Fast algorithm for finding the eigenvalue distribution of very large matrices

Hams, Anthony; Raedt, de Hans

doi:10.1103/physreve.62.4365

Cited by 203 publications

(261 citation statements)

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“…Based on similar arguments as given in Ref. [19], we find that the degree of decoherence of S decreases as 1/ √ D E , where D E is the dimension of the Hilbert space of the environment if the state of the whole system is chosen uniformly at random from the unit hypersphere in the Hilbert space. In this paper, we denote states chosen uniformly at random from the unit hypersphere in the Hilbert space of the whole system by "X" and of the environment by "Y .…”

Section: Introductionmentioning

confidence: 97%

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Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

et al. 2013

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We consider the decoherence of a quantum system S coupled to a quantum environment E. For states chosen uniformly at random from the unit hypersphere in the Hilbert space of the closed system S + E we derive a scaling relationship for the sum of the off-diagonal elements of the reduced density matrix of S as a function of the size D E of the Hilbert space of E. This sum decreases as 1/ √ D E as long as D E 1. We test this scaling prediction by performing large-scale simulations which solve the time-dependent Schrödinger equation for a ring of spin-1/2 particles, four of them belonging to S and the others to E, and for this ring with small world bonds added in E and/or between S and E. The spin-1/2 particles experience nearest-neighbor interactions that are identical for the interactions within S and random for the interactions within E and between S and E, or that are all identical. Provided that the time evolution drives the whole system from the initial state toward a scaling state, a state which has similar properties as states belonging to the class of quantum states for which we derived the scaling relationship, the scaling prediction holds. We examine various interaction parameters and initial states for our model system to find whether or not the time evolution reaches the class of states that have the scaling property. For the homogeneous ring we find that the evolution for select initial states does not reach these scaling states. This conclusion is not modified if we add some homogeneous random connections. For a ring we find that some randomness in the interaction parameters is required so that most initial configurations are driven toward the scaling state. Furthermore, if the amount of randomness is small the time required to reach the scaling states may be very large. For the case of all random interactions in E the ring is driven toward the scaling state. Adding small world bonds between S and E with random interaction strengths may decrease the time required to reach the scaling state or may prevent the scaling state from being reached. For the latter case we show that increasing the complexity of the environment by adding extra connections within the environment suffices to observe the predicted scaling behavior.

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Section: Introductionmentioning

confidence: 97%

“…(A8), (A12), and (A23) of Ref. [19] are used. We introduce the following shorthand notation for the sum over the off-diagonal elements, …”

Section: Scaling Property Of σmentioning

confidence: 99%

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

et al. 2013

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“…We incorporate finite temperatures β = 0 by introducing |ψ N (β) = exp(−βH/2) |ψ N and rewriting the current autocorrelation function in Eq. (10) in the form [33,47,68,69,76,77] (skipping the index σ for clarity)…”

Section: Dynamical Quantum Typicality a Conceptmentioning

confidence: 99%

Finite-temperature charge transport in the one-dimensional Hubbard model

et al. 2015

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We study the charge conductivity of the one-dimensional repulsive Hubbard model at finite temperature using the method of dynamical quantum typicality, focusing at half filling. This numerical approach allows us to obtain current autocorrelation functions from systems with as many as 18 sites, way beyond the range of standard exact diagonalization. Our data clearly suggest that the charge Drude weight vanishes with a power law as a function of system size. The low-frequency dependence of the conductivity is consistent with a finite dc value and thus with diffusion, despite large finite-size effects. Furthermore, we consider the mass-imbalanced Hubbard model for which the charge Drude weight decays exponentially with system size, as expected for a non-integrable model. We analyze the conductivity and diffusion constant as a function of the mass imbalance and we observe that the conductivity of the lighter component decreases exponentially fast with the mass-imbalance ratio. While in the extreme limit of immobile heavy particles, the Falicov-Kimball model, there is an effective Anderson-localization mechanism leading to a vanishing conductivity of the lighter species, we resolve finite conductivities for an inverse mass ratio of η 0.3.

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“…The (fast) Fourier transform of f t then yields the LRDOS. By using an unconditionally stable algorithm [14,18], we do not have to solve the eigenvalue problem for H [21]. This is an important advantage over methods that compute the L(R)DOS by integrating the modulus of the electric field in the FBZ.…”

Section: 120401 (2006) P H Y S I C a L R E V I E W L E T T E R S mentioning

confidence: 98%