Spectrum of Majorana Quantum Mechanics with $O(4)^3$ Symmetry

We study the quantum mechanics of 3-index Majorana fermions $\psi^{abc}$ governed by a quartic Hamiltonian with $O(N)^3$ symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large $N$ limit dominated by the melonic diagrams. For $N=4$ the total number of states is $2^{32}$, but they naturally break up into distinct sectors according to the charges under the $U(1)\times U(1)$ Cartan subgroup of one of the $O(4)$ groups. The biggest sector has vanishing charges and contains over $165$ million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is non-degenerate. If the $SO(4)^3$ symmetry is gauged, it is known from earlier work that the model has $36$ states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.


Introduction
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 [1,2]. Similarly to the Sachdev-Ye-Kitaev model [3][4][5], these models have solvable large N limits dominated by the so-called melonic diagrams [6][7][8]. In this limit they become solvable with the use of Schwinger-Dyson equations as were derived earlier for the SYK-like models [2,4,5,[9][10][11][12].
While this spectrum of eigenstates is discrete and bounded for finite N , the low-lying states become dense for large N leading to the (nearly) conformal behavior where it makes sense to calculate the operator scaling dimensions. In the SYK model, the number of states is 2 N SYK /2 , and numerical calculations of spectra have been carried out for rather large values of N SYK [13][14][15]. They reveal a smooth distribution of energy eigenvalues, which typically has no degeneracies and is almost symmetric under E → −E.
The corresponding studies of spectra in the tensor models of [1] and [2] have been carried out in [16][17][18][19][20][21][22][23], but in these cases the numerical limitations have been more severe -the number of states grows as 2 N 3 /2 in the O(N ) 3 symmetric model of [2] and as 2 2N 3 in the O(N ) 6 symmetric Gurau-Witten (GW) model [1]. The results have shown an interesting structure. For example, for the N = 2 GW model the exact values of the 140 SO(2) 6 invariant energies were found [22]. Due to the discrete symmetries, there are only 5 distinct E < 0 eigenvalues and each one squares to an integer (the singlet spectrum also contains 50 zero-energy states). For N = 2 there are only two gauge singlet states with E = ±8. For N = 3, as for any odd N , there are none [23]. While the complete spectra of (1.2) can be calculated for N = 2 and 3 using a laptop, this is no longer true for N = 4, where the total number of states is 2 32 .
However, they split into smaller sectors according to the charges (Q 0 , Q 1 ) of the U (1) × U (1) Cartan subgroup of one of the SO(4) groups. The most complicated and interesting is the (0, 0) sector; it is the part of the 32 qubit spectrum at the "half-half-filling," i.e. where the first 16 qubits contain 8 zeros and 8 ones, and the same applies to the remaining 16 qubits.
In particular, all the SO(4) 3 invariant states are in this subsector; their number, 36, was found using the gauged version of the free fermion theory [23]. Since there are over 165 million states at half-half-filling, the spectrum cannot be determined completely. However, using a Lanczos algorithm, we will be able to determine a number of low-lying eigenstates.
We will also be able to find the complete spectrum of the 36 gauge singlet states, including their transformation properties under the residual discrete symmetries of the model where the SO(4) 3 symmetry is gauged. Thus, our work reveals the spectrum of a finite-N system without disorder, which is nearly conformal and solvable in the large-N limit, and identifies the discrete symmetries crucial for efficient numerical studies of such finite systems.

Discrete symmetries acting on the gauge singlets
For any even N , if we gauge the SO(N ) 3 symmetry, there remain some gauge singlet states [23], which are annihilated by the symmetry charges These states may still have degeneracies due to the residual discrete symmetries. Indeed, each O(N ) group contains a Z 2 parity symmetry which is an axis reflection. For example, inside O(N ) 1 there is parity symmetry P 1 which send ψ 0bc → −ψ 0bc for all b, c and leaves all other components invariant. The corresponding generator is One can indeed check that where n = N 2 (N − 1)/2 is the number of fields in the product. They satisfy These permutations flip the sign of H [23,25]: This explains why the spectrum is symmetric under E → −E.
We now define the cyclic permutation operator P = P 12 P 23 such that P ψ abc P † = ψ cab , P HP † = H , Thus, P is the generator of the Z 3 symmetry of the Hamiltonian. Applying the Z 3 symmetry to the parity reflections P i we see that P P 1 P † = P 2 , P P 2 P † = P 3 , P P 3 P † = P 1 .
Forming all the possible products of I, P, P 1 , P 2 , P 3 , we find that the full discrete symmetry group contains 24 elements. Using the explicit representation (2.2) for P 1 , and the analogous ones for P 2 and P 3 , we note that the three parity operators commute with each other. Furthermore, Therefore, Π commutes with all the group elements, so that the group has a Z 2 factor with elements I and Π. The symmetry group turns out to be A 4 × Z 2 , and the 12 elements of the alternating group A 4 are I , P 1 , P 2 , P 1 P 2 , P , P 2 , P 1 P , P 2 P , P 1 P 2 P , P 1 P 2 , P 2 P 2 , P 1 P 2 P 2 . (2.10) Each of them can be associated with a sign preserving permutation of 4 ordered elements, and the action is Let us discuss the representations of A 4 in more detail. Using a reference eigenstate |ψ 0 not invariant under the Z 3 subgroup I, P, P 2 , we can form a triplet of states If the parities (P 1 , P 2 , P 3 ) of the state |ψ 0 are the same, then we can form a linear combination which transforms trivially under the Z 3 , while the remaining 2 linear combination form the degree 2 representation of Z 3 , where Because of this, some eigenstates have degeneracy 2. If the parities (P 1 , P 2 , P 3 ) of the state |ψ 0 are not equal, then the triplet representation (2.12) of the full discrete group is irreducible. For example for (P 1 , P 2 , P 3 ) = (+, +, −), i.e.
we find that the parities of the states P |ψ 0 and P 2 |ψ 0 are given by the cyclic permutations of (+, +, −). Indeed, using (2.8), we find that the parities of the state P |ψ 0 are Thus, each of the states in the triplet (2.12) has a distinct set of parities. Then it is impossible to form linear combinations which are eigenstates of the parities, and we have an irreducible representation of A 4 of degree 3. In this situation we find that an energy eigenvalue has degeneracy 3.
We also note the relations and their cyclic permutations. Since an operator P ij maps an eigenstate of energy E into an eigenstate of energy −E, we see that such mirror states have the same parities when N/2 is even, but opposite parities when N/2 is odd.
For the states at zero energy, the discrete symmetry group is enhanced to 48 elements because the permutation generators P ij map them into themselves. Using the relations (2.17) we find which implies that Π = P 1 P 2 P 3 commutes or anti-commutes with other elements depending on the value of N . Focusing on the case where N (N 2 − 1)/2 is even and the sign above is positive (this includes N = 4 which is our main interest in this paper), we find that Π commutes with all other generators, so that the group has a Z 2 factor with elements I and Π. The symmetry group for E = 0 turns out to be S 4 × Z 2 , and the 24 elements of the group S 4 are formed out of the products of I, P 1 , P 2 , P 12 , P 23 , P 13 . The parity generators are realized in the same way as in (2.11), while the permutations act by the natural embedding The degrees of the irreducible representations of S 4 are 1, 1, 2, 3, 3.

A basis for operators and states
The Majorana fermions ψ abc may be thought of as generators of the Clifford algebra in N 3dimensional Euclidean space. Restricting to the cases where N is even, the dimension of the Hilbert space is 2 N 3 /2 , and the states may be represented by series of N 3 /2 "qubits" |s , where s = 0 or 1. For example, for N = 4 we can use the explicit representation in terms of direct products of 32 2 × 2 matrices: where X, Y, Z stand for the Pauli matrices σ x , σ y , σ z . Their action on a qubit is It is convenient to introduce operators [20,23]  states is still very large, they turn out to be tractable numerically because the matrix we need to diagonalize is rather sparse. This has allowed us to study the low-lying eigenvalues of H, which occur in various representations of SO(4) 3 . To find the gauge singlet energies, we study the operator proportional to H + 100 3 i=1 C i 2 , where the quadratic Casimir of the SO(N ) 1 symmetry is and analogously for SO(N ) 2 and SO(N ) 3 . The Lanczos algorithm allows us to identify the lowest eigenvalues of this operator, which all correspond to SO(4) 3 invariant states; the non-singlets receive large additive contributions due to the second term.
The expressions for the parity operators are, omitting the direct product signs, The operator P 3 implements, up to a sign, the particle-hole conjugation on the first 16 qubits.
These parity operators may be used only on the SO(4) 3 invariant states. For example, a rotated form of P 3 = 2 8 ab ψ ab0 is It has the same eigenvalues as P 3 on the singlets becausẽ which is equal to 1 when acting on the singlet states, where the first 16 qubits are half-filled.
where a i are unequal small coefficients. 4 The biggest degeneracy is found for the eight E = 0 states; it corresponds to the 2 3 independent choices of the three parities. Since the discrete group acting on the E = 0 states is S 4 × Z 2 , it appears that we find two different irreducible representations of S 4 : the trivial one of degree 1 and the standard one of degree 3. The energies of the gauge singlet states and their degeneracies are plotted in figure 1.
Some of the energies agree within the available precision with square roots of integers: 277. The ratio of E 0 and E bound can be calculated in the large N limit using the exact propagator, and it is found to be ≈ 0.41 [23]. Since we find E 0 /E bound ≈ 0.96, this suggests that large N approximations cannot be applied for N = 4.
The list of all the low-lying energy levels in the (0, 0) sector, singlets and non-singlets, and the corresponding values of quadratic Casimirs C i 2 , is shown in table 2. In order to identify the values of C i 2 , we have calculated the low-lying spectrum of H + 3 i=1 a i C i 2 where a i are unequal small coefficients. When the C i 2 are not all equal, there are also states of the same energy with their values obtained by a cyclic permutation. For example, at E = −136.559039 we find states with (C 1 2 , C 2 2 , C 3 2 ) = (0, 4, 8), (4,8,0), (8,0,4). We may infer the (j 1 , j 2 ) representation of SO(4) ∼ SU (2) × SU (2) from the formula C 2 (j 1 , j 2 ) = 2 j 1 (j 1 + 1) + j 2 (j 2 + 1) . (4.7) For example, C 2 = 4 corresponds to the (1, 0)+(0, 1) irrep of dimension 6; C 2 = 8 corresponds to the (1, 1) irrep of dimension 9; C 2 = 12 corresponds to the (2, 0) + (0, 2) irrep of dimension 10; etc. of SO(4). Let us proceed to the sectors adjacent to one-particle and one-hole sectors, (±1, 0) and (0, ±1), which contain some of the additional representations, including the (4, 4, 4) of SO(4) 3 . The refined bound [23] for this representation gives |E (4,4,4) | < 72 √ 5 ≈ 160.997, Table 3: The low-lying states in the sectors (±1, 0) and (0, ±1), i.e. with one extra hole (h) or particle (p) added to half-half-filling The energies are the same within the accuracy shown, which is a good test of our diagonalization procedure. When the C i 2 are not all equal, there are additional states of the same energy with their values obtained by a cyclic permutation. We have also calculated the energies in other charge sectors. We find that the absolute ground state lies in the (0, 0) sector: as the magnitudes of charges increase, the energies tend to get closer to 0. For example, in the (−7, −7) sector, which contains 256 states c ab0ca b 1 |vac , only the second line in the Hamiltonian (4.1) acts non-trivially, and we find E = ±16 with multiplicity 15, and E = 0 with multiplicity 226. Due to the conjugation symmetry, noted below (3.5), the same spectrum is found in the (−7, 7) sector. In each of the (±6, ±6) sectors, some of the energies are square roots of integers, including the ground state E = −24 √ 5. Finally, let us note that in each sector of the form (q, ±8) or (±8, q) the Hamiltonian is isomorphic to that of the O(4) 2 × O(2) model, which was solved in [23], and therefore has the same integer spectrum. The lowest and highest energies, occurring for q = 0, are ±64.
part by the US NSF under Grant No. PHY-1620059. The work of GT was supported in part by the MURI grant W911NF-14-1-0003 from ARO and by DOE grant de-sc0007870.