Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots

Grabsch, Aurélien; Majumdar, Satya N.; Texier, Christophe

doi:10.1007/s10955-017-1780-4

Cited by 17 publications

(31 citation statements)

References 56 publications

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“…refs. [59][60][61][62][63][64] for reviews of general methods). In particular, it was found to control the classical limit of the XXX −1/2 Heisenberg spin chain [65,66].…”

Section: Spin-chain Mappingmentioning

confidence: 99%

Fishnet four-point integrals: integrable representations and thermodynamic limits

Basso

Dixon

Kosower

et al. 2021

J. High Energ. Phys.

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We consider four-point integrals arising in the planar limit of the conformal “fishnet” theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-point equations are found to match known two-cut singular equations arising in matrix models, enabling us to obtain a concise parametric expression for the free-energy density in terms of complete elliptic integrals. Interestingly, the latter depends non-trivially on the fishnet aspect ratio and differs from a scaling formula due to Zamolodchikov for large periodic fishnets, suggesting a strong sensitivity to the boundary conditions. We also find an intriguing connection between the saddle-point equation and the equation describing the Frolov-Tseytlin spinning string in AdS3 × S1, in a generalized scaling combining the thermodynamic and short-distance limits.

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“…refs. [59][60][61][62][63][64] for reviews of general methods). In particular, it was found to control the classical limit of the XXX −1/2 Heisenberg spin chain [65,66].…”

Section: Spin-chain Mappingmentioning

confidence: 99%

Fishnet four-point integrals: integrable representations and thermodynamic limits

Basso

Dixon

Kosower

et al. 2021

J. High Energ. Phys.

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“…(iii) The distribution of the truncated sum K a=1 τ a of proper times, with K < N , was studied in Ref. [100].…”

Section: Wigner Time Delay τ Wmentioning

confidence: 99%

Wigner time delay and related concepts: Application to transport in coherent conductors

Texier

2016

Physica E: Low-dimensional Systems and Nanostructures

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The concepts of Wigner time delay and Wigner-Smith matrix allow to characterize temporal aspects of a quantum scattering process. The article reviews the statistical properties of the Wigner time delay for disordered systems ; the case of disorder in 1D with a chiral symmetry is discussed and the relation with exponential functionals of the Brownian motion underlined. Another approach for the analysis of time delay statistics is the random matrix approach, from which we review few results. As a pratical illustration, we briefly outline a theory of non-linear transport and AC transport developed by Büttiker and coworkers, where the concept of Wigner-Smith time delay matrix is a central piece allowing to describe screening properties in out-of-equilibrium coherent conductors.The published version has been updated.

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“…As mentioned earlier, for the α = 0 case (corresponding to cases (a)-(c)), the analytical scaling function is given in eq. (24). For the α = 1 (corresponding to case (d)), we invert the Laplace transform in eq.…”

mentioning

confidence: 99%

“…recently introduced and studied for large N under the name of truncated linear statistics (TLS) [23,24]. The ground-state energy in eq.…”

mentioning

confidence: 99%

Ground-state energy of noninteracting fermions with a random energy spectrum

Schawe¹,

Hartmann²,

Majumdar³

et al. 2018

EPL

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PACS 05.10.Ln -Monte Carlo methods PACS 75.10.Nr -Spin-glass and other random models PACS 05.20.-y -Classical statistical mechanicsAbstract -We derive analytically the full distribution of the ground-state energy of K noninteracting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of N i.i.d. random energy levels with distribution p(ε) (with ε ≥ 0), in the same spirit as the "Random Energy Model". We show that for each fixed K, the distribution PK,N (E0) of the ground-state energy E0 has a universal scaling form in the limit of large N . We compute this universal scaling function and show that it depends only on K and the exponent α characterizing the small ε behaviour of p(ε) ∼ ε α . We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as 10 −160 . We found asymptotically a very good agreement between analytical predictions and numerical results.The celebrated "Random Energy Model" (REM) of Derrida [1] has continued to play a central role in understanding different aspects of classical disordered systems, including spin-glasses, directed polymers in random media and many other systems. In the REM, one typically has N energy levels which are considered to be independent and identically distributed (i.i.d.) random variables, each drawn from a probability distribution function (PDF) p(ε). Typical observables of interest are the partition function, free energy, etc. The REM can also be useful as a toy model in quantum disordered systems. For example, let us consider a single quantum particle in a disordered medium with the Hamiltonianĥ. We will assume that the spectrum of the operatorĥ has a finite number of states N (for instance a quantum particle on a lattice of finite size and a random onsite potential, as in the Anderson model). In general, solving exactly the spectrum of such an operator is hard, for a generic random potential. One possible approximation, in the spirit of the REM in classical disordered systems, would be to consider the toy model where one replaces the spectrum of the actual Hamiltonian by N ordered i.i.d. energy levels ε 1 ≤ ε 2 ≤ · · · ≤ ε N each drawn from the common PDF p(ε). Without loss of generality, we will also assume that the Hamiltonianĥ has only positive eigenvalues. This would mean that, in the corresponding toy model, the PDF p(ε) is supported on [0, +∞). It is well known that, in a strongly disordered quantum system, where all single-particle eigenfunctions are localised in space, the energy levels can be approximated by i.i.d. random variables (see e.g. [2]). Therefore the p-1 arXiv:1808.09246v1 [cond-mat.dis-nn]

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Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots

Cited by 17 publications

References 56 publications

Fishnet four-point integrals: integrable representations and thermodynamic limits

Fishnet four-point integrals: integrable representations and thermodynamic limits

Wigner time delay and related concepts: Application to transport in coherent conductors

Ground-state energy of noninteracting fermions with a random energy spectrum

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