Edge states and thermodynamics of rotating relativistic fermions under magnetic field
EEdge states and thermodynamics of rotating relativistic fermions under magnetic field
M. N. Chernodub
1, 2 and Shinya Gongyo Laboratoire de Math´ematiques et Physique Th´eorique UMR 7350, Universit´e de Tours, Tours 37200 France Laboratory of Physics of Living Matter, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia Theoretical Research Division, Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan (Dated: March 5, 2018)We discuss free Dirac fermions rotating uniformly inside a cylindrical cavity in the presence ofbackground magnetic field parallel to the cylinder axis. We show that in addition to the known bulkstates the system contains massive edge states with the masses inversely proportional to the radiusof the cylinder. The edge states appear at quantized threshold values of the fermion mass. In thelimit of infinite fermion mass the masses of the edge states remain finite but, generally, nonzero ascontrasted to the bulk states whose masses become infinite. The presence of magnetic field affectsthe spectrum of both bulk and edge modes, and the masses of the edge states may vanish at certainvalues of magnetic field. The moment of inertia of Dirac fermions is non-monotonically increasing,oscillating function of magnetic field. The oscillations are well pronounced in a low-temperaturedomain and they disappear at high temperatures.
I. INTRODUCTION
Rotating systems of relativistic fermions appear in var-ious physical settings characterized by different energyscales. The examples include interior of rapidly spin-ning neutron stars [1], quark-gluon plasma in noncentralheavy-ion collisions [2], and anomalous chiral transportphenomena [3] applied both to neutrino fluxes in rotat-ing astrophysical environments [4, 5] and to semimetalmaterials in solid state applications [6].Rotation changes the spectrum of free fermions [7–11]and, consequently, affects the mass gap generation in in-teracting fermionic systems. For example, the criticaltemperature of chiral symmetry restoration T c is a di-minishing function of the rotational angular frequency Ω[12–16]. The rotational effects have been studied undersimplifying assumption that the rotation is globally uni-form, so that the angular velocity does not depend onthe distance to the rotational axis. A uniformly rotatingrelativistic system should be bounded in the transversedirections with respect to the axis of rotation in orderto comply with the causality principle. The latter re-quires that the velocity of particles should not exceed thespeed of light to avoid pathological effects [9, 17]. Thepresence of the boundary implies a dependence of thechiral restoration temperature T c = T c (Ω) on geometri-cal features, in particular, on the type of the boundarycondition [18]. The uniform rotation in magnetic fieldbackground but in an unrestricted transverse geometryhas been studied in Ref. [13].In this paper we generalize the results of Refs. [10, 18]in threefold way. Firstly, we show that in addition to thebulk modes the spectrum of free massive Dirac fermionscontains the edge states localized at the boundary of thecylinder. Secondly, we discuss the spectrum of both bulkand edge modes in the presence of external magneticfield. Finally, we illustrate the importance of the edgemodes for thermodynamics of free Dirac fermions andfor its rotational properties such as moment of inertiawhich exhibits curious oscillating behavior as a function of magnetic field.Notice that possible effects of the edge states were notaccounted for in existing studies of phase structure of theinteracting rotating fermions [13–16, 18]. In Ref. [13, 14]rotational properties were investigated in the transver-sally unrestricted geometry which questions the consis-tency with the requirement of relativistic causality un-der uniform rotation and, simultaneously, does not allowfor the presence of the edge states. The existence of theedge states, found in the present paper, definitely callsfor a re-estimation of the phase diagram of interactingfermions under uniform rotation.We would like to mention that in solid state termsthe system of Dirac fermions considered in this articlecorresponds to a non-topological insulator as it is char-acterized by the presence of gapped bulk modes and theabsence of symmetry-protected boundary (edge) stateswith zero mass. The edge states are generally massiveand their mass is proportional to the mean curvature ofthe cylinder surface. This statement is not surprising be-cause the Dirac equation alone is not enough to describetopological insulators [23], where the presence of zero-mass edge states is guaranteed by topological reasons ofunderlying lattice Hamiltonians [24].The structure of this paper is as follows. In Sect. II wereview, following Ref. [10], known bulk solutions for theDirac fermions in the cylinder with the MIT boundaryconditions in the absence of magnetic field. We also dis-cuss particularities of the spectrum for the chiral bound-ary conditions [18]. In the same section we find the edgestates of the system and describe their properties. InSect. III we discuss properties of bulk and edge solutionsin the magnetic field background. Section IV is devotedto studies of rotational properties of the system in thelimit of (negative) infinite fermion mass. In this limitthe thermodynamics of the system is given by the edgestates only, allowing us to highlight the importance of theedge states. The last section is devoted to conclusions. a r X i v : . [ h e p - t h ] J un II. BULK AND EDGE SOLUTIONS IN THEABSENCE OF MAGNETIC FIELD
In this section we discuss solutions of massive rigidlyrotating Dirac fermions confined in a cylindrical geome-try in the absence of magnetic field. We start from theknown bulk states that were already described in Ref. [10](see also Ref. [15]) and then we demonstrate that the sys-tem contains also certain new (edge) states which possessrather peculiar properties.
A. Dirac equation in the cylinder
We consider a system of free fermions which is rigidlyrotating with the angular frequency Ω about the axis ofthe infinitely long cylinder of the radius R .Given the geometry of the system it is convenient towork in the cylindrical coordinates, x ≡ ( x , x , x , x ) =( t, ρ cos ϕ, ρ sin ϕ, z ). There are two natural referenceframes in this problem: the inertial laboratory frameand non-inertial corotating frame. The former one cor-responds to a rest frame while the latter one is rigidlyfixed with the rotating system. The coordinates t , ρ and z in the corotating reference frame coincide with the cor-responding coordinates of the laboratory frame: t = t lab , ρ = ρ lab and z = z lab . The angular variables in theseframes are related as follows: ϕ = [ ϕ lab − Ω t ] π , (1)where [ . . . ] π means “modulo 2 π ”. The simple relationbetween angular variables (1) leads, nevertheless, to quitenontrivial metric in the corotating frame: g µν = − ( x + y )Ω y Ω − x Ω 0 y Ω − − x Ω 0 − − , (2)which corresponds to the line element ds = g µν dx µ dx ν = η ˆ µ ˆ ν dx ˆ µ dx ˆ ν (3)= (cid:0) − ρ Ω (cid:1) dt − ρ Ω dtdϕ − dρ − ρ dϕ − dz , where η ˆ µ ˆ ν = diag (1 , − , − , −
1) is the flat metric. Herewe adopt the convention that ˆ i, ˆ j · · · = ˆ t, ˆ x, ˆ y, ˆ z and µ, ν · · · = t, x, y, z refer to the local coordinates in thelaboratory frame and the corotating frame, respectively.We use the units in which the speed of light and thereduced Planck constant are equal to unity, c = (cid:126) = 1.The spectrum of the fermions is described by the eigen-functions of the free Dirac equation in the corotating ref-erence frame: [ iγ µ ( ∂ µ + Γ µ ) − M ] ψ = 0 , (4)where the Dirac matrices in the curved corotating space-time γ µ ( x ) = e µ ˆ i ( x ) γ ˆ i are connected to the matrices inthe laboratory frame γ ˆ i via the vierbein e µ ˆ i . The vierbein is a “square root” of the metric η ˆ i ˆ j = g µν e µ ˆ i e ν ˆ j . In thecase of metric (2) the vierbein may be chosen in the form e t ˆ t = e x ˆ x = e y ˆ y = e y ˆ y = 1 , e x ˆ t = y Ω , e y ˆ t = − x Ω , (5)with all other components of η ˆ i ˆ j being zero.In Eq. (4) the spin connection Γ µ in the metric (2) hasonly one nonzero component:Γ t = − i σ ˆ x ˆ y , (6)where σ ˆ x ˆ y ≡ Σ z = (cid:18) σ σ (cid:19) (7)in the Dirac representation of the gamma matrices: γ ˆ t = (cid:18)
1l 00 − (cid:19) , γ ˆ i = (cid:18) σ i − σ i (cid:19) , γ = (cid:18) (cid:19) . (8)Equation (4) is supplemented with the MIT boundarycondition at the boundary of the cylinder: iγ µ n µ ( ϕ ) ψ ( t, z, R, ϕ ) = ψ ( t, z, R, ϕ ) . (9)where the spatial vector n µ ( ϕ ) = (0 , R cos ϕ, − R sin ϕ, j n ≡ − j µ n µ of the fermionic cur-rent j µ = ¯ ψγ µ ψ to vanish at the surface of the cylinder j n ( ρ = R ) = 0. B. Bulk states
A general solution of the Dirac equation (4) in the(co)rotating reference frame (1) has the following form: U λj = 12 π e − i (cid:101) Et + ikz u λj ( ρ, ϕ ) , (10)where u λj is an eigenspinor characterized by the eigen-state helicity λ = ± /
2, the z -component of momentum k ≡ k z ∈ R , the projection of the quantized angular mo-mentum m ≡ m z ∈ Z onto the z axis, and the radialquantum number l = 1 , , . . . which describes the be-havior of the solution in terms of the radial ρ coordinate.The helicity λ of the state is the eigenvalue of the helicityoperator ˆ W = ˆ P · ˆ J /p ,ˆ W U λEk z m = λU λEk z m , (11)where ˆ P = − i ∂ is the momentum operator and ˆ J is theangular momentum operator. In the absence of magneticfield the helicity operator ˆ W has the following simpleform: ˆ W = (cid:18) ˆ h
00 ˆ h (cid:19) , ˆ h = σ · ˆ P p , (12)where p ≡ √ E − M > ˆ P U λj = p U λj . (13)Here the notation j = ( k z , m, l ) is used to denote a set ofquantum numbers [10].The energy in the corotating frame (cid:101) E j is related to theenergy E j in the laboratory frame as follows: (cid:101) E j = E j − Ω (cid:16) m + 12 (cid:17) ≡ E j − Ω µ m , (14)where µ m can be identified with the quantized value ofthe z -component of the total angular momentumˆ J z ψ = µ m ψ , µ m = m + 12 , (15)which comprises the orbital and spin parts:ˆ J z = − i∂ ϕ + 12 Σ z , (16)where the matrix Σ z is given in Eq. (7).The solutions of the Dirac equation which satisfy theMIT boundary conditions (9) are linear combinations ofpositive and negative helicity spinors: U MIT j = C MIT j (cid:2) b U + Ekm + U − Ekm (cid:3) , (17)where the four-spinors with a definite helicity λu λj ( ρ, ϕ ) = 1 √ (cid:18) E + φ λj λE | E | E − φ λj (cid:19) (18)are expressed with the two-spinors φ λj ( ρ, ϕ ) = 1 √ (cid:18) p λ e imϕ J m ( qρ/R )2 iλ p − λ e i ( m +1) ϕ J m +1 ( qρ/R ) (cid:19) , (19) which are eigenspinors of the two-component helicity op-erator ˆ h (12): (cid:18) k j ˆ P − ˆ P + − k j (cid:19) φ j ( ρ, φ )2 p j = λ j φ j ( ρ, φ ) (20)with ˆ P ± = ˆ P x ± i ˆ P y = − ie ± iϕ (cid:0) ∂ ρ ± iρ − ∂ ϕ (cid:1) .In the eigenfunctions (17) the degree of mixing betweenpositive and negative helicity states is determined by theparameter b = E + p + + E − p − j ml sign( E ) E + p − + E − p + j ml sign( E ) , (21)where p ± ≡ p ± / = (cid:115) ± k z p , E ± ≡ E ± / = (cid:114) ± ME , (22)are, respectively, the momentum- and energy-relatedquantities which depend explicitly on the helicity of theeigenmodes, and p = (cid:114) k z + q R (23)is (the modulus of) the effective momentum which in-corporates the longitudinal continuous momentum k z and the transverse (radial) discrete momentum number q ≡ q ml . We also used the notation [10]j ml = J m ( q ml ) J m +1 ( q ml ) , (24)where J m ( x ) is the Bessel function.The dimensionless real-valued and positive quantity q ml is the l th real-valued positive root ( l = 1 , , . . . ) ofthe following equation: J m ( q ) + 2 M Rq J m ( q ) J m +1 ( q ) − J m +1 ( q ) = 0 . (25)The normalization coefficient C MIT j = 1 R | J m +1 ( q m,l R ) | · (cid:118)(cid:117)(cid:117)(cid:116) p − + p j ml (j ml + 1)(j ml − (2 m + 1) j ml q m,l R + 1) − (j ml − j ml q m,l R . (26)ensures that these modes are orthonormalized (cid:10) U MIT j , U MIT j (cid:48) (cid:11) = δ ( k j − k j (cid:48) ) δ m j ,m j (cid:48) δ l j ,l j (cid:48) θ ( E j E j (cid:48) ) , (27)with respect to the inner Dirac product: (cid:104) ψ, χ (cid:105) = (cid:90) + ∞−∞ dz (cid:90) π dϕ (cid:90) R dρ ρ ψ † ( x ) χ ( x ) . (28) The energies of the eigenmodes in the laboratory frameare as follows: E j ≡ E ml ( k z , M ) = ± (cid:114) k z + M + q ml R , (29)where the plus (minus) sign corresponds to the particle(antiparticle) modes.The density ¯ ψγ ψ ≡ ψ † ψ of the wavefunctions (18) isnot localized at the boundary of the cylinder, and there-fore we refer to these solutions as to the “bulk eigen-modes”. They should be discriminated from the “edge”solutions (to be discussed below) for which the densityis concentrated at the boundary of the cylinder. FromEq. (29) we conclude that the masses of the bulk states M bulk defined as M bulk ml = (cid:114) M + q ml R (30)are higher than or equal to the mass of the fermion M .The reflection m → − − m , corresponding to the signflips of the total angular momentum (15) µ m → − µ m ,leaves the q ml solutions unchanged, q ml → q − − m,l ≡ q ml . (31)This property implies that the mass spectrum (30) and,consequently, the energy spectrum of the bulk modes isinvariant under the flips µ m → − µ m . q l q l q l q l - - MRq ml (a) m = m = m = m = - - - - MR ν m e dg e - - - - MR ( ν m e dg e ) (b)FIG. 1. (a) Solutions of the eigenvalue equation (25) as thefunction of the fermion mass M : (a) the real-valued solu-tions q (cid:62) q edge = iν edge with ν (cid:62) ν edge ) vs M . A couple of real-valued solutions q ml of Eq. (25) areshown in Fig. 1(a) as a function of the fermion mass M .As the mass M decreases, the lowest ( l = 1) real-valuedmodes q ml (cid:62) q = 0 axis and disappear one byone at the critical values of the (negative) fermion mass: M ( m ) c = − R (cid:18) | µ m | + 12 (cid:19) ≡ (cid:40) − mR , m (cid:62) , mR , m < . (32)As the values q m, and q − − m, coincide with each otherdue to the reflection invariance (31), the real-valued q will disappear in pairs at the critical mass points (32).Contrary to the ground state with l = 1, the excited l (cid:62) n µ in the MIT boundarycondition (9), then the mass critical values (32) wouldalso change the sign, M ( m ) c ( − n µ ) = − M ( m ) c ( n µ ) so thatdisappearance of the ground state ( l = 1) modes wouldthen happen at the positive fermion masses, M ( m ) c > (cid:2) iγ µ n µ ( ϕ ) − e − i Θ γ (cid:3) ψ ( t, z, ρ, ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = R = 0 , (33)the critical masses becomes as follows [18]: M ( m ) c (Θ) = M ( m ) c (0)cos Θ ≡ (cid:40) − m cos Θ 1 R , m (cid:62) , m cos Θ 1 R , m < . (34)In particular, at the specific values of the chiral angleΘ = ± π/ C. Edge states
Besides the bulk eigenfunctions with real-valued so-lution q = q ml the system contains also quite peculiareigenstates which are localized at the boundary of thecylinder. These are the edge states which correspond topurely imaginary solutions of Eq. (25): q edge m = iν edge m , (35)with a real ν edge m (cid:62) Using the relation J m ( ix ) = i m I m ( x ) we get fromEq. (25) the following equation which determines ν : I m ( ν ) + 2 M Rν I m ( ν ) I m +1 ( ν ) + I m +1 ( ν ) = 0 , (36)where I m ( x ) is the modified Bessel function. As in the case of the bulk modes, the solutions ν edge m and − ν edge m correspond to the same eigenmode. In Fig. 1(b) we show the solutions of Eq. (36) as thefunction of the fermion mass M . First of all we noticethat there is only one edge eigenmode for each value ofthe orbital momentum m . Moreover, the edge modes ν edge m appear at the critical mass points (32) where thelowest bulk modes q m, disappear (as the fermion mass M diminishes). Therefore we conclude that at the criticalmass points (32) the lowest bulk modes (32) are trans-formed into the edge modes and vise versa.The energy E of the edge states in the laboratory frameis as follows: E edge m = ± (cid:112) p + M ≡ ± (cid:113) k + ( M edge m ) , (37)where p = (cid:114) k − ν m R , (38)is an analogue of momenta. The plus (minus) sign inEq. (37) corresponds to the particle (antiparticle) modessimilarly to the bulk modes (29).Equation (37) implies that contrary to the masses ofthe bulk states (30) the masses of the edge states aresmaller or equal to the mass of the fermion M : M edge m = (cid:114) M − ν m R . (39)Notice that due to the inequality | ν m | < M R themasses (39) of the edge states and their energies E edge always remain real numbers while the effective momen-tum p may take become purely imaginary for longitudinalmomenta | k | < ν/R . In other words, for the edge modes ν > k > E > p may take bothpositive and negative values.In the rotating frame the energy of the edge mode fol-lows from Eq. (14): (cid:101) E edge m = E edge m − Ω µ m . (40)where E edge m is the energy of the edge modes in the lab-oratory frame (37).In Fig. 2 we show the mass spectrum both for the bulkmodes (30) and for the edge modes (39). This figureclearly demonstrates that the ground state l = 1 becomesthe edge mode as the critical point (32) is passed for eachfixed m .Similarly to the bulk modes (31), a reflection in thesign of the total angular momentum (15), µ m → − µ m ,leaves the ν ml eigenvalues unchanged, q ml → q − − m,l ≡ q ml . (41)Therefore the energy spectrum of the edge modes is sym-metric with respect to the flips µ m → − µ m . Both bulkand edge modes are degenerate in the absence of externalmagnetic field.The two-spinors of the edge eigenmode with definitehelicity λ is given the spinor φ λ = C φ ˇ φ λ , ˇ φ λ = (cid:18) ( k + 2 pλ ) e imϕ I m ( νρ/R ) − iνe i ( m +1) ϕ I m +1 ( νρ/R ) (cid:19) , (42) Bulk modes with l > B u l k m o d e s w i t h l = m = , - m = , - m = , - m = , - - - - - - MR M m e dg e R , M m , l bu l k R FIG. 2. The masses (39) of the edge states (the solid bluelines) and the masses (30) of the lowest ( l = 1) bulk states(the dashed magenta lines) as the function of the fermionmass M in the absence of magnetic field, B = 0. Four loweststates are shown. The critical points (32) are marked by thered dots (and the thin gray lines). The asymptotic masses ofthe edge states (69) in the limit M → −∞ are shown by thegreen arrows. Four lowest l > where C φ is a normalization constant and we implied thatthe check mark over a spinor means that this spinor isnot normalized.In order to prove that the spinor (42) is the eigenmodeof the helicity operator (20) we used the following usefulrelations:ˆ P + (cid:104) e imϕ I m (cid:16) ν ρR (cid:17)(cid:105) = νiR e i ( m +1) ϕ I m +1 (cid:16) ν ρR (cid:17) , (43)ˆ P − (cid:104) e i ( m +1) ϕ I m +1 (cid:16) ν ρR (cid:17)(cid:105) = νiR e imϕ I m (cid:16) ν ρR (cid:17) , (44)for the operatorsˆ P ± = − ie ± iϕ (cid:0) ∂ ρ ± iρ − ∂ ϕ (cid:1) . (45)The two-spinors for the bulk modes (19) were normal-ized using the condition [10]: (cid:88) m ∈ Z φ λ, † Ekm φ λEkm = 1 , (46)which utilized the convenient summation property of theBessel functions: (cid:88) m ∈ Z J m ( x ) = 1 . (47)The edge states (42) depend on modified, rather thenusual, Bessel functions that possess a different summa-tion rule: (cid:88) m ∈ Z ( − m I m ( x ) = 1 . (48)This equations suggests that the edge eigenmodes (42)should be normalized according to another normalizationrelation: (cid:88) m ∈ Z ( − m φ edge , † Ekm φ edge Ekm = 1 , (49)which has a less clear physical sense. Nevertheless, for thesake of completeness, we give the value of the prefactor C φ corresponding to the normalization (49): C φ = 1 √ (cid:112) k + 2 λk Re p + Im p = 1 √ · (cid:40) ( k + 2 λkp ) − / , p > , /ν, p < . (50)In the corotating reference frame the Dirac equation,if expressed via the corotating coordinates, has the sameform as the standard Dirac equation in the laboratoryframe in the absence of rotation. Using the explicit rep-resentation of the γ matrices (8) the Dirac equation (4)in the corotating frame can be rewritten as follows (cid:0) i /∂ − M (cid:1) U edge j,λ ≡ (cid:18) E − M − p ˆ h p ˆ h − ( E + M ) (cid:19) U edge j,λ = 0 , or (cid:18) E − M − pλ pλ − ( E + M ) (cid:19) Ψ edge j,λ = 0 , (51)where we set U edge j,λ = 12 π e − i (cid:101) E edge j t + ik j z Ψ edge j,λ , Ψ edge j,λ = (cid:32) C up ˇ φ λj C down ˇ φ λj (cid:33) , (52)and then used the fact that the two-spinors ˇ φ λj , Eq. (42),are the eigenfunctions of the helicity operator ˆ h , Eq. (12).The self-consistency of the Dirac equation for the edgemodes (51) gives us the expression for their energy (37)and fixes the coefficients C up and C down in Eq. (52) upto the overall normalization factor (set to unity in thisexpression): ˇΨ edge j,λ = (cid:18) ( E + M ) ˇ φ λj λp ˇ φ λj (cid:19) , (53)Denoting Ψ edge j = (Ψ ↑ , Ψ ↓ ) T , the MIT boundary con-ditions (9) may be explicitly written as follows:( i/n − edge j = − (cid:18) iσ ρ − iσ ρ (cid:19) (cid:18) Ψ ↑ Ψ ↓ (cid:19) = 0 , (54)where we set ρ = R and defined σ ρ = σ cos ϕ + σ sin ϕ . (55)The four-spinor solutions satisfying these conditionsshould involve both λ = ± / edge j ≡ Ψ edge j, MIT = (cid:88) λ = ± C λj ˇΨ edge j,λ ≡ (cid:18) ( E + M ) (cid:0) C + j ˇ φ + j + C − j ˇ φ − j (cid:1) p (cid:0) C + j ˇ φ + j − C − j ˇ φ − j (cid:1) (cid:19) . (56) because the MIT boundary condition (9) breaks the he-licity conservation.The self-consistency requirement for the MIT condi-tion (54) and (56) gives us the relation (36) which deter-mines the value of the parameter ν .From Eq. (36) it follows that the nontrivial solutionsfor ν = ν m exist if and only if M <
0. Solving Eq. (36)as a quadratic equation we get: i m ≡ I m +1 ( ν m ) I m ( ν m ) = − M R + sign( µ m ) M edge m Rν m ≡ − MR + M edge m Rν m , m (cid:62) , − MR − M edge m Rν m , m < , (57)where the angular momentum µ m and the mass of theedge state M edge m are given in Eqs. (15) and (39), respec-tively.The coefficients in Eq. (56) satisfy the relation: (cid:88) λ = ± / C λj (1 + 2 λκ ) = 0 , (58)where κ m ( k ) = p [ E m ( k ) + M + νi m /R ] k [ E m ( k ) + M ]= p (cid:2) E m ( k ) − sign( µ m ) M edge m (cid:3) k [ E m ( k ) + M ] ≡ k [ E m ( k ) − M ] p (cid:104) E m ( k ) + sign( µ m ) M edge m (cid:105) , (59)and we adopted the usual convention C ± / j ≡ C ± j . Onecan also rewrite the last expression in the following ex-plicit form: κ m ( k ) = p (cid:18)(cid:113) k + M − ν m R − sign( µ m ) (cid:113) M − ν m R (cid:19) k (cid:18)(cid:113) k + M − ν m R + M (cid:19) . Combining (58) and (56) we get the edge eigenmodein the explicit form:Ψ j = C ( E + M )( κk − p ) e imϕ I m (cid:0) ν m ρR (cid:1) ν m iR ( E + M ) κe i ( m +1) ϕ I m +1 (cid:0) ν m ρR (cid:1) p ( pκ − k ) e imϕ I m (cid:0) ν m ρR (cid:1) ip ν m R e i ( m +1) ϕ I m +1 (cid:0) ν m ρR (cid:1) . (60)The overall constant C is determined by the orthonormalization condition given in Eq.(27). For the edge mode,the Dirac inner product is given by (cid:68) U edge j , U edge j (cid:48) (cid:69) = δ ( k − k (cid:48) ) δ mm (cid:48) θ ( E j E j (cid:48) ) | C | × (cid:34)(cid:26) ( E j + M ) ( κ m k − p ) + p ( pκ m − k ) + (cid:18) ν m R ( E j + M ) κ m + p ν m R (cid:19)(cid:27) I + m +1 / + (cid:26) ( E j + M ) ( κ m k − p ) + p ( pκ m − k ) − (cid:18) ν m R ( E j + M ) κ m + p ν m R (cid:19)(cid:27) I − m +1 / (cid:35) , (61)where I ± m +1 / is defined as I + m +1 / ( ν m ) = (cid:90) R dρ ρ I m ( ν m ρR ) + I m +1 ( ν m ρR )2 = R ν m I m ( ν m ) I m +1 ( ν m ) , I − m +1 / ( ν m ) = (cid:90) R dρ ρ I m ( ν m ρR ) − I m +1 ( ν m ρR )2 = R (cid:104) I m ( ν m ) − m + 1 ν m I m ( ν m ) I m +1 ( ν m ) − I m +1 ( ν m ) (cid:105) . (62)Thus, the normalization coefficient C is given by the following expression: C = 1 | I m ( ν m ) | √ kp √ ν m (cid:34)(cid:20)(cid:26) k + ( E − M ) + ν R (cid:27) i m + 4 νMR i m + (cid:26) k + ( E + M ) + ν R (cid:27)(cid:21) i m + (cid:20) E ( E − M ) i m − νER i m − E ( E + M ) (cid:21) (cid:18) − m + 1 ν m i m − i m (cid:19)(cid:35) − / . (63)The special case k ≡ k z = 0 one gets the followingexplicit expression of the edge eigenmode:Ψ j = C θ ( µ m ) (cid:0) M + M edge m (cid:1) e imϕ I m (cid:0) ν m ρR (cid:1) θ ( − µ m ) (cid:0) M + M edge m (cid:1) e i ( m +1) ϕ I m (cid:0) ν m ρR (cid:1) − iν m θ ( − µ m ) e imϕ I m (cid:0) ν m ρR (cid:1) − iν m θ ( µ m ) e i ( m +1) ϕ I m (cid:0) ν m ρR (cid:1) . The normalization coefficient (for m (cid:62) C = 1 | I m ( ν m ) | (cid:16) γ (cid:104) ν + 2 mγM edge − γ (cid:0) M edge (cid:1) (cid:105)(cid:17) − , where γ m = − ( M + M edge ) Rν . (64)Notice that 0 < γ < ψγ ψ ≡ ψ † ψ grows exponentially asone approaches the edge of the cylinder at ρ = R , Fig. 3.Since all modified Bessel functions I n grow exponentiallyat large values of its argument, the localization length ofthe edge states (60) at the boundary of the cylinder isdetermined by the length scale ξ edge m = Rν m . (65) � = � � = � �� =- � �� =- � �� =- �� ρ / R ψ † ψ ( ρ ) / ψ † ψ ( R ) FIG. 3. An example of the density of the edge modes for k = 0and m = 0 at various fermionic masses M in the absence ofmagnetic field ( B = 0). Thus, the edge modes are characterized by two dimen-sionful parameters, their mass (39) and the localizationlength (65). Notice that the former may be expressed viathe latter: M edge m = (cid:113) M − (cid:0) ξ edge m (cid:1) − , (66)Now, let us consider the behavior of the masses of theedge modes M edge in the limit of a large fermion mass M . For a large positive real z (cid:29) I m ( z ) = e z √ πz (cid:18) − m z + O (cid:0) z − (cid:1)(cid:19) . (67)Substituting Eq. (67) into the relation (57) we get that inthe limit of a large negative mass M the solutions ν edge behave as follows: ν edge m = | M | R − µ m | M | R + O (cid:16)(cid:0) M R (cid:1) − (cid:17) , (68)where µ m is the total angular momentum of themode (15).Therefore, in the limit of the infinite fermionic massthe masses of the edge modes remans finite contrary tothe bulk modes M edge ml which become infinitely massivein this limit (30) and therefore decouple from the system.Moreover, in the limit of large (negative) fermion massthe mass spectrum of the edge modes may be computedanalytically: M edge ∞ ,m = lim M →−∞ M edge m = | µ m | R . (69)We find that the masses of the edge modes (69) are(i) finite, (ii) quantized and (iii) independent of thefermion mass M . According to Eq. (68) the localizationlength (65) tends to zero in this limit. The edge statesare double-degenerate as the modes with opposite angu-lar momenta ( µ m and µ − − m ≡ − µ m ) possess the samemass. We also stress that in the absence of magneticfield there are no massless edge modes in the spectrumin a cylinder of a finite radius R . The modes eventuallybecome massless in the limit of a large radius R → ∞ .In conclusion of this section we would like to noticethat the physical particle-antiparticle interpretation ofthe fermionic modes in the second-quantization formal-ism depends on the presence of the modes for which E (cid:101) E <
0. The physical meaning of such modes is am-biguous (see Refs. [4, 7] as well as the detailed discussionin Ref. [10]), and therefore the absence of such modes inthe spectrum makes the theory well defined. In short,the modes
E >
E <
0) in laboratory frame are in-terpreted as particle (antiparticle) states in the Vilenkinquantization [4] while the modes with (cid:101)
E > (cid:101)
E < E j (cid:101) E j > | Ω | R < E j (cid:101) E j > E edge m (cid:101) E edge m > , (70)provided they rotate within the light cylinder, | Ω | R <
1. Since the energy for k (cid:54) = 0 is grater than the one for k = 0, we focus on the energy for k = 0, | E edge m | = M edge m . (71)The derivative of E edge m with respect to M is given by d | E edge m | dM = M − ν m R dν m dM | E edge m | . (72)The derivative can be also expressed via Eq. (57):1 R dν m dM I m +1 ( ν m ) I m ( ν m ) (cid:20) ν m I (cid:48) m +1 I m +1 − ν m I (cid:48) m I m (cid:21) = − − sign( µ m ) d | E edge m | dM , (73)with I (cid:48) m ( ν m ) = dI m ( ν m ) /dν m . Using the following prop-erties of the modified Bessel functions, I (cid:48) m ( z ) = mz I m ( z ) + I m +1 ( z ) , (74) I (cid:48) m +1 ( z ) = I m ( z ) − m + 1 z I m +1 ( z ) , (75)the derivative can be rewritten as d | E edge m | dM = 2 | µ m | M − | E edge m | M R − | E edge m | | µ m || E edge m | − E edge m ) R − M (76)If there is a local minimum at M = M < −| µ m | − / | E edge m | R = 2 M R M R | µ m | . (77)Due to the nonnegativity of the left hand side of theabove equation, the local minimum can exist only for M R < − /
2. In this region, the inequality | E edge m | R > | µ m | is satisfied, and thus E edge m (cid:101) E edge m > R < > M → −| µ m | − / M → −∞ . At such points, theenergies are given by E edge m = ( | µ m | +1 / /R and E edge m = | µ m | /R , respectively. Therefore, in the region of M R ∈ ( −∞ , −| µ m |− / E edge m (cid:101) E edge m > R < III. BULK AND EDGE SOLUTIONS IN THEMAGNETIC FIELD BACKGROUND
In this section we derive, following the general lineof the previous section, the eigenspectrum of the Diracfermions in the background of magnetic field.
A. Dirac equation in rotating spacetime in theuniform magnetic field
In the presence of an external magnetic field parallelto the cylinder axis B = (0 , , B z ≡ B ) the Dirac equa-tion (4) is modified:[ iγ µ ( D µ + Γ µ ) − M ] ψ = 0 , (78)where D µ = ∂ µ − ieA µ is the covariant derivative. Inthe laboratory frame the corresponding gauge field can be chosen in the symmetric form A ˆ i = (cid:18) , By , − Bx , (cid:19) . (79)In the corotating frame the background gauge field is asfollows: A µ = (cid:18) − B Ω r , By , − Bx , (cid:19) . (80)The Dirac equation (78) can be explicitly written asfollows: (cid:20) iγ ˆ t (cid:18) ∂ t + y Ω ∂ x − x Ω ∂ y − i σ ˆ x ˆ y (cid:19) + iγ ˆ x (cid:18) ∂ x + ieBy (cid:19) + iγ ˆ y (cid:18) ∂ y − ieBx (cid:19) + iγ ˆ z ∂ z − M (cid:21) ψ = 0 , (81)As in the absence of magnetic field the eigenvectors ofthe Dirac equation (81) are labeled by the eigenvaluesof commuting operators { ˆ (cid:101) H, ˆ P z , ˆ J z , ˆ W } , where ˆ (cid:101) H is thecorotating Hamiltonian, ˆ P z is the z -component of themomentum operator, ˆ J z is the z -component of the totalangular momentum (16), and ˆ W is the helicity operator.In the presence of magnetic field these operators coincidewith the ones given in Section II with the substitution ˆ P → ˆ P + e ˆ A which accounts for the gauge invarianceof these operators. In the presence of magnetic field thecorotating energy (cid:101) E j is related to the laboratory energy E j according to Eq. (14).Notice that Eq. (81) is gauge invariant because ofthe identity which holds for usual ∂ µ and covariant D µ derivatives in the corotating reference frame: ∂ t + y Ω ∂ x − x Ω ∂ y ≡ D t + y Ω D x − x Ω D y . (82)Here we used the fact that in the rotating frame thegauge field (80) acquires the compensating time compo-nent A = − B Ω r / B or not. Indeed, in order to maintain the gauge invari-ance the usual derivatives ∂ µ in the presence of magneticfield in all physical operators should transform to the co-variant derivatives D µ = ∂ µ − ieA µ . In particular, theangular momentum operator (16) should become as fol-lows ˆ J z ( A ) = J z − ieA ϕ ≡ − i∂ ϕ + 12 Σ z − eBr , (83)where J z ≡ J z ( A = 0). Therefore we could naturallyexpect that in the presence of magnetic field the crucial corotating-laboratory energy relation (14) could also bemodified. In order to clarify this issue we notice that therelation (14) comes from the relation between Hamilto-nians in the rotating ( ˆ (cid:101) H = i∂ t ) and laboratory ( ˆ H = i∂ ˆ t )reference frames ˆ (cid:101) H = ˆ H − Ω ˆ J z , (84)which has been used so far at vanishing magnetic field.However, in the presence of magnetic field the gauge-covariant Hamiltonian in corotating frame is given by H = iD t ≡ i∂ t + eA t , (85)[while the Hamiltonian in the laboratory frame ˆ H ≡ iD ˆ t remains untouched as A ˆ t ≡ iD t ψ = (cid:2) ˆ H − Ω ˆ J z ( A ) (cid:3) ψ . (86)However, taking into account in the rotating frame A t =Ω A ϕ ≡ − B Ω r / i∂ t ψ = (cid:16) ˆ H − Ω J z (cid:17) ψ . (87)Next, we notice that the energy in the corotating frameenters the wavefunction as ψ ( t, x ) = exp {− i (cid:101) E j t } ψ ( x )and therefore one gets from Eq. (87): (cid:101) E j ψ ( x ) = (cid:16) ˆ H − Ω J z (cid:17) ψ ( x ) , (88)which agrees with Eq. (84) which, in turn, leads to therelation in question (14). Thus we conclude that therelation (14) between the energies in the corotating (cid:101) E and laboratory E frames is still valid in the presence ofthe magnetic field background.0 B. Solutions
A general solution of the Dirac equation (81) has thefollowing form, U j ( t, z, ρ, ϕ ) = 12 π e − i (cid:101) E j t + ik z z u j ( ρ, ϕ ) , (89)where u j is an eigenspinor. The diagonal forms of ˆ J z andˆ W allow us to express the eigenspinor u j as follows u j ( ρ, ϕ ) = (cid:18) C up j φ j ( ρ, φ ) C down j φ j ( ρ, φ ) (cid:19) , (90) where the two-spinor φ j ( ρ, φ ) = (cid:18) e im j ϕ χ − j ( ρ ) e i ( m j +1) ϕ χ + j ( ρ ) (cid:19) , (91)is defined via two scalar functions χ ± j of the radial coor-dinate ρ . The helicity eigenvalue equation, ˆ W U j = λ j U j ,is reduced to the following relation, (cid:18) k j ˆ P − + e ˆ A − ˆ P + + e ˆ A + − k j (cid:19) φ j ( ρ, φ )2 (cid:113) E j − M = λ j φ j ( ρ, φ ) , (92)with ˆ P ± + e ˆ A ± = − ie ± iϕ (cid:0) ∂ ρ ± iρ − ∂ ϕ ± eBρ/ (cid:1) . Theequations for χ ± j are written as follows: (cid:34) ∂ ρ + ∂ ρ ρ − (cid:18) m j + 1 ρ (cid:19) + m j eB − e B ρ + (cid:0) E j − M − k j (cid:1)(cid:35) χ + j = 0 , (93) (cid:34) ∂ ρ + ∂ ρ ρ − (cid:18) m j ρ (cid:19) + ( m j + 1) eB − e B ρ + (cid:0) E j − M − k j (cid:1)(cid:35) χ − j = 0 . (94)Using the substitution ξ ≡ eB ρ , the above equations arereduced, respectively, to a simpler set of relations ξ ( χ + j ) (cid:48)(cid:48) + ( χ + j ) (cid:48) + (cid:16) − ξ + β + − ( m +1) ξ (cid:17) χ + j = 0 ,ξ ( χ − j ) (cid:48)(cid:48) + ( χ − j ) (cid:48) + (cid:16) − ξ + β − − m ξ (cid:17) χ − j = 0 , (95)where β ± = 2 µ m ∓
14 + 12 eB (cid:0) E j − M − k j (cid:1) , (96)and the angular momentum µ m is given in Eq. (15).The normalizable (regular in the origin) solutionsare given by the confluent hypergeometric function M ( a, b ; z ) ≡ F ( a, b ; z ) [13, 19] χ + j = N + j ρ | m j +1 | e − eB ρ M + , (97) χ − j = N − j ρ | m j | e − eB ρ M − , (98)where M ± is defined as M + ≡ M (cid:18) a + j , | m j + 1 | + 1 , eB ρ (cid:19) , (99) M − ≡ M (cid:18) a − j , | m j | + 1 , eB ρ (cid:19) , (100)and a ± j is defined as a + j = 12 ( | m j + 1 | − m j + 1) − eBR (cid:0) q Bj (cid:1) , (101) a − j = 12 ( | m j | − m j ) − eBR (cid:0) q Bj (cid:1) (102) with q Bj ≡ (cid:113) E j − M − k j R . The coefficient N + j can berelated to the coefficient N − j by a substitution of Eq.(97)and Eq.(98) into the helicity equation Eq.(92): N + j = + i ( E j − M − k j ) ( k j +2 λ j √ E j − M ) ( m j +1) N − j , m j ≥ , N + j = im j k j +2 λ j √ E j − M N − j , m j < . (103)The two spinors φ λj with the helicity λ are written asfollows φ λj ( ρ, ϕ ) = α j (cid:18) f λj − M − j f λj + M + j (cid:19) , (104)where α j is an overall constant and the two-spinor( f λj − f λj + ) T is defined as (cid:18) f λj − f λj + (cid:19) = (cid:18) m j + 1) G m j ( ρ, ϕ )2 iλ j p Bj (cid:0) p B − λ (cid:1) G m j +1 ( ρ, ϕ ) (cid:19) , m j ≥ , (cid:18) p Bj (cid:0) p Bλ (cid:1) G m j ( ρ, ϕ )4 iλ j m j G m j +1 ( ρ, ϕ ) (cid:19) , m j < , (105)with G m ( ρ, ϕ ) = e imϕ ρ | m | e − eB ρ , (106)and p B ± = (cid:115) ± k z p Bj , p Bj = (cid:113) E j − M . (107)1Next, we use the Dirac equation Eq.(81) determine theconstraint between C up j and C down j : E j − M − λ j (cid:113) E j − M λ j (cid:113) E j − M − E j − M u j ( ρ, φ ) = 0 , (108)or C up j = (cid:112) E j + M λ j E j | E j | (cid:112) E j − M C down j . (109)Consequently, the spinor u λj with the helicity λ can bewritten as follows u λj ( ρ, ϕ ) = C j (cid:32) E + φ λj λ j E j | E j | E − φ λj (cid:33) (110)with E ± = (cid:113) ± ME and an overall constant C j , which is determined by an orthogonal condition. Notice that theprefactor α j in Eq.(104) is absorbed into C j .The spinor u j which satisfies the MIT boundary con-dition (9) can be constructed in terms of the linear com-bination u j ( ρ, ϕ ) = b + j u + j ( ρ, ϕ ) + b − j u − j ( ρ, ϕ ) . (111)Substituting the eigenmode (89) and (111) into theboundary condition (9) as ψ ≡ U j and using the explicitform of the eigenspinors (110) we get a matrix equationfor the coefficients b ± with the solution (111): E + (cid:0) b + j φ + j + b − j φ − j (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ρ = R = − iE | E | E − (cid:0) b + j σ ρ φ + j − b − j σ ρ φ − j (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = R (112)where σ ρ is given in Eq. (55). The matrix equation (112)can also be represented in the form: i E | E | E − e − iϕ f + j + M + + E + f + j − M − − i E | E | E − e − iϕ f − j + M + + E + f − j − M − i E | E | E − e iϕ f + j − M − + E + f + j + M + − i E | E | E − e iϕ f − j − M − + E + f − j + M + b + b − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = R =0 . (113)We find that Eq. (113) has a nontrivial solution for b ± if the quantity q B = (cid:112) E − M − k R (114)satisfies the following relation: (cid:40) ( q B ) ( M + R ) − m + 1) M R M − R M + R − m + 1) ( M − R ) = 0 , m ≥ , ( q B ) ( M − R ) − mM R M − R M + R − m ( M + R ) = 0 , m < , (115)where M + R ≡ M + (cid:12)(cid:12)(cid:12) ρ = R = M (cid:0) a + j , | m j + 1 | + 1 , φ B /φ (cid:1) , M − R ≡ M − (cid:12)(cid:12)(cid:12) ρ = R = M (cid:0) a − j , | m j | + 1 , φ B /φ (cid:1) . (116)The magnetic field enters the spectrum in terms of theratio φ B φ ≡ eBR . (117)of the magnetic flux the crosssection of the cylinder φ B = πBR , (118)and the elementary magnetic flux φ = 2 πe , (119)(we remind that in our units (cid:126) = 1).Since the dimensionless quantity q B is discretized inaccordance with effects of both the boundary condition and the Landau quantization, it can be labeled by theangular momentum number m and the root number l =1 , , , . . . , i.e. q Bml .The zero solutions of Eq.(115), q Bml = 0 are achievedat specific values of the fermion masses M = M ( m ) c with M ( m ) c = − m +1 R M (1 ,m +2 ,φ B /φ ) , m ≥ , mR e φB/φ M ( − m, − m +1 ,φ B /φ ) , m < , (120)where we used the properties M (0 , b, z ) = 1 and M ( a, a, z ) = e z . In the limit of vanishing magnetic field, eB → , we can recover the result (32) for M ( m ) c usingthe property M ( a, b, z ) = 1 + O ( z ) valid for z →
0. Inthe limit of strong magnetic field, eB → ∞ , the massbecomes M ( m ) c = − e − φB/φ ( φ B /φ ) m +1 Rm ! → , m ≥ , − R φ B φ → −∞ , m < , (121)2where we used the asymptotic expansion M ( a, b, z ) ∼ (Γ( b ) / Γ( a )) e z z a − b valid at z → ∞ for all values of a except for non-positive integer a .We can recover Eq. (25) from Eqs. (115) and (116) inthe limit of vanishing magnetic field eB → q B (cid:54) = 0 and n ≥ a ± j eB → −−−−→ − q B,j eBR , (122)lim x → F (cid:18) − y x , n + 1; x (cid:19) = n ! (cid:18) y (cid:19) n J n ( y ) , (123)and J − m ( x ) = ( − m J m ( x ).The masses of the bulk and the edge states are givenby the same formulae (30) and, respectively, (39) as inthe case of the B = 0 states (with the obvious change q ml → q Bml ). The quantity ν Bm for the edge states in thebackground of magnetic field is defined similarly to the B = 0 definition in Eq. (35): q Bm = iν Bm . (124) C. Properties of the solutions
In order to obtain the spectrum of free fermions inthe cylinder in the presence of external magnetic field wesolve Eqs. (115) and (116) numerically.In Figs. 4 and 5 we show the behavior of, respectively,the bulk solutions q Bml and the edge solutions ν Bm for theorbital angular momentum m = 0 (which represents thequalitative behavior of all µ m > m = − µ m <
0) at nonzero magnetic field. These quantitiesat zero magnetic field were shown in Fig. 1.We notice the following effects of background magneticfield on the bulk modes:(i)
Critical mass: at zero magnetic field the groundstates ( l = 1) disappear at the quantized criticalmasses M c given in Eq. (32). As the magnetic fieldbecomes stronger the critical masses M c deviatefrom their B = 0 values: for eB > µ m > M c →
0, whilethe critical masses of the µ m < M c → − ( φ B /φ ) /R . The behav-iors are consistent with the analytical results givenin Eq.(121). One can show that at eB < µ m > µ m < M c → −∞ for the former and M c → Level degeneracy: at large positive or negative val-ues of the fermion mass, M → ±∞ , the levels aregrouping into pairs. This is a natural consequenceof growing mass of the bulk levels (30). As the massbecome large, the bulk states become more local-ized in space and they become less sensitive to the ���� ����� � = � ϕ � / ϕ � = ��� � = � � = � � = � � = � � = � � = � - - - - - MRq l (a) ���� ����� � =- � ϕ � / ϕ � = ��� � = � � = � � = � � = � � = � - - - - - MRq - l (b)FIG. 4. The bulk q Bml solutions of Eqs. (115) and (116) vs.the fermion mass M in the background of magnetic flux (117) φ B = 7 . φ for (a) m = 0 and (b) m = − l . presence of the boundary of the cylinder. Then theenergy spectrum shares a natural similarity withthe Landau levels in a boundless space where thespin-up and spin-down states of the excited levelsare double-degenerate in energy.The behavior of the edge modes ν Bm at values of mag-netic field – or, equivalently, the magnetic flux φ B ,Eq. (118) – is shown in Fig. 5. The mentioned prop-erties of the critical mass is well consistent with the onesfor the bulk modes, as expected. As the fermion mass M decreases the quantities ν m become linear functionsof the mass M .In Fig. 6 we show the masses of the lowest ( l = 1) bulkmodes (30) and the edge modes (39) as the functions ofthe fermion mass M at various values of magnetic field B . We notice the following remarkable properties of thesequantities:(i) Masses for the modes with negative angular mo-menta µ m (i.e. with m = − , − , . . . ) behave reg-ularly as the l = 1 bulk modes are transformedinto the edge modes at certain critical masses M = M ( m ) c ( B ). These critical masses are growing in ab-solute value (and negatively-valued) functions ofmagnetic field. At large enough strengths of the3 ���� ����� � = � ϕ � / ϕ � = ��� � ��� � - - - - - MR ν (a) ���� ����� � =- � ϕ � / ϕ � = ��� � ��� � - - - - - MR ν - (b)FIG. 5. The edge ν Bm solutions (124) of Eqs. (115) and (116)vs. the fermion mass M in the background of different mag-netic fluxes (117) φ B for (a) m = 0 and (b) m = − background magnetic field the masses of the bulkmodes experience, as functions of the fermion mass M , a global minimum.(ii) At positive values of the angular momenta µ m (i.e.at m = 0 , , , . . . ) the masses of the edge modesbehave rather irregularly. In particular, they vanishat certain mass M = M ( m ) c ( B ), M edge m (cid:16) M ( m ) c ( B ) (cid:17) = 0 , m (cid:62) . (125)In Fig. 7 we plot, for a few values of m , the massesof fermions M = M ( m ) c ( B ) at which the mass ofthe edge mode become zero (effectively, the mas-sive edge mode becomes the zero mode). Thesemasses are growing (in absolute value) negative-valued functions of the magnetic field B .(iii) At large negative values of the fermion mass, M →−∞ , the masses of the edge states become qualita-tively independent on the fermionic mass M .Notice that all these properties are valid for positive mag-netic field eB >
0. For the negative magnetic field, eB <
0, the modes with positive and negative magneticmomenta µ m swap their places. As in the absence of magnetic field, in the limit of alarge (negative) fermionic mass M → −∞ the masses ofthe edge modes remain finite contrary to the excited l (cid:62) M edge ∞ ,m ( B ) = lim M →−∞ M edge m ( B ) = (cid:12)(cid:12)(cid:12)(cid:12) µ m − φ B φ (cid:12)(cid:12)(cid:12)(cid:12) R . (126)In fact, we can obtain the result (126) analytically byusing the large a expansion of M ( a, b, z ) [25]: M ( a, b, z )= ( z/a ) (1 − b ) / e z/ Γ (1 + a − b ) Γ ( b )Γ ( a ) (cid:34) I b − (cid:0) √ az (cid:1) − (cid:114) za I b (cid:0) √ az (cid:1) (cid:18) b − z (cid:19) + O ( a − ) (cid:35) (127)= ( z/a ) (1 − b ) / e z/ Γ (1 + a − b ) Γ ( b )Γ ( a ) (cid:34) e √ az (cid:112) π √ az × (cid:18) − b − √ az − (cid:114) za (cid:18) b − z (cid:19) + O ( a − ) (cid:19)(cid:35) . Substituting Eqs. (127) and (67) into Eq. (115), we ob-tain the solution of ν m in terms of the expansion of alarge negative mass M : ν edge m = | M | R − ( µ m − φ B φ ) | M | R + O (cid:16)(cid:0) M R (cid:1) − (cid:17) , (128)which leads to Eq.(126).The masses of the edge states depend on the angularmagnetic moment µ m of the mode and the Aharonov–Bohm phase ϑ = φ B /φ . In the limit of vanishing mag-netic field, φ B = 0, Eq. (126) matches with the B = 0result (69). The mass spectrum of the edge states (126)in the M → ∞ limit is shown in Fig. 8. IV. EDGE MODES AND ROTATIONA. Zero magnetic field
In the limit of infinite negative mass M the thermo-dynamic and rotational properties of the system are de-termined only by the edge modes. Indeed, the masses ofthe edge modes remain finite (69) while the masses of thebulk modes tend to infinity implying that the latter donot contribute to the dynamics of the system. In the ab-sence of magnetic field the energy of the edge modes (37)4 m = Bulk modes m = l = Edge modes with m = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = - - - - - MR M e dg e R , M , l k R m = Bulk modes m = l = Edge modes with m = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = - - - - - MR M e dg e R , M , l k R (a) (b) m =- Bulk modes m =- l = m =- ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = - - - - - MR M - e dg e R , M - , l k R m =- Bulk modes m =- l = m =- ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = ϕ B / ϕ = - - - - - MR M - e dg e R , M - , l k R (c) (d)FIG. 6. The masses of the lowest bulk ( l = 1) and edge states vs. the mass of the fermion M for various values orbital angularmomenta m and magnetic field B . The bulk (edge) modes are shown by the thicker (thinner) lines while the positions wherethe bulk modes are converted to the corresponding edge modes are marked by the red points. � = � � = � � = � � = � � = � � = � - - - - - - - ϕ B / ϕ M c ( m ) R FIG. 7. The values of the fermion masses M = M ( m ) c at whichthe masses of the edge modes vanish (125) vs the magneticflux φ B for various values of orbital momentum m . � = - � � = - � � = - � � = � � = � � = � � = � � = � � = � � = � ϕ B / ϕ M m e dg e R FIG. 8. The masses of the edge modes (69) as functions ofmagnetic field B in the limit M → −∞ . E edge m ( k z ) = (cid:114) k z + µ m R , (129)where µ m is the angular momentum of the edgemode (15) and m ∈ Z .The thermodynamic effects of the edge modes are de-termined by the thermodynamic potential defined in thecorotating, as opposed to the laboratory, reference frame(the latter fact is stressed by the tilde sign in (cid:101) F ): (cid:101) F edge ( σ ; T, Ω) = − TπR (cid:88) m ∈ Z (cid:90) dk z π (130) (cid:20) ln (cid:18) e − E edge m ( kz ) − Ω µmT (cid:19) + (Ω → − Ω) (cid:21) . Below we omit the superscript “edge” in all our notations.The angular momentum density is given by the deriva-tive of the thermodynamic potential in the corotatingreference frame [20]: L = − (cid:32) ∂ ˜ F∂ Ω (cid:33) T . (131)Since the rotation axis Ω = Ω e z coincides with the sym-metry axis of the cylinder e z , the angular momentum hasonly one nonzero component, L = (0 , , L z ).It is convenient to consider the density of the angularmomentum per unit height of the cylinder: L z (Ω) ≡ πR L z (Ω) (132)= (cid:90) ∞−∞ k z π (cid:88) m ∈ Z µ m [ f m,k z (Ω , T ) − f m,k z ( − Ω , T )] , where f m,k z (Ω , T ) = 1 e Em ( kz ) − Ω µmT + 1 , (133)is the occupation number of the fermionic edge mode.The moment of inertia per unit height is related to thedensity of the angular momentum (132) as follows: I z (Ω) = L z (Ω)Ω . (134)The angular momentum (132) and the moment of iner-tia (134) at zero magnetic field are both shown in Fig. 9.These quantities are, respectively, odd and even func-tions with respect to the flips of the direction of rotation,Ω → − Ω, because the thermodynamic potential (130) isan even function of Ω.In Fig. 10 we show the density of the moment of inertiaat zero angular momentum. The moment of inertia is agrowing function of temperature because as temperatureincreases the heavier (energetic) modes may participatein rotation of the system. In this section we consider only the positively defined branchof the energy eigenmodes E = + | E | which corresponds to theparticle edge states (37) both for vanishing (129) and nonvanish-ing (135) magnetic field. �� = ��� �� = � �� = � � = � - - - Ω R L R - - - - - (a) � � = � � � � � = � � � = � � = � - Ω RI (b)FIG. 9. Densities of (a) the angular momentum (132) and (b)moment of inertia (134) of the cylinder in the limit an infinitefermion mass M → ∞ as the function of angular frequency Ωat various temperatures T and zero magnetic field. � = � TR I ( Ω = ) FIG. 10. Density of the moment of inertia (134) at Ω = 0 vs.temperature T at vanishing magnetic field B = 0. B. Effects of magnetic field
In the presence of magnetic field the energy dispersionof the edge modes (in the limit of an infinite fermion mass M → ∞ ) is given by the following formula: E edge m ( k z ) = (cid:115) k z + 1 R (cid:18) µ m − φ B φ (cid:19) , (135)where µ m is the angular momentum of the edgemode (15) with m ∈ Z .6 (a)(b)(c)FIG. 11. Angular momentum L of the edge modes per unitheight of cylinder vs. angular frequency Ω and magneticflux φ B at temperatures T R = 0 . , . , M → −∞ (the bulk modes are absent). The angular momentum (131) can be readily cal-culated using the partition function (130) and disper-sion (135). In Fig. 11 we show the angular momentum L in the magnetic field - angular frequency ( B, Ω) plane fortemperatures
T R = 0 . , . ,
1. Naturally, the angularmomentum is an increasing function of the angular fre-quency Ω for every fixed value of magnetic flux φ B andfor all temperatures T . At low temperatures T R (cid:46) . ∼ − /R ) the angular momentum L exhibits oscil-lating, but nonperiodic dependence on the value of mag-netic flux, as it is clearly seen in Fig. 11(a) and (b). Thelocal minima and maxima of L approximately correspondto the integer and, respectively, half-integer values of theratio of magnetic flux φ B and the elementary flux (117).Apart from these oscillations, the value of L slowly in-creases with strength of the background magnetic field.This quantum behavior is seen at sufficiently low tem-peratures: the lower temperature, the more pronouncedoscillations. There is also certain small correlation be-tween the magnetic field and the angular frequency seenin the range of middle frequencies, Ω R ∼ . T R ∼
1, shown in Fig. 11(c),the magnetic-field induced oscillations of the angular mo-mentum disappear completely. At sufficiently fast rota-tions the oscillations disappear for all temperatures. Inthese cases the angular momentum is an increasing func-tion of both magnetic field B and angular frequency Ω. �� = ��� ��� ��� ��� ��� ϕ B / ϕ I / ( T R ) FIG. 12. Moment of inertia (divided by temperature squared)per unit height of cylinder vs. the flux φ B of the backgroundmagnetic field at zero angular frequency Ω = 0. In Fig. 12 we show the dependence of the moment ofinertia (normalized by the temperature squared) at van-ishing angular frequency Ω = 0 vs. normalized mag-netic flux (117). We clearly see that that with increaseof temperature the moment of inertia of the edge modesincreases in agreement with zero-field behavior shown inFig. 10. Similarly to the angular momentum, the mo-ment of inertia experiences (nonperiodic) oscillations asa function of magnetic field. The local minima (maxima)approximately correspond to the integer (half-integer)values of the magnetic flux [calculated in units of the el-ementary flux (117)]. The oscillatory quantum behavioris well pronounced at low temperatures while at highertemperatures the dependence of the moment of inertiaon the magnetic flux reduces to a monotonically increas-ing function. These features are also well visible in theplot (13) which shows the moment of inertia I vs. bothmagnetic flux φ B and temperature T .The fact that both the moment of inertia and the an-gular momentum are not periodic function of magnetic7 FIG. 13. Moment of inertia per unit height of cylinder vs.magnetic flux φ B and temperature T at zero angular fre-quency Ω = 0. field is a natural consequence of non-equivalence of mag-netic field and rotation in relativistic domain. Indeed, inmany non-relativistic quantum-mechanical applicationsa (slow) rotation may be treated as a (weak) magneticfield. This fact is used, for example, in characterizing thespectrum of rotation optical lattices of cold atoms [21].The equivalence is no more true in the case of a fast rel-ativistic rotation: the effects of rotation and magneticfield in this case are very different [13, 15]. In order tohighlight the difference between rotation and magneticfield we mention that the ground state degeneracy is in-dependent of the value of the angular frequency contraryto case of magnetic field [15]. Moreover, the phenomenaof dimensional reduction, which govern many interestingeffects in magnetic field background, does not exist in thecase of rotation [15]. V. CONCLUSIONS
We study a uniformly rotating relativistic system offree Dirac fermions in the background of a constant mag-netic field directed along the axis of rotation. The systemmust be bounded in any plane perpendicular to the ro-tation axis in order to respect the relativistic causalityaccording to requirement that the rotational velocity ofparticles does not exceed the speed of light. Thereforewe enclose the system into an infinitely high cylinder ofradius R and restrict the angular frequency Ω of rotationto the subluminal domain: Ω R <
1. At the surface ofthe cylinder we impose either the MIT boundary condi-tion (9) or its chiral generalization (33) which is charac-terized by the chiral angle Θ. Both these conditions forcethe normal component of the fermionic current to vanishat cylinder’s surface thus conserving the global fermionicnumber inside the rotating cylinder.In general, the spectrum of fermions in a finite ge-ometry contains two types of solutions: bulk solutionsconcentrated in the interior of the system and the edge states which are localized at the boundary. The bulkstates in cylindrical geometry were already discussed inthe literature. In the absence of magnetic field the bulkspectrum of fermions was obtained in Ref. [10] where thecylinder with the MIT boundary conditions (9) was stud-ied. The bulk spectrum with the chiral MIT boundaryconditions (33) was found later in Ref. [18]. In our paperwe extend these results in various directions.Firstly, we find that the system possesses the edgemodes at certain region of the parameter space. Sec-ondly, we extend the results for the edge and bulk modesto the case of nonzero magnetic field parallel to the axisof the cylinder (so that the magnetic flux is a constantquantity along the axis of the cylinder). Thirdly, we im-plement the uniform rotation of the whole system andinvestigate the interplay between rotation and magneticfield in thermodynamical properties of free fermions. Fourthly, we highlight the role of the edge states thatwere neglected so far in the analysis of thermodynamicsof rotating fermionic systems.We found the following features of the system:1. The boundary condition is important for the edgestates. The mass spectrum and the very existenceof the edge modes depend on the values of thefermion mass M , magnetic field B and the chiralΘ angle at the boundary. For example, there areno edge states at the chiral angle Θ = π/ M crosses, for each fixed valueof the angular momentum (15), a certain thresholdmass. In the absence of magnetic field the thresholdmasses (32) are given, for the MIT boundary condi-tions (9), by M c = − n/R with n = 1 , , . . . . Theydiffer from the threshold masses for the fermionswith the chiral boundary conditions (34). Thethreshold masses for the MIT boundary conditionsare changed to Eq.(120) in the case of nonzero mag-netic field.3. The edge states are massive so that in the solid-state language the system may be associated witha non-topological insulator.4. The masses of the edge states are finite for B = 0.In the absence of magnetic field the spectrum isdegenerate with respect to the sign flips of the an-gular momentum, µ m → − µ m , see Fig. 2. Themasses of the bulk (edge) modes rise (fall) with in-crease of the absolute value of the fermion mass M .In the limit of a negative infinite fermionic mass, Uniformly rotating fermions in magnetic field were also studiedin Ref. [13] in a transversally unrestricted geometry which doesnot possess the edge modes. M → −∞ , the bulk modes become infinitely heavyso that they decouple from the dynamics of the sys-tem and disappear. On the contrary, in this limitthe masses of the edge modes remain finite (69).They are proportional to the mean curvature of thecylinder’s surface, 1 /R .5. The masses of the edge states may vanish for B (cid:54) = 0.Nonzero magnetic field lifts out the µ m → − µ m degeneracy of the mass spectrum of both the bulkstates and the edge states, see Fig. 6. For example,the edge states with sign( µ m eB ) > µ m eB ) > M → −∞ while the masses of theedge states exhibit a periodic dependence on themagnetic flux, see Fig. 8, described by the simpleformula (126).6. Moment of inertia oscillates with magnetic field. The presence of magnetic field affects drasticallythe rotational properties of the system. For exam-ple, in the domain of low temperatures in the limitof infinitely large negative fermion mass – wherethe thermodynamics is given by the edge modesonly – the angular momentum (Fig. 11) and, conse-quently, the moment of inertia (Fig. 12) experiencequasi-periodic (quantum) oscillations as functionsof magnetic flux φ B . The local minima (maxima)of the moment of inertia correspond to the inte-ger (half-integer) values of the magnetic flux φ B inunits of the elementary flux φ , Eq. (117). At hightemperature the oscillations disappear, see Fig. 12. ACKNOWLEDGMENTS
The authors are grateful to Pavel Buividovich, MarkGoerbig and Mar´ıa Vozmediano for discussions. Thework of S. G. was supported by the Special PostdoctoralResearchers Program of RIKEN. [1] G. B. Cook, S. L. Shapiro and S. A. Teukolsky, “Rapidlyrotating neutron stars in general relativity: Realisticequations of state,” Astrophys. J. , 823 (1994).[2] L. P. Csernai, V. K. Magas and D. J. Wang, “Flow Vor-ticity in Peripheral High Energy Heavy Ion Collisions,”Phys. Rev. C , no. 3, 034906 (2013) [arXiv:1302.5310[nucl-th]]; F. Becattini et al. , “A study of vorticity for-mation in high energy nuclear collisions,” Eur. Phys.J. C , no. 9, 406 (2015) [arXiv:1501.04468 [nucl-th]];Y. Jiang, Z. W. Lin and J. Liao, “Rotating quark-gluonplasma in relativistic heavy ion collisions,” Phys. Rev.C , no. 4, 044910 (2016) [arXiv:1602.06580 [hep-ph]];W. T. Deng and X. G. Huang, “Vorticity in Heavy-Ion Collisions,” Phys. Rev. C , no. 6, 064907 (2016)[arXiv:1603.06117 [nucl-th]].[3] D. T. Son and A. R. Zhitnitsky, “Quantum anomalies indense matter,” Phys. Rev. D , 074018 (2004) [hep-ph/0405216]; D. T. Son and P. Surowka, “Hydrody-namics with Triangle Anomalies,” Phys. Rev. Lett. ,191601 (2009) [arXiv:0906.5044 [hep-th]].[4] A. Vilenkin, “Quantum Field Theory At Finite Temper-ature In A Rotating System,” Phys. Rev. D , 2260(1980). doi:10.1103/PhysRevD.21.2260[5] A. Vilenkin, “Parity Violating Currents in Thermal Ra-diation,” Phys. Lett. , 150 (1978); “Macroscopic Par-ity Violating Effects: Neutrino Fluxes From RotatingBlack Holes And In Rotating Thermal Radiation,” Phys.Rev. D , 1807 (1979); M. Kaminski, C. F. Uhlemann,M. Bleicher and J. Schaffner-Bielich, “Anomalous hy-drodynamics kicks neutron stars,” Phys. Lett. B ,170 (2016) [arXiv:1410.3833 [nucl-th]]; N. Yamamoto,“Chiral transport of neutrinos in supernovae: Neutrino-induced fluid helicity and helical plasma instability,”Phys. Rev. D , no. 6, 065017 (2016) [arXiv:1511.00933[astro-ph.HE]]. [6] G. Basar, D. E. Kharzeev and H. U. Yee, “Triangleanomaly in Weyl semimetals,” Phys. Rev. B , no.3, 035142 (2014) [arXiv:1305.6338 [hep-th]]; K. Land-steiner, “Anomalous transport of Weyl fermions in Weylsemimetals,” Phys. Rev. B , no. 7, 075124 (2014)[arXiv:1306.4932 [hep-th]]; M. N. Chernodub, A. Cortijo,A. G. Grushin, K. Landsteiner and M. A. H. Vozmedi-ano, “Condensed matter realization of the axial mag-netic effect,” Phys. Rev. B , no. 8, 081407 (2014)[arXiv:1311.0878 [hep-th]].[7] B. R. Iyer, “Dirac Field Theory In Rotating Coordi-nates,” Phys. Rev. D , 1900 (1982).[8] F. Becattini and F. Piccinini, “The Ideal relativistic spin-ning gas: Polarization and spectra,” Annals Phys. ,2452 (2008) [arXiv:0710.5694 [nucl-th]].[9] V. E. Ambru¸s and E. Winstanley, “Rotating quantumstates,” Phys. Lett. B , 296 (2014) [arXiv:1401.6388[hep-th]].[10] V. E. Ambru¸s and E. Winstanley, “Rotating fermionsinside a cylindrical boundary,” Phys. Rev. D , no. 10,104014 (2016) [arXiv:1512.05239 [hep-th]].[11] A. Manning, “Fermions in Rotating Reference Frames,”arXiv:1512.00579 [hep-th].[12] B. McInnes, “Angular Momentum in QGP Holography,”Nucl. Phys. B , 246 (2014) [arXiv:1403.3258 [hep-th]]; B. McInnes, “Inverse Magnetic/Shear Catalysis,”Nucl. Phys. B , 40 (2016) [arXiv:1511.05293 [hep-th]]; B. McInnes, “A rotation/magnetism analogy forthe quark–gluon plasma,” Nucl. Phys. B , 173 (2016)[arXiv:1604.03669 [hep-th]].[13] H. L. Chen, K. Fukushima, X. G. Huang and K. Mameda,“Analogy between rotation and density for Diracfermions in a magnetic field,” Phys. Rev. D , no. 10,104052 (2016) [arXiv:1512.08974 [hep-ph]].[14] Y. Jiang and J. Liao, “Pairing Phase Transitions of Mat-ter under Rotation,” Phys. Rev. Lett. , no. 19, 192302 (2016) [arXiv:1606.03808 [hep-ph]].[15] M. N. Chernodub and S. Gongyo, “Interacting fermionsin rotation: chiral symmetry restoration, moment of in-ertia and thermodynamics,” JHEP , 136 (2017)[arXiv:1611.02598 [hep-th]].[16] S. Ebihara, K. Fukushima and K. Mameda, “Bound-ary effects and gapped dispersion in rotating fermionicmatter,” Phys. Lett. B , 94 (2017) [arXiv:1608.00336[hep-ph]].[17] G. Duffy and A. C. Ottewill, “The Rotating quan-tum thermal distribution,” Phys. Rev. D , 044002(2003) [hep-th/0211096]; P. C. W. Davies, T. Dray andC. A. Manogue, “The Rotating quantum vacuum,” Phys.Rev. D , 4382 (1996) [gr-qc/9601034]; O. Levin, Y. Pe-leg and A. Peres, “Unruh effect for circular motion in acavity”, J. Phys. A , 3001 (1993).[18] M. N. Chernodub and S. Gongyo, Phys. Rev. D , no.9, 096006 (2017) [arXiv:1702.08266 [hep-th]].[19] L. D. Landau, E. M. Lifshitz, “Quantum mechanics: non-relativistic theory,” (Butterworth-Heinemann, 1981) [20] L. D. Landau and E. M. Lifshitz, “Statistical Physics,Part 1: Volume 5”, (Butterworth-Heinemann, Oxword,1980).[21] D. Jaksch, P. Zoller, “Creation of effective magnetic fieldsin optical lattices: The Hofstadter butterfly for cold neu-tral atoms”, New Journal of Physics , 56 (2003).[22] C. A. Lutken and F. Ravndal, “Fermionic Vacuum Fluc-tuations Between Chiral Plates,” J. Phys. G , 123(1984).[23] Shun-Qing Shen, Wen-Yu Shan, Hai-Zhou Lu, “Topolog-ical insulator and the Dirac equation,” SPIN , 33 (2011)[arXiv:1009.5502 [cond-mat.mes-hall]].[24] J. E. Moore, “The birth of topological insulators,” Na-ture (London) , 194 (2010); M. Z. Hasan and C.L. Kane, “Topological insulators”, Rev. Mod. Phys.82