\perp}=0$ (for $R=1$), we obtain

What seems surprising in this expression is that the parallel components of the fermions cancel out from the interactions after all.

As for $\tilde{V}_{B}$ and $\tilde{V}_{BF}$ in (23), by using the results of the computations of $E_{2}$ and $E_{3}$ in Appendix A, we obtain

We again find the surprising fact that no parallel components appear in $\tilde{V}_{B}$ and $\tilde{V}_{BF}$.

Because the parallel components only exist in $\tilde{K}_{B},\tilde{K}_{F}$ and do not interact, these can trivially be integrated out, that generates the overall factors of $\pi$ and $-1$ for the bosons and fermions, respectively. Therefore we finally obtain

where

and $\tilde{V}_{F},\tilde{V}_{B},\tilde{V}_{BF}$ are given in (24) and (25).

In the expression (26) the variables projected to the transverse directions, i.e., ${\bar{\psi}}_{\perp}$, etc., just represent $N-1$-dimensional degrees of freedom with no other restrictions. Therefore we can simply regard them as $N-1$-dimensional variables from the beginning. The external variable, the vector $v$, only appears in the coupling constants of the four-interactions through $|v|$. Therefore we can simply write

where all the variables are $N-1$-dimensional, and

Note that, for simplicity, we are abusively using the same notations of the variables as those in Section 3 with a different dimension, and, to indicate this difference, the symbol $N^{*}$ is attached to the integral symbol in (28).

To compute $\rho(v)$ explicitly, we first perform the fermionic integrations. The integrand in (28) can be expanded in the fermions. A useful property in this expansion is that the expansion in $V_{BF}$ stops at the fourth order:

This can be proven as follows. The fermions in $V_{BF}$ are projected to $\phi$ or $\sigma$. Therefore $V_{BF}$ contains only eight independent fermions in total, i.e., ${\bar{\psi}}\cdot\phi,{\bar{\psi}}\cdot\sigma,\psi\cdot\phi,\psi\cdot\sigma,\bar{\varphi}\cdot\phi,\bar{\varphi}\cdot\sigma,{\varphi}\cdot\phi,{\varphi}\cdot\sigma$. Since each term of $V_{BF}$ contains two of these fermions and products of more than eight of these fermions vanish, we obtain

We also notice that, when $\sigma=\pm\phi$, only four of these fermions are independent. Therefore,

Now we want to determine the functional forms of the fermionic integrals of each summand in (30):

From the form of $V_{BF}$ and the $O(N-1)$ symmetry, $\beta_{n}$ should be a polynomial function of $\sigma^{2},\phi^{2},\sigma\cdot\phi$, and its order in $\sigma,\phi$ should be $2n$. In addition, each term of the polynomial function should contain equal numbers of $\sigma$ and $\phi$, and the polynomial function should also be invariant under the interchange $\sigma\leftrightarrow\phi$ as a whole.

From the above considerations, we uniquely obtain for $n=1$

Here $a_{1}$ is a function of $\epsilon$ and $|v|^{2}/\alpha$, and all $a_{i}$ below are also so.

For $n=2$, we have two possibilities,

As for $n=3$, $\beta_{3}$ must vanish for $\sigma=\pm\phi$ due to (32). Considering also the other conditions mentioned above, we uniquely obtain

As for $n=4$, we need not only (32) but also the following property: Taking the derivatives of $(V_{BF})^{4}$ with respect to $\sigma,\phi$ three times or less must vanish, when $\sigma=\pm\phi$. This can be proven as follows. Obviously, $(V_{BF})^{4}$ is proportional to the product of the eight projected fermions, ${\bar{\psi}}\cdot\phi,{\bar{\psi}}\cdot\sigma,\ldots$. After taking the derivatives of this three times or less with respect to $\sigma,\phi$, each term contains at least five of the eight projected fermions. Therefore all these terms vanish for $\sigma=\pm\phi$ because of the same reason for (32). Now using this property, we can uniquely determine

Collecting all the results above, we conclude that the fermionic integrations of the partition function has the following general form,

Here $a_{i}$ are generally functions of $\epsilon$ and $|v|^{2}/\alpha$, but $\epsilon$ in them turn out to simply disappear in the $\epsilon\rightarrow+0$ limit in (28) without any essential roles. Therefore in the following sections we ignore it by just putting $\epsilon=0$ in $a_{i}$.

Assuming the form (38) of the fermionic integrations, we perform the bosonic integrations but one in this section.

Let us first generalize the bosonic kinetic term by introducing new parameters $\epsilon_{i}\ (i=1,2,3)$:

where $\epsilon_{1},\epsilon_{3}>0$ are assumed for the convergence of the bosonic integration. The original kinetic term in (29) corresponds to $\epsilon_{1}=\epsilon_{2}=1,\ \epsilon_{3}=\epsilon$.

Using this new kinetic term, (28), and (38), $\rho(v)$ can be expressed as

where

with $D_{1},D_{2}$ being the following partial derivative operators,

The bosonic integration in (41) does not seem to be fully integrable, but it has a simpler expression. Since $\sigma$ appears at most quadratically in $K_{B}^{\epsilon}+V_{B}$, the $\sigma$ integration can be performed:

where for brevity we have introduced

(43) does not depend on the angular directions of $\phi$, and therefore the integration over these produces the spherical volume, $2\pi^{(N-1)/2}|\phi|^{N-2}/\Gamma[(N-1)/2]$. Then, after applying the derivative operators in (41) and performing a replacement of variable, $|\phi|=\sqrt{x}/\sqrt{1-8zx}$, we obtain

where we have used

to delete $a_{5}$. As we will see in Section 6, (46) holds for all the cases we consider (namely, $N\leq 8$). In fact this relation is essentially important, because otherwise the integrand has an extra factor $1/(1-8zx)$, and the feasibilty of the integration over $x$ which will be performed in Section 6 would become unclear.

In this section, we explicitly compute the integration (45) for each case of $N\leq 8$ to obtain exact analytic expressions. What is surprising is that the seemingly difficult integrations can be done explicitly. The apparent reason is that the integrand of (45) turns out to be a sum of a total derivative and a simple integrable function.