arXiv subjects

Alexandre Faribault

Publications and source records attributed to Alexandre Faribault.

24 records · Page 2Linked to original sources

arXiv2012

Bethe Ansatz and Ordinary Differential Equation Correspondence for Degenerate Gaudin Models

In this work, we generalize the numerical approach to Gaudin models developed earlier by us 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 to write the relevant equations in the form of sparse matrix equations. Moreover, we introduce a new inversion method based on a basis of barycentric polynomials which 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.

cond-mat.mes-hall↗

arXiv2011

Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new 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 generalized Dicke model.

cond-mat.mes-hall↗

arXiv2010

Dynamical correlation functions of the mesoscopic pairing model

We study the dynamical correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. We use the Algebraic Bethe Ansatz formalism, which gives exact expressions for the form factors of the most important observables. By summing these form factors over a relevant set of states, we obtain very precise estimates of the correlation functions, as confirmed by global sum-rules (saturation above 99% in all cases considered). Unlike the case of many other Bethe Ansatz solvable theories, simple two-particle states are sufficient to achieve such saturations, even in the thermodynamic limit. We provide explicit results at half-filling, and discuss their finite-size scaling behavior.

cond-mat.str-el↗

arXiv2009

Bethe Ansatz approach to quench dynamics in the Richardson model

By instantaneously changing a global parameter in an extended quantum system, an initially equilibrated state will afterwards undergo a complex non-equilibrium unitary evolution whose description is extremely challenging. A non-perturbative 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.

cond-mat.str-el↗

arXiv2008

Quantum quenches from integrability: the fermionic pairing model

Understanding the non-equilibrium dynamics of extended quantum systems after the trigger of a sudden, global perturbation (quench) represents a daunting challenge, especially in the presence of interactions. The main difficulties stem from both the vanishing time scale of the quench event, which can thus create arbitrarily high energy modes, and its non-local nature, which curtails the utility of local excitation bases. We here show that nonperturbative methods based on integrability can prove sufficiently powerful to completely characterize quantum quenches: we illustrate this using a model of fermions with pairing interactions (Richardson's model). The effects of simple (and multiple) quenches on the dynamics of various important observables are discussed. Many of the features we find are expected to be universal to all kinds of quench situations in atomic physics and condensed matter.

cond-mat.str-el↗

arXiv2007

Exact mesoscopic correlation functions of the pairing model

We study the static correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. Making use of the Algebraic Bethe Ansatz formalism, we obtain exact expressions which are easily evaluated numerically for any value of the pairing strength up to large numbers of particles. We provide explicit results at half-filling and extensively discuss their finite-size scaling behavior.

cond-mat.mes-hall↗