SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry

We introduce a family of tensor quantum-mechanical models based on irreducible rank-$3$ representations of $\mathrm{Sp}(N)$. In contrast to irreducible tensor models with $\mathrm{O}(N)$ symmetry, the fermionic tetrahedral interaction does not vanish and can therefore support a melonic large $N$ limit. The strongly-coupled regime has a very analogous structure as in the complex SYK model or in $\mathrm{U}(N)\times\mathrm{O}(N)\times\mathrm{U}(N)$ tensor quantum mechanics, the main difference being that the states are now singlets under $\mathrm{Sp}(N)$. We introduce character formulas that enumerate such singlets as a function of $N$, and compute their first values. We conclude with an explicit numerical diagonalization of the Hamiltonian in two simple examples: the symmetric model at $N=1$, and the antisymmetric traceless model at $N=3$.


Introduction
The large N limit of tensor models has found applications in a growing list of different subjects over the years. It was initially discovered in the context of discrete approaches to quantum gravity in dimension d ≥ 3 [1,2,3,4,5], where it triggered a number of developments, on topics such as: combinatorial aspects and refinements of the large N expansion [6,7,8,9,10,11,12], probability and random geometry [13,14], non-local field theories [15,16,17,18], group field theory [19,20,21,22,23], statistical physics [24,25,26], or functional renormalization group methods [27,28,29]. In this line of thought, research efforts have been primarily focused on a particular brand of tensor models -going by the name of colored [30] and uncolored [6] tensor models 3 -because of their nice relationship to combinatorial topology and simplicial geometry [31,32,33].
More recently, tensor models have found very interesting applications in the more familiar context of quantum mechanics and local quantum field theory. Motivated by the SYK model [34,35,36,37,38,39], Witten [40] and Klebanov-Tarnopolsky [41] have introduced tensor quantum-mechanical models which develop an emergent conformal symmetry in a suitable large N and strong-coupling regime. These two models have since then been investigated in detail and generalized in a number of directions. These include works on: properties of the large N expansion [42,43,44], properties of the spectrum (including at small N ) [45,46,47,48,49,50,51], the infrared structure of such theories [52,53], generalizations to d ≥ 2 [54,55,56,57], multi-matrix models with similar properties [58,59,60,61], and connections to higher-spin theories [62]. We refer to the recent TASI lectures [63] for a more exhaustive list of current research topics. The key feature of the large N limit underlying these recent developments is that it is generically dominated by melon diagrams [4,10,64]: this family of Feynman graphs turns out to be tractable enough to be of practical use, and rich enough to capture the characteristic bilocal effects of SYK-like strongly coupled phases. On the other hand, colors (better referred to as flavors in these examples) do not seem to play any fundamental role in this context. This has motivated new work extending the domain of validity of the large N expansion, from models in which no symmetry at all is assumed among the indices of the tensors -as is for instance the case in rank-3 tensor models with O(N ) 3 symmetry [10,41] -, to models based on irreducible rank-3 tensor representations [65,66,67,68].
In view of these new developments, it is natural to wonder whether it is possible to construct SYKlike tensor models with a single flavour. In this paper we provide three examples, based on the three irreducible rank-3 representations of Sp(N ). The motivation for working with the symplectic group is that, unlike O(N ) [41,68,63], it allows to write a non-zero tetrahedral interaction for fermions in d = 1. The price to pay is that one needs to work with complex fermions, which leads to a family of tensor model analogues of the complex SYK model [34,69].
Finally, we stress that the large N structure of symplectic tensor models that we explicitly describe in this paper is not tied to the particular d = 1 fermionic theory we choose to focus on, and could therefore be taken advantage of in other contexts.
The paper is organized as follows. We introduce the models and their symmetries in section 2. In section 3 we describe their large N and strong-coupling features. We then move on to the enumeration of singlet states by means of Sp(N ) character integrals (section 4), and we finally conclude our study by an explicit diagonalization of the two simplest instances of our models (section 5). Conventions as well as various technical details are relegated to the Appendix, which will be referred to whenever necessary.

Fock space, Hamiltonian and action
Let us consider a tensorial Fermionic algebra of operators of the form: where the tensor indices a, b, c . . . take value in {1, . . . , 2N } and P is some symmetric kernel. We furthermore assume that Γ † abc transforms as a tensor product of three fundamental representations of Sp(N ) = U(2N ) ∩ Sp(2N, C), namely 4 : We ensure that this action extends to an automorphism of the fermionic algebra by assuming P to be the orthogonal projector associated to some Sp(N ) rank-3 tensor representation. In order to guarantee the existence of a melonic large N limit, it is furthermore crucial to eliminate all vector modes from this representation. This leaves us with only three inequivalent choices of irreducible representation: 1. P = P (S) is the orthogonal projector onto completely symmetric tensors; 2. P = P (A) is the orthogonal projector onto completely antisymmetric traceless tensors; 3. P = P (M ) is the orthogonal projector onto mixed 5 traceless tensors.
A detailed construction of these representations, together with explicit expressions for P (S) , P (A) and P (M ) , are provided in the Appendix B. In the remainder of the paper, P will denote any one of these three projectors; we will reserve the use of superscripts for investigations of specific features of the models (S), (A) and (M ). We also denote by V the image of P in the vector space of complex rank-3 tensors. V has (complex) dimension n := Tr P, with: The Fock space generated by the algebra (1) In F we consider the two-body Hamiltonian: where is the invertible skew-symmetric matrix entering the definition of Sp(N ) (which one may take to be (72)), and g is a real coupling constant. H is invariant under Sp(N ), as an immediate consequence of the fact that: • pairs of indices a and b belonging to operators of different types (i.e. one creation operator and one annihilation operator) are contracted with the U(2N ) invariant bilinear δ ab ; • pairs of contracted indices belonging to operators of the same type are contracted with the Sp(N, C) invariant bilinear ab . 4 Summation over repeated indices is assumed throughout the paper, unless specified otherwise. 5 By mixed symmetry tensor we mean any tensor transforming under the two-dimensional irreducible representation of S3 associated to the Young diagram .
In the path-integral formulation, the dynamics is encoded in time-dependent Grassmann variables {ψ abc (t), ψ abc (t)} governed by the action: Importantly, the tensors ψ abc andψ abc must be confined to the irreducible vector space associated to P; the total number of field components being summed over in the path-integral is therefore 2 TrP = 2n. Equivalently, one can view ψ abc andψ abc as generic tensor fields, with bare propagator given by: which automatically projects the degrees of freedom down to the appropriate subspace. The kernel of the Sp(N ) invariant interaction term will be denoted V a,b,c,d with the shorthand notation a = a 1 a 2 a 3 , b = b 1 b 2 b 3 etc. Since the index contractions follow the combinatorial pattern of a tetrahedron, we will follow the literature and refer to it as a tetrahedral interaction. We will furthermore use the following two graphical representations where the arrows in the first representation keep track of the ordering of indices in contractions, while the arrows in the second encode the type of field (by convention, ψ's are attached to ingoing arrows, andψ's to outgoing ones). The kernel V has an obvious symmetry under rotation by a π angle: Since the associated permutation is even, the interaction itself is invariant. In addition, the latter changes sign under a reflection about a vertical (or horizontal) axis: where we have used that ψ a 1 a 2 a 3 = ψ a 3 a 2 a 1 for the S representation, and ψ a 1 a 2 a 3 = −ψ a 3 a 2 a 1 for the A and M representations. Note at this stage how important it is to work with fermions: the bosonic theory with complex tensors φ abc andφ abc (in any of the representations S, A or M ) has a vanishing tetrahedral interaction: Instead, in our fermionic model the statistics and the antisymmetry of conspire to make the interaction non-trivial. In fact, comparing with the tensor quantum mechanics of O(N ) irreducible tensors, we find O(N ) irreducible tensors Sp(N ) irreducible tensors Bosonic statstics = 0 = 0 Fermionic statistics = 0 = 0 that the situation is precisely reversed: if the two 's in V are replaced by δ's, a similar argument lead to opposite signs and as a result the interaction vanishes for fermions [41,68,63]. These elementary considerations are summarized in Table 1.
We find an additional subtlety in the mixed representation M : because of the two-dimensional nature of the S 3 representation associated to the Young tableau 1 2 3 , the tetrahedral interaction is not unique. The space of tetrahedral interactions turns out to be itself two-dimensional, and the most general action we will consider in this case is: where the second term is simply obtained by the action of the permutation (12). The fact that no other independent term can be constructed is a consequence of the identity in the representation 1 2 3 .
Finally, we note that the generalized vertex (18) has the same rotational symmetry as V under rotations by π. On the other hand it is not symmetric under reflections: in contrast to the S and A representations, the field does not transform by a simple sign under the permutation (12), and as a consequence the term proportional to g 2 in (18) is not invariant. This means that, strictly speaking, the amplitudes of this model cannot unambiguously be labelled by graphs. One must instead carefully keep track of the local embedding of the interaction vertices. It is convenient in such situation to use the language of embedded graphs, also known as combinatorial maps, and to speak of Feynman maps rather than Feynman graphs. We refer to [68] for more detail in the similar context of mixed traceless O(N ) tensor models. For clarity of the exposition and given that this subtlety only affects the representation M , we will largely ignore this point and pretend that the Feynman amplitudes can be unambiguously labelled by 4-regular diagrams with oriented lines and vertices as in (11).

Symmetries and charges
The action (7) is invariant under the global symmetry group Sp(N ) × U(1). The U(1) symmetry ψ → e iθ ψ implies the conservation of the fermionic number operator: As for the N (2N + 1) charges associated to the Sp(N ) symmetry (2), they can be inferred from the action of the generators constructed in the Appendix (A). One obtains: indicates a sum of two terms obtained by cyclic permutation of the indices 6 . The quadratic Casimir operator is then (proportional to) In holographic applications of tensor models, the states of interest are singlets, a restriction to which can be enforced by a gauging procedure [41]. We will ignore this point in most of the text, except in section 4 when we will enumerate those singlets.

Existence of the large N expansion
The models (90) (or (18)) admit a large N expansion with respect to the 't Hooft coupling: Its existence can be proven by means of a general method developed for (anti)-symmetric (traceless) O(N ) tensors in [67], and generalized to mixed traceless tensors in [68]. We will not reproduce the full construction here, but only briefly explain why it applies equally well in the symplectic context, as long as one makes sure to work with an irreducible representation. Two classes of diagrams play a central role in this discussion: tadpoles and melons. The elementary tadpole and melon diagrams shown in Figure 1 recursively generate the family of melon-tadpole diagrams, while the elementary melon alone generates the subclass of melon diagrams. A salient feature of tensor models in general is that they are typically dominated by melon diagrams in the large N limit [4,5]. Each Feynman diagram can be decomposed into a sum of stranded graphs resulting from the representations (10) (or (18)) of the vertex, and the representations (99), (100) or (101) of the projector P. Just like in O(N ) models [67], a 2-point stranded graph G can be shown to scale as: 6 Those three terms are identical in the representations A and S, but not necessarily in the representation M .
where V is the number of vertices, B is the number of broken propagators (i.e. terms involving contractions in (99), (100) or (101)), and F is the number of faces (i.e. the number of closed cycles formed by the strands). The only difference is that, due to the antisymmetry of , any stranded configuration containing a face involving an odd number of contractions vanishes. The proof itself can then proceed in two main steps: 1. it is shown that ω(G) > 0 for any stranded configuration with neither tadpoles nor melons, thus demonstrating that such diagrams are always suppressed in the large N limit; 2. it is then proven that melon-tadpoles can be resummed in a controlled manner, in such a way that only melon diagrams survive at leading order in N .
The first claim results from purely combinatorial considerations [67] and automatically applies to Sp(N ) since: a) the set of stranded configurations generated by Sp(N ) models is strictly contained in the set of O(N ) stranded graphs; b) the Sp(N ) and O(N ) power countings (24) agree in these two sets.
The second claim relies on precise cancellations between the certain stranded configurations of certain generalized tadpole diagrams, which would otherwise generate unbounded scalings. A nice feature of the argument is that such cancellations automatically occur in an irreducible tensor representation [68].
We conclude this section with the main combinatorial identities relevant to the computation of melonic amplitudes in the large N limit. Ignoring the time dependence, the contraction of projectors associated to an elementary melon diagram is: The irreducibility of the tensor representation implies -by Schur's lemma -that this contraction is proportional to P itself: P More explicitly, we can compute the coefficient of proportionality M N and check that it does indeed scale as O(1) at large N . Exact expressions can be found in the Appendix C. In the rest of the text, we will only make use of the limits: In the representation M , one may construct a second inequivalent melonic 2-point map, namely: and we compute For later use, we finally define the effective infrared coupling constant:

Two-point function in the infrared limit
As is now standard, the two-point function of a melonic quantum mechanics typically develops an emergent conformal invariance at strong coupling 7 . Being complex, our model is more closely related to the complex SYK model [69], or to the SU(N ) × O(N ) × SU(N ) tensor quantum mechanics reviewed in [41,63]. By virtue of the Sp(N ) invariance and the irreduciblity of the chosen tensor representation, the full two-point function must take the form: where G(−t) = −G(t). In the large N limit, it furthermore reduces to an infinite sum of melon diagrams. Given the simple recursive structure of the latter, this leads to the closed Schwinger-Dyson equation: where * is the convolution product . As one subsequently takes the infrared limit, only the right-hand side survives and one finally obtains: The remarkable feature of this equation is that it is (formally) invariant under diffeomorphisms t → f (t) (which is of course the root cause of the solvable nature of SYK-like theories): The (time-translation invariant) solution of (35) is then conformal with dimension ∆ = 1/4 [41,43]:

Four-point function and conformal spectrum
The conformal spectrum of bilinear operators in the infrared regime can be extracted from the 4-point correlator, which at large N decomposes as: where we remind the reader that n = Tr P ∼ N 3 . The second term is a sum of ladder diagrams as illustrated in the top half of Figure 2. As a result, it has the structure 7 By which we mean the regime |λ 2 eff /ω| 1 where ω is the Fourier conjugate of t. It can therefore be equivalently understood as an infrared limit. with: and where K is the kernel associated to the operator that adds a rung to a ladder diagram. In the present model there are two ways of doing so, as represented in the bottom half of Figure 2, which yields The combinatorial factor 2 in the first term is a consequence of the opposite relative orientations of the two lines making up the rung, while the minus sign in the second is due to the fermionic statistics.
Plugging the solution (37) in, the dependence in the effective coupling constant λ 2 eff drops out and we obtain: The rung operator (43) is identical to that found in the SU(N ) × O(N ) × SU(N ) tensor quantum mechanics discussed in [41,63]. As a result we find the same conformal spectrum as derived there, for all bilinear primary operators of the form: Let us briefly summarize those findings. Following [36,37,70], the conformal dimensions h k can be determined from an analogue of the Bethe-Salpeter equation. It consists in the eigenvalue equation for the three-point correlator This self-consistency equation is independent of the OPE coefficient c k , and since we also know the rung operator (43), the function g k may be explicitly computed. One finds [63]: The solutions of g even (h) = 1 and g odd (h) = 1 are the conformal dimensions: See Figure 3. The two exact solutions h 0 and h 1 are modes respectively associated to the U(1) and diffeomorphism symmetries [69,36,71].
Thanks to the 2PI formalism introduced in [53], an effective action for tensor models that reproduces the key features of the bilocal collective field theory of the SYK model can be devised. As anticipated in [52,45] and confirmed in [53], tensor models turn out to have an important extra feature: in addition to the pseudo-Goldstone modes already present in SYK (the h 0 and h 1 modes), the global symmetry of tensor models is responsible for the presence of a large number of extra pseudo-Goldstone modes. Specifying to our model, we find N (2N + 1) zero modes associated to the enhancement of the global Sp(N ) symmetry into a gauge symmetry in the infinite coupling limit. The effective dynamics of these infrared modes can then be determined by evaluating the leading-order inverse coupling correction, which breaks the emergent local symmetry. Following [52,53,36], the resulting effective field theory is a Sp(N ) non-linear sigma model.

Enumeration of singlet states
In this section we count the number of singlets by means of standard group-theoretic techniques. In particular, we follow the same approach as in [72,73,46]. Other combinatorial methods for enumerating tensor invariants can be found in [74,75].

Generating function of singlet states
The number of Sp(N ) singlets in the Fock space F may be inferred from character integrals. Let us denote by χ = χ ρ the character of the irreducible representation ρ associated to P. From (99), (100) and (101), one finds: The character of the representation ∧(ρ) induced by ρ in F is: The number I N of Sp(N ) singlets in F is the integral of χ ∧(ρ) with respect to the Haar measure, namely: The integrand being a class function, this expression reduces to a N -dimensional integral of the form (88), and can be evaluated by numerical methods when N is not too large. More generally, the generating function of singlet operators is:

Symmetric sector
Given U ∈ Sp(N ) with eigenvalues {e ±iθ k , k = 1, . . . , N }, let us define The matrix ρ (S) (U ) has then eigenvalues {e i(θ k +θ l +θm) , 1 ≤ k ≤ l ≤ m ≤ 2N }, and hence: Henceforth where the measure dµ is given by (89) in Appendix A. After some algebra (see the Appendix D), this formula can finally be reorganized as Upon (numerical) integration we find: or more explicitly We evaluated the first two non-zero values of I

Mixed traceless sector
From (54) and (51), one may directly infer that: which, by means of the elementary manipulations laid out in the Appendix D, can be expressed in the form The first non-zero values of I

Summary
We summarize our findings in Table 2. We notice that in all three representations, only a handful of singlets are found in the examples which are most easily diagonalizable on a computer: N = 1 or 2 in the S representation; N = 3 in the A representation; and N = 2 in the M representation.
Even though the number of singlets grows extremely quickly with N -as is to be expected in a tensor model [74,52,73,46] -we remark that this number remains rather reasonable in the symmetric Sp(3) model: we find ∼ 10 6 states, which remains small compared to the ∼ 6.10 8 states found in O(6) tensor quantum mechanics [46]. As a result, the Sp(3) model could perhaps provide a more tractable example for future investigations of tensor models at small N .

Explicit diagonalization at small N
We conclude with a numerical investigation of the energy spectrum at small N . We restrict our attention to the two examples which are most easily solvable on a standard laptop: the symmetric model at N = 1, and the antisymmetric traceless model at N = 3. Even though both of these systems are too crude to exhibit large N effects, they make up for a good warm-up exercise and provide non-trivial checks of our enumeration results. We leave more in-depth studies in the line of [46,47,48,49,50,51] for future work. For convenience, we will center the energy spectrum around 0. We therefore consider the traceless version of the original Hamiltonian (6) where In practice, we may build upH from any concrete realization of the O(2n) Clifford algebra (n = Tr P): {γ i , γ j } = 2δ ij , i, j = 1, . . . , n .
This matrix can be straightforwardly diagonalized on a computer, and the resulting spectrum is shown on Figure 4. Furthermore, the Casimir operator (22) can be used to identify the Sp(1) singlets: in agreement with the enumeration result (59), we find exactly three such states, with energies −2g, 2g 3 and 4g 3 . They also assume distinct values of the U(1) charge Q, which we summarize in the following table:

N = 3 antisymmetric traceless model
The traceless Hamiltonian isH = H + 7 3 1l. Following the same method as in the symmetric case, one can represent the generators of the algebra (1) in terms of 28 γ matrices (69) of size 16384. The main new ingredient in this construction is the traceless condition, which can be straightforwardly implemented by a suitable choice of basis.
We obtain the spectrum plotted in Figure 5. We find in particular 8 singlet states, which nicely agrees with the counting (62). We were also able to determine the charge Q of each of the 8 singlets, which we summarize in the following table:  We note that the degeneracy of the three energy doublets is lifted by the charge. Furthermore, it turns out that all the singlets have distinct charges in this simple model.

Conclusion
In this paper, we have extended the scope of the melonic large N expansion to Sp(N ) rank-3 tensors.
Analogously to what happens with O(N ) (anti)symmetrized tensor models [65,67,68], the existence of the expansion is guaranteed provided that one works with irreducible tensors. In rank three, this leaves us with three inequivalent choices of representations: completely symmetric tensors, completely antisymmetric traceless tensors, and traceless tensors with mixed symmetry (where the trace operation refers to the skew-symmetric matrix (72)). Interestingly, and in contrast to O(N ), the Sp(N ) symmetry makes it possible to write down a nonvanishing tetrahdral interaction for fermions in dimension 1. This has allowed us to construct three examples of tensor quantum mechanics exhibiting non-trivial SYK-like properties at strong coupling, and without the need to introduce multiple flavours of the gauge group. More precisely, these models can be seen as tensor analogues of the complex SYK model [69].
Having enumerated the singlet states and explicitly diagonalized the simplest small N realizations of our models, we hope to have prepared the ground for further numerical investigations, which we leave for future work.

Definition
The group Sp(N ) = U(2N ) ∩ Sp(2n, C) can be realized as the space of matrices U ∈ M 2N (C) such that: where is an invertible skew-symmetric matrix: In particular 2 = −1l, so that defines a complex structure. Without loss of generality, we will represent by the 2N × 2N block-diagonal matrix Sp(N ) is a compact and simply connected group of (real) dimension N (2N + 1). To emphasize the distinction with the symplectic group Sp(2N, C), which is non-compact, it is sometimes called compact symplectic group and denoted USp(2N ). Sp(N ) can also be interpreted as the quaternionic unitary group and is therefore quite analogous to O(N ) (the real unitary group) and U(N ) (the complex unitary group).

Lie algebra and generators
A 2N × 2N complex matrix A lies in the Lie algebra sp(N ) if and only if: With our choice of representation of , it is convenient to write such an A as an N × N matrix of 2 × 2 blocks A kl : In view of the relation between Sp(N ) and the quaternions, each block A kl can advantageously be parametrized as: where σ = (σ 1 , σ 2 , σ 3 ) is a vector of Pauli matrices 8 . The conditions (73) are then equivalent to At this stage it is convenient to introduce the elementary matrices E k,l , defined as having a one in position (k, l) and zeroes elsewhere: The parametrization of sp(N ) we have just described leads to the following basis of generators: In other words It is easy to check that this basis is orthogonal with respect to the inner product X, Y = −tr (XY ), which is itself proportional to the Killing form on sp(N ). Moreover: Integration formula for class functions Any matrix U ∈ Sp(N ) can be diagonalized by another symplectic matrix P : where θ 1 , . . . , θ N are real angles. Let us denote by dU the normalized Haar measure on Sp(N ). Any class function f on Sp(N ) reduces to a function of the eigenvalues f (θ 1 , . . . , θ N ) := f (diag(e iθ 1 , e −iθ 1 , e iθ 2 , e −iθ 2 . . .)). There is then a useful integration formula (see e.g. [76]): where dµ(θ 1 , . . . , θ N ) : (89) 8 For definiteness:

B Irreducible Sp(N ) tensors and projectors
A rank-3 complex tensor T a 1 a 2 a 3 transforms under U ∈ Sp(N ) as: This action commutes with the action of the permutations σ ∈ S 3 : and can therefore be decomposed into S 3 irreducible subrepresentations. In the language of Young tableaux, one obtains: The first sector is completely symmetric, the second one completely antisymmetric. The last two contain mixed symmetry tensors and yield two equivalent representations; for definiteness, we will only consider mixed tensors of the form 1 2 3 . The orthogonal projectors on each of the three representations are: The symmetric representation is already irreducible, the three others are not. To see this, remark that the action (90) commutes with the trace operations: Except for completely symmetric tensors, all sectors contain a non-trivial trace and can therefore be reduced further. To guarantee the existence of a rich large N limit, it is particularly important to remove such vector modes from the representation. This is achieved by acting with the orthogonal projector onto traceless tensors: The second and third lines correspond to trace removal contributions. Strands connecting two indices on a same side of the drawing encode contractions, with the convention that indices are cyclically ordered as 1 → 2 → 3: for instance, a strand connecting indices i 1 and i 2 is associated to a tensor i 1 i 2 , while a strand connecting j 1 and j 3 encodes a contraction with j 3 j 1 = − j 1 j 3 . Finally, the mixed symmetry projector is: We however obtain much longer expressions in the mixed case:

D Character formulas
We start by explaining how to get (58) from (57). Using the shorthand f klm := cosθ k +θ l +θm 2 the integrand of the latter can be decomposed as Remembering that theθ's take value ±θ 1 , . . . , ±θ N , each factor can be computed separately, yielding: All in all, we obtain the desired integrand: Formulas (61) and (64) follow similarly from (60) and (63); in the mixed case one may use the factorization: 1≤k,l,m≤2N