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Title: Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 96; Journal Issue: 8; Related Information: CHORUS Timestamp: 2017-08-08 10:09:15; Journal ID: ISSN 2469-9950
American Physical Society
Country of Publication:
United States

Citation Formats

He, Zhuoran, and Millis, Andrew J. Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics. United States: N. p., 2017. Web. doi:10.1103/PhysRevB.96.085107.
He, Zhuoran, & Millis, Andrew J. Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics. United States. doi:10.1103/PhysRevB.96.085107.
He, Zhuoran, and Millis, Andrew J. 2017. "Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics". United States. doi:10.1103/PhysRevB.96.085107.
title = {Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics},
author = {He, Zhuoran and Millis, Andrew J.},
abstractNote = {},
doi = {10.1103/PhysRevB.96.085107},
journal = {Physical Review B},
number = 8,
volume = 96,
place = {United States},
year = 2017,
month = 8

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Publisher's Accepted Manuscript

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  • Cited by 2
  • Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time {tau}{sub Q} across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length {xi} proportional to themore » square root of the quench time {tau}{sub Q}: {xi}={radical}({tau}{sub Q}). The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same {xi} controls the entropy of entanglement, e.g., entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at (1/6) log{sub 2} {xi}, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L{approx_equal}{xi}, but -- in the subsequent evolution in the ferromagnetic phase -- a somewhat larger length scale l={radical}({tau}{sub Q}) ln {tau}{sub Q} develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when L{approx_equal}l. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length {xi}, but (as for entropy of entanglement) in the following evolution length scale l gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by {xi}. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.« less
  • It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of L spins with the rest of the system scales logarithmically with L with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with L butmore » the prefactor depends on the quench time. This puts a constraint on the block size L so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.« less
  • In this work, using the non-equilibrium Keldysh formalism, we study the effects of the electron–electron interaction and the electron-spin correlation on the non-equilibrium Kondo effect and the transport properties of the symmetric single impurity Anderson model (SIAM) at zero temperature by generalizing the self-consistent method of Singwi, Tosi, Land, and Sjolander (STLS) for a single-band tight-binding model with Hubbard type interaction to out of equilibrium steady-states. We at first determine in a self-consistent manner the non-equilibrium spin correlation function, the effective Hubbard interaction, and the double-occupancy at the impurity site. Then, using the non-equilibrium STLS spin polarization function in themore » non-equilibrium formalism of the iterative perturbation theory (IPT) of Yosida and Yamada, and Horvatic and Zlatic, we compute the spectral density, the current–voltage characteristics and the differential conductance as functions of the applied bias and the strength of on-site Hubbard interaction. We compare our spectral densities at zero bias with the results of numerical renormalization group (NRG) and depict the effects of the electron–electron interaction and electron-spin correlation at the impurity site on the aforementioned properties by comparing our numerical result with the order U{sup 2} IPT. Finally, we show that the obtained numerical results on the differential conductance have a quadratic universal scaling behavior and the resulting Kondo temperature shows an exponential behavior. -- Highlights: •We introduce for the first time the non-equilibrium method of STLS for Hubbard type models. •We determine the transport properties of SIAM using the non-equilibrium STLS method. •We compare our results with order-U2 IPT and NRG. •We show that non-equilibrium STLS, contrary to the GW and self-consistent RPA, produces the two Hubbard peaks in DOS. •We show that the method keeps the universal scaling behavior and correct exponential behavior of Kondo temperature.« less