Weyl-Gauge Symmetry of Graphene

Weyl-Gauge Symmetry of Graphene

Abstract

The conformal invariance of the low energy limit theory governing the electronic properties of graphene is explored. In particular, it is noted that the massless Dirac theory in point enjoys local Weyl symmetry, a very large symmetry. Exploiting this symmetry in the two spatial dimensions and in the associated three dimensional spacetime, we find the geometric constraints that correspond to specific shapes of the graphene sheet for which the electronic density of states is the same as that for planar graphene, provided the measurements are made in accordance to the inner reference frame of the electronic system. These results rely on the (surprising) general relativistic-like behavior of the graphene system arising from the combination of its well known special relativistic-like behavior with the less explored Weyl symmetry. Mathematical structures, such as the Virasoro algebra and the Liouville equation, naturally arise in this three-dimensional context and can be related to specific profiles of the graphene sheet. Speculations on possible applications of three-dimensional gravity are also proposed.


PACS No.: 11.30.-j, 04.62.+v, 72.80.Vp

Keywords: Symmetry and conservation laws, Quantum fields in curved spacetime, Electronic transport in graphene


1 Introduction

In the low energy limit the electronic properties of graphene are well described by massless Dirac spinors in two space and one time dimensions, hence an analog of a relativistic system but with characteristic velocity given by the Fermi velocity rather than the speed of light [1]

(1.1)

where is a (flat) spacetime index. This action is scale and conformally invariant1.

The main messages of this paper are two, one general and one practical. The general suggestion is that intrinsic curvature of the two-dimensional graphene sheet within this special relativistic-like behavior naturally leads to a general relativistic-like description2 of regarded as a Dirac field in a curved three-dimensional spacetime back-ground

(1.2)

where is the general covariant derivative, an important role being played by the local Weyl symmetry enjoyed by the electronic system. This suggestion, we shall see, comes very naturally when time is included in the analysis. The second, practical, main message of this paper is that the local Weyl symmetry of the graphene action can simplify the analysis of its electronic properties to the extent of allowing to find specific shapes of the graphene sheet for which the electronic local density of states (LDOS) is the same as that for a planar sheet, provided times and lengths are measured in accordance to the inner reference frame of the electronic system.

We find that when the bending of the graphene sheet gives rise to a constant intrinsic two-dimensional curvature, the three-dimensional metric (the one that includes time) is truly curved also in the time direction, although, for the cases specifically considered here, only in a very special way (conformally flat). Once such a general relativistic-like spacetime is accepted as physically meaningful, a special role is played by these conformally flat spacetimes

(1.3)

because in that case the curved space action (1.2) is equal to the flat space action (1.1), provided the fields are related by a Weyl transformation

(1.4)

hence the Euler-Lagrange (Dirac) equations for in the curved case are equal to those for in the flat case. As we shall see, this is so because of the special status of the massless Dirac action in any dimension. The three dimensions play a crucial role for other important matters, like for instance the particular form for the constraints for conformal flatness (it is the Cotton tensor rather than the Weyl tensor that needs to vanish) but this particular instance, , is independent from the dimensions. This fact is known in field theory on curved spaces as an instance of conformal triviality [3]. We shall revisit that instance and shall show that it has the consequence of an invariant electronic LDOS. Our results do not seem to contradict existing studies of the nontrivial effects of curvature on the LDOS, such as, e.g, those reported in [4], for two reasons: first, the ad hoc choice of the shape made there can well be outside the class of Weyl invariant choices (surely it is not within the class identified here); second, the measuring procedure adopted there seems to be a non-invariant one (we shall be more explicit later on this point).

The paper is organized as follows. In Section 2 we shall recall the concepts of scale, conformal and Weyl symmetries in general and for the case of a massless Dirac field, and discuss the special case of conformal triviality and its impact on the Green functions. In Section 3 we shall relate the above to the case in point of the electronic properties of a graphene sheet and shall discuss the impact on the LDOS. In Section 4 we shall see which two-dimensional bending of the graphene sheet produces a case of conformal triviality in the three-dimensional spacetime and learn that these conditions are quite natural, i.e. constant intrinsic curvature of the two-dimensional sheet. This means three possibilities: i) the sheet is a plane; ii) the sheet is flat but not planar (we find that this corresponds to two-dimensional conformal factors that are harmonic functions); iii) the bent sheet is truly curved with a constant intrinsic curvature (we find that this corresponds to two-dimensional conformal factors that are solutions of the Liouville equation). There we notice that, although the system is truly a three-dimensional one, due to the nature of the metric used, two-dimensional conformal field theory structures, such as the Virasoro algebra or the Liouville equation naturally appear. Section 5 is devoted to collect our results and to discuss various possible directions for future investigations, in particular whether it makes sense for the physics of graphene to use the rich landscape of solutions of three-dimensional gravity, including black-holes.

2 Scale, Conformal and Weyl Symmetries of the Dirac Action

Scale, conformal, and Weyl symmetries are intimately related but different. For the general case of theories in any dimension and with arbitrary spin the issue was discussed at length in [2]. In this paper we want to focus on the special case of the massless Dirac field in two space and one time dimensions because it is relevant for graphene. To fully appreciate the peculiarities of this case, though, and for the sake of introducing notation and concepts let us start this Section by reviewing the results of [2] and by putting them in the right perspective for the applications we intend to pursue.

Consider an action in dimensions for the field , where stands for a generic spin index, invariant under general coordinate transformations (diffeomorphisms) and local Lorentz transformations

(2.1)

This action could be in flat space in curvilinear coordinates or in a truly curved geometry. We implicitly introduced the Vielbein and its inverse

(2.2)

the indices that respond to diffeomorphisms (Einstein indices), while respond to flat space transformations (Lorentz indices), , and the diffeomorphic covariant derivative that is

(2.3)

with , , the appropriate form of the generators of the Lorentz transformations3 and

(2.4)

is the spin connection obtained by requiring the full covariant derivative of the Vielbein to be zero (metricity condition)

(2.5)

where is the Christoffel connection. Torsion is defined as where the expression in terms of Vielbein is , see the Appendix. At this point it is not required that and to keep track of that might become important if topological defects described by a Cartan-Einstein model of gravity with torsion (dislocations) need to be included [5], [6].

Due to diffeomorphism invariance of (2.1) a rigid scaling of the coordinates can be transferred to the Vielbein4 (or, equivalently, to the metric). It is then clear then that when the flat space limit of in (2.1), say

(2.7)

is symmetric under rigid scaling

(2.8)

where is the scale dimension of the field dictated by the kinetic term, then is symmetric under

(2.9)

The transformations (2.9) are called rigid (being constant) Weyl transformations. They are called local Weyl transformations if .

Let us now recall that often Poincarè and scale invariant actions are also invariant under the full conformal group of transformations, namely, on top of the Poincarè and scale transformations, they are invariant under those transformations for which the scaling parameter is5

(2.10)

where is a constant vector parameter, and SC stands for special conformal transformations. This is the case of the massless Dirac action of our concern here but it is not always true.

The exact criterion to establish when it is true that scale invariance of the flat space action implies full conformal invariance of the latter invokes the local Weyl invariance of the curvilinear action, as one might suspect by noticing that is coordinate dependent. The general procedure is presented in detail in [2] and we shall not do it again here but it is crucial for us to understand the special status of the Dirac action in any dimension with respect to fields of integer spin.

Thus, let us consider the case of a general -dependent scaling parameter , i.e. not constrained to be . For the Weyl transformations (2.9) are not symmetries of in (2.1), just like the scale transformations (2.8) when , in general, are no longer symmetries of in (2.7). To make invariant we need to gauge it à la Weyl (these considerations now presume torsionless connections)

(2.11)

where

(2.12)

and

(2.13)

with

(2.14)

under (2.9) with . Here and . As for usual gauge theories, the action is modified to a new one that is invariant: under (2.9) with . If and only if this produces an where only appears in a combination, say it , such that in the flat space limit has in (2.10) as the only solution, then the flat space action (2.7) is conformal invariant. In general this only happens for certain combinations of the spin of the fields and of the dimension of the spacetime.

What just said needs only a little adjustment in relation to the two-dimensional case, where the s are many more than just (2.10) as all harmonic s (i.e. the infinite solutions of ) correspond to conformal transformations. The procedure in this case gives as necessary and sufficient condition for a scale symmetric to be fully conformal invariant that enters as . Special attention is needed for the scalar fields, as pointed out in [7].

In this respect, the status of the massless Dirac action6 in any dimension is very special because7

(2.15)

due to

(2.16)

where we used and the definition of the Lorentz generators . This means that not only the Dirac action in flat space

(2.17)

is conformal invariant but that the curvilinear or truly curved space action

(2.18)

is local Weyl invariant, as can be also seen directly by transforming (2.18) under (2.9) with obtaining

(2.19)

Here it was used . This and other relevant formulae are in the Appendix.

In any dimension this is a very large symmetry that enjoys because there are no restrictions to of any sort. This happens for as it stands which means that the theory is, we may say, “intrinsically gauged”: . We shall exploit this symmetry in what follows for the graphene action where a crucial role is played by the dimensionality of the problem, which is two space and one time dimensions. Let us see here what this symmetry means in physical terms in any dimensions, including the case of graphene.

When two metrics are related as the classical physics for the field in is precisely the same as the classical physics for the field in because . Furthermore, we can choose

(2.20)

and obtain

(2.21)

when

(2.22)

where we used the simple algebra illustrated in the Appendix to write the second form of the action.

We are then dealing with a conformally invariant field in a conformally flat spacetime, a case sometimes referred to as conformal triviality8 [3]. This means that the effects of curvature are null on the classical physics of a massless Dirac field as in (2.22) if the spacetime is only curved in a conformally flat fashion: if the metrics can only be the conformally flat we know from here that the classical physics is once and for all governed by the flat space action. Of course this is not always true as there is no reason in general to consider only conformally flat metrics, but when this happens there is the “magic” consequence that although the spacetime is curved this has no effects whatsoever on the classical physics of the system, provided the field is properly redefined.

What for the quantum case? If we define the flat propagator as usual

(2.23)

where

(2.24)

we can see the effect of the Weyl transformations by considering the unitary operator that implements the transformations quantum mechanically, i.e.

(2.25)

By writing and by using , and from

(2.26)

we easily obtain

(2.27)

i.e., and . Notice also that , hence and . The vacuum associated to the Weyl transformed fields is of course

(2.28)

that shows the typical condensate structure of a quantum field in a curved (classical) background [8]. Now, as discussed at length, e.g., in [8], we can do two things: either we consider

(2.29)
(2.30)
(2.31)

or we consider

(2.32)
(2.33)

notice that when , as it should be from . Summarizing:

(2.34)
(2.35)

with the Green function for the curved Dirac operator

(2.36)

as proved in the Appendix.

The physical meaning of the two procedures is clear. In the first case, Eqs. (2.29)-(2.31), the measurements are made within the frame of reference that is co-moving with the particles described by the field , hence the effects of curvature are completely removed also at the quantum level. In the second case, Eqs. (2.32) and (2.33), the measurements are made by an observer that sees the (quantum) effects of curvature in the form of a condensate in the vacuum, Eq. (2.28).

We faced this sort of situation in [8] where we dealt with vacuum expectations of the kind (or viceversa ), with and , where is the vacuum for a scalar field, e.g., in a Schwarzschild spacetime, with a function related to the acceleration or to the surface gravity (near the horizon the space-time is always Rindler, see, e.g., [9]). The number operator is that of the same field in an inertial frame, i.e. a frame that is not freely falling into the black-hole with acceleration . In other words, represents a particles’ “counter” placed asymptotically far-away from the horizon in a region that can be approximated as flat. It is this counter that sees the condensate structure of

(2.37)

where is the four momentum in the curved frame, is the inverse temperature, the descends from the algebraic structure of and the relation of the latter to the Bose distribution comes from the minimization of the free energy [8]. If, on the other hand, we let our particle counter fall into the black hole, this will not see any horizon, nor there will be a way for it to detect the acceleration, as the principle of equivalence dictates. Mathematically this translates to

(2.38)

that is precisely what we get in the absence of any gravitational/acceleration effect, .

Returning to our case, if we think of as our “counter”, we can apply the same logic just illustrated, i.e. means that we see no effects of curvature, while contains the information on the quantum vacuum condensate. Two instances are important here. First, the physical picture can only become fully clear when the particular (conformally flat) metric has been specified. When we can do that we can give meaning to lengths and times measured according to the line element , hence we can appreciate the meaning of the invariance . What this means practically (i.e. what needs to be done to the measuring apparatus) depends upon . Second, in [8] we have been dealing with a truly field theoretical setting, i.e. with an infinite number of degrees of freedom and the related infinite volume limit. In that case the Bogoliubov operators were not well defined in the limit and unitarity was lost (on this see, e.g., [10]). For the applications to graphene we can safely take the view that unitarity of is never lost because the size of the sample is always finite and the number of degrees of freedom cannot be infinite because to excite more and more degrees of freedom we would need higher and higher energy than that necessary for the “Dirac-like” approximation to hold (see next Section).

A last remark on the quantum regime is related to the fact that what just said seems to apply to any , i.e. not just to the s in (2.27) that generate the Weyl transformations hence are symmetries of the classical theory. To select which s are to be used one can look at the path integral version of the Green functions (see, e.g., [11]) and use the classical Weyl-symmetry of the action

(2.39)
(2.40)
(2.41)

Here, since is an external field (see later), we used , similarly for , hence, in the ratio the overall factor cancels. From the above we see that the Green function computed via standard path integral methods, , is indeed in (2.32). This is so because we can as well read (2.40) as

(2.42)

Nonetheless, after having exploited the classical symmetries in the path integral Green functions we can then use the operator approach to find the s corresponding to these symmetries, for us here the s in (2.27), and as well construct the invariant Green function as in (2.30).

Let us now compare Weyl gauge symmetry to standard gauge symmetry. To this end consider the Dirac action

(2.43)

invariant under

(2.44)

where is the field that needs to be introduced to adsorb the derivative of the gauge parameter . The invariance means that , i.e. we can use the fields and or the gauge transformed ones and and we would not be able to see any difference because the physical effects of a nonzero are null for this form of the action. For this part the analogy with the Weyl symmetry we just discussed is total: we can use the fields and or the Weyl transformed ones and and we would not be able to see any difference. Furthermore, for the standard gauge theory (2.43) there is no physical meaning we can ascribe to : it is just the parameter of a kind of rotation of the field point by point. This is the reason why, besides topological non-triviality, the gauge field itself is not observable. Here the analogy with the Weyl case breaks down twice, once because is directly related to the metric hence indeed it has a physical meaning on its own, and once because, for the case in point of an intrinsically Weyl-gauged action, the Weyl-gauge field is absent hence it is not observable because it is not there in the first place. Thus we can indeed take the view that the non-transformed field is performing a curved space physics in a metric , but identical physical results are obtained by considering the Weyl-transformed field performing a flat space physics.

Weyl symmetry is an internal symmetry hence, as such, the associated Noether current is of the form , where, as usual, . Now we have to decide how to treat in (2.21), namely if we need to include it in the current as one of the fields or not. To properly treat as a dynamical field we would need to include a kinetic term for it, just like we do for standard gauge theories where we add to the gauged action a further term, usually of the form , where is the field strength. But the Dirac action in point needs not be Weyl-gauged, that is why there is no field, hence the associated kinetic term, if any, can only be relative to the geometric quantity , hence to the metric.

Thus, if no dynamical meaning can be ascribed to , i.e. is an external field, the Weyl current is

(2.45)

The action (2.21) is also invariant under standard rigid gauge transformations, i.e. (2.44) with , whose associated Noether current is

(2.46)

The way we interpret this result is that we have to choose which current to retain as physical9, and we may as well choose . With this choice no further conserved current is introduced when rigid Weyl symmetry is present. The action is not invariant under local gauge transformation, but it is under local Weyl and the physical meaning of this symmetry we have explained earlier.

It might look surprising that the Weyl current has nothing to do with the energy-momentum tensor but this is as it should be. The Weyl transformations are abelian internal transformations hence no (a current associated to the non-abelian spatiotemporal group ) could ever appear this way. The way to obtain the latter is through

(2.47)

whose trace is zero on-shell () with no needs to add improvement terms. The absence of such improvement terms is the effect on we have to expect from local Weyl symmetry.

3 Weyl-Gauge Symmetry of Graphene

Graphene is a two-dimensional honeycomb lattice of carbon atoms arranged in two triangular sub-lattices, say and , whose electrons in the -bonds belonging to one sublattice can hop to the nearest neighbor sites of the other sublattice. The electronic properties of graphene are ascribed to these electrons. The elastic properties, instead, are ascribed to the -bonds and involve an energy orders of magnitude stronger than that relative to the -bonds. Let us focus on the electronic properties which, in the low-energy approximation, are described by the Hamiltonian (we set )

(3.1)

where  eV is the nearest neighbors hopping parameter [12] (the next-to-nearest neighbors hopping parameter is taken to be zero, and that is the low-energy approximation), () and () are anti-commuting annihilation (creation) operators for an electron in the sub-lattice and in the sub-lattice, respectively, and all vectors are two-dimensional and , as described in Fig. 1. The relevant anticommutation relations are

(3.2)
Figure 1: The honeycomb graphene lattice.

First one notices that due to the geometry of the hexagonal lattice the basis vectors of the sublattice are

(3.3)

and the vectors moving from a site of to the three nearest neighbors sites are

(3.4)

In the following we shall set the lattice spacing to 1. Taking the Fourier transform, , etc, the Hamiltonian (3.1) becomes

(3.5)

where

(3.6)

The 1-particle spectrum being given by

(3.7)

the modes with zero 1-particle energy are easily found as solutions of , i.e. , where the superscript stands for (points). If we linearize around , , and , , , then in (3.1) can be written as

(3.8)

where is the Fermi velocity (at ) that we shall now set to 1. If we define

(3.9)

and , with

(3.10)

the usual Pauli matrices, and we Fourier-transform back to configuration space, then the Hamiltonian (3.8) becomes

(3.11)

The spin structure here is entirely due to the lattice, that is why it is often called pseudospin, but we shall treat it as a proper spin, as suggested, e.g., in [13]. With the introduction of the four component Dirac spinor10

(3.12)

of the following form for the Dirac matrices

(3.13)

which lead to the usual definition of -matrices

(3.14)

satisfying

(3.15)

with , the “ relativistic” Dirac Hamiltonian in (3.11) can be further compacted to

(3.16)

where .

Thus, starting from the instance that the energy dispersion relations (3.7) become linear at the Dirac points, taking into account only the low energy contributions, including the effect of the honeycomb structure of Fig. 1 and considering situations that do not include defects or impurities, the effective description of the electronic properties of graphene is given by a special relativistic-like Hamiltonian from which through customary Legendre transformation, , a Lagrangian density can be derived. What is obtained is precisely the scale- and conformal-symmetric massless Dirac action in flat space we discussed in the previous Section with

(3.17)

Now, let us suppose that the graphene sheet is bent. By this we mean that the two-dimensional sheet of graphene is not planar, hence curvilinear coordinates are better suited. A typical situation we have in mind is, of course, that of the ripples [14], [15], [16] (see also the review [17]), that are found in experiments and that are the subject of much theoretical and experimental investigations on their origins and their impact on the electronic properties of graphene, but other geometries are also within the reach of the following analysis. The graphene action to consider is then

(3.18)

where is a three-dimensional metric that bears the information about the bending in the two spatial dimensions (we shall be more precise about this in the next Section). Thus the graphene action, in the limits recalled above, is a particular instance of the “intrinsically Weyl-gauged” action for massless Dirac fields we have discussed before.

Every bending of the two-dimensional sheet produces a conformally flat two-dimensional metric , for the simple reason that every two-dimensional metric is conformally flat. It is matter of finding the coordinates system where this is evident, but for a well known general argument this is always possible. These conformally flat s describe graphene sheets of which: (i) one is planar, hence intrinsically (and extrinsically) flat; (ii) some are non-planar but are in fact intrinsically flat; (iii) some have constant intrinsic curvature; (iv) some have non-constant intrinsic curvature. Due to the local Weyl invariance in any dimension that we discussed in the previous Section, if the dimensionality of the problem was indeed two we would have immediately the result that in each one of the cases (i)-(iv) the electronic properties of graphene are described by a conformally trivial system and we could take advantage from this every time.

On the other hand, the correct dimensionality is three because there is time. In the next Section we shall see that the presence of time, the special relativistic-like form of the action and the two-dimensional conformal flatness of the graphene sheet naturally lead to a general relativistic-like metric, namely a 2+1 dimensional conformally flat metric, including the truly curved. We resort to such a discovery while asking the question: is there a particular shape of the two-dimensional sheet that gives raise to a conformally trivial Dirac system also in three dimensions, i.e. including time? We postpone the answer to the next Section as we want to come back now to the effect of local Weyl invariance on the density of states of graphene.

For graphene, the quantum version of the local Weyl invariance discussed in general in the previous Section can have striking consequences. First we notice that the electronic LDOS can be written in terms of the two-point function [18] (see also [19] and [4])

(3.19)

where stands for a generic Green function (, , or of the previous Section) and to obtain its energy dependence it is customary to move to the Hamiltonian form of the Dirac propagator, formally: . Then we need to apply the general discussion to the case of graphene, by first asking whether it makes sense to have a truly 2+1 dimensional conformally flat situation for graphene. As said we shall face this problem in detail in the following Section but we anticipate that the answer is “yes”. Thus, what is left to do is to identify the metric and define the corresponding measuring procedure such that

(3.20)

and the LDOS for the planar sheet is the same as that for a curved sheet when the two-dimensional curvature is such that the 2+1 dimensional metric is conformally flat.

4 The “nearly two-dimensional” Ansatz for the metric

The two-dimensional sheet in immersed in a three-dimensional space . One can easily set the frame so that the profile of the sheet is . Including time then we have the extrinsic frame and the intrinsic frame , where flat () and curved () indices need to be noticed. The (standard) induced metric procedure gives for the 2 + 1 dimensional system the metric

(4.1)

where the choice of four dimensional flat metric, , is dictated by the special relativistic-like form of the action, we just need to remember that the limiting speed for the electronic system is and not , all the rest goes through. With this choice for and the electrons on graphene see a spacetime metric of the form