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

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 (Elze et al., 1986; Vasak et al., 1987). 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 (Gao et al., 2012; Yang et al., 2020). The pair production in parallel electric and magnetic fields with finite temperature and chemical potential from Wigner function approach is also investigated recently (Sheng et al., 2019). 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 (Buzzegoli et al., 2017; Buzzegoli and Becattini, 2018; Palermo et al., 2021). For a uniformly rotating massless fermion system, the analytic expressions of the charge current and the energy-momentum tensor are obtained (Ambruş and Winstanley, 2014). 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 (Cangemi and Dunne, 1996; Zhang et al., 2020a; Fang et al., 2021). There are also some investigations on the system of free fermion gas, quark matter or hadronic matter, with pure rotation (Chen et al., 2021; Fujimoto et al., 2021; Becattini et al., 2021), or with the coexistence of rotation and magnetic field (Chen et al., 2016; Fukushima et al., 2020), or with specific boundary conditions (Chernodub and Gongyo, 2017a, b, c; Zhang et al., 2020b; Yang et al., 2021). The quantum superfluid phenomena of Dirac fermions in the background of magnetic field and rotation are discussed recently (Liu and Zahed, 2018; Mottola and Sadofyev, 2021).

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 (Vilenkin, 1978, 1979, 1980), 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 $B=0$, $\Omega=0$ and $\mu=0$, where the lower orders are consistent with that from the approaches of thermal field theory and Wigner function respectively (Buzzegoli et al., 2017; Buzzegoli and Becattini, 2018; Yang et al., 2020), 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 (Felipe et al., 2008; Itokazu et al., 2018). 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 (Hiscock and Lindblom, 1983, 1985, 1987), i.e., the group speed of some transport coefficients, such as heat conductivity, would exceed the speed of light (Denicol et al., 2008). 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 (Jimenez-Alba and Yee, 2015; Hattori and Yin, 2016), thermal field theory (Buzzegoli et al., 2017; Buzzegoli and Becattini, 2018), Wigner function (Yang et al., 2020, 2022), 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 $\Omega$ and the chemical potential $\mu$, expanded around $B=0$, $\Omega=0$ and $\mu=0$, 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 $\hbar=c=k_{B}=1$. We use the Heaviside-Lorentz convention for electromagnetism and the chiral representation for gamma matrixes where $\gamma^{5}=\mathrm{diag\,(-1,-1,+1,+1)}$, which is the same as Peskin and Schroeder (Peskin and Schroeder, 1995).

In this section we briefly introduce the Dirac equation in curved spacetime (Parker and Toms, 2009), which is applied to a uniformly rotating frame (Chen et al., 2016).

In curved spacetime, under the background of the electromagnetic field, the Dirac equation for a single chiral fermion is

where the covariant derivative $D_{\mu}$ and gamma matrices $\underline{\gamma}^{\mu}$ are defined as

The underline in $\underline{\gamma}^{\mu}$ is used to distinguish the spacetime-dependent gamma matrices $\underline{\gamma}^{\mu}$ from the constant gamma matrices $\gamma^{a}$, and $\Gamma_{\mu}=\frac{1}{8}\omega_{\mu ab}[\gamma^{a},\gamma^{b}]$ is the affine connection. The definitions of vierbein $e_{a}^{\ \mu}$, metric tensor $g_{\mu\nu}$, and spin connection $\omega_{\mu ab}$ are listed as follows,

where $\eta_{ab}=\mathrm{diag}\,(+1,-1,-1,-1)$ is the metric tensor in Minkowski space, $X^{a}$ and $x^{\mu}$ are the coordinates in a local Lorentz frame and in a general frame, respectively.

In curved spacetime, the vector $J_{V}^{\mu}$, axial vector $J_{A}^{\mu}$ and symmetric energy-momentum tensor $T^{\mu\nu}$ become

where $D^{\mu}$, $\underline{\gamma}^{\mu}$ in curved spacetime have replaced $\partial^{a}$, $\gamma^{a}$ in flat spacetime.

Now we consider a frame $\mathcal{K}$ rotating uniformly with angular velocity $\boldsymbol{\Omega}=\Omega\mathbf{e}_{z}$ relative to an inertial frame $K$. The coordinates in $\mathcal{K}$ and $K$ are denoted as $x^{\mu}=(t,x,y,z)$ 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 $\Omega$ can not be too large, otherwise the synchronous condition in Eq. (8) can not be satisfied. According to Eq. (3), the metric tensor $g_{\mu\nu}$ and its inverse are

Keeping $g_{\mu\nu}$ unchanged, the vierbein $e_{\ \mu}^{a}$ still has a freedom degree of an arbitrary local Lorentz transformation. We can choose $e_{\ \mu}^{a}$ as

with zeros for other components.

Now we consider a single chiral fermion in a uniformly rotating frame under the background of a uniform magnetic field $\mathbf{B}=B\mathbf{e}_{z}$, and we choose the gauge potential in the inertial frame as $A^{a}=(0,\mathbf{A})$ with $\mathbf{B}=\nabla\times\mathbf{A}$. The covariant derivative $D_{\mu}$ and gamma matrices $\underline{\gamma}^{\mu}$ become

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, $J_{z}=\frac{1}{2}\Sigma_{3}-i(x\partial_{y}-y\partial_{x})$ is the $z$-component of the total angular momentum $\mathbf{J}$, and the term $-\Omega J_{z}$ can be naturally explained as the coupling of the angular momentum $\mathbf{J}$ and the angular velocity $\boldsymbol{\Omega}$.

In the chiral representation of gamma matrixes, where $\gamma^{5}=\mathrm{diag\,(-1,-1,+1,+1)}$, we can divide the chiral fermion field into left- and right-handed fermion fields, i.e. $\psi=(\psi_{L},\psi_{R})^{T}$. Since the equations of motion for $\psi_{L}$ and $\psi_{R}$ 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 $H$, $J_{R,z}=\frac{1}{2}\sigma_{3}-i(x\partial_{y}-y\partial_{x})$ 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 $\mathbf{A}$, i.e. $\mathbf{A}=(-\frac{1}{2}By,\frac{1}{2}Bx,0)$. Then the explicit form of the Hamiltonian is

It can be proved that, these three Hermitian operators, $H,\hat{p}_{z}=-i\partial_{z},J_{R,z}$, 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 $z,r,\phi$) as follows:

When $m=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots$,

When $m=-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\cdots$,

where $\rho=\frac{1}{2}eBr^{2}$, $L_{n}^{\mu}(\rho)$ is general Laguerre polynomial as introduced in Appendix B, $m$ is the magnetic quantum number, $\lambda=\pm 1$ represent the states with positive and negative energy respectively, and $n=0,1,2,\cdots$ represent different Landau levels. The eigenfunctions $\psi_{\lambda nmp_{z}}$ are denoted by the group of good quantum numbers ($\lambda,n,m,p_{z}$), which are normalized according to

In this section we consider a right-handed fermion system under the background of a uniform magnetic field $\mathbf{B}=B\mathbf{e}_{z}$, and the system is rotating uniformly with angular velocity $\boldsymbol{\Omega}=\Omega\mathbf{e}_{z}$. 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 $T=1/\beta$ and constant chemical potential $\mu$.

In this section, we will calculate the energy-momentum tensor $\mathcal{T}^{\mu\nu}$ (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 $\mathcal{T}^{\mu\nu}$:

The possible nonzero components of $\mathcal{T}^{\mu\nu}$ are $\mathcal{T}^{00}$, $\mathcal{T}^{11}=\mathcal{T}^{22}$, $\mathcal{T}^{33}$, and $\mathcal{T}^{03}$.

For the right-handed fermion system, the field operator of the symmetric energy-momentum tensor at $r=0$ is

with $\sigma^{\mu}=(1,\boldsymbol{\sigma})$ and the right-handed covariant derivative $D_{R}^{\mu}$ defined as

The macroscopic energy-momentum tensor $\mathcal{T}^{\mu\nu}$ can be calculated from $T^{\mu\nu}$ as follows,