Superconductivity in Ultrasmall Grains: Introduction to Richardson’s Exact Solution

Delft, Jan von; Braun, Fabian

doi:10.1007/978-94-011-4327-1_24

Cited by 10 publications

(17 citation statements)

References 21 publications

(36 reference statements)

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“…A simpler proof, also due to Richardson [119], may be found in Refs. [40,120] and in App. B; its strategy is to verify that (Ĥ U − E n )|Ψ n = 0 by simply commutingĤ U past the B † ν operators in (83).…”

Section: General Eigenstatesmentioning

confidence: 98%

“…A series of more sophisticated canonical approaches [33][34][35][36][37][38][39][40][41] (summarized in Sec. 5.2) confirmed the parity dependence of pairing correlations, but established that the abrupt vanishing of pairing correlations at d BCS p/2 is an artifact of g.c.…”

Section: Superconductivity: Crossover From the Bulk To The Limit Of Amentioning

confidence: 99%

“…5.2) confirmed the parity dependence of pairing correlations, but established that the abrupt vanishing of pairing correlations at d BCS p/2 is an artifact of g.c. treatments: pairing correlations do persist, in the form of so-called fluctuations, to arbitrarily large level spacings [28], and the crossover between the bulk superconducting (SC) regime (d ≪∆) and the fluctuation-dominated (FD) regime (d ≫∆) is completely smooth [36][37][38][39][40]. Nevertheless, these two regimes are qualitatively very different [35][36][37][38][39][40]: the condensation energy, e.g., is an extensive function of volume in the former and almost intensive in the latter, and pairing correlations are quite strongly localized around the Fermi energy ε F , or more spread out in energy, respectively.…”

Section: Superconductivity: Crossover From the Bulk To The Limit Of Amentioning

confidence: 99%

“…We begin in App. B.1 by deriving in pedagogical detail, following [40], some of the key results of Richardson's exact solution of the discrete BCS model of Sec. 4 …”

Section: B Richardson's Exact Solution Of Discrete Bcs Modelmentioning

confidence: 99%

“…Describing this crossover constituted a conceptual challenge, since the standard grand-canonical mean-field BCS treatment of pairing correlations [21][22][23][24][25][26][27][28] breaks down for d ∆ . This challenge elicited a series of increasingly sophisticated canonical treatments of pairing correlations [33][34][35][36][37][38][39][40][41], based on a simple reduced BCS-Hamiltonian for discrete energy levels, which showed that the crossover is completely smooth, but, interestingly, depends on the parity of the number of electrons on the grain, as pointed out by von Delft et al [21]. Very recently, the main conclusions of these works were confirmed [39] using an exact solution of the reduced BCS model, discovered by Richardson in the context of nuclear physics in the 1960s [46][47][48][49][50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

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Spectroscopy of discrete energy levels in ultrasmall metallic grains

2001

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We review recent experimental and theoretical work on ultrasmall metallic grains, i.e. grains sufficiently small that the conduction electron energy spectrum becomes discrete. The discrete excitation spectrum of an individual grain can be measured by the technique of single-electron tunneling spectroscopy: the spectrum is extracted from the current-voltage characteristics of a single-electron transistor containing the grain as central island. We review experiments studying the influence on the discrete spectrum of superconductivity, nonequilibrium excitations, spin-orbit scattering and ferromagnetism. We also review the theoretical descriptions of these phenomena in ultrasmall grains, which require modifications or extensions of the standard bulk theories to include the effects of level discreteness.

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Section: General Eigenstatesmentioning

confidence: 98%

Section: Superconductivity: Crossover From the Bulk To The Limit Of Amentioning

confidence: 99%

Section: Superconductivity: Crossover From the Bulk To The Limit Of Amentioning

confidence: 99%

“…We begin in App. B.1 by deriving in pedagogical detail, following [40], some of the key results of Richardson's exact solution of the discrete BCS model of Sec. 4 …”

Section: B Richardson's Exact Solution Of Discrete Bcs Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectroscopy of discrete energy levels in ultrasmall metallic grains

2001

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Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Piątek

Nazmitdinov

Puente

et al. 2022

J. High Energ. Phys.

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Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, $$ \mathcal{N} $$ N = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.

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Arrays of Cooper pair boxes coupled to a superconducting reservoir: superradiance and revival

Rodrigues

Györffy

Spiller

2004

J. Phys.: Condens. Matter

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We consider an array of l b Cooper Pair Boxes, each of which is coupled to a superconducting reservoir by a capacitive tunnel junction. We discuss two effects that probe not just the quantum nature of the islands, but also of the superconducting reservoir coupled to them. These are analogues to the well-known quantum optical effects 'superradiance,' and 'revival.' When revival is extended to multiple systems, we find that 'entanglement revival' can also be observed. In order to study the above effects, we utilise a highly simplified model for these systems in which all the singleelectron energy eigenvalues are set to be the same (the strong coupling limit), as are the charging energies of the Cooper Pair Boxes, allowing the whole system to be represented by two coupled quantum spins, one finite, which represents the array of boxes, and one representing the reservoir, which we consider in the limit of infinite size. Although this simplification is drastic, the model retains the main features necessary to capture the phenomena of interest. Given the progress in superconducting box experiments over recent years, it is possible that experiments to investigate both of these interesting quantum coherent phenomena could be performed in the forseeable future. § Current address:

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Superconductivity in Ultrasmall Grains: Introduction to Richardson’s Exact Solution

Cited by 10 publications

References 21 publications

Spectroscopy of discrete energy levels in ultrasmall metallic grains

Spectroscopy of discrete energy levels in ultrasmall metallic grains

Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Arrays of Cooper pair boxes coupled to a superconducting reservoir: superradiance and revival

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