Thermodynamics of Chiral Fermion System in a Uniform Magnetic Field

We construct the grand partition function of the system of chiral fermions in a uniform magnetic field from Landau levels, through which all thermodynamic quantities can be obtained. Taking use of Abel-Plana formula, these thermodynamic quantities can be expanded as series with respect to a dimensionless variable $b=2eB/T^{2}$. We find that the series expansions of energy density, pressure, magnetization intensity and magnetic susceptibility contain a singular term with $\ln b^{2}$, while particle number density, entropy density and heat capacity are power series of $b^{2}$. The asymptotic behaviors of these thermodynamic quantities in extreme conditions are also discussed.


I. INTRODUCTION
The properties of matter under the electromagnetic field have been studied extensively these years in many fields of physics. It is well known that strong electric field can lead to pair production of fermions in QED vacuum, which is called Schwinger mechanism [1][2][3]. Recently the effect of magnetic field on Schwinger mechanism is studied through the approaches of equal-time Wigner function and AdS/CFT correspondence [4][5][6]. In astronomy, compact stars, such as white dwarfs, neutron stars, and quark stars, often rotate very rapidly, which can produce magnetic field as strong as 10 12 ∼ 10 15 Gauss [7,8]. This strong magnetic field may have great impact on the state of compact stars [9,10]. In high energy physics, such as peripheral high energy heavy ion collisions, there also produce strong magnetic field in the collision region [11][12][13][14], and may induce the currents of charged particles along the direction of the magnetic field, which is called chiral magnetic effect [15][16][17][18][19]. In condensed matter physics, the strong magnetic field can reduce chiral condensate, which is called magnetic catalysis [20][21][22][23]. Meson condensation under background of the magnetic field together with the electric field or rotation is also discussed in [24][25][26][27]. In relativistic hydrodynamics, the Wigner function approach is often used to study the hydrodynamics of fermion system in general electromagnetic field [17,[28][29][30][31]. The chiral kinetic theory in the electromagnetic field is also studied recently [32,33].
In this article, we study the influence of magnetic field on the thermodynamics of the chiral fermion system, where we ignore the interaction among the fermions. Since the equations of motion for left-handed and right-handed fermions decouple, we will only consider the case of right-handed fermions in this article, and all results can be generalised to the left-handed case directly. In the previous work [34] by some of us, the electric current of the right-handed fermion system along the magnetic field, which is explained as chiral magnetic effect, has been obtained through the ensemble average of normal ordering of corresponding operator.
In this article we will use the method of grand partition function instead, from which we can obtain all knowledge of the thermodynamic system. Through solving the stationary Schroedinger equation of a single right-handed fermion in a uniform magnetic field, we can obtain a series of Landau levels, from which we can construct the grand partition function.
According to the standard procedure in quantum statistical mechanics, all thermodynamic quantities can be obtained by the grand partition function. In the expression of the grand partition function there is a summation over all Landau levels, which is difficult to be dealt with analytically. Fortunately, there is a Abel-Plana formula which can transform the discrete summation into integrations. Then we can express the grand partition function as a two-dimensional integration, which can be expanded as a series with respect to a dimensionless variable b = 2eB/T 2 , with e the electric charge of the right-handed fermion, B the magnetic field and T the temperature of the system. In the series, besides the terms with b 2n , there is an additional singular term b 2 ln b 2 , which indicates that the grand partition function is not analytic at b = 0, leading to the non-analyticity of some thermodynamic quantities at b = 0. We investigate the asymptotic behaviors of these thermodynamic quantities in the extreme conditions, such as weak/strong magnetic field limits and high/low temperature limits. Our study of the effect of the magnetic field on the thermodynamics of the chiral fermion system may have important theoretical meaning for the research on the quark gluon plasma state which can be produced in high energy heavy ion collisions.
The rest of this article is organised as follows. In Sec. II, the Landau levels of a single right-handed fermion in a uniform magnetic field is briefly listed. In Sec. III, we construct the grand partition function from Landau levels and express all thermodynamic quantities by the grand partition function. In sec. IV, all thermodynamic quantities are expanded as series with respect to a dimensionless variable b. In Sec. V, we study the asymptotic behaviors of these thermodynamic quantities in extreme conditions. In Sec. VI, all elements of the energy-momentum tensor are calculated. This article is summarized in Sec. VII.
Throughout this article we adopt natural units where = c = k B = 1. The convention for the metric tensor is g µν = diag (+1, −1, −1, −1). We use the Heaviside-Lorentz convention for electromagnetism and the chiral representation for gamma matrixes where , which is the same as Peskin and Schroeder [35].

II. LANDAU LEVELS FOR A SINGLE RIGHT-HANDED FERMION IN A UNIFORM MAGNETIC FIELD
The Lagrangian of a chiral (massless) fermion field ψ under the background of a uniform where D µ = ∂ µ + ieA µ , with e the electric charge of the fermion and A µ the gauge potential chose as A µ = (0, 0, Bx, 0). In this article we set eB > 0 for simplicity. The results of all thermodynamic quantities in this article can be extended to the range eB < 0.
In the chiral representation of gamma matrixes, we can write ψ = (ψ L , ψ R ) T , where the two-component spinors ψ L and ψ R are called left-handed and right-handed fermion fields.
Euler-Lagrange equation of the Lagrangian in Eq. (1) gives where σ = (σ 1 , σ 2 , σ 3 ) are Pauli matrixes and D = (−∂ x , −∂ y + ieBx, −∂ z ). 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.
The stationary Schroedinger equation iσ · Dψ R = Eψ R gives a series of Landau levels and eigenfunctions as follows [34], |c nλ | 2 = 1/(1 + F 2 nλ ), and ϕ n (ξ) is the n-th harmonic oscillator function along x-axis whose center is x = k y /eB. We have assumed that the eigenfunctions are set up in a box with sides of lengths L, i.e., 0 < x, y, z < L, and satisfy periodic boundary conditions in the y-axis and z-axis, i.e., k y = 2πn y /L, k z = 2πn z /L, (n y , n z = −∞, · · · , ∞). The condition that the center of the oscillation along x-axis is inside the box leads to 0 < n y < eBL 2 /(2π). Since E n (k z ) are independent of k y , the degeneracy of each Landau level is eBL 2 /(2π).

III. GRAND PARTITION FUNCTION AND THERMODYNAMIC QUANTITIES
We consider a system of right-handed fermions in a uniform magnetic field B = Be z , which is in equilibrium with a reservoir with temperature T and chemical potential µ R . The interaction among the fermions in this system is ignored for simplicity. From the Landau levels for a single right-handed fermion in Sec. II, we can construct the grand partition function ln Ξ of this system as follows, where β = 1/T , a = βµ R . The two theta functions θ(k z ) and θ(−k z ) in ln Ξ is necessary, as discussed in [34]. The vacuum terms in ln Ξ have been thrown away. The summations for k y and k z in Eq. (5) can be replaced by the degeneracy factor eBL 2 /(2π) and the integral (L/2π)´dk z respectively. Defining a dimensionless variable b = 2eBβ 2 , ln Ξ can be written where V = L 3 and g(a, b) is defined as From the grand partition function ln Ξ, the thermodynamic quantities of the system, such as particle number N = V n, energy U = V ε, pressure p, entropy S = V s and magnetization intensity M, can be expressed as Taking use of Eq. (6), all intensive quantities, such as particle number density n, energy density ε, pressure p, entropy density s, magnetization intensity M, magnetic susceptibility χ = ∂M/∂B and heat capacity c T = ∂ε/∂T , can be expressed by g(a, b) as follows,

IV. EXPANSIONS OF INTENSIVE QUANTITIES WITH RESPECT TO b
To study the influence of the magnetic field on the thermodynamics of the right-handed fermion system, in this section we will expand all thermodynamic quantities as series with respect to b = 2eBβ 2 .
Defining an auxiliary function f (a, x) as then g(a, b) in Eq. (7) becomes In Appendix A, we have proven that, when −π < a < π, the summation over Landau levels in the integrand in Eq. (21) can be transformed into integrations by following Abel-Plana formula [36,37], Then g(a, b) can be expressed as where the first term comes from the first integration on the right-hand side of Eq. (22), In Appendix B, we obtained the series expansion of g(a, b) at b = 0 as follows, where G = 1.282427 is the Glaisher number, B n are Bernoulli numbers, and C 2n+1 (a) (n 0) In the series of g(a, b), besides the terms with b 2n , there is also a single singular term b 2 ln b 2 , which indicates that g(a, b) is not analytic at b = 0. When a = 0, as calculated in Appendix D, the integration in Eq. (26) can be analytically integrated out, Taking use of Eqs. (13)- (19), all intensive quantities, such as particle number density n, energy density ε, pressure p, entropy density s, magnetization intensity M, magnetic susceptibility χ and heat capacity c T , can be expressed as series of b at b = 0 in the following, where we have used the temperature factor β and the electric charge e to cancle the dimensions of all intensive quantities. We can see that in the expansions of energy density ε, pressure p, magnetization intensity M and magnetic susceptibility χ, besides the terms with b n , there is also a term involving ln b 2 , which is singular at b = 0. Nevertheless, the expansions of particle number density n, entropy density s and heat capacity c T are just power series of b 2 , i.e. they are analytic at b = 0.
For left-handed fermion system, we can simply replace a = βµ R with βµ L in the expressions obtained above for right-handed fermion system, and all intensive quantities of chiral fermion system can be obtained as a summation of the results of left-handed and right-handed fermion systems.

V. EFFECT OF MAGNETIC FIELD ON THERMODYNAMIC QUANTITIES
In this section, we will investigate the effect of magnetic field on the thermodynamics of the chiral fermion system. Since the dimensionless variables a = µ R /T and b = 2eB/T 2 are proportional to the chemical potential µ R and the magnetic field B respectively, we will often use a and b instead of µ R and B to represent the chemical potential and the magnetic field in following discussion. For a fixed chemical potential a, we take Q(a, b) as the thermodynamic quantity with nonzero b and Q(a, 0) as the one with b = 0.
According to Eqs. (13)- (19) and the integration expression of g(a, b) in Eq. (23), we can plot the curves of all thermodynamic quantities as functions of b and a. The asymptotic behaviors of all thermodynamic quantities as b → 0 can be obtained directly by Eqs. (28)- (34). For b → ∞ limit, we must take use of following asymptotic formula of g(a, b), as Then Eqs. (13)-(19) can give the leading order term for all thermodynamic quantities as b → ∞.

A. Particle number density
The asymptotic behavior of particle number density ratio n(a, b)/n(a, 0) as b → 0 is .
Taking use of Eqs. (D9) and (C9) in the appendix, the coefficient of b 2 in Eq. (36) tends to be −7ζ ′ (−2)/(8π 2 ) = 0.0027 as a → 0 and tends to be zero as a → ∞, which indicates that in weak magnetic field limit the enhancement of particle number density is smaller for larger chemical potential. It is worth pointing out that Eq. (36) is consistent with the second order result of particle number density in weak electromagnetic field expansion by some of us in [31], where the Wigner function approach is used.
which increases linearly as b → ∞ and increases more slowly for larger a.
The asymptotic behavior of n(a, b)/n(a, 0) as a → 0 is listed as follows, In Figure 1, we plot the curves of n(a, b)/n(a, 0) with respect to b for a = 1, 2 and a → 0.
We can see that the existence of magnetic field can considerably enhance the particle number density, and the enhancement is larger when b is stronger and a is smaller. The trends of the curves are consistent with our asymptotic analysis. From the expressions of the grand partition function ln Ξ and energy U in Eqs. (5) and (9), we can express the energy density ε by following form The physical meaning of Eq. (39) is very clear: the whole energy of the system is jointly determined by the energy and the particle number of every quantum state and the degeneracy factor eBL 2 /(2π) of every Landau level. As the magnetic field increases, the particle number of every state will decrease rapidly as e − √ b due to Fermi-Dirac distribution, meanwhile the product of the energy and the degeneracy of every Landau level increases as b 3/2 , which indicates the existence of an extremum of energy density as b increases.
In Figure 2, we plot the curves of energy density ratio ε(a, b)/ε(a, 0) with respect to b for a = 0, 1, 2. We can see that the curves decline first and then rise, leading to the appearance of extremum as expected. When a = 0, the extremum locates at about (5.89, 0.9864), and ε(a, b)/ε(a, 0) = 1 at b ≈ 10.25. When a increases, the extremum moves to the lower right corner.
Since b 2 ln b 2 < 0 as b < 1, the energy density in weak magnetic field limit is smaller than the one without magnetic field.
The asymptotic behavior of ε(a, b) as b → ∞ is which increases linearly as b → ∞ and increases more slowly for larger a. The trend of the curves in Figure 2 are consistent with our asymptotic analysis.
If we take high temperature limit, i.e. b ≪ 1 and a ≪ 1, then the energy density becomes We emphasize that the ln T term for high temperature limit in Eq. (42) is also consistent with the second order result of energy density in [31].
If we take low temperature limit and a = 0, then the energy density becomes which decreases to be zero as T → 0. We can see that the existence of magnetic field makes the asymptotic behavior of the energy density become T 2 instead of T 4 as T → 0.
In Figure 3, we plot the curves of heat capacity ratio c T (a, b)/c T (a, 0) with respect to b for a = 0, 1, 2, where there are also extremum. For high temperature limit, the heat capacity becomes For low temperature limit with a = 0, the heat capacity becomes which tends to be zero as T → 0. Similar to the case of energy density, the existence of magnetic field makes the asymptotic behavior of the heat capacity become T instead of T 3 as T → 0. For high temperature limit, the pressure and entropy density become For low temperature limit with a = 0, the pressure and entropy density become which both tend to be zero as T → 0. Similar to the cases of energy density and heat capacity, the asymptotic behaviors of the pressure and the entropy density as T → 0 are also changed due to the existence of magnetic field. Since the magnetization intensity M(a, b) becomes saturation as b → ∞ , the magnetic susceptibility χ(a, b) tends to be zero as b → ∞. In Figure 7, we plot the curves of magnetic susceptibility χ(a, b) with respect to b for a = 0, 1, 2, where the curves tend to be zero as b → ∞ and tend to be divergent as b → 0. In fact, the magnetic susceptibility in weak magnetic field limit or high temperature limit becomes which is logarithmically divergent as B → 0 or T → ∞. The ln T term in Eq. (51) is also derived in a recent article [38], where the authors have calculated the high temperature expansion of the magnetic susceptibility of QCD matter with physical quark mass and the leading order term is just ln T with the same coefficient as ours.
In low temperature limit, i.e. b → ∞, the magnetic susceptibility tends to be zero, indicating that the system reaches the magnetic saturation state. This is consistent with the numerical result of lattice method for QCD matter system [38,39].
In Figure 7, we can see that χ is always positive, so the chiral fermion system is a paramagnetic system.

E. A brief summary
We have studied the effect of magnetic field on the thermodynamics of the right-handed fermion system. All results calculated above can be directly generalised to the left-handed case. In Table I, we list the leading order terms of all intensive quantities of chiral fermion system as b → 0 and b → ∞, with a = 0. In summary, we conclude that the magnetic field has great influence on the thermodynamics of chiral fermion system. In high energy heavy ion collisions, a new matter state "Quark Gluon Plasma" can be created [40][41][42], where the quarks and gluons are deconfined from the interior of the hadrons due to the high temperature and the chiral symmetry may be recovered [43,44]. For peripheral collisions, there will produce huge magnetic field in the collision region [11][12][13][14], which may considerably change the properties of the deconfined matter state and hence influence the distributions in phase space of the final particles from the collision.

VI. ENERGY-MOMENTUM TENSOR
In this section, we will show that how the nonzero components of T µν of the right-handed fermion system are related to g(a, b). The 00-component of T µν is just the energy density ε, which has been calculated through grand partition function in Secs. (III) and (IV). Now we want to calculate other components of T µν .
Since the thermodynamic system is located in a uniform magnetic field B = Be z , all thermodynamic quantities will be unchanged under the rotation along z-axis. The rotation by an angle φ along z-axis can be represented by following matrix, In order to keep T µν unchanged under the rotation matrix Λ µ ν (φ) for any angle φ, i.e. T µν = Λ µ α Λ ν β T αβ , the components of T µν must satisfy these conditions, i.e. the nonzero components are T 00 , T 33 , T 11 = T 22 , T 03 . According to Eq. (3), the trace of T µν is zero, so the independent components are T 00 , T 11 , T 03 .
We can not directly calculate T 11 and T 03 by the grand partition function. Alternatively, we appeal to ensemble average approach, in which all macroscopic quantities are the ensemble average of normal ordering of the corresponding operators. The symmetric and gauge invariant energy-momentum tensor is where σ µ = (1, σ), D µ = (∂ t , −∂ x , −∂ y + ieBx, −∂ z ), the angular brackets means statistical average, and the double dots enclosing the field operators means normal ordering as adopted in [34,45]. (In Appendix F, we have compared the results of normal ordering and un-normal ordering descriptions.) In order to calculate Eq. (54), the field operator ψ R (x) must be expanded by the orthonormal and complete eigenfunctions in Eq. (4) as follows [34], where a n , a † n , b n , b † n are annihilation and creation operators for fermions and antifermions. As calculated in [34] , the ensemble average of normal ordering for a † n a n and b n b † n is : : a † n (k y , k z )a n (k y , k z ) : = Substituting Eq. (55) into Eq. (54) and taking use of Eqs. (56)-(59) gives We can see that only the lowest Landau level (n = 0) contributes to T 03 , while T 11 comes from higher Landau levels (n > 0). Further calculation gives We can see that T 03 is only linear in b. In fact, the space component of the particle number current is ab/(8π 2 β 3 ), also linear in b as calculated in [34], which is called chiral magnetic effect. When a = 0, i.e. the chemical potential is zero, the particle number current disappears, but a nonzero term eBT 2 /24 remains in T 03 , which is reasonable since the same amount of fermions and antifermions move in the same direction along z-axis as discussed in [34], resulting in a zero value of the net particle number current and a nonzero value of energy current.
For left-handed fermion system, T 11 can be obtained by simply replacing a = βµ R with βµ L in Eq. (63). However, T 03 for left-handed fermion system can be obtained by space inversion in Eq. (62) and the result is The total T 03 of the chiral fermion system is the summation of the results of left-handed and right-handed fermion systems, and the result is eBµµ 5 /(2π 2 ) with µ = 1 2 (µ R + µ L ) and µ 5 = 1 2 (µ R − µ L ), which is consistent with [17]. From the series expansion of g(a, b), we can obtain Since g µν T µν = 0 and T 11 = T 22 , the result for T 33 is so we conclude that T 33 is just the pressure p of the system.

VII. SUMMARY
In this article, we have studied the thermodynamics of the chiral fermion system in a uniform magnetic field, where we ignored the interaction among all fermions. Since the equations of motion for left-handed fermions and right-handed fermions decouple, we did all the calculations for the case of right-handed fermion, which can be generalised to the left-handed case directly. From the Landau levels of a single right-handed fermion in a uniform magnetic field, we construct the grand partition function of this thermodynamic system, through which all intensive quantities can be obtained as a summation over all Landau levels. Taking use of Abel-Plana formula, the summation over all Landau levels can be transformed into integrations, which can be more readily dealt with analytically. We expanded these thermodynamic quantities as series with respect to a dimensionless variable b = 2eB/T 2 . We find that, the series expansions of energy density, pressure, magnetization intensity, magnetic susceptibility contain a singular term with ln b 2 , which indicates that these thermodynamic quantities are not analytic at b = 0. Meanwhile, the series expansions of particle number density, entropy density and heat capacity are power series of b 2 , which indicates the analyticity of them at b = 0. We plot the curves of these thermodynamic quantities with respect to b with zero and finite chemical potentials respectively, and discuss the asymptotic behaviors of these quantities in strong/weak magnetic field limits and low/high temperature limits. All elements of energy-momentum tensor are also calculated.
We conclude that the magnetic field can have important influence on the thermodynamics of chiral fermion system, which may be helpful to study the properties of the quark gluon plasma state created in high energy heavy ion collisions.

VIII. ACKNOWLEDGMENTS
RHF thanks Michael Peskin for helpful discussion. This work was supported by the Na- In mathematics, the Abel-Plana formula is a summation formula which is discovered by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820) independently [37].
It states that: If a function F (z) is analytic at Re z 0 and F (z) → 0 as |z| → ∞ along positive real axis, then In Sec. IV, we meet with following function, where b > 0 and f (a, x) is defined as We can see that the function F (z) = f (a, √ zb + s 2 ) satisfies F (z) → 0 as |z| → ∞ along positive real axis. In order to apply the Abel-Plana formula to the summation in the integrand in Eq. (A2), the function F (z) = f (a, √ zb + s 2 ) must be analytic at Re z 0.
The possible poles appear in F (z) when which implies In order to make F (z) analytic at Re z 0, the poles in Eq. (A5) must lie in the range Re z < 0, i.e. the inequation a 2 − (2n + 1) 2 π 2 − s 2 < 0 must be satisfied for any n and s, which indicates −π < a < π.
Appendix B: Expansion of g(a, b) at b = 0 Define an auxiliary function F (a, x) as where f (a, x) = ln(1 + e a−x ) + ln(1 + e −a−x ) as defined in Eq. (20). Then g(a, b) in Eq. (23) becomes Now we will expand F (a, x) at x = 0 in the following. By the variable transformation F (a, x) can be written as The factor y/ y 2 − x 2 in the integrand in F (a, x) can be replaced by following Taylor expansion, y where we have defined (−1)!! = 0!! = 1. Then F (a, x) becomes where we have defined d n (a, x) as The derivative of d n (a, x) with respect to x is Note that the factor f (a, x) + x in Eq. (B7) is an even function of x and can be expanded where C n (a) are independent of x which will be calculated later.
Taking use of following two identities, we can obtain F (a, x) in Eq. (B5) as In the following we will extract C n (a) in d n (a, x). When n = 0, we have C 0 (a) = d 0 (a, 0) = π 2 /6 + a 2 /2. When n > 0, through integration by parts we have Taking use of Eq. (B8), we obtain From Eq. (B2), we can see that the terms with x 4n (n 0) in F (a, x) do not contribute to g(a, b), so we can express F (a, x) as follows, Substituting Eq. (B15) into Eq. (B2), we have where we have used following integrations, with Glaisher number G = 1.282427... and Bernoulli numbers B n defined as We list some Bernoulli number B 2n+2 as follows, To study the aymptotic behavior of C 2n+1 (a) as a → ∞, we rewrite C 2n+1 (a) as Taking use of following integration formula, we have where Li s (z) is the polylogarithm function defined as This definition of Li s (z) is valid for arbitrary complex number s and for |z| < 1, and it can be extended to |z| 1 by the process of analytic continuation. Taking use of following asymptotic formulas for Li s (z), where we have used Γ ′ (2) = 1 − γ. So we obtain which shows that C 1 (a) → −∞ and C 2n+1 (a) → 0 (n > 0) as a → ∞. To expand C 2n+1 (a) at a = 0, we use following expression of C 2n+1 (a), C 2n+1 (a) = −δ n,0 + 1 (4n + 1)!ˆ∞ 0 dy ln y d 4n+1 dy 4n+1 1 e y+a + 1 + 1 e y−a + 1 .
This UNO description is also used in recent articles [46,47]. Now the ensemble average of the energy density operator becomes where the subscript "un" of ε un means un-normal ordering. We can see that UNO description leads to the infinite vacuum energy in ε un . Comparing Eq. (39) and Eq. (F3), it is equivalent to do following regularization, Taking use of two dimensionless variables a = βµ R and b = 2eBβ 2 instead of µ R , β, eB, Eq.
(F4) can be rewritten as and ε 0 is just the familiar infinite vacuum energy, as in the free field theory [35,48]. This infinite vacuum energy can not be detected experimentally, since only the energy difference from the ground state can be observable. It is worth pointing out that the term b 2 ln b 2 in ε disappears in ε un and the b 2 order of ε un is consistent with [31].
In Eq. (C8), we obtained the asymptotic behavior of C 2n+1 (a) as a → ∞, Note that C un 1 (a) is a positive infinity which is expected since C 1 (a) → −∞ as a → ∞. This expression of C un 1 (a) can also be obtained in [31] where the authors adopt dimensional regularization to deal with the divergence.
Since only the derivative of C 2n+1 (a) appears in the expression of the particle number density n in Eq. (28), and C ′ 2n+1 (a) → 0 as a → ∞ as calculated in Eq. (C9), i.e. C un′ 2n+1 (a) = C ′ 2n+1 (a), so the UNO and NO descriptions can obtain the same expression of the particle number density n, which is also consistent with [31].