Abstract:

This thesis mainly deals with integrable quantum critical systems that exhibit peculiar features such as non-unitarity or non-compactness, through the technology of Bethe ansatz. These features arise in non-local statistical physics models such as percolation, but also in for example disordered systems. The manuscript both presents detailed studies of the continuum limit of ﬁnite-size lattice integrable models, and develops new techniques to study this correspondance. In a ﬁrst part we study in great detail the continuum limit of non-unitary (and sometimes non-compact) super spin chains with orthosymplectic symmetry which is shown to be supersphere sigma models, by computing their spectrum from ﬁeld theory, from the Bethe ansatz, and numerically. The non-unitarity allows for a spontenous symmetry breaking usually forbidden by the Mermin-Wagner theorem. The fact that they are marginal perturbations of a Logarithmic Conformal Field Theory is particularly investigated. We also establish a precise correspondance between the spectrum and intersecting loops conﬁgurations, and derive new critical exponents for fully-packed trails, as well as their multiplicative logarithmic corrections. During this study we developed a new method to compute the excitation spectrum of a critical quantum spin chain from the Bethe ansatz, together with their logarithmic corrections, that is also applicable in presence of so-called ’strings’, and that avoids Wiener-Hopf and Non-Linear Integral Equations. In a second part we address the problem of the behaviour of a spin chain in a magnetic ﬁeld, and show that one can derive convergent series for several physical quantities such as the acquired magnetization or the critical exponents, whose coeﬃcients can be eﬃciently and explicitely computed recursively using only algebraic manipulations. The structure of the recurrence relations permits to study generically the excitation spectrum content - moreover they are applicable even to some cases where the Bethe roots lie on a curve in the complex plane. It is our hope that the analytic continuation of such series might be helpful the study non-compact spin chains, for which we give some ﬂavour. Besides, we show that the ﬂuctuations within the arctic curve of the six-vertex model with domain-wall boundary conditions are captured by a Gaussian free ﬁeld with space-dependent coupling constant that can be computed from the free energy of the periodic XXZ spin chain with an imaginary twist and in a magnetic ﬁeld.

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