Probabilistic approach to Richardson equations Part I W. V. Pogosov, Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, - презентация

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:

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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

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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

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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