Emergent Weyl spinors in multi-fermion systems

G.E Volovik,M.A Zubkov

doi:10.1016/j.nuclphysb.2014.02.018

Abstract

In Ref. [1] Hořava suggested, that the multi-fermion many-body system with topologically stable Fermi surfaces may effectively be described (in a vicinity of the Fermi surface) by the theory with coarse-grained fermions. The number of the components of these coarse-grained fermions is reduced compared to the original system. Here we consider the 3+1 D system and concentrate on the particular case when the Fermi surface has co-dimension p=3, i.e. it represents the Fermi point in momentum space. First we demonstrate explicitly that in agreement with Hořava conjecture, in the vicinity of the Fermi point the original system is reduced to the model with two-component Weyl spinors. Next, we generalize the construction of Hořava to the situation, when the original 3+1 D theory contains multi-component Majorana spinors. In this case the system is also reduced to the model of the two-component Weyl fermions in the vicinity of the topologically stable Fermi point. Those fermions experience the emergent gauge field and the gravitational field given by the emergent vierbein. Both these fields (the emergent gauge field and the emergent gravitational field) originate from certain collective excitations of the original system. We speculate, that the given construction may be relevant for the high energy physics in the paradigm, in which the Lorentz symmetry as well as the gravitational and gauge fields are the emergent phenomena, i.e. they appear dynamically in the low energy approximation of the underlined high energy theory.

Highlights

As in particle physics, the condensed matter systems are described by the multi-component fermionic fields
On the high-energy side the application of the given pattern may be related to the unification of interactions in the paradigm, in which at the extremely high energies the Lorentz-invariance as well as the general covariance are lost. In this paradigm Lorentz symmetry, the two-component Weyl fermions that belong to its spinor representation, the gravitational and gauge fields appear at low energies as certain collective excitations of the microscopic theory
When the interaction between the original fermions in this system is taken into account, the partition function of the low energy effective theory receives the form of Eq (18) with the effective action Eq (19)

Summary

Introduction

The condensed matter systems are described by the multi-component fermionic fields. On the high-energy side the application of the given pattern may be related to the unification of interactions in the paradigm, in which at the extremely high energies the Lorentz-invariance as well as the general covariance are lost In this paradigm Lorentz symmetry, the two-component Weyl fermions that belong to its spinor representation, the gravitational and gauge fields appear at low energies as certain collective excitations of the microscopic theory. We consider the situation, when Fermi energy coincides with the value of energy at the branches crossing It was suggested by Froggatt and Nielsen in their random dynamics theory, that this case may be distinguished due to the specific decrease of particle density as follows from the Hubble expansion [19]. The interpretation of quantity fkj in terms of the emergent gravitational field will be given in the subsection

Taking into account interaction between the fermions

Path integral for Majorana fermions

The reduced 4-component spinors

Introduction of new coordinate space

Interaction between the fermions

Conclusions