This thesis is devoted to the study of ghost theories out of the critical point, in two dimensions. The first chapter offers a bird's eye view of the most important applications of ghosts to condensed matter physics. After a brief exposition of the basic (and less basic) facts concerning ghosts at the critical point, an outline of the non-perturbative methods, used in Part I and Part II, is furnished. Essentially, they rely on the so-called integrable approach, which is based on the possibility of describing all the states of an integrable quantum field theory in terms of pseudo-particles in a Hilbert space. The scattering properties of such excitations are encoded into a matrix (the S-matrix), which can be exactly determined by imposing a set of stringent constraints, and which allows to specify completely the particle content of the theory (masses, multiplicities, bound states) [38, 39]. The knowledge of the scattering amplitude, then, permits to extract all the thermodynamic quantities of the system (free energy) by means of the Thermodynamic Bethe ansatz (TBA) technique [40] and, at least in principle, to determine the off-critical correlation functions of the local operators of the theory, thanks to the so-called Form Factor bootstrap approach [41-43]. Part I contains the simplest examples of off-critical ghost theories, namely the massive versions of the conformal free bosonic and fermionic ones [44]. Despite their non-interacting nature, still there are non-local sectors of the models, which exhibit a highly interacting behavior. Correlation functions of operators belonging to these sectors are computed exactly and a comparison with the massive ordinary counterparts is performed. Afterwards, the effects;produced by the introduction of impurities are considered [45]. At the moment, such models lack a physical realization, but they are important as 'prototype' systems, shedding light on some crucial basic aspects (e.g. the choice of the most convenient basis for the space of states). Part II deals with a deceptively simple representative of the aforementioned nonlinear sigma models defined on supersymmetric manifolds, where the vector field, with one commuting component and two anticommuting ones, transforms under the global symmetry OSP(ll2). This system has a simple physical realization in terms of a dense loop model, where crossings of loops are allowed [28, 46]. At long wavelength, the theory is gapless and the Goldstone excitations are nothing but free fermionic ghosts [25, 28]. We propose the exact S-matrix for this system and present TBA calculations, supporting such conjecture. The bootstrap form factor approach is outlined, including a detailed discussion about the symmetry properties of the model and the explicit derivation of some basic objects, such as the minimal form factors. Moreover, we compute explicitly the two-point correlation function of a suitably chosen operator of the theory, comparing its large distance limit with the result expected on the basis of conformal field theory considerations. Since the work is still in progress [47], we conclude sketching the main goals and the route we intend to take, in order to pursue them.

Ghost Models in Two-Dimensional Condensed Matter Physics / Mosconi, Paola. - (2003 Oct 24).

Ghost Models in Two-Dimensional Condensed Matter Physics

Mosconi, Paola
2003-10-24

Abstract

This thesis is devoted to the study of ghost theories out of the critical point, in two dimensions. The first chapter offers a bird's eye view of the most important applications of ghosts to condensed matter physics. After a brief exposition of the basic (and less basic) facts concerning ghosts at the critical point, an outline of the non-perturbative methods, used in Part I and Part II, is furnished. Essentially, they rely on the so-called integrable approach, which is based on the possibility of describing all the states of an integrable quantum field theory in terms of pseudo-particles in a Hilbert space. The scattering properties of such excitations are encoded into a matrix (the S-matrix), which can be exactly determined by imposing a set of stringent constraints, and which allows to specify completely the particle content of the theory (masses, multiplicities, bound states) [38, 39]. The knowledge of the scattering amplitude, then, permits to extract all the thermodynamic quantities of the system (free energy) by means of the Thermodynamic Bethe ansatz (TBA) technique [40] and, at least in principle, to determine the off-critical correlation functions of the local operators of the theory, thanks to the so-called Form Factor bootstrap approach [41-43]. Part I contains the simplest examples of off-critical ghost theories, namely the massive versions of the conformal free bosonic and fermionic ones [44]. Despite their non-interacting nature, still there are non-local sectors of the models, which exhibit a highly interacting behavior. Correlation functions of operators belonging to these sectors are computed exactly and a comparison with the massive ordinary counterparts is performed. Afterwards, the effects;produced by the introduction of impurities are considered [45]. At the moment, such models lack a physical realization, but they are important as 'prototype' systems, shedding light on some crucial basic aspects (e.g. the choice of the most convenient basis for the space of states). Part II deals with a deceptively simple representative of the aforementioned nonlinear sigma models defined on supersymmetric manifolds, where the vector field, with one commuting component and two anticommuting ones, transforms under the global symmetry OSP(ll2). This system has a simple physical realization in terms of a dense loop model, where crossings of loops are allowed [28, 46]. At long wavelength, the theory is gapless and the Goldstone excitations are nothing but free fermionic ghosts [25, 28]. We propose the exact S-matrix for this system and present TBA calculations, supporting such conjecture. The bootstrap form factor approach is outlined, including a detailed discussion about the symmetry properties of the model and the explicit derivation of some basic objects, such as the minimal form factors. Moreover, we compute explicitly the two-point correlation function of a suitably chosen operator of the theory, comparing its large distance limit with the result expected on the basis of conformal field theory considerations. Since the work is still in progress [47], we conclude sketching the main goals and the route we intend to take, in order to pursue them.
Mussardo, Giuseppe
Mosconi, Paola
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/4205
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