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arXiv:1508.05975 · Journal of Statistical Mechanics Theory and Experiment

Bound states of Majorana fermions in semi-classical approximation

Giuseppe Mussardo

Published Dec 10, 2015• 6 citations
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Abstract

We derive a semi-classical formula for computing the spectrum of bound states made of Majorana fermions in a generic non-integrable two-dimensional (2D) quantum field theory with a set of degenerate vacua. We illustrate the application of the formula in a series of cases, including an asymmetric well potential where the spectra of bosons and fermions may have some curious features. We also discuss the merging of fermionic and bosonic spectra in the presence of supersymmetry. Finally, we use the semi-classical formula to analyse the evolution of the particle spectra in a class of non-integrable supersymmetry models.

DOI
10.1088/1742-5468/2015/12/p12003

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We derive a semi-classical formula for computing the spectrum of bound states made of Majorana fermions in a generic non-integrable two-dimensional (2D) quantum field theory with a set of degenerate vacua. We illustrate the application of the formula in a series of cases, including an asymmetric well potential where the spectra of bosons and fermions may have some curious features. We also discuss the merging of fermionic and bosonic spectra in the presence of supersymmetry. Finally, we use the semi-classical formula to analyse the evolution of the particle spectra in a class of non-integrable supersymmetry models.
I consider a class of one-dimensional models where Majorana fermions interact with bosonic fields. Contrary to a more familiar situation where bosonic degrees of freedom are phonons and as such form a slow subsystem, I consider fast bosons. Such situation exists when the bosonic modes appear as collective excitations of interacting electrons as, for instance, in superconductors or carbon nanotubes. It is shown that an entire new class of excitations emerge, namely bound states of solitons and Majorana fermions. The latter bound states are not topological and their existence and number depend on the interactions and the soliton's velocity. Intriguingly the number of bound states increases with the soliton's velocity.
The stationary states of a few bosons in a one-dimensional harmonic trap are investigated throughout the crossover from weak to strongly attractive interactions. For sufficient attraction, three different classes of states emerge: (i) N -body bound states, (ii) bound states of smaller fragments and (iii) gas-like states that fermionize, that is, map to ideal fermions in the limit of infinite attraction. The two-body correlations and momentum spectra characteristic of the three classes are discussed, and the results are illustrated using the soluble two-particle model.
We derive a condition for the existence of completely or exponentially localised Majorana bound states (with the potential for non-Abelian statistics) in a generic many-body system. We discuss the relationship between the existence of these operators and the protection of the ground state degeneracy from local perturbations. We use our conditions to study the exponential decay of the Majoranas in the Kitaev chain, finding complete agreement with single-particle results, and in a strongly interacting model of two quantum wires, finding again exponential decay. We give a method for deriving the exact form of the operators corresponding to a Majorana bound state in small systems, and apply this to a strongly correlated model of two spinful quantum dots.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.
This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.
Mk_kk​ models: the field theory connection
The M _k k models for 1D lattice fermions are characterised by {\cal N}=2 supersymmetry and by an order- k k clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of {\cal N}=2 supersymmetric conformal field theory (CFT) - we analyse finite size spectra on open chains with a variety of supersymmetry preserving boundary conditions. Specific staggering perturbations lead to a gapped regime corresponding to massive {\cal N}=2 supersymmetric QFT with Chebyshev superpotentials. At ‘extreme staggering’ we uncover a simple physical picture with degenerate supersymmetric vacua and mobile kinks. We connect this kink-picture to the Chebyshev QFTs and use it to derive novel CFT character formulas. For clarity the focus in this paper is on the simplest models, M _1 1 , M _2 2 and M _3 3 .