Interacting fermions in rotation: chiral symmetry restoration, moment of inertia and thermodynamics
PPrepared for submission to JHEP
Interacting fermions in rotation: chiral symmetryrestoration, moment of inertia and thermodynamics
M. N. Chernodub a,b and Shinya Gongyo a,c a CNRS, Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e de Tours, France b Laboratory of Physics of Living Matter, Far Eastern Federal University, Vladivostok, Russia c Theoretical Research Division, Nishina Center, RIKEN, Saitama, Japan
Abstract:
We study rotating fermionic matter at finite temperature in the framework ofthe Nambu–Jona-Lasinio model. In order to respect causality the rigidly rotating systemmust be bound by a cylindrical boundary with appropriate boundary conditions that con-fine the fermions inside the cylinder. We show the finite geometry with the MIT boundaryconditions affects strongly the phase structure of the model leading to three distinct regionscharacterized by explicitly broken (gapped), partially restored (nearly gapless) and spon-taneously broken (gapped) phases at, respectively, small, moderate and large radius of thecylinder. The presence of the boundary leads to specific steplike irregularities of the chiralcondensate as functions of coupling constant, temperature and angular frequency. Thesesteplike features have the same nature as the Shubnikov–de Haas oscillations with the cru-cial difference that they occur in the absence of both external magnetic field and Fermisurface. At finite temperature the rotation leads to restoration of spontaneously broken chi-ral symmetry while the vacuum at zero temperature is insensitive to rotation (“cold vacuumcannot rotate”). As the temperature increases the critical angular frequency decreases andthe transition becomes softer. A phase diagram in angular frequency-temperature planeis presented. We also show that at fixed temperature the fermion matter in the chirallyrestored (gapless) phase has a higher moment of inertia compared to the one in the chirallybroken (gapped) phase. a r X i v : . [ h e p - t h ] N o v ontents Recently much interest has been attracted to rigidly rotating systems of interacting rela-tivistic fermions. At the particle physics side the interest is heated by the fact that thenoncentral heavy-ion collisions produce rapidly rotating quark-gluon plasma which shouldcarry large global angular momentum [1–4]. The rotating relativistic systems experienceanomalous transport phenomena such as, for example, the chiral vortical effect [6] whichalso has been discussed in an astrophysical context earlier [7]. In the condensed matterthe relativistic fermions appear in Weyl/Dirac semimetals that may also be sensitive torotation due to the anomalous transport [8–10].Theoretically, the problem of rotating free fermion states has recently been addressedin the Refs. [11, 12] and the system of interacting fermions has been studied both inunbounded [13, 14] and bounded [15] geometries in effective field-theoretical models, aswell as in the holographic approaches [16–18]. Properties of rotating strongly interactingmatter were also probed in lattice simulations of euclidean QCD [19].In our paper we study how rotation affects phase structure and thermodynamics of asystem of interacting fermions described by the Nambu–Jona-Lasinio model [20]. We pointout that a rigidly rotating system must be bounded in directions transverse to the axis torotation in order to avoid the causality violation. The unbounded rotating systems may– 1 –ave several pathologies related to instabilities and excitations by rotation when a regionof the rotating space exceeds the speed of light [11, 21, 22]. Thus the rotating space shouldbe bounded which immediately implies that the physics of the system must be dependenton the type of the boundary conditions that are imposed in the finite space directions. Theimportance of the boundary effects has also been recently noted in a different approachin Ref. [15]. In our study we consider a system of rigidly rotating fermions in a regionbounded by a cylindrical shell at which the fermions are subjected to the MIT boundaryconditions.The structure of this paper is as follows. First, in Sect. 2 we discuss free massiverigidly rotating fermions in a cylinder of a finite radius: in Sect. 2.1 we review the detailsof the spectrum obtained in Ref. [12], then in Sect. 2.2 we describe specific features of thestructure of the energy spectrum and, based on these results, we discuss similarities anddifferences between global rotation and external magnetic field in Sect. 2.3. Our studysupports the idea that the properties of the energy spectrum of the relativistic rotatingfermions imply that the rotation in a relativistic system cannot be associated with thepresence of an effective magnetic field background.
Figure 1 . The fermionic medium is uni-formly rotating with constant angular veloc-ity Ω inside the cylinder of fixed radius R . In Section 3 we describe rotating interactingfermions in the framework of the Nambu–Jona-Lasinio model. We stress that a rigid rotationof a relativistic system in thermal equilibriumis necessarily a finite-geometry problem. Thephase diagram of the system at zero tempera-ture is calculated in Sect. 4.1. In Section 4.2we demonstrate that the cold vacuum cannotrotate. The properties of the rotating fermionicmatter at finite temperature are discussed in de-tails in Sect. 5. We find the finite-temperaturephase diagram in Sect. 5.1, discuss the angularmomentum and moment of inertia in differentphases (Sect. 5.2), and calculate energy and en-tropy densities of rotating fermions (Sect. 5.3).The last Section is devoted to discussions andconclusions.
In this Section we describe certain properties of solutions to the Dirac equation for uni-formly rotating massive fermions inside a cylindrical cavity, Fig. 1, following closely Ref. [12].We consider a fermionic system which uniformly rotates with the constant angularvelocity Ω about the fixed z ≡ x axis. We assume that all spatial regions of the systemhave the same angular velocity so that the rotation is rigid. The rigid nature of rotationimmediately implies that the system must have a finite size in the plane perpendicular to– 2 –he axis of rotation. Indeed, the absolute value of velocity of a point located at the distance ρ from the axis of rotation, v = Ω ρ should not exceed the speed of light to preserve thecausality ρ Ω (cid:54)
1. Without loss of generality we assume that the rotation is always goinginto the counterclockwise direction so that Ω (cid:62) x ≡ ( x , x , x , x ) = ( t, ρ sin ϕ, ρ cos ϕ, z ). The coordinates t , ρ and z in the corotatingreference frame (which rotates together with the system) coincide with the correspondingcoordinates of the laboratory frame: t = t lab , ρ = ρ lab and z = z lab . The angular variablesin these frames are related as follows: ϕ = [ ϕ lab − Ω t ] π , (2.1)where [ . . . ] π means “modulo 2 π ”. In both frames the boundary of the cylindric volume isgiven by ρ = R . The requirement of causality implies that product of the angular velocityand the radius of rigidly rotating cylinder is bound:Ω R (cid:54) . (2.2)In order to preserve the number of fermions inside the cylindrical cavity it is natu-ral to impose on the fermion wavefunctions the MIT conditions at the boundary of thecylinder ρ = R , (cid:2) iγ µ n µ ( ϕ ) − (cid:3) ψ ( t, z, ρ, ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = R = 0 , [MIT b.c.] (2.3)where n µ ( ϕ ) = (0 , R cos ϕ, − R sin ϕ,
0) is a vector normal to the cylinder surface and γ µ arethe Dirac matrices. The boundary condition (2.3) confines the fermions inside the cavitybecause Eq. (2.3) forces the normal component of the fermionic current, j µ = ¯ ψγ µ ψ , (2.4)to vanish at the surface of the cylinder ( ρ = R ): j n ≡ jn ≡ − j µ n µ = 0 at ρ = R . (2.5)The rotating frame (2.1) is a curvilinear reference frame with the metric g µν = − ( x + y )Ω y Ω − x Ω 0 y Ω − − x Ω 0 − − , (2.6)which corresponds to the line element ds ≡ g µν dx µ dx ν = (cid:0) − ρ Ω (cid:1) dt − ρ Ω dtdϕ − dρ − ρ dϕ − dz . (2.7)– 3 –ere we 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 theCartesian coordinate in the local rest frame, corresponding to the laboratory frame, andthe general coordinate in the rotating frame, respectively.Consequently, a fermion with the mass M is described by the Dirac equation in thecurved spacetime [ iγ µ ( ∂ µ + Γ µ ) − M ] ψ = 0 . (2.8)where the affine connection Γ µ is defined byΓ µ = − i ω µij σ ij , ω µ ˆ i ˆ j = g αβ e α ˆ i (cid:16) ∂ µ e β ˆ j + Γ βνµ e µ ˆ j (cid:17) , σ ˆ i ˆ j = i (cid:104) γ ˆ i , γ ˆ j (cid:105) , (2.9)with the Christoffel connection, Γ λµν = g λσ ( g σν,µ + g µσ,ν − g µν,σ ), and the gamma matrixin curved space-time, γ µ = e µ ˆ i γ ˆ i , fulfills the anti-commutation relation, { γ µ , γ ν } = 2 g µν . (2.10)The vierbein e µ ˆ i is written in the Cartesian gauge [11], which connects the generalcoordinate with the Cartesian coordinate in the local rest frame, x µ = e µ ˆ i x ˆ i , is given by 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 , (2.11)and the other components being zero. This leads to the metric η ˆ i ˆ j = g µν e µ ˆ i e ν ˆ j .In the rotating case, the nonzero components of the Christoffel connection areΓ ytx = Γ yxt = Ω , Γ xty = Γ xyt = − Ω , Γ xtt = − x Ω , Γ ytt = − y Ω , (2.12)and the other components are zero. Thus, the only nonzero component of Γ µ is given byΓ t = − i σ ˆ x ˆ y . (2.13)The gamma matrix in the rotating case is given by γ t = γ ˆ t , γ x = y Ω γ ˆ t + γ ˆ x , γ y = − x Ω γ ˆ t + γ ˆ y , γ z = γ ˆ z . (2.14)The Dirac equation is rewritten as (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 ∂ x + iγ ˆ y ∂ y + iγ ˆ z ∂ z − M (cid:21) ψ = 0 . (2.15)In the Dirac representation, σ ˆ x ˆ y = (cid:32) σ σ (cid:33) , (2.16)and the Dirac equation is reduced to (cid:104) γ ˆ t ( i∂ t + Ω J z ) + iγ ˆ x ∂ x + iγ ˆ y ∂ y + iγ ˆ z ∂ z − M (cid:105) ψ = 0 . (2.17)– 4 –ith ˆ J z being the z-component angular momentum,ˆ J z = − i ( − y∂ x + x∂ y ) + 12 (cid:32) σ σ (cid:33) ≡ − i∂ ϕ + 12 (cid:32) σ σ (cid:33) . (2.18)Alternatively, the Dirac equation in the rotating reference frame may also be obtainedby shifting the free Hamiltonian in the laboratory frame as follows H lab = (cid:90) d x lab H lab (cid:16) ψ ( x lab ) , ψ † ( x lab ) (cid:17) → H rot ≡ (cid:90) d x H lab (cid:16) ψ ( x ) , ψ † ( x ) (cid:17) − Ω J , (2.19)where J is the z –component of the fermionic angular momentum operator: J = M ≡ (cid:90) d xψ † ( x ) (cid:20) i ( − x∂ y + y∂ x ) + 12 σ ˆ x ˆ y (cid:21) ψ ( x ) . (2.20)The Hamiltonian in the rotating reference frame (2.19) is thus given by H rot = (cid:90) d xψ † (cid:20)(cid:16) − iα ˆ i ∂ i + M β (cid:17) − Ω (cid:18) i ( − x∂ y + y∂ x ) + 12 σ ˆ x ˆ y (cid:19)(cid:21) ψ (2.21)with α ˆ i = γ γ ˆ i , and β = γ , and α ˆ i ∂ i = α ˆ x ∂ x + α ˆ y ∂ y + α ˆ z ∂ z . The Dirac equation (2.15) isobtained as a Heisenberg equation, using the equal-time anticommutation relation obtainedfrom the canonical quantization.A general solution of the Dirac equation (2.15) with the boundary conditions (2.3) inthe rotating reference frame (2.1) has the following form: U j = 12 π e − i (cid:101) Et + ik z z u j ( ρ, ϕ ) , (2.22)where u j is an eigenspinor characterized by the set of quantum numbers, j = ( k z , m, l, sign( E )) , m ∈ Z , l = 1 , , . . . , k z ∈ R , (2.23) k z is the momentum of the fermion along the z axis, m is the quantized angular momentumwith respect to the z axis, and l is the radial quantum number which describes the behaviorof the solution in terms of the radial ρ coordinate.The energy (cid:101) E j in the corotating frame, (cid:101) E j = E j − Ω (cid:16) m + 12 (cid:17) ≡ E j − Ω µ m , (2.24)is related to the energy E j in the laboratory frame: E j ≡ E ml ( k z , M ) = ± (cid:115) k z + q ml R + M , (2.25)where the dimensionless quantity q ml is the l th positive root ( l = 1 , , . . . ) of the followingequation: j m ( q ) + 2 M Rq j m ( q ) − , (2.26)– 5 –ith j m ( x ) = J m ( x ) J m +1 ( x ) , (2.27)and J m ( x ) is the Bessel function.The quantity µ m in Eq. (2.24) is the eigenvalueˆ J z ψ = µ m ψ , µ m = m + 12 , (2.28)of the z -component of the total angular momentum operator which comprises the orbitaland spin parts (2.18). Thus the quantiry µ m can be identified with the quantized value ofthe total angular momentum.In the cylindric volume the integration measure over the momentum is modified withrespect to the measure in an unbounded space: (cid:90) d k (2 π ) → (cid:88) j ≡ πR ∞ (cid:88) l =1 ∞ (cid:88) m = −∞ (cid:90) dk z π . (2.29)The integration over continuous 3-momentum is replaced by the integration over the mo-mentum along the axis of rotation k z and sums over the projection of the angular momen-tum on the z axis and over the radial excitation number l . For a detailed derivation of theenergy spectrum we refer the reader to Ref. [12]. The spectrum of rotating fermions contains certain interesting and, sometimes, quite un-expected features.In the laboratory frame the fermionic eigenenergy (2.25) does not depend on the an-gular frequency Ω. However, in the corotating frame the eigenenergy is a linear function ofthe angular frequency Ω. The frequency Ω plays the role of a chemical potential associatedwith the total angular momentum µ m .The change of the orbital number m → − m − q m,l ofEq. (2.26) while the total angular momentum µ m flips its sign: q m,l = q − m − ,l , µ m = − µ − m − , (2.30)Therefore the densities of states with positive and negative total angular momenta µ m areequal to each other. Equation (2.30) also implies that the energy spectrum (2.24) is doubledegenerate at Ω = 0.In Fig. 2 we show the energy spectrum in the corotating frame as the function of thetotal angular momentum µ m for positive and negative values of masses M and for variousvalues of the angular frequency Ω. We plot only the lowest mode with zero momentumalong the rotation axis k z = 0 and with the lowest radial excitation number l = 1. Themodes with k z (cid:54) = 0 and higher values of l (cid:62) k z = 0 and l = 1) eigenmode and they qualitatively resemble the structure of the lowest– 6 – R =- . Ω R = - . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = - . Ω R = - . M = R - l = k z = - - μ m E m R Ω R = - . Ω R = - . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = - . Ω R = - . M = R - l = k z = - - μ m E m R (a) (b) Ω R =- . Ω R = - . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = - . Ω R = - . M =- R - l = k z = - - μ m E m R Ω R = - . Ω R = - . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = . Ω R = - . Ω R = - . M =- R - l = k z = - - μ m E m R (c) (d) Figure 2 . Lowest energy eigenmodes (with l = 1 and k z = 0) in the corotating frame (2.24) vs.the total angular momentum µ m , Eq. (2.28), for fixed positive (the upper panel) and negative (thelower panel) masses M for various values of the rotation frequency Ω. eigenmode. We study both positive and negative values of the mass M since in the NJLapproach the mass gap is the dynamical variable which may take any real value.The energy spectra in the corotating frame, for all values of the fermion masses M and for all nonzero angular momenta Ω (cid:54) = 0 are asymmetric functions with respect toinversion of the total angular momentum µ m → − µ m . The asymmetry appears due to thepresence of the linear term in the energy in the corotating frame (2.24). The presence of theasymmetry indicates that the clockwise and counterclockwise rotations are not equivalentfor each particular mode. However, the spectrum is invariant under the simultaneous flips µ m → − µ m and Ω → − Ω. Since the partition function includes the sum over all values ofthe angular momentum, Eqs. (3.2) and (3.11), then for the whole system the clockwise andcounterclockwise rotation are equivalent. For Ω = 0 the spectrum is obviously symmetricunder the inversion of the total angular momentum µ m → − µ m .At large angular momenta µ m , the corotating energy spectrum is approaching a linearfunction. The slopes of this function differ for positive and negative frequencies Ω (cid:54) = 0.For positive angular frequencies Ω > < µ m the slope of the energy in the rotating frame has the following universallimit ( cf. Fig. 2): ∂∂µ m (cid:101) E m,l,k z (Ω) (cid:12)(cid:12)(cid:12)(cid:12) µ m →±∞ = 1 ∓ Ω R . (2.31)As one can see from Fig. 2, certain features of the energy spectrum at positive andnegative masses M are quite different from each other.At positive values of the fermion mass M > µ m (or, equivalently, of theangular momentum number m ). A change in the value of the mass does not affect thespectrum qualitatively. Indeed, at low mass the energy spectrum bends sharply at µ m = 0,Fig. 2(a). As the mass M increases, the behavior of the energy as the function of the totalangular momentum µ m smoothens at µ m = 0, Fig. 2(b), while all other features stay thesame.At negative values of mass, M <
0, the energy spectrum becomes more involved. Thedependence of energy on the total angular momentum µ m becomes generally non-convex.At low negative mass the particularities of the spectrum are observed at µ m ∼
0, Fig. 2(c),while at large negative mass the spectrum experiences a sharp discontinuity-like feature atthe angular momentum µ m (cid:39) ± M R , Fig. 2(d). As we will see below, these features of theenergy spectrum will have an interesting effect on the ground state of the system. q l q l q l q l - - MRq ml Figure 3 . Lowest l = 1 , , . . . solutions q ml ofEq. (2.26) for the angular quantum numbers m = 0(the blue dashed lines), m = 1 (the green solid lines), m = 2 (the red dot-dashed lines) and m = 3 (the or-ange dotted lines) vs the normalized mass M R . In order to understand the ori-gin of the discontinuities in the energyspectrum at negative mass M we no-tice that the solution q ml of Eq. (2.26)enters the energy eigenvalue (2.25) asan effective momentum which, in turn,depends on the mass M via Eq. (2.26).In Fig. 3, we plot the solutions q ml fora few values of the angular momentum, m = 0 , , , M R . As expected, aspositive values of mass
M > q ml are smooth functions of themass M . However, at negative valuesof M the branches of q ml solutions ter-minate at certain quantized values ofthe mass M where these solutions touch the q = 0 axis. More precisely, the q ml solutionswith fixed l = 1 and variable m = 0 , , . . . terminate at the quantized values of the masses, Notice that positive and negative m are related by the symmetry (2.30) with respect to the flips of theorbital number m → − m − – 8 – R = − − m : q m ( M ) (cid:12)(cid:12)(cid:12)(cid:12) MR = − − m = 0 . (2.32)In fact, the existence of the solutions are proved analytically by expanding the Besselfunction around x ∼
0. At a slightly lower mass the lowest- q solution becomes q m thuscausing a set of the discontinuities in the lowest-energy eigenvalue (2.25) that we observein Fig. 2(d). Notice that the solutions with l (cid:62) l = 1 modes. MR = − MR = 0 MR = 5 - - Ω R Ε (cid:1) m R - - Ω R Ε (cid:1) m R - - Ω R Ε (cid:1) m R (a) (b) (c) Figure 4 . The energy spectrum in the corotating frame (2.24) at vanishing longitudinal momentum k z = 0 vs. the rotation frequency Ω for (a) positive, (b) zero and (c) negative masses M . The colorsvary gradually from blue [the total angular momentum (2.28) is negative µ m <
0] via green/yellow( µ m ∼
0) to red ( µ m > l = 1 , , . . . . The first tower ( l = 1) is shown by thicker lines. Figure 3 illustrates the asymmetry of the fermionic spectrum in the cylindrical cavitywith respect to a change of the sign of the fermion mass M → − M . This asymmetry is aparticular feature of the fermionic modes in the cylindric finite-volume geometry with theMIT boundary condition at cylinder’s surface (2.3). The MIT boundary condition (2.3)breaks down the chiral symmetry explicitly as it is not invariant under the chiral rotations: ψ → e − iθγ ψ , ¯ ψ → ¯ ψe − iθγ . (2.33)In the Dirac equation (2.8) the chiral transformations (2.33) lead to the modification ofthe mass term: M → e − iθγ M e − iθγ . (2.34)In particular, the choice θ = π/ M → − M . Thus, thechirally non-invariant boundary conditions lead to asymmetry of the fermionic spectrumwith respect to the flip of the fermion mass. As we will show below, this asymmetry isessential for the dynamical symmetry breaking at small radii of the static cylinder and itbecomes less important at larger radii. However, we will also point out that the asymmetryis quite essential for the rotating cylinder. – 9 –inally, we would like to notice that there exists a “chiral” version of the MIT boundaryconditions (2.3) which is given by a simple flip of a sign in Eq. (2.3): (cid:2) iγ µ n µ ( ϕ ) + 1 (cid:3) ψ ( t, z, ρ, ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = R = 0 . [chiral MIT b.c.] (2.35)This boundary condition in the cylindrical geometry was considered in Ref. [12].The MIT boundary condition (2.3) may be transformed into the chiral MIT boundarycondition (2.35) and vice-versa by the chiral transformation (2.33) with the chiral angle θ = π/
2. The same transformation flips the sign of the mass term, M → − M . Thereforeall features of the system (i.e. mass spectra, Figs. 2 and 4, etc) of the fermions of positivemass subjected to the MIT conditions are the same for the negative-mass fermions satis-fying the chiral MIT conditions. In other words, the would-be inequivalence of features offermions with positive and negative masses is caused by the boundary conditions whichbreak explicitly the flip symmetry M → − M . In the next section we will make use of the fermionic spectrum (2.24) to study the phasestructure of the interacting fermions in the rotating environment. However, before finishingthis Section we would like to make a comment on possibility, in the present relativisticcontext, of existence of a relation between rotation and (effective) magnetic field whichwas discussed in the literature recently in Refs. [13, 18].In the nonrelativistic quantum mechanics the effects of rotation may be representedby an effective uniform magnetic field with the axis parallel to the axis of rotation. Theparticles in the system then become “electrically” charged with respect to the correspondingfictitious “electromagnetic” field that represents the rotation. In this Section we show thatthe effect of a rigid rotation on the spectrum of a relativistic fermion system is generallynot equivalent to the effect of the background magnetic field.In Fig. 4 we show the (co)rotating energy spectrum (2.24) as a function of the rotationfrequency Ω at vanishing longitudinal momentum k z = 0. From the behavior of the energyspectrum one can figure out at least two arguments against the identification of rotationwith external magnetic field:1. Dimensional reduction and energy gap . In a background magnetic field B the energy gap between the ground state [the LowestLandau Level (LLL)] and the first excited level becomes wider as the strength of themagnetic field increases. This leads to the effect of dimensional reduction of particle’smotion: in strong magnetic field the particle reside at the LLL as it cannot be excitedto the higher energy level due to large energy gap. Since a particle at the LLL maypropagate only along the axis of magnetic field, the restriction of particle’s energylevel to the LLL leads to the dimensional reduction of its physical motion. In a rotating system , on the contrary, the energy levels of free fermions do not showdimensional reduction with increasing angular frequency Ω. Indeed, according toEq. (2.24) the energy gap between the levels labelled by radial excitation number l – 10 –oes not depend on the angular frequency Ω at fixed angular momentum m . FromFig. 4 we also see that the energy gap between the ground state and the first excitedstate does not increase. On the contrary, the general tendency is that the energy gapreduces for all fermion’s masses M as the rotation frequency increases.2. Ground state degeneracy . In a background magnetic field B the energy levels are degenerate. The degeneracyfactor – defined as the density of states per unit area of surface transverse to magneticfield – is a linearly increasing function of the magnetic field strength, | eB | / (2 π ). In a rotating system , on the contrary, there is no degeneracy apart from occasionallevel crossing, Fig. 4. Moreover, the rotation lists off the double degeneracy of the en-ergy spectrum (2.24) that emerges between the states with opposite total momentum ± µ m due to relation (2.30).We would like to stress that the rotation leads to the rotational steplike features whichare similar to the Shubnikov–de Haas oscillations in magnetic field. This similarity is adirect consequence of the finite-sized geometry and it is not directly linked to the rotation(see Sections 4 and 5.1 for more details). It is also important to note that the rotationalSdH steps occur in the absence of both external magnetic field and Fermi surface contraryto the conventional magnetic SdH oscillations.As the radius R increases the rotational SdH steps become smaller. Figure 7 suggeststhat in the limit R → ∞ all discontinuities eventually disappear.Below we show that at zero-temperature and zero chemical potential the system pro-vides no response to the global rigid rotation, in agreement with Ref. [15]. The absence ofany T = 0 response with respect to the rotation comes in sharp contrast with the effectof magnetic field which leads to an enhancement of the chiral symmetry breaking in thevacuum at zero temperature (this effect is known as the magnetic catalysis, Ref. [23]). How-ever, at finite temperature the overall influence of global rotation on the chiral transitionis similar to the one of the inverse magnetic catalysis [23]: the critical temperature of thechiral transition is a decreasing function of both temperature and angular frequency [18] .A qualitatively similar effect on the critical temperature may also be produced by a finitechemical potential [13], so that the analogy of the rotation and magnetic field at finitetemperature is not clear.Thus, we conclude that the energy spectrum shows that the rotation in the relativisticfermion system cannot identically be associated with an effective fictitious magnetic fieldcontrary to the rotation of nonrelativistic fermions as there are strong qualitative differencesin responses of the fermionic system with respect to the rotation and magnetic field. A simplest description of interacting fermions is given by the Nambu–Jona-Lasinio (NJL)model. In the rotating frame the NJL Lagrangian for the single fermion species of the mass A relevant phase diagram at finite radius of the cylinder is plotted below in Fig. 11. It has also beendiscussed in unbounded space in Ref. [14]. – 11 – is given by the following formula: S NJL = (cid:90) V d x (cid:113) − det ( g µν ) L NJL (cid:0) ¯ ψ, ψ (cid:1) , L NJL = ¯ ψ [ iγ µ ( ∂ µ + Γ µ ) − m ] ψ + G (cid:104)(cid:0) ¯ ψψ (cid:1) + (cid:0) ¯ ψiγ ψ (cid:1) (cid:105) , (3.1)where g µν corresponds to the metric (2.7) in the rotating reference frame (2.1) and Γ µ isthe connection (2.13).Similarly to the case of free fermions, we consider the infinitely long cylinder of ra-dius R , Fig. 1. For self-consistency the radius of the cylinder should lie within the lightcylinder, R Ω ≤
1, so that the rotational velocity at the boundary should not exceedthe speed of light. At the cylinder, the fermion modes satisfy the MIT boundary condi-tions (2.3). The system rotates as a rigid body so that the rotation can be described by asingle frequency Ω. Below we assume that the phase is uniform, so that the ground statecharacteristics (for example, the quark condensate) are coordinate-independent quantities.The partition function of the NJL model (3.1), Z = (cid:90) DψD ¯ ψ exp (cid:26) i (cid:90) V d x L NJL (cid:27) , (3.2)can be partially bosonized by inserting the identity1 = (cid:90) DπDσ exp (cid:26) − i G (cid:90) V d x (cid:104)(cid:0) σ + G ¯ ψψ (cid:1) + (cid:0) π + G ¯ ψiγ ψ (cid:1) (cid:105)(cid:27) , (3.3)into Eq. (3.2) and performing the integral over the fermionic fields with the following result: Z = (cid:90) DπDσ exp (cid:26) − i G (cid:90) V d x (cid:0) σ + π (cid:1) + ln Det (cid:2) iγ µ ( ∂ µ + Γ µ ) − m − ( σ + iγ π ) (cid:3)(cid:27) . (3.4)In order to study effects of rotation on the dynamical symmetry breaking we put thefermionic current mass – which breaks the chiral symmetry explicitly – to zero, m = 0.Moreover, we notice that we can always chirally rotate, with a suitably chosen chiral angle θ ,the expression under the fermionic determinant in the partition function (3.4):Det [ iγ µ ( ∂ µ + Γ µ ) − ( σ + iγ π )] = Det (cid:104) e − iγ θ { iγ µ ( ∂ µ + Γ µ ) − ( σ + iγ π ) } e − iγ θ (cid:105) = Det (cid:104) iγ µ ( ∂ µ + Γ µ ) − e − iγ θ ( σ + iγ π ) e − iγ θ (cid:105) = Det [ iγ µ ( ∂ µ + Γ µ ) − ˜ σ ] , (3.5)where we applied the chiral transformations (2.33) and used the fact that det (cid:0) e − iγ θ (cid:1) = 1.Next, we rotate the scalar-pseudoscalar combination σ + iγ π → e − iγ θ ( σ + iγ π ) e − iγ θ = ˜ σ ( σ, π ) , (3.6)into the purely scalar direction determined by new scalar field ˜ σ which has an arbitrarysign, ˜ σ = ±| ˜ σ | , and the absolute value | ˜ σ | = √ σ + π .– 12 –e stress that the sign of the field ˜ σ in Eq. (3.6) should be kept arbitrary becausethe field ˜ σ plays the role of the fermionic mass as it is seen from the expression for thefermion determinant (3.5). Indeed, the structure of the energy spectrum of free rotatingfermions depends – as we mentioned in Section 2.2 – on the sign of the fermion mass. Forthe system of interacting fermions the mass (i.e. its absolute value and sign) will be chosendynamically. As we will see below, in the chirally symmetric phase the system wouldunexpectedly prefer to choose the mass with a negative sign due to the MIT boundaryconditions (2.3).The measure of integration in the partition function (3.4) is invariant under the trans-formation (3.6). Since the rotation does not couple specifically to the pion mode π , then inEq. (3.4) we may always rotate the combination of the pion fields σ and π to the σ directionusing the chiral rotation (3.6). In the mean-field approximation we neglect fluctuations ofthe pion fields, so that we may always set π = 0 and ascribe the effects of the mass gapgeneration to the constant, coordinate-independent mean-field field σ . This homogeneousapproximation should work for small values of the rotation frequency Ω.The density of the Helmholtz free energy (the thermodynamic potential) of the systemin the (co)rotating frame can be read off from the partition function (3.4): (cid:101) F ( σ, π ) = σ G + V (cid:0) σ (cid:1) , (3.7)where V ( σ ) = − i Vol ln Det [ iγ µ ( ∂ µ + Γ µ ) − σ ] ≡ − i Vol tr ln (cid:104)(cid:0) i∂ t + Ω ˆ J z (cid:1) + (cid:126)∂ − σ (cid:105) , (3.8)is the potential induced by the vacuum fermion loop and Vol = (cid:82) V d x is the volume of the(3+1) dimensional space-time. The fermionic determinant in Eq. (3.8) has been rewrittenusing the following chain of identities:Det [ iγ µ ( ∂ µ + Γ µ ) − σ ] = (Det [ iγ µ ( ∂ µ + Γ µ ) − σ ] Det [ iγ µ ( ∂ µ + Γ µ ) − σ ]) = (Det [ iγ µ ( ∂ µ + Γ µ ) − σ ] Det γ [ iγ µ ( ∂ µ + Γ µ ) − σγ ]) = (Det [ iγ µ ( ∂ µ + Γ µ ) − σ ] Det [ iγ µ ( ∂ µ + Γ µ ) + σ ]) = (cid:16) Det (cid:104)(cid:0) i∂ t + Ω ˆ J z (cid:1) + (cid:126)∂ − σ (cid:105)(cid:17) , (3.9)where (cid:126)∂ is the spatial Laplacian. We also used the identity ln det ≡ tr ln in Eq. (3.8).In an unbounded (3+1) dimensional space the trace of the logarithm in Eq. (3.8) cangenerally be represented as follows:tr ln ˆ O = (cid:90) dt (cid:90) d x (cid:90) dk π (cid:90) d k (2 π ) ln O k , k , (3.10)where O k , k are the eigenvalues of the operator ˆ O . However, in the space bounded by thecylindrical surface the integration over the momentum subspace of the phase space (3.10)is modified according to Eq. (2.29):tr ln ˆ O = (cid:90) dt (cid:90) dz (cid:90) dk π (cid:90) dk z π ∞ (cid:88) l =1 ∞ (cid:88) m = −∞ ln O k ,k z ,l,m . (3.11)– 13 –onsequently, we get for the potential (3.8) the following expression: V ( σ ) = − iπR (cid:90) dk π (cid:90) dk z π ∞ (cid:88) l =1 ∞ (cid:88) m = −∞ ln (cid:34) − (cid:18) k + Ω (cid:18) m + 12 (cid:19)(cid:19) + k z + q m,l ( σ ) R + σ (cid:35) . (3.12)We see that the effect of rotation is to introduce an effective chemical potential Ω µ m foreach rotational mode labeled by the rotational quantum number m . The effective chemicalpotential is the product of the total angular momentum µ m , Eq. (2.28), and the angularfrequency Ω. Although the term Ω µ m is identified with “the chemical potential”, it is notab initio clear that the effect of the potential on the rotation is the same as the effect ofthe standard chemical potential because of the summation over m . Notice also that inEq. (3.12) the effect of rotation cannot be nullified by a simple shift k → k − µ m foreach rotational mode m since the integration over the momentum k is carried out over aninfinite interval.It is also important to mention that the effective potential (3.12) is generally not sym-metric under the reflection σ → − σ . Indeed, the condensate σ enters the potential (3.12)explicitly, via σ term, and implicitly, via the eigenvalues q m,l given by the solution ofEq. (2.26). The condensate σ plays a role of the (dynamically generated) mass of thefermion, M ≡ σ , which affects the spectrum of the eigenvalues q m,l in a nontrivial way,Fig. 3. Thus, in general, q m,l ( σ ) (cid:54) = q m,l ( − σ ) and, consequently, V ( σ ) (cid:54) = V ( − σ ).The system at finite temperature is obtained by performing the Wick rotation inEq. (3.12): k → iω n , (cid:90) dk π → iT (cid:88) n ∈ Z , ω n = πT (cid:18) n + 12 (cid:19) , (3.13)where ω n is the Matsubara frequency for fermions. Then we get in Eq. (3.12): V ( σ ) = − TπR (cid:88) m ∈ Z (cid:88) n ∈ Z ∞ (cid:88) l =1 (cid:90) dk z π ln ( ω n − i Ω µ m ) + E ml ( k z , σ ) T , (3.14)where µ m = m + 1 / E ml ( k z , σ ) in the laboratoryframe is given in Eq. (2.25).Next, we take into account that the sum over the fermionic Matsubara frequen-cies (3.13) using the standard formula: (cid:88) n ∈ Z ln ( ω n − i Ω µ m ) + E ml T = 12 (cid:88) n ∈ Z (cid:32) ln ω n + ( E ml − Ω µ m ) T + ln ω n + ( E ml + Ω µ m ) T (cid:33) = εT + ln (cid:16) e − ( E ml − Ω µ m ) /T (cid:17) + ln (cid:16) e − ( E ml +Ω µ m ) /T (cid:17) + const , (3.15)which highlights the interpretation of the product of the total angular momentum (2.28)and the angular frequency Ω as a chemical potential Ω µ m for each eigenmode m .– 14 –e substitute Eq. (3.15) into Eq. (3.14) and disregard in Eq. (3.15) the last term whichdoes not include the condensate σ and the angular frequency Ω. The resulting effectivepotential has two parts: V ( σ ; T, Ω) = V vac ( σ ) + V rot ( σ ; T, Ω) , (3.16)where the first term is the divergent vacuum energy which depends neither on temperaturenor on rotation velocity: V vac ( σ ) = − πR (cid:88) m ∈ Z ∞ (cid:88) l =1 (cid:90) dk z π E ml ( k z , σ ) . (3.17)The second term in Eq. (3.16) captures the rotational and temperature effects: V rot ( σ ; T, Ω) = − TπR (cid:88) m ∈ Z ∞ (cid:88) l =1 (cid:90) dk z π (cid:20) ln (cid:16) e − Eml ( kz,σ ) − Ω µmT (cid:17) + ln (cid:16) e − Eml ( kz,σ )+Ω µmT (cid:17)(cid:21) . (3.18)Generally, the effect of rotation on the particle properties may be understood fromthe fact that at finite temperature the thermal occupation number of fermionic particles isdetermined by the energy of the particles calculated in the rotating frame (2.25), n ( T, Ω) = (cid:16) e − (cid:101) ET (cid:17) − ≡ (cid:16) e − E − Ω µmT (cid:17) − , (3.19)and not by the energy defined in the laboratory frame (2.24).The vacuum part of the potential (3.17) is divergent in the ultraviolet limit and there-fore it has to be regularized: V vac ( σ ) = − πR (cid:88) m ∈ Z ∞ (cid:88) l =1 (cid:90) dk z π f Λ (cid:115) k z + q ml R E ml ( k z , σ ) . (3.20)where f Λ is a cutoff function. For this function one may use the exponential cutoff [13, 24]: f expΛ ( ε ) = sinh(Λ /δ Λ)cosh( ε/δ
Λ) + cosh(Λ /δ Λ) , (3.21)where one takes phenomenologically δ Λ = 0 .
05Λ following Ref. [13].Summarizing, the free energy of the gas of rigidly rotating fermions in the cylinder isgiven by Eq. (3.7): (cid:101) F ( σ ) = σ G + V vac ( σ ) + V rot ( σ ; T, Ω) , (3.22)where the vacuum and (rotational) thermal parts of the total potential are given byEqs. (3.20) and (3.18), respectively.We stress that the free energy (3.22) is defined in the (co)rotating frame which rotateswith the angular velocity that matches exactly the angular velocity of rotating fermionmedium. One can see that the free energy (3.22) is determined with respect to the ro-tating frame since the thermodynamic potential (3.18) involves only the energy levels (cid:101) E determined in the rotating frame . Moreover, the densities of the particles that are as- A rotation-dependent contribution to the vacuum energy (3.20) cancels out exactly. – 15 –ociated with the free energy (3.22) have the Fermi distribution in the rotating frame, n = ( e (cid:101) E/T + 1) − . We discuss the (free) energies in the laboratory and (co)rotating framesin Section 5.2. Let us consider the properties of the vacuum potential (3.20) which is regularized with thehelp of the cutoff function (3.21). In Fig. 5, we show the regularized vacuum potential asthe function of the condensate σ at different radii of the cylinder R . R = Λ - - - - - - - - σ / Λ V v ac / Λ R = Λ - - - - - - - - - σ / Λ V v ac / Λ (a) (b) R = Λ - R = Λ - R →∞ - - - - - - - - - σ / Λ V v ac / Λ (c) Figure 5 . Vacuum energy (3.20) with the cutoff function (3.21) at (a) small, (b) moderate and (c)large values of the cylinder radius R . All dimensional quantities are given in terms of the cutoffparameter Λ. First of all, we notice that at finite radius of the cylinder the vacuum potential isa smoothly diminishing function at all positive values of the condensate σ . However atnegative σ the potential exhibits infinite series of local minima which become progressivelydeeper as the value of the condensate σ decreases. One can observe that the potential is– 16 – sum of two parts: a smooth potential which is symmetric with respect to the reflectionof the potential, σ → − σ , and a saw-like part at negative values of the condensate σ . Atsmall radius R the smooth part is very small so that the saw-like part dominates, Fig. 5(a).As the radius increases, both parts of the potential becomes comparable with each other,Fig. 5(b) and, finally, at large radius the potential is dominated by the smooth component.In the limit R → ∞ the potential becomes a smooth function of σ and the symmetry withrespect to the flips σ → − σ is restored, Fig. 5(c).The unusual behavior of the vacuum potential observed at a finite radius of the cylin-der is an indirect consequence of the presence of the regularizing function (3.21) whicheffectively suppresses the energies E ml , Eq. (2.25), that are higher than the value of thecutoff parameter Λ. This procedure works rather well at positive values of the condensate σ because the energy is a regular, increasing function of the condensate. However at negativevalues of σ the energy (2.25) becomes a non-monotonic function of the condensate σ dueto the successive vanishing of the eigenvalues q ml of Eq. (2.26) with increase of the valueof the condensate σ , Fig. 3. We stress that this particular feature is a specific propertyof the MIT boundary conditions imposed at the cylindrical surface (2.3). If instead of theMIT boundary conditions we would choose its chiral analog (2.35) then the vacuum energyin Fig. 5 would appear in a mirrored form, σ → − σ , and consequently the nonmonotonicbehavior of the potential would be observed at the positive values of the chiral condensate.The vacuum part (3.20) of the potential plays an important role in our discussionsbelow as it defines the ground state of the theory together with the thermal part (3.18)that captures the effects of rotation. The regularization of the vacuum potential by a cutofffunction is a very standard procedure which is always implemented in the NJL model inthe thermodynamic limit. We thoroughly follow this standard procedure in our case whenthe physical volume of the model is restricted by the cylinder surface in two dimensions butstill stays infinite due to unbounded third dimension along the axis of the cylinder. Theimplementation of the cutoff is an obligatory requirement, which is, basically, associatedwith the nonrenormalizability of the NJL model. Notice that the saw-like behavior ofthe vacuum potential does not depend qualitatively on the particular form of the cutofffunction which in our case is given by Eq. (3.21). Therefore we consider this property ofthe vacuum potential as a physical feature. We hope that it may eventually be clarified inother, renormalizable models.In the limit of large radius R the invariance of the potential V ( σ ) on the sign flips σ → − σ is restored, as expected. In Fig. 6 we show the ground-state condensate σ ≡ (cid:104) σ (cid:105) as a function of the coupling constant G . The dynamically broken phase is realized at G > G c , where the critical value of the NJL coupling constant is as follows: G c ≡ G c ( R → ∞ ) = 19 .
578 Λ − . (4.1)The free energy (3.22) of fermions in the nonrotating cylinder of finite radius R =20 Λ − at zero temperature is shown in Fig. 7(a) at various values of the coupling constant G . One notices that the saw-like nature of the potential becomes will-pronounced at thefinite radius as negative ( σ <
0) and positive ( σ >
0) parts of the potential are visibly– 17 – =
10 15 20 25 30 35 40 450.00.20.40.60.81.01.2 G / Λ | σ | / Λ Figure 6 . The condensate as a function of the coupling constant G in the ground state in thermo-dynamic limit R → ∞ at zero temperature. The inset zooms in on the critical region around G s . different from each other. The direct consequence of this asymmetry is that at finite radiusthe system prefers to develop a negative condensate σ < σ is not a ground state of the model. [Noticethat if we would choose the chiral analog of the MIT boundary conditions (2.35) then thecorresponding free energy would be given by Fig. (3.22) with the sign flip σ → − σ and thesystem would develop a positive condensate σ > T = Ω = Λ = G Λ = G Λ
10 20304050 - - - σ / Λ F ( σ ) / Λ ∞
50 20 12 8 R Λ = T = Ω =
010 15 20 25 30 35 40 - - - - - - G / Λ σ / Λ (a) (b) Figure 7 . (a) Free energy (3.22) of the nonrotating cylinder at zero temperature as the functionof the field σ at various values of the coupling constant G . (b) The ground-state condensate σ asfunction of the coupling constant G at various fixed radii R . The negative branch of the dynamicallygenerated condensate in the infinite volume R → ∞ is also shown for comparison (taken from Fig. 6). The condensation of nonrotating fermions at finite radius R of the cylinder has a fewinteresting features which are readily seen in Fig. 7(b).1. Steplike discontinuities in the condensate. Contrary to the smooth behavior of thecondensate in the thermodynamic limit, the condensate σ at a finite radius R showsmultiple steplike features signaling that the condensate (and, consequently, mass gap)changes discontinuously at (infinite) set of values of the coupling constant G . These– 18 –redicted steplike features have the same nature as the Shubnikov–de Haas (SdH)oscillations [25] because they occur due to the presence of the discrete energy levels.Although the discreteness of the levels is a natural feature of finite-volume systems,these rotational SdH-like behaviors evolve nontrivially with the angular frequency Ω[see, for example, Fig. 11(a) below] thus suggesting certain similarity of rotation withthe presence of external magnetic field (see, however, our discussion in Sect. 4.2).It is crucial to highlight that the rotational SdH steps occur in the absence of bothexternal magnetic field and Fermi surface contrary to conventional SdH oscillations.As the radius R increases the rotational SdH steps become smaller. Figure 7 suggeststhat in the limit R → ∞ all discontinuities eventually disappear.2. Delayed broken phase. The critical value G c of the coupling constant becomes largeras the radius R decreases. In other words, the dynamically broken phase in a cylinderof a finite radius emerges at higher values of G compared to the thermodynamic limit.3. Weaker dynamical symmetry breaking. At finite R the dynamical mass gap genera-tion is smaller compared to the infinite-volume limit. The finite geometry suppressesthe condensate σ .4. Explicit symmetry breaking due to finite radius. There is no dynamical symmetrybreaking small coupling constant G < G c . However, we observe the effect of explicitsymmetry breaking: as radius R decreases the condensate σ becomes larger. As wewill see later this natural effect is associated with the MIT boundary conditions (2.3)which violate the chiral symmetry (2.33) explicitly.Thus, in nonrotating cylinder at zero temperature we expect to observe the existenceof three different regions in the phase diagram. At small radius of the cylinder the MITboundary conditions break the chiral symmetry explicitly by inducing a large nonzero neg-ative value of the condensate σ that increases with decrease of the radius R . At moderateradii, as the boundary effect becomes weak, the value of the condensate almost disappearsand the chiral symmetry is partially restored. At larger radii the chiral symmetry getsbroken again, at this case dynamically. The dynamical chiral symmetry breaking at finitecylinder’s radius is smaller compared to the thermodynamic limit. All these features areillustrated in Figs. 8(a) and (b).The zero-temperature phase diagram in the R − G plane is shown in Fig. 9. Thecritical coupling of the chiral transition G c ( R ) at finite R is higher than its value in thethermodynamic limit (4.1). As the radius R increases the critical coupling G c ( R ) becomessmaller. At small radii the chiral symmetry is broken explicitly by the MIT boundaryconditions (not shown in Fig. 9). At the end of this Section let us consider the response of the rotating fermionic mediumin the zero-temperature limit. As we have already seen, the vacuum part (3.20) doesnot depend on the rotating frequency Ω, and therefore the rotational effects may only be– 19 – Λ = T = Ω = R →∞ explicitly broken symmetrypartially restored symmetrydynamically broken symmetry - - - - R Λ σ / Λ G = G c ( R )- T = Ω = - - - - R Λ σ / Λ (a) (b) Figure 8 . (a) Ground-state condensate in a nonrotating cylinder at the coupling G = 42 Λ − >G c ( R → ∞ ) as the function of the radius R of the cylinder. The value of the condensate in thethermodynamic limit R → ∞ is shown by the arrow. At the bottom of the figure we highlight theexplicitly broken, partially restored and dynamically broken phases (regions) which appear, respec-tively, at small, moderate and large radii of the cylinder. (b) The explicitly induced condensate σ which appears in the interior of the cylinder due to the MIT boundary conditions imposed at itsfinite radius R at G = G c ( R ) −
0. At small radii R ∼ (a few)Λ − the condensate is large whileas the radius of the cylinder increases the explicitly broken chiral symmetry gets restored and thecondensate vanishes. Dynamically broken phaseUnbroken phase T = Ω = R →∞
10 20 30 40 502025303540 R Λ G c Λ Figure 9 . Phase diagram in a cylinder of the finite radius R at zero temperature. We do notshow a stretch at small values of the cylinder radius R ∼ (a few) / Λ where the chiral symmetry isexplicitly broken by the MIT boundary conditions. captured by the rotational part of the effective potential (3.18). Using the identity,lim T → T ln (cid:16) e − ε − µT (cid:17) = ( µ − ε ) θ ( µ − ε ) , (4.2)– 20 –e rewrite the rotational potential (3.18) as follows: V ( σ ; Ω) ≡ lim T → V rot ( σ ; T, Ω) = − πR (cid:88) m ∈ Z ∞ (cid:88) l =1 (cid:90) dk z π ( | µ m | − E ml ) θ ( | µ m | − E ml )= − πR ∞ (cid:88) m =0 ∞ (cid:88) l =1 (cid:90) dk z π ( µ m − E ml ) θ ( µ m − E ml ) , (4.3)where we used the flip symmetry of Eq. (2.30) to simplify the sum over angular momentumvariable m . An explicit integration gives: V = − ∞ (cid:88) m =0 ∞ (cid:88) l =1 q ml + σ R π R (cid:20) ν ml (cid:113) ν ml − (cid:18) ν ml + (cid:113) ν ml − (cid:19)(cid:21) θ ( ν ml − , (4.4)where ν ml = ( m + 1 / R (cid:113) q ml + σ R . (4.5)Notice that due to the step function, the sum over the integer variable l in Eq. (4.4) isalways finite. The quantity (4.5) can be expressed via energies (cid:101) E ml and E ml in, respectively,the rotating frame (2.24) and the laboratory frame (2.25) as follows: ν ml − E ml − (cid:101) E ml E ml − − (cid:101) E ml E ml . (4.6)However it is known that (cid:101) EE > | Ω | R < ν ml − < , and we come to the conclusion that at zero temperature the rotationalcontribution to the thermodynamic potential is exactly zero. In other words the coldvacuum does not rotate .The rotational response of the system may be qualified in terms of the angular mo-mentum and the moment of inertia that will be discussed in details in Sect. 5.2. Here wenotice that the insensitivity of the condensed medium to rotation in the zero temperaturelimit agrees well with the result obtained in the cold atom systems where it was shownthat the condensed (superfluid) fraction of the system does not contribute to the momentof inertia [27]. In this Section we discuss properties of rotating fermionic matter at finite temperature.As an example, we take large values of the coupling constant G = 42 / Λ and the cylinder In Ref. [12], the q ml = 0 solutions of Eq. (2.26), which may potentially give nonzero contributionsto Eq. (4.4), have not been discussed explicitly. We checked that there is no normalizable solution withvanishing q ml that are satisfying the MIT boundary conditions (2.3). – 21 –adius R = 20 Λ − . According to Fig. 8(a) at these values the T = 0 ground state is quiteclose to the thermodynamic limit.In Fig. 10(a) we show the free energy (3.22) as the function of the field σ for a set of theangular frequencies Ω at fixed temperature T = 0 . σ ) of the potential exhibits a saw-like behavior while the right part (positive σ ) is a smoothfunction of σ . The ground-state value of the condensate is determined by a minimum of thefree energy F ( σ ) which – similarly to the zero-temperature case – takes place at negative σ ’s for all studied values of the angular frequencies Ω. At this temperature the minimum ofthe free energy at Ω = 0 corresponds to the chirally broken phase characterized by a largenegative σ , the minimum of the free energy at Ω R = 0 . R = 0 . Ω R R Λ = G Λ = T / Λ = - - - - - - - - σ / Λ F ~ ( σ ) / Λ Ω R R Λ =
20, G Λ =
42, T / Λ = - - - - - - - σ / Λ [ F ~ ( σ ) - V v ac ( σ ) ] / Λ (a) (b) Figure 10 . (a) Free energy (defined in the corotating frame) for a rotating fermion matter inside thecylinder of the radius R = 20 Λ − at the temperature T = 0 . G = 42 / Λ for a set of angular frequencies Ω. (b) The same but for the free energy excluding the regularizedvacuum part. At this point we would like to stress that the saw-like behavior is not a direct con-sequence of the regularization of the vacuum part of the energy as the irregularities arealso naturally present at the thermal part of the free energy. In order to demonstrate thelatter, we show in Fig. 10(b) the free energy (3.22) with the vacuum term subtracted sothat only the only thermal part and the smooth σ / G term contribute to this quantity.The thermal potential leads to the saw-like dependence of the free energy although a bitless pronounced compared to the vacuum part.In Fig. 11(a) we show the ground-state condensate (mass gap) σ as the function of theangular frequency Ω at a set of fixed temperatures T . The chiral symmetry breaking inthe rotating environment has a few interesting features:1. The rotation restores (dynamically) broken chiral symmetry at all temperatures.2. The restoration of the chiral symmetry happens abruptly, and generally in a seriesof discontinuous steps. At certain temperatures the restoration happens in smaller– 22 –teps, but is always discontinuous. Therefore the restoration transition is of the firstorder.3. The rotation-induced steplike changes in the condensate are similar to the conven-tional Shubnikov–de Haas oscillations that are caused by external magnetic field. Aswe have already discussed, this similarity is a direct consequence of the finite-sizedgeometry of the rigidly rotating system. It is also important to stress that the ro-tational SdH steps occur in the absence of both external magnetic field and Fermisurface contrary to the conventional magnetic SdH oscillations.4. The chirally broken and chirally restored values of the condensate – respectively,before and after the step-like transition – are almost [apart from a small step-likediscontinuities in the broken region, visible in in Fig. 11(a)] independent of the angularfrequency Ω and temperature T . The critical temperature of the transition itself doesdepend on frequency, T c = T c (Ω), see Fig. 11(b) below. In the restored phase thevalue of the condensate is small but it is still nonzero due to the explicit breakingof the chiral symmetry by the MIT boundary conditions. The residual condensatecorresponds to the largest (rightmost) ”tooth” in the saw-like negative part of the freeenergy, Fig. 10(a), the position of which is independent of the angular frequency Ω.We would like to remind that the negative value of the ground state condensate σ isnot a universal feature as it is a direct consequence of the MIT boundary conditions (2.3)which break the chiral symmetry explicitly. For example, the so-called chiral analogue ofthe MIT boundary condition (2.35) would lead to the exactly same results but with thesign flip in the condensate σ → − σ .In Fig. 11(b) we present the phase diagram of the rotating fermionic matter in the T − Ω plane. We find that the critical temperature of the chiral symmetry restoration T c drops down with increase of the angular frequency Ω. The behavior of the condensatesuggests that the transition in between the chirally broken and chirally restored phasesis of the first order regardless of temperature because the discontinuity in the condensate(mass gap) σ at the transition does not depend on T . We will show below that despiteof the T –independent discontinuity in the mass gap σ , the free energy behavior softensacross the transition as temperature increases. Notice that as shown in Fig. 11(a), ourfinite-geometry result for the condensate qualitatively disagrees with the result of Ref. [14]which was obtained in rotating but unbounded space, where the behavior of the condensatedepends strongly on temperature T . However, as shown in Fig. 11(b), the qualitative formof the phase diagram obtained in the bounded space agrees with the result of Ref. [14]. A mechanical response of a physical system with respect to a global rotation can be quan-tified in terms of the angular momentum L and an associated moment of inertia I of thesystem as a whole. Since the rotation leads to the chiral symmetry restoration of thefermionic liquid in seemingly first order phase transition, one may naturally expect that– 23 – .3050.3090.3130.315 T / Λ = R Λ = Λ = - - - - Ω R σ / Λ Dynamically broken phaseUnbroken phase R Λ = G Λ = Ω R T c / Λ (a) (b) Figure 11 . (a) Condensate (mass gap) σ for the fermionic matter which is rigidly rotating withthe angular frequency Ω inside the cylinder of the radius R = 20 Λ − at the coupling G = 42 / Λ for a set of fixed temperatures T . (b) The phase diagram of the rotating fermionic matter in the T -Ω plane. both the angular momentum and the moment of inertia of the rotating fermionic mediummay exhibit a non-analytic behavior across the phase transition.We have already seen that for each fermionic level the eigenenergy in the corotatingframe (cid:101) E is expressed via the eigenenergy in the laboratory frame E and the correspondingangular momentum µ m with the help of a linear relation (2.24). Averaging Eq. (2.24) overthe whole thermodynamical ensemble in the rotating frame we get the following relationbetween the corresponding thermodynamic quantities [26]: (cid:101) E = E − L Ω , (5.1)which imply that L = − (cid:32) ∂ ˜ E∂ Ω (cid:33) S , (5.2)and, consequently, d (cid:101) E = T dS − L d Ω . For the free energy in the rotating frame (cid:101) F = (cid:101) E − T S , (5.3)one then gets d (cid:101) F = − SdT − L d Ω , which immediately leads to the following useful iden-tieis [26]: L = − (cid:32) ∂ ˜ F∂ Ω (cid:33) T , S = − (cid:32) ∂ ˜ F∂T (cid:33) Ω . (5.4)For completeness we notice that the free energy in the laboratory frame, F = E − T S thesame considerations lead to the relation dF = − SdT + Ω d L which imply that independentvariables in this case are the temperature T and the angular momentum L . One gets,consequently, the following relation associated with the angular momentum, Ω = (cid:18) ∂F∂ L (cid:19) T , (5.5)– 24 –hich is much less useful for our purposes compared to Eq. (5.4) that involves the freeenergy in the corotating frame.The moment of inertia I = I (Ω) may naturally be defined as a linear response of theangular momentum L of the system with respect to the angular velocity Ω , Ref. [27]: L = I (Ω) Ω . (5.6)Then, according to Eq. (5.4) the moment of inertia can be expressed as follows: I (Ω) ≡ L (Ω)Ω = − (cid:32) ∂ ˜ F∂ Ω (cid:33) T , (5.7)where we reduced the vector notations Ω = Ω e by setting that the rotation goes about afixed axis e .In Fig. 12 we show the ground-state free energy in the rotating frame as a function ofthe angular frequency Ω at a fixed set of temperatures. The temperature range is chosento cover the region of the phase transition shown previously in Fig. 11(b). As above, wefix the coupling constant G = 42Λ − and radius R = 20Λ − of the fermionic ensemble.Notice that the free energy shown in Fig. 12 is evaluated at the ground state σ whichis determined as a minimum of the free energy (3.22) for a given fixed set of parameters Ωand T (therefore we refer to this quantity as the “ground-state free energy”). The phasetransition is clearly visible as a change of the slope which appears at distinct angularfrequencies Ω for different temperatures T . The changes in the slope indicate that therotational properties of the fermionic medium are different in chirally broken and chirallyrestored phases. R Λ = G Λ = T / Λ T / Λ = T / Λ = - - - Ω R F ~ ( Ω , T ) / Λ Figure 12 . The ground-state free energy (3.22) determined in the rotating frame vs the angularfrequency Ω for the fixed coupling G = 42Λ − , fixed radius R = 20Λ − and various temperatures T ranging from T = 0 . T = 0 . The angular momentum of the rotating fermions (5.4) is shown in Figs. 13(a). Theangular momentum in the chirally broken phase is lower than the angular momentum in– 25 –he chirally restored phase. The two phases are separated by a sharp increase of the angularmomentum indicating, counterintuitively, that the chirally restored, gapless phase is morerotationally “massive” as compared to the chirally broken, gapped phase. R Λ = G Λ = T / Λ T / Λ = T / Λ = Ω R L ( Ω , T ) / Λ R Λ = G Λ = T / Λ = T / Λ = symmetry breaking symmetry restoration Ω R I ( Ω , T ) / Λ (a) (b) Figure 13 . (a) The angular momentum (5.4) and (b) the moment of inertia (5.7) of the rotatingfermionic matter across the phase transition line. The parameters are the same as in Fig. 12.
The same observation is confirmed by the behavior of the moment of inertia (5.7)which is shown in Figs. 13(b) for the same set of parameters. The moment of inertia I (Ω)of the fermionic matter in the chirally broken phase (lower Ω) is lower compared to therotating matter in the chirally restored phase (higher Ω). The moment of inertia is not aconstant function of the frequency as it growth up in both phases, and exhibits a rapidincrease in the transition region. According to Eqs. (5.1) and (5.3) the energy density in the laboratory frame is given bythe following formula: E = (cid:101) F + L Ω + T S , (5.8)where the angular momentum L and the entropy S can be obtained with the help ofEq. (5.4). We plot the entropy and energy densities in the laboratory frame in Fig. 14.Both entropy and energy (as determined in the laboratory frame) of the rotatingfermions experience a visible change with increase of the rotational frequency Ω as thesystem passes from the chirally broken region to the chirally restored region. The entropyand energy are smaller at the chirally broken region at low Ω compared to their values inchirally restored region the at higher Ω. Moreover, the transition between these regionsbecomes substantially smoother with increase of temperature. The rotational SdH steplikefeatures are also seen in Fig. 14 at low temperatures. In our paper we concentrated on properties of rotating systems of interacting fermions inthe framework of the Nambu–Jona-Lasinio model. Before summarizing the results of our– 26 – Λ = G Λ = T / Λ = T / Λ = Ω R S ( Ω , T ) / Λ R Λ = G Λ = T / Λ = T / Λ = Ω R E ( Ω , T ) / Λ (a) (b) Figure 14 . (a) Entropy density (5.4) and (b) energy density in the laboratory frame (5.8) as thefunction of rotation frequency Ω of the fermionic matter. The parameters are the same as in Fig. 12. studies we would like to stress that the problem of relativistic rigid rotation should alwaysbe studied in a spatially bounded physical volume and thus should depend on specifics ofconditions imposed on fermion fields at the boundaries of the volume.Indeed, the velocity of any particle in a rotating system should not exceed the speedof light. Therefore, a rigid rotation of relativistic matter in thermodynamic equilibriummust always be considered in a volume which is bounded in directions perpendicular tothe axis of rotation. The finite geometry implies that physical properties of the systemshould generally depend on conditions that are imposed on the fields at the boundariesof the volume. Thus, we come to the conclusion that physical systems in rotation shouldalways be considered in conjunction with appropriate boundary conditions. Moreover, afaster rotation implies a smaller size of the system in the direction transverse to the rota-tion axis. The latter inevitably leads to a stronger dependence of properties of the systemon the boundary conditions. One may look to this problem from a different perspective:the boundary conditions are becoming increasingly irrelevant for larger volumes for which,however, the rotation must be constrained to smaller angular frequencies. Thus, the prop-erties of any rigidly rotating system should always be formulated in finite geometries withappropriate boundary conditions.In our studies we considered the system of rotating fermions bounded by cylindricalshell with the MIT boundary conditions that physically confine the fermions inside thecylinder of the fixed radius R . Our main results can be summarized as follows:1. Boundaries are very important.
The MIT boundary conditions affect stronglythe phase structure of interacting fermions in cylinder. The finite boundaries brakethe chiral symmetry explicitly in the interior of the cylinders with the small ra-dius R (cid:46) (2 . . . − . They lead to the restoration of the chiral symmetry in themoderately-sized cylinders thus not allowing the dynamical chiral symmetry to occurat radii (2 . . . − (cid:46) R < R c ( G ). The dynamical condensation allowed at relativelylarge radius of the cylinder, R > R c ( G ), where the critical condensation radius R c ( R )is a rising function of the coupling G . A typical behavior of chiral condensate as the– 27 –unction of the radius of the cylinder is shown in Fig. 8(a). The zero-temperaturephase diagram in the radius–coupling constant plane is shown in Fig. 92. Boundary conditions break explicitly the reflection symmetry of the massgap , σ → − σ , in general. For the MIT boundary conditions (2.3) the mean-fieldvalue of the condensate is negative, σ <
0, while for the chiral MIT boundary con-ditions (2.35) the condensate takes positive values, σ >
0. The two boundary con-ditions are related to each other by the chiral transformation (2.33) with the chiralangle θ = π/
2. In the infinite volume, R → ∞ , the reflection symmetry σ → − σ isrestored, Fig. 5.3. Rotational Shubnikov–de Haas-like effect . The presence of the boundary leadsto specific steplike irregularities of the chiral condensate and other quantities asfunctions of coupling constant G , Fig. 7(b), temperature T and radius R , Fig. 8(a),and, most importantly, in angular frequency Ω, Fig. 11(a). We argued that thesesteplike features occur due to discreteness of the energy levels of fermions whichresult, for example, in well-pronounced steplike behavior of free energy both at zerotemperature, Fig. 7 and at finite temperature, Fig. 10. The predicted steplike featureshave the same nature as the Shubnikov–de Haas oscillations in the conductivity of amaterial with the crucial, however, difference that they occur in the absence of bothexternal magnetic field and Fermi surface.4. Cold vacuum cannot rotate . The vacuum at zero temperature vacuum is insensi-tive to rotation, Sect. 4.2, in agreement with a similar result obtained in nonrelativis-tic bosonic cold atom systems [27]. The same property has been also noted recentlyfor relativistic fermionic systems in Ref. [15].5.
Rotation cannot be associated with fictitious magnetic field.
We have pro-vided the arguments – which are based on the energy level structure, associateddensity of states and levels’ degeneracy – that the rigid rotation of a relativistic sys-tem cannot be associated with an external magnetic field thus supporting a similarstatement made recently in Ref. [13].6.
At finite temperature the rotation leads to restoration of spontaneouslybroken chiral symmetry.
The phase diagram in the angular frequency-temperatureplane for a cylinder with a finite radius is shown in Fig. 11(b). It agrees qualitativelywith calculation of Ref. [14] made in an unbounded space.7.
Softening of the transition strength with increase temperature.
As thetemperature increases, the critical angular frequency decreases and the transitionbecomes softer according to behavior of the free energy density in rotating frame,Fig. 12 (the softening is well seen in entropy density and energy density in the lab-oratory frame, Fig. 14). However in our approach – contrary to the calculations ofRef. [14], where the boundary effects are not taken into account – the discontinuity ofthe mass gap across the phase transition does not depend on the angular frequency Ωif one takes boundary conditions into account, Fig. 11(a).– 28 –.
Inequivalence of angular momentum and moment of inertia in differentphases.
At fixed temperature the fermion matter in the chirally restored (higherΩ) region has a higher angular momentum and higher moment of inertia comparedto the ones in the chirally broken (lower Ω) region, Fig. 14. In the transition regionboth quantities grow significantly as the angular frequency Ω is increasing.
Acknowledgments
The work of S. G. was supported by a grant from La Region Centre (France).
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