Emergent Weyl spinors in multi - fermion systems
In Ref. Horava2005 () 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 D system and concentrate on the particular case when the Fermi surface has co-dimension , 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 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 veirbein. 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.
As in particle physics, the condensed matter systems are described by the multi - component fermionic fields. In addition to spin they may have Bogoliubov spin, layer index in the multilayered systems, etc. In crystals the band indices are added, and the spinor acquires infinite number of components. In the low energy corner the effective number of degrees of freedom is essentially reduced. The gapped (massive) degrees of freedom are frozen out and only gapless states survive. The gaplessness is the fragile property, since it can be violated by interaction between femions. However, there exist fermionic systems, in which the gaplessness (masslessness) is robust to interaction. These are the topological materials, where stability of nodes in the energy spectrum with respect to deformations is protected by the conservation of topological invariants of different types Horava2005 ().
Examples of topologically protected zeroes in fermionic spectrum are: Fermi surface in metals VolovikBook (); Fermi points in D Weyl superfluid He-A VolovikBook () and in D Weyl semimetals Abrikosov1971 (); Abrikosov1998 (); Burkov2011 (); XiangangWan2011 (); Weylsemimetal (); Dirac points in graphene Semenoff:1984dq (); CastroNeto:2009zz (); fermionic edge modes on the surface and interfaces of the fully gapped topological insulators VolkovPankratov1985 (); HasanKane2010 (); Xiao-LiangQi2011 () and superfuids SalomaaVolovik1988 (); Volovik2009 ().
The Fermi or Weyl points represent the exceptional (conical, diabolic) points of level crossing, which avoid the level repulsion NeumannWigner (). Topological invariants for points at which the branches of spectrum merge were introduced by Novikov Novikov1981 (). In our case crossing points occur in momentum space Avron1983 (); Volovik1987 ().
The spectrum near the point nodes typically acquires the relativistic form, which is the consequence of the Atyah - Bott - Shapiro construction applied to the nodes with unit value of topological invariant Horava2005 (). This results in emergence of effective gauge and gravitational field as collective Bose modes Froggatt1991 (); VolovikBook (); Volovik1986A (); Volovik2011 (). This means, that the fermionic excitations reside in curved space - time. The geometry of this space - time is given by the vierbein formed by certain collective excitations of the microscopic system.
The higher values of topological invariant give rise to exotic Weyl or Dirac fermions, with nonlinear touching points of positive and negative energy branches. The D example of such system is given by the multilayer graphene with the stacking multilayer (). The nonlinear Dirac spectrum results in the effective gravitational and gauge field theories, which obey anisotropic scaling of Hořava type HoravaPRL2009 (); HoravaPRD2009 (); Horava2008 (); Horava2010 (), see KatsnelsonVolovik2012 (); KatsnelsonVolovikZubkov2013a (); KatsnelsonVolovikZubkov2013b (); VZ2013gr (). The multilayer graphene also demonstrates the reduction of the degrees of freedom at low energy. The original tight - binding model may be described by the field theory with the multi - component fermionic field, which carries the spin, pseudospin, and layer indices. Due to the specific interaction between the fermions that belong to different layers, in the emergent low energy theory the layer index drops out. The final effective theory operates with the two - component spinors existing in the vicinity of each of the two Fermi points. These spinors also carry the flavor index that corresponds to the real spin.
The general theory that describes reduction of the fermion components and the emergent gravity experienced by the reduced fermions is not developed so far in sufficient details. The main progress in this direction has been made by Hořava Horava2005 (), who considered the general case of dimensional condensed matter system with dimensional Fermi surface ( dimensional manifold of zeroes in the dimensional momentum space). The classification of the fully gapped topological materials Schnyder2008 (); Kitaev2009 () can be obtained from Hořava classification by dimensional reduction (see examples in VolovikBook ()).
We are interested in the particular case of the D systems with Fermi - point, i.e. with the node of co-dimension . In this particular case it follows from the statement of Horava2005 (), that in the vicinity of the Fermi - point the system is effectively described by the two - component fermion field . The action of this two - component field is given by
where is the integration measure over momentum and frequency . It was claimed in Horava2005 (), that operator contains the construction of Atyah - Bott - Shapiro that enters the expression for the topological invariant corresponding to the nontrivial , where is the original number of the fermion components:
Here is - momentum; is an emergent vierbein; is the position of the Fermi point, whose space - time variation gives rise to the effective dynamical gauge field ; and dots mean the subdominant terms, which include the emergent spin connection .
The emergence of this equation (2) has been advocated by Froggatt and Nielsen in their random dynamics theory, where the infinite number of degrees of freedom is reduced to subspace of Hermitian matrices (see page 147 in the book Froggatt1991 ()). In superfluid He-A this equation (2) has been explicitly obtained by expansion of the Bogoliubov-de Gennes Hamiltonian near the Weyl point Volovik1986A (); for the expansion near Dirac point in D graphene see Ref. VK2010 (); Manes2013 (); VZ2013gr (). In both cases the complicated atomic structure of liquid and electronic structure in crystals are reduced to the description in terms of the effective two-component spinors, and this supports the conjecture of Froggatt and Nielsen and the Hořava approach.
The emergence of Weyl spinor has important consequences both in the condensed matter physics and in the high energy physics. This is because the Weyl fermions represent the building blocks of the Standard Model of particle physics (SM). Emergence of Weyl fermions in condensed matter together with Lorentz invariance, effective gravity and gauge fields and the topological stability of emergent phenomena suggest that SM and Einstein theory of gravitational field (GR) may have the status of effective theories. The chiral elementary particles (quarks and leptons), gauge and Higgs bosons, and the dynamical vierbein field may naturally emerge in the low-energy corner of the quantum vacuum, provided the vacuum has topologically protected Weyl points.
When considering the possible emergence of SM and GR, one should resolve between the symmetries which emerge in the low energy corner (Lorentz invariance, gauge symmetry, etc.) and the underlying symmetry of the microscopic system – the quantum vacuum. The discrete and continuous global symmetries of the underlying microscopic systems influence the topological classification producing the additional classes of system, which are protected by the combined action of symmetry and topology VolovikBook (); Schnyder2008 (); Kitaev2009 (); Fu2011 (); Makhlin2013 (). They also determine the effective symmetries emerging at low energy, such as gauge symmetry in He-A, which follows from the discrete symmetry of the underlying high-energy theory VolovikBook ().
Especially we are interested in the case, when the original multi - fermion system consists of real fermions. i.e. it is the system of the underlying Majorana fermions of general type not obeying Lorentz invariance. This case may be related both to emergent gravity and to the foundations of quantum mechanics. The equations of ordinary quantum mechanics are described in terms of complex numbers. These are the Weyl equation; the Dirac equation obtained after electroweak symmetry breaking, when particles acquire Dirac masses; and finally the Schrödinger equation obtained for energies below the mass parameters. As is known, Schrödinger strongly resisted to introduce into his wave equations (see Yang Yang ()). The imaginary unit is the product of human mind, which is mathematically convenient. However, all the physical quantities are real, which implies that the imaginary unit should not enter any physical equation.
This suggests that the underlying microscopic physics is described solely in terms of the real numbers, while the complexification occurs on the way from microscopic to macroscopic physics, i.e. complexification of quantum mechanics (and of the quantum field theory) is the emergent phenomenon that appears at low energies. To see that we start with underlying microscopic system described in terms of the real - valued multi - component spinor, whose evolution is governed by the differential equation with real coefficients. We find that if the vacuum is toplogically nontrivial, the low energy phenomena will be described by the emergent Weyl quantum mechanics, which is expressed in terms of the emergent complex numbers.
The quantum dynamics of the corresponding field system is described by the integral over the - component Grassmann variables that does not contain imaginary unity. In the low energy approximation the multi - component Majorana fermions are reduced to the two - component Weyl fermions, which descisription is given in terms of the complex - valued two component wave function. The functional integral of is over the two sets of - component Grassmann variables and , where is the action for the emergent Weyl fermions (and the conjugated fermions ) in the presence of the emergent vierbein and emergent gauge field.
It is worth mentioning that in most of the cases the main symmetry of the gravitational theory (invariance under the diffeomorphisms) does not arise. For emergence of the diffeomorphism invariance the Lorentz violation scale must be much higher than the Planck scale. If this hierarchy of scale is not obeyed, in addition to Eq. (1) the effective action contains the terms that do not depend on but depend on , and directly. These terms are, in general case, not invariant under the diffeomorphisms. That’s why in the majority of cases we may speak of the gravity only as the geometry experienced by fermionic quasiparticles. The fluctuations of the fields , , and themselves are not governed by the diffeomorphism - invariant theory.
We shall demonstrate, that under certain reasonable assumptions the emergent spin connection in the considered systems is absent. This means, that we deal with the emergent teleparallel gravity, i.e. the theory of the varying Weitzenbock geometry 111The Riemann - Cartan space is defined by the translational connection (the vierbein) and the Lorentz group connection. There are two important particular cases. space is called Riemannian if the translational curvature (torsion) vanishes. If the Lorentz group curvature vanishes, it is called Weitzenbock space..
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.
The paper is organized as follows. In section 2 we describe the original construction of Hořava Horava2005 () and give its proof for the particular case of the D system, in which Fermi - surface is reduced to Fermi - point. In section 3 we generalize the construction of section 2 to the case, when the original system contains multi - component Majorana fermions. In section 4 we end with the conclusions.
2 Emergent Weyl spinors in the system with multi - component fermions (Hořava construction)
2.1 The reduction of the original multi - fermion model to the model with minimal number of spinor components
Following Horava2005 () we consider the condensed matter model with - component spinors . The partition function has the form:
Here the Hamiltonian is the Hermitian matrix function of momentum . We introduced here the symbol of the summation over the points of coordinate space. This symbol is to be understood as the integral over for continuous coordinate space. First, we consider the particular case, when there is no interaction between the fermions and the coefficients in the expansion of in powers of do not depend on coordinates. We know, that there is the ”repulsion” between the energy levels in ordinary quantum mechanics. Similar situation takes place for the spectrum of . The eigenvalues of are the real - valued functions of .
Several branches of spectrum for the Hermitian operator repel each other, i.e. any small perturbation pushes apart the two crossed branches. That’s why only the minimal number of branches of its spectrum may cross each other. This minimal number is fixed by the topology of momentum space that is the space of parameters .
Let us consider the position of the crossing of branches of . There exists the Hermitian matrix such that the matrix is diagonal. In this matrix the first block corresponds to the crossed branches (i.e. all eigenvalues of coincide at ). The remaining block of matrix corresponds to the ”massive” branches. The functional integral can be represented as the product of the functional integral over ”massive” modes and the integral over reduced fermion components
Here is the projector to space spanned on the first components. Let us denote the remaining components of by
Let us denote the only eigenvalue of by . The transformation , leaves the expression in exponent of Eq. (3) unchanged. That’s why we can always consider the matrix equal to zero at the position of the branches crossing . We are left with the following expression for the partition function:
Spectrum of operator has exceptional properties around vanishing eigenvalues. The corresponding eigenfunctions do not depend on time. The key point is that at low energy the integral over dominates. The other components contribute the physical quantities with the fast oscillating factors, and, therefore, may be neglected in the description of the long - wavelength dynamics. As a result at the low energies we may deal with the theory that has the following partition function:
Here 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 Froggatt1991 ().
2.2 Momentum space topology, and the two - component spinors
This consideration allows to prove the Hořava’s conjecture presented in Horava2005 (). According to this conjecture any condensed matter theory with fermions and with the topologically protected Fermi - points may be reduced at low energies to the theory described by the two - component Weyl spinors. The remaining part of the proof is the consideration of momentum space topology. It protects the zeros of (i.e. it is robust to deformations) only when there is the corresponding nontrivial invariant in momentum space. The minimal number of fermion components that admits nontrivial topology is two. This reduces the partition function to
where functions are real - valued.
Let us, in addition, impose the CP symmetry generated by and followed by the change . Its action on the spinors is:
It prohibits the term with . Thus operator can be represented as
The topologically nontrivial situation arises when has the hedgehog singularity. The hedgehog point zero is described by the topological invariant
where is the surface around the point.
For the topological invariant in Eq.(11) the expansion near the hedgehog point at in -space gives
Here by we denote the coefficients in the expansion. It will be seen below, that these constants are related to the emergent vierbein. As a result, Eq. (16) has the form:
In the absence of the mentioned above symmetry we have, in addition, the function that is to be expanded around : , . The new quantities are introduced here. So, in general case we arrive at the expression for the partition function of Eq. (13), in which the sum is over , and while . The situation here becomes much more complicated, than in the presence of the symmetry. Namely, when , we have the more powerful zeros of the Hamiltonian (better to say, - of its determinant). For there is the conical Fermi - surface of co-dimension given by the equation
There exists the choice of coordinates, such that on this Fermi surface the energy of one of the two branches of spectrum of is equal to zero. The energy corresponding to the second branch vanishes at only, where the two branches intersect each other. However, in this situation the first branch dominates the dynamics, and we already do not deal with the Fermi - point scenario of the effective low energy theory. That’s why the - invariance is important because it protects the system from the appearance of the Fermi surface in the vicinity of the branches crossing. It is worth mentioning, that in the marginal case we deal with the line of zeros of the Hamiltonian (Fermi - surface of co-dimension ). We do not consider here the other marginal cases, such as that, in which .
In the following we shall imply, that there is the additional symmetry (like the mentioned above symmetry) that protects the system from the appearance of the more powerful zeros in the spectrum of the Hamiltonian (i.e. Fermi surfaces and Fermi lines). The symmetry may be approximate instead of exact, i.e. it may be violated by small perturbations and the interactions. The approximate symmetry is enough to provide the inequality that restricts the appearance of the Fermi - surfaces of co-dimension and . In this case we may apply Lorentz transformation (boost) that brings the system to the reference frame, in which for . In the following the value of may be interpreted as the external vector potential. The interpretation of quantity in terms of the emergent gravitational field will be given in the next subsection.
2.3 Taking into account interaction between the fermions
Next, we should consider the situation, when the coefficients of expansion of in powers of , depend on coordinates and fluctuate. The original partition function for the fermions with the interaction between them can be written as follows:
Here the new fields that provide the interaction between the fermions are denoted by . is some function of these fields. Now operator also depends on these fields. In mean field approximation, when the values of are set to their ”mean” values we come back to the consideration of the previous subsections. However, at the end of the consideration the fluctuations of the fields are to be taken into account via the fluctuations of the field and the Fermi - point position .
Let us consider for the simplicity the low energy effective theory with only one emergent Weyl fermion. The interaction between the particles appears when the fluctuations of and are taken into account. We assume, that these fluctuations are long - wave, so that the corresponding variables should be considered as if they would not depend on coordinates. Nevertheless, in the presence of the varied field the time reversal symmetry is broken. As a result the partition function of the theory receives the form
The appearance of the field reflects, that in the presence of interaction the value of energy at the position of the crossing ot several branches of spectrum may differ from zero. We represented the quantity of Eq. (12) (that depends now on the coordinates) as , where the fluctuating long - wave fields depend on the primary fields . This representation for is chosen in this way in order to interpret the field as the vierbein. We require for , and . Here is equal to the determinant of the vierbein . In the mean field approximation, is set to its mean value , while , and , where variable was introduced in sect. 2.2. It is implied (see remark 2.1), that the approximate CP symmetry is present, that may be slightly violated by the interactions. This means, that the values of are suppressed compared to the values of for . This allows to keep the Fermi point in the presence of interactions.
As a result, the partition function of the model may be rewritten as:
where the sum is over while , and is the covariant derivative that includes the gauge field . is the part of the effective action that depends on and only.
It is worth mentioning, that to write the expressions for the functional integral Eq. (18) and the expression for the action Eq. (19) is not enough to define the field system. Besides, we are to impose boundary conditions on the fields. Typically, the anti - periodic in time boundary conditions are imposed on the spinor fields in quantum field theory. These boundary conditions correspond to the choice of vacuum, in which all states with negative energy are occupied. This is important to point out the reference frame, in which these anti - periodic boundary conditions in time are applied. Here and below we always imply, that these boundary conditions are imposed in the synchronous reference frame, i.e. in the one, in which the mean values vanish for .
Eq. (19) is reduced to Eq. (16) with given by Eq. (17) if the particular gauge (of the emergent ) is fixed. In this gauge for . Besides, we rescale time in such a way, that . This means, that the term contains the corresponding gauge fixing term. Even modulo this gauge fixing the theory given by Eq. (19) is not diffeomorphism - invariant. The fermionic term alone would become diffeomorphism - invariant if the spin connection of zero curvature is added. Then, in addition Eq. (19) is to be understood as the result of the gauge fixing corresponding to vanishing spin connection. In some cases may be neglected, and only the second term of Eq. (19) contributes the dynamics. Then the fields and may be identified with the true gravitational field (vierbein) and the true gauge field correspondingly (modulo mentioned above gauge fixing). Their effective action is obtained as a result of the integration over the fermions. It is worth mentioning, that in most of the known condensed matter systems with Fermi - points (say, in He-A) we cannot neglect the term . That’s why the given opportunity in the condensed matter theory remains hypothetical.
Recall, that we have considered the long - wavelength fluctuations of the emergent fields and . That is we neglected the derivatives of these fields. In the fermion part of the action in Eq. (19) there are no dimensional parameters. The only modification of this action that is analytical in and their derivatives and that does not contain the dimensional parameters is if the covariant derivative receives the contribution proportional to the derivative of . That’s why, even for the non - homogenious variations of and in low energy approximation we are left with effective action of the form of Eq. (19) if the value of the emergent electromagnetic field is much larger than the order of magnitude of quantity . Such a situation takes place, for example for the consideration of the emergent gravity in graphene VZ2013gr ().
Let us formalize the consideration of the given section as the following theorem.
The multi-fermion system without interaction between the particles in the vicinity of the Fermi - point (Fermi surface of co-dimension ) is reduced to the model that is described by the two - component Weyl fermions described by partition function Eq. (16). In addition, we require, that the (approximate) CP symmetry is present. This symmetry prohibits the appearance of the Fermi surfaces of co-dimension and and results in the suppression of the values of compared to the values of . The nontrivial momentum space topology with the topological invariant of Eq. (11) equal to unity provides that the effective low energy theory has the partition function of Eq. (13) with some constants that depend on the underlying microscopic theory.
When the interaction between the original fermions in this system is taken into account (while momentum space topology remains the same as in the non - interacting theory), the partition function of the low energy effective theory receives the form of Eq. (18) with the effective action Eq. (19). This is the partition function of Weyl fermion in the presence of the emergent vierbein and the emergent gauge field . Both these fields represent certain collective excitations of the microscopic theory. (It is assumed, that the value of the emergent electromagnetic field is much larger than the order of magnitude of quantity .)
One can see, that in the considered long wave approximation the emergent spin connection does not arise. That’s why we deal with the emergent teleparallel gravity described by the veirbein only.
The given theorem represents the main statement given without proof in Horava2005 () in a more detailed and elaborated form (for the particular case of D system with Fermi - surface reduced to the Fermi - point). We considered only one Fermi point. This case also corresponds to the situation, when there exist several Fermi points, but the corresponding collective excitations do not correlate with each other. The situation, when the correlation is present is more involved. We make a remark on it at the end of section 3.
3 Emergent Weyl spinors in the system of multi - component Majorana fermions
In this section we consider the generalization of the problem considered in the previous section to the case, when the original system contains multi - component Majorana fermions.
3.1 Path integral for Majorana fermions
On the language of functional integral the evolution in time of the field system is given by the correlations of various combinations of the given fields. The lagrangian density for - component Majorana fermions can be written in the form:
where is the arbitrary operator that may be highly non - local. First, we consider the situation, when there is no interaction between the original Majorana fermions. This means, that operator does not depend on the other fields. As a result the partition function is represented as
Various correlators of the field are given by
Here is the - component anti - commuting variable. The Majorana nature of the fermions is reflected by the absence of the conjugated set of variables and the absence of the imaginary unit in the exponent. The dynamics of the system is completely described by various correlators of the type of Eq. (22). It is worth mentioning that the complex numbers do not enter the dynamics described by Eq. (22). It can be easily seen, that if is linear in the spacial derivatives, and is represented by the product , where do not depend on coordinates, then should be symmetric. (For the anti - symmetric expression vanishes.) We feel this instructive to give the representation of the partition function of Eq.(21) in terms of the analogues of the energy levels.
We consider the functional integral over real fermions basing on the analogy with the integral over complex fermions (see PhysRevD.12.2443 ()). We start from the partition function of Eq. (21). In lattice discretization the differential operator is represented as the skew - symmetric matrix, where is the total number of the lattice points while is the number of the components of the spinor . As a result there exists the orthogonal transformation that brings matrix to the block - diagonal form with the blocks of the form
with some real values . We represent as , where , and has the above block - diagonal form in the basis of . These vectors are normalized to unity (). Further, we represent
where the system is considered with the anti-periodic in time boundary conditions: . We use the decomposition
Integrating out the Grassmann variables we come to:
The values depend on the parameters of the Hamiltonian, with the index enumerating these values. Eq. (26) is derived as follows. Recall that in (25) the summation is over . The product over can be calculated as in PhysRevD.12.2443 ():
Formally the partition function may be rewritten as
The explanation that the square root of the determinant appears is that operator itself being discrcetized becomes the skew - symmetric matrix. Via the orthogonal transformations it may be made block - diagonal with the elementary blocks. In the latter form the functional integral is obviously equal to the square root of the determinant because for the 2 - component spinor
We get (see also PhysRevD.12.2443 ()):
Following PhysRevD.12.2443 (), we interpret Eq. (30) as follows. represents the number of occupied states with the energy . These numbers may be or . The term vanishes if values come in pairs with the opposite signs (this occurs when the time reversal symmetry takes place). We can rewrite the last expression in the form, when the integer numbers represent the numbers of occupied states of positive energy and the holes in the sea of occupied negative energy states:
After the Wick rotation we arrive at
where is temperature. This shows, that in equilibrium the configuration dominates with the vanishing numbers . This corresponds to the situation, when all states with negative energy are occupied. This form of vacuum is intimately related with the anti - periodic in time boundary conditions imposed on . The other boundary conditions would lead to the other prescription for the occupied states in vacuum.
The values are given by the solution of the system of equations
for the pair of the real - valued - component wave functions. Alternatively, we may solve equation
Here the the complex - valued - component wave function has the particular dependence on time . However, Eq. (34) does not contain imaginary unity. Therefore, we may consider its real - valued solutions. These solutions may be interpreted as the time - dependent real - valued spinor wave functions of Majorana fermions. It is worth mentioning, that there are no such real valued wave functions that would correspond to definite energy.
3.2 Repulsion of fermion branches the reduced number of fermion species at low energy.
The notion of energy in the theory described by operator may be based on the definition of the values given above. Besides, we may introduce the notion of energy scale as the typical factor in the dependence of various dimensionless physical quantities on time: , where is a certain dimensionless function of dimensionless argument such that its derivatives are of the order of unity. With this definition of energy it can be shown, that at low energies only the minimal number of fermion components effectively contributes the dynamics. Below we make this statement explicit and present the sketch of its proof.
As it is explained in Sect. 3.1, operator in lattice discretization is given by the skew - symmetric matrix, where is the total number of the lattice points while is the number of the components of the spinor . As a result there exists the orthogonal lattice transformation that brings matrix to the block - diagonal form with the blocks of the form with some real values . In the continuum language matrix becomes the operator that acts as a matrix, whose components are the operators acting on the coordinates. There are several branches of the values of . Each branch is parametrized by the continuum parameters. Several branches of spectrum of repel each other because they are the eigenvalues of the Hermitian operator. This repulsion means, that any small perturbation pushes apart the two crossed branches. That’s why only the minimal number of branches of its spectrum may cross each other. This minimal number is fixed by the topology of momentum space (see below, sect. 3.3.4).
As it was mentioned, there exists the orthogonal operator (it conserves the norm ) such that the operator
is given by the block - diagonal matrix with the elementary blocks:
Here we denote by the three - dimensional vector that parametrizes the branches of spectrum and the basis vector functions that correspond to the given form of . The first values coincide at . This value is denoted by . The first block corresponds to the crossed branches. The remaining block of matrix corresponds to the ”massive” branches. The functional integral can be represented as the product of the functional integral over ”massive” modes and the integral over reduced fermion components
Here by we denote matrix , while is the projector to space spanned on the first components. Let us denote the remaining components of by
We arrive at
where while .
The exponent in Eq. (39) contains the following term that corresponds to the contribution of the fermion fields defined in a vicinity of :
We have the analogue of the hamiltonian that vanishes at . Following Sect. 3.1 we come to the conclusion, that in the expression for the partition function Eq. (31) the small values of energies dominate (when the negative energy states are occupied), and these energies correspond to the reduced fermions . It is important, that in order to deal with vacuum, in which negative energy states for Eq. (40) are occupied we need to impose the antiperiodic boundary conditions in time on (not on the original fermion field ). The other components contribute the physical quantities with the fast oscillating factors because they are ”massive”, i.e. do not give rise to the values of from the vicinity of zero. Therefore, these degrees of freedom may be neglected in the description of the long - wavelength dynamics.
Any basis of the wave functions is related via an orthogonal operator to the basis of the wave functions, in which has the form of the block - diagonal matrix (Eq. (36)). We require, that commutes with for the transformation to the basis associated with the observed low energy coordinates. This observed coordinate space may differ from the primary one, so that is not equal to of Eq. (35). This new coordinate space in not the primary notion, but the secondary one. is the requirement, imposed on the representation of the theory, that allows to recover the usual Weyl spinors and the conventional quantum mechanics with complex - valued wave functions (see the next subsection). We denote the new coordinates by to distinguish them from the original coordinates , in which the partition function of Eq. (21) is written. In this new basis is given by the differential operator. It is expressed as a series in powers of derivatives with real - valued matrices as coefficients. From it follows, that in this basis .
3.3 The reduced - component spinors
3.3.1 Analytical dependence of on
In Section 3.2 it was argued that the number of fermion components at low energies should be even. The minimal even number that admits nontrivial momentum space topology (see below) is . That’s why we consider the effective low energy four - component spinors. This corresponds to the crossing of the two branches of the energy.
The values coincide at . The corresponding value of is denoted by . The first block of Eq. (36) corresponds to the crossed branches. The remaining block of matrix corresponds to the ”massive” branches. The Fermi point appears at if chemical potential is equal to . Then the four reduced components dominate the functional integral while the remaining ”massive” components decouple and do not influence the dynamics. The form of the reduced matrix is exceptional. It is related by the orthogonal transformation that commutes with with the matrix of a more general form. In this form also commutes with . In general case the dependence of on is analytical. This is typical for the functions that are encountered in physics. The non - analytical functions represent the set of vanishing measure in space of functions. However, this is not so for the exceptional block - diagonal form in case of non - trivial topology that protects the levels crossing.
Let us illustrate this by the example, in which
Here the three real - valued - matrices form the basis of the algebra and have the representation in terms of the three complex Pauli matrices:
There exists the orthogonal matrix that brings to the block - diagonal form:
One can see, that in the form of Eq. (41) the matrix is analytical at while in the block - diagonal representation it is not.
In the following, speaking of the low energy dynamics, we shall always imply, that is discussed, and shall omit the superscript ””. We shall refer to space of parameters as to generalized momentum space. The zeros of in this space should be topologically protected; i.e. they must be robust to deformations.
3.3.2 Introduction of new coordinate space
Let us identify the quantities with the eigenvalues of operator . Here by we denote the new coordinates. They do not coincide with the original coordinates . This means, that the fields local in coordinates are not local in coordinates and vice versa.
We illustrate the appearance of the new coordinates by the following simple example. Let us consider the two - component Majorana spinors in dimensions with original non - local operator given by
where is parameter while the integral operator is given by
with some odd function . This operator is well - defined for the functions that tend to zero at infinity sufficiently fast.
Our aim is to find the two representations:
1)Generalized momentum space, where for a certain function of generalized momenta .
2)New space with coordinates , related to momentum space via identification .
This aim is achieved via the following operator
It is orthogonal and brings to the form corresponding to the new coordinates :
This defines the new coordinates , in which operator is proportional to . Space of coordinates differs from space of coordinates just like conventional momentum space differs from the conventional coordinate space: the functions local in one space are not local in another one and vice versa. In generalized momentum space operator receives the form with .
3.3.3 How the fermion number conservation reduces the general form of for D Majorana fermions
It was argued, that for the low energy effective fermion fields in new coordinate space operator has the form of the series in powers of the derivatives with the real valued constant matrices as coefficients. Moreover, the reduced operator commutes with . The latter condition may be identified with the fermion number conservation, that is rather restrictive. Below we describe the general form of the operator that may be expanded in powers of derivatives with real - valued constant matrices as coefficients. It may always be considered as skew - symmetric ( for real - valued spinors , i.e. ) because the combination vanishes for any symmetric operator and Grassmann valued fields . We shall demonstrate how the fermion number conservation reduces the general form of such skew - symmetric operator. Let us introduce the two commuting momentum operators:
The two commuting operators and have common real - valued eigenvectors corresponding to their real - valued eigenvalues. Matrix can be represented as the analytical function
More specifically, it can be represented as
while are real - valued functions of the momenta . Functions are odd; and are the generators of the two groups; and are real antisymmetric matrices that commute with all (or ) correspondingly; are the matrices that commute with but do not commute with either of and . (Notice, that . That’s why odd part of the function may be set equal to zero. )
According to our condition operator commutes with matrix . The coordinates of new emergent coordinate space are denoted by . Matrix anticommutes with , and , . Yet another way to look at this symmetry is to require, that the momentum defined as is conserved, i.e. commutes with . This requirement reduces the partition function to