Krylov Complexity of Supersymmetric SYK Models

Abstract

We study the effect of supersymmetry breaking on Krylov complexity in the $\mathcal{N}=2$ SYK model under irrelevant and mass deformations of the Hamiltonian. The irrelevant deformation breaks $\mathcal{N}=2$ supersymmetry down to $\mathcal{N}=1$, while the mass deformation breaks supersymmetry completely. Using Krylov subspace methods, we analyze the Lanczos sequence, Krylov dimension, complexity, and entropy at finite system size as functions of deformation strength. Both deformations enlarge the Krylov space, but the mass deformation has a stronger effect. Krylov complexity exhibits initial quadratic growth, followed by linear growth across both deformations. We observe that both deformations increase the quadratic and linear growth rates of Krylov complexity at early times. At late times, the irrelevant deformation increases the saturation complexity as a fraction of the Krylov dimension, while the mass deformation decreases it. This reveals distinct signatures of how supersymmetry breaking impacts quantum complexity.

Figures (24)

• Figure 1: The spectral density $\rho(E)$ for different SYK models. Figure (a) resembles the Wigner semicircle distribution, typical of chaotic systems, while figure (b) resembles a Gaussian, which is characteristic of integrable systems. Figure (c) exhibits a large zero-energy ground state degeneracy and a gap to the first excited state. Each plot corresponds to 500 realizations of the Hamiltonian with $N=10$, $F=5$, and $J=1$.
• Figure 2: The level spacing density $p(s)$ for different SYK models. Figure (a) is peaked around $s>0$ because the energy levels repel each other, while figure (b) is a Poisson distribution peaked at $s=0$. Figure (c) has a peak at $E=0$ because of the zero-energy degeneracy, but otherwise peaks at $s>0$ similar to figure (a). Each plot corresponds to 500 realizations of the Hamiltonian with $N=10$, $F=5$, and $J=1$.
• Figure 3: The spectral density $\rho(E)$ and level spacing density $p(s)$ for $\mathcal{N}=2$ SYK as the strength $\epsilon$ of an irrelevant deformation varies. The BPS degeneracy lifts, while the energy gap closes. The level spacings shift to the right before flattening out. Each plot corresponds to 500 realizations of the Hamiltonian with $N=10$, $q=3$, and $J=1$.
• Figure 4: The average gap ratio $\langle r \rangle$ as the strength $\epsilon$ of both deformations varies. The undeformed model has $\langle r \rangle \approx 0.48$. The irrelevant deformation increases the gap ratio to a level consistent with chaotic systems. The mass deformation decreases the ratio to a value in line with integrable systems. Each curve corresponds to the average of $500$ realizations of the Hamiltonian with $N=10$, $q=3$, and $J=1$. The shaded regions indicate the standard deviation. To account for degeneracies, spacings less than $10^{-10}$ were discarded.
• Figure 5: The spectral density $\rho(E)$ and level spacing density $p(s)$ for $\mathcal{N}=2$ SYK as the strength $\epsilon$ of a mass deformation varies. The BPS degeneracy and energy gap are eliminated, and the energy is allowed to go negative. The level spacing density resembles a Poisson distribution when the deformation is turned on. Each plot corresponds to 500 realizations of the Hamiltonian with $N=10$, $q=3$, and $J=1$.