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Презентация была опубликована 3 года назад пользователемТамара Лызлова

1 Probabilistic approach to Richardson equations Part I W. V. Pogosov, Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia arXiv: arXiv: , submitted to Journal of Physics: Condensed Matter

2 Part I Motivation/Introduction General formulation Ground state energy through the binomial sum Ground state energy through the Nörlund- Rice integral Summary Outline

3 Part II Excited states Finite systems Summary следующий вторник (?)

4 Мотивация/Введение Проблема перехода БЭК-БКШ (ультрахолодные газы, ВТСП) -Предел локальных пар поверхность Ферми размыта - Предел БКШ плотность пар очень велика, есть поверхность Ферми - Как описать переход? Проблема обсуждалась еще Шриффером в связи с переходом от двухчастичной модели Купера к многочастичной модели БКШ. Ω c 2ω = Ω ? переход

5 Gedanken experiment: let us add more and more pairs to the potential layer, until it becomes half-filled ( toy model of density-induced BEC-BCS crossover ) Ground state energy Solution of Richardson equations in the dilute regime Alternative understanding of BCS theory c 2ω = Ω M. Crouzeix and M. Combescot, PRL 2012

6 Richardson equations (also derivable from Bethe ansatz)

7 БКШ Энергия сверхпроводящего состояния: Сверхпроводящая щель: Утверждение Шриффера: пары перекрыты так сильно, что концепция изолированной пары не имеет смысла (has a little meaning) - вводятся «виртуальные» пары с энергией = щели - сконцентрированы вблизи поверхности Ферми - отличаются от «сверхтекучих» пар из волновой функции БКШ - их число гораздо меньше числа пар в слое - вводятся не ab initio, а для понимания результата, «руками» В настоящее время под куперовскими парами в БКШ обычно понимаются как раз виртуальные пары (см., например, Walecka- Fetter) c 2ω = Ω !

8 Мотивация: - Установить возможную связь между куперовскими парами в обоих пределах - Попытаться описать переход, выходя за рамки обобщенной теории БКШ Альтернативное представление : c 2ω = Ω

9 General formulation Hamiltonian c 2ω = Ω

10 Thermodynamical limit

11 Richardson equations 3 pairs: N enters through the number of equations = Bethe ansatz equations

12 Electrostatic analogy charges of free particles: charges of fixed particles: magnitude of external force:

14 Probabilistic approach Probability: Analogies with the square of Laughlin wave function

15 Saddle point is very sharp! One can find a position of the saddle point without solving Richardson equations explicitly, but using integration Can be extended to the case of many variables

16 Single pair problem Partial-fraction decomposition

17 1 2 3 inverse problem, Radon transform, topology We reconstruct information about saddle point using nonlocal properties of S. Equilibrium is not stable

18 Large parameter is N partition function! thermodynamics similarities with: A. Zabrodin & P. Wiegmann (2006) – Dyson gas 1 2 3

19 Z has a form of a Coulomb integral (or integral of Selberg type). Conformal field theory, random matrices (Dyson gas), 2D gravitation, etc. Richardson equations is a special case of Kniznik- Zamolodchikov equations appearing in conformal field theory Why Laughlin wave function? -- Chern-Simons-Witten theory describes topological order in fraction quantum Hall effect CFT/AdS correspondence Quantum inverse scattering method

20 At the same time, it is an integral of Nörlund-Rice type Canonical form:

21 Electron-hole duality Creation and destruction operators for holes

22 Ground state energy through a binomial sum

23 Useful identities-I Pochhammer symbol

24 Useful identities-II

25 Vandermonde matrix

26 Transformation of Vandermonde matrix

29 More formal way of writing

30 Qualitative understanding

31 Factorization of probability Single pair in the environment with bands of states removed Similarities with Hubbard-Stratonovich transformation, sign-change problem

32 Ground state energy through Nörlund-Rice integral New variables r

34 Single-pair saddle point

35 Rescaling

36 In new variables Integrating by parts

37 Derivative in the integrand Substitute back

38 Derivative in the integrand

39 Energy delta couples with N

40 How to prove that remaining terms are underextensive? We keep integrating by parts

41 First magic cancellation:

42 Second magic cancellation Energy as a continued fraction?

43 -A new method for the analytical evaluation of Richardson equations in the thermodynamical limit. We introduce a probability of the system of charges to occupy certain states in a configurational space and a partition function (Coulomb integral), from which energy can be found. -For the model with constant density of states, we calculated a ground state energy, which is given by a single expression all over from the dilute to dense regime of pairs. -The method is rather generic and can be applied to other pairing Hamiltonians, which have an electrostatic analogy (Bethe anzats?). -Rich math structure as well as numerous links with other topics of modern theoretical physics Summary

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