Anastasia Doikou
Home  Research  Brief CV  Publications  Teaching  Links
Main Research Interests
Integrable Models
Two dimensional integrable models have consistently attracted a great deal of research interest in recent years primarily because their study provides exact results, without recourse to perturbation theory or other approximation schemes as is the case when studying most physical problems. The main challenge when investigating a physical system is to develop exact, nonperturbative means in order to tackle physically relevant questions in the most indubitable manner possible. Integrability offers such an exact framework, and this is one of its great appeals. Beyond their physical significance, such models are also of great mathematical interest since their investigations entail the development of intriguing algebraic and geometric structures, whose study has increasingly grown over the last decades.
Factorizable ScatteringQuantum algebras
Integrable models display in general particlelike excitations, whose interactions are described by the bulk scattering Smatrix satisfying a collection of algebraic constraints, known as the YangBaxter equation. From the physical viewpoint, as advocated in the works of ZamolodchikovZamolodchikov and FaddeevTakhtakjan, the YangBaxter equation describes the factorization of multiparticle scattering, a unique feature displayed by 2d integrable systems. From a mathematical viewpoint Jimbo and Drinfeld argued independently, that the algebraic structures underlying the YangBaxter equation may be seen as deformations of the usual Lie algebras or their infinite dimensional extensions, the KacMoody algebras. Such deformed algebraic structures are endowed with a non trivial coproduct and are known as quantum groups or quantum algebras; they also turn out to have deep connections with the so called braid group.
Quantum spin chain
Quantum Spin ChainsBethe Ansatz
Many ideas in quantum integrability have their origins in statistical physics dating back in the seminal works of Bethe and Onsager, who solved the Heisenberg and Ising models respectively, as well as the celebrated solution of the XYZ model discovered by Baxter. A more recent approach on the resolution of the spectrum of 1d statistical models is the Bethe ansatz method, an elegant algebraic technique developed mainly by the St. Petersburg group, yielding also one of the main paths towards formulating the quantum algebras. In the frame of integrable quantum spin chains the first step towards the derivation of observables is the diagonalization of the corresponding Hamiltonian, by means of the quantum inverse scattering method, which is based on the algebra underlying the model.
Hamiltonian FormulationLax pair
In the context of classical integrable systems Lax representation of classical dynamical evolution equations is one key ingredient together with the associated notion of classical rmatrix as suggested by Sklyanin and SemenovTianShansky. It takes the generic form of an isospectral evolution equation. The spectrum of the Lax matrix or its extension or equivalently the invariant coefficients of the characteristic determinant provide automatically candidates to realize the hierarchy of Poissoncommuting Hamiltonians required by Liouville's theorem. Existence of the classical rmatrix guarantees Poisson commutativity of these natural dynamical quantities taken as generators of the algebra of classical conserved charges.
Past and present collaborators
Recent Selected Talks
Suggested Reading
 R. Baxter, "Exactly Solved Models in Statistical Mechanics", Academic Press (1982).
 T. Deguchi, "Introduction to solvable lattice models in statistical and mathematical physics", Institute of Physics Publishing, condmat/0304309.
 A. Doikou, S. Evangelisti, G. Feverati, N. Karaiskos, "Introduction to Quantum Integrability", Int. J. Mod. Phys. A25 (2010) 33073351, arXiv:0912.3350.
 A. Doikou, "Selected Topics in Classical Integrability", Int. J. Mod. Phys. A27 (2012) 1230003, arXiv:1110.4235.
 L.D. Faddeev, "How Algebraic Bethe Ansatz works for integrable models", LesHouches lectures,
hepth/9605187; "Algebraic Aspects of BetheAnsatz", Lectures at SUNY Stony Brook, hepth/9404013.
 L.D. Faddeev and L.A. Takhtajan, "Hamiltonian Methods in the Theory of Solitons',
(1987) SpringerVerlag.
 M. Jimbo and T. Miwa, "Algebraic analysis of solvable lattice models", CBMS vol 85, AMS (1995).
 A. Klu"mper, "Integrability of quantum chains: theory and applications to the spin1/2 XXZ chain", Lect. Notes Phys. 645 (2004) 349379, condmat/0502431.
 V.E. Korepin, G. Izergin, N.M. Bogoliubov, "Quantum Inverse Scattering Method,
Correlation Functions and Algebraic Bethe Ansatz", Cambridge University Press, (1993).
 B.M. McCoy, "The 1999 Heineman Prize Address Integrable models in statistical mechanics: The hidden field with unsolved problems", mathph/9904003; "The Baxter Revolution", J. Stat. Phys. 102 (2001) 375, condmat/0001256.
 R.I. Nepomechie, "A Spin Chain Primer", Int. J. Mod. Phys. B13 (1999) 29732986, hepth/9810032.
 H. Saleur, "Lectures on Non Perturbative Field Theory and Quantum Impurity Problems", condmat/9812110.
 E.K. Sklyanin, "Quantum Inverse Scattering Method. Selected Topics", hepth/9211111.
