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
Part I Motivation/Introduction General formulation Ground state energy through the binomial sum Ground state energy through the Nörlund- Rice integral Summary Outline
Part II Excited states Finite systems Summary следующий вторник (?)
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
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
Richardson equations (also derivable from Bethe ansatz)
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 !
Мотивация: - Установить возможную связь между куперовскими парами в обоих пределах - Попытаться описать переход, выходя за рамки обобщенной теории БКШ Альтернативное представление : c 2ω = Ω
General formulation Hamiltonian c 2ω = Ω
Thermodynamical limit
Richardson equations 3 pairs: N enters through the number of equations = Bethe ansatz equations
Electrostatic analogy charges of free particles: charges of fixed particles: magnitude of external force:
Probabilistic approach Probability: Analogies with the square of Laughlin wave function
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
Single pair problem Partial-fraction decomposition
1 2 3 inverse problem, Radon transform, topology We reconstruct information about saddle point using nonlocal properties of S. Equilibrium is not stable
Large parameter is N partition function! thermodynamics similarities with: A. Zabrodin & P. Wiegmann (2006) – Dyson gas 1 2 3
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
At the same time, it is an integral of Nörlund-Rice type Canonical form:
Electron-hole duality Creation and destruction operators for holes
Ground state energy through a binomial sum
Useful identities-I Pochhammer symbol
Useful identities-II
Vandermonde matrix
Transformation of Vandermonde matrix
More formal way of writing
Qualitative understanding
Factorization of probability Single pair in the environment with bands of states removed Similarities with Hubbard-Stratonovich transformation, sign-change problem
Ground state energy through Nörlund-Rice integral New variables r
Single-pair saddle point
Rescaling
In new variables Integrating by parts
Derivative in the integrand Substitute back
Derivative in the integrand
Energy delta couples with N
How to prove that remaining terms are underextensive? We keep integrating by parts
First magic cancellation:
Second magic cancellation Energy as a continued fraction?
-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