Tunable Quantum Chaos in the Sachdev-Ye-Kitaev Model Coupled to a Thermal Bath

The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with $N$ Majorana fermion modes coupled to another SYK model with $N^2$ Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from $2\pi/\beta$ ($\beta=1/(k_BT)$, $T$ is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a $\sqrt{T}$-dependence at relatively high temperature to a linear $T$-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.Comment: 28 pages, 9 figures. References adde

Numerical Simulation and Erosion Prediction for an Electrical Submersible Pump

Electrical Submersible Pumps (ESP) are widely used in the oil industry to lift oil and gas at the same time with high efficiency. Many ESPs operate with multiphase flow – liquid, gas and a low concentration of sand, thus having problems of pressure degradation and erosion. To investigate these problems, a numerical method can be used, which provides details of the inner flow field. Using the commercial software ANSYS Fluent, 3D transient multiphase simulations are conducted for a Baker Hughes made ESP MVP-G400. The simulations focus on three parts: secondary flow path in multiphase flow, pressure degradation due to gas volume fraction and erosion prediction. Aside from the main flow path, the clearances and balancing holes inside the ESP create a secondary path which enables the flow to recirculate. Although the volume flow rate in this path is low compared with the main flow path, the erosion in the secondary path cannot be neglected and can result in pump failure. In this research, a water-air-sand three phase simulation is performed on dual stages of the ESP, with all secondary path included. Second part of this research focuses on the pressure degradation due to the presence of a gas phase, especially at the first stage near the pump inlet. The compressibility of gas and the bubble break-up and coalescence effects are considered using the Population Balancing Module in ANSYS Fluent. The last part is the erosion prediction. A low concentration of sand is often inevitable during the operation of an ESP, causing erosion and reducing the life span of the ESP. This erosion becomes more severe with the existence a of gas phase. The three-phase simulations with both Eulerian multiphase and particle tracking explain the erosion and the role of gas in this process, giving a reasonable qualitative prediction on the erosion of the ESP

Spectral form factor for free large NN gauge theory and strings

We investigate the spectral form factor in two different systems, free large $N$ gauge theories and highly excited string gas. In both cases, after a rapid decay of the spectral form factor at early time, new contributions come in, preventing the spectral form factor from ever becoming exponentially small. We consider $U(N)$ gauge theories with only adjoint matter and compute the spectral form factor using a matrix integral of the thermal holonomy $U$. The new saddles differ from the early time saddle by preserving certain subgroups of the center symmetry. For a gas of strings, the short time decay of the spectral form factor is governed by the continuous Hagedorn density of states, which can be associated to the thermal winding mode with winding number $\pm 1$. We show that the rise of the spectral form factor comes from other winding modes that also carry momentum along the time direction. We speculate on the existence of a family of classical solutions for these string modes, similar to the Horowitz-Polchinski solution. We review a similar problem for black holes. In particular, we examine the Kontsevich-Segal criterion on complex black holes that contribute to the spectral form factor. In the canonical ensemble quantity $Z(\beta+it)$, the black hole becomes unallowed at $t\sim \mathcal{O}(\beta)$. A way to avoid this is to consider the microcanonical ensemble, where the black hole stays allowable.Comment: 36 pages, 15 figures. v2: added references. v3: matched published versio