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Seniority-zero geminal wavefunctions are known to capture bond-breaking correlation. Among this class of wavefunctions, Richardson–Gaudin states stand out as they are eigenvectors of a model Hamiltonian. This provides a clear physical picture, clean expressions for reduced density matrix (RDM) elements, and systematic improvement (with a complete set of eigenvectors). Known expressions for the RDM elements require the computation of rapidities, which are obtained by first solving for the so-called eigenvalue based variables (EBV) and then root-finding a Lagrange interpolation polynomial. In this paper, we obtain expressions for the RDM elements directly in terms of the EBV. The final expressions can be computed at the same cost as the rapidity expressions. Therefore, except, in particular, circumstances, it is entirely unnecessary to compute rapidities at all. The RDM elements require numerically inverting a matrix, and while this is usually undesirable, we demonstrate that it is stable, except when there is degeneracy in the single-particle energies. In such cases, a different construction would be required. [ABSTRACT FROM AUTHOR]

Eigenvectors of the reduced Bardeen–Cooper–Schrieffer (BCS) Hamiltonian, Richardson–Gaudin (RG) states, are used as a variational wavefunction ansatz for strongly correlated electronic systems. These states are geminal products whose coefficients are solutions of non-linear equations. Previous results showed an un-physical apparent avoided crossing in ground state dissociation curves for hydrogen chains. In this paper, it is shown that each seniority-zero state of the molecular Coulomb Hamiltonian corresponds directly to an RG state. However, the seniority-zero ground state does not correspond to the ground state of a reduced BCS Hamiltonian. The difficulty is in choosing the correct RG state. The systems studied showed a clear choice, and we expect that it should always be possible to reason physically which state to choose. [ABSTRACT FROM AUTHOR]

In this work we construct the eigenstates of the most general spin-1/2 Richardsonâ€"Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the S i x S j x , S i y S j y and S i z S j z terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r -matrices. The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U (1) symmetry so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided. [ABSTRACT FROM AUTHOR]

We study the decoherence of a spin in a quantum dot due to its hyperfine coupling to a randomly fluctuating bath of nuclear spins. The system is modeled by the central spin model with the spin bath initially being at infinite temperature. We calculate the spectrum and time evolution of the coherence factor using a Monte Carlo sampling of the exact eigenstates obtained via the algebraic Bethe ansatz. The exactness of the obtained eigenstates allows us to study the nonperturbative regime of weak magnetic fields in a full quantum mechanical treatment. In particular, we find a large nondecaying fraction in the zero-field limit. The crossover from strong to weak fields is similar to the decoherence starting from a pure initial bath state treated previously. We compare our results to a simple semiclassical picture [ Merkulov et al. Phys. Rev. B 65 205309 (2002)] and find surprisingly good agreement. Finally, we discuss the effect of weakly coupled spins and show that they will eventually lead to complete decoherence. [ABSTRACT FROM AUTHOR]

Using the algebraic Bethe ansatz in conjunction with a simple Monte Carlo sampling technique, we study the problem of the decoherence of a central spin coupled to a nuclear spin bath. We describe in detail the full crossover from strong to weak external magnetic field, a limit where a large nondecaying coherence factor is found. This feature is explained by Bose-Einstein-condensate-like physics which also allows us to argue that the corresponding zero frequency peak would not be broadened by statistical or ensemble averaging. [ABSTRACT FROM AUTHOR]

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnov's determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models. [ABSTRACT FROM AUTHOR]

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a different set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root-finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model, and the generalized Dicke model. [ABSTRACT FROM AUTHOR]

By instantaneously changing a global parameter in an extended quantum system, an initially equilibrated state will afterwards undergo a complex nonequilibrium unitary evolution whose description is extremely challenging. A nonperturbative method giving a controlled error in the long time limit remained highly desirable to understand general features of the quench induced quantum dynamics. In this paper we show how integrability (via the algebraic Bethe ansatz) gives one numerical access, in a nearly exact manner, to the dynamics resulting from a global interaction quench of an ensemble of fermions with pairing interactions (Richardson’s model). This possibility is deeply linked to the specific structure of this particular integrable model which gives simple expressions for the scalar product of eigenstates of two different Hamiltonians. We show how, despite the fact that a sudden quench can create excitations at any frequency, a drastic truncation of the Hilbert space can be carried out therefore allowing access to large systems. The small truncation error which results does not change with time and consequently the method grants access to a controlled description of the long time behavior which is a hard to reach limit with other numerical approaches. [ABSTRACT FROM AUTHOR]

We report on the calculation of the frequency-dependent conductivity of the quantum Hall stripes and bubble crystals that form in high Landau level. We use the replica and Gaussian variational methods (GVM) with a dynamical matrix obtained from the time-dependent Hartree-Fock approximation. In the stripe state, we go beyond the semiclassical approximation of the saddle point equations obtained with the GVM and demonstrate the existence of a quantum depinning transition as a function of filling factor. Below a critical filling factor, the pinned state is described by a replica symmetry breaking (RSB) solution that gives resonant peaks in the frequency-dependent conductivity in both directions, parallel and perpendicular to the stripes orientation. These peaks shift to zero frequency as the critical filling is approached. Above the critical filling, we find a depinned stripe state described by a partial replica symmetry breaking solution in which there is free sliding only along the stripe direction. The transition has a Kosterlitz-Thouless character and includes a jump in the low-frequency exponent of the dynamical conductivity. In the bubble crystals a semiclassical approximation yields a pinning peak frequency and resonance width that generally decrease with increasing filling factor, in accordance with recent microwave absorption experiments. [ABSTRACT FROM AUTHOR]

Using the exact eigenstates of the inhomogeneous Dicke model obtained by numerically solving the Bethe equations, we study the decay of bosonic excitations due to the coupling of the mode to an ensemble of two-level (spin 1/2) systems. We compare the quantum time evolution of the bosonic mode population with the mean-field description which, for a few bosons, agree up to a relatively long Ehrenfest time. We demonstrate that additional excitations lead to a dramatic shortening of the period of validity of the mean-field analysis. However, even in the limit where the number of bosons equal the number of spins, the initial instability remains adequately described by the mean-field approach leading to a finite, albeit short, Ehrenfest time. Through finite size analysis we also present indications that the mean-field approach could still provide an adequate description for thermodynamically large systems even at long times. However, for mesoscopic systems one cannot expect it to capture the behavior beyond the initial decay stage in the limit of an extremely large number of excitations. [ABSTRACT FROM AUTHOR]

In this work, we generalize the numerical approach to Gaudin models developed earlier by us [Faribault, El Araby, Sträter, and Gritsev, Phys. Rev. B 83, 235124 (2011)] to degenerate systems, showing that their treatment is surprisingly convenient from a numerical point of view. In fact, high degeneracies not only reduce the number of relevant states in the Hilbert space by a non-negligible fraction, they also allow us to write the relevant equations in the form of sparse matrix equations. Moreover, we introduce an inversion method based on a basis of barycentric polynomials that leads to a more stable and efficient root extraction, which most importantly avoids the necessity of working with arbitrary precision. As an example, we show the results of our procedure applied to the Richardson model on a square lattice. [ABSTRACT FROM AUTHOR]

The decohering environment of a quantum bit is often described by the coupling to a large bath of spins. The quantum bit itself can be seen as a spin S=1/2 which is commonly called the central spin. The resulting central spin model describes an important mechanism of decoherence. We provide mathematically rigorous bounds for a persisting magnetization of the central spin in this model with and without magnetic field. In particular, we show that there is a well-defined limit of infinite number of bath spins. Only if the fraction of very weakly coupled bath spins tends to 100% does no magnetization persist. [ABSTRACT FROM AUTHOR]