Chaos thresholds in finite Fermi systems

Silvestrov, P. G.

doi:10.1103/physreve.58.5629

Cited by 36 publications

(49 citation statements)

References 17 publications

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“…This was the approach taken in Refs. [6,8,[11][12][13][14], focusing on two quantities: The distribution of the energy level spacings and the inverse participation ratio (IPR) of the wave functions in Fock space. Both quantities can serve as a signature for chaotic behavior, the spacing distribution by comparing with Wigner's distribution [1] and the IPR by comparing with the golden rule (according to which the IPR is the mean spacing δ of the many-particle states divided by the mean decay rate Γ of a non-interacting manyparticle state [12]).…”

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confidence: 99%

Scaling at the Chaos Threshold for Interacting Electrons in a Quantum Dot

2000

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The chaotic mixing by random two-body interactions of many-electron Fock states in a confined geometry is investigated numerically and compared with analytical predictions. Two distinct regimes are found in the dependence of the inverse participation ratio in Fock space I on the dimensionless conductance of the quantum dot g and the excitation energy ε. In both regimes I ≫ 1, but only the small-g regime is described by the golden rule. The crossover region is characterized by a maximum in a scaling function that becomes more pronounced with increasing excitation energy. The scaling parameter that governs the transition is (ε/g) ln g.PACS numbers: 05.45.+b, 71.10.-w, 73.23.-b The highly excited atomic nucleus was the first example of a quantum chaotic system, although the interpretation of Wigner's distribution of level spacings [1] as a signature of quantum chaos came many years later, from the study of electron billiards [2]. While the spectral statistics of the nucleus and the billiard is basically the same, the origin of the chaotic behavior is entirely different [3]: In the billiard chaos appears in the single-particle spectrum as a result of boundary scattering, while in the nucleus chaos appears in the many-particle spectrum as a result of interactions.The study of the interaction-induced transition to chaos entered condensed matter physics with the realization that a semiconductor quantum dot could be seen as an artificial atom or compound nucleus [4]. A particularly influential paper by Altshuler, Gefen, Kamenev, and Levitov [5] studied the interaction-induced decay of a quasiparticle in a quantum dot and interpreted the broadening of the peaks in the single-particle density of states as a delocalization transition in Fock space. Different scenario's leading to a smooth rather than an abrupt transition from localized to extended states were considered later [6][7][8]. Recent computer simulations [9,10] also confirm the smooth crossover from localized to delocalized regime for quasiparticle decay.As emphasized by Altshuler et al. [5], the delocalized regime in the quasiparticle decay problem is not yet chaotic because the states do not extend uniformly over the Fock space. One may study the transition to chaos in the single-particle density of states, but theoretically it is easier to consider instead the mixing by interactions of arbitrary many-particle states. This was the approach taken in Refs. [6,8,[11][12][13][14], focusing on two quantities: The distribution of the energy level spacings and the inverse participation ratio (IPR) of the wave functions in Fock space. Both quantities can serve as a signature for chaotic behavior, the spacing distribution by comparing with Wigner's distribution [1] and the IPR by comparing with the golden rule (according to which the IPR is the mean spacing δ of the many-particle states divided by the mean decay rate Γ of a non-interacting manyparticle state [12]). Two fundamental questions in these investigations are: 1) What is the scaling parameter that governs th...

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confidence: 99%

Scaling at the Chaos Threshold for Interacting Electrons in a Quantum Dot

2000

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“…This means that, by expanding |φ i over the quantum register states, |φ i = α c α |α , the coefficient c α are randomly fluctuating and have amplitudes |c α | ∼ 1/ √ N . In this regime, the complexity of the eigenstates, characterized by the entropy S i = − α |c α | 2 log |c α | 2 , is maximal, that is S i ≈ n. We note that indeed the maximum entropy criterion S i ≈ n gives the ergodicity threshold J E ∼ δ [11]. Due to quantum ergodicity, for times t >> t E c ∼ 1/J the wave function |Ψ(t) is a random superposition of the quantum register states: |Ψ(t) = α a α (t)|α , where the coefficients a α (t) have amplitudes ∼ 1/ √ N and random phases.…”

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Dynamics of Entanglement in Quantum Computers with Imperfections

2003

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The dynamics of the pairwise entanglement in a qubit lattice in the presence of static imperfections exhibits different regimes. We show that there is a transition from a perturbative region, where the entanglement is stable against imperfections, to the ergodic regime, in which a pair of qubits becomes entangled with the rest of the lattice and the pairwise entanglement drops to zero. The transition is almost independent of the size of the quantum computer. We consider both the case of an initial maximally entangled and separable state. In this last case there is a broad crossover region in which the computer imperfections can be used to create a significant amount of pairwise entanglement.PACS numbers: 03.67. Lx, 03.67.Mn, 24.10.Cn Any practical implementation of a quantum computer will have to face errors, due to the inevitable coupling to the environment [1,2] or to device imperfections [3]. The effect of the environment is to introduce decoherence which sets the time scale over which quantum computation is no longer possible. The presence of static imperfections, although not leading to any decoherence, may be also decremental for the implementation of any quantum computational task. A small inaccuracy in the coupling constants induces errors in gating or an unwanted time evolution in the case in which the Hamiltonian cannot be switched exactly to zero. If the imperfection strength increases, new phenomena occur and above a certain threshold the core of the computer can even "melt" due to the setting in of chaotic behavior [3]. Previous investigations of the stability of quantum information processing in the presence of such effects mainly studied the fidelity of the quantum evolution as an indicator of the quality of the computation [3,4,5]. In particular, in Ref.[3] the fidelity was used to measure the stability of the quantum memory, that is of a state loaded on a quantum computer with imperfections. A more complete characterization of the stability of a quantum computation requires a deeper investigation of the state of the system. Fidelity is one of the ways to characterize it. In this Letter we discuss the behavior of entanglement on approaching the transition to quantum chaos. Entanglement is not only one of the most intriguing features predicted by the quantum theory but also a fundamental resource for quantum computation and communication [6]. Therefore studies of the stability of entanglement under decoherence and imperfection effects are, in our opinion, of interest.We model the quantum computer as a lattice of interacting spins (qubits) where, due to the unavoidable presence of imperfections, the spacing between the up and down states (external field) and the couplings between the qubits (exchange interactions) are both random. The entanglement properties in spin systems has recently attracted attention and we refer to [7] for a more detailed introduction. In this work we consider, for the first time, the effect of disorder in the couplings.We consider n qubits on a two-dimensional lattice, d...

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“…4 suggested an approximate mapping between the problem of Fock-space localization in a quantum dot and a single-particle localization on a Bethe lattice. (See also subsequent works [5][6][7][8][9][10][11][12][13][14][15][16] .) Later, these ideas were extended to explore the manybody-localization (MBL) in spatially extended systems with localized single-particle states and with short-range interaction 17,18 .…”

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confidence: 99%

Multifractality of wave functions on a Cayley tree: From root to leaves

2017

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We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments Pq = N |ψ| 2q (i) exhibit a multifractal scaling Pq ∝ N −τq with the volume (number of sites) N at N → ∞. The multifractality spectrum τq depends on the strength of disorder and on the parameter s characterizing the position of the observation point i on the lattice. Specifically, s = r/R, where r is the distance from the observation point to the root, and R is the "radius" of the lattice. We demonstrate that the exponents τq depend linearly on s and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the n-orbital model with n 1 that can be mapped onto a supersymmetric σ model. These results are supported by numerical simulations (exact diagonalization) of the conventional (n = 1) Anderson tight-binding model.

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Chaos thresholds in finite Fermi systems

Cited by 36 publications

References 17 publications

Scaling at the Chaos Threshold for Interacting Electrons in a Quantum Dot

Scaling at the Chaos Threshold for Interacting Electrons in a Quantum Dot

Dynamics of Entanglement in Quantum Computers with Imperfections

Multifractality of wave functions on a Cayley tree: From root to leaves

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