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1 Probabilistic approach to Richardson equations W. V. Pogosov, Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia W. V. Pogosov, J. Phys.: Condens. Matter 24, (2012).

2 Motivation / Introduction General formulation Large-sample limit Small-sized systems Summary Outline

3 Motivation / Introduction -BCS theory plays a fundamental role -BCS Hamiltonian is exactly solvable through the Richardson approach -Richardson equations can be used to study small-sized systems (nano-scale), as well as delicate phenomena like BEC-BCS crossover

4 Richardson equations (also derivable from the algebraic Bethe-ansatz approach) Analytical solution in general case is an open problem Numerical methods are widely used Analytical methods are highly desirable -- system energy

5 General formulation BCS Hamiltonian fermions of two sorts

6 Richardson wave function N = 1: N = 2: where and so on… R. W. Richardson, Phys. Lett. 3, 277 (1963)

7 Richardson equations 3 pairs: N enters through the number of equations – nontrivial ! = Bethe ansatz equations* *J. von Delft and R. Poghossian, PRB (2002). T = 0

8 - Arbitrary filling of window (toy model of density-induced BEC-BCS crossover, related to systems with low carrier density) Configuration - Equally-spaced model: energy levels are distributed equidistantly within the Debye window - Interaction within the Debye window, between two cutoffs W. P., M. Combescot, and M. Crouzeix, PRB 2010; W. P., M. Combescot, Письма в ЖЭТФ 2010, M. Combescot and M. Crouzeix, PRL 2011.

9 Thermodynamical limit - density of states - interaction amplitude - dimensionless interaction constant -Debye window & Fermi energy of frozen electrons (lower cutoff) - number of states in the Debye window - number of pairs - filling factor of the window (1 / 2 in BCS) - volume

10 Electrostatic analogy * charges of free particles: charges of fixed particles: magnitude of the external force: * by Gaudin and Richardson Remarkable example of quantum-to-classical correspondence

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

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

13 Single-pair problem Partial-fraction decomposition - binomial coefficient

14 Problem: equilibrium is not stable. No confining potential. Saddle point. 1 2 Line 1: steepest descent of the energy, 1D integration instead of 2D However, the position of the saddle point is unknown! Z

15 - Since the probability is a meromorphic function, we can use various paths (Cauchy theorem) -Thus, we reconstruct an information about the saddle point using the nonlocal nature of S. Known result for N=1 (one-pair problem) -- nonanalytic function, typical for BCS topology of an integration path is of importance

16 Many pairs partition function thermodynamics similarities with: A. Zabrodin & P. Wiegmann (2006) – Dyson gas

17 Quantum-mechanical energy = minus logarithmic derivative of the classical partition function An interesting example of quantum-to-classical correspondence

18 Z has a form of the integral of Selberg type Conformal field theory, random matrices (Dyson gas), 2D gravitation, etc. Richardson equations are linked to Kniznik-Zamolodchikov equations appearing in conformal field theory Why Laughlin wave function? -- Chern-Simons-Witten theory describes topological order in fractional quantum Hall effect

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

20 Electron-hole duality Creation and destruction operators for holes

21 Large-sample limit Probability

22 Partition function (after the integration of probability)

23 Vandermonde matrix

24 Useful identities-I Pochhammer symbol (or falling factorial)

25 Transformation of the Vandermonde matrix

26 Useful identities-II

28 Full agreement with BCS-like treatment for the whole crossover from BEC to BCS. Pair binding energy as an energy scale. Any observables?

29 Coefficient A superfactorial

30 More formal derivation (through the Levi-Civita symbol)

31 Mean-square deviation (estimate of the error) - negligible

32 Factorization of probability

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

34 New variables r Energy by the saddle-point method

35 - Iterative integration by parts – tree-like procedure - Energy density as an expansion in pair density (virial expansion) - Third and fourth terms are exactly zero - Difficult to proceed with higher-order terms similar to our method with M. Combescot

36 Single-pair saddle point

37 Rescaling

38 In new variables Integrating by parts

39 Derivative in the integrand Substitute back

40 Derivative in the integrand

41 Energy delta couples with N

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

43 First magic cancellation:

44 Second magic cancellation Energy as a continued fraction?

45 Small-sized systems Condensation energy: II – nonanalytic dependence on v I – simply proportional to v, How to describe a crossover from superconducting to fluctuation- dominated regime? In collaboration with V. Misko & N. Lin

46 window filling as an extra degree of freedom e-h symmetry information about half-filling

47 Hamiltonian in terms of holes Ground state energy Creation and destruction operators for holes Functional equation

48 N is a discrete variable

49 Conjecture Consequences --- boundary condition in the space of discrete N

50 Solvable limits Regime I Regime II From analyticity to nonanalyticity

51 Analytics vs numerics vs g.c. BCS (a) N = 5 (b) N = 25 (c) N = 50

52 Pair binding energy – another energy scale? BCS theory fails at It is easy to see that For the thermodynamical limit * * W. V. Pogosov, M. Combescot, Письма в ЖЭТФ 92, 534 (2010); M. Crouzeix and M. Combescot, PRL 2011.

53 -A new method for the analytical evaluation of Richardson equations. Basic ingredients are the occupation probability and the partition function. -Energy in the thermodynamical limit. -Rich math structure as well as numerous links with other topics of modern theoretical physics. -Small-sized systems – analytical expression for the ground state energy -Another energy scale? Summary

54 Волновая функция БКШ Проекция на состояние с фиксированным N амплитуда вероятности того, что два состояния заняты = произведению амплитуд вероятностей для индивидуальных ф-й.

55 «пайроны»

56 Двухчастичная корреляционная функция: разложение: в разреженном пределе: обычные волновые функции пары. Обобщим на произвольный случай. «аномальная» корреляционная функция:

57 энергия основного состояния + квазичастицы + их взаимодействие

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