Emergent symmetry in Brownian SYK models and charge dependent scrambling

Abstract

In this work, we introduce a symmetry-based approach to study the scrambling and operator dynamics of Brownian SYK models at large finite N and in the infinite N limit. We compute the out-of-time-ordered correlator (OTOC) in the Majorana model without charge conservation and the complex model with charge conservation, and demonstrate that in both models taking the random average of the couplings gives rise to emergent symmetry structures. The random averaging exactly maps the operator dynamics of the Majorana model and the complex model to the imaginary time dynamics of an SU(2) spin and an SU(4) spin respectively, which become solvable in the large N limit. Furthermore, the symmetry structure drastically reduces the size of the Hilbert space required to calculate the OTOC from exponential to linear in N, providing full access to the operator dynamics at all times for large finite N. In the case of the complex model with charge conservation, using this approach, we obtain the OTOC within each charge sector both numerically at finite N and analytically in the large N limit. We find that the time scale of the scrambling dynamics for all times and in each sector is characterized by the charge density. Furthermore, after proper rescaling, the OTOC corresponding to different finite charge densities collapses into a single curve at large finite N. In the large N limit, the rescaled OTOCs at finite density are described by the same hydrodynamic equation as in the Majorana case.

Charge transport, information scrambling and quantum operator-coherence in a many-body system with U(1) symmetry

“…• Just as there are multiple states within each charge sector, there are also multiple operators [49]. The variable ξ controls the choice of the operator once the charge is fixed.…”

“…This is justifiable for two reasons, firstly, it is expected that in the case of scrambling dynamics, the non-symmetric operator-strings will decay exponentially fast, leaving behind only the symmetric strings after a short time. Secondly, it has also been shown that features such as Lyapunov growth and the scrambled steady-state are entirely contained within this symmetric irrep, making it the only relevant sector to study the chaotic behavior of the model [49]. For the Brownian model, starting from a symmetric operator-string completely restricts the dynamics to the symmetric sector, which can be seen from the emergent algebra structure.…”

“…There is extensive literature on operator-dynamics [43][44][45][46][47][48][49][50][51][52][53][54] and spectral form factors [55][56][57][58][59][60] in the presence of symmetries. The primary mechanism through which charge, or other conserved quantities, can influence operator growth is by confining the access of growing operators to a specific (symmetry) sector of the Hilbert space.…”

“…SYK models, which are all-to-all interacting models with random couplings [61], are well-known for their connection to black-hole physics [62,63] and for their exactly solvable nature in the large-N limit [17][18][19][20]. Their Brownian limit, which considers time-dependent couplings, has also been studied in various contexts [47,49,[64][65][66][67][68]. The dynamics of Brownian models can be mapped to a stochastic process [39,67,69], or the imaginary time dynamics of bosonic-models [65].…”

“…The dynamics of Brownian models can be mapped to a stochastic process [39,67,69], or the imaginary time dynamics of bosonic-models [65]. It has also been shown that Brownian SYK models, in particular, give rise to emergent symmetry structures after the disorder averaging procedure which can be used to compute the OTOC both at large finite N and in the infinite-N limit [49]. In this work, we show that the extended complex Brownian model is mapped to a quantum SU(4) spin chain with inter-site Heisenberg coupling and intra-site interaction.…”