‘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, 075701 (2012).

•Motivation / Introduction

•General formulation

•Large-sample limit

•Small-sized systems

•Summary

OutlineOutline

Motivation / IntroductionMotivation / 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

• 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

General formulationGeneral formulation

BCS Hamiltonian

fermions of two sorts

• Richardson wave function

N = 1:

N = 2:

where

and so on…

• 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

- Arbitrary filling of “window”(toy model of density-induced BEC-BCS crossover, related to systems with low carrier density)

Configuration

- Equally-spaced model: energy levelsare 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.

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

• 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

‘Probabilistic’ approach

‘Probability’:

Analogies with the square of Laughlin wave function

factorizable

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”

Single-pair problem

Partial-fraction decomposition

- binomial coefficient

• Problem: equilibrium is not stable. No confining potential. Saddle point.

                

1 2Line 1: steepest descent of the “energy”,1D integration instead of 2D

However, the position of the saddle point is unknown!

Z

- 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

Many pairs

‘partition function’thermodynamics

similarities with: A. Zabrodin & P. Wiegmann (2006) – Dyson gas

               

Quantum-mechanical energy =

minus logarithmic derivative of the classical ‘partition function’

An interesting example of quantum-to-classical correspondence

• 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

• At the same time, it is an integral of Nörlund-Rice type

Canonical form:

• Electron-hole duality

Creation and destruction operators for holes

Large-sample limit Large-sample limit

‘Probability’

‘Partition function’(after the integration of ‘probability’)

Vandermonde matrix

• Useful identities-I

Pochhammer symbol (or “falling factorial”)

• Transformation of the Vandermonde matrix

• Useful identities-II

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

• Coefficient A

superfactorial

More formal derivation (through the Levi-Civita symbol)

• Mean-square deviation(estimate of the error)

- negligible

• Factorization of “probability”

• Single pair in the environment with bands of states removed

Similarities with Hubbard-Stratonovich transformation,sign-change problem

                

New variables r

Energy by the saddle-point method

- 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

• 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?

Small-sized systemsSmall-sized systems

Condensation energy:II – nonanalytic dependence on vI – simply proportional to v,How to describe a crossover fromsuperconducting to fluctuation-dominated regime?

In collaboration with V. Misko & N. Lin

• “window” filling as an extra degree of freedom

• e-h symmetry

information about half-filling

• Hamiltonian in terms of holes

Ground state energy

Creation and destruction operators for holes

Functional equation

• N is a discrete variable

• Conjecture

• Consequences

--- “boundary condition” in the space of discrete N

• Solvable limits

Regime I

Regime II

From analyticity to nonanalyticity

Analytics vs numerics vs g.c. BCS

(a) N = 5

(b) N = 25

(c) N = 50

• 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.

- 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?

SummarySummary

• Волновая функция БКШ

• Проекция на состояние с фиксированным N

амплитуда вероятности того, что два состояния заняты = произведению амплитуд вероятностей для индивидуальных ф-й.

• «пайроны»

• Двухчастичная корреляционная функция:

разложение:

в разреженном пределе: обычные волновые функции пары. Обобщим напроизвольный случай.

«аномальная» корреляционная функция:

энергия основного состояния + квазичастицы + их взаимодействие