^{1,2}

^{1}

^{1}

^{3}

^{3}.

We study the

In recent literature there has been considerable interest in the quantum mechanical models where the degrees of freedom are fermionic tensors of rank 3 or higher

At the same time, there are significant differences between the tensor and SYK-like models. An early hint was the different scaling of the corrections to t

The absence of disorder and the presence of the continuous symmetry groups in the tensor models endows them with a number of theoretical advantages, but also makes them quite difficult to study. In the tensor models any invariant operator should be meaningful and be assigned a definite scaling dimension in the large

Beyond the operator spectrum, it is interesting to investigate the spectrum of eigenstates of the Hamiltonian. While this spectrum is discrete and bounded for finite

The corresponding studies of spectra in the GW model

In

The goal of this paper is to improve our understanding of energy spectra in the tensor models. We will mostly focus on the simplest tensor model with

If the global symmetry of the quantum mechanical model is gauged, this simply truncates the spectrum to the

Number of singlet states in the

The

The

One can easily compute

We can make some general restrictions on the possible values of the energies. Operators

The discrete symmetries of the theory depend on whether some of the ranks are equal. In a

Let us construct the operator which implements the interchange

Let us now discuss the case when all three ranks are equal and we have

The

The antiunitary time reversal symmetry

Since the Hilbert space of the model is finite dimensional, it is interesting to put an upper bound on the absolute value of the energy eigenvalues in each representation of the symmetry group. In this section we address this question in two different ways. We first derive a basic linear relation between the Hamiltonian, a quadratic Casimir operator, and a square of a Hermitian operator which is positive definite. This gives bounds which are useful for the representations where the quadratic Casimir of one of the orthogonal groups is near its maximum allowed value. We also find that the bounds are exactly saturated for

To derive an energy bound we introduce the Hermitian tensor

The Hamiltonian may be written as

An interesting special case, which we will consider in Sec.

More generally, if at least one of the ranks is even (we will call it

For any even

Specifying the bound

For the singlet states

In the large

In this section we present another approach to deriving energy bounds for the

Consider an arbitrary singlet density matrix

Because the density matrix

For the case when

We can try to estimate how close the singlet ground state

In the large

It was argued in

For clarity, we have omitted the indices in the

In the strong coupling limit

The effective action

Suppose we have a free fermionic system of

One can compute the partition function because the integral over

Let us apply this approach to the case when Majorana fermions live in the fundamental representation of several orthogonal groups. It is important to distinguish between

Now we discuss the case where the gauge group is the product of three orthogonal groups

For the

Direct diagonalization of the Hamiltonian for

Using similar methods, the number of singlets can be calculated in the

Number of singlet states in the

We may similarly calculate the number of singlets for the

Number of singlet states in the

In this section we will estimate the number of singlets in the

Since we are studying fermions on a compact space

If the gauge group is a product

We do not find any more anomalies: using the long exact sequence in homotopy groups one can show that the fundamental group of

One has to divide by

If

If only one of

Finally, when both

When

Setting

Spectra of the

For

In

If we set

This Hamiltonian is related to that in Sec. IV of

The

The absence of singlets for other values of

Using

Spectra of the

Setting

To construct the Hilbert space, we define the operators

Setting

There is a “Clifford vacuum” state, which satisfies

The states with vanishing

For low values of

Spectrum of the

Spectra of the

A remarkable feature of the spectra is that all the eigenvalues of

Now we take the difference between Eq.

Using

To calculate the energies of all states, we need to first decompose the Hilbert space into

Actually, it is not necessary to compute the above integral for various representations. It is very well-known that characters of

After finding the representations under

Let us exhibit this method to find the spectrum of the

Here we are using the notation multiplicity

As a further check, in Appendix

We are grateful to Ksenia Bulycheva for collaboration at the early stages of this project. We also thank Dio Anninos, Andrei Bernevig, Sylvain Carrozza, Chethan Krishnan, Juan Maldacena, Daniel Roberts, Douglas Stanford and Edward Witten for useful discussions. The work of IRK was supported in part by the U.S. NSF under Grant No. PHY-1620059. The work of G. T. was supported in part by the Multidisciplinary University Research Initiative (MURI) Grant No. W911NF-14-1-0003 from U.S. Army Research Office (ARO) and by DOE Grant No. de-sc0007870.

In this Appendix we describe the value of quadratic Casimir operator for the representations of

For the representation of the group

We will also need an explicit expression for the quadratic Casimir of

Let us list the allowed representations for some low values of

For the

For the

For the

For the

For the

For the

For the

Due to the relation

As was described in the main text, first we have to find

Let us list the explicit form of quadratic Casimirs. For

The spectrum can be found in Table

Energy spectrum of the

The construction of singlet states for the

For example, for the

Generalizing to any

For the

Defining the antisymmetric matrix

By analogy with

More generally, for