W. V. Pogosov
,
Institute for Theoretical and Applied Electrodynamics,
Russian Academy of Sciences, Moscow, Russia arXiv:1111.2907
, submitted to Journal of Physics: Condensed Matter

• Motivation/Introduction
• General formulation
• Ground state energy through the binomial sum
• Ground state energy through the Nörlund-
Rice integral
• Summary

• Excited states
• Finite systems
• Summary следующий вторник (?)

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.
2ω
=
Ω
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
)
Solution of Richardson equations in the dilute regime
Ground state energy
Alternative understanding of BCS theory
M. Crouzeix and M. Combescot, PRL 2012

• Richardson equations
(also derivable from Bethe ansatz)

БКШ
Энергия сверхпроводящего состояния:
2ω = Ω
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)

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

• Hamiltonian

• 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
Equilibrium is not stable
We reconstruct information about saddle point using “nonlocal” properties of S .
inverse problem, Radon transform, topology

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

• 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

• 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

New variables r

• Single-pair saddle point

• Rescaling

• In new variables
• Integrating by parts

• Derivative in the integrand
• Substitute back

• Derivative in the integrand

delta couples with N
• Energy

• 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