Thermodynamics for a rotating chiral fermion system in the uniform magnetic field

We study the thermodynamics for a uniformly rotating system of chiral fermions under the uniform magnetic field. Then we obtain the mathematical expressions of some thermodynamic quantities in terms of the series with respect to the external magnetic field $B$, the angular velocity $\Omega$ and the chemical potential $\mu$, expanded around $B=0$, $\Omega=0$ and $\mu=0$. Our results given by such series are a generalization of the expressions available in the references simply corresponding to the lower-order terms of our findings. The zero-temperature limit of our results is also discussed.


I. INTRODUCTION
The properties of Dirac fermion system have been investigated from many aspects for a long time. For a hydrodynamic system consisting of Dirac fermions under the background of electromagnetic fields, Wigner function is an appropriate tool, which can provide a covariant and gauge invariant formalism [1,2]. It is worth pointing out that, although Wigner function defined in 8-dimensional phase space is not always non-negative, one can always obtain non-negative probability density when the 4-dimensional momentum is integrated out. For massless (or chiral) fermion system with uniform vorticity and electromagnetic fields, the charge current and the energy-momentum tensor up to the second order have been obtained from Wigner function approach, including chiral anomaly equation, chiral magnetic and vortical effects [3,4]. The pair production in parallel electric and magnetic fields with finite temperature and chemical potential from Wigner function approach is also investigated recently [5]. Without external electromagnetic fields, the energy-momentum tensor and charge current of the massless fermion system up to second order in vorticity have been obtained from thermal field theory [6][7][8]. For a uniformly rotating massless fermion system, the analytic expressions of the charge current and the energy-momentum tensor are obtained [9]. For the massive and massless fermion systems under the background of a uniform magnetic field, the general expansions with respect to fermion mass, magnetic field and chemical potential are derived by the approaches of proper-time and grand partition function [10][11][12]. There are also some investigations on the system of free fermion gas, quark matter or hadronic matter, with pure rotation [13][14][15], or with the coexistence of rotation and magnetic field [16,17], or with specific boundary conditions [18][19][20][21][22]. The quantum superfluid phenomena of Dirac fermions in the background of magnetic field and rotation are discussed recently [23,24].
In this article, we consider a uniformly rotating chiral fermion system in a uniform magnetic field, where we ignore the interaction among the fermions and the directions of the angular velocity and the magnetic field are chosen to be parallel. In this article we will adopt the approach of normal ordering and ensemble average to calculate the thermodynamical quantities of the system. Firstly we briefly derive the Dirac equation in a rotating frame under the background of a uniform magnetic field from the Dirac equation in curved space.
Then through solving the eigenvalue equation of the Hamiltonian in cylindrical coordinates, we can obtain a series of Landau levels, from which one can calculate the expectation value of corresponding thermodynamical quantities for each eigenstate. From the approach of ensemble average used in [25][26][27], the macroscopic thermodynamical quantities can be expressed as the summation over the product of the particle number (Fermi-Dirac distribution) and the expectation value in each eigenstate. We expand all thermodynamical quantities as threefold series at B = 0, Ω = 0 and µ = 0, where the lower orders are consistent with that from the approaches of thermal field theory and Wigner function respectively [4,6,7], and to our knowledge the general orders have not been obtained before. We also calculate all quantities in zero temperature limit, and obtain the equality of partcile/energy density and corresponding currents along z-axis, which can provide qualitative reference for the thermodynamics of compact stars in astrophysics, such as neutron star and magnetic star, since the magnitudes of the magnetic field and rotational speed are huge compared to the temperature of the compact stars [28,29]. In this article all thermodynamical quantities will be calculated at the rotating axis (r = 0), so the boundary condition at the speed-of-light surface will not affect our results.
From the point of view of hydrodynamics, it has been pointed out that the relativistic hydrodynamical equations with only first order term does not obey the causality [30][31][32], i.e., the group speed of some transport coefficients, such as heat conductivity, would exceed the speed of light [33]. Therefore, the high order terms in hydrodynamics are necessary, which indeed repair the issue of causality. There have been some earlier work to study the second order terms of transport phenomena, such as Kubo formula from quantum field theory [34,35], thermal field theory [6,7], Wigner function [4,36], etc. All of these work are perturbation theory essentially, from which the general order terms have not been obtained. In this article, we consider a special configuration for the electromagnetic field and vorticity field in hydrodynamics, i.e. with a pure homogeneous magnetic field parallel to a homogeneous vorticity field, and obtain the general order terms of all thermodynamic quantities, which is important to study the analytic behavior of hydrodynamics in mathematics.
The rest of this article is organized as follows. In Sec. II and III, we briefly derive the Dirac equation in a uniformly rotating frame and list the Landau levels and corresponding eigenfunctions of a single right-handed fermion, which are just reference review. In Sec.
IV and V, we obtain the expressions of some thermodynamic quantities in terms of the series with respect to the external magnetic field B, the angular velocity Ω and the chemical potential µ, expanded around B = 0, Ω = 0 and µ = 0, which are our main results. In Sec. VI, the zero temperature limit of the thermodynamical quantities is discussed. This article is summarized in Sec. VII.
Throughout this article we adopt natural units where = c = k B = 1. We use the Heaviside-Lorentz convention for electromagnetism and the chiral representation for gamma matrixes where γ 5 = diag (−1, −1, +1, +1), which is the same as Peskin and Schroeder [37].

II. DIRAC EQUATION IN A UNIFORMLY ROTATING FRAME
In this section we briefly introduce the Dirac equation in curved spacetime [38], which is applied to a uniformly rotating frame [16].
In curved spacetime, under the background of the electromagnetic field, the Dirac equation for a single chiral fermion is where the covariant derivative D µ and gamma matrices γ µ are defined as The underline in γ µ is used to distinguish the spacetime-dependent gamma matrices γ µ from the constant gamma matrices γ a , and Γ µ = 1 8 ω µab [γ a , γ b ] is the affine connection. The definitions of vierbein e µ a , metric tensor g µν , and spin connection ω µab are listed as follows, where η ab = diag (+1, −1, −1, −1) is the metric tensor in Minkowski space, X a and x µ are the coordinates in a local Lorentz frame and in a general frame, respectively.
In curved spacetime, the vector J µ V , axial vector J µ A and symmetric energy-momentum tensor T µν become where D µ , γ µ in curved spacetime have replaced ∂ a , γ a in flat spacetime. Now we consider a frame K rotating uniformly with angular velocity Ω = Ωe z relative to an inertial frame K. The coordinates in K and K are denoted as x µ = (t, x, y, z) and X a = (T, X, Y, Z) respectively, which are related to each other by following transformations, It should be pointed out that the rotational angular velocity Ω can not be too large, otherwise the synchronous condition in Eq. (8) can not be satisfied. According to Eq. (3), the metric tensor g µν and its inverse are Keeping g µν unchanged, the vierbein e a µ still has a freedom degree of an arbitrary local Lorentz transformation. We can choose e a µ as with zeros for other components.
and in this case the Dirac equation for a single chiral fermion can be written as where e is the charge of the chiral fermion, is the z-component of the total angular momentum J, and the term −ΩJ z can be naturally explained as the coupling of the angular momentum J and the angular velocity Ω.
Since the equations of motion for ψ L and ψ R decouple, we only discuss right-handed fermion field in this article. All results can be directly generalised to the left-handed case. In the following, we set eB > 0 for simplicity.
The right-handed part of Eq. (14) is where H, J R,z = 1 2 σ 3 − i(x∂ y − y∂ x ) is Hamiltonian and the z-component of the total angular momentum of the right-handed fermion. In this article we shall choose the symmetric gauge for A, i.e. A = (− 1 2 By, 1 2 Bx, 0). Then the explicit form of the Hamiltonian is It can be proved that, these three Hermitian operators, H,p z = −i∂ z , J R,z , are commutative with each other, then we can construct the common eigenfunctions of them. According to the calculations for Landau levels in Appendix A, we list the common eigenfunctions and corresponding energy in cylindrical coordinate system (where the three coordinate variables are z, r, φ) as follows: where ρ = 1 2 eBr 2 , L µ n (ρ) is general Laguerre polynomial as introduced in Appendix B, m is the magnetic quantum number, λ = ±1 represent the states with positive and negative energy respectively, and n = 0, 1, 2, · · · represent different Landau levels. The eigenfunctions ψ λnmpz are denoted by the group of good quantum numbers (λ, n, m, p z ), which are normalized according tô

IV. PARTICLE CURRENT
In this section we consider a right-handed fermion system under the background of a uniform magnetic field B = Be z , and the system is rotating uniformly with angular velocity Ω = Ωe z . The interaction among the fermions in this system is ignored. We assume that this rotating system is in equilibrium with a reservoir, which keeps constant temperature T = 1/β and constant chemical potential µ.

A. Ensemble average
We will calculate the macroscopic particle current of the system at the rotation axis (i.e. at r = 0) through ensemble average approach, in which all macroscopic thermodynamical quantities are the ensemble average of the normal ordering of the corresponding field operators.
The forms of the eigenfunctions in Eqs. (18,20) at r = 0 or ρ = 0 are simplified to which are to be used in the following calculations of ensemble average. We find that the zcomponent m of the total angular momentum can only take values ±1/2 due to the absence of the orbital angular momentum at r = 0.
We can see that the macroscopic particle current J µ consists of the summation over the product of the particle number (Fermi-Dirac distribution) and the expectation value in each mode described by the quantum numbers (λ, n, m, p z ).

B. Particle number density
Firstly we calculate the particle number density ρ ≡ J 0 of the system. Making use of and from Eq. (25) one can obtain where we have defined three dimensionless quantities a = βµ, b = 2eBβ 2 , ω = βΩ, and have defined g(x, b) as In a recent article [11], making use of Abel-Plana formula, the authors obtained the asymptotic expansion of g(x, b) at b = 0 as follows where G = 1.28242... is the Glaisher number, B n are Bernoulli numbers, and C 2n+1 (x) is defined and expanded at x = 0 in the following, C 2n+1 (x) = −δ n,0 + 1 (4n + 1)!ˆ∞ 0 dy ln y d 4n+1 dy 4n+1 Plugging Eqs. (29,30) into Eq. (27), one can get the threefold series expansion of the particle number density at a = 0, b = 0, ω = 0 or µ = 0, B = 0, Ω = 0 as follows, The lower orders O(b 2 , ω 2 , bω) in Eq. (31) are consistent with the perturbative results in [4,6,7], where the authors used the approaches of thermal field theory and Wigner function respectively.

V. ENERGY-MOMENTUM TENSOR
In this section, we will calculate the energy-momentum tensor T µν (at r = 0) of the right-handed fermion system as described in Sec. IV. According to the rotation symmetry along z-axis, the energy-momentum tensor at r = 0 are unchanged under the rotation along z-axis, which leads to following constraints on T µν : The possible nonzero components of T µν are T 00 , T 11 = T 22 , T 33 , and T 03 .
For the right-handed fermion system, the field operator of the symmetric energymomentum tensor at r = 0 is with σ µ = (1, σ) and the right-handed covariant derivative D µ R defined as The macroscopic energy-momentum tensor T µν can be calculated from T µν as follows,

A. Energy density
Firstly we calculate the energy density ε ≡ T 00 of the system. Making use of and from Eq. (25) one can obtain which can be expanded as the threefold series at a = 0, b = 0, ω = 0 or µ = 0, B = 0, Ω = 0 as follows, where the logarithmic term b 2 ln b 2 has been discussed in detail in [11], and its coefficient is independent of ω in this work. It is worth noting that there would be no such logarithmic term if the un-normal ordering description of field operators was adopted [4,51].

B. Pressure
The pressure P of the system is T 33 . Making use of and from Eq. (25) one can obtain which can be expanded as the threefold series at a = 0, b = 0, ω = 0 or µ = 0, B = 0, Ω = 0 as follows, One can obtain T 11 from the traceless condition for energy-momentum tensor, T 00 = 2T 11 + T 33 .

C. Energy current
The energy current along z-axis is T 03 . Making use of and from Eq. (25) one can obtain which can be expanded as the threefold series at a = 0, b = 0, ω = 0 or µ = 0, B = 0, Ω = 0 as follows, Up to now, we have obtained all thermodynamical quantities of the right-handed fermion system. For left-handed fermion system, one can derive corresponding quantities from the right-handed case through space inversion: , Ω → Ω, where the subscripts R, L are used to distinguish the quantities in right-handed case from that in left-handed case.

VI. ZERO TEMPERATURE LIMIT
Now we turn to the thermodynamics of the system at zero temperature limit. When the temperature tends to be zero, with chemical potential µ, magnetic field B, and angular velocity Ω fixed, then the three dimensionless quantities a = βµ, b = 2eBβ 2 , ω = βΩ all tend to be infinity. The asymptotic behavior of g(x, b) as x → ∞ and b → ∞ has been obtained in [11], From Eqs. (27,33), one can derive the expressions of the particle density ρ and the current J z at zero temperature limit as follows, At zero temperature limit, due to the coupling of the spin with the magnetic field and the angular velocity, the spin alignment of all particles and antiparticles will be along z-axis of the system. Since these particles are right-handed, they will move along z-axis with the speed of light c (c = 1 in natural unit), so it is reasonable that the particle density ρ equals to the z-component current J z at zero temperature limit.
From Eqs. (40,43,46), the expressions of energy density ε, pressure P and energy current T 03 at zero temperature limit are The movements of the particles and antiparticles with the speed of light along z-axis leads to the equality of the energy density ε and the energy current T 03 . Since there is no energy current along the direction of the x-and y-axis, then T 11 and T 22 vanish in this system, which results in the equality of the energy density ε and the pressure P .

VII. SUMMARY
In this article, we have investigated the thermodynamics of the uniformly rotating right- We also calculate all quantities in zero temperature limit, and obtain the equality of partcile/energy density and corresponding currents along z-axis. Since for the chiral fermion the right-handed part decouples from the left-handed part, in this article we only considered the case of the right-handed fermion system, which can be directly generalized to the left-handed case through space inversion. In this article, the currents and energy-momentum tensor are calculated at the rotating axis (r = 0), so the boundary condition at the speed-of-light surface will not affect our results. The calculations for these quantities off or far from the rotating axis (r = 0) as well as with the boundary condition at the speed-of-light surface may be investigated in the future. The Hamiltonian for a right-handed fermion under the background of the uniform mag- where we have chosen A = (− 1 2 By, 1 2 Bx, 0) for the gauge potential. One can refer to [39,51,52] for other choices of the gauge potential.
When γ / ∈ Z, there are two independent solutions as follows, y 1 = F (α, γ, z), where F (α, γ, z) is the confluent hypergeometric function defined as The asymptotic behavior of F (α, γ, z) as z → ∞ is the same as e z . When α is a non-positive integer, then F (α, γ, z) becomes a polynomial.