Particle-hole duality in Richardson solution of pairing Hamiltonian W. V. Pogosov, (1) N.L. Dukhov All-Russia Research Institute of Automatics, Moscow; (2) Institute of Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow; (3) Moscow Institute for Physics and Technology, Dolgoprudny, Moscow region Russian Quantum Center,

1. Motivation / Introduction 2. Particle-hole duality 3. Crossover from BCS limit to fluctuation- dominated regime 4. Integrability and new equations 5. Other applications 6. Results Outline

1. Motivation / Introduction Theory: universal Hamiltonian (disordered metallic grains) In Cooper channel: similarities with the reduced Hamiltonian of BCS theory of superconductivity n +, n - -- time-reversed states (plane waves in macroscopically large samples in the clean limit) Minimal model: one-particle energy levels are distributed equidistantly (equally-spaced model) «Debye window»: - half filled in usual BCS theory Superconductivity at nanoscale: recent experimental progress

Some possible solutions -Mean-field approximation. However, it can give incorrect results -Richardson approach. (fixed particle number + exact solution of Schrödinger equation) M-pair wave function (R. W. Richardson, 1963): Singly-populated states to be excluded from the sum

Richardson equations: --- the equation for E j - energy of the system = Bethe ansatz equations (quasiclassical limit of XXX model, P. P. Kulish and E. K. Sklyanin, 1982); Pair operators – Pauli operators (two-level systems)

An example (M = 3, L = 6) Multiple solutions { E j }: different Hamiltonian eigenstates

g = 0 limit: all rapidities populate one-particle energy levels. Noninteracting electrons. Pictorial representation. Empty and occupied states. Simple structure of the wave function. Ground stateAn excited state g increases: rapidities decouple from one-particle levels and start moving in a complex plane. Complicated transformations. Singularities. Complicated structure of the wave function. Ground stateAn excited state

Difficulties with Richardson equations Resolution is a formidable task Very few explicit results Difficulties with computing correlation functions No appropriate finite-T formalism New tools / approaches are highly desirable ! Interdisciplinary importance: Connections with various other exactly solvable models (especially, Gaudin-like models), conformal field theories, random matrix theories etc.

Hole picture Hole creation/destruction operators Hole-pair and hole-number operators Hamiltonian in terms of hole operators 2. Particle-hole duality

Hamiltonian in the hole representation is the same as in the particle representation up to the replacement of some scalars Bethe ansatz techniques are also applicable for this hole Hamiltonian The same state can be expressed through Bethe vectors both in the electron and hole pictures, but sets of Bethe roots in these two pictures are different W. V. Pogosov, J. Phys.: Condens. Matter 24, (2012): Probabilistic approach

Particle picture: Hole picture:

Various hidden relations for Bethe roots follow from this observation. Such relations can hardly be obtained staying on the level of Bethe equations only, but they can be helpful W. V. Pogosov, J. Phys.: Condens. Matter 24, (2012). the simplest situation: …various other relations can be derived

Interaction energy: II – non-analytical function of (non-perturbative result); extensive quantity I – proportional to ; Intensive quantity How a crossover from BCS condensate (II) to fluctuation-dominated regime (I) can be explicitly described? (emergence of macroscopic properties) Richardson equations at half-filling: solvable limits -dimensionless interaction constant d is an interlevel spacing in the Debye window 3. Crossover from BCS limit to fluctuation regime M Minimal model: equally-spaced distribution of levels

Trick: let us treat filling as an additional degree of freedom Functional equation for the ground state energy: … and use a conjecture for it: This simple dependence on the additional degree of freedom M appears in all exactly solvable limits, while the dependencies on g change drastically – from analytical function to nonanalytical function We extract a relevant contribution

--- boundary condition in the space of discrete M Consequence: is digamma function

Interaction energy per pair: Comparison with numerics and BCS theory Solid lines – our result, Dashed lines – numerics, Dotted lines – BCS approximation. W. V. Pogosov, N. S. Lin, V. R. Misko, Eur. Phys. J. B (2013) M = 50 M = 25 M = 5

Under what conditions BCS theory becomes inaccurate (as g or system size decrease)? -- when it is not allowed to replace the summation by an integration: -- single-pair binding energy, as found by Cooper A conventional energy scale is a superconducting gap: Cooper pair binding energy is another energy scale

However, by a direct substitution it is readily seen that is a single-pair binding energy Expansion from limit I (very weak coupling): Main result: BCS fails to describe correctly an interaction energy, when --- puzzling quantity with unknown physical meaning W. V. Pogosov, Solid State Comm. (2015). Possible experimental investigation: spin magnetization Conclusion: There are probably at least two criteria: 1. for the gap in the spectrum (relatively easy to probe in experiments) 2. for the ground state energy itself (not so easy to probe, but still possible)

Electron quantum invariants (commute with the Hamiltonian) Their eigenvalues 4. Integrability and new equations each operator is related to a one-particle energy level all rapidities enter symmetrically Can be derived from the algebraic Bethe ansatz

Relation between eigenvalues of electron and hole quantum invariants A. Faribault and D. Schuricht (2012): Richardson-Gaudin models H. Tschirhart and A. Faribault (2014): XXX model L. Bork and W. Pogosov (2014): Russian doll BCS model P. Claeys, S. De Baerdemacker, M. Van Raemdonck, and D. Van Neck (2015) : XXZ model Each equation is now related to one-particle energy and is symmetrical with respect to rapidities

Example (M = 1, L = 3) Particle representation:Hole representation: Connection:

Equally-spaced model at half-filling (M = L - M) Dual (alpha and beta are different) and self-dual (alpha = beta) solutions. This classification holds along the whole crossover from g = 0 to g = infinity. Simple rules to understand if a given solution in g = 0 limit is dual or self-dual and to establish a partner for a dual solution. These new equations can be used instead of the initial Richardson equations. There are some advantages. Ground state is always self-dual. Mismatches between singularities for dual solutions. Applies also for any other distribution, mirror-symmetrical with respect to the Fermi level L. V. Bork and W. V. Pogosov, submitted to Nucl. Phys. B (2015). Trick: changing an order of summation in new equations (reflection of bare kinetic energies does not change a character of a distribution) Richardson equations in both pictures are the same up to the shift of rapidities

Examples (numerics)

Other applications G. Gorohovsky and E. Bettelheim, J. Phys. A: Math. Theor. (2014) Coherence Factors Beyond the BCS Expressions -- A Derivation Highly excited states in the thermodynamical limit, when BCS fails, fully analytical calculations of matrix elements J. Links, I. Marquette, and A. Moghaddam, arXiv: Exact solution of the p+ip Hamiltonian revisited: duality relations in the hole-pair picture Internal structure of the wave function: assymetry between particle-pair and hole-pair pictures A. Faribault and D. Schuricht, J. Phys. A: Math. Theor. (2012) On the determinant representations of Gaudin models scalar products and form factors Numerical computations of scalar products and form factors.

H. Tschirhart, A. Faribault, J. Phys. A: Math. Theor. (2014) Algebraic Bethe Ansätze and eigenvalue-based determinants for Dicke- Jaynes-Cummings-Gaudin quantum integrable models some analogy of particle-hole duality Inhomogeneous Dicke model. Superconducting metamaterials. Qubits. Bethe equations, very similar to Richardson equations:

Results Particle-hole duality as a new tool to deal with some exactly solvable models Superconducting systems along the crossover from BCS limit to fluctuation-dominated regime New equations instead of Richardson equations Insights on the correspondence between different solutions of the same system of Richardson equations (different states of the spectrum)

Algebraic Bethe ansatz R-matrix Yang-Baxter equation Monodromy matrix

Transfer matrix It satisfies an equation