Supersymmetric SYK models
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The model with a single supercharge has unbroken supersymmetry at large , but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large equation, and also by performing numerical computations for small values of . We also compute the large spectrum of “singlet” operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
The Sachdev-Ye-Kitaev (SYK) models (or their variants) realize non-Fermi liquid states of matter without quasiparticle excitations Sachdev and Ye (1993); Parcollet and Georges (1999); Georges et al. (2000, 2001). They also have features in common with black holes with AdS horizons Sachdev (2010a, b), and this connection has been significantly sharpened in recent work Kitaev (2015); Almheiri and Polchinski (2015); Almheiri and Kang (2016); Sachdev (2015); Hosur et al. (2016); Polchinski and Rosenhaus (2016); You et al. (2016); Fu and Sachdev (2016); Jevicki et al. (2016); Jevicki and Suzuki (2016); Maldacena and Stanford (2016); Maldacena et al. (2016); Danshita et al. (2016); García-Álvarez et al. (2016); Engelsöy et al. (2016); Jensen (2016); Bagrets et al. (2016); Cvetič and Papadimitriou (2016); Gu et al. (2016).
In this paper, we introduce supersymmetric generalizations of the SYK models. Like previous models, the supersymmetric models have random all-to-all interactions between fermions on sites. There are no canonical bosons in the underlying Hamiltonian, and in this respect, our models are similar to the supersymmetric lattice models in Refs. Fendley et al. (2003a, b); Fendley and Schoutens (2005); Huijse et al. (2008); Huijse and Schoutens (2010); Huijse et al. (2011, 2012). As we describe below, certain structures in the correlations of the random couplings of our models lead to and supersymmetry. Supersymmetric models with random couplings that include both bosons and fermions were considered in Anninos et al. (2016).
Let us discuss now the model with supersymmetry, and defer presentation of the case to Section V. For the case, we introduce the supercharge
where are Majorana fermions on sites ,
and is a fixed real antisymmetric tensor so that is Hermitian. We will take the to be independent gaussian random variables, with zero mean and variance specified by the constant :
where is positive and has units of energy. As is the case in supersymmetric theories, the Hamiltonian is the square of the supercharge
with representing all possible anti-symmetric permutations. Note that the are not independent gaussian random variables, and this is formally the only difference from the Hamiltonian of the non-supersymmetric SYK models Kitaev (2015); Sachdev (2015); Hosur et al. (2016); Polchinski and Rosenhaus (2016); You et al. (2016); Fu and Sachdev (2016); Jevicki et al. (2016); Jevicki and Suzuki (2016); Maldacena and Stanford (2016); Danshita et al. (2016); Bagrets et al. (2016). These particular correlations change the structure of the large equations and lead to a solution where the fermion has dimension . In addition, there is a supersymmetric partner of this operator which is bosonic and has dimension . This large solution has unbroken supersymmetry, and we have checked this numerically by comparing with exact diagonalization of the Hamiltonain. We have also computed the large ground state entropy from a complete numerical solution of the saddle-point equations. In the exact diagonalization we find that the lowest energy state has non-zero energy, and therefore, broken supersymmetry. However, this energy is estimated to be of order where is a numerical constant. We have also generalized the model to include a supercharge of the schematic form , and we also solved this model in the large limit. We also formulated the model in superspace, and show that the large equations have a super-reparameterization invariance, which is both spontaneously as well as explicitly broken by the appearance of a superschwarzian action, which we describe in detail.
We have also analyzed the eigenvalues of the ladder kernel which appears in the computation of the four point function. There are both bosonic and fermionic operators that can propagate on this ladder. There is a particular eigenvalue of the kernel which is a zero mode and corresponds to the degrees of freedom described by the Schwarzian. They are a bosonic mode with dimension and a fermionic one with . The other eigenvalues of the kernel should describe operators appearing in the OPE. These also come in boson-fermion pairs and have a structure similar to the usual SYK case. One interesting feature is the appearance of a boson fermion pair with dimensions and , which is associated to an additional symmetry of the low energy equations. These do not give rise to extra zero modes but simply correspond to other operators in the theory.
We have also analyzed the version of the theory. In this case we can also compute a kind of Witten index. More precisely, the model has a discrete global symmetry that commutes with supersymmetry, so that we can include the corresponding discrete chemical potential in the Witten index, which turns out to be non-zero. These are generically expected to be lower bounds on the large ground state entropy; it turns out that the largest Witten index is, in fact, equal to the large ground state entropy. The model also has a symmetry. The exact diagonalization analysis also suggests a conjecture for number of ground states for each value of the charge. For the case, they are concentrated at very small values of the R-charge, within .
This paper is organized as follows. In section II we define the supersymmetric model, write the large effective action and the corresponding classical equations. We determine the dimensions of the operators in the IR and we derive a constraints imposed by unbroken supersymmetry on the correlators. We also present a generalization of the model where the supercharge is a product of fermions and solve the whole flow in the limit. In section III we present some results on exact numerical diagonalization of the Hamiltonian. This includes results on the ground state energy and two point correlation functions. In section IV we discuss the physics of the low energy degrees of freedom associated to the spontaneously and explicitly broken super-reparameterization symmetry of the theory. In section V we define and study a model with supersymmetry. We compute the Witten index and use it to argue that the model has a large exact degeneracy at zero energy. We also discuss the superspace and super-reparameterization symmetry in this case. In section VI we discuss the ladder diagrams that contribute to the four point function. We use them to determine the eigenvalues of the ladder kernel and use it to determine the spectrum of dimensions of composite operators.
Ii Definition of the model and the large effective action
To set up a path integral formulation of , we first note that the supercharge acts on the fermion as
We introduce a non-dynamical auxiliary boson to linearize the supersymmetry transformation and realize the supersymmetry algebra off-shell. The Lagrangian describing is
Under the transformation , it changes as
This implies that the action is invariant as long as the structure constants in (7) are totally anti-symmetric.
Now we proceed to obtain the effective action. This can be done by averaging over the Gaussian random variables in the replica formalism. In this model, as in SYK, the interaction between replicas is suppressed by , so that we can simply average over disorder by treating it as an additional field with time indepedent two point functions as in (3). Averaging over disorder, we obtain
Note that this action contains terms in which the bosons and fermions carry the same index, and which should be omitted e.g. ; however they are subdominant in the large limit, and so we ignore this issue.
Notice further that the relative coefficient between the last two terms is determined by the supersymmetry requirement that the structure constants are totally anti-symmetric, so that
The purpose of this section is to discuss the large saddle-point equations for the diagonal Green’s functions
where we have a sum over . We will thus drop the last term in (9), which only affects the saddle-point equations for the off-diagonal Green’s functions
We will restore it in a later section IV, where we write the saddle-point equations in a manifestly super-symmetric fashion.
We introduce the Lagrange multipliers
As the notation suggests, these Lagrange multipliers will eventually become the self energies. Inserting these factors of 1 in the fermion path integral with the action (9), using the delta functions implied by the integration over to express the interaction terms in (9), and integrating out the fermions we obtain
which becomes a classical action when is large. Let us look at the classical equations for the action in (19). Taking derivatives with respect to and , we obtain
Taking derivatives with respect to and , assuming time translation symmetry and going to Fourier space, we obtain
which confirms that are the self energies.
In temporal space, the saddle point equations take the form
These equations can be solved numerically, and we can see some plots in figure (5). Once we find a solution to these equations, we can compute the on-shell action, which can be written as
where are the Matsubara fequencies for the fermion and boson cases. From this we can compute the entropy through the usual thermodynamic formula. A plot of the entropy as a function of the temperature can be found in figure (1).
where and are the scaling dimensions of the fermion and the boson. We then insert (25) into (20), (21) in order to fix the values of and . Matching the power-laws in the saddle point equations yields only the single constraint
As we will see later the dimension can be determined by looking at the constant coefficients. Before showing this, let us discuss a simpler way to obtain another condition.
ii.1 Supersymmetry constraints
Further analytic progress can be made if we assume that the solutions of the saddle point equations (20), (21) preserve supersymmetry. With such an assumption, we now show that the scaling dimensions and can be easily determined. Again, we refer the reader to a later section IV for a full discussion of the supersymmetry properties of the saddle point equations.
If supersymmetry is not spontaneously broken, then
This relationship together with
is compatible with the saddle-point equations in Section II.
ii.2 Simple generalization
We now show how to derive the constraint directly from the saddle point equations without assuming that the solution preserves supersymmetry.
It is useful to consider a simple generalization of Eq. (1) to case where the supercharge is the sum over products of fermions111 In detail , with .. The Hamiltonian involves sums of terms with up to fermions. corresponds to the case discussed above (1).
The large equations are (21) and
We can explore them at low energy by making the ansatz
where , are some constants.
Again, if we assume supersymmetry we immediately derive and . Doing so without that assumption requires us to look at the equations for and .
Using the Fourier transforms for symmetric and antisymmetric functions
The following relations are useful
Matching the frequency dependent part we get the condition . The equations for the coefficients reduce to
The ratio between the two equations gives another condition for , with one rational solution obeying , which is also independently implied by supersymmetry, see (28). In the range where and are both positive there is a second, irrational solution to the equations which has higher . This second solution breaks supersymmetry, since it does not obey (29). It would be nice to understand it further, but we leave that to the future.
We also see that the low energy equations have a symmetry
In fact, the supersymmetry relation (28) also fixes this freedom of rescaling, by setting .
In the end this fixes the coefficients to
This coefficient (for ) is used in the plot of figure 4. Of course the finite temperature version is
This generalization makes it easy to compute the ground state entropy. In principle this can be done by inserting these solutions into the effective action
It is slighly simpler to take the derivative with respect to , ignoring any term that involves derivatives of since those terms vanish by the equations of motion. This gives
Where we inserted (42) and used (41). The constant term includes UV divergencies which are independent. This term contributes to the ground state energy222If we computed it using the exact solution (as opposed to the conformal soution) of the equations we expect the ground state energy to vanish due to supersymmetry., but not to the entropy. Integrating (44) we obtain the ground state entropy
where in integrating we used the boundary condition that the entropy should be the entropy of free fermion system at , a fact we will check below. For this matches the numerical answer, see figure 1.
ii.3 The large limit
It is interesting to take the large limit of the model since then we can find an exact solution interpolating between the short and long distance behavior. The analysis is very similar to the one in Maldacena and Stanford (2016). We expand the functions as follows
where we neglected higher order terms in the expansion. We can then Fourier transform, compute , to first order in the expansion. This gives , and . Replacing this the equations (31) we find
where we take the large limit keeping fixed. The solution obeying the boundary conditions is
where are integration constants fixed by the boundary conditions. It is interesting to note that the UV supersymmetry condition is only approximately true at short distances, distances shorter than the temperature.
It is also interesting to compute the free energy. Again, this is conveniently done by taking a derivative with respect to and using the equations of motion.
where the first equality holds in general and the second only for large . Expressing it in term of the parameters in (48) we get
we can also easily compute the small expansion, which, as expected, goes in powers of . We have used the entropy of the free fermion system, at , as an integration constant in going from (49) to (50). The constant term in the large expansion agrees with the large expansion of the ground state entropy (45). The term can also be obtained form the Schwarzian and this can serve as a way to fix the coeffiicient of the Schwarzian action at large . The linear term in represents the ground state energy and it should be subtracted off.
All these results have the same form as the large limit of the usual SYK model Maldacena and Stanford (2016). This is not a coincidence. What happens is that the leading boson propagator is simply the delta function in (46) which collapses the diagrams to those of the large limit of the usual SYK model.
Iii Exact diagonalization
This section presents results from the exact numerical diagonalization of the Hamiltonian in Eq. (4). We examined samples with up to sites, and averaged over 100 or more realizations of disorder. This exact diagonalization allows us to check the validity of the answer we obtained using large methods.
An important purpose of the numerical study was to examine whether supersymmetry was unbroken in the limit. In Fig. 2 we test the basic relationship in Eq. (27) between the fermion and boson Green’s functions.
The agreement between the boson Green’s function and the time derivative of the fermion Green’s function is evidently excellent.
We also computed the value of the ground state energy . Supersymmetry is unbroken if an only if . We have found that is non-zero in the exact theory, but it becomes very small for large . Indeed fig. 3 shows that does become very small, and the approach to zero is compatible with an exponential decrease of with . This is then compatible with a supersymmetric large solution, supersymmetry is then broken non-perturbatively in the expansion. The combination of Figs. 2 and 3 is strong numerical evidence for the preservation of supersymmetry in the limit (with suppersymmetry breaking at finite ). The ground state energy can be fitted well by with , which is compatible with . Here is the ground state entropy, (45) . This is smaller that the naive estimate for the interparticle level spacing which is .
Note that the breaking of supersymmetry is also compatible with the Witten index of this model which is . This can be easily computed in the free theory. For odd we defined the Hilbert space by adding an extra Majorana mode that is decoupled from the ones appearing in the Hamiltonian.
As in Ref. You et al. (2016), we found a ground state degeneracy pattern that depended upon (mod 8). The pattern in our case is (for )
For odd this degeneracy includes all the states in the Hilbert space defined by adding an extra decoupled fermion. We also found that the value of has structure dependent upon (mod 8), as is clear from Fig. 3.
We also compared our numerical results for the Green’s functions with the conformal scaling structure expected at long times and low temperatures. From Eq (41), with , we expect that at and large
We also extended this comparison to , where we expect the generalization of Eq. (53) to
The comparison of these results with the numerical data appears in Fig. 5.
Iv Superspace and super-reparameterization
So far, we have seen that the main consequence of supersymmetry was the relationship Eq. (27) between the boson and fermion Green’s functions at . However, as is clear from Eq. (54), this simple relationship does not extend to . Of course, this is not surprising, since finite temperature breaks supersymmetry.
Previous work on the SYK models has highlighted reparameterization and conformal symmetries Parcollet and Georges (1999); Kitaev (2015); Sachdev (2015); Maldacena and Stanford (2016) which allow one to map zero and non-zero temperature correlators. This section will describe how supersymmetry and reparameterizations combine to yield super-reparameterization symmetries, and the consequences for the correlators.
As in the SYK model, most of this super-reparameterization symmetry is spontaneously broken. There is, however, a part of it that is left unbroken by (54). This unbroken part includes both a bosonic group as well as two fermionic generators, giving an global super-conformal group. These super-symmetry generators are emergent, and are different from the original supersymmetry of the model. In particular, they square to general conformal transformations of the thermal circle rather than time translations. We will come back to this point more explicitly later.
Superspace offers a simple way to package together the degrees of freedom and equations of motion for Eq. (4) while making supersymmetry manifest. Concretely, we define a super-field
which is a function of both time and an auxiliary anticommuting variable .
Supersymmetry transformations combine with translations into a group of super-translations
It is well-known that a Grassman integral of the form
for some function of and is invariant under super-translation: if we expand then, by definition,
and we find that
are the same up to total derivatives.
The Lagrangian Eq. (7) can be written in a manifestly supersymmetric form
by introducing the super-derivative operator
which is invariant under super-translations. Indeed,
We can now derive the super equations of motion. Let us define
This super-field includes both the bosonic bilinears and and the fermionic bilinears and . The equations of motions of the disorder-averaged Lagrangian can now be expressed in a manifestly supersymmetric way as
The right hand side is the supersymmetric generalization of the delta function:
Some useful super-translation invariant combinations are and , which satisfies . In a translation-invariant, supersymmetric vacuum of definite fermion number, the solution must take the form
If we use a vacuum that does not have definite fermion number, supersymmetry imposes that , so in a translation-invariant, supersymmetric vacuum (without definite fermion number) we have
Of course, the whole derivation of the effective action can be re-cast in superspace, starting from