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.
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.
Abstract In this work, we derive an exact hydrodynamical description for the coupled, charge and operator dynamics, in a quantum many-body system with U(1) symmetry. Using an emergent symmetry in the complex Brownian SYK model with charge conservation, we map the operator dynamics in the model to the imaginary-time dynamics of an SU(4) spin-chain. We utilize the emergent SU(4) description to demonstrate that the U(1) symmetry causes quantum-coherence to persist even after disorder-averaging, in sharp contrast to models without symmetries. In line with this property, we write down a ‘restricted’ Fokker-Planck equation for the out-of-time ordered correlator (OTOC) in the large-N limit, which permits a classical probability description strictly in the incoherent sector of the global operator-space. We then exploit this feature to describe the OTOC in terms of a Fisher-Kolmogorov-Petrovsky-Piskun (FKPP)-equation which couples the operator with the charge and is valid at all time-scales and for arbitrary charge-density profiles. The coupled equations obtained belong to a class of models also used to describe the population dynamics of bacteria embedded in a diffusive media. We simulate them to explore operator-dynamics in a background of non-uniform charge configuration, which reveals that the charge transport can strongly affect dynamics of operators, including those that have no overlap with the charge.
Operator scrambling is commonly diagnosed by the growth of out-of-time-ordered correlators (OTOCs), yet a general symmetry principle underlying their effective dynamics has remained elusive. For Brownian or short-time-correlated large-N Majorana systems, we develop a symmetry-based effective field theory for operator scrambling, organized by a strong-to-weak U(1) symmetry breaking in operator space. The key observation is that, in the noninteracting fermion limit, the four-fold Keldysh contour representation of an OTOC admits an emergent strong U(1) symmetry in a doubled Hilbert-space description, even when the original system has no ordinary conserved quantity. The associated slow mode is the phase of the strong-charge creation operator, whose conjugate density is identified with the local operator size. Generic interactions explicitly break the strong symmetry and generate a mass term at lowest order for the would-be Goldstone mode, thereby converting diffusive operator spreading into chaotic growth. We further show that higher-order symmetry breaking terms are tightly constrained by an emergent duality that combines time reversal with contour permutation. This duality fixes the effective action up to quadratic order in the response field, relates the multiplicative noise strength directly to the Lyapunov exponent, and makes the positivity of the Lyapunov exponent a consequence of convergence of the real-time path integral. The resulting OTOC dynamics is governed by a noisy FKPP equation, which captures within a unified framework the early-time exponential growth, ballistic propagation, nonlinear saturation, and stochastic front broadening of operator scrambling. We verify this construction in a Brownian SYK chain, where a direct saddle-point expansion reproduces the symmetry-based effective action. Our results reveal a symmetry origin of operator-size hydrodynamics and scrambling.
A bstract We consider the Brownian SYK model of N interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the Rényi-2 tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of N qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in terms of bosonic collective modes, we show that for the quantities of interest the dynamics takes place in a subspace of the effective Hilbert space whose dimension grows either linearly or quadratically with N , allowing us to perform numerically exact calculations up to N = 10 6 . We analyze in detail the interesting features of the OTOCs, including their dependence on the chosen observables, and of the tripartite information. We observe explicitly the emergence of a scrambling time t ∗ ∼ ln N controlling the onset of both chaotic and scrambling behavior, after which we characterize the exponential decay of the quantities of interest to the corresponding Haar scrambled values.
Abstract Quantum scrambling plays an important role in understanding thermalization in closed quantum systems. By this effect, quantum information spreads throughout the system and becomes hidden in the form of non-local correlations. Alternatively, it can be described in terms of the increase in complexity and spatial support of operators in the Heisenberg picture, a phenomenon known as operator growth. In this work, we study the disordered fully-connected Sachdev-Ye-Kitaev (SYK) model, and we demonstrate that scrambling is absent for disorder-averaged expectation values of observables. In detail, we adopt a formalism typical of open quantum systems to show that, on average and within charge-conserved sectors, operators evolve in a relatively simple way which is governed by their operator size. This feature only affects single-time correlation functions, and in particular it does not hold for out-of-time-order correlators, which are well-known to show scrambling behavior. Making use of these findings, we develop a cumulant expansion approach to approximate the evolution of equal-time observables. We employ this scheme to obtain analytic results that apply to arbitrary system size, and we benchmark its effectiveness by exact numerics. Our findings shed light on the structure of the dynamics of observables in the SYK model, and provide an approximate numerical description that overcomes the limitation to small systems of standard methods.
Abstract We study operator dynamics in Brownian quantum many-body models with q -local interactions. The operator dynamics are characterized by the time-dependent size distribution, for which we derive an exact master equation in both the Brownian Majorana Sachdev–Ye–Kitaev model and the spin model for general q . This equation can be solved numerically for large systems. Additionally, we obtain the analytical size distribution in the large N limit for arbitrary initial conditions and q . The distributions for both models take the same form, related to the χ -squared distribution by a change of variable, and strongly depend on the initial condition. For small initial sizes, the operator dynamics are characterized by a broad distribution that narrows as the initial size increases. When the initial operator size is below q − 2 for the Majorana model or q − 1 for the spin model, the distribution diverges in the small size limit at all times. The mean size of all operators, which can be directly measured by the out-of-time ordered correlator, grows exponentially during the early time. In the late time regime, the mean size for a single Majorana or Pauli operator for all q decays exponentially as t e − t , much slower than all other operators, which decay as e − t . At finite N , the size distribution exhibits modulo-dependent branching within a symmetry sector for the q ⩾ 8 Majorana model and the q ⩾ 4 spin model. Our results reveal universal features of operator dynamics in q -local quantum many-body systems.
A bstract Under the Heisenberg evolution in chaotic quantum systems, initially simple operators evolve into complicated ones and ultimately cover the whole operator space. We study the growth of the operator “size” in this process, which is related to the out-of-time-order correlator (OTOC). We derive the full time evolution of the size distribution in large N quantum mechanics of Majorana fermions. As examples, we apply the formalism to the Brownian SYK model (infinite temperature) and the large q SYK model (finite temperature).