2 Hamiltonian and the Dirac equation in (1 + 2)-dimensions

Physica E: Low-dimensional Systems and Nanostructures 115 (2020) 113685

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Fractional versions of the spin 1/2 Hamiltonian and the Dirac equation in (1 + 2)-dimensions

T

C.A. Dartora∗, Fernando Zanella Electrical Engineering Department, Federal University of Paraná (UFPR), Brazil

A R T I C LE I N FO

A B S T R A C T

Keywords: Pauli matrices Spin Hamiltonian Fractional order Dirac's equation

The prototypical example of a quantum mechanical system is the spin 1/2 Hamiltonian, which describes a vast class of two level systems, including the well known Zeeman interaction between a spin 1/2 particle and a magnetic field. As a pure quantum mechanical system, it conserves probability. In the present contribution a simple extension to this problem is put forward, by taking the α-th fractional power of the spin-1/2 Hamiltonian, leading to a dissipative system. It can be used to describe a vast class of quantum mechanical problems displaying relaxation, like a spin 1/2 in the presence of a thermal bath, or the asymmetry in energy spectrum between electrons and holes in a semiconductor. The mean-field theory of a fractional magnet is explored and a fractional version of Dirac's equation in 1 + 2 dimensions is also examined.

1. Introduction The spin 1/2 hamiltonian can be considered the finest example of a quantum mechanical system, displaying its most striking features, such as the physical configurations being given by state vectors in a Hilbert space, the quantization of the energy spectrum and the Copenhagen probabilistic interpretation of measurements. It can be used to describe a wide range of physical problems that can be mapped into a two level system, at least approximately, such as a two level laser, the Zeeman interaction of a spin 1/2 particle, and so on. The Pauli spin matrices and their algebra are the fundamental mathematical objects within that framework. Furthermore, by using a suitable definition, one can use the Pauli matrices to computationally implement the fermionic anti-commuting algebra. These matrices also appear in the standard model of elementary particles, associated with the SU(2) gauge group describing weak isospin [1,2]. For instance, the relativistic theory of fermions in 1 + d spacetime dimensions, being d the number of spatial dimensions are constructed using the Dirac algebra of gamma matrices [1,2]:

{γ μ, γ ν} = γ μγ ν + γ ν, γ μ = 2g μν , μ

(1)

where γ are the Dirac matrices, μ = 0, 1, 2 … d are the spacetime indices and gμν = diag(1, −1, −1…) is the Minkowski metric tensor. It is well known that the Pauli matrices provide exact representations of the gamma matrices in d = 1 and d = 2 spatial dimensions, but they also appear as sub-matrices of the gamma matrices in spatial

∗

dimensions d > 2. Considering integer dimensions and integer order differential operators in the Dirac equation, matter and anti-matter are related by CPT symmetries. The mean lifetime of particles and antiparticles are usually the same at a quantum level, but no one knows the mechanism behind the prevalence of matter over anti-matter in the observed universe. Also, quantum field theories using only integer order operators run into trouble when physical quantities are calculated due to infrared and/or ultravioled divergences. Despite the enormous successes achieved through the agreement between calculated quantities and experimental observations after the so-called renormalization techniques are applied, the present status of quantum field theories is not free of critiques, being Dirac himself one of the most famous critics. This way, the search for divergence-free alternatives to renormalization is an active research field. The particle quantization, also known as first quantization, leads to a conservative version of quantum mechanics using integer order differential operators. On the other hand, non-locality and memory effects are a necessary feature to describe many natural phenomena, arising quite naturally in fractional calculus, which has been applied to many branches of physics and engineering (see Ref. [3]). Inspired by the ideas of Gelfand and Shilov, early attempts aiming regularization and removal of divergences in quantum field theories were made by C.G. Bollini et al. [4], introducing fractional powers in the denominator of Green's functions. Following the same lines, canonical quantization of relativistic non-local scalar and gauge fields, obtained by taking arbitrary powers of the D'Alembertian operator, which introduces a non-

Corresponding author. E-mail address: [email protected] (C.A. Dartora).

https://doi.org/10.1016/j.physe.2019.113685 Received 14 December 2018; Received in revised form 6 July 2019; Accepted 9 August 2019 Available online 17 August 2019 1386-9477/ © 2019 Elsevier B.V. All rights reserved.

Physica E: Low-dimensional Systems and Nanostructures 115 (2020) 113685

C.A. Dartora and F. Zanella

local kinetic term in the lagrangian, was discussed in the work of D.G. Barci et al. [5]. Within the canonical formalism, it was shown that nonlocal gauge theory has a continuum mass spectrum and the zero mass modes ensure gauge invariance. Lorentz invariance of fractional order Klein-Gordon equation has also been was considered in details in Ref. [6]. In another direction, the theory of Levy flights was used to generalize quantum mechanics to non-integer order differential operators, leading to what is known as fractional quantum mechanics, which displays memory, non-locality and dissipation effects [7-12]. Fractional versions of the Schrödinger equation allowed to solve the fractional Hydrogen atom and the fractional harmonic oscillator energy spectra [9–11]. The Dirac equation has also been extended to the fractional order domain [13]. By relaxing the need for a hermitian hamiltonian operator, one gets a complex valued energy spectrum, leading to quantum mechanical decayment of eigenstates. For instance, it would be interesting in the modeling of electron/hole distinct lifetimes in semiconductors, or eventually describe the prevalence of matter over anti-matter as a decaying process of anti-matter states. The aim of this letter is to contribute to the recent advances towards fractional versions of the fundamental problems in quantum mechanics, introducing a fractional version of the spin 1/2 hamiltonian, by taking the α-th fractional power of spin 1/2 Hamiltonian, leading to a dissipative spin-1/2 system. It can be used to describe a vast class of quantum mechanical problems displaying relaxation, like a spin 1/2 in the presence of a thermal bath, or the asymmetry in energy spectrum between electrons and holes in a semiconductor. A fractional version of Dirac's equation in 1 + 2 dimensions is also examined. A simple way to recover hermiticity, when necessary, is also considered.

Ĥ = B0α e−i

απ απ Ĥ = B0α ⎡cos2 ⎛ ⎞ + sin2 ⎛ ⎞ σ ⋅nˆ ⎤ . ⎝ 2 ⎠ ⎝ 2 ⎠ ⎣ ⎦

Irreversibility and dissipation are essential ingredients in the description of the real classical world. They also emerge in a quantum system coupled to a thermal bath [14], as successfully demonstrated by A.O. Caldeira and A.J. Legget for a non-relativistic quantum particle coupled to a thermal bath sonsisting of an infinite sum of harmonic oscillators [15]. In their model, using the functional formalism, memory effects can be included through the bath spectral function (or density of states), which can take the form of a general power law ωs, being omega the bath modal frequencies (or energies) and s assume any value. For a memoryless system s = 1 and dissipation takes the form of Ohm's law. A particular case of the Caldeira-Legget model is the twolevel system or spin-boson hamiltonian (for a review see Ref. [16]), which is relevant for the study of dissipation and decoherence in quantum computation. While in the Caldeira-Legget model for the spin-1/2 system dissipation comes from the coupling to a thermal bath and the two-level system is described by a hermitian hamiltonian, the fractional hamiltonian presented here in its non-hermitian form (8) leads to an asymmetric dissipation, even in the absence of a thermal bath. Considering the Schrödinger equation to be valid, i.e., iℏd∣ψ〉/dt = H∣ψ〉, notice that the in our case the evolution operator is no longer unitary. Assuming a constant B0, one has U(t) = e−iHt/ℏ. For an arbitrary initial state ∣ψ0〉 = c1∣1〉 + c2∣2〉 at t = 0, being c1 and c2 the usual probability amplitudes, the evolved state ∣ψ(t)〉 = U(t)∣ψ0〉, at a later time t, reads:

(2)

where is an energy scale, σ = (σx, σy, σz) are the Pauli spin matrices, nˆ = (n x , n y , nz ) is the unit vector pointing towards the direction of the (pseudo)magnetic field, whose magnitude is B0 and α can take values in the complex numbers set, except zero, in which case we are lead to a trivial problem and effectively spinless system. In order to calculate the α-th power of the matrix σ ⋅nˆ , it is convenient to consider an extension of the Euler's formula for Pauli matrices, which reads:

∣ψ (t ) 〉 = c1 e−iω1 t ∣1〉 + c2 e−iω2 t e−Γt ∣2〉 , (3)

(4)

As a next step, taking the α-th power of σ ⋅nˆ , one gets: απ 2

⎡cos ⎛ απ ⎞ + i sin ⎛ απ ⎞ σ ⋅nˆ ⎤ . ⎝ 2 ⎠ ⎝ 2 ⎠ ⎣ ⎦

(5)

It can be easily demonstrated that the eigenvalues of the operator (σ ⋅nˆ)α are λ+ = 1 and λ2 = e−iαπ, with the corresponding eigenstates being given by:

∣1〉 = cos θ∣↑〉 + eiφ sin θ∣↓〉 , ∣2〉 = sin θ∣↑〉 −

eiφ

cos θ∣↓〉 ,

(10)

where ω1 = B0α /ℏ , ω2 = B0α cos(απ )/ℏ and Γ = B0α sin(απ )/ℏ . Notice that the state ∣2〉 will be exponentially damped, leading to a probability nonconservation, meaning that for t → ∞ the probability of measuring the system at state ∣2〉 vanishes. The attenuation constant Γ sets the time scale for the exponential decay, while the resonant frequency of the system will be given by ω0 = ω1 − ω2 = 2B0α sin2 (απ /2)/ℏ . The extreme cases are α = 0 and α = 1. The later case is the usual spin 1/2 hamiltonian, which yields the so-called Larmor resonant frequency ω0 = 2B0/ℏ for a spin 1/2 particle immersed in a magnetic field and no dissipation (Γ → 0), while in the former the hamiltonian reduces to a doubly degenerate energy state and ω0 = 0, i.e., the system becomes effectively spinless. Therefore, it is reason to exclude the scenario in which α = 0, by imposing the constraint that |α| > 0. For α > 1 the state ∣2〉 evolves in time with an exponential growth, which is usually undesired from a physical viewpoint, but could be interesting in effectively describing the pumping of a two-level system by means of an external source, which constantly populates one of the energy levels. On the other hand, if exponential growth must be avoided for any value of α > 1, one must consider the negative of (8) or change its hermitian version. A fractional spin 1/2 hamiltonian with dissipation would be seen as

Making θ = π in the above equation one can straightforwardly show the following identity:

(σ ⋅nˆ)α = e−i

(9)

2.1. Spin dynamics with dissipation

B0α

ˆ /2 σ ⋅nˆ = −ieiσ ⋅ nπ .

(8)

The energy states of the hermitian version (9) have the values E1 = B0α and E2 = B0α cos(απ ) , but the eigenstates are unchanged, being given by (6) and (7). Before going to the main result of the present contribution, which is the fractional Dirac equation in 1 + 2 dimensions, two straightforward applications, the spin dynamics in a dissipative scenario and the fractional paramagnetic susceptibility, will be briefly discussed.

In what follows, consider as the starting point the following hamiltonian:

θ θ ˆ /2 eiσ ⋅ nθ = cos ⎛ ⎞ + iσ ⋅nˆ sin ⎛ ⎞ . ⎝2⎠ ⎝2⎠

⎡cos ⎛ απ ⎞ + i sin ⎛ απ ⎞ σ ⋅nˆ ⎤ . ⎝ 2 ⎠ ⎝ 2 ⎠ ⎣ ⎦

The above hamiltonian is inherently non-hermitian, but a hermitian version can be promptly obtained by adding the hermitian conjugate in the form H′ = (H + H†)/2, yielding:

2. General framework for fractional spin 1/2 Hamiltonian

Ĥ = B0α (σ ⋅nˆ)α ,

απ 2

(6) (7)

respectively, where (θ, φ) are the angles in spherical coordinates which define nx = sin θ cos φ, ny = sin θ sin φ and nz = cos θ. Notice that the eigenvectors ∣1〉 and ∣2〉 are also the eigenstates of σ ⋅nˆ having eigenvalues +1 and −1, respectively. This way the hamiltonian (2) can be written as follows: 2

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an alternative description of the Caldeira-Legget model describing a two level sub-system. However, contrasting to the Caldeira-Legget model, where the two levels are coupled to the thermal bath, here only one of the energy levels is effectively coupled to a collection of (implicit) energy levels or to a thermal bath, leading to an asymmetric dissipation in the two-level subset. In the Caldeira-Legget model, fractional exponents can also be introduced by means of the spectral function of the thermal reservoir, while in the present model the exponent arises from the fractional quantum hamiltonian itself.

Ĥ =

− ⎡B0α sin2 ⎣

In the present Section, the hermitian version (9) of the fractional spin 1/2 hamiltonian will be considered. Using the formalism of density matrix to obtain the magnetic susceptibility, it can be shown that the partition function in the canonical ensemble reads: α

=

α

∑n = 1,2 e−βEn = e−βB0 + e−βB0 α 2 2e−βB0 cos (απ /2) cosh (βB α

0

( ) + nJ sin (απ ) ⎤⎦ ∑ σ , απ 2

2

⎡ n sin2 (απ ) − cos(απ )sin2 m = tanh ⎢ T / Tc ⎢ ⎣

(11)

sin2 (απ /2))

i zi

(15)

,

ns μB α sin2 (απ /2) α − 1 ∂M = B0 . ∂B0 kB T

( ) m ⎤⎥ , απ 2

⎥ ⎦

(16)

where m = M/Ms ≤ 1 is the normalized magnetization, in modulus, and Tc is critical temperature depending on Ms and J. By solving (16) at any given temperature T one finds the spontaneous magnetization of the system. Fig. 1 displays the behavior of the spontaneous magnetization as a function of normalized temperature. The spontaneous magnetization exactly vanishes at the critical temperature TcK only for the usual case α = 1. In this special case the usual dependence m ∝ Tc − T is found. Experimentally, however, the correct power law describing the magnetization as a function of temperature differs from the usual CurieWeiss law and also from more sophisticated predictions made using the methods of renormalization group for the Ising (or Heisenberg) models [17]. Introducing a fractional exponent in the spins of the Ising model would be interpreted, roughly speaking, as an indicative of deviations from crystalline order in a material as well as to some degree of fractalization. Disordered (but nearly self-similar) material is composed of atoms and small molecules at a very microscopic level, but at a larger distance it can be coarse-grained and the macromolecules or spin chains will have an effective spin which scales with some exponent α. This argument can be put into correspondence with the lines of reasoning presented in Ref. [18], which considered the scaling of the size of a polymeric chain. Therefore, here we interpret α as the fractal dimension which scales the elementary spin block.

(12)

where ns is the magnetic moment density and μB is the Bohr magneton. In the limit βB0α < < 1, we have M ≈ ns μB βB0α sin2 (απ /2) and the magnetic susceptibility of a paramagnetic material will take the form below:

χ=

σ σ i, j = i ± 1 zi zj

( )

where β = 1/(kBT) is the reciprocal of the absolute temperature T. The magnetization of the system along the direction nˆ is given by:

M = ns μB ⟨σ ⋅nˆ ⟩ =

απ 2

cos(απ )

sin2 (απ /2)) ,

ns μB tanh (βB0α

( )∑

where a constant energy term is been omitted and n is the number of nearest neighbours. Notice that the fractional system is mapped to a conventional Ising model, but for α ≠ 1, even at vanishing externally applied field, B0 = 0, the spins experience an effective internal field given by nJ sin2(απ). Restricting our attention to the range 0.5 < α < 1.5 in the above model, the first term of the above hamiltonian, responsible for spin-spin interactions, will be negative, representing a ferromagnetic fractional material. In the spirit of a CurieWeiss mean field theory, we can replace the externally applied field B0α in equation (12) by an effective internal field which depends on the απ magnetization of the sample, as nJ sin2 (απ ) − J cos(απ )sin2 2 m , being m proportional to the averaged value 〈σzj〉, leading to the following equation:

2.2. Fractional paramagnetic susceptibility of a spin 1/2 system

Z =

J cos(απ )sin2

(13)

Notice that the magnetic susceptibility in this limit is non-linear on the applied magnetic field B0, contrasting to the usual case for which α = 1, but still depends on the inverse of the absolute temperature. For α < 1 in the limit of vanishing applied field, B0 → 0, the susceptibility diverges, which would imply a permanent magnetization at zero-field, regardless of the temperature value, contradicting (12), which predicts M → 0 for B0 = 0. A possible explanation would be that the usual definition of magnetic susceptibility no longer applies for B0 → 0 and a more suitable definition would correspond to a fractal susceptibility, in the form χ = ∂M / ∂B0α , which restores the independence of the susceptibility w.r.t to the applied field. However, in the light of equation (12) we know that the magnetization vanishes at zero field (there is no spontaneous magnetization for any given finite temperature). Keeping the usual definition (13) for the susceptibility implies that at small applied field the system is highly responsive, with the actual value of the susceptibility limited other physical mechanisms, not captured in the above model, the susceptibility decreasing as the applied field is increased. It can be used to describe a soft ferromagnet, which can be easily magnetized (high susceptibility at small fields) and hysteresis is ideally absent. 2.3. Fractional mean-field equations The starting point is the hermitian fractional version of the Ising model describing fractional ferromagnetic interactions, given below:

Ĥ=−

J 2

∑ σziα σzjα − i, j ≠ i

B0α 2

∑ σziα + h. c. i

(14)

where the possible eigenvalues of σzi are ± 1, J is an effective exchange coupling between fractional the spins. Using (5), we can express σzi = e−iπα/2[ cos(απ/2) + i sin(α/2)σzi], which after a little bit of algebra allows one to recast (14) into the following expression:

Fig. 1. Behavior of the normalized spontaneous magnetization as a function of normalized temperature T/Tc, for n = 2 and distinct values of α. 3

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C.A. Dartora and F. Zanella

fermions. In the following, for the sake of convenience, we will consider a first order time derivative operator, at the expense of loosing Lorentz invariance. This way, the eigenvalues (19) and (20) are the energy states already, and the energy-momentum dispersion acquires a fractional exponent, which seems to be more interesting.

The scenario presented here can be useful to describe experimental results in ferromagnetic polymers, which effectively present fractal dimension. 3. The fractional Dirac equation in 1 + 2 dimensions

3.1. Fractional massive fermions

The Dirac equation can be written in any number of spacetime dimensions d ≥ 2, provided a suitable set of Dirac gamma matrices. The dimensionality of the Dirac matrices are related to d as 2⌊d/2⌋, where ⌊d/ 2⌋ is the floor function. In d = 1 + 1 and d + 1 + 2 spacetime dimensions the Dirac matrices have size 2 × 2 and by a fortuitous coincidence a representation for the Dirac algebra is provided by the Pauli matrices, making γ0 = σz e γ1 = iσy, γ2 = −iσx [19]. Therefore, for the sake of simplicity, in the present Section, the fractional version of the Dirac equation will be obtained in 1 + 2 spacetime dimensions. The Dirac equation in 1 + 2 dimensions has been studied for a long time, but only in recent years it attracted the attention of a broad audience in the scientific community, due to the experimental production of a truly two-dimensional material known as graphene, in which electrons behave as massless Dirac fermions, living in an effectively 1 + 2 dimensional world [20–27]. It is a well known fact that the Dirac equation is the correct solution to the problem of finding first order differential operators which satisfy the relativistic dispersion relation, E2 = p2 + m2 in 1 + d spacetime dimensions (natural units, in which ℏ = 1 and c = 1, are being used), where m is the invariant mass term under the Lorentz-Poincarè symmetry group. Roughly, it corresponds to calculate the square root of p2 + m2, which is related to the Klein-Gordon operator. In 1 + 2 dimensions, it leads to the Dirac hamiltonian in the form HD = σxpx + σypy + σzm. Now, one can ask if there is a solution to the problem of finding the root of order α/2 of the relativistic dispersion, being α any number. The (non-unique) answer to the posed question is a fractional version of the Dirac hamiltonian, taking the form below:

respectively. The following features are noticeable: i) dispersion relation is no longer parabolic, being proportional to |p|2α, which can be promptly related to the fractional Riesz derivative [9]; ii) the bandgap is given by 2mα sin2(απ/2) and particle's mass cannot be defined, because the usual non-relativistic quantum mechanical definition meff = 1/(∂2E/∂p2) diverges at |p| = 0. It is a consequence of the McLaurin series failure for fractional exponents at |p| = 0, which means integer derivatives will lead to vanishing or divergent results. As a matter of fact, another way of defining mass is through the poles of

Ĥ = (σx px + σy py + σz m)α .

3.2. Fractional massless Dirac fermions

In the non-relativistic limit (m ≫ |p|), the conduction and valence energy dispersion relations can be given approximately by:

Ec (p) ≈ mα + α

(20)

cos(απ ) ,

(22)

For vanishing mass, (m = 0 in the relativistic dispersion relation), the energy eigenvalues are given by Ec = |p|α and Ev = cos(απ)|p|α. At small momentum and α ≈ 1, the positive and negative energy solutions have velocities v = 1 and v = |cos(απ)|, respectively, which slightly differ. Also, the densities of states for electrons and holes are given by:

where E0 = px2 + py2 + m2 and nˆ = (px , py , m)/ E0 . The eigenvalues of the fractional Dirac hamiltonian are given explicitly below:

Ev (p) = (px2 + py2 + m2)α /2 cos(απ ) ,

|p|2α 2m2 − α

|p|2α

(18)

(19)

(21)

Green's functions, but here G (E , p) = (E − mα − α 2m2 − α )−1, which has poles but branch cuts (due to the fractional exponents) in the complex plane. This way, mass becomes a continuous parameter [28] and quasiparticles with a definite mass are absent in the usual sense. However, it is possible to define a fractional pseudo-mass from the coefficients multiplying |p|2α. For the fractional electron-like and hole-like energy bands, we have mα and mα/| cos(απ)|1/(2−α), respectively, which take distinct values for α ≠ 1.

(17)

Ec (p) = (px2 + py2 + m2)α /2 ,

,

Ev (p) ≈ mα cos(απ ) + α

Clearly, it is equivalent to the coupling of a pseudo-spin to a pseudomagnetic field B = (px, py, m), allowing one to rewrite the equation as follows:

Ĥ = E0α (σ ⋅nˆ)α ,

|p|2α 2m2 − α

Dc (E ) =

D0 2/ α − 1 E α

,

Dv (E ) =

D0 α | cos(απ )|

(

(23) 2/ α − 1 E | cos(απ )|

)

,

(24)

respectively. Assuming the Boltzmann transport approximation to be valid, the electric conductivity σ is proportional to the density of charge V carriers (n = ∫0 g D (E ) dE ) at a given applied gate voltage Vg, in the form σ = σ0n, where σ0 is a constant related to the elementary electric charge e and the mean free path (lF) of the carriers (for α = 1 one has σ0 = lFe2/(vFm), being vF the Fermi velocity). Usually, the gate voltage is applied to the graphene sheet by means of an additional electrode mounted on the bottom of the substrate. For negative (positive) values of Vg the hole (electron) energy band is populated. This way, one can obtain the following expressions:

representing the energy of fractional electrons and holes (positrons) in a (1 + 2)-dimensional spacetime, if time derivative is considered to be of first order (which destroys Lorentz invariance). Notice that, for uniform plane waves (eigenstates of the energy and momentum operators, simultaneously), the above eigenvalues can only be obtained from a fractional differential operator. We point out that to ensure Lorentz invariance one must consider a fractional order temporal derivative, by taking (i∂/∂t)α in the Schrödinger equation. In the hermitian representation of this Dirac hamiltonian, by a coincident related to the properties of Pauli matrices, which provide a representation of the integer order Dirac matrices in 1 + 2 dimensions and also appear in the fractionalized version, the fractional Dirac gamma matrices can be kept unchanged from the integer versions, replacing only the order of the spacetime differential operators. However, in higher number of spacetime dimensions the fractional Dirac matrices must be modified, but this goes beyond the scope of the present contribution. Next, we will discuss two limiting cases of special interest: i) fractional massive non-relativistic fermions and ii) fractional massless Dirac

σ (Vg ) = σ (Vg ) =

σ0 2/ α V 2 g

, Vg > 0 ,

σ0 2 | cos(απ )|2/ α − 1

Vg2/ α , Vg < 0 .

(25) (26)

Fig. 2 shows the behavior of the conductivity as a function of the applied gate voltage, for distinct values of α. Notice that the conductivity is asymmetric for α ≠ 1. Disorder in graphene has many distinct origins, being classified mainly according to the length scale (short and long-range disorder) and to the type as extrinsic (coming from the substrate and chemical 4

Physica E: Low-dimensional Systems and Nanostructures 115 (2020) 113685

C.A. Dartora and F. Zanella

Finally, a fractional version of Dirac's equation in 1 + 2 dimensions and its consequences for the graphene conductivity was also examined. Conflicts of interest None. Acknowledgements C.A. Dartora would like to thank the Brazilian agency CNPq for partial financial support through scholarship CNPq 301848/2017-3. References [1] L.H. Ryder, Quantum Field Theory, 2nd. ed., Cambridge University Press, 1996. [2] S. Weinberg, The Quantum Teory of Fields vol. I and vol. II, Cambridge University Press, 1996 and references therein. [3] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Netherlands, 2007. [4] C.G. Bollini, J.J. Giambiagi, A. Gonzáles Domínguez, Il Nuovo Cimento 31 (1964) 550. [5] D.G. Barci, L.E. Oxman, M. Rocca, Int. J. Mod. Phys. A11 (1996) 2111. [6] D.G. Barci, C.G. Bollini, L.E. Oxman, M.C. Rocca, Int. J. Theor. Phys. 37 (1998) 3015. [7] Fred Riewe, Phys. Rev. E 53 (1996) 1890. [8] V. Zaburdaev, S. Denisov, J. Klafter, Rev. Mod. Phys. 87 (2015) 483. [9] N. Laskin, Phys. Rev. E 66 (2002) 056108. [10] N. Laskin, Phys. Lett. A 268 (2000) 298. [11] N. Laskin, Phys. Rev. E 62 (2000) 3135. [12] Bruce J. West, Rev. Mod. Phys. 86 (2014) 1169. [13] Everton M.C. Abreu, Cresus F.L. Godinho, Phys. Rev. E 84 (2011) 026698. [14] R.P. Feynman, F.L. Vernon, Ann. Phys. 24 (1963) 118. [15] A.O. Caldeira, A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211. [16] (a) A.J. Leggett, S. Chakravarty, A.T. Dorsey, Matthew P.A. Fisher, Anupam Garg, W. Zwerger, Rev. Mod. Phys. 59 (1987) 1; (b) A.J. Leggett, S. Chakravarty, A.T. Dorsey, Matthew P.A. Fisher, Anupam Garg, W. Zwerger, Erratum Rev. Mod. Phys. 67 (1995) 725. [17] Jean Zinn-Justin, Phase Transitions and Renormalization Group, Oxford University Press, 2013. [18] S. Havlin, D. Ben-Avraham, J. Phys. A Math. Gen. 15 (1982) L311. [19] C.A. Dartora, G.G. Cabrera, Phys. Rev. B 87 (2013) 165416. [20] P.R. Wallace, Phys. Rev. 71 (1947) 622. [21] J.W. McLure, Phys. Rev. 108 (1957) 612. [22] G.W. Semenoff, Phys. Rev. Lett. 53 (1984) 2449. [23] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [24] M.I. Kastnelson, K.S. Novoselov, A.K. Geim, Nat. Phys. 2 (2006) 620. [25] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [26] C.A. Dartora, G.G. Cabrera, Phys. Lett. A 377 (2013) 907–909. [27] Miguel S. Jimenez, C.A. Dartora, Physica E 59 (2014) 1–5. [28] D.G. Barci, L.E. Oxman, Mod. Phys. Lett. A12 (1997) 493. [29] E.R. Mucciolo, C.H. Lewenkopf, J. Phys. Condens. Matter 22 (2010) 273201. [30] S. Adam, P.W. Brouwer, S. Das Sarma, Phys. Rev. B 79 (2009) 201404 R.

Fig. 2. Conductivity of a fractional graphene sheet as a function of the applied gate voltage Vg, for distinct values of the fractional parameter α. Notice that, for α ≠ 1, the curves are asymmetric with respect to the sign of Vg.

contamination, for instance) or intrinsic. It is accepted that graphene has few lattice defects, weakening the intrinsic disorder. In Ref. [29] a detailed theoretical account of the disorder effects is provided, with brief discussions of experimental aspects. It has been found that the conductivity scales with n3/2, considering a Gaussian disorder model in the standard Boltzmann transport [30], i.e., the exponent of n is 3/2. Here, the approach is to include disorder through the fractional exponent α in the energy bands, while keeping the conductivity proportional to n. As a final remark in the present Section, we point out that the asymmetry in the conductivity w.r.t. the applied gate voltage is actually verified in experiment [25], which is in no contradiction with the fractional model presented here. 4. Conclusion In summary, in the present contribution a simple extension to the spin 1/2 hamiltonian was put forward, by taking the α-th fractional power of the spin-1/2 Hamiltonian, leading to a dissipative system in its non-hermitian version. The obtained hamiltonian can be used to describe a vast class of quantum mechanical problems displaying relaxation, like a spin 1/2 in the presence of a thermal bath, or the asymmetry in energy spectrum between electrons and holes in a semiconductor. Using the fractional hamiltonian in the context of a Curie-Weiss meanfield theory for a fractional magnet, the normalized spontaneous magnetization as a function of the absolute temperature was explored.

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