# The full Lorentz-violating vacuum polarization tensor: low and high energy
limits

J. C. C. Felipe (a<EMAIL_ADDRESS>A. Yu. Petrov (b)
<EMAIL_ADDRESS>A. P. Baêta Scarpelli (c<EMAIL_ADDRESS>A. R.
Vieira (d<EMAIL_ADDRESS>(a) Instituto de Engenharia, Ciência e
Tecnologia, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Avenida
Um, 4050 - 39447-790 -Cidade Universitária - Janaúba - MG - Brasil  (b)
Departamento de Física, Universidade Federal da Paraíba, João Pessoa - PB -
Brasil  (c) Centro Federal de Educação Tecnológica - MG, Avenida Amazonas,
7675 - 30510-000 - Nova Gameleira - Belo Horizonte - MG - Brasil  (d)
Universidade Federal do Triângulo Mineiro, Campus Iturama, Iturama - MG -
Brasil

###### Abstract

We compute the full vacuum polarization tensor in the fermion sector of
Lorentz-violating QED. It turns out to be that even if we assume momentum
routing invariance of the Feynman diagrams, it is not possible to fix all
surface terms and find an ambiguity-free vacuum polarization tensor. The high
and low energy limits of this tensor are obtained explicitly. In the high
energy limit, only $c_{\mu\nu}$ coefficients contribute to the result. In the
low energy limit, we find that Lorentz-violating induced terms depend on
$b_{\mu}$, $c_{\mu\nu}$ and $g_{\mu\nu\lambda}$ coefficients and vanish at
$p=0$. At small $p$, we succeeded to obtain implications for condensed matter
systems, explicitly, for the Hall effect in Weyl semi-metals.

###### pacs:

11.10.Gh, 11.30.Cp,11.30.-j

##  I Introduction

Lorentz and CPT symmetries are known to be among the main criteria to
formulate field theory models. To test how good these symmetries are, the
Standard Model Extension (SME) [1] presents itself as the usual Standard Model
extended by adding all possible Lorentz and CPT-violating terms. One
possibility of the origin of these terms could be their emergence due to a
spontaneous symmetry breaking that occurs at the Planck Scale [2], explicit
symmetry breaking or noncommutative field theory [3]. On the other hand, even
if Lorentz and CPT symmetries are in fact exact at low energies, the question
would be on what precision one can say that they are indeed valid.

The tree level SME brings consequences to low energy physical models like
quantum mechanical systems, and it can be used as a framework to test Lorentz
and CPT symmetries in that limit. By the way, most of the searches on Lorentz
and CPT violation are based on non-relativistic Hamiltonians allowing us to
see how SME coefficients affect usual quantum mechanics. Some examples include
spectroscopy [4] and condensed matter systems [5] (for other studies of
Lorentz symmetry breaking within the condensed matter context see also f.e.
[6]).

Beyond tree level, the minimal SME is one-loop renormalizable, both in the
electroweak [7, 8] and the strong sectors [9, 10]. There is also a recent
investigation concerning Weyl semi-metals and terms induced by quantum
corrections [12]. However, results for finite quantum corrections coming from
loop diagrams are usually controversial. There was a long standing debate
concerning the issue of radiatively induced Carroll-Field-Jackiw (CFJ) terms
[13]-[23] suggesting that loop computations are in general regularization
dependent. The good old Dimensional Regularization [24, 25] can be used for
computing the divergent part of the diagrams, as it was used in the proof of
one-loop renormalizability. Unfortunately, it is not appropriate in some cases
due to the presence of some Lorentz and CPT violating terms which contain
objects which are well-defined only in specific dimensions, like Levi-Civita
symbols and $\gamma_{5}$ matrices. In this case, computing the finite part of
the amplitudes can be a delicate problem. The $\gamma_{5}$ issue can be
avoided in certain situations [26], and there are also some recipes to treat
it within traces involving Dirac matrices [27, 28]. Besides this, the question
what LV parameters contribute to perturbative corrections is interesting on
its own.

In this work, we compute the full Lorentz-violating vacuum polarization
tensor. We perform the computation of loop corrections within a four-
dimensional implicit regularization [29] framework, which does not assume any
explicit regulator. The regularization-depended objects are mapped in surface
terms. They manifest themselves as differences between integrals with the same
degree of divergence and so their value can yield any number including
infinity. They are also the ones which can cause the breaking of symmetries of
the model in a spurious way if explicitly computed. Therefore, we keep these
terms intact till the end of the calculation and then require the fulfillment
of a Ward-Takahashi or a Slavnov-Taylor identity. In this case, we guarantee
gauge symmetry beyond tree level and at the same time find conditions on
surface terms. As a consequence, not only the induced CFJ term, but also other
radiatively induced terms are arbitrary.

Another feature that can set values for surface terms is the momentum routing
invariance (MRI) of the loop diagrams. For gauge field theories, there is a
one-to-one diagrammatic relation between gauge and MRI which is regularization
independent. Some regularization schemes that are by construction momentum
routing invariant, like dimensional regularization, automatically fulfill
gauge invariance. These conditions could be considered as another trial for
finding equations that could fix arbitrary surface terms. However, requiring
MRI leads to the same relationships between the surface terms obtained when
requiring gauge invariance and therefore at least one surface term remains
making the result arbitrary. All these reasoning on MRI do not cause any
problem with the momentum routing diagrammatic computation of the chiral
anomaly. Choosing the internal routing in order to fulfill the desired gauge
Ward identity does not necessarily mean that momentum routing invariance was
broken. Requiring MRI fixes the relation between surfaces terms and
automatically fulfill gauge Ward-Takahashi identities and also reproduce the
breaking of the axial current [30], the Adler-Bell-Jackiw anomaly. Although
arbitrary and regularization dependent, the full Lorentz-violating vacuum
polarization tensor involves only pieces proportional to $b_{\mu}$,
$c_{\mu\nu}$ and $g_{\mu\nu\lambda}$ coefficients from the matter sector that
affect the photon sector in the renormalization process.

The structure of the paper looks as follows: in section II, we list the
relevant one-loop Feynman diagrams. In section III, we present the framework
of the regularization. In section IV, we calculate and present the quantum
corrections. In section V, we discuss applications of our results to condensed
matter. We present a summary in section VI and a list of the relevant
integrals in the Appendix.

##  II The framework and the one-loop diagrams

We consider the fermion sector of the standard minimal LV QED Lagrangian [31]:

$\displaystyle\mathcal{L}=\frac{1}{2}i\bar{\psi}\Gamma^{\mu}\overleftrightarrow{D}_{\mu}\psi-\bar{\psi}M\psi-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{4}(k_{F})_{\kappa\lambda\mu\nu}F^{\mu\nu}F^{\kappa\lambda}+\frac{1}{2}(k_{AF})^{\kappa}\epsilon_{\kappa\lambda\mu\nu}A^{\lambda}F^{\mu\nu},$
(1)

where $D_{\mu}\equiv\partial_{\mu}+iqA_{\mu}$ is the usual covariant
derivative which couples the gauge field with matter,

$\displaystyle\Gamma^{\nu}=\gamma^{\nu}+\Gamma^{\nu}_{1},$
$\displaystyle\Gamma^{\nu}_{1}=c^{\mu\nu}\gamma_{\mu}+d^{\mu\nu}\gamma_{5}\gamma_{\mu}+e^{\nu}+if^{\nu}\gamma_{5}+\frac{1}{2}g^{\lambda\mu\nu}\sigma_{\lambda\mu}$
(2)

and

$\displaystyle M=m+M_{1},$ $\displaystyle
M_{1}=im_{5}\gamma_{5}+a^{\mu}\gamma_{\mu}+b_{\mu}\gamma_{5}\gamma^{\mu}+\frac{1}{2}H_{\mu\nu}\sigma^{\mu\nu}.$
(3)

The coefficients $a_{\mu}$, $b_{\mu}$, $c_{\mu\nu}$, $d_{\mu\nu}$, $e_{\mu}$,
$f_{\mu}$, $g_{\lambda\mu\nu}$, $H_{\mu\nu}$, $(k_{F})_{\kappa\lambda\mu\nu}$
and $(k_{AF})_{\kappa}$ break Lorentz symmetry and only the coefficients
$a_{\mu}$, $b_{\mu}$, $e_{\mu}$, $f_{\mu}$, $g_{\lambda\mu\nu}$ and
$(k_{AF})_{\kappa}$ break CPT symmetry since their number of indices are odd.
There is a particular subtleness in the $f_{\mu}$ coefficient because, unlike
$a_{\mu}$ and $e_{\mu}$ ones, its time component is odd under all spacial
reflections [32].

Figure 1: Feynman rules of the Lorentz-violating QED.

The Feynman rules corresponding to the Lagrangian in eq. (1) are listed in
Fig. 1. We see that this Lagrangian involves a general vertex with
$\Gamma_{\mu}$ instead of just $\gamma_{\mu}$ and a general fermion propagator
$\frac{i}{p_{\mu}\Gamma^{\mu}-M}$. However, considering this complete
propagator for a generic $\Gamma^{\mu}$ is a difficult task. Therefore, the
$\bullet$ and the $\times$ insertions in Fig. 1 denote the leading order in
Lorentz and CPT violation in the fermion propagator.

Figure 2: One-loop 2-point functions of the Lorentz-violating QED.

The one-loop diagrams are depicted in Fig. 2, and their amplitudes are:

$\Pi^{\mu\nu}_{(a)}=-q^{2}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left[\Gamma_{1}^{\nu}\frac{1}{\not{k}-m}\gamma^{\mu}\frac{1}{\not{k}-\not{p}-m}\right],$
(4)

$\Pi^{\mu\nu}_{(b)}=-q^{2}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left[\gamma^{\nu}\frac{1}{\not{k}-m}\Gamma_{1}^{\mu}\frac{1}{\not{k}-\not{p}-m}\right],$
(5)

$\Pi^{\mu\nu}_{(c)}=q^{2}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left[\gamma^{\nu}\frac{1}{\not{k}-m}\Gamma_{1}^{\lambda}k_{\lambda}\frac{1}{\not{k}-m}\gamma^{\mu}\frac{1}{\not{k}-\not{p}-m}\right],$
(6)

$\Pi^{\mu\nu}_{(d)}=q^{2}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left[\gamma^{\nu}\frac{1}{\not{k}-m}\gamma^{\mu}\frac{1}{\not{k}-\not{p}-m}\Gamma_{1}^{\lambda}(k_{\lambda}-p_{\lambda})\frac{1}{\not{k}-\not{p}-m}\right],$
(7)

$\Pi^{\mu\nu}_{(e)}=-q^{2}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left[\gamma^{\nu}\frac{1}{\not{k}-m}M_{1}\frac{1}{\not{k}-m}\gamma^{\mu}\frac{1}{\not{k}-\not{p}-m}\right],$
(8)

$\Pi^{\mu\nu}_{(f)}=-q^{2}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left[\gamma^{\nu}\frac{1}{\not{k}-m}\gamma^{\mu}\frac{1}{\not{k}-\not{p}-m}M_{1}\frac{1}{\not{k}-\not{p}-m}\right].$
(9)

Choosing of the regularization scheme to be applied is a subtle task because
some LV terms involve objects defined only in the four-dimensional space-time,
namely, the Levi-Civita symbol and $\gamma_{5}$ matrices. Thus, the inadequate
choice of a regulator may cause spurious terms in these amplitudes and affect
the conclusions below. On the other hand, assuming that an implicit regulator
exists allows us to manipulate the integrand and also to stay in $4$
dimensions and not to be concerned about spurious breaking terms in the
process of renormalization.

##  III Basic Divergent Integrals and Surface Terms

Here we briefly describe the framework [29] (for a recent review see [33])
allowing us to handle the divergent integrals in four dimensions which appear
in the amplitudes of the previous section and establish some notation. In this
scheme, we assume that integrals are regularized by an implicit regulator
$\Lambda$ just to justify algebraic operations within the integrands. We then
use, for instance, the following identity

$\int_{k}\frac{1}{(k+p)^{2}-m^{2}}=\int_{k}\frac{1}{k^{2}-m^{2}}-\int_{k}\frac{(p^{2}+2p\cdot
k)}{(k^{2}-m^{2})[(k+p)^{2}-m^{2}]},$ (10)

where $\int_{k}\equiv\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}$, in order to
separate basic divergent integrals from the finite part. The former ones are
defined as follows:

$I^{\mu_{1}\cdots\mu_{2n}}_{log}(m^{2})\equiv\int_{k}\frac{k^{\mu_{1}}\cdots
k^{\mu_{2n}}}{(k^{2}-m^{2})^{2+n}}$ (11)

and

$I^{\mu_{1}\cdots\mu_{2n}}_{quad}(m^{2})\equiv\int_{k}\frac{k^{\mu_{1}}\cdots
k^{\mu_{2n}}}{(k^{2}-m^{2})^{1+n}}.$ (12)

The basic divergences with Lorentz indices can be represented as differences
between integrals with the same superficial degree of divergence, according to
the equations below, which define the surface terms 111The Lorentz indices
between brackets stand for permutations, i.e.,
$A^{\\{\alpha_{1}\cdots\alpha_{n}}B^{\beta_{1}\cdots\beta_{n}\\}}=A^{\alpha_{1}\cdots\alpha_{n}}B^{\beta_{1}\cdots\beta_{n}}$
\+ sum over permutations between the two sets of indices
$\alpha_{1}\cdots\alpha_{n}$ and $\beta_{1}\cdots\beta_{n}$.:

$\displaystyle\Upsilon^{\mu\nu}_{2w}=g^{\mu\nu}I_{2w}(m^{2})-2(2-w)I^{\mu\nu}_{2w}(m^{2})=\upsilon_{2w}g^{\mu\nu},$
(13)
$\displaystyle\Xi^{\mu\nu\alpha\beta}_{2w}=g^{\\{\mu\nu}g^{\alpha\beta\\}}I_{2w}(m^{2})-4(3-w)(2-w)I^{\mu\nu\alpha\beta}_{2w}(m^{2})=\xi_{2w}g^{\\{{\mu\nu}}g^{{\alpha\beta}\\}},$
(14)
$\displaystyle\Sigma^{\mu\nu\alpha\beta\gamma\delta}_{2w}=g^{\\{\mu\nu}g^{\alpha\beta}g^{\gamma\delta\\}}I_{2w}(m^{2})-8(4-w)(3-w)(2-w)I^{\mu\nu\alpha\beta\gamma\delta}_{2w}(m^{2})=\sigma_{2w}g^{\\{\mu\nu}g^{\alpha\beta}g^{\gamma\delta\\}}.$
(15)

In the expressions above, $2w$ is the degree of divergence of the integrals
and for the sake of brevity, we substitute the subscripts $log$ and $quad$ by
$0$ and $2$, respectively. Surface terms can be conveniently written as
integrals of total derivatives, namely

$\displaystyle\upsilon_{2w}g^{\mu\nu}=\int_{k}\frac{\partial}{\partial
k_{\nu}}\frac{k^{\mu}}{(k^{2}-m^{2})^{2-w}},$ (16)
$\displaystyle(\xi_{2w}-v_{2w})g^{\\{{\mu\nu}}g^{{\alpha\beta}\\}}=\int_{k}\frac{\partial}{\partial
k_{\nu}}\frac{2(2-w)k^{\mu}k^{\alpha}k^{\beta}}{(k^{2}-m^{2})^{3-w}}.$ (17)

and

$\displaystyle(\sigma_{2w}-\xi_{2w})g^{\\{\mu\nu}g^{\alpha\beta}g^{\gamma\delta\\}}=\int_{k}\frac{\partial}{\partial
k_{\nu}}\frac{4(3-w)(2-w)k^{\mu}k^{\alpha}k^{\beta}k^{\gamma}k^{\delta}}{(k^{2}-m^{2})^{4-w}}.$
(18)

Equations (13)-(15) are in principle arbitrary and regularization dependent.
From the mathematical point of view, a surface term can be any number since it
is a difference between two infinities. They can be shown to vanish in usual
dimensional regularization and they can be finite or infinite if computed with
a sharp cutoff. In general, we leave these terms unevaluated until the end of
the calculation to be fixed on symmetry grounds or phenomenology, when it is
the case [11].

To illustrate this method, it is instructive to consider first the usual
vacuum polarization tensor [34]:

$\displaystyle\Pi^{\mu\nu}(p)=\frac{4}{3}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})I_{log}(m^{2})-4\upsilon_{2}\eta^{\mu\nu}+\frac{4}{3}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\upsilon_{0}-$
$\displaystyle-\frac{4}{3}(p^{2}\eta^{\mu\nu}+2p^{\mu}p^{\nu})(\xi_{0}-2\upsilon_{0})-\frac{8i}{(4\pi)^{2}}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\int^{1}_{0}x(1-x)\ln\frac{m^{2}-p^{2}x(1-x)}{m^{2}},$
(19)

where $\upsilon_{0}$ and $\xi_{0}$ are logarithmic surface terms. Note that if
we use a Ward identity, the surface term $\upsilon_{2}$ is fixed to be equal
to zero, and we get a relation between $\xi_{0}$ and $\upsilon_{0}$. At the
same time, it is not possible to fix the $\upsilon_{0}$ term because this term
is already gauge invariant. This surface term is the just same that appears in
the CFJ induced term. In that case it is not possible to fix it as well
because it is proportional to a Levi-Civita symbol.

The use of eq. (10) is not the only one possible since the implicit regulator
assumed allows any other operation with the integrands. It is, however, the
equation we use the most in order to separate divergent from the finite part,
because the second term in the r.h.s. of this equation is less divergent than
the first. Disadvantages of doing this include the high number of powers in
$k$ that can appear in the numerator of the integrals, which makes them
difficult to compute (by the way, one manner to do this is proposed in [34]),
and the surface terms that cannot be completely fixed by symmetries if only
eq. (10) is used.

##  IV Evaluation of the diagrams

After taking the traces in eqs. (4)-(9), regularizing the integrals and
summing all the diagrams we find the full one-loop vacuum polarization tensor.
We list all integrals in the appendix.

$\displaystyle\Pi^{\mu\nu}_{LV}=\frac{8}{3}q^{2}\left\\{\left(c^{\mu\alpha}p^{\nu}+c^{\nu\alpha}p^{\mu}\right)p_{\alpha}-c^{\alpha\beta}p_{\alpha}p_{\beta}\eta^{\mu\nu}-p^{2}c^{\mu\nu}\right\\}\left\\{I_{log}(m^{2})-\frac{i}{16\pi^{2}}\left[\frac{(p^{2}+2m^{2})}{p^{2}}Z_{0}+\frac{1}{3}\right]+\frac{\upsilon_{0}}{2}\right\\}+$
$\displaystyle+\frac{i}{16\pi^{2}}q^{2}c^{\alpha\beta}p_{\alpha}p_{\beta}\frac{1}{p^{2}}\left(p^{\mu}p^{\nu}-p^{2}\eta^{\mu\nu}\right)\left\\{p^{2}\iota_{0}+\frac{(p^{2}+4m^{2})}{p^{2}}Z_{0}+\frac{2}{3}\right\\}+$
$\displaystyle-\frac{mq^{2}}{2\pi^{2}}p_{\lambda}\left\\{p^{2}g^{\mu\nu\lambda}+p_{\beta}\left(g^{\nu\beta\lambda}p^{\mu}-g^{\mu\beta\lambda}p^{\nu}\right)\right\\}\iota_{1}-4imq^{2}p_{\alpha}\left(g^{\nu\mu\alpha}-g^{\alpha\mu\nu}+g^{\alpha\nu\mu}\right)\upsilon_{0}+$
$\displaystyle+q^{2}\left(-\frac{m^{2}}{\pi^{2}}\iota_{0}+4i\upsilon_{0}\right)b_{\alpha}p_{\beta}\epsilon^{\alpha\beta\mu\nu},$
(20)

in which $Z_{n}=\int_{0}^{1}dx\
x^{n}\ln\left[\frac{m^{2}-p^{2}x(1-x)}{m^{2}}\right]$ and
$\iota_{n}=\int_{0}^{1}dx\frac{x^{n}(1-x)}{m^{2}-p^{2}x(1-x)}$. Besides, we
note that $c^{\mu\nu}$ is symmetric, and $g^{\mu\nu\alpha}$ is antisymmetric
in the two first indices, as expected, since only these parts of the above
mentioned tensors can contribute to observables. In order to obtain the result
given by eq. (20), the following relations involving the integrals in the
Feynman parameters were used:

$Z_{k}=\frac{1}{k+1}\left\\{kZ_{k-1}-(k-1)\frac{m^{2}}{p^{2}}Z_{k-2}-\frac{k-1}{k(k+1)}\right\\},$
(21)
$\iota_{k+1}=\frac{1}{2}\left\\{\iota_{k}-\frac{1}{p^{2}}\left[kZ_{k-1}-(k+1)Z_{k}\right]\right\\}.$
(22)

In eq. (20), we have constrained the surface terms such that the result in eq.
(20) is transverse. This procedure fixes $\xi_{0}=2\upsilon_{0}$,
$\sigma_{0}=3\upsilon_{0}$ and $\upsilon_{2}=\xi_{2}=0$. With these relations
between the surface terms, the contributions proportional to the vectors
$a^{\mu}$ and $e^{\mu}$ turned out to be zero. We see that gauge invariance of
the action is not sufficient to determine all the surface terms, which implies
the ambiguity of the induced CFJ term [35]. It is important to notice that the
contributions due to the parameter $\upsilon_{0}$ in the terms with the
tensors $c^{\mu\nu}$ and $g^{\mu\nu\alpha}$ are irrelevant since they can be
eliminated through renormalization or by imposing some normalization
condition.

One could try to determine $\upsilon_{0}$ by enforcing momentum routing
invariance in the diagrams. One possible procedure would be to calculate the
amplitude with arbitrary routing, parameterized by an alpha constant, and then
require the result to not depend on such a parameter. It can be shown, for a
QED amplitude $T^{\mu_{1}\mu_{2}\cdots\mu_{n}}$, with $n$ external photon
legs, that its transversality only can be respected if a relative shift is
allowed between the remaining $(n-1)$-point functions which result from the
contraction of the external momentum $p_{\mu_{i}}$ with
$T^{\mu_{1}\mu_{2}\cdots\mu_{n}}$. This relative shift is only allowed if some
relation between surface terms are established. Here, we show this for $n=2$.

Figure 3: Gauge and momentum routing invariance relation for a two leg
diagram. Figure 4: Momentum routing invariance of a tadpole.

We attribute a general loop momentum in the diagrams of Fig. 3, respecting the
energy-momentum conservation in the vertices. When the external momentum
$p_{\mu}$ is contracted with the diagrams, each one of the graphs, after the
contraction is decomposed in a difference of two identical tadpole diagrams
with different loop momentum. Considering the six two-point amplitudes, after
the contraction with $p_{\mu}$, only four tadpole diagrams survive, which are
shown in Fig. 4, in which $l$ is an general routing that is proportional to
the external momentum, i.e., $l=\alpha p$. The tadpoles are functions of $l$
($l^{\prime}$), $\tau^{\nu}(l)$. The result of the calculation represented by
Fig. 4 is given by:

$\displaystyle\tau^{\nu}_{(a)}(l)-\tau^{\nu}_{(a)}(l^{\prime})+\tau^{\nu}_{(b)}(l)-\tau^{\nu}_{(b)}(l^{\prime})=8qc_{\mu\alpha}\left\\{(\alpha^{3}-\alpha^{\prime
3})\left[p^{\mu}p^{\alpha}p^{\nu}+p^{2}p^{\alpha}\eta^{\mu\nu}\right](-v_{0}+2\xi_{0}-\sigma_{0})+\right.$
$\displaystyle\left.+p^{\alpha}\eta^{\mu\nu}(2\upsilon_{2}-\xi_{2})\right\\}+4q(\alpha^{2}-\alpha^{\prime
2})(p^{2}me^{\nu}+2mp^{\nu}e\cdot p)(2\upsilon_{0}-\xi_{0}).$ (23)

Since $\alpha\neq\alpha^{\prime}$ by definition, the only possible solution
for preserving gauge invariance (the tranversality of the photon polarization
tensor) is the same that assures momentum routing invariance. In fact, the
explicit calculation which results in eq. (23) enforces that
$\xi_{0}=2\upsilon_{0}$ and $\sigma_{0}=3\upsilon_{0}$. If we take into
account the contributions from the traditional QED, we also obtain that
$\upsilon_{2}=\xi_{2}=0$. It is interesting to remark that, although we have
considered an arbitrary loop momentum in the vacuum polarization tensor, this
result implies momentum routing invariance of the tadpole diagram, which
results in transversality of the photon two-point function. In general, the
transversality of an amplitude with $n$ external legs will result in routing
independence of an amplitude with $n-1$ outer legs. This condition is weaker
than the independence of loop-momentum of the original amplitude. Also, MRI
looks like a symmetry of the Feynman diagrams, as in Fig. 4. But is not a
symmetry in terms of an action, i. e. transformations of the fields that make
this action invariant.

The piece $I_{log}(m^{2})$ is a logarithmically divergent integral, namely
$\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-m^{2})^{2}}$, introduced in
section III. We do not have to evaluate it explicitly. It gives rise to
$1/\epsilon+...$ in dimensional regularization, for example. We now take the
low energy limit ($m^{2}>>p^{2}$) in each term and integral of eq. (20). For
instance, $Z_{1}(m^{2}>>p^{2})\approx-\frac{p^{2}}{12m^{2}}$ in this limit.
Thus, we find the low energy limit of the vacuum polarization tensor
($m^{2}>>p^{2}$):

$\displaystyle\Pi^{\mu\nu}_{LV}(p)=\frac{8}{3}q^{2}\left\\{c^{\mu
p}p^{\nu}+c^{\nu
p}p^{\mu}-c^{pp}\eta^{\mu\nu}-p^{2}c^{\mu\nu}\right\\}\left(I_{log}(m^{2})+\frac{i}{16\pi^{2}}\frac{p^{2}}{6m^{2}}+\frac{\upsilon_{0}}{2}\right)+$
$\displaystyle+\frac{i}{16\pi^{2}}\frac{q^{2}}{3m^{2}}c^{pp}\left(p^{\mu}p^{\nu}-p^{2}\eta^{\mu\nu}\right)-\frac{mq^{2}}{12\pi^{2}}\frac{p^{2}}{m^{2}}\left\\{g^{\mu\nu
p}+\frac{1}{p^{2}}(p^{\mu}g^{\nu pp}-p^{\nu}g^{\mu pp})\right\\}+$
$\displaystyle-4imq^{2}p_{\alpha}\left(g^{\nu\mu\alpha}-g^{\alpha\mu\nu}+g^{\alpha\nu\mu}\right)\upsilon_{0}-\left(\frac{q^{2}}{2\pi^{2}}\right)\epsilon^{bp\mu\nu}\left(1-8i\pi^{2}\upsilon_{0}\right)$
$\displaystyle\approx\frac{8}{3}q^{2}\left\\{c^{\mu p}p^{\nu}+c^{\nu
p}p^{\mu}-c^{pp}\eta^{\mu\nu}-p^{2}c^{\mu\nu}\right\\}I_{log}(m^{2})+\frac{i}{16\pi^{2}}\frac{q^{2}}{3m^{2}}c^{pp}\left(p^{\mu}p^{\nu}-p^{2}\eta^{\mu\nu}\right)+$
$\displaystyle-\left(\frac{q^{2}}{2\pi^{2}}\right)\epsilon^{bp\mu\nu}\left(1-8i\pi^{2}\upsilon_{0}\right),$
(24)

where $c^{p\nu}$ stands for $c^{\mu\nu}p_{\mu}$.

The result in eq. (24) shows that only $b_{\mu}$, $c_{\mu\nu}$ and
$g_{\mu\nu\lambda}$ affect usual QED at low energies. The one-loop LV
contribution to spectroscopy or condensed matter physics would be due only to
these terms. In particular, the radiatively induced CFJ term
$\epsilon^{\alpha\beta\mu\nu}b_{\alpha}p_{\beta}$ is the one most studied
within various contexts and it is dominant compared to terms proportional to
$g^{\mu\nu\lambda}$.

On the other hand, in the high-energy limit ($m^{2}<<p^{2}$) we find that
vacuum polarization tensor is affected only by $c_{\mu\nu}$ coefficients:

$\displaystyle\Pi^{\mu\nu}_{LV}(p)=\frac{8}{3}q^{2}\left\\{c^{\mu
p}p^{\nu}+c^{\nu
p}p^{\mu}-c^{pp}\eta^{\mu\nu}-p^{2}c^{\mu\nu}\right\\}\left\\{I_{log}(m^{2})-\frac{i}{16\pi^{2}}\left[\ln{\left(-\frac{p^{2}}{m^{2}}\right)}-\frac{5}{3}\right]+\frac{\upsilon_{0}}{2}\right\\}+$
$\displaystyle-\frac{i}{16\pi^{2}}\frac{4}{3}q^{2}c^{pp}\left(\frac{p^{\mu}p^{\nu}}{p^{2}}-\eta^{\mu\nu}\right),$
(25)

where pieces proportional to $p_{\mu}$ can be disregarded since they couple
with currents, and contributions proportional to $\partial_{\mu}J^{\mu}$
vanish due to the gauge symmetry.

##  V The parameter $b_{\mu}$ and applications to condensed matter

The bridge between high-energy physics and condensed matter has shown
promising in recent years and the question of studies on the renormalization
group has been the great trump card of this relationship [36]. With the
emergence of graphene, low-dimensional systems could be described via the
massless Dirac equation in ($2+1$)-dimensional space-time. The application of
field theory in low-dimensional systems has become a reality, with regard to
theoretical studies, such as the model that describes how electrons propagate
over a sheet of graphene, from the point of view of the renormalization group
[37]. Another interesting application consists in considering curved space
effects in graphene [38], showing how useful is the application of field
theory techniques to low-dimensional electronic systems.

The proposal that particles presenting a relativistic scattering relationship
can be considered as quasi-particles in condensed matter models has been known
for some time [39]. In some models, the dispersion relation can be linearized,
by means of an expansion around the Fermi energy, which ends up with
relativistic energy-momentum relations. In this sense, Dirac’s fermions gained
some prominence with the discovery of graphene which, being an essentially
($2+1$)-dimensional system, is nothing more than a sheet composed of carbon
atoms in a hexagonal lattice [40]. Thus, electrons moving over this sheet of
carbon interact with the potential of this lattice, giving rise to conical
structures close to Fermi energy. Near this region (called a node), the
scattering relationships of electrons on the graphene sheet turn out to be
linear and the Hamiltonian that describes the system is given by the non-
massive Dirac equation, where the propagation velocity is the Fermi velocity
($v_{f}$) and the spin of the particle gives rise to a pseudo-spin, which is
related to the sub-lattice of the system.

After the observation of Dirac fermions, it was realized [41] that some
materials whose band structure present nodes around the excitation of the
material are Weyl fermions, such materials being called Weyl semi-metals [42].
Initially, the Weyl semi-metals proposals were based on studies of pyrochlore
iridium molecules, topological insulators and heterostructures [43]. Such
descriptions paved the way for the distance between condensed matter and high
energy physics to become smaller with regard to the description of certain
phenomena, more specifically the emergence of field theory according to Anti-
de Sitter-conformal field theory (AdS-CFT) correspondence or Anti-de Sitter-
condensed matter theory correspondence (AdS-CMT) [44]. In the case of Weyl
semi-metals, this connection occurs through the chiral anomaly, which can be
translated into condensed matter models. The chiral anomaly basically tells us
that the conservation laws of the vector current ($\partial_{\mu}j^{\mu}=0$)
and that of the chiral current ($\partial_{\mu}j^{\mu}_{5}=0$) cannot be
satisfied at the same time. Therefore, if we enforce vector current
conservation, the chiral current cannot be conserved, which leads to the well-
known chiral symmetry breaking. From the point of view of Weyl semi-metals, it
can be written in terms of the electromagnetic fields as well as the number of
fermions with right-left chirality [45].

Thus, from this perspective, one might wonder whether the inclusion of terms
causing the violation of Lorentz symmetry would also lead to interesting
results, considering this relationship with condensed matter. In this sense,
one can apply QED to a class of materials that can be considered as Weyl semi-
metals [46, 47, 48, 49, 50]. With the proper choice of physical parameters,
these materials can be represented by the massive Dirac equation in (3 +
1)-dimensions (at low energies, it is considered to describe quasi-particles),
which is related to the following action (corresponding to only the $b_{\mu}$
term in equation (1))

$S=\int
d^{4}x\bar{\psi}(i\cancel{\partial}-m-\cancel{b}\gamma_{5}-e\cancel{A})\psi,$
(26)

with $b_{\mu}$ being a constant four-vector.

This model has been studied with great enthusiasm, since it can generate a CFJ
term [13]-[23] that can be finite but undeterminated [11]. However, starting
from eq. (26), it is possible to radiatively induce the CFJ term and determine
it unambiguously, since condensed matter models can be verified
experimentally.

The corresponding action of eq. (26) for the condensed matter model in the
momentum space is given by the expression

$S=\int\frac{d^{4}k}{(2\pi)^{4}}\bar{\psi}(\gamma_{\mu}M^{\mu}_{\nu}k^{\nu}-m-\cancel{b}\gamma_{5})\psi,$
(27)

with $M^{\mu}_{\nu}=(v_{f},v_{f},\tilde{v}_{f})$ being a diagonal matrix which
is necessary due to the anisotropy introduced by the Fermi velocity (in some
materials, we consider $v_{f}=c/300$).

The amplitudes and propagators that come from eq. (27) are similar to the
usual QED. The complete propagator is given by the expression

$G(k,b)=\frac{i}{(\cancel{k}-m-\cancel{b}\gamma_{5})},$ (28)

with the polarization tensor $\Pi^{\mu\nu}$ given by the following expression,
already adapted due to the velocity of propagation of the charge carriers on
the graphene sheet being the Fermi velocity $v_{f}$,

$\Pi^{\mu\nu}(b,p)=\frac{e^{2}}{v_{f}\tilde{v}_{f}}\int\frac{d^{4}k}{(2\pi)^{4}}\
Tr[\gamma^{\mu}G(k,b)\gamma^{\nu}G(k+p,b)].$ (29)

Equation (29) provides all the necessary information regarding the
relationship between the Lorentz breaking and the Weyl semi-metals. It is now
sufficient to perform a direct calculation to obtain information about the
consequences of the parameter $b$. Thus, taking the trace and making the
necessary calculations, we found the result for the amplitude given by
equation (20), in which only the $b_{\mu}$ parameter contributes to the result
for conductivity. From the point of view of implicit regularization, which was
discussed in detail in Section III, the parameter $\upsilon_{0}$, even when is
required gauge (and/or momentum routing) invariance, is undetermined. In this
sense, the process of fixing the parameter should be by phenomenology, like
building an experimental apparatus that can measure the 4-current $j_{\mu}$
associated with the fermion current adapted to a condensed matter model (an
example of measurement to fix such a parameter is an analysis of the Hall
effect from the perspective of Weyl semi-metals, where it is possible to
experimentally obtain the value of the so-called Hall conductivity [5]). On
the other hand, from a theory point of view, some results agree that the
parameter $\upsilon_{0}$ can be determined unambiguously for massless theories
[51, 52].

After contracting the $A_{\mu}$ field with the finite piece that remains in
eq. (24), we find out the following current

$j^{\nu}=\frac{q^{2}}{2\pi^{2}v_{f}\tilde{v}_{f}}(1-8\pi^{2}\upsilon_{0})b_{m}\epsilon^{m\nu\alpha\beta}\partial_{\alpha}A_{\beta},$
(30)

considering the spatial part and

$j^{\nu}=\frac{q^{2}}{2\pi^{2}v_{f}\tilde{v}_{f}}(1-8\pi^{2}\upsilon_{0})b_{0}\epsilon^{0\nu\alpha\beta}\partial_{\alpha}A_{\beta},$
(31)

when we consider the temporal part. The spatial part presented by (30) gives
rise to the anomalous Hall effect, whose conductivity is proportional to the
separation term between the Weyl nodes.

$\sigma^{xy}=\frac{1}{2\pi^{2}v_{f}\tilde{v}_{f}}\epsilon^{xyl}(1-8\pi^{2}\upsilon_{0})|\vec{b}|\hat{b}_{l}.$
(32)

The second equation, given by (31), describes the so-called magnetic effect,
which sometimes implies the equilibrium of currents in the presence of the
chiral magnetic field [53]. However, this is a naive statement, because the
chiral anomaly in condensed matter models takes place near the Weyl nodes.
Therefore, one must take a great care to predict chiral anomaly effects
directly for a Weyl semi-metal (for more details on Weyl semi-metals, see
[54]). Nevertheless, we see that Weyl semi-metals are interesting for studies
in high energy physics, more specifically, considering that they can be
analyzed from the point of view of Lorentz symmetry breaking, when the
conductivity can be generated by a $\cancel{b}\gamma_{5}$ type term in the LV
action.

A comment here is necessary. The $\Pi^{\mu\nu}$ polarization tensor modifies
Maxwell equations. The even part is related to the characteristics of the
electrical permeability and magnetic permeability constants of the medium
where the electrons propagate. The odd part, on the other hand, can add new
terms to Maxwell equations, which can modify the response of the material
medium to the propagation of electrons in a Weyl semi-metal. Considering
$j=\rho=0$ (absence of sources), the equation of the wave propagating in a
Weyl semi-metal is modified, leading to the effect of vacuum birefringence
associated with it. This effect is exclusively associated with the CFJ term
induced radiatively and such observation is one example of how to fix the
parameter $\upsilon_{0}$ by phenomenology. Some other interesting effects on
Weyl semi-metals are described in [55] (Repulsive Casimir Effect) and [56]
(Axionic Electrodynamics) 222A discussion about Weyl semi-metals theory and
their relation with induced CFJ term and another models in high energy physics
can be found in [50].

##  VI Summary

We calculated the polarization tensor of the Abelian gauge field in a minimal
LV extension of QED involving all terms listed in [31]. Within our studies,
the main attention was paid, first, to divergent contributions, while most of
previous studies dealt with finite ones, the most known of them is the CFJ
term, second, to the infrared-leading parts of finite contributions. The
importance of these terms is justified by the fact that they play a special
role within condensed matter studies where the LV effects attract essential
attention. In this study, we followed this line and calculated the anomalous
Hall conductivity and discussed other possible applications of our results to
the condensed matter, especially, to Weyl semi-metals.

A natural continuation of this study, besides of study of other applications
of Lorentz symmetry breaking within the condensed matter context, could
consist in treating the low-energy impacts of higher-derivative LV terms. We
are planning to do this study in a forthcoming paper.

## Acknowledgments

The work of A. Yu. P. has been partially supported by the CNPq project No.
301562/2019-9.

## Appendix

All integrals needed after taking the traces in eqs. (4)-(9) can be obtained
from integrals below:

$\displaystyle\int_{k}\frac{1}{[(k-p)^{2}-m^{2}]}=I_{quad}(m^{2})-p^{2}\upsilon_{0};$
(33)
$\displaystyle\int_{k}\frac{k^{\alpha}}{[(k-p)^{2}-m^{2}]}=p^{\alpha}(I_{quad}(m^{2})-\upsilon_{2})-p^{2}p^{\alpha}(\xi_{0}-\upsilon_{0});$
(34)
$\displaystyle\int_{k}\frac{k^{\alpha}}{[(k-p)^{2}-m^{2}]^{2}}=p^{\alpha}(I_{log}(m^{2})-\upsilon_{0});$
(35)
$\displaystyle\int_{k}\frac{k^{2}}{[(k-p)^{2}-m^{2}]^{2}}=I_{quad}(m^{2})+(m^{2}+p^{2})I_{log}(m^{2})-3p^{2}\upsilon_{0};$
(36)
$\displaystyle\int_{k}\frac{k^{\alpha}k^{\beta}}{[(k-p)^{2}-m^{2}]^{2}}=\frac{1}{2}g^{\alpha\beta}(I_{quad}(m^{2})-\upsilon_{2})+p^{\alpha}p^{\beta}(I_{log}(m^{2})-\xi_{0})-\frac{1}{2}p^{2}g^{\alpha\beta}(\xi_{0}-\upsilon_{0});$
(37)
$\displaystyle\int_{k}\frac{k^{2}k^{\alpha}}{[(k-p)^{2}-m^{2}]^{2}}=2p^{\alpha}(I_{quad}(m^{2})-\upsilon_{2})+p^{\alpha}(m^{2}+p^{2})I_{log}(m^{2})+p^{\alpha}(3p^{2}-m^{2})\upsilon_{0}-4p^{2}p^{\alpha}\xi_{0};$
(38)
$\displaystyle\int_{k}\frac{k^{\alpha}k^{\beta}k^{\gamma}}{[(k-p)^{2}-m^{2}]^{2}}=\frac{1}{2}p^{\\{\alpha}g^{\gamma\beta\\}}\left[I_{quad}(m^{2})-\xi_{2}\right]-\frac{1}{2}p^{2}p^{\\{\alpha}g^{\gamma\beta\\}}\left[I_{log}(m^{2})-\xi_{0}\right]+$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\frac{1}{2}(p^{2}p^{\\{\alpha}g^{\gamma\beta\\}}+2p^{\alpha}p^{\beta}p^{\gamma})\left[I_{log}(m^{2})-\sigma_{0}\right];$
(39)

$\displaystyle I=\int_{k}\frac{1}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=-b\
\int^{1}_{0}dx\frac{(1-x)}{\Delta^{2}};$ (40) $\displaystyle
I_{1}^{\beta}=\int_{k}\frac{k^{\beta}}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=b\
p^{\beta}\ \int^{1}_{0}dx\frac{x(1-x)}{\Delta^{2}};$ (41) $\displaystyle
J_{1}^{\beta}=\int_{k}\frac{k^{\beta}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]^{2}}=I_{1}^{\beta}+p^{\beta}I;$
(42) $\displaystyle
I_{2}=\int_{k}\frac{k^{2}}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=I_{log}(m^{2})-b\
Z_{0}(p^{2},m^{2})-b\ m^{2}\ \int^{1}_{0}dx\frac{(1-x)}{\Delta^{2}}$ (43)
$\displaystyle
J_{2}=\int_{k}\frac{k^{2}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]^{2}}=I_{2}+p^{2}I+2p_{\beta}I_{1}^{\beta};$
(44) $\displaystyle
I_{2}^{\beta\nu}=\int_{k}\frac{k^{\beta}k^{\nu}}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=\frac{1}{4}g^{\beta\nu}(I_{log}(m^{2})-\upsilon_{0})-\frac{1}{2}b\
g^{\beta\nu}[Z_{0}(p^{2},m^{2})-Z_{1}(p^{2},m^{2})]-$ $\displaystyle-b\
p^{\beta}p^{\nu}\ \int^{1}_{0}dx\frac{x^{2}(1-x)}{\Delta^{2}};$ (45)
$\displaystyle
J_{2}^{\beta\nu}=\int_{k}\frac{k^{\beta}k^{\nu}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]^{2}}=I_{2}^{\beta\nu}+p^{\beta}p^{\nu}I+I_{1}^{\beta}p^{\nu}+I_{1}^{\nu}p^{\beta};$
(46) $\displaystyle
I_{3}^{\nu}=\int_{k}\frac{k^{2}k^{\nu}}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=-\frac{1}{2}p^{\nu}(I_{log}(m^{2})-\upsilon_{0})+b\
p^{\nu}Z_{1}(p^{2},m^{2})+b\ m^{2}\
p^{\nu}\int^{1}_{0}dx\frac{x(1-x)}{\Delta^{2}};$ (47) $\displaystyle
J_{3}^{\nu}=\int_{k}\frac{k^{2}k^{\nu}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]^{2}}=I_{3}^{\nu}+p^{\nu}I_{2}+p^{2}I_{1}^{\nu}+p^{2}p^{\nu}I+2p_{\gamma}I_{2}^{\gamma\nu}+2p_{\gamma}p^{\nu}I_{1}^{\gamma}-p^{\nu}\upsilon_{0};$
(48) $\displaystyle
I_{5}^{\beta\nu\alpha}=\int_{k}\frac{k^{\beta}k^{\nu}k^{\alpha}}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=\frac{-1}{12}bp^{\\{\alpha}g^{\beta\nu\\}}(I_{log}(m^{2})-\xi_{0})+b\
p^{\alpha}p^{\beta}p^{\nu}\int^{1}_{0}dx\frac{x^{3}(1-x)}{\Delta^{2}}+$
$\displaystyle+\frac{1}{2}b\
p^{\\{\alpha}g^{\beta\nu\\}}[Z_{1}(p^{2},m^{2})-Z_{2}(p^{2},m^{2})];$ (49)
$\displaystyle
J_{5}^{\beta\nu\alpha}=\int_{k}\frac{k^{\beta}k^{\nu}k^{\alpha}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]^{2}}=I_{5}^{\beta\nu\alpha}+p^{\beta}p^{\nu}p^{\alpha}I+p^{\\{\nu}p^{\beta}I^{\alpha\\}}_{1}+p^{\\{\beta}I^{\nu\alpha\\}}_{2}+\frac{1}{4}p^{\\{\alpha}g^{\beta\nu\\}}(\upsilon_{0}-\xi_{0})$
(50) $\displaystyle
I_{4}^{\beta\nu}=\int_{k}\frac{k^{2}k^{\beta}k^{\nu}}{(k^{2}-m^{2})^{2}[(k+p)^{2}-m^{2}]}=\frac{1}{2}g^{\beta\nu}(I_{quad}(m^{2})-\upsilon_{2})+\frac{1}{4}(m^{2}-p^{2})g^{\beta\nu}(I_{log}(m^{2})-\upsilon_{0})+$
$\displaystyle+\frac{1}{6}(p^{2}g^{\nu\beta}+2p^{\beta}p^{\nu})(I_{log}(m^{2})-\xi_{0})-b\
(-g^{\beta\nu}p^{2}+p^{\nu}p^{\beta})Z_{2}(p^{2},m^{2})+\frac{1}{2}\
b(m^{2}-3p^{2})\ g^{\beta\nu}Z_{1}(p^{2},m^{2})+$
$\displaystyle+\frac{1}{2}b(p^{2}-m^{2})g^{\beta\nu}Z_{0}(p^{2},m^{2})-b\
p^{\beta}p^{\nu}m^{2}\ \int^{1}_{0}dx\frac{x^{2}(1-x)}{\Delta^{2}};$ (51)
$\displaystyle
J_{4}^{\beta\nu}=\int_{k}\frac{k^{2}k^{\beta}k^{\nu}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]^{2}}=\frac{1}{2}g^{\nu\beta}(I_{quad}(m^{2})-\upsilon_{2})+(m^{2}-p^{2})J_{2}^{\beta\nu}+2p_{\lambda}J_{5}^{\beta\nu\lambda}-2p_{\lambda}I_{5}^{\beta\nu\lambda}-p^{2}I_{2}^{\beta\nu},$
(52)

where $b\equiv\frac{i}{(4\pi)^{2}}$, $Z_{k}(p^{2},m^{2})$ and $\Delta^{2}$ are
defined as

$\displaystyle
Z_{k}(p^{2},m^{2})=\int^{1}_{0}dzz^{k}\ln\frac{m^{2}-p^{2}z(1-z)}{m^{2}},$
(53) $\displaystyle\Delta^{2}=m^{2}-p^{2}x(1-x).$ (54)

The basic divergent integrals $I_{log}(m^{2})$ and $I_{quad}(m^{2})$ and the
surface terms $\upsilon_{0}$, $\upsilon_{2}$ and $\xi_{0}$ are defined in
section III.

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Towards Morphologically Induced Anisotropy in Thermally Hysteretic Dielectric
Properties of Vanadium Dioxide

Tom G. Mackay***E–mail<EMAIL_ADDRESS>
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh, Edinburgh EH9 3FD, UK
and
NanoMM — Nanoengineered Metamaterials Group
Department of Engineering Science and Mechanics
Pennsylvania State University, University Park, PA 16802–6812, USA
Akhlesh Lakhtakia
NanoM — Nanoengineered Metamaterials Group
Department of Engineering Science and Mechanics
Pennsylvania State University, University Park, PA 16802–6812, USA

Abstract

The Bruggeman homogenization formalism was used to numerically investigate the
dielectric properties of a columnar thin film (CTF) made from vanadium
dioxide. For visible and near-infrared wavelengths, the CTF is
electromagnetically equivalent to a homogeneous orthorhombic material. Over
the $58\,^{\circ}{\rm C}$–$72\,^{\circ}{\rm C}$ temperature range, the
eigenvalues of the CTF’s relative permittivity dyadic are highly sensitive to
temperature, and vary according to whether the CTF is being heated or cooled.
The anisotropy revealed through the eigenvalues, and the anisotropy of the
associated hysteresis, were investigated in relation to temperature for CTFs
of different porosities and columnar cross sections. When the free-space
wavelength is 800 nm, the CTF is a dissipative dielectric material that
exhibits temperature-dependent anisotropy and anisotropic hysteresis. In
contrast, when the free-space wavelength is 1550 nm, the CTF can be either a
dissipative dielectric material, a hyperbolic material or a metal-like
material, depending on the temperature and the porosity of the CTF. As the
porosity of the CTF decreases from $0.55$ to $0.3$, the anisotropy of the CTF
becomes more pronounced, as does the anisotropy of the hysteresis. Only
relatively modest variations in anisotropy and hysteresis arise in response to
varying the columnar cross-sectional shape, as compared to the variations
induced by varying the porosity.

Keywords: Thermal hysteresis; columnar thin film; hyperbolic material;
Bruggeman homogenization formalism; anisotropy

##  1 Introduction

Materials, metamaterials, and metasurfaces with thermally controllable
dielectric properties are of interest for numerous applications involving
reconfigurable and multifunctional devices [1, 2, 4, 3]. Vanadium dioxide
(VO2) is especially interesting in this context as it undergoes a thermally-
induced hysteretic phase change over the temperature range $58\,^{\circ}{\rm
C}$–$72\,^{\circ}{\rm C}$, in the visible, infrared, and terahertz spectral
regimes [5, 6, 7, 8, 9, 10, 11].

The electromagnetic response of VO2 is isotropic, as encapsulated by its
(complex-valued) relative permittivity $\varepsilon_{\text{VO}_{2}}$. However,
the crystal structure of VO2 is not cubic — instead, it is monoclinic at
temperatures below $58\,^{\circ}{\rm C}$ and tetragonal at temperatures above
$72\,^{\circ}{\rm C}$ [12]. Both monoclinic and tetragonal crystals coexist in
varying proportions in the $58\,^{\circ}{\rm C}$–$72\,^{\circ}{\rm C}$
temperature range. Furthermore, the value of $\varepsilon_{\text{VO}_{2}}$ for
monoclinic VO2 is significantly different to its value for tetragonal VO2, at
a given free-space wavelength $\lambda_{\scriptscriptstyle 0}$.

The transition from monoclinic VO2 to tetragonal VO2 can be achieved by
heating and it can be reversed by cooling. And this transition is hysteretic.
That is, at a temperature in the $58\,^{\circ}{\rm C}$–$72\,^{\circ}{\rm C}$
range, the value of $\varepsilon_{\text{VO}_{2}}$ depends on whether VO2 was
heated up or cooled down to reach that temperature [13]. Since VO2 is
hysteretically biphasic, $\varepsilon_{\text{VO}_{2}}$ may be regarded as a
double-valued function of temperature; hence, we write
$\varepsilon_{\text{VO}_{2}}=\varepsilon^{\rm{heat}}_{\text{VO}_{2}}$ for the
heating phase and
$\varepsilon_{\text{VO}_{2}}=\varepsilon^{\rm{cool}}_{\text{VO}_{2}}$ for the
cooling phase.

The prospect of an anisotropic material, metamaterial or metasurface whose
dielectric properties for propagation in different directions are thermally
controllable presents opportunities for directional control and tuning in
applications involving reconfigurable and multifunctional devices. How could
such a prospect be realized? A practical means of realizing thermal
directional control is to use VO2 to fabricate a columnar thin film (CTF) [14,
15, 16]. A CTF is a forest of parallel columns, inclined at a certain angle to
a substrate upon which they are deposited by a technique such as oblique angle
deposition. Macroscopically, a CTF is electromagnetically equivalent to a
homogeneous anisotropic material with orthorhombic symmetry.

In the following sections, the Bruggeman homogenization formalism [17, 18, 19]
is employed to estimate the relative permittivity parameters of a CTF made
from VO2, and the effects of temperature on anisotropy and on the anisotropy
of hysteresis are thereby investigated. The following notation is adopted:
3-vectors are underlined once and 3$\times$3 dyadics are underlined twice. The
triad of Cartesian basis vectors is denoted as
$\left\\{\hat{\underline{u}}_{\rm x},\hat{\underline{u}}_{\rm
y},\hat{\underline{u}}_{\rm z}\right\\}$.

##  2 Homogenization model of columnar thin film

###  2.1 Preliminaries

A bulk material is evaporated and a collimated vapor flux is directed in a
low-pressure chamber towards a planar substrate to form a CTF [14, 15, 20,
21]. Suppose that the planar substrate is parallel to the $xy$ plane, and the
bulk material is $\text{VO}_{2}$ with relative permittivity
$\varepsilon_{\text{VO}_{2}}$. Provided that wavelengths are much greater than
the cross-sectional dimensions of the CTF columns, the CTF may be regarded as
a homogeneous orthorhombic material which is characterized by the frequency-
domain constitutive relation [14, 15, 21]

$\underline{D}=\varepsilon_{\scriptscriptstyle
0}\,\underline{\underline{\varepsilon}}_{\,\rm{CTF}}\mbox{
\tiny{${}^{\bullet}$} }\underline{E}\,.$ (1)

Herein, the CTF’s relative permittivity dyadic is

$\underline{\underline{\varepsilon}}_{\,\rm{CTF}}=\underline{\underline{S}}_{\rm{y}}(\chi)\mbox{
\tiny{${}^{\bullet}$} }\left(\varepsilon_{a}\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z}+\varepsilon_{b}\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm x}\,+\,\varepsilon_{c}\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm y}\right)\mbox{ \tiny{${}^{\bullet}$}
}\underline{\underline{S}}_{\rm{y}}^{-1}(\chi)\,,$ (2)

which has eigenvalues
$\left\\{\varepsilon_{a},\varepsilon_{b},\varepsilon_{c}\right\\}$, and
$\varepsilon_{\scriptscriptstyle 0}$ is the permittivity of free space. The
parallel columns of the CTF are inclined to the $xy$ plane at an angle
$\chi\in(0,\pi/2]$, as prescribed by the inclination dyadic

$\underline{\underline{S}}_{\rm{y}}(\chi)=\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm y}+(\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm x}+\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z})\,\cos\chi+(\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm x}-\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm z})\,\sin\chi\,.$ (3)

Note that $\varepsilon_{a,b,c}$ and $\chi$ depend on the direction of the
collimated vapor flux with respect to the $xy$ plane [14], and that
$\varepsilon_{a,b,c}$ also depend on $\lambda_{\scriptscriptstyle 0}$.

From the perspective of homogenization theory, each column of the CTF can be
viewed as a string of elongated ellipsoidal inclusions [15, 22]. All
inclusions have the same orientation, and the same shape as prescribed by the
shape dyadic

$\underline{u}_{\rm n}\,\underline{u}_{\rm
n}+\gamma_{\tau}\,\underline{u}_{\tau}\,\underline{u}_{\tau}+\gamma_{\rm{b}}\,\underline{u}_{\rm
b}\,\underline{u}_{\rm b},$ (4)

wherein

$\left.\begin{array}[]{l}\underline{u}_{\rm n}=-\hat{\underline{u}}_{\rm
x}\,\sin\chi+\hat{\underline{u}}_{\rm z}\,\cos\chi\vspace{4pt}\\\
\underline{u}_{\tau}=\hat{\underline{u}}_{\rm
x}\,\cos\chi+\hat{\underline{u}}_{\rm z}\,\sin\chi\vspace{4pt}\\\
\underline{u}_{\rm b}=-\hat{\underline{u}}_{\rm y}\end{array}\right\\}$ (5)

are the normal, tangential, and binormal basis vectors, respectively. The
columns of the CTF are needlelike. Accordingly, the shape parameters
$\gamma_{\rm{b}}\gtrsim 1$ and $\gamma_{\tau}\gg 1$. For slim inclusions,
increasing $\gamma_{\tau}$ beyond 10 has negligible effect; hence,
$\gamma_{\tau}=15$ is assumed. A schematic representation of such a CTF is
presented in Fig. 1.

The void region between columns of the CTF is filled with air, which is taken
to be electromagnetically equivalent to free space (or vacuum). The fraction
of the CTF’s volume that is occupied by ellipsoidal inclusions is denoted $f$.
Since $f\in\left(0,1\right)$, the porosity of the CTF is given as $1-f$.

###  2.2 Bruggeman homogenization formalism

The eigenvalues
$\left\\{\varepsilon_{a},\varepsilon_{b},\varepsilon_{c}\right\\}$ of
$\underline{\underline{\varepsilon}}_{\,CTF}$ that characterize the CTF at
macroscopic length scales can be estimated from the parameters
$\varepsilon_{\text{VO}_{2}}$, $f$, and $\gamma_{\rm b}$ that characterize the
CTF at microscopic length scales by means of the Bruggeman homogenization
formalism. The Bruggeman formalism has the advantage over other commonly used
homogenization formalisms, e.g., the Maxwell Garnett formalism [18], insofar
as it treats the deposited material region and the void region symmetrically.
Consequently, there are no restrictions on the value of
$f\in\left(0,1\right)$.

The polarizability density dyadic relevant to the deposited material (i.e.,
$\varepsilon_{\text{VO}_{2}}$) is [19]

$\displaystyle\underline{\underline{a}}_{d}$ $\displaystyle=$
$\displaystyle\left[\varepsilon_{\text{VO}_{2}}\,\underline{\underline{I}}-\left(\varepsilon_{a}\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z}+\varepsilon_{b}\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm x}\,+\,\varepsilon_{c}\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm y}\right)\right]$ (6) $\displaystyle\mbox{
\tiny{${}^{\bullet}$}
}\left\\{\underline{\underline{I}}+\,\underline{\underline{D}}_{d}\mbox{
\tiny{${}^{\bullet}$}
}\left[\varepsilon_{\text{VO}_{2}}\,\underline{\underline{I}}-\left(\varepsilon_{a}\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z}+\varepsilon_{b}\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm x}\,+\,\varepsilon_{c}\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm y}\right)\right]\right\\}^{-1}\,,$

while the polarizability density dyadic relevant to the void medium (i.e.,
air) is

$\displaystyle\underline{\underline{a}}_{v}$ $\displaystyle=$
$\displaystyle\left[\underline{\underline{I}}-\left(\varepsilon_{a}\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z}+\varepsilon_{b}\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm x}\,+\,\varepsilon_{c}\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm y}\right)\right]$ (7) $\displaystyle\mbox{
\tiny{${}^{\bullet}$}
}\left\\{\underline{\underline{I}}+\underline{\underline{D}}_{v}\mbox{
\tiny{${}^{\bullet}$}
}\left[\underline{\underline{I}}-\left(\varepsilon_{a}\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z}+\varepsilon_{b}\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm x}\,+\,\varepsilon_{c}\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm y}\right)\right]\right\\}^{-1}\,,$

wherein the corresponding depolarization dyadics

$\underline{\underline{D}}_{d}=\,\frac{2}{\pi}\,\int_{\phi=0}^{\pi/2}\,d\phi\int_{\theta=0}^{\pi/2}\,d\theta\,\sin\theta\,\,\frac{\frac{\cos^{2}\theta}{\gamma_{\tau}^{2}}\,\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm
x}+\sin^{2}\theta\left(\cos^{2}\phi\,\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm
z}+\frac{\sin^{2}\phi}{\gamma_{b}^{2}}\,\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm
y}\right)}{\epsilon_{b}\,\,\frac{\cos^{2}\theta}{\gamma_{\tau}^{2}}+\sin^{2}\theta\left(\epsilon_{a}\cos^{2}\phi+\epsilon_{c}\,\frac{\sin^{2}\phi}{\gamma_{b}^{2}}\right)}\,$
(8)

and

$\underline{\underline{D}}_{v}=\,\frac{2}{\pi}\,\int_{\phi=0}^{\pi/2}\,d\phi\int_{\theta=0}^{\pi/2}\,d\theta\,\sin\theta\,\,\frac{{\cos^{2}\theta}{}\,\,\hat{\underline{u}}_{\rm
x}\hat{\underline{u}}_{\rm
x}+\sin^{2}\theta\left(\cos^{2}\phi\,\,\hat{\underline{u}}_{\rm
z}\hat{\underline{u}}_{\rm z}+{\sin^{2}\phi}{}\,\,\hat{\underline{u}}_{\rm
y}\hat{\underline{u}}_{\rm
y}\right)}{\epsilon_{b}\,\,{\cos^{2}\theta}{}+\sin^{2}\theta\left(\epsilon_{a}\cos^{2}\phi+\epsilon_{c}\,{\sin^{2}\phi}{}\right)}.$
(9)

The essence of the Bruggeman formalism is that the volume-fraction-weighted
sum of the two polarizability density dyadics is null valued [19]; i.e.,

$f\,\underline{\underline{a}}_{d}+(1-f)\,\,\underline{\underline{a}}_{v}=\underline{\underline{0}}\,,$
(10)

with $\underline{\underline{0}}$ being the 3$\times$3 null dyadic. The
eigenvalues $\left\\{\varepsilon_{a},\varepsilon_{b},\varepsilon_{c}\right\\}$
of $\underline{\underline{\varepsilon}}_{\,CTF}$ can be extracted from the
Bruggeman eqn. (10) by numerical means, such as the Jacobi method [23],
provided that $\varepsilon_{\text{VO}_{2}}$, $f$, and $\gamma_{\rm{b}}$ are
known [22].

##  3 Numerical studies

The refractive index of $\text{VO}_{2}$ was determined by Cormier _et al._
[13] for heating and cooling phases over the temperature range
$55\,^{\circ}{\rm C}$–$75\,^{\circ}{\rm C}$, for $\lambda_{\scriptscriptstyle
0}\in\left\\{800,1550\right\\}$ nm. Plots of the corresponding real and
imaginary parts of $\varepsilon_{\text{VO}_{2}}$ versus temperature are
provided in Fig. 2, with $\varepsilon^{\rm{heat}}_{\text{VO}_{2}}$ and
$\varepsilon^{\rm{cool}}_{\text{VO}_{2}}$ denoting the values of
$\varepsilon_{\text{VO}_{2}}$ for the heating and cooling phases,
respectively. Let $\varepsilon^{55\,^{\circ}{\rm C}}_{\text{VO}_{2}}$ and
$\varepsilon^{75\,^{\circ}{\rm C}}_{\text{VO}_{2}}$ denote the values of
$\varepsilon_{\text{VO}_{2}}$ at $55\,^{\circ}{\rm C}$ and $75\,^{\circ}{\rm
C}$, respectively. Plots of the measures of hysteresis

$\left.\begin{array}[]{l}\Delta^{\text{Re}}_{\text{VO}_{2}}=\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}-\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}\vspace{8pt}\\\
\Delta^{\text{Im}}_{\text{VO}_{2}}=\mbox{Im}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}-\mbox{Im}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}\end{array}\right\\}$
(11)

and relative measures of hysteresis

$\left.\begin{array}[]{l}\hat{\Delta}^{\text{Re}}_{\text{VO}_{2}}=\displaystyle{\frac{\Delta^{\text{Re}}_{\text{VO}_{2}}}{\left|\mbox{Re}\left\\{\varepsilon^{75\,^{\circ}{\rm
C}}_{\text{VO}_{2}}\right\\}-\mbox{Re}\left\\{\varepsilon^{55\,^{\circ}{\rm
C}}_{\text{VO}_{2}}\right\\}\right|}\vspace{8pt}}\\\
\hat{\Delta}^{\text{Im}}_{\text{VO}_{2}}=\displaystyle{\frac{\Delta^{\text{Im}}_{\text{VO}_{2}}}{\left|\mbox{Im}\left\\{\varepsilon^{75\,^{\circ}{\rm
C}}_{\text{VO}_{2}}\right\\}-\mbox{Im}\left\\{\varepsilon^{55\,^{\circ}{\rm
C}}_{\text{VO}_{2}}\right\\}\right|}}\end{array}\right\\}$ (12)

versus temperature are also provided in Fig. 2.

For $\lambda_{\scriptscriptstyle 0}=$ 800 nm,
$\mbox{Re}\left\\{\varepsilon_{\text{VO}_{2}}\right\\}$ and
$\mbox{Im}\left\\{\varepsilon_{\text{VO}_{2}}\right\\}$ exhibit hysteresis
over the approximate temperature range $60\,^{\circ}{\rm C}$–$71\,^{\circ}{\rm
C}$. In the thermal-hysteresis regime,
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}>\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}$
whereas
$\mbox{Im}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}<\mbox{Im}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}$.
Also, the maximum magnitude of $\Delta^{\text{Re}}_{\text{VO}_{2}}$ is
substantially larger than the maximum magnitude of
$\Delta^{\text{Im}}_{\text{VO}_{2}}$ but the maximum magnitude of
$\hat{\Delta}^{\text{Re}}_{\text{VO}_{2}}$ is substantially smaller than the
maximum magnitude of $\hat{\Delta}^{\text{Im}}_{\text{VO}_{2}}$. That is, the
hysteresis is more pronounced for the real part of
$\varepsilon_{\text{VO}_{2}}$ than for the imaginary part of
$\varepsilon_{\text{VO}_{2}}$, but the relative hysteresis is more pronounced
for the imaginary part of $\varepsilon_{\text{VO}_{2}}$ than for the real part
of $\varepsilon_{\text{VO}_{2}}$.

The nature of the thermal hysteresis for $\lambda_{\scriptscriptstyle 0}=$
1550 nm is qualitatively similar to that for $\lambda_{\scriptscriptstyle 0}=$
800 nm, but

  * •

the thermal-hysteresis regime is slightly wider;

  * •

the maximum magnitudes of $\Delta^{\text{Re}}_{\text{VO}_{2}}$ and
$\Delta^{\text{Im}}_{\text{VO}_{2}}$ are substantially larger;

  * •

the maximum magnitude of $\hat{\Delta}^{\text{Re}}_{\text{VO}_{2}}$ is
slightly larger and the maximum magnitude of
$\hat{\Delta}^{\text{Im}}_{\text{VO}_{2}}$ is substantially smaller

for $\lambda_{\scriptscriptstyle 0}=$ 1550 nm than for
$\lambda_{\scriptscriptstyle 0}=$ 800 nm. Also, for
$\lambda_{\scriptscriptstyle 0}=$ 1550 nm, the maximum magnitude of
$\Delta^{\text{Re}}_{\text{VO}_{2}}$ is approximately the same as that of
$\Delta^{\text{Im}}_{\text{VO}_{2}}$ and likewise the maximum magnitude of
$\hat{\Delta}^{\text{Re}}_{\text{VO}_{2}}$ is approximately the same as that
of $\hat{\Delta}^{\text{Im}}_{\text{VO}_{2}}$; i.e., the hysteresis and
relative hysteresis are approximately equally pronounced for the real and
imaginary parts of $\varepsilon_{\text{VO}_{2}}$. Another notable difference
is that $\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}>0$
and $\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}>0$
across the entire temperature range considered for
$\lambda_{\scriptscriptstyle 0}=$ 800 nm but, for $\lambda_{\scriptscriptstyle
0}=$ 1550 nm,
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}>0$ in the
temperature range $55\,^{\circ}{\rm C}$–$69.5\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}<0$ at
higher temperatures, and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}>0$ in the
temperature range $55\,^{\circ}{\rm C}$–$63.5\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}<0$ at
higher temperatures. That is, VO2 is a dissipative dielectric material across
the entire temperature range considered for $\lambda_{\scriptscriptstyle 0}=$
800 nm, but it is a dissipative dielectric material at low temperatures and
becomes metal-like at higher temperatures for $\lambda_{\scriptscriptstyle
0}=$ 1550 nm. Furthermore, the temperature at which the transition between as
dissipative dielectric material and a metal-like material occurs depends upon
whether the material is being heated or cooled.

Next we turn to the VO2 CTFs. We begin by setting $\gamma_{\rm{b}}=1.5$ and
$f=0.6$, which are typical values for CTFs made from a range of different
materials [22]; the effects of varying $\gamma_{\rm{b}}$ and $f$ are explored
later in this section. For $\lambda_{\scriptscriptstyle 0}=$ 800 nm, plots of
the Bruggeman estimates of the real and imaginary parts of $\varepsilon_{a}$,
$\varepsilon_{b}$, and $\varepsilon_{c}$ (for both heating and cooling phases)
versus temperature are presented in Fig. 3. Therein,
$\varepsilon^{\rm{heat}}_{a,b,c}$ and $\varepsilon^{\rm{cool}}_{a,b,c}$ denote
the values of $\varepsilon_{a,b,c}$ for the heating and cooling phases,
respectively.

The orthorhombic symmetry of the CTF is apparent since, for both heating and
cooling phases, the values of $\varepsilon_{b}$ are clearly different from the
values of $\varepsilon_{a}$ and $\varepsilon_{c}$, and there are smaller
differences between the values of $\varepsilon_{a}$ and $\varepsilon_{c}$. In
addition, both the real and imaginary parts of $\varepsilon_{a}$,
$\varepsilon_{b}$, and $\varepsilon_{c}$ exhibit thermal hysteresis to varying
degrees. Plots of the measures of hysteresis

$\left.\begin{array}[]{l}\Delta^{\text{Re}}_{\ell}=\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\ell}\right\\}-\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\ell}\right\\}\vspace{8pt}\\\
\Delta^{\text{Im}}_{\ell}=\mbox{Im}\left\\{\varepsilon^{\rm{heat}}_{\ell}\right\\}-\mbox{Im}\left\\{\varepsilon^{\rm{cool}}_{\ell}\right\\}\end{array}\right\\},\qquad\ell\in\left\\{a,b,c\right\\}$
(13)

and relative measures of hysteresis

$\left.\begin{array}[]{l}\hat{\Delta}^{\text{Re}}_{\ell}=\displaystyle{\frac{\Delta^{\text{Re}}_{\ell}}{\left|\mbox{Re}\left\\{\varepsilon^{75\,^{\circ}{\rm
C}}_{\ell}\right\\}-\mbox{Re}\left\\{\varepsilon^{55\,^{\circ}{\rm
C}}_{\ell}\right\\}\right|}\vspace{8pt}}\\\
\hat{\Delta}^{\text{Im}}_{\ell}=\displaystyle{\frac{\Delta^{\text{Im}}_{\ell}}{\left|\mbox{Im}\left\\{\varepsilon^{75\,^{\circ}{\rm
C}}_{\ell}\right\\}-\mbox{Im}\left\\{\varepsilon^{55\,^{\circ}{\rm
C}}_{\ell}\right\\}\right|}}\end{array}\right\\},\qquad\ell\in\left\\{a,b,c\right\\}$
(14)

versus temperature are also presented in Fig. 3. Herein,
$\varepsilon^{55\,^{\circ}{\rm C}}_{\ell}$ and $\varepsilon^{75\,^{\circ}{\rm
C}}_{\ell}$ denote the values of $\varepsilon_{\ell}$ at $55\,^{\circ}{\rm C}$
and $75\,^{\circ}{\rm C}$, respectively, with $\ell\in\left\\{a,b,c\right\\}$.
The maximum magnitude of $\Delta^{\text{Re}}_{b}$ is substantially larger than
the maximum magnitude of $\Delta^{\text{Re}}_{c}$ which, in turn, is
substantially larger than the maximum magnitude of $\Delta^{\text{Re}}_{a}$.
Also, the maximum magnitude of $\Delta^{\text{Im}}_{b}$ is slightly larger
than the maximum magnitude of $\Delta^{\text{Im}}_{c}$ which, in turn, is
slightly larger than the maximum magnitude of $\Delta^{\text{Im}}_{a}$. As
regards the relative measures of hysteresis, the maximum of
$|\hat{\Delta}^{\text{Im}}_{b}|$ is substantially larger than the maximum of
$|\hat{\Delta}^{\text{Im}}_{c}|$ which, in turn, is substantially larger than
the maximum of $|\hat{\Delta}^{\text{Re}}_{a}|$; in contrast, the plots for
$\hat{\Delta}^{\text{Re}}_{a}$ , $\hat{\Delta}^{\text{Re}}_{b}$, and
$\hat{\Delta}^{\text{Re}}_{c}$ are all similar. Thus, thermal hysteresis has
been transformed from isotropic for bulk VO2 to anisotropic for the VO2 CTF.

The computations for Fig. 3 were repeated for $\lambda_{\scriptscriptstyle
0}=$ 1550 nm and the results plotted in Fig. 4. As regards anisotropy and
hysteresis, the results for $\lambda_{\scriptscriptstyle 0}=$ 1550 nm
displayed in Fig. 4 are qualitatively similar to those for
$\lambda_{\scriptscriptstyle 0}=$ 800 nm displayed in Fig. 3, but

  * •

the anisotropy of the CTF is substantially more pronounced and

  * •

the anisotropy of the thermal hysteresis, but not of the relative thermal
hysteresis, is substantially more pronounced

for $\lambda_{\scriptscriptstyle 0}=$ 1550 nm.

However, there is an important difference in the nature of the VO2 CTF for
$\lambda_{\scriptscriptstyle 0}=$ 1550 nm as compared to its nature for
$\lambda_{\scriptscriptstyle 0}=$ 800 nm. According to Fig. 3, the real parts
of $\varepsilon_{a}$, $\varepsilon_{b}$, and $\varepsilon_{c}$ (for both
heating and cooling phases) are all positive valued for
$\lambda_{\scriptscriptstyle 0}=$ 800 nm across the entire temperature range
considered. According to Fig. 4, the same is true for $\varepsilon_{a}$ and
$\varepsilon_{c}$ for $\lambda_{\scriptscriptstyle 0}=$ 1550 nm, but
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{b}\right\\}$ is positive valued
only for the temperature range $55\,^{\circ}{\rm C}$–$71\,^{\circ}{\rm C}$ and
negative valued at higher temperatures, and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{b}\right\\}$ is positive valued
only for the temperature range $55\,^{\circ}{\rm C}$–$65\,^{\circ}{\rm C}$ and
negative valued at higher temperatures. That is, the CTF is a dissipative
dielectric material for $\lambda_{\scriptscriptstyle 0}=$ 800 nm across the
entire temperature range considered, it is a dissipative dielectric material
only at low temperatures and becomes a hyperbolic material [24] at higher
temperatures for $\lambda_{\scriptscriptstyle 0}=$ 1550 nm. Furthermore, the
temperature at which the dissipative dielectric material/hyperbolic material
transition occurs depends upon whether the material is being heated or cooled.

Now let us explore the effect of varying $f$. In order to appreciate more
easily the effects of the variation on the anisotropy of the CTF,
$\lambda_{\scriptscriptstyle 0}$ was fixed at 1550 nm. Since there can be no
anisotropy for $f\in\left\\{0,1\right\\}$, we confined our attention to mid-
range values of $f$. The computations for Fig. 4 were repeated for
$f\in\left\\{0.45,0.66,0.7\right\\}$ and the results plotted in Fig. 5. As
regards anisotropy and hysteresis, the results for
$f\in\left\\{0.45,0.66,0.7\right\\}$ displayed in Fig. 5 are qualitatively
similar to those for $f=0.6$ displayed in Fig. 4, but certain trends are
apparent: the anisotropy of the CTF becomes substantially more pronounced as
the value of $f$ increases and the anisotropy of the hysteresis, but not of
the relative hysteresis, becomes substantially more pronounced as the value of
$f$ increases. In addition, there are important differences in the natures of
the CTFs for each of $f=0.45$, $0.6$, $0.66$, and $0.7$:

  * (i)

for $f=0.45$, $\mbox{Re}\left\\{\varepsilon_{a}\right\\}>0$,
$\mbox{Re}\left\\{\varepsilon_{b}\right\\}>0$, and
$\mbox{Re}\left\\{\varepsilon_{c}\right\\}>0$ across the entire temperature
range considered;

  * (ii)

for $f=0.6$, as discussed earlier in connection with Fig. 4,
$\mbox{Re}\left\\{\varepsilon_{a}\right\\}>0$ and
$\mbox{Re}\left\\{\varepsilon_{c}\right\\}>0$ across the entire temperature
range considered but $\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{b}\right\\}>0$
only for the temperature range $55\,^{\circ}{\rm C}$–$71\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{b}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$65\,^{\circ}{\rm C}$;

  * (iii)

for $f=0.66$, $\mbox{Re}\left\\{\varepsilon_{a}\right\\}>0$ across the entire
temperature range considered but
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{b}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$70.5\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{b}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$64\,^{\circ}{\rm C}$; and
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{c}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$72\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{c}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$66\,^{\circ}{\rm C}$;

  * (iv)

for $f=0.7$, $\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{a}\right\\}>0$ only
for the temperature range $55\,^{\circ}{\rm C}$–$71\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{a}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$65.5\,^{\circ}{\rm C}$;
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{b}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$70\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{b}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$64\,^{\circ}{\rm C}$; and
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{c}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$70.5\,^{\circ}{\rm C}$ and
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{c}\right\\}>0$ only for the
temperature range $55\,^{\circ}{\rm C}$–$64.5\,^{\circ}{\rm C}$.

In other words, the CTF is always a dissipative dielectric material at
temperatures close to $55\,^{\circ}{\rm C}$, for all values of $f$ considered;
but at temperatures close to $75\,^{\circ}{\rm C}$ the CTF can be (a) a
dissipative dielectric material, (b) a hyperbolic material, or (c) a metal-
like material, depending on the value of $f$. Furthermore, the temperature at
which the dissipative dielectric material/hyperbolic material transition or
the hyperbolic material/metal-like material transition occurs depends upon
whether the material is being heated or cooled.

Lastly let us explore the effect of varying $\gamma_{\rm{b}}$. As for Fig. 5,
$\lambda_{\scriptscriptstyle 0}$ was fixed at 1550 nm in order to appreciate
more easily the effects of the variation on the anisotropy of the CTF. The
computations for Fig. 4 were repeated for
$\gamma_{\rm{b}}\in\left\\{1,5\right\\}$ and the results plotted in Fig. 6.
The columns of the CTF are circular in cross section when $\gamma_{b}=1$ but
like blades when $\gamma_{b}=5$ [25]. As regards anisotropy and hysteresis,
the results for $\gamma_{\rm{b}}\in\left\\{1,5\right\\}$ displayed in Fig. 6
are qualitatively similar to those for $\gamma_{\rm{b}}=1.5$ displayed in Fig.
4, but there are certain notable features: When $\gamma_{\rm{b}}=1$,
$\varepsilon_{a}=\varepsilon_{c}$ for both heating and cooling phases;
correspondingly, the CTF is a uniaxial material with distinguished axis
parallel to $\underline{u}_{\tau}$. When $\gamma_{\rm{b}}=5$,
$\varepsilon_{b}\approx\varepsilon_{\rm{c}}$ for both heating and cooling
phases; correspondingly, the CTF is very close to being a uniaxial material
with distinguished axis parallel to $\underline{u}_{\rm n}$. Only relatively
modest variations in anisotropy and hysteresis arise as $\gamma_{\rm{b}}$
increases from 1 to 5, as compared to the variations induced by varying $f$.

##  4 Closing remarks

The Bruggeman homogenization formalism was used to numerically investigate the
anisotropic dielectric properties of a CTF made from VO2. In the visible and
near-infrared spectral regimes, the CTF is electromagnetically equivalent to a
homogeneous orthorhombic material, characterized by the relative permittivity
dyadic $\underline{\underline{\varepsilon}}_{\,\rm{CTF}}$ with eigenvalues
$\left\\{\varepsilon_{a},\varepsilon_{b},\varepsilon_{c}\right\\}$. Over the
temperature range $58\,^{\circ}{\rm C}$–$72\,^{\circ}{\rm C}$,
$\left\\{\varepsilon_{a},\varepsilon_{b},\varepsilon_{c}\right\\}$ are highly
sensitive to temperature, and vary according to whether the CTF is being
heated or cooled. The anisotropy revealed through
$\left\\{\varepsilon_{a},\varepsilon_{b},\varepsilon_{c}\right\\}$, and the
anisotropy of the associated hysteresis, were investigated as a function of
temperature for CTFs of different porosities and of different column shapes.
The CTF is a dissipative dielectric material that exhibits temperature-
dependent anisotropy and anisotropic hysteresis, for
$\lambda_{\scriptscriptstyle 0}=800$ nm. In contrast, for
$\lambda_{\scriptscriptstyle 0}=1550$ nm, the CTF can be either a dissipative
dielectric material, a hyperbolic material or a metal-like material, depending
on the temperature and on the porosity of the CTF.

The ability to modulate the anisotropy of a CTF made from VO2 with temperature
is an attractive feature when considering directional control and tuning for
applications involving reconfigurable and multifunctional devices.
Furthermore, the ability to modulate anisotropic thermal hysteresis may be an
attractive feature for directional control in optical/infrared switching
applications. And the prospect of thermally controlling whether a CTF is a
dissipative dielectric material, a hyperbolic material or a metal-like
material should present additional opportunities [26, 27].

Acknowledgments: TGM was supported by EPSRC (grant number EP/V046322/1). AL
was supported by the US National Science Foundation (grant number DMS-1619901)
as well as the Charles Godfrey Binder Endowment at Penn State.

## References

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Figure 1:  Schematic representation of a CTF with columns inclined at angle
$\chi$ to the planar substrate.

Figure 2:  Real and imaginary parts of
$\varepsilon^{\rm{heat,cool}}_{\text{VO}_{2}}$ plotted against temperature
(derived from Fig. 4 of Ref. 13), for $\lambda_{\scriptscriptstyle
0}\in\left\\{800,1550\right\\}$ nm. Also plotted against temperature are
$\Delta^{\text{Re}}_{\text{VO}_{2}}$,
$\hat{\Delta}^{\text{Re}}_{\text{VO}_{2}}$,
$\Delta^{\text{Im}}_{\text{VO}_{2}}$, and
$\hat{\Delta}^{\text{Im}}_{\text{VO}_{2}}$. Key:
$\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}$: thick
green solid curve;
$\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}$: thick
blue dashed curve;
$\mbox{Im}\left\\{\varepsilon^{\rm{heat}}_{\text{VO}_{2}}\right\\}$: thin red
solid curve;
$\mbox{Im}\left\\{\varepsilon^{\rm{cool}}_{\text{VO}_{2}}\right\\}$: thin
black dashed curve; $\Delta^{\text{Re}}_{\text{VO}_{2}}$: thick black dashed
curve; $\hat{\Delta}^{\text{Re}}_{\text{VO}_{2}}$: thick green dashed curve;
$\Delta^{\text{Im}}_{\text{VO}_{2}}$ : thin black solid curve;
$\hat{\Delta}^{\text{Im}}_{\text{VO}_{2}}$: thin green solid curve;

Figure 3:  Real and imaginary parts of the CTF’s relative permittivity scalars
$\varepsilon^{\rm{heat,cool}}_{a,b,c}$ estimated using the Bruggeman
homogenization formalism plotted against temperature for
$\lambda_{\scriptscriptstyle 0}=800$ nm, with $\gamma_{\tau}=15$,
$\gamma_{b}=1.5$, and $f=0.6$. Also plotted against temperature are
$\Delta^{\text{Re}}_{a,b,c}$, $\Delta^{\text{Im}}_{a,b,c}$,
$\hat{\Delta}^{\text{Re}}_{a,b,c}$ and $\hat{\Delta}^{\text{Im}}_{a,b,c}$.
Key: $\mbox{Re}\left\\{\varepsilon^{\rm{heat}}_{a,b,c}\right\\}$: thick green
solid curve; $\mbox{Re}\left\\{\varepsilon^{\rm{cool}}_{a,b,c}\right\\}$:
thick blue dashed curve;
$\mbox{Im}\left\\{\varepsilon^{\rm{heat}}_{a,b,c}\right\\}$: thin red solid
curve; $\mbox{Im}\left\\{\varepsilon^{\rm{cool}}_{a,b,c}\right\\}$: thin black
dashed curve; $\Delta^{\text{Re}}_{a}$, $\hat{\Delta}^{\text{Re}}_{a}$: thick
blue broken dashed curve; $\Delta^{\text{Re}}_{b}$,
$\hat{\Delta}^{\text{Re}}_{b}$: thick black solid curve;
$\Delta^{\text{Re}}_{c}$, $\hat{\Delta}^{\text{Re}}_{c}$: thick green dashed
curve; $\Delta^{\text{Im}}_{a}$, $\hat{\Delta}^{\text{Im}}_{a}$: thin blue
broken dashed curve; $\Delta^{\text{Im}}_{b}$, $\hat{\Delta}^{\text{Im}}_{b}$:
thin black solid curve; $\Delta^{\text{Im}}_{c}$,
$\hat{\Delta}^{\text{Im}}_{c}$: thin green dashed curve.

Figure 4:  As Fig. 3 but for $\lambda_{\scriptscriptstyle 0}=1550$ nm.

Figure 5:  As Fig. 4 but for $f\in\left\\{0.45,0.66,0.7\right\\}$.

Figure 6:  As Fig. 4 but for $\gamma_{b}\in\left\\{1,5\right\\}$.