We construct the supersymmetric SYK model on a 1D (Euclidean time) lattice. One nilpotent supersymmetry is exactly realized on the lattice in the use of the cyclic Leibniz rule.

B10B17B38Grants-in-Aid for Scientific Research2528704915K0505525400260Japanese Ministry of Education, Science, Sports and Culture

The Sachdev–Ye–Kitaev (SYK) model [1,2] and its generalizations have been attracting much attention in several contexts.^{1} In these analyses, the large- limit is fully utilized to obtain an effective theory in the IR region. Finite- analysis, however, is also interesting to see, e.g., some sort of stringy corrections for AdS/CFT-type correspondence.

For finite- analysis beyond perturbation, one approach is the exact diagonalization of the Hamiltonian [5] based on the realization of fermion operators by gamma matrices. Another approach may be lattice formulation on Euclidean time. The latter seems convenient for the calculation of multi-point correlation functions of the operators with distinct times, especially for comparison with results from the effective field theory of bi-local collective modes. Also, if higher-dimensional extension becomes possible, a Monte Carlo simulation will be numerically less expensive than exact diagonalization of the Hamiltonian. So we concentrate on lattice formulation in the present letter. Among others, we will focus on lattice formulation of the supersymmetric generalization [6] of the SYK model. This is actually highly non-trivial, since realizing supersymmetry on a lattice is a very difficult task [7].

The present authors have been studying a way for realizing the nilpotent subalgebra of supersymmetry on a lattice by using the cyclic Leibniz rule (CLR) [8,9]. In the present letter, as an application of the CLR, we will construct an supersymmetric SYK model [6] on the lattice. As will be seen, one of the two supersymmetries is exactly realized on the lattice thanks to the CLR.

Let us consider the supersymmetric SYK model [6] whose Hamiltonian is given by the anti-commutator of two nilpotent supercharges and :
where supercharges are defined by complex fermions and complex random couplings with totally anti-symmetric indices:

These fermions satisfy the following anti-commutation relations:

The random couplings have non-zero second moments under a quenched average:
with a characteristic constant , which controls the strength of the interaction.

The above supercharges are cubic in fermions, and thereby the Hamiltonian has quartic interactions. This can be generalized by taking any odd number of fermions in the supercharges. Hereafter we use generalized supercharges with fermions. The corresponding action in Euclidean time is defined by

Here we have introduced complex auxiliary variables , in order to realize the supersymmetry linearly and make the action off-shell invariant. and are -index generalizations of the random couplings. The supersymmetry transformation for each variable is
where we omit the transformation parameters, so that and should be treated as Grassmann-odd quantities.

Now let us make a lattice version of Eqs. (5) and (6). First, we replace the variables , , , and with the lattice variables , , , and , where stands for a lattice site. Then the supersymmetry transformation should become
where is an appropriate difference operator. We use superscript in order to distinguish it from the difference operator in the action for which we use .

Next, we construct a lattice action in the following form:
where repeated lattice site indices are summed. and are complex coefficients that define the multiple product of variables. Note that the last site indices of and are totally symmetric. This action gives Eq. (5) in the naive continuum limit as long as and go to 1 and goes to . Requiring the invariance of the action under either transformation or in Eq. (7), we obtain the conditions that should be satisfied by , , , and . For example, if we require invariance, then we have

The first condition (12) is satisfied if the difference operator meets

The second condition (13) is satisfied if we ensure that

The third condition (14) is satisfied if meets

The summation in Eq. (17) can be reduced to that in the only cyclic permutation due to the totally symmetric nature of the last indices. Thus this relation is nothing but the cyclic Leibniz rule (CLR) [8,9].

It should be stressed that the CLR (17) is not an abstract relation but has many concrete solutions. For example, if we simply take , then
is one of the ultralocal^{2} solutions of Eqs. (15), (16), and (17) for . Systematic construction of the solutions for the CLR with a symmetric difference operator can be found in Ref. [10].

Although this might be the simplest solution, we would have a species doubler with it. So we propose an alternative solution:

Here is a real parameter and the kinetic action with this contains the so-called Wilson term ( corresponds to the Wilson term coefficient), which lifts doublers up with a cutoff-scale mass.

For the case, which corresponds to the forward difference operator, we can write down solutions with generic :
where stands for the summation over all permutations of , and is understood.

Thus we have a concrete way to construct the -invariant lattice action for the supersymmetric SYK model. If you need -invariant action, just exchange the roles of and .

A few remarks are in order, as follows.

We cannot require both and invariance because these simultaneously impose the CLR relation and totally symmetric indices on and , but there is no local solution of CLR with symmetric or [8].

For invariance, is totally symmetric, but is not because of CLR and locality. Therefore, is not a complex conjugate of , so that the resulting action is not Hermitian. This “could-be sign problem” is of the order of where is the lattice constant, and disappears at least in the naive continuum limit.

It seems that the CLR approach is a unique way to realize supersymmetry for models of this type. The other approaches, like the method using the Nicolai map or the method of finding a nilpotent transformation without a difference operator [11–13], may not work for the model. In particular, the last term of the action (5) is -invariant but not -exact, and therefore the topological field theoretic approach cannot be applied. This supports the uniqueness of the CLR approach.

Acknowledgements

This work is supported in part by Grants-in-Aid for Scientific Research No. 25287049 (M.K.), No. 15K05055 (M.S.), and No. 25400260 (H.S.) by the Japanese Ministry of Education, Science, Sports and Culture.

Funding

Open Access funding: SCOAP.

Footnotes

^{1}For recent reviews see, e.g., Refs. [3,4] and references therein.

^{2}Here ultralocal means the operator defined in a region with a finite extent on a lattice. Local operators on the lattice include not only ultralocal ones but also ones with infinite extent that decay exponentially.

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