Monomial integrals on the classical groups

Gorin, T.; López, Gustavo V.

doi:10.1063/1.2830520

Cited by 7 publications

(18 citation statements)

References 25 publications

(39 reference statements)

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“…Moments of this kind have attracted much attention in the past, and several approaches have been proposed to evaluate them [27,28,29,30,31,32,33,34,35,36]. Table 1 gives the values of the moments A U , B U , C U and D U for the unitary group U(N ) (see (3. Using the results of table 1, we obtain…”

Section: Random Matrix Theory Approachmentioning

confidence: 99%

Equilibration properties of small quantum systems: further examples

Luck¹

2017

J. Phys. A: Math. Theor.

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It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace T of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension N of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of T can be investigated by analytical means. We first study the statistics of T when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U(N ) or O(N ). We then investigate a quantum spin S in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -such as the tilt angle or the electric field -through some scaling regime induces a continuous transition from good to bad equilibration properties.

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Section: Random Matrix Theory Approachmentioning

confidence: 99%

Equilibration properties of small quantum systems: further examples

Luck¹

2017

J. Phys. A: Math. Theor.

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“…We may write these ensemble averages as integrals over the flat space of all matrix elements, implementing the orthonormality conditions on the column vectors by additional delta functions. Originally, this idea is due to Ullah [23,24] and more recently it had been used to calculate group averages of monomials [25,16].…”

Section: Definitions and Notationmentioning

confidence: 99%

“…Averages over monomials are considered in [14,15,16]. The statistical properties of eigenvalues of M×M sub-matrices (M < N) in [17,18].…”

Section: Introductionmentioning

confidence: 99%

Joint probability distributions for projection probabilities of random orthonormal states

Alonso¹,

Gorin²

2016

J. Phys. A: Math. Theor.

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Abstract. A finite dimensional quantum system for which the quantum chaos conjecture applies has eigenstates, which show the same statistical properties than the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.

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“…without local terms) [1] it was possible to obtain an exact analytical solution, here we have to resort to a linear response approximation [12]. Even then, the analytical solution is quite involved, as it requires the calculation of a large number of monomial integrals over the unitary group (for simplicity, we will assume the absence of any symmetries, including anti-unitary ones) [13,14].…”

Section: Introductionmentioning

confidence: 99%

Transition from non-Markovian to Markovian dynamics for generic environments

2016

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Using random matrices, we study the reduced dynamics of a two-level system interacting with a generic environment. In the weak-coupling limit, the result can be obtained directly from known results for purity decay, and result in Markovian dynamics. We then focus on the case of strong coupling, when the dynamics is known to be non-Markovian. In this regime, the coupling dominates over the local parts of the Hamiltonian, and thus we treat the latter as a perturbation of the former. With the help of a linear response approximation, this allows us to obtain an analytical description of the reduced dynamics. Finally, we find a transition from non-Markovian to Markovian dynamics at a point where the coupling and the local Hamiltonian are comparable in size.

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Monomial integrals on the classical groups

Cited by 7 publications

References 25 publications

Equilibration properties of small quantum systems: further examples

Equilibration properties of small quantum systems: further examples

Joint probability distributions for projection probabilities of random orthonormal states

Transition from non-Markovian to Markovian dynamics for generic environments

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