IPM/P-2024/17

Entanglement in Lifshitz Fermion Theories

Mohammad Javad Vaslia,b, Komeil Babaei Velnia, M. Reza Mohammadi Mozaffara and
Ali Mollabashib

a Department of Physics, University of Guilan, P.O. Box 41335-1914, Rasht,
Iran
b School of Physics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5531, Tehran, Iran

E-mails: vasli@phd.guilan.ac.ir,{babaeivelni<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the static entanglement structure in (1+1)-dimensional free Dirac-
fermion theory with Lifshitz symmetry and arbitrary integer dynamical critical
exponent. This model is different from the one introduced in [Hartmann et al.,
SciPost Phys. 11, no.2, 031 (2021)] due to a proper treatment of the square
Laplace operator. Dirac fermion Lifshitz theory is local as opposed to its
scalar counterpart which strongly affects its entanglement structure. We show
that there is quantum entanglement across arbitrary subregions in various pure
(including the vacuum) and mixed states of this theory for arbitrary integer
values of the dynamical critical exponent. Our numerical investigations show
that quantum entanglement in this theory is tightly bounded from above. Such a
bound and other physical properties of quantum entanglement are carefully
explained from the correlation structure in these theories. A generalization
to (2+1)-dimensions where the entanglement structure is seriously different is
addressed.

###### Contents

  1. 1 Introduction
  2. 2 Fermionic Lifshitz Theory
    1. 2.1 $\mathcal{L}_{\mathrm{I}}$ and $\mathcal{L}_{\mathrm{II}}$ Theories
    2. 2.2 Lattice Regularization
    3. 2.3 $\mathcal{L}_{\mathrm{I}}$ versus $\mathcal{L}_{\mathrm{II}}$ at the Fixed Point
    4. 2.4 Two-point Function: Symmetry Argument
  3. 3 Numerical Results
    1. 3.1 Entanglement Entropy
    2. 3.2 Logarithmic Negativity
    3. 3.3 Entropic $c$-function
  4. 4 Conclusions and discussions
  5. A Lifshitz fermionic model in $(2+1)$-dimensions

##  1 Introduction

With regard to the question about bipartite entanglement structure in quantum
field theories and many-body systems, we study unknown features of bipartite
entanglement in free fermionic Lifshitz theories. Lifshitz theories are non-
relativistic theories that are invariant under

$t\to\lambda^{z}t\;\;\;\;\;\;\;,\;\;\;\;\;\;\;\vec{x}\to\lambda\vec{x}\;,$ (1)

where $\lambda>0$ and $z\neq 1$ ($z=1$ corresponds to relativistic theories).

Lifshitz symmetry algebra encompasses generators for the aforementioned
scaling symmetry together with rotations and translations (see e.g. [1]).
Compared to relativistic scale invariance, there is no counterpart for the
boost generator in Lifshitz algebra. Hence, in contrast with the relativistic
case in (1+1)-dimensions (2$d$) (see e.g. [2]), the symmetry is not generally
powerful enough to lead to universal entanglement structures. However people
have utilized various methods to study the bipartite entanglement structure in
such theories in (2+1)-dimensions (3$d$) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14,
13], in 2$d$ [15, 11, 16, 12, 17, 14, 18, 19], as well as generalizations to
higher dimensions [20].111See also [21, 22, 24, 25, 23] for related works.

Specifically, entanglement entropy of the ground state of a proposed free
Dirac-Lifshitz fermion theory has been partially studied in [18]. The proposed
theory in [18] leads to a product state as the vacuum state of Lifshitz
fermions for even values of the dynamical critical exponent222For a similar
symmetry-based argument in Galilean field theories see [26]., while for odd
values of $z$ they find $z$-independent entanglement entropy inherited from
the $z$-independence of the two-point correlation functions.

In this paper, we carefully consider a revisited version for free Dirac-
Lifshitz fermion theory (see Eq. (10))333This theory has been previously
introduced in [19], though the lattice regularization in this paper is
different. which includes a negative momentum mode corresponding to all
positive momentum modes for any integer value of the dynamical critical
exponent. Indeed, for odd values of $z$, this revisited theory coincides with
that in [18]. On the other hand, for even values of $z$, these theories are
drastically different in the sense that the Fermi surface is empty in the one
introduced in [18] but is populated in the revisited version. As a result, we
find non-trivial two-point functions and hence quantum and classical
correlations in the vacuum state. We study the entanglement structure in
certain pure state (basically the vacuum state) and mixed states in this
family of theories.

The rest of the paper is organized as follows: In the next subsection we
review our calculation method for entanglement measures stressing on the less
well-known part of the numerical method corresponding to logarithmic
negativity for fermionic Gaussian states. In section 2 we introduce the
revisited free Dirac-Lifshitz fermion theory in 2$d$ and discuss the structure
of equal-time two-point functions. In section 3 we present our numerical
results for entanglement entropy, logarithmic negativity, and address
generalizations to higher dimensions which seriously changes the entanglement
structure. We also discuss about entropic $c$-theorem counterparts for these
non-relativistic theories. The last section is devoted to concluding remarks.

### Technical Preliminaries

Through this paper we will compute different entanglement and correlation
measures employing the correlator method, sometimes referred to as the
fermionic covariance matrix formalism [27, 28, 29, 30, 31]. The first measure
is entanglement entropy associated to a subregion $A$ which is defined as the
von Neumann entropy $S_{A}=-\mathrm{Tr}\left(\rho_{A}\log\rho_{A}\right)$
where $\rho_{A}$ is the reduced density matrix of subregion $A$ obtained by
tracing out the degrees of freedom of the complement of this region. Within
the correlator method formalism this quantity can be computed as

$\displaystyle
S_{A}=-\sum_{n=0}^{n_{A}}(1-\nu_{n})\log(1-\nu_{n})+\nu_{n}\log\nu_{n},$ (2)

where $n_{A}$ is the number of lattice sites included in region $A$ and
$\nu_{n}$’s are eigenvalues of the fermionic two point function

$\displaystyle C_{rs}=\langle\Psi_{r}\Psi^{\dagger}_{s}\rangle,$ (3)

and $\Psi_{r}$ denotes the fermionic field at site $r$. Moreover, all
quantities which are defined as combinations of von Neumann entropies such as
mutual information between two subregions $A$ and $B$ defined as

$\displaystyle I(A:B)=S_{A}+S_{B}-S_{A\cup B},$ (4)

can be computed using Eq. (2).

Although the von Neumann entropy is a unique measure for quantum correlations
in pure states, the story is complicated for mixed states. It has been shown
by [32, 33] that the negative eigenvalues of the partial transposed density
matrix measure quantum correlations for mixed states. Following this
observation negativities were introduced as computable measures for quantum
correlations for mixed states [34]. Later it has been shown that this measure
is not ideal but a good measure [35, 36] though widely studied in the
literature.

We are interested in logarithmic negativity that is defined as
$\mathcal{E}(A:B)=\log||\rho^{T_{A}}_{AB}||$ where $\rho_{AB}^{T_{A}}$ is the
partial transposed density matrix with respect to $A$ and $||\bullet||$
denotes the trace norm. In conformal field theories that the case of $z=1$
would be an example of, the replica trick and conformal symmetry has been used
to compute this quantity for certain configurations, namely a single interval
and two adjacent intervals in the vacuum state in [39, 37] (for extension to
massive case see [38]) and a single interval in finite temperature state in
[40].

In the following, our focus is on Gaussian states in free fermion theories.
Let us mention that for bosonic Gaussian states the partial transposition has
a simple realization in terms of the covariance matrix [41] which leads to
straightforward calculation of logarithmic negativity in lattice bosonic
models (see e.g. [39, 42, 12]). In contrast, for fermionic Gaussian states a
similar calculation has been a long standing challenge since the partial
transposition of a Gaussian density matrix does not preserve the Gaussianity
of the state [43]. There has been several attempts to get around this problem
[43, 44, 45, 46, 47]444Prior to these developments, Monte-Carlo and tensor
network methods have been utilized to calculate (logarithmic) negativity in
one dimensional Ising model [48, 49, 50]..

The authors of [47] suggested a path integral estimation to calculate
negativities for free fermions. Later on the authors of [51, 53] introduced a
method which can be thought of as a sharp justification for the estimation of
[47], based on concerns about the separability criteria for fermionic states.
It has been shown in [51] that taking the fermion number conservation into
account leads to a slightly different separability criteria for fermionic
states.555We thank Ken Shiozaki and Hassan Shapourian for fruitful discussions
on this point. From the separability criteria point of view, this leads to a
different partial transposition operation which is denoted by partial time-
reversal transpose or fermionic partial transpose (for details see [51, 52]).
Here we use this definition to calculate logarithmic negativity for fermionic
Gaussian states in our quadratic fermionic theories.

Suppose $A=A_{1}\cup A_{2}$ and we are considering the fermion partial
transpose over $A_{1}(A_{2})$. The recipe for this calculation is formulated
in terms of the covariance matrix given by

$\displaystyle\Gamma=\mathbf{1}-2C=\Gamma=\begin{pmatrix}\Gamma^{11}&\Gamma^{12}\\\
\Gamma^{21}&\Gamma^{22}\\\ \end{pmatrix}.$ (5)

The fermionic partial transposed correlation matrix $C_{\Xi}$ can be evaluated
as

$\displaystyle
C_{\Xi}=\frac{1}{2}(\mathbf{1}-(\mathbf{1}+\Gamma_{+}\Gamma_{-})^{-1}(\Gamma_{+}+\Gamma_{-})),$
(6)

where $\Gamma_{+}$ and $\Gamma_{-}$ are defined by

$\displaystyle\Gamma_{\pm}=\begin{pmatrix}-\Gamma^{11}&\pm i\Gamma^{12}\\\ \pm
i\Gamma^{21}&\Gamma^{22}\\\ \end{pmatrix}.$ (7)

Finally, the logarithmic negativity can be computed as

$\displaystyle\mathcal{E}(A_{1}:A_{2})=\sum_{n}\ln(\xi^{\frac{1}{2}}_{j}+(1-\xi_{j})^{\frac{1}{2}})+\frac{1}{2}\sum_{j}\ln(\zeta_{j}^{2}+(1-\zeta_{j})^{2}),$
(8)

where $\zeta_{j}$ and $\xi_{j}$ are eigenvalues of $C$ and $C_{\Xi}$
respectively.

##  2 Fermionic Lifshitz Theory

In this section we will introduce a revisited free Dirac-Lifshitz fermion
theory in 2$d$ and investigate its discrete version using a lattice
regularization. We also compare our model with the previous fermionic theories
and study the structure of equal-time two-point functions in this setup.

###  2.1 $\mathcal{L}_{\mathrm{I}}$ and $\mathcal{L}_{\mathrm{II}}$ Theories

A 2$d$ free fermionic version of Lifshitz theory for generic integer values of
$z$ has been introduced in [18] and later on a similar family of theories for
generic $d$ was introduced in [19]. Hereafter we will denote these two family
of theories with index I and II respectively. Theory II is well-defined for
odd values of $z$ while for even $z$ careful consideration is needed. Our
focus in this section and in most of this paper is on 2$d$ where the extension
of theory II to even values of $z$ is straightforward. The Lagransian density
for these theories are given by

$\displaystyle\mathcal{L}_{\mathrm{I}}$
$\displaystyle=\bar{\Psi}(i\gamma^{0}\partial_{0}+\gamma^{1}(i\partial_{1})^{z}-m^{z})\Psi,$
(9) $\displaystyle\mathcal{L}_{\mathrm{II}}$
$\displaystyle=\bar{\Psi}(i\gamma^{0}\partial_{0}+i\gamma^{1}T^{z-1}\partial_{1}-m^{z})\Psi,$
(10)

where $T=\sqrt{-\partial_{1}\partial^{1}}$.

Both $\mathcal{L}_{\mathrm{I}}$ and $\mathcal{L}_{\mathrm{II}}$ lead to the
desired dispersion relation for Lifshitz theories
$\omega(k)=\sqrt{k^{2z}+m^{2z}}$. Furthermore, as it has been addressed in
[18], $\mathcal{L}_{\mathrm{I}}$ leads to a strange property for the vacuum
state of theories corresponding to even values of $z$, namely the vacuum state
in this case turns out to be a product state in the position basis. This leads
for instance to vanishing entanglement between spatial subregions in these
theories. To the best of our knowledge there is no physical justification for
such a feature and we believe that this odd behavior arises from the asymmetry
between odd and even values of $z$ in (9), namely for even values of $z$ the
imaginary unit in the spatial derivative vanishes. One important massage of
this paper is a solution to this problem via a careful treatment of the square
root of the square Laplace operator, as is introduced in (10).

The $T$ operator introduced in [54] has the following properties

$\displaystyle T\partial_{i}$ $\displaystyle=\partial_{i}T,$ (11)
$\displaystyle T^{2n}$
$\displaystyle=\left(\sqrt{-\partial_{1}^{2}}\right)^{2n}=\left(-\partial_{1}^{2}\right)^{n},$
(12) $\displaystyle T^{2n+1}$
$\displaystyle=\left(\sqrt{-\partial_{1}^{2}}\right)^{2n+1}=\left(\sqrt{-\partial_{1}^{2}}\right)^{2n}\left(\sqrt{-\partial_{1}^{2}}\right)=\left(-\partial_{1}^{2}\right)^{n}T.$
(13)

Moreover, this operator acts in momentum space as

$\displaystyle T^{2n}\exp(ikx)$ $\displaystyle=k^{2n}\exp(ikx),$ (14)
$\displaystyle T^{n}\exp(ikx)$ $\displaystyle=|k|^{n}\exp(ikx).$ (15)

In what follows, for our 2$d$ theory we will use the following choice for the
gamma matrices

$\displaystyle\gamma^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\gamma^{1}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\qquad\gamma^{2}\equiv\gamma^{0}\gamma^{1}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (16)

For future convenience, the Hamiltonian density corresponding to
$\mathcal{L}_{\mathrm{II}}$ is found to be

$\displaystyle\mathcal{H}_{\mathrm{II}}$
$\displaystyle=\pi_{\Psi}\dot{\Psi}-\mathcal{L}=-\bar{\Psi}(i\gamma^{1}T^{z-1}\partial_{1}-m^{z})\Psi.$
(17)

Note that we will suppress the index II in the following.

###  2.2 Lattice Regularization

In order to find the discretized version of (10) we need to express higher
derivative terms on a lattice. To do so, we choose the following definition
for the first derivative, which preserves the hermiticity of the lattice
Hamiltonian

$\partial_{1}\Psi_{n}=\frac{\Psi_{n+1}-\Psi_{n-1}}{2}.$ (18)

Due to the presence of the $T$ operator for even values of $z$, the definition
of this operator on the lattice is challenging. To overcome this challenge, we
use momentum space representation which can be obtained by using the Fourier
transform as $\Psi_{n}=\frac{1}{L}\sum_{p=0}^{L}c_{k}e^{-ikn}$ with
$k=\frac{2\pi}{L}p$ and $p\in\\{0,\cdots,L-1\\}$. Hence, we can rewrite the
corresponding expression for the first derivative on the lattice as

$\displaystyle\begin{split}\partial_{1}\Psi_{n}=\frac{\Psi_{n+1}-\Psi_{n-1}}{2}&=\frac{1}{2}\left(c_{k}(\omega)e^{i((n+1)k+\omega
k)}-c_{k}(\omega)e^{i((n-1)k+\omega k)}\right)\\\
&=\frac{1}{2}\left(e^{ik}-e^{-ik}\right)\Psi_{n}=i\sin(k)\Psi_{n}\;.\end{split}$
(19)

Similarly for the second derivative we have

$\displaystyle\begin{split}\partial_{1}(\partial_{1}\Psi_{n})=\frac{\partial_{1}\Psi_{n+1}}{2}-\frac{\partial\Psi_{n-1}}{2}&=\frac{\Psi_{n+2}-\Psi_{n}}{4}-\frac{\Psi_{n}-\Psi_{n-2}}{4}\\\
&=\left(\frac{1}{4}\left(e^{2ik}+e^{-2ik}\right)-\frac{1}{2}\right)\Psi_{n}=-\sin^{2}(k)\Psi_{n}.\end{split}$
(20)

In general, the $z$-th derivative yields

$(\partial_{1})^{z}\Psi_{n}=i^{-z}(-\sin(k))^{z}\Psi_{n}.$ (21)

Using the above relations, we can simply find the action of $T$ operator in
momentum space as follows

$\displaystyle T^{2}\Psi_{n}=-\partial_{1}^{2}\Psi_{n}=\sin^{2}(k)\Psi_{n},$
$\displaystyle T\Psi_{n}=|\sin(k)|\Psi_{n}.$

Now it is straightforward to find the corresponding expression for the
Hamiltonian on the lattice

$H_{\mathrm{II}}=\sum_{n}\Psi_{n}^{\dagger}\left(-\gamma^{2}f_{\mathrm{II}}(k)+m^{z}\gamma^{0}\right)\Psi_{n},$
(22)

where

$\displaystyle f_{\mathrm{II,odd}}(k)=-(\sin(k))^{z},\qquad\qquad
f_{\mathrm{II,even}}(k)=-|\sin(k)|(-\sin(k))^{z-1}.$ (23)

The hermiticity of the Hamiltonian is guaranteed by
${\gamma^{0}}^{\dagger}=\gamma^{0}$, ${\gamma^{2}}^{\dagger}=\gamma^{2}$, and
$f_{\mathrm{II}}(k)$ being a real function. Next, we diagonalize this model by
treating the two components of the fermionic field separately as
$\psi_{k}=\left(u_{k},d_{k}\right)^{T}.$ In this representation, the
Hamiltonian takes the following form

$\displaystyle H_{\mathrm{II}}=$ $\displaystyle\int
dk\left((d_{k}^{\dagger}d_{k}-u_{k}^{\dagger}u_{k})f_{\mathrm{II}}(k)+m^{z}(u_{k}^{\dagger}d_{k}+d_{k}^{\dagger}u_{k})\right).$
(24)

We consider the following Bogoliubov transformations

$u_{k}=\cos\frac{\theta_{k}}{2}b_{k}+i\sin\frac{\theta_{k}}{2}b^{\dagger}_{-k},\qquad\qquad
d_{k}=\sin\frac{\theta_{k}}{2}b_{k}-i\cos\frac{\theta_{k}}{2}b_{-k}^{\dagger},$
(25)

with

$\tan\theta_{k}=\frac{m^{z}}{-f_{\mathrm{II}}(k)},\qquad\qquad\cos\theta_{k}=-\frac{f_{\mathrm{II}}(k)}{\sqrt{m^{2z}+f_{\mathrm{II}}(k)^{2}}},$
(26)

where $\\{b_{k}^{\dagger},b_{k^{\prime}}\\}=\delta_{kk^{\prime}}$. Finally,
the diagonal Hamiltonian can find as below

$H_{\mathrm{II}}=\int
dk\sqrt{m^{2z}+f_{\mathrm{II}}(k)^{2}}\left(b_{k}^{\dagger}b_{k}+b_{-k}^{\dagger}b_{-k}\right).$
(27)

It is worth to note that this discretization introduced in Eq. (18) is
different from what has been introduced in reference [19]. In that reference
forward difference operator has been used for all derivative orders together
with the standard procedure to end up with a Hermitian Hamiltonian. The
diagonalization follows from the same aforementioned steps by replacing
$f_{\mathrm{II}}$ by $f^{*}_{\mathrm{II}}$ where

$f^{*}_{\mathrm{II}}(k)=\left(2\sin\frac{k}{2}\right)^{z}\cos\frac{zk}{2}.$
(28)

Let us also mention that for the case of (9), the authors of [18] have used
the discretization Eq. (18). The Hamiltonian has the same form as (27) with
$f_{\mathrm{II}}(k)$ replaced by

$f_{\mathrm{I}}(k)=(-\sin(k))^{z}\,.$ (29)

Note that $f_{\mathrm{I}}$, $f_{\mathrm{II}}$, and $f^{*}_{\mathrm{II}}$ all
lead to the desired Lifshitz dispersion relation,
$\omega(k)=\sqrt{k^{2z}+m^{2z}}$, in the continuum limit.

Since we are interested in the two point correlators to compute entanglement
measures, hereafter the expressions with $f(k)$ are valid for all three
$f_{\mathrm{I}}(k)$ corresponding to $\mathcal{L}_{\mathrm{I}}$, and
$f_{\mathrm{II}}(k)$, $f^{*}_{\mathrm{II}}(k)$ corresponding to different
discretizations of $\mathcal{L}_{\mathrm{II}}$.

### Two-point Function

It is relatively straightforward to find the fermionic field’s two-point
function as follows

$\displaystyle
C_{rs}=\langle\Psi_{s}\Psi_{r}^{\dagger}\rangle=\frac{\delta_{rs}}{2}\mathbb{1}+\frac{1}{4\pi}\int_{-\pi}^{\pi}\frac{dk}{\omega_{k}}e^{ik(r-s)}\begin{pmatrix}-f(k)&m^{z}\\\
m^{z}&f(k)\end{pmatrix}.$ (30)

In the following we will mainly investigate the behavior of entanglement
measures for $f_{\mathrm{II}}(k)$, while we present and compare the results
with $f_{\mathrm{I}}(k)$ and $f^{*}_{\mathrm{II}}(k)$ to highlight the
corresponding physical features.

Moreover, we also examine the behavior of logarithmic negativity for finite
temperature state where the corresponding two-point function is given as
follows

$\displaystyle
C_{rs}=\frac{\delta_{rs}}{2}\mathbb{1}+\frac{1}{4\pi}\int_{-\pi}^{\pi}\frac{dk}{\omega_{k}}e^{ik(r-s)}\begin{pmatrix}-f(k)&m^{z}\\\
m^{z}&f(k)\end{pmatrix}\tanh{\frac{\beta\omega_{k}}{2}}.$ (31)

As we will see studying the above expression in the low and high temperature
limits will be illuminating. Indeed, a simple calculation gives the two-point
function in the high temperature limit for odd values of $z$ as follows

$\displaystyle
C^{\mathrm{odd}}_{rs}=\frac{\delta_{rs}}{2}\mathbb{1}+\frac{\beta}{2}\frac{1}{4\pi}\frac{2i^{z}z!\sin(\pi(r-s))}{2^{z}(r-s-z)\left(\frac{1}{2}(r-s-z+2)\right)_{z}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}+\cdots$ (32)

and for the even values of $z$

$\displaystyle
C^{\mathrm{even}}_{rs}=\frac{\delta_{rs}}{2}\mathbb{1}+\frac{\beta}{2}\frac{1}{4\pi}\frac{2i^{z+1}z!(\cos(\pi(r-s))-1)}{2^{z}(r-s-z)\left(\frac{1}{2}(r-s-z+2)\right)_{z}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}+\cdots$ (33)

where $(\bullet)_{x}$ stands for the Pochhammer symbol.

We are now basically equipped with what we need to study the entanglement
measures which will come in section 3. Before getting into our numerical
results in the next subsections we first further analyse the difference
between $\mathcal{L}_{\mathrm{I}}$ and $\mathcal{L}_{\mathrm{II}}$ and proceed
by presenting a symmetry based argument which will be used to justify some of
our results for Lifshitz fermions.

###  2.3 $\mathcal{L}_{\mathrm{I}}$ versus $\mathcal{L}_{\mathrm{II}}$ at the
Fixed Point

Before examining the full $z$ dependence of entanglement measures, we would
like to study their specific behaviors in the massless regime where the theory
respects scaling symmetry. As we will see this study plays an important role
in our analysis in what follows. Indeed $f_{\mathrm{I}}(k)$,
$f_{\mathrm{II}}(k)$, and $f^{*}_{\mathrm{II}}(k)$ have distinct behaviors in
the massless regime which reflect the main features of each theory and the
corresponding regularization.

Eq. (30) for $m=0$ simplifies to

$\displaystyle
C_{rs}=\frac{\delta_{rs}}{2}\mathbb{1}+\frac{1}{4\pi}\int_{-\pi}^{\pi}dk\;e^{ik(r-s)}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\frac{f(k)}{|f(k)|}.$ (34)

We can see that, not the exact value of $f(k)$, but its sign between the zeros
of it matters for this correlation function. The authors of [18] have shown
that for the case of $f_{\mathrm{I}}(k)$, for odd values of $z$ this sign
changes at $k=n\pi$, thus in this case a straightforward calculation gives

$\displaystyle
C_{rs}=\frac{1}{2}\begin{pmatrix}\delta_{rs}-\frac{i\left(e^{i\pi(r-s)}-1\right)}{2\pi(r-s)}&0\\\
0&\delta_{rs}+\frac{i\left(e^{i\pi(r-s)}-1\right)}{2\pi(r-s)}\end{pmatrix}\,.$
(35)

On the other hand, for even values of $z$ the corresponding two-point function
simplifies as follows

$\displaystyle C_{rs}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}\delta_{rs}\,.$
(36)

Interestingly, the eigenvalues of the above matrix are either 0 or 1 and hence
using Eq. (2), the entanglement entropy literally vanishes for even values of
the dynamical exponent. Moreover, for $f_{\mathrm{II}}(k)$, using Eq. (30) we
see that the result for the odd and even values of $z$ are the same as Eq.
(35). In figure 1, we compare the behavior of $f(k)$ for
$\mathcal{L}_{\mathrm{I}}$ and $\mathcal{L}_{\mathrm{II}}$ where it manifestly
shows that the Fermi surface ($f(k)=0$) is empty for even values of $z$ for
$\mathcal{L}_{\mathrm{I}}$ but quite non-trivial for
$\mathcal{L}_{\mathrm{II}}$.666We thank Masaki Oshikawa for bringing this
point to our attention. This structure manifestly shows why
$\mathcal{L}_{\mathrm{I}}$ leads to a product state for the vacuum state of
even values of $z$, while there is no counterpart for $+k$ modes inside the
Fermi surface to be entangled with. It is worth noting that although this
result correspond to the lattice regularized theories, exactly the same
structure holds in the continuum limit of $\mathcal{L}_{\mathrm{I}}$ and
$\mathcal{L}_{\mathrm{II}}$.

Figure 1: The behavior of $f_{\mathrm{I}}(k)$ (left) and $f_{\mathrm{II}}(k)$
(right) for several values of the dynamical exponent. For odd $z$ both models
have the same behavior and so is the two-point functions. $f_{\mathrm{I}}(k)$
is always positive for even values of $z$ leading to an empty Fermi surface.
Moreover the zeros of $f_{\mathrm{II}}(k)$ are the same for all values of $z$,
and hence the two-point function in the massless case is $z$-independent.

An important comment is about the complicated structure of
$f^{*}_{\mathrm{II}}(k)$. Unlike $f_{\mathrm{I}}(k)$ and $f_{\mathrm{II}}(k)$
which have a single zero at $k=0$, $f^{*}_{\mathrm{II}}(k)$ accommodates
$2(z-1)$ extra zeros. As a result the this regularization is distinct from
$f_{\mathrm{I}}(k)$ and $f_{\mathrm{II}}$ by being explicitly $z$-dependent
and not being bounded by the $z=1$ result. In this case the Fermi surface is
also non-trivial and the vacuum state is an entangled state. We will address
the implications of this point in section 3.

Our results show that for integer values of $z$ the two-point functions are
all the same as the relativistic case. This structure is very different from
the similar structure for Lifshitz invariant scalar theories [11]. As we show
in the next section, this structure also interestingly affects the
entanglement structure. In the next section we try to further investigate the
physics behind this specific behavior of the fermionic two-point function from
a symmetry viewpoint.

###  2.4 Two-point Function: Symmetry Argument

It is a well-known fact that at a field theory fixed-point the typical scaling
is given by Lifshitz scaling, i.e., Eq. (1). The goal of this subsection is to
show that using this scaling symmetry with a reasonable assumption fixes the
form of the two-point function of a fermionic field in Lifshitz theory.

Various features of nonrelativistic symmetries have been studied in the
literature, e.g., see [57, 58, 59, 1, 60]. At a Lifshitz fixed point the
symmetry algebra of the field theory will contain the generators of rotations
$M_{ij}$, spatial translations $P_{i}$, time translation $H$ and dilatation
$D$. In particular in 2$d$ we have

$\displaystyle H=\partial_{t},\qquad P=\partial_{x},\qquad
D=-zt\partial_{t}-x\partial_{x}\,.$ (37)

The corresponding nontrivial commutation relations are

$\displaystyle[D,P]=P,\hskip 56.9055pt[D,H]=zH.$ (38)

Moreover, under infinitesimal coordinate transformations, the fields transform
as follows

$\displaystyle\begin{split}[H,\phi(x,t)]&=\partial_{t}\phi(x,t),\\\
[P,\phi(x,t)]&=\partial_{x}\phi(x,t),\\\
[D,\phi(x,t)]&=zt\frac{\partial\phi(x,t)}{\partial
t}+\Delta\phi(x,t)+x\frac{\partial\phi(x,t)}{\partial x},\end{split}$ (39)

where we have used
$\phi^{\prime}(x^{\prime},t^{\prime})=\lambda^{-\Delta}\phi(x,t)$. Our goal is
to apply these transformations to the (massless) fermionic fields. Indeed in
this case the corresponding two-point function splits for the independent
components of the fermionic field that each can be treated as a scalar field.
Now we define

$\displaystyle
F=F(x_{a},x_{b};t_{a},t_{b})=\langle\phi_{a}(x_{a},t_{a})\phi_{b}^{*}(x_{b},t_{b})\rangle,$
(40)

where $\phi_{a}(x_{a},t_{a})$ denotes a chiral component of the fermion field.
Translation invariance implies that $F=F(x,\tau)$ where $x=x_{a}-x_{b}$ and
$\tau=t_{a}-t_{b}$. Further, scale invariance which is generated by $D$
requires that

$\displaystyle\left(zt_{a}\partial_{a}+x_{a}D_{a}+\Delta_{a}+zt_{b}\partial_{b}+x_{b}D_{b}+\Delta_{b}\right)F(x;\tau)$
$\displaystyle=0.$ (41)

We would like to recast the above equation in a more simplified form. We do so
by defining $\partial_{a}=-\partial_{b}=\partial_{t}$ and $D_{a}=-D_{b}=D_{x}$
which yields

$\displaystyle\left(z\tau\partial_{\tau}+xD_{x}+\Delta_{a}+\Delta_{b}\right)F(x;\tau)=0.$
(42)

The general form of the two-point function can be found by solving the above
equation. Of course, from dimensional analysis we can deduce

$\displaystyle
F(x,\tau)=x^{-\left(\Delta_{a}+\Delta_{b}\right)}G\left(\frac{x^{z}}{\tau}\right),$
(43)

where $G(u)$ is an arbitrary function. In what follows for simplicity we
consider the equal time two-point function and thus $t_{b}=t_{a}$. We also
assume $\Delta_{b}=\Delta_{a}\equiv\Delta$.777Indeed, this choice is
consistent with our expectations from the general results for the two-point
functions in the relativistic limit corresponds to $z=1$. Under these
assumptions, eq. (42) yields

$\displaystyle\left(xD_{x}+2\Delta\right)F(x)=0,$ (44)

and the corresponding solution becomes

$\displaystyle F(x)=\frac{C}{x^{2\Delta}},$ (45)

where $C$ is a constant which can be set to unity after a proper
normalization. Hence, the equal time two-point function depends on the
dynamical exponent through the scaling dimension. Indeed, for the bosonic
field, where $\Delta=\frac{1-z}{2}$, the two-point function becomes

$\displaystyle F(x)=C\,x^{z-1},$ (46)

with a nontrivial dependence on the dynamical exponent. On the other hand, for
the fermionic field, where $\Delta=\frac{1}{2}$, the two-point function
becomes

$\displaystyle F(x)=\frac{C}{x},$ (47)

which is independent of the dynamical exponent. This behavior corresponds to
both theories defined in Eq. (9) and (10). It is also possible to write the
above two-point function as the form reported in the previous section.
Although, this expression is not well defined for $x_{b}=x_{a}$, we can find a
more general expression which holds even in this specific case. To do so, we
note that the delta function is a solution to eq. (42). Indeed, a
straightforward calculation gives888Here we use the properties of delta
function to show that
$x\delta(x)=0\longrightarrow\partial_{x}(x\delta(x))=0=\delta(x)+x\partial_{x}\delta(x)\longrightarrow\delta(x)=-x\partial_{x}\delta(x).$

$\displaystyle\left(xD_{x}+2\Delta\right)\delta(x)=xD_{x}\delta(x)+2\Delta\delta(x)=-\delta(x)+2\Delta\delta(x)=0$
(48)

where in the last equality we used $\Delta_{a}=\Delta_{b}=\frac{1}{2}$. Now
the general solution of Eq. (42) up to a coefficient would be a linear
combination of the solution given in Eq. (47) and the delta function as

$\displaystyle F(x_{a}-x_{b})=c\,\delta(x_{a}-x_{b})+\frac{d}{x_{a}-x_{b}},$
(49)

where $c$ and $d$ are constants. The above expression shares the same form
with the two-point function we found in Eq. (30).

##  3 Numerical Results

In this section, we numerically investigate the entanglement measures mainly
in 2$d$ for the aforementioned models. We provide a detailed numerical
analysis and examine how entanglement measures depend on the dynamical
exponent for different states and subregion configurations. We compare these
measures across various aspects which enables us to gain insight into distinct
features of our models. We address the possibility of introducing a c-function
defined in terms of entanglement entropy.

Through this section we always consider translational invariance in our
lattice models by either considering an infinite lattice or applying periodic
boundary conditions on a finite lattice. We would like to note that an
appropriate factor to avoid fermion doubling problem is considered all over
this section.

###  3.1 Entanglement Entropy

Plugging the eigenvalues of the corresponding covariance matrix into Eq. (2)
we numerically evaluate the entanglement entropy.

#### Vacuum State of $\mathcal{L}_{\mathrm{II}}$

Figure 2 shows the entanglement entropy as a function of the length of the
entangling region for model $f_{\mathrm{II}}(k)$ in $L\rightarrow\infty$
limit. The left panel shows that the entanglement entropy does not depend on
the dynamical exponent in the vanishingly small mass limit. As a result there
is no difference between relativistic and nonrelativistic fermions in the
massless (scale-invariant) regime.

In contrast, in Lifshitz scalar theories entanglement entropy monotonically
increases with $z$ [11, 12]. The dynamical exponent can be interpreted as a
measure for the correlation length, inspired from the interaction terms in the
lattice model. This picture explains the observed behavior in the scalar
theories. On the lattice this effect may lead to the emergence of volume law
behavior for entanglement entropy when $z\gg n_{A}$.

With regard to figure 2:

  * •

The behavior of massless $f_{\mathrm{II}}(k)$ fermions in the left panel is
consistent with what we expect in the continuum limit from the structure of
the two-point function (see section 2.3). Specifically, the $z$-independence
of the entanglement entropy is inherited from the two-point function.

  * •

Massless $f_{\mathrm{II}}(k)$ fermions do not follow the bosonic picture, that
monotonically increases by $z$, but rather experience a strong upper bound
preventing the amount of quantum correlations to exceed that of the $z=1$
case, namely $\frac{1}{3}\log n_{A}$. This upper bound is much smaller than
the intrinsic upper bound for fermions, namely $n_{A}\log 2$ for $n_{A}\gg 1$.

  * •

The inset of the left panel shows that the massless $f^{*}_{\mathrm{II}}(k)$
fermions do follow the bosonic picture and there is no upper bound for the
entanglement entropy. To get more insight about the difference between
$f^{*}_{\mathrm{II}}(k)$ and $f_{\mathrm{II}}(k)$ one should note that in the
latter $|f_{\mathrm{II}}(k)|\leq 1$ but it is easy to check that such a bound
does not exist for $f^{*}_{\mathrm{II}}(k)$. The larger the value of the
dynamical exponent, the larger $|f^{*}_{\mathrm{II}}(k)|$. Based on this key
difference, regularization $f^{*}_{\mathrm{II}}(k)$ does not reproduce the
$z$-independent structure in the massless continuum limit.

  * •

The right panel shows the well-understood fact that the mass parameter
decreases the entanglement entropy. Further, for the massive case increasing
the dynamical exponent results in increase of this measure. This behavior is
consistent with the massive scalar fields.

  * •

The inset of right panel shows a trade off between the mass parameter and the
dynamical exponent for $f_{\mathrm{II}}(k)$ fermions. The mass term decreases
the entanglement entropy due to the decrease of the correlation length, while
$z$ increases the entanglement entropy through increasing the correlation
strength. The $z$-independent scale invariance result plays the role of an
upper bound for this quantity in massive theories.

Figure 2: Left: Entanglement entropy for a single interval in the massless
regime as a function of the interval length $\ell$ for $f_{\mathrm{II}}(k)$.
The inset shows the same plot corresponding to $f^{*}_{\mathrm{II}}(k)$ where
there is no upper bound for EE. Right: Entanglement entropy in the massless
regime as a function of $\ell$ for $f_{\mathrm{II}}(k)$. The inset shows how
the massless result plays the role of an upper bound in massive theories for
larger values of the dynamical exponent. All results correspond to infinite
spatial direction.

#### Vacuum State of $\mathcal{L}_{\mathrm{I}}$ versus
$\mathcal{L}_{\mathrm{II}}$

Let us now turn to compare the entanglement entropy from $f_{\mathrm{II}}(k)$
with the previously known results from $f_{\mathrm{I}}(k)$. The left panel of
figure 3 corresponds to $m\to 0$ limit. For odd values of the dynamical
exponent, there is no surprise that entanglement entropy for
$f_{\mathrm{I}}(k)$ coincides with that for $f_{\mathrm{II}}(k)$ illustrated
in figure 2. However, for even values of $z$, there is a peculiar property. As
mentioned in [18], the entanglement entropy in the massless limit becomes zero
due to the structure of the vacuum state which is a product state for spatial
degrees of freedom (see Eq. (36)). The right panel corresponds to the massive
case where entanglement entropy is non trivial for both even and odd values of
the dynamical exponent. Moreover, for odd $z$ we show that the effect of the
mass parameter is qualitatively similar to what we find in figure 2, but for
even values of $z$ the entanglement entropy counterintuitively increases with
the mass parameter.

Figure 3: Entanglement entropy for $f_{\mathrm{I}}(k)$ in the massless (left)
and massive (right) regimes for several values of the dynamical exponent.

#### Thermal State

Let us now examine the behavior of von Neumann entropy for a single interval
in thermal state of $\mathcal{L}_{\mathrm{II}}$. Clearly this is not expected
to be a quantum correlation measure. As we have already noted in this case we
employ Eq. (31) to find the corresponding two-point function for thermal
states. The numerical results in the massless limit are summarized in figure
4. Let us make a number of observations regarding to these plots. First, we
note that in both plots, at finite temperature, entanglement entropy increases
with the dynamical exponent. Next, at low temperature the dependence on $z$
becomes less pronounced such that in $\beta\rightarrow\infty$ limit, the
entanglement entropy is independent of $z$ as expected. Moreover, the right
panel is showing that the entropy of the subregion is saturating at some value
$\sim n_{A}\log 2$, the natural upper bound on entropy.

Figure 4: Left: $S(\ell)$ for several values of the dynamical exponent at
finite temperature. Right: Entanglement entropy as a function of $z$ for
different values of $\beta$. In both panels we consider a finite lattice with
$L$ sites and the massless regime.

#### 3$d$ versus 2$d$: $z$-dependence of the Entanglement Entropy

In this part we digress our main focus to look at the behavior of
$\mathcal{L}_{\mathrm{II}}$ in 3$d$. The goal is to shed light on the unique
properties of the fermionic Lifshitz theory in 2$d$. In two spatial
dimensions, the two-point functions can be found as (see appendix A)

$\displaystyle\langle\Psi^{\dagger}_{i,k}\Psi_{j,l}\rangle=\frac{1}{2}\delta_{ij}\delta_{kl}-\frac{1}{2L^{2}}\sum_{k_{1},k_{2}}\frac{m\gamma^{0}+f(k_{1})\gamma^{0}\gamma^{1}+f(k_{2})\gamma^{0}\gamma^{2}}{\sqrt{m^{2}+f(k_{1})^{2}+f(k_{2})^{2}}}e^{i(k_{1}(i-j)+k_{2}(k-l))}.$
(50)

In this case, we choose the following representation for the gamma matrices

$\gamma^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\gamma^{1}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\qquad\gamma^{2}=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}.$
(51)

With this choice the two-point function takes following form

$\displaystyle\langle\Psi^{\dagger}_{i,k}\Psi_{j,l}\rangle=\frac{1}{2}\delta_{ij}\delta_{kl}-\frac{1}{2L^{2}}\sum_{k_{1},k_{2}}\frac{e^{i(k_{1}(i-j)+k_{2}(k-l))}}{\omega_{k}}\begin{pmatrix}-f(k_{1})&&m^{z}-if(k_{2})\\\
m^{z}+if(k_{2})&&f(k_{1})\\\ \end{pmatrix},$ (52)

where $\omega_{k}=\sqrt{m^{2z}+f(k_{1})^{2}+f(k_{2})^{2}}$ and we consider
$f(k_{i})=f_{\mathrm{II}}(k_{i})$ in our numerical analysis.

By employing the generalization of the correlator method to higher dimensions
we work out the entanglement entropy in this case were the results are shown
in figure 5 for a square entangling region in the $m\to 0$ limit.
Interestingly, we see that entanglement entropy monotonically increases with
the dynamical exponent. From Eq. (52), we see that in 3$d$, the corresponding
two point function depends on $z$ even in the massless regime which is a
strong evidence that $z$-independence is an intriguing feature related to the
scaling dimension of the fermion field peculiar to 2$d$. From the data
presented in figure 5, one can verify from the quadratic $\ell$-dependence of
the entanglement entropy that for large values of $z$, namely when $z\gg
n_{A}$ where $n_{A}$ is the number of sites on a side of the square region,
entanglement entropy scales with the volume of the subregion.

However, we should note that numerical results in a 3$d$ generalization of
$\mathcal{L}_{\mathrm{I}}$ indicate that although entanglement entropy is not
zero for even values of the dynamical exponent, but it is not a monotonic
function of $z$. Indeed, we find $S(z_{\rm even})<S(z_{\rm odd})$ for $z_{\rm
even}>z_{\rm odd}$ for certain regions in the parameter space.

Figure 5: Entanglement entropy for a square entangling region with side length
$\ell$ in the $m\to 0$ regime.

###  3.2 Logarithmic Negativity

In this section we mainly consider logarithmic negativity for mixed states as
well as mutual information which were defined in section 1. We calculate
logarithmic negativity for fermionic Gaussian states using fermionic partial
transpose by Eq. (8) and also address the behavior of mutual information that
can be calculated by plugging Eq. (2) into Eq. (4).

#### Vacuum State

We consider two intervals with same width $\ell$ and separation $d$ on the
vacuum state and trace out the complement. Figure 6 shows logarithmic
negativity between these symmetric intervals. With regard to this figure:

  * •

The left panel shows that logarithmic negativity for adjacent intervals does
not depend on $z$ in the massless regime similar to the entanglement entropy
for a connected interval in the vacuum state. Here we again recall that it was
shown in [12] that in a Lifshitz scalar theory $\mathcal{E}$ monotonically
increases with the dynamical exponent which is again in contrast with the
current results for the fermions.

  * •

The inset of this panel corresponds to a configuration that these symmetric
intervals are at a finite distance $d$. The inset also includes the results
for the mutual information in the same configuration that is indicated by
dashed lines. Both these measures are again independent of $z$ and decreases
as one increases the separation between the intervals, expected in local
theories.

  * •

In the right panel, we have plotted the massive regime for adjacent intervals.
In this case logarithmic negativity grows with the region width and saturates
to a finite value in the large $\ell$ limit. Moreover, logarithmic negativity
increases with $z$ but there is still an upper bound that saturates to.

Figure 6: Logarithmic negativity for two adjacent subregions with equal width
$\ell$ for (left) massless and (right) massive regime. The inset of the left
panel shows the results for both logarithmic negativity and mutual information
for disjoint symmetric subregions. Here we consider $L\rightarrow\infty$ limit
and the full state is the vacuum state.

#### Thermal State

Let us now examine the behavior of logarithmic negativity at finite
temperature states. In order to interpret our results, we would like to recall
that negativities take their maximum value for pure states (e.g., logarithmic
negativity for a pure state is equal to Renyi entropy with index $n=1/2$ which
is even larger than the von Neumann entropy) and vanish for maximally mixed
states. A quick extrapolation of this property leads us to expect negativities
to be decreasing as the mixedness of the state increases. For instance it is
well known that the temperature corrections to negativities always take a
negative sign, see g.e. [40].

In figure 7 we illustrate our numerical results where we have considered a
single interval in a finite temperature state and the massless regime. With
regard to figure 7:

  * •

From the left plot $\mathcal{E}(\ell)$ is a monotonically increasing function
that almost saturates to some finite value. We believe this saturation happens
due to the locality of the theory and the correlation scheme which is
$z$-independent.

  * •

The right panel shows that as expected from physical grounds, logarithmic
negativity decreases with the temperature for a fixed value of $z$. As the
temperature increases, the number of relevant terms in the expansion of the
thermal state increases. This increases the mixedness of the state though
logarithmic negativity decreases.

  * •

The right panel also shows that logarithmic negativity is decreasing as a
function of $z$.999For different aspects about the effect of $z$ in quantum
correlations in terms of odd entropy see [61]. The $z$-dependence behavior is
rather complicated compared to $\beta$-dependence. Although $\omega(k)$ is an
increasing function of $z$, the eigenvalues of the mixed state also depend on
the fermionic occupation number. Strictly speaking, focusing on mode $k$,
increasing $z$ decreases the eigenvalue corresponding to the unoccupied state
while increases the one corresponding to the occupied state. The net effect of
all modes forms the the number of relevant terms in the thermal state
expansion. This can be explained with the behavior of the two-point function.
In this fermionic local theory at finite temperature, as one can easily verify
from Eqs. (32) and (33), the net effect of increasing $z$ on the thermal
correlation function is decreasing the correlation. Though it is not
surprising that quantum correlations captured by logarithmic negativity should
also decrease.

Figure 7: Left: Logarithmic negativity as a function of subregion width for
several values of the dynamical exponent with $\beta=10$. The inset of the
left panel shows the results for $f^{*}_{\mathrm{II}}(k)$ . Right: Logarithmic
negativity as a function of the dynamical exponent for several values of the
temperature. The inset of the right panel shows the results for
$f_{\mathrm{I}}(k)$ . : In all panels we consider single intervals in small
mass regime with $L=100$.

###  3.3 Entropic $c$-function

In this section we consider another interesting quantity which estimates the
monotonic readjustment of entanglement under the RG flow, the so-called
(entropic) $c$-function [56, 30]. In a 2$d$ relativistic theory, this quantity
is defined as follows

$\displaystyle c(\ell)=\ell\frac{dS}{d\ell},$ (53)

which at the corresponding fixed points coincides with the coefficient of the
universal logarithmic term in the expansion of the entanglement entropy for a
subsystem of width $\ell$. Indeed, this function monotonically decreases along
the RG flow from UV to IR which is a manifestation of $c$-theorem. Because the
mass term is relevant we expect that $c$-function decreases as the mass
increases. Here we assume that the same definition as Eq. (53) holds101010A
justification for this assumption: the definition of (53) is based on two
assumptions, 1) the existence of a light-cone structure in the theory, 2) a
logarithmic leading term for the entanglement entropy. The existence of the
light-cone structure has been addressed in Lifshitz invariant free scalar
theories in [17] for lattice models and in [19] for the continuum limit. The
same argument holds for free fermions as the argument is solely based on the
form of the dispersion relation. With regard to the second assumption, for
fermionic theories at the UV fixed point the entanglement entropy is
$z$-independent and the relativistic result is valid for any $z$. For bosonic
theories as long the entangling region (in the units of the UV cut off) is
much larger than the dynamical exponent, the leading term of the entanglement
entropy is logarithmic [11, 16]. for these non-relativistic theories and
evaluate the corresponding $c$-function numerically. The results are collected
in figure 8 (left panel) where we show the $c$-function as a function of mass
parameter. For any $z$, it starts at the same value of the relativistic case
given by $c_{\rm rel}=\frac{1}{3}$, then decreases monotonically and saturates
to zero in the large mass limit. A curious feature that we found is that the
$c$-function has a plateau at some $m_{\star}$. Here we observe that as $z$
increases, the $c$-function approaches a constant value $\frac{1}{3}$.

The right panel of figure 8 illustrates the behavior of the $c$-function in
the Lifshitz scalar theory for several values of the dynamical exponent. Our
numerical results show that the number of effective number of degrees of
freedom is increasing with $z$ while for a fixed $z$, as for the $z=1$ case,
the theory is flowing to a massive theory in IR limit where the effective
number of degrees of freedom gets arbitrarily small as the the IR scale is
decreased.

In the Dirac fermion case for $z=1$, the behavior of the $c$-function
coincides with that of the $z=1$ scalar theory. For larger values of the
dynamical exponent the UV value of the $c$-function is $z$-independent, as the
entropy was, and behavior through the flow is qualitatively similar to the
scalar case. For arbitrary non-vanishing IR scale, the effective number of
degrees of freedom gets is an increasing function of $z$ subject to the upper
bound at the UV scale. However, as a natural result of the existence of such
an upper bound, in the $z\rightarrow\infty$ limit the flow would be trivial
since the introduced $c$-function would take a constant value $c=\frac{1}{3}$
all over the flow.

Figure 8: $c$-function as a function of mass for different values of the
dynamical exponent for fermionic (left) and scalar (right) fields.

##  4 Conclusions and discussions

In this paper, we explored the entanglement structure in a new
$(1+1)$-dimensional fermionic field theory whose UV fixed point respects
Lifshitz scaling symmetry. We have carefully examined the behavior of several
entanglement measures in static cases both for vacuum and mixed states
including finite temperature states. We now proceed to summarize some
interesting implications of our results and also some potential future
directions that would be interesting to pursue.

Considering the entanglement entropy for a massless fermionic field
(corresponds to the UV fixed point of the theory) in the vacuum state for
$f_{\mathrm{II}}$, we have found that this measure is $z$-independent. Indeed,
this behavior differs from the corresponding results for $f^{*}_{\mathrm{II}}$
which are previously reported in [19]. We expect that this intriguing result
can be explained in terms of the $z$-dependence of the fermionic field’s two-
point function in Eq. (30). Figure 9 illustrates the eigenvalue of the
correlation matrix as a function of distance between two lattice sites. As we
can verify from this figure (left panel), the correlation function as a
function of distance in $f_{\mathrm{II}}$ regularization follows the same
pattern of the continuum limit in the sense that there is no $z$-dependence in
the massless limit. On the other hand, from the right panel corresponding to
$f^{*}_{\mathrm{II}}$, we see that there is a $z$-dependence behavior such
that increasing the dynamical exponent, enhances the correlation. Further,
this figure shows that there is no upper bound on the correlations in the
$f^{*}_{\mathrm{II}}$ regularization.

Figure 9: Comparison of correlation strength in (left) $f_{\mathrm{II}}$ and
(right) $f^{*}_{\mathrm{II}}$. The eigenvalue for the correlation matrix as a
function of distance between two points $x$. There are a bunch of points with
vanishing $\nu$ which correspond to some values with even value of $x$,
suppressed in $f^{*}_{\mathrm{II}}$ as $z$ is increased. For a more clear
comparison, in the insets we show the moving averaged value $\bar{\nu}$ for
the same plot.

Additionally, based on a symmetry argument and some physical assumptions we
have shown that the two-point function for fermionic fields does not depend on
the dynamical exponent and thus the $z$-independence of the entanglement
entropy is inherited from this particular behavior. Further, our approach can
be easily extended to higher dimensions where we have found that entanglement
entropy monotonically increases with the dynamical exponent even in the
massless regime. Thus we have shown that $z$-independence is an fascinating
feature which particularly appears in $(1+1)$-dimensions.

Moreover, in order to gain further insights into certain properties of the
entanglement structure of mixed states in fermionic theories, we also studied
logarithmic negativity and an entropic $c$-function in a specific regime of
the parameter space. Focusing on the $z$-dependence of logarithmic negativity
between two subregions when the whole systems is in the vacuum state, we show
that it does not depend on the dynamical exponent in the massless regime
similar to the entanglement entropy for a single interval. Again, we can
understand this intuitively in terms of the behavior of the fermionic two-
point function. This behavior is in contrast with the previous results for the
Lifshitz scalar theory where $\mathcal{E}$ monotonically increases with the
dynamical exponent. On the other hand, in the massive case the logarithmic
negativity increases with $z$ and then saturates from below to a finite value,
depending on the mixedness of the state.

Further, considering a single interval at finite temperature in the massless
regime, $\mathcal{E}(\ell)$ is a monotonically increasing function that more
or less saturates to a finite value in the large $\ell$ limit. Indeed, this
specific behavior is due to the locality of the theory and the correlation
scheme which is $z$-independent. Remarkably, our numerical results show that
the logarithmic negativity is a monotonically decreasing function of the
dynamical exponent. We have argued that this peculiar behavior is due to a
competition between the $z$-dependence of $\omega(k)$ and the fermionic
occupation number such that the net effect of increasing the dynamical
exponent on the thermal correlation function is decreasing the total
correlation. Hence, logarithmic negativity which captures quantum correlations
also decreases. Moreover, regarding the $z$-dependence of the mutual
information, our numerical analysis supply results qualitatively similar to
observations made for the entanglement entropy as expected.

We have also observed that employing the same relativistic definition yields a
convenient (entropic) $c$-function which decreases along the RG flow.
Specifically, our numerical results show that for arbitrary integer value of
the dynamical exponent, the proposed function starts at the same value of the
relativistic case and decreases monotonically through the flow before in the
large mass limit. A curious feature that we have found is that the
$c$-function has a plateau at some value of the mass parameter.

The results of this paper are restricted to entanglement and correlation
measures with Lifshitz fermions for static states. A straightforward extension
of these results would be studying entanglement and correlation measures with
Lifshitz fermions in out-of-equilibrium states, e.g., states followed by by
quantum quenches. Another interesting question would be examining the behavior
of other entanglement measures including entanglement of purification and odd
entropy.

## Acknowledgements

We would like to thank Masaki Oshikawa, Saleh Rahimi-Keshari, Shinsei Ryu,
Hassan Shapourian, Ken Shiozaki, and Erik Tonni for fruitful discussions. MV
would like to thank the School of Physics of IPM for hospitality during this
work. MV is also grateful to Mohsen Alishahiha for his supports. MV is
supported by Iran National Science Foundation (INSF) under project No.4023620.
AM would like to acknowledge support from ICTP through the Associates
Programme (2023-2028) and for hospitality during the final stages of this
work. Some numerical calculations of this work has been carried on facilities
provided by the School of Physics, IPM.

##  Appendix A Lifshitz fermionic model in $(2+1)$-dimensions

The Lagrangian of the lifshitz fermionic model in $(2+1)$-dimensions is given
as follows

$\displaystyle\mathcal{L}=\bar{\Psi}(i\gamma^{0}\partial_{0}+i\gamma^{x}T^{z-1}_{(x)}\partial_{x}+i\gamma^{y}T^{z-1}_{(y)}\partial_{y}-m)\Psi,$
(54)

where $T_{(x)}=\sqrt{-\partial_{x}^{2}}$ and
$T_{(y)}=\sqrt{-\partial_{y}^{2}}$. It is straightforward to show that in this
case the corresponding Hamiltonian becomes

$\displaystyle\mathcal{H}$
$\displaystyle=\pi_{\Psi}\dot{\Psi}-\mathcal{L}=-\bar{\Psi}(i\gamma^{x}T^{z-1}_{(x)}\partial_{x}+i\gamma^{y}T^{z-1}_{(y)}\partial_{y}-m)\Psi.$
(55)

In $(1+2)$-dimensions, we choose the following majorana representation for the
gamma matrices

$\gamma^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\gamma^{1}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\qquad\gamma^{2}=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}.$
(56)

A similar derivation to the one presented for $(1+1)$-dimensions gives the
Hamiltonian density on the square lattice as follows

$\displaystyle\mathcal{H}=-\sum_{n_{x}=0,n_{y}=0}^{N}\bar{\Psi}_{\vec{n}}(iT^{z-1}(\gamma^{1}\partial_{1}+\gamma^{2}\partial_{2})-m)\Psi_{\vec{n}}.$
(57)

Now using the Fourier transformations we can find the momentum space
representation which yields

$\displaystyle\mathcal{H}=\sum_{k_{1},k_{2}=0}\psi^{\dagger}_{k}\begin{pmatrix}f(k_{1})&m+if(k_{2})\\\
m-if(k_{2})&-f(k_{1})\end{pmatrix}\psi_{k},$ (58)

where $f(k)$ is the same as eq. (23). Further, we can diagonalize the above
Hamiltonian by treating the two components of the fermionic field separately
as $\psi_{k}=\begin{pmatrix}u_{k}\\\ d_{k}\end{pmatrix}$. Using eq. (25), the
diagonalization constraint gives

$\displaystyle if(k_{1})\sin\theta_{k}-f(k_{2})-im\cos\theta_{k}=0.$ (59)

Moreover, solving the above equation we can find $\sin\theta_{k}$ and
$\cos\theta_{k}$ as follows

$\displaystyle\cos\theta_{k}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{f(k_{2})^{2}\left(f(k_{1})^{2}+f(k_{2})^{2}+m^{2}\right)}+imf(k_{1})}{f(k_{1})^{2}+f(k_{2})^{2}},$
$\displaystyle\sin\theta_{k}$ $\displaystyle=$
$\displaystyle\frac{-\sqrt{f(k_{1})^{2}\left(f(k_{1})^{2}+f(k_{2})^{2}+m^{2}\right)}+imf(k_{2})}{f(k_{1})^{2}+f(k_{2})^{2}}.$
(60)

Finally, the diagonal Hamiltonian becomes

$H=\sum_{k_{1},k_{2}}\sqrt{m^{2z}+f(k_{1})^{2z}+f(k_{2})^{2z}}\left(b_{\vec{k}}^{\dagger}b_{\vec{k}}+b_{-{k}}^{\dagger}b_{-\vec{k}}\right)$
(61)

In the next step, we can find the two-point function using the following
equation

$\langle\Psi_{s}^{\dagger}\Psi_{r}\rangle=\frac{1}{L^{2}}\sum_{k,k^{\prime}}e^{ikr}e^{-ik^{\prime}s}\langle\psi_{k^{\prime}}^{\dagger}\psi_{k}\rangle.$

Employing the definition of Bogoliubov transformation, we can write the two
point function in terms of $\sin\theta_{k}$ and $\cos\theta_{k}$ as follows

$\displaystyle\langle\Psi_{s}^{\dagger}\Psi_{r}\rangle$ $\displaystyle=$
$\displaystyle\frac{\delta_{rs}}{2}\mathbb{1}-\frac{1}{2L^{2}}\sum_{\vec{k}}e^{i\vec{k}(\vec{r}-\vec{s})}\begin{pmatrix}\cos\theta_{k}&\sin\theta_{k}\\\
\sin\theta_{k}&-\cos\theta_{k}\end{pmatrix}.$ (62)

Now, we can use eq. (A) to find the two-point function which yields

$\displaystyle\langle\Psi^{\dagger}_{i,k}\Psi_{j,l}\rangle=\frac{1}{2}\delta_{ij}\delta_{kl}-\frac{1}{2L^{2}}\sum_{k_{1},k_{2}}\frac{m\gamma^{0}+f(k_{1})\gamma^{0}\gamma^{1}+f(k_{2})\gamma^{0}\gamma^{2}}{\sqrt{m^{2}+f(k_{1})^{2}+f(k_{2})^{2}}}e^{i(k_{1}(i-j)+k_{2}(k-l))}.$
(63)

## References

  * [1] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,” Class. Quant. Grav. 26 (2009), 224002 [arXiv:0903.3246 [hep-th]].
  * [2] P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory,” J. Phys. A 42 (2009), 504005 [arXiv:0905.4013 [cond-mat.stat-mech]].
  * [3] E. Ardonne, P. Fendley and E. Fradkin, “Topological order and conformal quantum critical points,” Annals Phys. 310 (2004), 493-551 [arXiv:cond-mat/0311466 [cond-mat]].
  * [4] E. Fradkin and J. E. Moore, “Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum,” Phys. Rev. Lett. 97 (2006), 050404 [arXiv:cond-mat/0605683 [cond-mat.str-el]].
  * [5] B. Hsu, M. Mulligan, E. Fradkin and E. A. Kim, “Universal entanglement entropy in 2D conformal quantum critical points,” Phys. Rev. B 79 (2009), 115421 doi:10.1103/PhysRevB.79.115421 [arXiv:0812.0203 [cond-mat.stat-mech]].
  * [6] E. Fradkin, “Scaling of Entanglement Entropy at 2D quantum Lifshitz fixed points and topological fluids,” J. Phys. A 42 (2009), 504011 [arXiv:0906.1569 [cond-mat.str-el]].
  * [7] M. Oshikawa, “Boundary Conformal Field Theory and Entanglement Entropy in Two-Dimensional Quantum Lifshitz Critical Point,” [arXiv:1007.3739 [cond-mat.stat-mech]].
  * [8] M. P. Zaletel, J. H. Bardarson and J. E. Moore, “Logarithmic terms in entanglement entropies of 2D quantum critical points and Shannon entropies of spin chains,” Phys. Rev. Lett. 107 (2011), 020402 [arXiv:1103.5452 [cond-mat.str-el]].
  * [9] T. Zhou, X. Chen, T. Faulkner and E. Fradkin, “Entanglement entropy and mutual information of circular entangling surfaces in the 2 + 1-dimensional quantum Lifshitz model,” J. Stat. Mech. 1609 (2016) no.9, 093101 [arXiv:1607.01771 [cond-mat.stat-mech]].
  * [10] X. Chen, W. Witczak-Krempa, T. Faulkner and E. Fradkin, “Two-cylinder entanglement entropy under a twist,” J. Stat. Mech. 1704 (2017) no.4, 043104 [arXiv:1611.01847 [cond-mat.str-el]].
  * [11] M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement in Lifshitz-type Quantum Field Theories,” JHEP 07, 120 (2017) doi:10.1007/JHEP07(2017)120 [arXiv:1705.00483 [hep-th]].
  * [12] M. R. Mohammadi Mozaffar and A. Mollabashi, “Logarithmic Negativity in Lifshitz Harmonic Models,” J. Stat. Mech. 1805, no.5, 053113 (2018) [arXiv:1712.03731 [hep-th]].
  * [13] J. Angel-Ramelli, “Entanglement Entropy of Excited States in the Quantum Lifshitz Model,” J. Stat. Mech. 2101 (2021), 013102 [arXiv:2009.02283 [hep-th]].
  * [14] J. Angel-Ramelli, C. Berthiere, V. G. M. Puletti and L. Thorlacius, “Logarithmic Negativity in Quantum Lifshitz Theories,” JHEP 09 (2020), 011 [arXiv:2002.05713 [hep-th]].
  * [15] X. Chen, E. Fradkin and W. Witczak-Krempa, “Quantum spin chains with multiple dynamics,” Phys. Rev. B 96 (2017) no.18, 180402 [arXiv:1706.02304 [cond-mat.str-el]].
  * [16] T. He, J. M. Magan and S. Vandoren, “Entanglement Entropy in Lifshitz Theories,” SciPost Phys. 3 (2017) no.5, 034 [arXiv:1705.01147 [hep-th]].
  * [17] M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement Evolution in Lifshitz-type Scalar Theories,” JHEP 01 (2019), 137 [arXiv:1811.11470 [hep-th]].
  * [18] D. Hartmann, K. Kavanagh and S. Vandoren, “Entanglement entropy with Lifshitz fermions,” SciPost Phys. 11, no.2, 031 (2021) [arXiv:2104.10913 [quant-ph]].
  * [19] M. R. M. Mozaffar and A. Mollabashi, “Time scaling of entanglement in integrable scale-invariant theories,” Phys. Rev. Res. 4, no.2, L022010 (2022) [arXiv:2106.14700 [hep-th]].
  * [20] J. Angel-Ramelli, V. G. M. Puletti and L. Thorlacius, “Entanglement Entropy in Generalised Quantum Lifshitz Models,” JHEP 08 (2019), 072 [arXiv:1906.08252 [hep-th]].
  * [21] S. N. Solodukhin, “Entanglement Entropy in Non-Relativistic Field Theories,” JHEP 04 (2010), 101 [arXiv:0909.0277 [hep-th]].
  * [22] M. A. Rajabpour and S. Sotiriadis, “Quantum quench in long-range field theories,” Phys. Rev. B 91 (2015) no.4, 045131 [arXiv:1409.6558 [cond-mat.str-el]].
  * [23] C. Boudreault, C. Berthiere and W. Witczak-Krempa, “Entanglement and separability in continuum Rokhsar-Kivelson states,” Phys. Rev. Res. 4 (2022) no.3, 033251 [arXiv:2110.04290 [cond-mat.str-el]].
  * [24] M. Mintchev, D. Pontello and E. Tonni, “Entanglement entropies of an interval in the free Schrödinger field theory on the half line,” JHEP 09 (2022), 090 [arXiv:2206.06187 [hep-th]].
  * [25] C. Berthiere, B. Chen and H. Chen, “Reflected entropy and Markov gap in Lifshitz theories,” JHEP 09 (2023), 160 [arXiv:2307.12247 [hep-th]].
  * [26] I. Hason, “Triviality of Entanglement Entropy in the Galilean Vacuum,” Phys. Lett. B 780 (2018), 149-151 doi:10.1016/j.physletb.2018.02.064 [arXiv:1708.08303 [hep-th]].
  * [27] I. Peschel, “Calculation of reduced density matrices from correlation functions,” J. Phys. A 36 (2003) no.14, L205 [arXiv:cond-mat/0212631 [cond-mat]].
  * [28] G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, “Entanglement in quantum critical phenomena,” Phys. Rev. Lett. 90 (2003), 227902 [arXiv:quant-ph/0211074 [quant-ph]].
  * [29] J. I. Latorre, E. Rico and G. Vidal, “Ground state entanglement in quantum spin chains,” Quant. Inf. Comput. 4 (2004) no.1, 48-92 [arXiv:quant-ph/0304098 [quant-ph]].
  * [30] H. Casini and M. Huerta, “Entanglement entropy in free quantum field theory,” J. Phys. A 42 (2009), 504007 [arXiv:0905.2562 [hep-th]].
  * [31] V. Eisler and I. Peschel, “Reduced density matrices and entanglement entropy in free lattice models,” J. Phys. A 42 (2009) no.50, 504003 [arXiv:0906.1663 [cond-mat.stat-mech]].
  * [32] A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77 (1996), 1413-1415 [arXiv:quant-ph/9604005 [quant-ph]].
  * [33] M. Horodecki, P. Horodecki and R. Horodecki, “On the necessary and sufficient conditions for separability of mixed quantum states,” Phys. Lett. A 223 (1996), 1 [arXiv:quant-ph/9605038 [quant-ph]].
  * [34] G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65 (2002), 032314 [arXiv:quant-ph/0102117 [quant-ph]].
  * [35] M. B. Plenio, “Logarithmic Negativity: A Full Entanglement Monotone That is not Convex,” Phys. Rev. Lett. 95 (2005), 090503 [arXiv:quant-ph/0505071 [quant-ph]].
  * [36] M. B. Plenio and S. Virmani, “An introduction to entanglement measures,” Quant. Inf. Comput. 7 (2007) no.1-2, 001-051 [arXiv:quant-ph/0504163 [quant-ph]].
  * [37] P. Calabrese, J. Cardy and E. Tonni, “Entanglement negativity in quantum field theory,” Phys. Rev. Lett. 109 (2012), 130502 [arXiv:1206.3092 [cond-mat.stat-mech]].
  * [38] O. Blondeau-Fournier, O. A. Castro-Alvaredo and B. Doyon, “Universal scaling of the logarithmic negativity in massive quantum field theory,” J. Phys. A 49 (2016) no.12, 125401 [arXiv:1508.04026 [hep-th]].
  * [39] P. Calabrese, J. Cardy and E. Tonni, “Entanglement negativity in extended systems: A field theoretical approach,” J. Stat. Mech. 1302 (2013), P02008 [arXiv:1210.5359 [cond-mat.stat-mech]].
  * [40] P. Calabrese, J. Cardy and E. Tonni, “Finite temperature entanglement negativity in conformal field theory,” J. Phys. A 48 (2015) no.1, 015006 [arXiv:1408.3043 [cond-mat.stat-mech]].
  * [41] R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. 84 (2000), 2726-2729 [arXiv:quant-ph/9909044 [quant-ph]].
  * [42] K. Audenaert, J. Eisert, M. B. Plenio and R. F. Werner, “Entanglement Properties of the Harmonic Chain,” Phys. Rev. A 66 (2002), 042327 [arXiv:quant-ph/0205025 [quant-ph]].
  * [43] V. Eisler and Z. Zimborás, “On the partial transpose of fermionic Gaussian states,” New J. Phys. 17 (2015) no.5, 053048 [arXiv:1502.01369 [cond-mat.stat-mech]].
  * [44] A. Coser, E. Tonni and P. Calabrese, “Partial transpose of two disjoint blocks in XY spin chains,” J. Stat. Mech. 1508 (2015) no.8, P08005 [arXiv:1503.09114 [cond-mat.stat-mech]].
  * [45] A. Coser, E. Tonni and P. Calabrese, “Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion,” J. Stat. Mech. 1603 (2016) no.3, 033116 [arXiv:1508.00811 [cond-mat.stat-mech]].
  * [46] A. Coser, E. Tonni and P. Calabrese, “Spin structures and entanglement of two disjoint intervals in conformal field theories,” J. Stat. Mech. 1605 (2016) no.5, 053109 [arXiv:1511.08328 [cond-mat.stat-mech]].
  * [47] C. P. Herzog and Y. Wang, “Estimation for Entanglement Negativity of Free Fermions,” J. Stat. Mech. 1607 (2016) no.7, 073102 doi:10.1088/1742-5468/2016/07/073102 [arXiv:1601.00678 [hep-th]].
  * [48] H. Wichterich, J. Molina-Vilaplana and S. Bose, “Scaling of entanglement between separated blocks in spin chains at criticality,” Phys. Rev. A 80 (2009) no.1, 010304 [arXiv:0811.1285 [quant-ph]].
  * [49] V. Alba, “Entanglement negativity and conformal field theory: a Monte Carlo study,” J. Stat. Mech. 1305 (2013), P05013 [arXiv:1302.1110 [cond-mat.stat-mech]].
  * [50] P. Calabrese, L. Tagliacozzo and E. Tonni, “Entanglement negativity in the critical Ising chain,” J. Stat. Mech. 1305 (2013), P05002 [arXiv:1302.1113 [cond-mat.stat-mech]].
  * [51] H. Shapourian and S. Ryu, “Entanglement negativity of fermions: monotonicity, separability criterion, and classification of few-mode states,” Phys. Rev. A 99 (2019) no.2, 022310 [arXiv:1804.08637 [quant-ph]].
  * [52] H. Shapourian and S. Ryu, “Finite-temperature entanglement negativity of free fermions,” J. Stat. Mech. 1904 (2019), 043106 doi:10.1088/1742-5468/ab11e0 [arXiv:1807.09808 [cond-mat.stat-mech]].
  * [53] H. Shapourian, K. Shiozaki and S. Ryu, “Partial time-reversal transformation and entanglement negativity in fermionic systems,” Phys. Rev. B 95 (2017) no.16, 165101 [arXiv:1611.07536 [cond-mat.str-el]].
  * [54] H. Montani and F. A. Schaposnik, “Lifshitz fermionic theories with z=2 anisotropic scaling,” Phys. Rev. D 86, 065024 (2012) [arXiv:1206.1027 [hep-th]].
  * [55] A. Jafarizadeh and M. A. Rajabpour, “Bipartite entanglement entropy of the excited states of free fermions and harmonic oscillators,” Phys. Rev. B 100, no.16, 165135 (2019) [arXiv:1907.09806 [cond-mat.str-el]].
  * [56] H. Casini and M. Huerta, “A c-theorem for the entanglement entropy,” J. Phys. A 40, 7031-7036 (2007) doi:10.1088/1751-8113/40/25/S57 [arXiv:cond-mat/0610375 [cond-mat]].
  * [57] M. Henkel, “Schrodinger invariance in strongly anisotropic critical systems,” J. Statist. Phys. 75, 1023-1061 (1994) doi:10.1007/BF02186756 [arXiv:hep-th/9310081 [hep-th]].
  * [58] M. Henkel and J. Unterberger, “Schrodinger invariance and space-time symmetries,” Nucl. Phys. B 660, 407-435 (2003) doi:10.1016/S0550-3213(03)00252-9 [arXiv:hep-th/0302187 [hep-th]].
  * [59] S. Stoimenov and M. Henkel, “Lie symmetries of semi-linear Schrodinger equations and applications,” J. Phys. Conf. Ser. 40, 144-149 (2006) doi:10.1088/1742-6596/40/1/018 [arXiv:math-ph/0512025 [math-ph]].
  * [60] J. Figueroa-O’Farrill, R. Grassie and S. Prohazka, “Lifshitz symmetry: Lie algebras, spacetimes and particles,” SciPost Phys. 14, no.3, 035 (2023) doi:10.21468/SciPostPhys.14.3.035 [arXiv:2206.11806 [hep-th]].
  * [61] A. Mollabashi and K. Tamaoka, JHEP 08 (2020), 078 doi:10.1007/JHEP08(2020)078 [arXiv:2004.04163 [hep-th]].