We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with p x + ip y symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parametrized by the filling fraction and the coupling constant. It consists of three phases that are denoted weak-coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are "dressed" versions of the ground states of the strong phase by zero energy ͑Moore-Read͒ pairs and characterized by a topological order parameter. DOI: 10.1103/PhysRevB.79.180501 PACS number͑s͒: 74.20.Fg, 74.20.Mn, 74.20.Rp In 1957, Bardeen, Cooper, and Schrieffer 1 ͑BCS͒ published an epoch defining paper giving a microscopic explanation of the properties of superconducting metals at low temperatures. The model was based on a reduced Hamiltonian which describes the pairing interaction between conduction electrons. The original study of the BCS model was formulated in the grand-canonical ensemble and solved with a mean-field approximation. In 1963 Richardson 2 derived the exact solution of the reduced BCS Hamiltonian with s-wave symmetry in the canonical ensemble. This solution was largely unnoticed until its rediscovery in the theoretical studies of ultrasmall metallic grains in the 1990s, where it was employed to understand the crossover between the fluctuation dominated regime and the fully developed superconducting regime ͑for a review see Ref. 3͒. The exact solution for the s-wave BCS model is related to the Gaudin spin Hamiltonians, and their integrability can be understood in the general framework of the quantum inverse scattering method.4,5 These later developments allowed for an exact computation of various correlators, 4,6,7 and led to generalizations of the Richardson-Gaudin models with applications to condensed matter and nuclear physics. 3,8 In this Rapid Communication we analyze the twodimensional ͑2D͒ BCS model where the symmetry of the pairing interaction is p x + ip y ͑hereafter referred to as p + ip͒. The Hamiltonian of the model is ͑1͒where c k , c k † are destruction and creation operators of 2D spinless or polarized fermions with momentum k, m is their mass, and G is a dimensionless coupling constant which is positive for an attractive interaction. The p + ip model has attracted considerable attention due to the connection with the Moore-Read ͑MR͒ pfaffian state arising in the quantum Hall effect at filling fraction 5/2, 9 which has been proposed to support non-Abelian anyons allowing for topological quantum computation. 10,11 Motivated by these considerations, concre...
In this review we demonstrate how the algebraic Bethe ansatz is used for the calculation of the energy spectra and form factors (operator matrix elements in the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As examples we apply the theory to several models of current interest in the study of Bose-Einstein condensates, which have been successfully created using ultracold dilute atomic gases. The first model we introduce describes Josephson tunneling between two coupled Bose-Einstein condensates. It can be used not only for the study of tunneling between condensates of atomic gases, but for solid state Josephson junctions and coupled Cooper pair boxes. The theory is also applicable to models of atomic-molecular Bose-Einstein condensates, with two examples given and analysed. Additionally, these same two models are relevant to studies in quantum optics. Finally, we discuss the model of Bardeen, Cooper and Schrieffer in this framework, which is appropriate for systems of ultracold fermionic atomic gases, as well as being applicable for the description of superconducting correlations in metallic grains with nanoscale dimensions. In applying all of the above models to physical situations, the need for an exact analysis of small scale systems is established due to large quantum fluctuations which render mean-field approaches inaccurate. *
Using the well-known trigonometric six-vertex solution of the Yang-Baxter equation we derive an integrable pairing Hamiltonian with anyonic degrees of freedom. The exact algebraic Bethe ansatz solution is obtained using standard techniques. From this model we obtain several limiting models, including the pairing Hamiltonian with p + ip-wave symmetry. An in-depth study of the p + ip model is then undertaken, including a mean-field analysis, analytic and numerical solution of the Bethe ansatz equations, and an investigation of the topological properties of the ground-state wavefunction. Our main result is that the ground-state phase diagram of the p + ip model consists of three phases. There is the known boundary line with gapless excitations that occurs for vanishing chemical potential, separating the topologically trivial strong pairing phase and the topologically non-trivial weak pairing phase. We argue that a second boundary line exists separating the weak pairing phase from a topologically trivial weak coupling BCS phase, which includes the Fermi sea in the limit of zero coupling. The ground state on this second boundary line is the Moore-Read state.Combining these we have a single determinant expression
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced BCS model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.PACS numbers: 71.24+q, 74.20Fg Due to recent advances in nanotechnology it has become possible to fabricate and characterise individual metallic grains with dimensions as small as a few nanometers [1]. They are sufficiently small that the spacing, d, of the discrete energy levels can be determined. A particularly interesting question concerns whether superconductivity can occur in a grain with d comparable to ∆, the energy gap in a bulk system. If d ≪ ∆, the superconducting correlations are well-described by a mean-field solution to the reduced pairing Hamiltonian (equation (1) below) due to Bardeen, Cooper, and Schrieffer (BCS) in the grand canonical ensemble with a variable number of electrons. However, if d ∼ ∆ recent numerical calculations have shown that when the number of electrons is fixed (as in the canonical ensemble) the superconducting fluctuations become large and approximate treatments become unreliable [1,2]. Thus, exact calculations of physical quantities are highly desirable. It has only recently been appreciated that the exact eigenstates and spectrum of the BCS model were found in the 1960's by Richardson, in the context of nuclear physics [1,3]. The model has subsequently been found to have a rich mathematical structure: it is integrable (i.e., has a complete set of conserved operators) [4], has a connection to conformal field theory [5], and is related to Gaudin's inhomogeneous spin-1/2 models [6][7][8][9].In this Letter we show how the BCS model can be solved using the algebraic Bethe ansatz (ABA) method. This result can be deduced from the observation that the conserved operators obtained in [4] were also obtained in [9] via the ABA, but in another context. However, the approach we adopt here is slightly different from [9], which facilitates the solution of the quantum inverse problem [10][11][12] to explicitly evaluate form factors (i.e., one point functions) and correlation functions. This completes the agenda recently set out by Amico, Falci, and Fazio [8]. We also readily obtain known results for eigenstates, the spectrum, and conserved operators. Our treatment is also applicable to superconductivity in fermionic atom traps [14,15] and can also be extended to a solvable model for condensate fragmentation in boson systems [16].The Hamiltonian for the reduced BCS model consists of a kinetic energy term and an interaction term which describes the attraction between electrons in time rev...
In this work we investigate the quantum dynamics of a model for two singlemode Bose-Einstein condensates which are coupled via Josephson tunneling. Using direct numerical diagonalisation of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalised and self-trapping phases. We show that these behaviours are dependent on both the initial state of the system as well as regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail.
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