A melonic quantum mechanical model without disorder

Abstract

We consider a quantum mechanical model involving interacting fermions without disorder that has the same low energy physics as the supersymmetric SYK model. The model is $SU(2)$ invariant, and the supercharge involves the $ SU(2) $ 3j symbol. We analyze various solvable corners, conceptually explain why it has a melonic expansion, and perform an exact diagonalization for small values of $N$. Expanded around the states with maximal angular momentum, the model is approximated by a two dimensional CFT. The BPS states have a simple description in that regime.

Figures (17)

• Figure 1: (a) Simplest melon diagram. (b) Melon vacuum diagram. (c) Simplest non-melonic vacuum diagram. (d) Simplest non-melonic diagram that appears in our theory, which involves oriented lines (we have not distinguished the bosonic vs fermionic lines). It gives a correction of order $(\log{j})/j$ and is proportional to the 12j symbol of the second kind.
• Figure 2: The 6j symbol appears when we use a "crossing relation" to rewrite the contractions of 3j symbols that appear in these diagrams. In other words, this diagram should be read as a relation between two sums that are quadratic in the 3j symbols.
• Figure 3: We use the crossing equation of figure \ref{['Crossing']} to simplify the tetrahedron diagram. We apply crossing to the subdiagram inside the dotted-lined circle. Then the bubble identity (\ref{['BubId']}) implies that $\ell' =j$ and leads to a final expression involving a 6j symbol with all entries equal to $j$. All lines without a label have angular momentum $j$.
• Figure 4: (a) Three unit vectors summing to zero determine an equilateral triangle. (b,c) In a bubble diagram, taking $\vec{n}_1$ fixed, the angular-momentum-conserving configurations for $\vec{n}_2$ and $\vec{n}_3$ sweep out a circle (highlighted red) on the surface of $S^2$, leading to an order $\sqrt j$ enhancement for such a bubble appearing in any diagram. (d) The non-melonic diagrams do not have as much freedom for the angular momenta of the internal lines and thus do not receive an enhancement compared to melonic diagrams at the same order in the coupling $\mathsf{J}$.
• Figure 5: (a) In black, a diagram that contributes to the four point function. The red lines indicate how the $SU(2)$ indices are contracted. In this case, they are contracted to a singlet with spin zero, $\ell=0$. The dotted lines represent fermions, and the solid lines represent bosons. In (b), we see another diagram. Here, the $SU(2)$ indices of the external lines are contracted into a spin $\ell$ representation of $SU(2)$. In (c) we see the pattern of contractions of $SU(2)$ indices for the diagram in (a). All the lines whose spins are not indicated carry spin $j$. In (d), the SU(2) pattern of contractions that arises from (b). In (e), we see the basic identity that we use to reduce the diagrams. After we add a rung, we can use the crossing relation in figure \ref{['Crossing']} to express the circled part of the diagram in the other channel, which produces the 6j symbol. Then we use the identity (\ref{['BubId']}) which gives the result for the bubble. In the end, the addition of the rung just results in a multiplication by an $\ell$-dependent numerical factor (\ref{['TilK']}).