Abstract In the theory of protoplanetary disk turbulence, a widely adopted ansatz, or assumption, is that the turnover frequency of the largest turbulent eddy, ΩL, is the local Keplerian frequency ΩK. In terms of the standard dimensionless Shakura–Sunyaevαparameter that quantifies turbulent viscosity or diffusivity, this assumption leads to characteristic length and velocity scales given respectively by and , in whichHandcare the local gas scale height and sound speed. However, this assumption is not applicable in cases when turbulence is forced numerically or driven by some natural processes such as vertical shear instability. Here, we explore the more general case where ΩL≥ ΩKand show that, under these conditions, the characteristic length and velocity scales are respectively and , where is twice the Rossby number. It follows that , where is the root-mean-square average of the turbulent velocities. Properly allowing for this effect naturally explains the reduced particle scale heights produced in shearing box simulations of particles in forced turbulence, and it may help with interpreting recent edge-on disk observations; more general implications for observations are also presented. For , the effective particle Stokes numbers are increased, which has implications for particle collision dynamics and growth, as well as for planetesimal formation.
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The Kondo effect in the quantum XX spin chain
Abstract We investigate the boundary phenomena that arise in a finite-sizeXXspin chain interacting through anXXinteraction with a spin impurity located at its edge. Upon Jordan–Wigner transformation, the model is described by a quadratic Fermionic Hamiltonian. Our work displays, within this ostensibly simple model, the emergence of the Kondo effect, a quintessential hallmark of strongly correlated physics. We also show how the Kondo cloud shrinks and turns into a single particle bound state as the impurity coupling increases beyond a critical value. In more detail, using bothBethe Ansatzandexact diagonalizationtechniques, we show that the local moment of the impurity is screened by different mechanisms depending on the ratio of the boundary and bulk coupling . When the ratio falls below the critical value , the impurity is screened via the Kondo effect. However, when the ratio between the coupling exceeds the critical value an exponentially localized bound mode is formed at the impurity site which screens the spin of the impurity in the ground state. We show that the boundary phase transition is reflected in local ground state properties by calculating the spinon density of states, the magnetization at the impurity site in the presence of a global magnetic field, and the finite temperature susceptibility of the impurity. We find that the spinon density of states in the Kondo phase has the characteristic Lorentzian peak that moves from the Fermi level to the maximum energy of the spinon as the impurity coupling is increased and becomes a localized bound mode in the bound mode phase. Moreover, the impurity magnetization and the finite temperature impurity susceptibility behave differently in the two phases. When the boundary coupling exceeds the critical value , the model is no longer boundary conformal invariant as a massive bound mode appears at the impurity site.
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- Award ID(s):
- 1941569
- PAR ID:
- 10643251
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Journal of Physics A: Mathematical and Theoretical
- Volume:
- 57
- Issue:
- 26
- ISSN:
- 1751-8113
- Page Range / eLocation ID:
- 265004
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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