Dr. Prosen Replies

We present results for the zero and finite temperature Drude weight D(T ) and for the Meissner fraction of the attractive and the repulsive Hubbard model, as well as for the model with next nearest neighbor repulsion. They are based on Quantum Monte Carlo studies and on the Bethe ansatz. We show that the Drude weight is well defined as an extrapolation on the imaginary frequency axis, even for finite temperature. The temperature, filling, and system size dependence of D is obtained. We find counterexamples to a conjectured connection of dissipationless transport and integrability of lattice models.

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“…(9) in the extended Hubbard model. A similar conclusion has been drawn for a related model recently [29].…”

Section: Extended Hubbard Modelsupporting

confidence: 86%

Transport properties of one-dimensional Hubbard models

1999

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“…For example, our finding has strong implication for the stability of quantum computation with respect to static imperfections (e.g. uncontrolable residual interaction among qubits) [15]. In other words, Eq.…”

mentioning

confidence: 70%

General relation between quantum ergodicity and fidelity of quantum dynamics

2002

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General relation is derived which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time-scale ∝ δ −2 , δ ∼ strength of perturbation, whereas faster, typically gaussian decay on shorter time scale ∝ δ −1 is predicted for integrable, or generally non-ergodic dynamics. This surprising result is demonstrated in quantum Ising spin-1/2 chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of non-ergodic and non-integrable motion in thermodynamic limit.PACS number: 03.65.Yz, 75.10.Jm The quantum signatures of various types of classical motion, ranging from integrable to ergodic, mixing and chaotic, are still lively debated issues (see e.g. [1]). Most controversial is the absence of exponential sensitivity to variation of initial condition in quantum mechanics which prevents direct definition of quantum chaos [2]. However, there is an alternative concept which can be used in classical as well as in quantum mechanics [3]: One can study stability of motion with respect to small variation in the Hamiltonian. Clearly, in classical mechanics this concept, when applied to individual trajectories, is equivalent to sensitivity to initial conditions. Integrable systems with regular orbits are stable against small variation in the hamiltonian (the statement of KAM theorem), wheres for chaotic orbits varying the hamiltonian has similar effect as varying the initial condition: exponential divergence of two orbits for two nearby chaotic hamiltonians.The quantity of the central interest here is the fidelity of quantum motion. Consider a unitary operator U being either (i) a short-time propagator, or (ii) a Floquet map U =T exp(−i p 0 dτ H(τ )/ ) of (periodically timedependent) Hamiltonian H (H(τ + p) = H(τ )), or (iii) a quantum Poincaré map. The influence of a small perturbation to the unitary evolution, which is generated by a hermitean operator A, U δ = U exp(−iAδ), δ being a small parameter, is described by the overlap ψ δ (t)|ψ(t) measuring the Hilbert space distance between exact and perturbed time evolution from the same initial pure state |ψ(t) = U t |ψ , |ψ δ (t) = U t δ |ψ , where integer t is a discrete time (in units of period p) [4]. This defines the fidelitywhere the average is performed either over a fixed pure state . = ψ|.|ψ , or, if convenient, as a uniform average over all possible initial states . = (1/N ) tr (.), N being the Hilbert space dimension. The quantity F (t) has already raised considerable interest, though under different names and interpretations: First, it has been proposed by Peres [3] as a measure of stability of quantum motion. Second, it is the Loschmidt echo measuring the dynamical irreversibility of quantum phases, used e.g. in spin-echo experiments [5] where one is interested in the ov...

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“…[22,23]). Moreover, they have shifted into the focus of attention due to recent advances in the understanding of the role of the Lyapunov exponents for quantum-chaotic wave propagation [24,25,26,27,28,29,30]: It has been observed that under certain conditions the Lyapunov exponent can be extracted from the decay of the overlap of two wavefunctions which are propagated by two slightly different Hamiltonians (the socalled Loschmidt echo). Since the overlap is studied as a function of time, the distribution of the finite-time Lyapunov exponent is directly relevant for these investigations.…”

Section: Introductionmentioning

confidence: 99%

Statistics of finite-time Lyapunov exponents in a random time-dependent potential

Schomerus

Titov

2002

Phys. Rev. E

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The sensitivity of trajectories over finite time intervals t to perturbations of the initial conditions can be associated with a finite-time Lyapunov exponent λ, obtained from the elements Mij of the stability matrix M . For globally chaotic dynamics λ tends to a unique value (the usual Lyapunov exponent λ∞) as t is sent to infinity, but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity. We compute for a particle moving in a random time-dependent potential how the distribution function P (λ; t) approaches the limiting distribution P (λ; ∞) = δ(λ − λ∞). Our method also applies to the tail of the distribution, which determines the growth rates of positive moments of Mij . The results are also applicable to the problem of wave-function localization in a disordered one-dimensional potential. 42.25.Dd, 72.15.Rn

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Dr. Prosen Replies

Cited by 14 publications

References 0 publications

Transport properties of one-dimensional Hubbard models

Transport properties of one-dimensional Hubbard models

General relation between quantum ergodicity and fidelity of quantum dynamics

Statistics of finite-time Lyapunov exponents in a random time-dependent potential

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