# 3D Lovelock Gravity and the Holographic c-theorem

Gökhan Alkaç∗<EMAIL_ADDRESS>and Gün Süer†<EMAIL_ADDRESS>
(November 21, 2022)

###### Abstract

We show that the scalar-tensor theory that arises in a rigorous $D\to 3$ limit
of Lovelock gravity up to cubic order admits a holographic c-theorem and
verify that the value of the c-function at the UV fixed point matches with the
Weyl anomaly coefficient (up to an irrelevant factor). The constructed
c-function is the same as one that follows from the “naive limit”, which is
obtained by scaling the couplings by a factor of $\frac{1}{D-3}$, and then
setting $D=3$.

Contents

toc

##  1 Introduction

Given that gravity should have a geometrical interpretation due to the
principles of equivalence and general covariance, it is possible to show the
uniqueness of Einstein’s theory in $D=4$ as follows [1]: The structure of non-
gravitational theories suggest that the metric tensor should be related to
matter variables by a second-order differential equation. Therefore, the
consistent coupling to a covariantly conserved energy-momentum tensor requires
a symmetric, covariantly conserved tensor with at most second-derivatives of
the metric tensor. Lovelock showed that the most general form of such a tensor
in $D$-dimensions follows from the variation of an action functional

$S=\int\mathrm{d}^{D}x\,\sqrt{-g}\,\mathcal{L},\qquad\qquad\mathcal{L}=\sum_{m}c_{m}\mathcal{L}_{m},$
(1.1)

where the $m$-th order Lovelock Lagrangian is given by [2, 3]

$\mathcal{L}_{m}=\frac{1}{2^{m}}\delta^{\mu_{1}\nu_{1}\cdots\mu_{m}\nu_{m}}_{\rho_{1}\sigma_{1}\cdots\rho_{m}\sigma_{m}}R^{\rho_{1}\sigma_{1}}_{\
\ \ \ \mu_{1}\nu_{1}}\cdots R^{\rho_{m}\sigma_{m}}_{\ \ \ \ \mu_{m}\nu_{m}}.$
(1.2)

In addition to satisfying these basic requirements, Lovelock gravity has a
unitary massless spin-2 graviton around any of its constant curvature vacua
[4] and is singled out among theories derived from a Lagrangian of the form
$\mathcal{L}\,[g_{\mu\nu},R_{\mu\nu\rho\sigma}]$ as the unique theory with
equivalent metric and Palatini formulations [5]. However; $\mathcal{L}_{m}$
vanishes when $D<2m$ because of the anti-symmetrization in the generalized
Kronecker delta symbol, and it is a topological invariant in $D=2m$, i.e., the
variation of the action yields a total derivative. Hence, a richer structure
can only be achieved for $D>2m$, which leaves the Einstein-Hilbert action as
the only possibility for having the above-mentioned properties in $D=4$.

Although this is somewhat disappointing for gravitational phenomenology, it is
hard to overemphasize the importance of Lovelock gravity in testing
theoretical ideas beyond Einstein’s theory and the applications of the AdS/CFT
correspondence [6, 7, 8], our holographic framework to probe the properties of
strongly coupled gauge theories, are no exception (see [9, 10, 11, 12, 13, 14,
15] for examples). In this work, our focus will be the holographic c-theorem
of [16], where it was shown for Einstein’s theory in $D$-dimensions that it is
possible to construct a monotonic function along an RG-flow induced by matter
fields satisfying the null-energy condition (NEC). In the UV
($r\rightarrow\infty$), the value of the function is proportional to the Weyl
anomaly coefficients of the boundary field theory, and as a result, one
obtains a holographic generalization of Zamolodchikov’s celebrated c-theorem
for 2D QFTs ($c_{\text{UV}}>c_{\text{IR}}$ in any RG-flow connecting two fixed
points) [17] to arbitrary (even) dimensions.

For a better understanding, a natural question needs to be answered: Although
it is possible to construct a monotonic function along an RG-flow in generic
$D$-dimensions, only even-dimensional CFTs have a Weyl anomaly. What should be
the interpretation of the value of the function at the UV in the case of odd-
dimensional CFTs? By studying Quasi-topological gravity (QTG) [18], a
particular higher-curvature gravity theory that allows distinguishing the
type-A and type-B anomaly coefficients (associated with the Euler density and
the independent Weyl invariants of weight $-d$ respectively), Myers and Sinha
formulated a c-theorem in arbitrary dimensions [19, 20]: Considering a
$d$-dimensional CFT on $S^{d-1}\times R$, one can calculate the entanglement
entropy of the ground state between two halves of the sphere, and find a
universal contribution. At the fixed points, the monotonic function is
associated with this universal contribution
($a^{*}_{\text{UV}}>a^{*}_{\text{IR}}$) and for even $d$, as expected, it
precisely matches with the A-type anomaly coefficient ($a^{*}=a$, and
therefore $a_{\text{UV}}>a_{\text{IR}}$)111Note that the anomaly coefficient
related to the Euler density is denoted by $c$ and $a$ for $d=2$ and $d\geq 4$
respectively. Therefore, the monotonic function is denoted by $c(r)$ and
$a(r)$, and the theorem is named as c-theorem and a-theorem accordingly.
Throughout this paper, we keep using the former terminology even when we
discuss the higher-dimensional cases since our main focus at the end will be
3D gravity theories with 2D CFT duals., providing further evidence for Cardy’s
conjecture in $d=4$ [21] that was later proven in [22].

Based on the importance of higher-curvature interactions in establishing these
results, in [23], Sinha investigated the analog of Lovelock Lagrangians (1.2)
that naturally admit a c-function but vanish in 3D except the Einstein-Hilbert
term ($m=1$). He showed that demanding the existence of a simple holographic
c-theorem up to quadratic curvature invariants leads to New Massive Gravity
(NMG), a non-linear completion of the Fierz-Pauli theory describing a unitary
excitation of massive spin-2 gravitons around the constant curvature vacua
[24, 25]. In [23], cubic and quartic invariants admitting a c-function were
also found and a prescription for obtaining such curvature invariants of
arbitrary order was given in [26]. Despite having higher-derivative field
equations unlike Lovelock gravity, NMG and its extensions exhibit many similar
properties especially in holography, and later a limit relating these three-
and higher-dimensional theories at the Lagrangian level was discovered [27].
Due to these similarities and also their own interesting properties, there has
been an ongoing investigation of the holographic properties of 3D higher-
curvature gravities [28, 29, 30, 31, 32, 33, 34].

In this paper, we will take a different route motivated by the recent
developments over the past two years that started with a proposal for
obtaining lower-dimensional ($D\leq 2m$) versions of Lovelock gravity by
Glavan and Lin, which can be summarized as follows [62]: inserting a
$D$-dimensional metric ansatz into field equations of Lovelock gravity gives
terms that are proportional to $D-p$ with $p\leq 2m$ so that they vanish when
$D=p$. However; by scaling the relevant couplings in (1.1) as
$c_{m}\rightarrow\frac{c_{m}}{D-p}$ and then setting $D=p$, one can obtain
non-trivial solutions. In their paper, they studied $D\rightarrow 4$ limit of
Einstein-Gauss-Bonnet theory ($m=1,2$ and $c_{2}$ is scaled by
$\frac{1}{D-4}$) and obtained various novel solutions (constant curvature
spacetimes, the cosmological spacetimes, the spherically symmetric black
holes, and the linearized fluctuations around maximally symmetric vacua).
Although this “naive limit” was later shown to be inconsistent on different
grounds [35, 36, 37, 38, 39, 46, 40, 41, 42, 43, 44, 45], it led to the
realization that rigorous procedures giving rise to well-defined theories
exist [46, 47, 48, 49, 50] and they result in scalar-tensor theories with two-
derivative field equations, i.e., examples of of Horndeski gravity [51, 52] or
generalized Galileons [53, 54]. Properties of this special class of scalar-
tensor theories have been investigated in [55, 57, 56, 58, 59].

Here we aim to find out whether the 3D scalar-tensor theory arising in a
rigorous limit of Lovelock gravity up to cubic order, which we will refer to
as 3D Lovelock gravity, admits a holographic c-theorem, and therefore
constitute an alternative to NMG and its extensions in this context. Although,
as we show in Section 2, that the naive limit suggests the existence of a
monotonic c-function along an RG-flow, it is known that some solutions do not
survive in the well-defined scalar-tensor theories. For example, it was shown
in [50] that 4D static black holes with spherical and hyperbolic horizons are
excluded in 4D cubic Lovelock gravity. Therefore, this is a nontrivial check
that needs to be performed for the evaluation of this theory in the
holographic context.

The outline of this paper is as follows: In Section 2, we construct the
holographic c-function and find its value at the UV fixed point of Lovelock
gravity in $D>6$. Then, we present the results obtained from the naive limit.
Section 3 is devoted to the study of holographic c-theorem in 3D Lovelock
gravity up to cubic term. We show that the theory admits a holographic
c-function, and this function and its value at the UV fixed point match with
the ones obtained from the naive limit. We also make a comparison with an
earlier study of Horndeski gravity involving up to only linear curvature terms
[60]. In Section 4, we comment on some important relations discovered in the
study of 3D higher-derivative theories and their applicability in 3D Lovelock
gravity. We end our paper with a summary of our results in Section 5.

##  2 c-theorem in Lovelock Gravity in $D>6$ and Its Naive Limit to $D=3$

We will construct a holographic c-function for the cubic Lovelock gravity in
$D>6$ by the method of [63]. The action that we consider is as follows:

$S=\int\mathrm{d}^{D}x\sqrt{-g}\left[R-2\Lambda_{0}+\alpha
L^{2}\mathcal{L}_{m=2}+\beta
L^{4}\mathcal{L}_{m=3}-\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\right],$
(2.1)

where the Lagrangians $\mathcal{L}_{m=2,3}$ can be found from the general
expression (1.2), and the free-scalar $\chi$ is introduced to impose the NEC
in a practical way. Considering the following domain-wall ansatz

$\mathrm{d}s^{2}=e^{2A(r)}\eta_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}+e^{2B(r)}\mathrm{d}r^{2},$
(2.2)

where $\eta_{ab}$ is the Minkowski metric and the Latin indices run from $0$
to $(D-2)$, and assuming a radial profile for the free-scalar, $\chi=\chi(r)$,
one can show that the NEC reduces to the positivity of the radial kinetic
energy of the scalar field, i.e., $\frac{1}{2}\chi^{\prime 2}\geq 0$. Using
this configuration in the action (2.1) gives an effective action for the
functions $\left[A(r),B(r),\chi(r)\right]$. After finding the corresponding
Euler-Lagrange equations, and then fixing the gauge by $B(r)=0$, an inequality
of the following form can be obtained by eliminating the bare cosmological
constant $\Lambda_{0}$ from the equations:

$\frac{1}{2}\chi^{\prime
2}=\mathcal{F}\left(A^{\prime},A^{\prime\prime},\ldots\right)\geq 0,$ (2.3)

where $\mathcal{F}$ is a polynomial of the derivatives of the function $A(r)$.
If it does not contain more than second-order derivatives of the function
$A(r)$, and has the form

$\mathcal{F}=\sum_{n}a_{n}{A^{\prime}}^{2n}A^{\prime\prime}\geq 0,$ (2.4)

a monotonically increasing c-function can be defined as [31]

$c(r):=\frac{1}{({\ell_{P}{A^{\prime}})}^{(D-2)}}\sum_{n}\frac{a_{n}}{2(n+1)-D}{A^{\prime}}^{2n}.$
(2.5)

While the monotonicity is evident for even $D$, it was also shown to be the
case by construction for odd $D$ in [20]. For the value of the c-function at
the UV fixed point, we set $A(r)=\frac{r}{L}$ and find

$c=\left(\frac{L}{\ell_{P}}\right)^{(D-2)}\sum_{n=0}^{D-1}\frac{a_{n}}{2(n+1)-D}.$
(2.6)

The application of this procedure to cubic Lovelock gravity in $D$-dimensions
yields the following inequality

$\mathcal{F}=(D-2)\left[-1+2\alpha(D-3)(D-4)L^{2}A^{\prime
2}-3\beta(D-3)(D-4)(D-5)(D-6)L^{4}A^{\prime 4}\right]A^{\prime\prime}\geq 0,$
(2.7)

which leads to the monotonically increasing c-function

$c(r)=\frac{1}{({\ell_{P}{A^{\prime}})}^{(D-2)}}\left[1-2\alpha(D-2)(D-3)L^{2}A^{\prime
2}+3\beta(D-2)(D-3)(D-4)(D-5)L^{4}A^{\prime 4}\right],$ (2.8)

whose value at the UV fixed point is given by

$c=\left(\frac{L}{{\ell_{P}}}\right)^{(D-2)}\left[1-2\alpha(D-2)(D-3)+3\beta(D-2)(D-3)(D-4)(D-5)\right].$
(2.9)

It is obvious that this whole structure allows a naive limit to $D=3$ by
scaling the couplings as $(\alpha,\beta)\to\dfrac{1}{D-3}\ (\alpha,\beta)$ and
then setting $D=3$. The resulting expressions are

$\displaystyle\mathcal{F}$ $\displaystyle=\left[-1-2\alpha L^{2}A^{\prime
2}+18\beta L^{4}A^{\prime 4}\right]A^{\prime\prime}\geq 0,$ (2.10)
$\displaystyle c(r)$
$\displaystyle=\frac{1}{{\ell_{P}{A^{\prime}}}}\left[1-2\alpha L^{2}A^{\prime
2}+6\beta L^{4}A^{\prime 4}\right],$ (2.11) $\displaystyle c$
$\displaystyle=\frac{L}{{\ell_{P}}}\left[1-2\alpha+6\beta\right].$ (2.12)

In the next section, we will show that these expressions are indeed preserved
in 3D Lovelock gravity up to the cubic term. In 3D, the value of the
c-function at the UV fixed point $c$ should be proportional to the Weyl
anomaly coefficient and the central charge of the boundary CFT.

##  3 c-theorem in 3D Lovelock Gravity

3D cubic Lovelock gravity is described by the action

$S=\int\mathrm{d}^{3}x\sqrt{-g}\left[R-2\Lambda_{0}+\alpha
L^{2}\mathcal{L}_{m=2}+\beta
L^{4}\mathcal{L}_{m=3}-\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\right],$
(3.1)

where we again introduce the free-scalar $\chi$ to impose the NEC and
$\mathcal{L}_{m=2,3}$ terms arising after a rigorous limit read

$\displaystyle\mathcal{L}_{m=2}$
$\displaystyle=4G^{\mu\nu}\phi_{\mu}\phi_{\nu}-4X\Box\phi+2X^{2}$ (3.2)
$\displaystyle\mathcal{L}_{m=3}$
$\displaystyle=-48R^{\mu\nu}\phi_{\mu\nu}X-48R^{\mu\nu}\phi_{\mu}\phi_{\nu}X+24RX\Box\phi+6RX^{2}$
$\displaystyle+96\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}\Box\phi+48\phi_{\mu\nu}\phi^{\mu\nu}\Box\phi-24\phi_{\mu\nu}\phi^{\mu\nu}X$
$\displaystyle-144\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}X-96\phi^{\mu}\phi^{\nu}\phi_{\mu\rho}\phi_{\
\nu}^{\rho}-32\phi^{\mu\nu}\phi_{\mu\rho}\phi_{\ \nu}^{\rho}$
$\displaystyle-16(\Box\phi)^{3}+24X(\Box\phi)^{2}-24X^{3},$ (3.3)

with the following definitions: $\phi_{\mu}\equiv\partial_{\mu}\phi$,
$\phi_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}\phi$ and
$X\equiv\partial_{\mu}\phi\partial^{\mu}\phi$. The $\mathcal{L}_{m=2}$ term is
found in [56, 49, 46] with the “Weyl trick” of [61] that was first used to
obtain the $D\rightarrow 2$ limit of general relativity. A more general
version was obtained in [47] by a regularized Kaluze-Klein reduction, where
one gets additional terms breaking the shift symmetry of the scalar $\phi$
that are proportional to the curvature of the maximally symmetric internal
space considered in the reduction. When the internal space is taken to be
flat, these different approaches give the same result up to field
redefinitions. The $\mathcal{L}_{m=3}$ term is found in [50] by using both
methods. We prefer the version arising from a flat internal space for
simplicity.

In order to establish a holographic c-theorem for this theory, we will use the
method described in the previous section by also assuming a radial profile for
the scalar field $\phi=\phi(r)$ in addition the free-scalar $\chi$ and employ
the domain wall ansatz (2.2) in 3D, which lead to an effective action for the
functions $\left[A(r),B(r),\phi(r),\chi(r)\right]$. This time, one arrives at
the following inequality

$\mathcal{F}(A^{\prime},A^{\prime\prime},\phi^{\prime},\phi^{\prime\prime})=-A^{\prime\prime}+\alpha\left(4A^{\prime
2}\phi^{\prime 2}+4\phi^{\prime 4}-2\phi^{\prime
2}(A^{\prime\prime}-2\phi^{\prime\prime})-4A^{\prime}\phi^{\prime}(2\phi^{\prime
2}+\phi^{\prime\prime})\right)\geq 0,$ (3.4)

which is not in the standard form (2.4). However; considering the Euler-
Lagrange equation of the function $\phi(r)$, which is given by

$\displaystyle 0$
$\displaystyle=(A^{\prime}-\phi^{\prime})\Big{[}A^{\prime}\left(2\phi^{\prime
2}\left(\alpha+27\beta\phi^{\prime\prime}\right)-\alpha\phi^{\prime\prime}-36\beta\phi^{\prime
4}\right)+\phi^{\prime}\left(3\phi^{\prime\prime}\left(\alpha-30\beta\phi^{\prime
2}\right)-2A^{\prime\prime}\left(\alpha-18\beta\phi^{\prime 2}\right)\right)$
$\displaystyle+A^{\prime 2}\left(36\beta\phi^{\prime
3}-2\alpha\phi^{\prime}\right)\Big{]},$ (3.5)

one sees that this equation has two branches, both of which are solved by
$\phi^{\prime}=A^{\prime}$. Inserting this into the original inequality (3.4),
we find exactly the inequality suggested by the naive limit (2.10). Therefore,
the 3D cubic Lovelock gravity admits a holographic c-function. The c-function
and its value at the UV boundary are as follows:

$\displaystyle c(r)$
$\displaystyle=\frac{1}{{\ell_{P}{A^{\prime}}}}\left[1-2\alpha L^{2}A^{\prime
2}+6\beta L^{4}A^{\prime 4}\right],$ (3.6) $\displaystyle c$
$\displaystyle=\frac{L}{{\ell_{P}}}\left[1-2\alpha+6\beta\right],$ (3.7)

which are the same expressions obtained from the naive limit (2.11,2.12).

The relation between the AdS3 radius $L$ and the bare cosmological constant
$\Lambda_{0}$ can be easily found by setting
$\phi^{\prime}=A^{\prime}=\frac{1}{L}$ and $\chi=0$ in the Euler-Lagrange
equations following from the effective action, which yields

$\Lambda_{0}=-\frac{1}{L^{2}}\left(1+\alpha-6\beta\right).$ (3.8)

With this at hand, we can show the agreement with the Weyl anomaly coefficient
[64] of the boundary CFT by considering the theory on $S^{2}$ with the
following Euclidean metric [65]

$\mathrm{d}s^{2}=\frac{\mathrm{d}r^{2}}{1+\frac{r^{2}}{L^{2}}}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}),$
(3.9)

which is a solution when $\phi(r)=\log\left(\frac{r}{L}\right)$. In the on-
shell action, we are interested in the logarithmic divergence since other
divergences are expected to be canceled by surface terms and appropriately
chosen counter terms. The relevant part of the on-shell action is as follows:

$S\propto\left[\Lambda_{0}+\frac{1}{L^{2}}(3-3\alpha+6\beta)\right]\ln\left(\frac{2R}{L}\right),$
(3.10)

where $R$ is the UV cut-off scale. Upon using the relation between the bare
cosmological constant $\Lambda_{0}$ and the AdS3 radius $L$ in (3.8), we see
that the anomaly coefficient is proportional to the value of the c-function at
the UV boundary (3.7), as expected.

Note that the c-theorem in Horndeski gravity involving up to only linear
curvature terms was investigated in [60]. With our conventions, the action
that the authors considered is as follows:

$S=\int\mathrm{d}^{D}x\,\sqrt{-g}\,\mathcal{L},\qquad\qquad\mathcal{L}=R-2\Lambda_{0}-\frac{1}{2}\left(\alpha
X-\gamma\,G^{\mu\nu}\phi_{\mu}\phi_{\nu}\right).$ (3.11)

For an AdS vacuum solution characterized by the effective cosmological
constant $\Lambda_{\text{eff}}=-\frac{D(D-1)}{2L^{2}}$ where $L$ is the AdS
radius, the effective kinetic term for the Horndeski scalar $\phi$ is

$\mathcal{L}_{\phi}=-\frac{1}{2}(\alpha+\gamma\Lambda_{\text{eff}})X.$ (3.12)

They found that a monotonic c-function is admitted only at the critical point
where the effective kinetic term (3.12) vanishes, i.e., the couplings are
related by $\gamma=\frac{2\alpha L^{2}}{D(D-1)}$. Therefore, having second-
order field equations is indeed not enough for the existence of a c-theorem. A
particular combination of Horndeski couplings are required, and we see that 3D
Lovelock gravity is such an example.

##  4 Comment on the Wald Formula and the Central Charge Formula of [66, 67]

Having shown the existence of a holographic c-theorem, we will now briefly
comment on two important relations that appeared in the study of 3D higher-
derivative theories described by the Lagrangians of the form222Note that this
is the most general possibility for a pure gravity theory since the Riemann
tensor can be written in terms of the metric, the Ricci tensor and the Ricci
scalar in 3D due to the vanishing of the Weyl tensor.
$\mathcal{L}\,[g_{\mu\nu},R_{\mu\nu}]$. In [23], an interesting relation
between the c-function and the Wald formula [68, 69] was observed: Evaluation
of the integrand of the Wald formula for the metric ansatz (2.2) in 3D with
$B(r)=0$ yields

$\frac{1}{2}g_{tt}g_{rr}\frac{\partial\mathcal{L}}{\partial
R_{trtr}}=c(r)A^{\prime}(r).$ (4.1)

Additionally, the central charge of the boundary CFT for higher-derivative
gravity theories, which is proportional to the anomaly coefficient, can be
computed by the formula [66, 67]

$c=\frac{L}{3\ell_{P}}\frac{\partial\mathcal{L}}{\partial
R_{\mu\nu}}g_{\mu\nu},$ (4.2)

where this quantity should be evaluated for the AdS3 solution. When these
relations are applied to 3D Lovelock gravity described by the action (3.1)
(without the $\chi$-term that is introduced for imposing the NEC), they yield
the following expressions

$\displaystyle c(r)$
$\displaystyle=\frac{1}{{\ell_{P}{A^{\prime}}}}\left[1+6\beta L^{4}A^{\prime
4}\right],$ (4.3) $\displaystyle c$
$\displaystyle=\frac{L}{{\ell_{P}}}\left[1+6\beta\right].$ (4.4)

Although there is no reason for expecting these relations to hold for an
arbitrary scalar-tensor theory, we see that the contribution from the
$\mathcal{L}_{m=3}$ term matches our previous results, while the
$\mathcal{L}_{m=2}$ term gives no contribution, which is incorrect.
Interestingly, the correct contribution arises from the following Lagrangian

$\displaystyle\mathcal{L}_{m=2}$
$\displaystyle=8R^{\mu\nu}\phi_{\mu}\phi_{\nu}+8R^{\mu\nu}\phi_{\mu\nu}+4\phi_{\mu\nu}\phi^{\mu\nu}+8\phi_{\mu\nu}\phi^{\mu}\phi^{\nu}$
$\displaystyle-4R\Box\phi-2RX-4(\Box\phi)^{2}+2X^{2},$ (4.5)

which is indeed the immediate result that is obtained after the application of
the Weyl trick or the regularized Kaluza-Klein reduction to the
$\mathcal{L}_{m=2}$ Lagrangian in $D$-dimensions. The Lagrangian given in
(3.2) is obtained after integration by parts and the use of Bianchi
identities. Recently, it was observed in [58] that when the Lagrangian
holographic relation obeyed by Lovelock gravity in higher dimensions is
demanded to be realized in the resulting scalar-tensor theories in lower
dimensions, it requires introducing certain boundary terms by hand. Together
with this finding, our results suggest that boundary terms play a crucial role
in comparing Lovelock gravity in lower dimensions with pure gravity theories.

##  5 Summary

In this paper, we studied the existence of a holographic c-function in 3D
Lovelock gravity described by the Lagrangian (3.1), a scalar-tensor theory
that arises after a rigorous $D\to 3$ limit of Lovelock gravity. Being an
example of a Horndeski theory, it has second-order field equations. However,
as shown in [60] before, this does not guarantee the formulation of the
holographic c-theorem. We have shown that the theory admits a holographic
c-function and its value at the UV fixed point agrees with the one predicted
by the Weyl anomaly in (3.10). Moreover, our findings match the results
obtained by the naive limit (2.10-2.12). By checking two important holographic
relations (4.1) and (4.2) of 3D higher-derivative gravity theories, we also
shed light on the important role that boundary terms play in the comparison of
lower-dimensional Lovelock gravity and pure gravity theories.

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

Ren-Hong Fang<EMAIL_ADDRESS>Key Laboratory of Particle Physics and
Particle Irradiation (MOE), Institute of Frontier and Interdisciplinary
Science, Shandong University, Qingdao, Shandong 266237, China  Theoretical
Physics Research and Innovation Team, College of Intelligent Systems Science
and Engineering, Hubei Minzu University, Enshi, Hubei 445000, China

###### Abstract

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
(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).

##  II Dirac equation in a uniformly rotating frame

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

$i\underline{\gamma}^{\mu}D_{\mu}\psi(x)=0,$ (1)

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

$D_{\mu}=\partial_{\mu}+ieA_{\mu}+\Gamma_{\mu},\ \ \
\underline{\gamma}^{\mu}=\gamma^{a}e_{a}^{\ \mu}.$ (2)

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,

$e_{a}^{\ \mu}=\frac{\partial x^{\mu}}{\partial X^{a}},\ \ \ e_{\
\mu}^{a}=\frac{\partial X^{a}}{\partial x^{\mu}},\ \ \
g_{\mu\nu}=\eta_{ab}e_{\ \mu}^{a}e_{\ \nu}^{b},$ (3) $\omega_{\mu
ab}=g_{\alpha\beta}e_{a}^{\ \alpha}(\partial_{\mu}e_{b}^{\
\beta}+\Gamma_{\mu\nu}^{\beta}e_{b}^{\ \nu}),$ (4)
$\Gamma_{\mu\nu}^{\beta}=\frac{1}{2}g^{\beta\sigma}(g_{\sigma\mu,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma}),$
(5)

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

$J_{V}^{\mu}=\overline{\psi}\underline{\gamma}^{\mu}\psi,\ \ \ \
J_{A}^{\mu}=\overline{\psi}\underline{\gamma}^{\mu}\gamma^{5}\psi,$ (6)
$T^{\mu\nu}=\frac{1}{4}\left(\overline{\psi}i\underline{\gamma}^{\mu}D^{\nu}\psi+\overline{\psi}i\underline{\gamma}^{\nu}D^{\mu}\psi+\mathrm{H.C.}\right),$
(7)

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,

$\left\\{\begin{array}[]{ccc}T&=&t\\\ X&=&x\cos\Omega t-y\sin\Omega t\\\
Y&=&x\sin\Omega t+y\cos\Omega t\\\ Z&=&z\end{array}\right..$ (8)

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

$g_{\mu\nu}=\left(\begin{array}[]{cccc}1-(x^{2}+y^{2})\Omega^{2}&y\Omega&-x\Omega&0\\\
y\Omega&-1&0&0\\\ -x\Omega&0&-1&0\\\ 0&0&0&-1\end{array}\right),$ (9)
$g^{\mu\nu}=\left(\begin{array}[]{cccc}1&y\Omega&-x\Omega&0\\\
y\Omega&y^{2}\Omega^{2}-1&-xy\Omega^{2}&0\\\
-x\Omega&-xy\Omega^{2}&x^{2}\Omega^{2}-1&0\\\ 0&0&0&-1\end{array}\right).$
(10)

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

$e_{\ 0}^{0}=e_{\ 1}^{1}=e_{\ 2}^{2}=e_{\ 3}^{3}=1,\ \ \ \ e_{\
0}^{1}=-y\Omega,\ \ \ \ e_{\ 0}^{2}=x\Omega,$ (11)

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

$D_{\mu}=\left(\partial_{t}-\frac{i}{2}\Omega\Sigma_{3}+\Omega(yA_{x}-xA_{y}),\partial_{x}-ieA_{x},\partial_{y}-ieA_{y},\partial_{z}-ieA_{z}\right),$
(12)
$\underline{\gamma}^{\mu}=\left(\gamma^{0},y\Omega\gamma^{0}+\gamma^{1},-x\Omega\gamma^{0}+\gamma^{2},\gamma^{3}\right),$
(13)

and in this case the Dirac equation for a single chiral fermion can be written
as

$i\frac{\partial}{\partial
t}\psi(x)=\left[-i\gamma^{0}\boldsymbol{\gamma}\cdot(\nabla-
ie\mathbf{A})-\Omega J_{z}\right]\psi(x),$ (14)

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 $\boldsymbol{\Omega}$.

##  III Landau levels for a single right-handed fermion in a rotating frame

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

$i\frac{\partial}{\partial t}\psi_{R}(x)=H\psi_{R}(x),$ (15)
$H=-i\boldsymbol{\sigma}\cdot(\nabla-ie\mathbf{A})-\Omega J_{R,z},$ (16)

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

$H=-i\boldsymbol{\sigma}\cdot\nabla+\frac{1}{2}eB\left(y\sigma_{1}-x\sigma_{2}\right)-\Omega
J_{R,z}.$ (17)

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

$\psi_{\lambda
nmp_{z}}=\sqrt{\frac{n!}{(n+m-\frac{1}{2})!}}\left(\begin{array}[]{c}\sqrt{\frac{eB(E+p_{z}+m\Omega)}{2(E+m\Omega)}}e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}-\frac{1}{4}}L_{n}^{m-\frac{1}{2}}(\rho)e^{i(m-\frac{1}{2})\phi}\\\
\frac{i\lambda
eB}{\sqrt{(E+m\Omega)(E+p_{z}+m\Omega)}}e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}+\frac{1}{4}}L_{n-1}^{m+\frac{1}{2}}(\rho)e^{i(m+\frac{1}{2})\phi}\end{array}\right)\frac{e^{-iEt+izp_{z}}}{2\pi},$
(18)
$E=\left\\{\begin{array}[]{cc}\lambda\sqrt{p_{z}^{2}+2eBn}-m\Omega,&n>0\\\
p_{z}-m\Omega,&n=0\end{array}\right..$ (19)

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

$\psi_{\lambda
nmp_{z}}=\sqrt{\frac{n!}{(n-m+\frac{1}{2})!}}\left(\begin{array}[]{c}\sqrt{\frac{eB(E+p_{z}+m\Omega)}{2(E+m\Omega)}}e^{-\frac{\rho}{2}}\rho^{\frac{1}{4}-\frac{m}{2}}L_{n}^{\frac{1}{2}-m}(\rho)e^{i(m-\frac{1}{2})\phi}\\\
-\frac{i\lambda
eB(n-m+\frac{1}{2})}{\sqrt{(E+m\Omega)(E+p_{z}+m\Omega)}}e^{-\frac{\rho}{2}}\rho^{-\frac{1}{4}-\frac{m}{2}}L_{n}^{-\frac{1}{2}-m}(\rho)e^{i(m+\frac{1}{2})\phi}\end{array}\right)\frac{e^{-iEt+izp_{z}}}{2\pi},$
(20) $E=\lambda\sqrt{p_{z}^{2}+2eB\left(n-m+\frac{1}{2}\right)}-m\Omega,$ (21)

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

$\int
dV\psi_{\lambda^{\prime}n^{\prime}m^{\prime}p_{z}^{\prime}}^{\dagger}\psi_{\lambda
nmp_{z}}=\delta_{\lambda^{\prime}\lambda}\delta_{n^{\prime}n}\delta_{m^{\prime}m}\delta(p_{z}^{\prime}-p_{z}).$
(22)

##  IV Particle current

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 $\mu$.

###  IV.1 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 $\rho=0$ are
simplified to

$\psi_{\lambda
nmp_{z}}=\frac{e^{-iEt+izp_{z}}}{2\pi}\left(\begin{array}[]{c}\sqrt{\frac{eB(E+p_{z}+\Omega/2)}{2(E+\Omega/2)}}\delta_{m,1/2}\\\
0\end{array}\right)+\frac{e^{-iEt+izp_{z}}}{2\pi}\left(\begin{array}[]{c}0\\\
-\frac{i\lambda
eB\sqrt{n+1}}{\sqrt{(E-\Omega/2)(E+p_{z}-\Omega/2)}}\delta_{m,-1/2}\end{array}\right),$
(23)

which are to be used in the following calculations of ensemble average. We
find that the $z$-component $m$ of the total angular momentum can only take
values $\pm 1/2$ due to the absence of the orbital angular momentum at $r=0$.

For the right-handed fermion system, the field operator of the particle
current at $r=0$ is

$J^{\mu}=\psi_{R}^{\dagger}\sigma^{\mu}\psi_{R},$ (24)

with $\sigma^{\mu}=(1,\boldsymbol{\sigma})$. From the approach of ensemble
average used in (Vilenkin, 1978, 1979, 1980), the macroscopic particle current
$\mathcal{J}^{\mu}$ can be calculated from $J^{\mu}$ as follows,

$\displaystyle\mathcal{J}^{\mu}$ $\displaystyle=$
$\displaystyle\langle:J^{\mu}:\rangle$ (25) $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\sum_{\lambda}\int_{-\infty}^{\infty}dp_{z}\frac{\lambda}{e^{\beta\left(E_{n}-\lambda\Omega/2-\lambda\mu\right)}+1}\psi_{\lambda,n,1/2,p_{z}}^{\dagger}\sigma^{\mu}\psi_{\lambda,n,1/2,p_{z}}$
$\displaystyle+\sum_{n=0}^{\infty}\sum_{\lambda}\int_{-\infty}^{\infty}dp_{z}\frac{\lambda}{e^{\beta\left(E_{n+1}+\lambda\Omega/2-\lambda\mu\right)}+1}\psi_{\lambda,n,-1/2,p_{z}}^{\dagger}\sigma^{\mu}\psi_{\lambda,n,-1/2,p_{z}}$
$\displaystyle+\sum_{\lambda}\int_{-\infty}^{\infty}dp_{z}\frac{\lambda\theta(\lambda
p_{z})}{e^{\beta\left(|p_{z}|-\lambda\Omega/2-\lambda\mu\right)}+1}\psi_{\lambda,0,1/2,p_{z}}^{\dagger}\sigma^{\mu}\psi_{\lambda,0,1/2,p_{z}},$

where $\langle:\cdots:\rangle$ means normal ordering and ensemble average of
corresponding field operator (Vasak et al., 1987; Dong et al., 2020),
$\theta(x)$ is the step function, and we have defined
$E_{n}=\sqrt{p_{z}^{2}+2eBn}$. The second, third, and fourth lines of Eq. (25)
represent the contributions of high Landau levels with $m=1/2$, all Landau
levels with $m=-1/2$, and the lowest Landau level with $m=1/2$, respectively.
We can see that the macroscopic particle current $\mathcal{J}^{\mu}$ 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 ($\lambda,n,m,p_{z}$).

###  IV.2 Particle number density

Firstly we calculate the particle number density $\rho\equiv\mathcal{J}^{0}$
of the system. Making use of

$\psi_{\lambda nmp_{z}}^{\dagger}\psi_{\lambda
nmp_{z}}=\left\\{\begin{array}[]{cc}\frac{eB}{4\pi^{2}}\frac{E+\Omega/2+p_{z}}{E+\Omega/2},&m=\frac{1}{2}\\\
\frac{eB}{4\pi^{2}}\frac{E-\Omega/2-p_{z}}{E-\Omega/2},&m=-\frac{1}{2}\end{array}\right.$
(26)

and from Eq. (25) one can obtain

$\rho\beta^{3}=\frac{b\omega}{16\pi^{2}}+\frac{1}{2}\sum_{s=\pm
1}\frac{\partial}{\partial a}g\left(a+\frac{1}{2}s\omega,b\right),$ (27)

where we have defined three dimensionless quantities $a=\beta\mu$,
$b=2eB\beta^{2}$, $\omega=\beta\Omega$, and have defined $g(x,b)$ as

$g(x,b)=\frac{b}{4\pi^{2}}\int_{0}^{\infty}dy\sum_{n=0}^{\infty}\sum_{s=\pm
1}\left(1-\frac{1}{2}\delta_{n,0}\right)\ln\left(1+e^{sx-\sqrt{nb+y^{2}}}\right).$
(28)

In a recent article (Zhang et al., 2020a), making use of Abel-Plana formula,
the authors obtained the asymptotic expansion of $g(x,b)$ at $b=0$ as follows

$\displaystyle g(x,b)$ $\displaystyle=$
$\displaystyle\left(\frac{7\pi^{2}}{360}+\frac{x^{2}}{12}+\frac{x^{4}}{24\pi^{2}}\right)-\frac{b^{2}\ln
b^{2}}{384\pi^{2}}-\frac{b^{2}}{96\pi^{2}}\ln\left(\frac{e}{2G^{6}}\right)$
(29)
$\displaystyle-\frac{1}{2\pi^{2}}\sum_{n=0}^{\infty}\frac{(4n+1)!!}{(4n+4)!!}\mathscr{B}_{2n+2}C_{2n+1}(x)b^{2n+2},$

where $G=1.28242...$ is the Glaisher number, $\mathscr{B}_{n}$ are Bernoulli
numbers, and $C_{2n+1}(x)$ is defined and expanded at $x=0$ in the following,

$\displaystyle C_{2n+1}(x)$ $\displaystyle=$
$\displaystyle-\delta_{n,0}+\frac{1}{(4n+1)!}\int_{0}^{\infty}dy\ln
y\frac{d^{4n+1}}{dy^{4n+1}}\left(\frac{1}{e^{y+x}+1}+\frac{1}{e^{y-x}+1}\right)$
(30) $\displaystyle=$ $\displaystyle(\ln
4+\gamma-1)\delta_{n,0}+\frac{2}{(4n+1)!}\sum_{k=0}^{\infty}\left(2^{4n+2k+1}-1\right)\zeta^{\prime}(-4n-2k)\frac{x^{2k}}{(2k)!}.$

Plugging Eqs. (29, 30) into Eq. (27), one can get the threefold series
expansion of the particle number density at $a=0$, $b=0$, $\omega=0$ or
$\mu=0$, $B=0$, $\Omega=0$ as follows,

$\displaystyle\rho\beta^{3}$ $\displaystyle=$
$\displaystyle\frac{a}{6}+\frac{a^{3}}{6\pi^{2}}+\frac{a\omega^{2}}{8\pi^{2}}+\frac{b\omega}{16\pi^{2}}$
(31)
$\displaystyle-\frac{1}{\pi^{2}}\sum_{n=0}^{\infty}\frac{\mathscr{B}_{2n+2}b^{2n+2}}{(4n+4)!!(4n)!!}\sum_{j=0}^{\infty}\frac{\omega^{2j}}{(2j)!2^{2j}}$
$\displaystyle\times\sum_{k=0}^{\infty}\left(2^{4n+2k+2j+3}-1\right)\zeta^{\prime}(-4n-2k-2j-2)\frac{a^{2k+1}}{(2k+1)!}.$

The lower orders $O(b^{2},\omega^{2},b\omega)$ in Eq. (31) are consistent with
the perturbative results in (Buzzegoli et al., 2017; Buzzegoli and Becattini,
2018; Yang et al., 2020), where the authors used the approaches of thermal
field theory and Wigner function respectively.

###  IV.3 Particle current along $z$-axis

Next we calculate the space components of the particle current
$\mathcal{J}^{\mu}$. According to the rotation symmetry along $z$-axis of the
system, the $x$\- and $y$-components of $\mathcal{J}^{\mu}$ vanish. The unique
nonzero component is $\mathcal{J}^{z}$. Making use of

$\psi_{\lambda nmp_{z}}^{\dagger}\sigma_{3}\psi_{\lambda
nmp_{z}}=\left\\{\begin{array}[]{cc}\frac{eB}{4\pi^{2}}\frac{E+\Omega/2+p_{z}}{E+\Omega/2},&m=\frac{1}{2}\\\
-\frac{eB}{4\pi^{2}}\frac{E-\Omega/2-p_{z}}{E-\Omega/2},&m=-\frac{1}{2}\end{array}\right.$
(32)

and from Eq. (25) one can obtain

$\mathcal{J}^{z}\beta^{3}=\frac{ab}{8\pi^{2}}+\frac{1}{2}\sum_{s=\pm
1}s\frac{\partial}{\partial a}g\left(a+\frac{1}{2}s\omega,b\right),$ (33)

which can be expanded as the threefold series at $a=0$, $b=0$, $\omega=0$ or
$\mu=0$, $B=0$, $\Omega=0$ as follows,

$\displaystyle\mathcal{J}^{z}\beta^{3}$ $\displaystyle=$
$\displaystyle\frac{ab}{8\pi^{2}}+\frac{\omega}{12}+\frac{\omega^{3}}{48\pi^{2}}+\frac{\omega
a^{2}}{4\pi^{2}}$ (34)
$\displaystyle-\frac{1}{\pi^{2}}\sum_{n=0}^{\infty}\frac{\mathscr{B}_{2n+2}b^{2n+2}}{(4n+4)!!(4n)!!}\sum_{j=0}^{\infty}\frac{\omega^{2j+1}}{(2j+1)!2^{2j+1}}$
$\displaystyle\times\sum_{k=0}^{\infty}\left(2^{4n+2k+2j+3}-1\right)\zeta^{\prime}(-4n-2k-2j-2)\frac{a^{2k}}{(2k)!}.$

When $\omega=0$ or $\Omega=0$ in Eq. (34), one can obtain
$\mathcal{J}^{z}\beta^{3}=\frac{ab}{8\pi^{2}}$, which is the chiral magnetic
effect (Kharzeev et al., 2008; Fukushima et al., 2008; Son and Surowka, 2009;
Kharzeev and Son, 2011; Son and Yamamoto, 2012); When $b=0$ or $B=0$ and
keeping the leading order of $\omega$ in Eq. (34), one can obtain
$\mathcal{J}^{z}\beta^{3}=\frac{\omega}{12}\left(1+\frac{3a^{2}}{\pi^{2}}\right)$,
which is the chiral vortical effect (Landsteiner et al., 2011; Golkar and Son,
2015; Hou et al., 2012; Lin and Yang, 2018; Gao et al., 2019; Shitade et al.,
2020).

##  V Energy-momentum tensor

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 $\mathcal{T}^{\mu\nu}$:

$\mathcal{T}^{01}=\mathcal{T}^{02}=\mathcal{T}^{12}=\mathcal{T}^{13}=\mathcal{T}^{23}=0,\
\ \ \mathcal{T}^{11}=\mathcal{T}^{22}.$ (35)

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

$T^{\mu\nu}=\frac{1}{4}\left(\psi_{R}^{\dagger}i\sigma^{\mu}D_{R}^{\nu}\psi_{R}+\psi_{R}^{\dagger}i\sigma^{\nu}D_{R}^{\mu}\psi_{R}+\mathrm{H.C.}\right),$
(36)

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

$D_{R}^{\mu}=\left(\partial_{t}-\frac{i}{2}\Omega\sigma_{3},-\partial_{x},-\partial_{y},-\partial_{z}\right).$
(37)

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

$\displaystyle\mathcal{T}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\langle:T^{\mu\nu}:\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{n=1}^{\infty}\sum_{\lambda}\int_{-\infty}^{\infty}dp_{z}\frac{\lambda}{e^{\beta\left(E_{n}-\lambda\Omega/2-\lambda\mu\right)}+1}\psi_{\lambda,n,1/2,p_{z}}^{\dagger}\left(i\sigma^{\mu}D_{R}^{\nu}+i\sigma^{\nu}D_{R}^{\mu}\right)\psi_{\lambda,n,1/2,p_{z}}$
$\displaystyle+\frac{1}{4}\sum_{n=0}^{\infty}\sum_{\lambda}\int_{-\infty}^{\infty}dp_{z}\frac{\lambda}{e^{\beta\left(E_{n+1}+\lambda\Omega/2-\lambda\mu\right)}+1}\psi_{\lambda,n,-1/2,p_{z}}^{\dagger}\left(i\sigma^{\mu}D_{R}^{\nu}+i\sigma^{\nu}D_{R}^{\mu}\right)\psi_{\lambda,n,-1/2,p_{z}}$
$\displaystyle+\frac{1}{4}\sum_{\lambda}\int_{-\infty}^{\infty}dp_{z}\frac{\lambda\theta(\lambda
p_{z})}{e^{\beta\left(|p_{z}|-\lambda\Omega/2-\lambda\mu\right)}+1}\psi_{\lambda,0,1/2,p_{z}}^{\dagger}\left(i\sigma^{\mu}D_{R}^{\nu}+i\sigma^{\nu}D_{R}^{\mu}\right)\psi_{\lambda,0,1/2,p_{z}}+\mathrm{H.C.}$

###  V.1 Energy density

Firstly we calculate the energy density $\varepsilon\equiv\mathcal{T}^{00}$ of
the system. Making use of

$\psi_{\lambda
nmp_{z}}^{\dagger}\left(i\partial_{t}+\frac{1}{2}\Omega\sigma_{3}\right)\psi_{\lambda
nmp_{z}}=\left\\{\begin{array}[]{cc}\frac{eB}{8\pi^{2}}(E+p_{z}+\Omega/2),&m=\frac{1}{2}\\\
\frac{eB}{8\pi^{2}}(E-p_{z}-\Omega/2),&m=-\frac{1}{2}\end{array}\right.$ (39)

and from Eq. (25) one can obtain

$\varepsilon\beta^{4}=\frac{ab\omega}{16\pi^{2}}+\sum_{s=\pm
1}\left(\frac{3}{2}-b\frac{\partial}{\partial
b}\right)g\left(a+\frac{1}{2}s\omega,b\right),$ (40)

which can be expanded as the threefold series at $a=0$, $b=0$, $\omega=0$ or
$\mu=0$, $B=0$, $\Omega=0$ as follows,

$\displaystyle\varepsilon\beta^{4}$ $\displaystyle=$
$\displaystyle\frac{7\pi^{2}}{120}+\frac{a^{2}}{4}+\frac{\omega^{2}}{16}+\frac{a^{4}}{8\pi^{2}}+\frac{3a^{2}\omega^{2}}{16\pi^{2}}+\frac{\omega^{4}}{128\pi^{2}}$
(41) $\displaystyle+\frac{ab\omega}{16\pi^{2}}+\frac{b^{2}\ln
b^{2}}{384\pi^{2}}+\frac{b^{2}}{96\pi^{2}}\ln\left(\frac{2e^{\gamma+1}}{G^{6}}\right)$
$\displaystyle+\frac{1}{\pi^{2}}\sum_{n=0}^{\infty}\frac{(4n+1)\mathscr{B}_{2n+2}b^{2n+2}}{(4n+4)!!(4n)!!}\sum_{j=0}^{\infty}\frac{\omega^{2j}}{(2j)!2^{2j}}$
$\displaystyle\times\sum_{k=0}^{\infty}\left(2^{4n+2k+2j+1}-1\right)\zeta^{\prime}(-4n-2k-2j)\frac{a^{2k}}{(2k)!},$

where the logarithmic term $b^{2}\ln b^{2}$ has been discussed in detail in
(Zhang et al., 2020a), and its coefficient is independent of $\omega$ 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 (Sheng et al.,
2018; Yang et al., 2020).

###  V.2 Pressure

The pressure $P$ of the system is $\mathcal{T}^{33}$. Making use of

$\psi_{\lambda nmp_{z}}^{\dagger}\sigma_{3}(-i\partial_{z})\psi_{\lambda
nmp_{z}}=\left\\{\begin{array}[]{cc}\frac{eB}{8\pi^{2}}\frac{(E+p_{z}+\Omega/2)p_{z}}{E+\Omega/2},&m=\frac{1}{2}\\\
-\frac{eB}{8\pi^{2}}\frac{(E-p_{z}-\Omega/2)p_{z}}{E-\Omega/2},&m=-\frac{1}{2}\end{array}\right.$
(42)

and from Eq. (25) one can obtain

$P\beta^{4}=\frac{ab\omega}{16\pi^{2}}+\frac{1}{2}\sum_{s=\pm
1}g\left(a+\frac{1}{2}s\omega,b\right),$ (43)

which can be expanded as the threefold series at $a=0$, $b=0$, $\omega=0$ or
$\mu=0$, $B=0$, $\Omega=0$ as follows,

$\displaystyle P\beta^{4}$ $\displaystyle=$
$\displaystyle\frac{7\pi^{2}}{360}+\frac{a^{2}}{12}+\frac{\omega^{2}}{48}+\frac{a^{4}}{24\pi^{2}}+\frac{a^{2}\omega^{2}}{16\pi^{2}}+\frac{\omega^{4}}{384\pi^{2}}$
(44) $\displaystyle+\frac{ab\omega}{16\pi^{2}}-\frac{b^{2}\ln
b^{2}}{384\pi^{2}}-\frac{b^{2}}{96\pi^{2}}\ln\left(\frac{2e^{\gamma}}{G^{6}}\right)$
$\displaystyle-\frac{1}{\pi^{2}}\sum_{n=0}^{\infty}\frac{\mathscr{B}_{2n+2}b^{2n+2}}{(4n+4)!!(4n)!!}\sum_{j=0}^{\infty}\frac{\omega^{2j}}{(2j)!2^{2j}}$
$\displaystyle\times\sum_{k=0}^{\infty}\left(2^{4n+2k+2j+1}-1\right)\zeta^{\prime}(-4n-2k-2j)\frac{a^{2k}}{(2k)!}.$

One can obtain $\mathcal{T}^{11}$ from the traceless condition for energy-
momentum tensor, $\mathcal{T}^{00}=2\mathcal{T}^{11}+\mathcal{T}^{33}$.

###  V.3 Energy current

The energy current along $z$-axis is $\mathcal{T}^{03}$. Making use of

$\psi_{\lambda
nmp_{z}}^{\dagger}\left(-i\partial_{z}+\sigma_{3}i\partial_{t}+\frac{1}{2}\Omega\right)\psi_{\lambda
nmp_{z}}=\left\\{\begin{array}[]{cc}\frac{eB}{8\pi^{2}}\frac{(E+p_{z}+\Omega/2)^{2}}{E+\Omega/2},&m=\frac{1}{2}\\\
-\frac{eB}{8\pi^{2}}\frac{(E-p_{z}-\Omega/2)^{2}}{E-\Omega/2},&m=-\frac{1}{2}\end{array}\right.$
(45)

and from Eq. (25) one can obtain

$\mathcal{T}^{03}\beta^{4}=\frac{b}{8\pi^{2}}\left(\frac{\pi^{2}}{6}+\frac{\omega^{2}}{8}+\frac{a^{2}}{2}\right)+\sum_{s=\pm
1}s\left(1-\frac{b}{2}\frac{\partial}{\partial
b}\right)g\left(a+\frac{1}{2}s\omega,b\right),$ (46)

which can be expanded as the threefold series at $a=0$, $b=0$, $\omega=0$ or
$\mu=0$, $B=0$, $\Omega=0$ as follows,

$\displaystyle\mathcal{T}^{03}\beta^{4}$ $\displaystyle=$
$\displaystyle\frac{b}{8\pi^{2}}\left(\frac{\pi^{2}}{6}+\frac{\omega^{2}}{8}+\frac{a^{2}}{2}\right)+\left(\frac{a\omega}{6}+\frac{a^{3}\omega}{6\pi^{2}}+\frac{a\omega^{3}}{24\pi^{2}}\right)$
(47)
$\displaystyle+\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\frac{n\mathscr{B}_{2n+2}b^{2n+2}}{(4n+4)!!(4n)!!}\sum_{j=0}^{\infty}\frac{\omega^{2j+1}}{(2j+1)!2^{2j+1}}$
$\displaystyle\times\sum_{k=0}^{\infty}\left(2^{4n+2k+2j+3}-1\right)\zeta^{\prime}(-4n-2k-2j-2)\frac{a^{2k+1}}{(2k+1)!}.$

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:
$\rho_{R}\rightarrow\rho_{L}$,
$\mathcal{J}_{R}^{z}\rightarrow-\mathcal{J}_{L}^{z}$,
$\varepsilon_{R}\rightarrow\varepsilon_{L}$, $P_{R}\rightarrow P_{L}$,
$\mathcal{T}_{R}^{03}\rightarrow-\mathcal{T}_{L}^{03}$,
$\mu_{R}\rightarrow\mu_{L}$, $B\rightarrow B$, $\Omega\rightarrow\Omega$,
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 $\mu$, magnetic
field $B$, and angular velocity $\Omega$ fixed, then the three dimensionless
quantities $a=\beta\mu$, $b=2eB\beta^{2}$, $\omega=\beta\Omega$ all tend to be
infinity. The asymptotic behavior of $g(x,b)$ as $x\rightarrow\infty$ and
$b\rightarrow\infty$ has been obtained in (Zhang et al., 2020a),

$\lim_{x,b\rightarrow\infty}g(x,b)=\frac{x^{2}b}{16\pi^{2}}.$ (48)

From Eqs. (27, 33), one can derive the expressions of the particle density
$\rho$ and the current $\mathcal{J}^{z}$ at zero temperature limit as follows,

$\rho=\mathcal{J}^{z}=\frac{eB}{4\pi^{2}}\left(\mu+\frac{\Omega}{2}\right).$
(49)

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 $\rho$ equals
to the $z$-component current $\mathcal{J}^{z}$ at zero temperature limit.

From Eqs. (40, 43, 46), the expressions of energy density $\varepsilon$,
pressure $P$ and energy current $\mathcal{T}^{03}$ at zero temperature limit
are

$\varepsilon=P=\mathcal{T}^{03}=\frac{eB}{8\pi^{2}}\left(\mu+\frac{\Omega}{2}\right)^{2}.$
(50)

The movements of the particles and antiparticles with the speed of light along
$z$-axis leads to the equality of the energy density $\varepsilon$ and the
energy current $\mathcal{T}^{03}$. Since there is no energy current along the
direction of the $x$\- and $y$-axis, then $\mathcal{T}^{11}$ and
$\mathcal{T}^{22}$ vanish in this system, which results in the equality of the
energy density $\varepsilon$ and the pressure $P$.

##  VII Summary

In this article, we have investigated the thermodynamics of the uniformly
rotating right-handed fermion system under the background of a uniform
magnetic field through the approach of normal ordering and ensemble average,
where all thermodynamical quantities are expanded as threefold series at
$B=0$, $\Omega=0$ and $\mu=0$. For these threefold series, our results at
lower orders are consistent with previous ones by other authors. It is worth
pointing out that the general orders of $B$ and $\Omega$ in the expressions of
the thermodynamical quantities are obtained for the first time and can provide
a useful reference for the high order calculations from several different
approaches, such as thermal field theory and Wigner function. 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 ($r\neq 0$) as well as with the
boundary condition at the speed-of-light surface may be investigated in the
future.

##  VIII Acknowledgments

I thank De-Fu Hou for helpful discussion. This work was supported by the
National Natural Science Foundation of China under Grants No. 11890713, and
No. 12073008.

##  Appendix A Landau levels for a single right-handed fermion

The Hamiltonian for a right-handed fermion under the background of the uniform
magnetic field $\mathbf{B}=B\mathbf{e}_{z}$ is

$H=-i\boldsymbol{\sigma}\cdot(\nabla-
ie\mathbf{A})=-i\boldsymbol{\sigma}\cdot\nabla+\frac{1}{2}eB(y\sigma_{1}-x\sigma_{2}),$
(51)

where we have chosen $\mathbf{A}=(-\frac{1}{2}By,\frac{1}{2}Bx,0)$ for the
gauge potential. One can refer to (Sheng et al., 2018; Sheng, 2019; Dong et
al., 2020) for other choices of the gauge potential.

In the following, we will solve the eigenvalue equation of $H$ in cylindrical
coordinates,

$H\psi=E\psi.$ (52)

We can see that the three Hermitian operators, $H$,
$\hat{p}_{z}=-i\partial_{z}$,
$\hat{J}_{z}=\frac{1}{2}\sigma_{3}+(x\hat{p}_{y}-y\hat{p}_{x})$ are
commutative with each other, so the eigenfunction $\psi$ can be chosen as

$\psi=\left(\begin{array}[]{c}f(r)e^{i(m-\frac{1}{2})\phi}\\\
ig(r)e^{i(m+\frac{1}{2})\phi}\end{array}\right)e^{izp_{z}},$ (53)

where $-\infty<p_{z}<\infty$ and $m=\pm 1/2,\pm 3/2,\pm 5/2,...$ are the
eigenvalues of $\hat{p}_{z}$ and $\hat{J}_{z}$ respectively. The explicit form
of the Hamiltonian $H$ in cylindrical coordinates is

$H=\left(\begin{array}[]{cc}-i\frac{\partial}{\partial
z}&e^{-i\phi}\left(-i\frac{\partial}{\partial
r}-\frac{1}{r}\frac{\partial}{\partial\phi}+\frac{i}{2}eBr\right)\\\
e^{i\phi}\left(-i\frac{\partial}{\partial
r}+\frac{1}{r}\frac{\partial}{\partial\phi}-\frac{i}{2}eBr\right)&i\frac{\partial}{\partial
z}\end{array}\right),$ (54)

then from Eq. (52) we can obtain two differential equations for $f(r),g(r)$ as
follows,

$\displaystyle(p_{z}-E)f(r)+\left(\frac{\partial}{\partial
r}+\frac{m+\frac{1}{2}}{r}-\frac{1}{2}eBr\right)g(r)$ $\displaystyle=$
$\displaystyle 0,$ (55) $\displaystyle\left(-\frac{\partial}{\partial
r}+\frac{m-\frac{1}{2}}{r}-\frac{1}{2}eBr\right)f(r)+(-p_{z}-E)g(r)$
$\displaystyle=$ $\displaystyle 0,$ (56)

which are equivalent to

$\left\\{\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{\left(m-\frac{1}{2}\right)^{2}}{r^{2}}-\left[p_{z}^{2}-E^{2}-eB\left(m+\frac{1}{2}\right)\right]-\frac{1}{4}e^{2}B^{2}r^{2}\right\\}f(r)=0$
(57) $g(r)=\frac{1}{p_{z}+E}\left(-\frac{\partial}{\partial
r}+\frac{m-\frac{1}{2}}{r}-\frac{1}{2}eBr\right)f(r)$ (58)

We can define a dimensionless variable $\rho=\frac{1}{2}eBr^{2}$, then

$\frac{d}{dr}=eBr\frac{d}{d\rho},\ \ \ \
\frac{d^{2}}{dr^{2}}=eB\frac{d}{d\rho}+2eB\rho\frac{d^{2}}{d\rho^{2}}.$ (59)

Now Eq. (57) becomes

$\left\\{\rho\frac{d^{2}}{d\rho^{2}}+\frac{d}{d\rho}-\frac{\left(m-\frac{1}{2}\right)^{2}}{4\rho}-\frac{1}{2eB}\left[p_{z}^{2}-E^{2}-eB\left(m+\frac{1}{2}\right)\right]-\frac{1}{4}\rho\right\\}f=0$
(60)

Next, we choose

$f=e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}-\frac{1}{4}}G(\rho),$ (61)

then Eqs. (58, 60) become

$g=-\frac{\sqrt{2eB}}{E+p_{z}}e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}+\frac{1}{4}}G^{\prime}(\rho),$
(62) $\rho
G^{\prime\prime}+\left[\left(m+\frac{1}{2}\right)-\rho\right]G^{\prime}-\frac{1}{2eB}\left(p_{z}^{2}-E^{2}\right)G=0.$
(63)

Define following two quantities,

$\gamma=m+\frac{1}{2},\ \ \ \
\alpha=\frac{1}{2eB}\left(p_{z}^{2}-E^{2}\right),$ (64)

then Eq. (63) becomes

$\rho G^{\prime\prime}+(\gamma-\rho)G^{\prime}-\alpha G=0,$ (65)

which is the confluent hypergeometric equation (Zeng, 2007). With the boundary
conditions, $|f(0)|,|f(\infty)|<\infty$, the solutions for $G(\rho)$,
$f(\rho)$, $g(\rho)$ can be chosen as:

(1) When $\gamma=0,-1,-2,...$, i.e. $m=-1/2,-3/2,-5/2$, …, the boundary
condition $|f(0)|<\infty$ requires that

$G(\rho)=\rho^{1-\gamma}F(\alpha-\gamma+1,2-\gamma,\rho)=\rho^{\frac{1}{2}-m}F\left(\alpha-m+\frac{1}{2},\frac{3}{2}-m,\rho\right),$
(66)

where $F(\alpha,\gamma,\rho)$ is the confluent hypergeometric function as
discussed in Appendix B. In addition, the boundary condition
$|f(\infty)|<\infty$ requires that

$\alpha-m+\frac{1}{2}=-n\ \ (n\in\mathbb{N}),\ \ \
E=\lambda\sqrt{p_{z}^{2}+2eB\left(n-m+\frac{1}{2}\right)}\ \ \ (\lambda=\pm
1),$ (67)
$G(\rho)=\rho^{\frac{1}{2}-m}F\left(-n,\frac{3}{2}-m,\rho\right)\sim\rho^{\frac{1}{2}-m}L_{n}^{\frac{1}{2}-m}(\rho),$
(68)

where $L_{n}^{k}(\rho)$ is the general Laguerre polynomial as discussed in
Appendix B. Then one obtain

$f(\rho)\sim
e^{-\frac{\rho}{2}}\rho^{\frac{1}{4}-\frac{m}{2}}L_{n}^{\frac{1}{2}-m}(\rho),\
\ \ \
g(\rho)\sim-\frac{\sqrt{2eB}}{E+p_{z}}\left(n-m+\frac{1}{2}\right)e^{-\frac{\rho}{2}}\rho^{-\frac{1}{4}-\frac{m}{2}}L_{n}^{-\frac{1}{2}-m}(\rho).$
(69)

(2) When $\gamma=1,2,3,...$, i.e. $m=1/2,3/2,5/2,...$, the boundary condition
$|f(0)|<\infty$ requires that

$G(\rho)=F(\alpha,\gamma,\rho)=F\left(\alpha,\frac{1}{2}+m,\rho\right).$ (70)

In addition, the boundary condition $|f(\infty)|<\infty$ requires that

$\alpha=-n\ \ (n\in\mathbb{N}),\ \ \ E=\lambda\sqrt{p_{z}^{2}+2eBn}\ \ \
(\lambda=\pm 1),$ (71) $G(\rho)=F\left(-n,\frac{1}{2}+m,\rho\right)\sim
L_{n}^{m-\frac{1}{2}}(\rho).$ (72)

Then one obtain

$f(\rho)\sim
e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}-\frac{1}{4}}L_{n}^{m-\frac{1}{2}}(\rho),\
\ \ \
g(\rho)\sim\frac{\sqrt{2eB}}{E+p_{z}}e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}+\frac{1}{4}}L_{n-1}^{m+\frac{1}{2}}(\rho).$
(73)

There is a special case we must point out here: When $m>0$, $n=0$, we must
choose $E=p_{z}$, in which case we have
$f(\rho)=e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}-\frac{1}{4}}$, $g(\rho)=0$.
There is no physical solution for $m>0$, $n=0$, $E=-p_{z}$.

Making use of the orthonormal relation of the general Laguerre polynomials,

$\int_{0}^{\infty}dxe^{-x}x^{\gamma}L_{m}^{\gamma}(x)L_{n}^{\gamma}(x)=\frac{\Gamma(n+\gamma+1)}{n!}\delta_{mn},$
(74)

we can obtain the normalized eigenfunctions as follows:

When $m<0$,

$\psi_{\lambda
nmp_{z}}=\sqrt{\frac{n!}{(n-m+\frac{1}{2})!}}\left(\begin{array}[]{c}\sqrt{\frac{eB(E+p_{z})}{2E}}e^{-\frac{\rho}{2}}\rho^{\frac{1}{4}-\frac{m}{2}}L_{n}^{\frac{1}{2}-m}e^{i(m-\frac{1}{2})\phi}\\\
-\frac{i\lambda
eB(n-m+\frac{1}{2})}{\sqrt{E(E+p_{z})}}e^{-\frac{\rho}{2}}\rho^{-\frac{1}{4}-\frac{m}{2}}L_{n}^{-\frac{1}{2}-m}e^{i(m+\frac{1}{2})\phi}\end{array}\right)\frac{e^{izp_{z}}}{2\pi},$
(75) $E=\lambda\sqrt{p_{z}^{2}+2eB\left(n-m+\frac{1}{2}\right)}.$ (76)

When $m>0$,

$\psi_{\lambda
nmp_{z}}=\sqrt{\frac{n!}{(n+m-\frac{1}{2})!}}\left(\begin{array}[]{c}\sqrt{\frac{eB(E+p_{z})}{2E}}e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}-\frac{1}{4}}L_{n}^{m-\frac{1}{2}}e^{i(m-\frac{1}{2})\phi}\\\
\frac{i\lambda
eB}{\sqrt{E(E+p_{z})}}e^{-\frac{\rho}{2}}\rho^{\frac{m}{2}+\frac{1}{4}}L_{n-1}^{m+\frac{1}{2}}e^{i(m+\frac{1}{2})\phi}\end{array}\right)\frac{e^{izp_{z}}}{2\pi},$
(77) $E=\lambda\sqrt{p_{z}^{2}+2eBn}.$ (78)

All normalized eigenfunctions are orthogonal with each other,

$\int
dV\psi_{\lambda^{\prime}n^{\prime}m^{\prime}p_{z}^{\prime}}^{\dagger}\psi_{\lambda
nmp_{z}}=\delta_{\lambda^{\prime}\lambda}\delta_{n^{\prime}n}\delta_{m^{\prime}m}\delta(p_{z}^{\prime}-p_{z}).$
(79)

##  Appendix B Confluent hypergeometric function and Laguerre polynomial

The confluent hypergeometric equation is (Zeng, 2007)

$zy^{\prime\prime}+(\gamma-z)y^{\prime}-\alpha y=0.$ (80)

When $\gamma\notin\mathbb{Z}$, there are two independent solutions as follows,

$\displaystyle y_{1}$ $\displaystyle=$ $\displaystyle F(\alpha,\gamma,z),$
$\displaystyle y_{2}$ $\displaystyle=$ $\displaystyle
z^{1-\gamma}F(\alpha-\gamma+1,2-\gamma,z),$ (81)

where $F(\alpha,\gamma,z)$ is the confluent hypergeometric function defined as

$F(\alpha,\gamma,z)=\sum_{k=0}^{\infty}\frac{(\alpha)_{k}}{(\gamma)_{k}}\frac{z^{k}}{k!}\equiv
1+\frac{\alpha}{\gamma}z+\frac{\alpha(\alpha+1)}{\gamma(\gamma+1)}\frac{z^{2}}{2!}+\frac{\alpha(\alpha+1)(\alpha+2)}{\gamma(\gamma+1)(\gamma+2)}\frac{z^{3}}{3!}+\cdots.$
(82)

The asymptotic behavior of $F(\alpha,\gamma,z)$ as $z\rightarrow\infty$ is the
same as $e^{z}$. When $\alpha$ is a non-positive integer, then
$F(\alpha,\gamma,z)$ becomes a polynomial.

The general Laguerre polynomial $L_{n}^{\gamma}(z)$ is defined from
$F(\alpha,\gamma,z)$ as follows (Gradshteyn and Ryzhik, 2014),

$L_{n}^{\gamma}(z)=\frac{\Gamma(\gamma+n+1)}{n!\Gamma(\gamma+1)}F(-n,\gamma+1,z)=\left(\begin{array}[]{c}\gamma+n\\\
n\end{array}\right)F(-n,\gamma+1,z),$ (83)

where $\gamma\in\mathbb{R}$ and $n\in\mathbb{N}$. Laguerre polynomial
$L_{n}^{\gamma}(z)$ satisfies following differential equation

$zy^{\prime\prime}+(\gamma+1-z)y^{\prime}+ny=0$ (84)

We can rewrite Eq. (84) as a type of Sturm-Liouvelle equation,

$\frac{d}{dz}\bigg{(}z^{\gamma+1}e^{-z}\frac{dy}{dz}\bigg{)}+nz^{\gamma}e^{-z}y=0,$
(85)

which gives the orthogonality of $L_{n}^{\gamma}(z)$,

$\int_{0}^{\infty}dze^{-z}z^{\gamma}L_{m}^{\gamma}(z)L_{n}^{\gamma}(z)=\frac{\Gamma(n+\gamma+1)}{n!}\delta_{mn}.$
(86)

When $\gamma=0$, then $L_{n}^{\gamma}(z)$ becomes the normal Laguerre
polynomial $L_{n}(z)$,

$L_{n}(z)=L_{n}^{0}(z)=F(-n,1,z).$ (87)

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