\section{Introduction}

There are now more than twenty brown dwarfs known in the solar neighborhood that have effective temperatures ($T_{\rm eff}$) lower than 500~K  (Cushing et al. 2011, 2014; Kirkpatrick et al. 2012; Liu et al. 2012; Luhman 2014; Luhman, Burgasser \& Bochanski 2011, Pinfield et al. 2014; Schneider et al. 2015; Tinney et al. 2012). These have been classified as Y dwarfs   (Cushing et al. 2011, Kirkpatrick et al. 2012). Evolutionary models   show that for $300 \leq T_{\rm eff}$~K $\leq 500$ and 0.2 $\leq$ age~Gyr $\leq$ 10 (as appropriate for the solar neighborhood) the mass range is 2 -- 30 Jupiter masses (Saumon \& Marley 2008). Hence this population of isolated brown dwarfs has a mass that is very planet-like.

Our group has an ongoing program measuring the photometric variability of Y dwarfs.
For warmer brown dwarfs variability is usually associated with inhomogeneous or variable cloud structure in the atmosphere (e.g. Radigan et al. 2012).
For  Y0 and Y1 dwarfs with $T_{\rm eff} \approx 400$~K the atmospheres are generally cloud-free, because most of the atmosphere is too cold for chloride or sulphide clouds, and too warm  for  water or ammonia clouds
(Burrows et al. 2003; Morley et al. 2012, 2014); in fact
cloud-free models can reproduce Y dwarf observations (Leggett et al. 2015, 2016).
Nevertheless variability may be seen at wavelengths where flux is emitted from very high and cold or low and warm layers where condensates can be present (Morley et al. 2012), or variability may be seen due to temperature variations across the brown dwarf surface (Showman \& Kaspi 2013).

In our first paper (Cushing et al. 2016, hereafter Cu16) we show that the Y0.5(pec) brown dwarf WISEPC J140518.40$+$553421.5  (W1405, Cushing et al. 2011) is variable at  mid-infrared wavelengths.
W1405 was observed with {\em Spitzer} using the IRAC camera (Fazio et al. 2004)
in the [3.6] and [4.5] filters. Variability was evident at [4.5] in the first epoch and at both [3.6] and [4.5] in the second epoch. The  second-epoch light curves have a period of about 8.5 hr, and semi-amplitudes of 3.5\%.
In the current paper we present the detection of variability at mid-infrared wavelengths in another Y0,
WISEP J173835.52$+$273258.9 (W1738, Cushing et al. 2011). We also present the tentative  detection
of variability at near-infrared wavelengths, at the $\lesssim 2 \sigma$ confidence level.
We extend to lower limits the work of Rajan et al. (2015) who  excluded any $J$-band variability larger than 20\% for this brown dwarf.

Leggett et al. (2016) compares near-infrared spectra and photometry for W1738 to recent models which include chemical disequilibrium driven by vertical transport (Tremblin et al. 2015). It was necessary to include mixing in order to reproduce the observations, and a  cloud-free  solar metallicity model with   $T_{\rm eff} = 425 \pm 25$K and log $g = 4.0 \pm 0.25$ fit the data best. This temperature and relatively low gravity imply that W1738 is a
3 -- 9 Jupiter mass object with an age of 0.15 -- 1 Gyr. Table 1 lists properties of W1738.

In \S 2 we present new observations of W1738 obtained with {\em Spitzer} and IRAC, and Gemini Observatory and its near-infrared imager NIRI (Hodapp et al. 2003). We obtained two epochs of  [3.6] and [4.5] data, separated by four months, as well as two epochs of near-infrared $Y$ and $J$ data, obtained between the  {\em Spitzer} observations and separated by one month.
\S 3 presents our analysis of the data, which we discuss in \S 4. Our conclusions are given in \S 5.

\section{Observations}

\subsection{Gemini NIRI Observations}

W1738 was observed as part of the Gemini North program GN-2013A-Q-21. The brown dwarf was imaged in the  $Y$ and $J$ filters using NIRI.  The NIRI $Y$ filter differs slightly from the MKO standard, where $Y_{\rm NIRI} - Y_{\rm MKO} = 0.17\pm 0.03$ magnitudes for late type T and early type Y dwarfs (Liu et al. 2012).

Long-duration observations were obtained on UT 2013 July 17 and 2013 August 23 in photometric conditions with typical seeing $0\farcs8$. Integration times for both filters were 60\,s, and offsets of about 12$\arcsec$ were used in dither patterns that were moved slightly on the detector through the observation, to minimise the impact of bad pixels and reduce any flat-fielding artifacts. The filters were alternated so that a five-position dither in $Y$ was executed, followed by a nine-position dither in $J$, followed by another five-position dither in $Y$, etc.

On 2013 July 17 the observation ran from UT 07:22:57 to 12:18:09, for a duration of 4.92 hours. Seventeen five-position dithers in $Y$ and sixteen nine-position
dithers in $J$ were obtained, for an on-source time of 1.4 hours in $Y$ and 2.4 hours in $J$. On 2013 August 23 the observation ran from UT 5:41:48 to 10:19:56,
for a duration of 4.64 hours. Sixteen five-position dithers in $Y$ and fifteen nine-position dithers in $J$ were obtained, for an on-source time of 1.33 hours in
$Y$ and 2.25 hours in $J$. Photometric standard FS 27 was observed before the W1738 observation, and FS 35 was observed after, on both nights. The data were
reduced in the standard way, using dark and flatfield images obtained with the on-telescope calibration unit, and Gemini IRAF routines.

The NIRI detector suffers from first-frame pattern noise, which occurs in the first frame following any filter change. Because of this we discarded all of the first frames in each set, and used sets of four $Y$ images and sets of eight $J$ images to first generate sky frames and then coadded images. Aperture photomery with annular skies and an aperture diameter of 2$\farcs$4 was performed.

Six stars (or point-like objects) with $Y$ and $J$ magnitudes between 17 and 18 provided the photometric baseline reference. We used five point sources with  $Y$ and $J$ magnitudes between 18.9 and 20.7 to further explore the precision of the photometry, in order to determine whether or not the similarly-bright W1738 is variable in these filters. The eleven sources were selected for brightness, and for being present in every coadded image with a clean annular sky region.
The uncertainty in the photometry obtained from the four-minute $Y$ and eight-minute $J$ coadded images was 1 -- 2\% for the brighter reference stars in both filters, and $\sim$5\% in $Y$ and $\sim$3\% in $J$ for W1738 and the five fainter point sources. Figure 1 shows examples of a
coadded four-minute $Y$ and eight-minute $J$ image, with W1738 and the eleven other sources identified. Table 2 gives the nightly average of the photometry for W1738  and the other sources. The last two coadded images for each filter on the night of 2013 August 23 suffered from poor seeing, and while the data were used for relative photometry, the absolute values in Table 2 exclude these two datapoints.

We compared the photometry of W1738 and the five fainter sources to the six brighter reference stars, and also compared each bright reference star to the other five reference stars as a check on data quality. Figures 2 and 3 show the resulting light curves for each night. These are discussed below in \S 3.

\subsection{{\em Spitzer} IRAC Observations}

W1738 was observed in both the [3.6] and [4.5] filters of the IRAC camera on {\em Spitzer}, with the filters observed consecutively. For each filter 12.0 hour
long observations were obtained, and the 24.0 hour long observation with the filter pair was repeated four months later. The first epoch ran from UT 2013 June 29
19:22:39.36 to 2013 June 30 07:23:48.48 for [3.6], and 2013 June 30 07:30:00.00 to 19:31:26.40 for [4.5]. The second epoch ran from UT 2013 October 29
18:29:22.56 to 2013 October 30 06:30:31.68 for [3.6], and 2013 October 30 06:36:25.92 to 18:37:52.32 for [4.5]. The data were obtained as part of program 90015
during campaigns 35100 and 35900. The data were obtained and reduced in the same way as that for W1405, as decribed in Cu16. Briefly, the
100-second images were obtained in ``staring" mode for each filter. Photometric analysis starts with the Basic Calibrated Data frames, which are converted from
units of MJy sr$^{-1}$ to electrons.
Aperture photometry with a radius of 3 pixels and a background annulus is then obtained using custom
Interactive Data Language (IDL) code. For W1738 the field was more crowded than that of W1405, and extra care had to be taken with placement of
the sky annuli. Data points that were extreme outliers, exceeding the median by 50 times the median absolute deviation, were removed.
Figure 4 shows the light curves obtained. The apparently bright data points are due to hot pixels which are caused by cosmic rays.

The average  brightness of the target did not change significantly between June and October 2013. We derive from the processed mosaics downloaded from the {\em Spitzer} archive [3.6] $=  16.89 \pm  0.08$ (where the uncertainty is dominated by the uncertainty in the sky value) and [4.5] $= 14.46 \pm 0.01$.

\section{Results: Variability  of W1738}

Figure 4 indicates that W1738 is variable in the [4.5] bandpass with an approximately six-hour period. We ran a simple  Fourier Transform analysis for each of the two [4.5] data sets, not taking into account the truncation of the time series. Figure 5 plots the result --- a  strong peak is found at a period of 6.0 hours for the 2013 June 30 data, and less significant
peaks at 3.0 and 6.0 hours for the 2013 October 30 dataset. We also ran a Lomb-Scargle periodogram analysis on both the [3.6] and [4.5] data, on the two epochs. A significant period was only found for the 2013 June 30 [4.5] data, with a broad
peak in power between 5 and 7 hours.

The near-infrared light curves shown in Figures 2 and 3 suggest that W1738 may also vary at near-infrared wavelengths.
In Figure 2,  both $Y$ and $J$ brighten at around 9 and 12 hours. Trends are less clear in Figure 3, but there
appears to be another three hour cycle present for both filters, with a minimum around 7.9 hours and maxima at 6.4 and 9.4 hours.  Table 3 compares the peak-to-peak variation (range) and standard deviation of the curves obtained for W1738 and the five faint point sources (numbered 2, 4, 8, 9, and 11  in Figure 1).  The stars are listed from faintest to brightest in the Table. As expected, the range and standard deviation generally increase with decreasing brightness.
Figures 2 and 3 show that the dispersion for W1738 is similar to that of the comparison star that is fainter, and larger than that of the similarly bright comparison star. The dispersion is largely driven by two or three  data points however and the Y dwarf should be monitored for a longer period of time to confirm the presence of any variability.
The standard deviation of the light curves for the five reference stars is on average $1.2 \times$ the error, while that for W1738 is  $1.9 \times$ the error. The fact that the three hour cycle is also seen in the mid-infrared data, and that models calculate a $1\, \mu$m variability amplitude $\sim 10 \times$ the $4\, \mu$m variability amplitude (\S 4.2), indicates that the near-infrared variability may be real. We interpret the data as implying that W1738 is variable at
$1\, \mu$m
at the  $\lesssim 2 \sigma$ confidence level.

Based on the Fourier transform results and our visual inspection of the light curves, three- and six-hour cycles are present in the W1738 data.
We therefore fit the [4.5] light curves assuming a double sinusoid model, where the second sinusoid has a period half the first. The amplitudes and phases of the two
sinusoids can vary freely. A possible physical explanation of this double sinusoid, where the second period is half the first, is presented in \S 4.3. The
model is: $$F(t) = A_1\sin\left(\frac{2\pi}{P}t+\phi_1\right) + A_2\sin\left(2\frac{2\pi}{P}t+\phi_2\right) + C$$ where $A$ is semi-amplitude in \%, $P$ is period in
hours, $t$ is time in hours, $\phi$ is phase in radians and $C$ is a constant.

The fitting procedure is described in detail in Cu16.
Briefly, we assume the uncertainties are gaussian and account for the bad data points following Hogg, Bovy \& Lang (2010) whereby we assume that they are generated from a normal distribution with a mean  $y_{\rm bad}$ and a variance of $\sigma_{\rm bad}$.  The joint posterior distribution of the parameters $A_1$, $A_2$, $\phi_1$, $\phi_2$, $P$, $C$, $\sigma_{\rm good}$, ${\cal P}_{bad}$ (the probability that a given data point is bad) is sampled using a Markov Chain Monte Carlo method. Distributions for each model parameter are computed by marginalizing over the other parameters.

The one and two dimensional projections of the posterior probability distributions of the model parameters for the 2013 June 30 [4.5] sequence is shown in Figure
6. Figures 7 and 8 show the resulting best fits to the [4.5] data for the two epochs, and the residuals between fit and data.  As was also indicated by the
initial Fourier transform, the data from 2013 June 30 is dominated by the longer-period sinusoid, while the 2013 October 30 data consists of two sinusoids with
almost equal amplitude.  A good fit could not be achieved with the noisy [3.6] data, however extrapolating the sinusoids to earlier times shows
that the [3.6] data are not inconsistent with variability at the same amplitude and phase as the [4.5] data (Figure 4). Table 4 gives semi-amplitude, period and
phase for each of the two sinusoids on each epoch, using the [4.5] data only.

\section{Discussion}

\subsection{Observed Variability in Brown Dwarfs and Giant Planets}

There are no other published studies of variability at both near-infrared and mid-infrared wavelengths for Y dwarfs at the time of writing. Nine T dwarfs do have
such data published and these results are summarized in Table 5. Variability is found in five of the nine T dwarfs.
Generally the periods are found to be the same at both near- and mid-infrared wavelengths, and the near-infrared amplitudes are similar to or larger than the mid-infrared amplitude
(although the near- and mid-infrared data are likely to not have been taken at the same time, and amplitudes are likely to vary with time).

The variability measurements presented here for W1738 are in general agreement with results from other brown dwarf studies.
Crossfield (2014) presents a multi-wavelength collation of variability amplitude and period, and projected rotational velocities, for late-type M, L and T dwarfs. The Crossfield sample of L and T dwarfs has variability periods of 1.5 -- 10 hours and  peak-to-peak amplitudes of 0.1 -- 120\%, although most have amplitudes of 1 -- 15 \%. The {\em Spitzer} study of 39 single L3 -- T8 dwarfs by Metchev et al. (2015) found that 19 varied; peak-to-peak amplitudes were 0.8 -- 4.6\% and periods were 1.6 -- 24 hours.
Recently Zhou et al. (2016) have determined a rotation period of about 11 hours for the young planetary-mass L dwarf 2MASSWJ 1207334-393254b, based on near-infrared variability with  peak-to-peak amplitudes of $\sim$ 2\%.
In Cu16 we monitored the Y0.5(pec) W1405 with  {\em Spitzer} and found  variability with  peak-to-peak amplitude of 7\% and a period of 8.5 hours.
Cu16 showed that the variability could be reproduced by a single bright spot model, with the light curve period equal to the rotational period, meaning that W1405  has the longest rotation period measured to date for spectral types later than T3.

Variability seen in the surface features of the solar system gas giants are also a useful reference point for cold Y dwarfs.  Gelino \& Marley (2000) used observations of Jupiter to show that, were it to be unresolved, 20\% variability would be observed at $4.8\, \mu$m with a period equal to the planet's rotation period of 9.9 hours. The variability in this case is predominantly due to Jupiter's Great Red Spot.
Sromovsky et al. (2012) present observations of episodic bright and dark spots on Uranus. One and then two bright spots were seen on the planet's surface in 2011, drifting at very different rates, and evolving over a period of months.
Simon et al. (2016) present a 49-day light curve for Neptune using the $Kepler$ Space Telescope. The data are compared to contemporaneous images taken with the Keck telescope at $1.65\, \mu$m and to {\em Hubble Space Telescope (HST)} visible imaging taken several months later. The authors find that a single large, long-lived, storm that is seen in their Keck images dominates the $Kepler$ and  $HST$ light curves. The periodicity of the long-term variability is consistent with the planet's rotation and surface wind speed at the latitude of the storm. The short-term variability  is interpreted as being due to smaller or fainter clouds.

\subsection{Models of Variability in Brown Dwarfs and Giant Planets}

Three-dimensional simulations of convection in brown dwarf and giant planet interiors by  Showman \& Kaspi (2013)
show that significant circulation is generated at both small and large scales.   Large-scale horizontal temperature variations of tens of K are produced, resulting in flux variations of a few percent on rotation timescales. Stratified turbulence can generate vortices and storms, and the circulation can support the formation of patchy clouds. These models
produce  vertical velocities consistent with the chemical mixing observed in brown dwarf atmospheres, plausible surface wind speeds, and time scales for the evolution of the light curve that are in agreement with observations.

It is likely that both thermal and cloud cover fluctuations are important sources of variability in giant planets and brown dwarfs (e.g. Morley et al. 2014, Robinson \& Marley 2014).
Although the $T_{\rm eff} \approx 400$ K atmospheres of Y dwarfs are expected to be essentially clear, Morley et al. (2012, their Figure 4) show that flux emitted from the $1\, \mu$m region originates deep enough in the atmosphere that it could be impacted by thin layers of KCl and Na$_2$S clouds. Morley et al. (2014) examine spectral variability that occurs if the clouds are patchy. Figure 1 of  Morley et al. (2014) shows the 0.7 -- $10\, \mu$m spectrum of a 400~K brown dwarf where one hemisphere has 30\% cloud cover and the other 70\%, and as a result also differ by 5~K or $> 1$\% of the $T_{\rm eff}$. The difference in the resulting spectra, which is the inferred peak-to-peak variability, is calculated to be $\sim$ 50\% at $1\,\mu$m ($Y$), 40\% at $1.2\,\mu$m ($J$) and 2\% at $4.5\, \mu$m. We ran a similar model with a covering fraction of 45\% on the Eastern hemisphere and 55\% on the Western hemisphere, which produced peak-to-peak variability of  7\% in $Y$, 5\%
in $J$, 0.5\% in [3.6] and 0.8\% in [4.5]. Thus, atmospheres with patchy clouds and small variations in surface temperature can reproduce the size of variability we observe at [4.5], and have tentatively observed at $Y$ and $J$.

\subsection{Interpretation of the W1738 Variability}

Physical constraints on the rotation period of brown dwarfs can be helpful in determining the source of any observed variability. If the variation is caused by
multiple surface features such as clouds or several discrete storms, then it can have a period smaller than the rotation period. If it is due to a single feature
such as Jupiter's Red Spot or the single bright spot model used to reproduce the W1405 variability, then the period will be equal to the rotation period. The
lower limit on the rotation period of any stable object can be estimated by assuming solid body rotation and constraining the surface velocity to be less than
the escape velocity. Adopting a radius equal to that of Jupiter's (which is approximately true for most brown dwarfs, see Burrows et al. 1997), we find that the
period $P > 2.1 / \sqrt M$ hours, where the mass of the brown dwarf is $M$ Jupiter masses. Marley \& Sengupta (2011, their Figure 2) more accurately derive a
breakup velocity, and show that for brown dwarfs or gas planets older than 0.1 Gyr the lower limit on rotation period is 5 hours or 1.8 hours for a 1 or a 10
Jupiter mass object, respectively. For W1738 with mass $\approx$ 5 Jupiter masses and age between 0.15 and 1.0 Gyr (Leggett et al. 2016), the rotational period
must be greater than about 3 hours. We have found two dominant periodicities, three hours and six hours, in our data sets. The three hour
component is very close to the break up speed
-- which is very unlikely -- suggesting that there are multiple features on the surface of the brown dwarf.

Metchev et al. (2015) found substantial power in periods approximately half of the best-fit period for three
of the 19 variable L and T dwarfs in their sample.   Studies of pre-main-sequence stars have also found photometric variability with periods separated by a factor of two (Tackett, Herbst \& Williams 2003; Herbst 1989).
The pre-main-sequence observations could be reproduced by a model where the
rotation period was equal to the longer of the two periods, the
star was viewed close to equator-on, and there were two similar spots, one on each hemisphere. The model showed that the shape and amplitude of the light curve evolved as the spots drifted in longitude (Herbst 1989).
We attempted to fit a model with two equally bright spots to the W1738 data, where the spots are circular and  have their own longitude, latitude and size. The first epoch, where there is a dominant sinusoid, could be reasonably well reproduced by a two spot model, but not the second epoch, where there are two sinusoids of different frequency of almost equal amplitude. We did not explore models with more than two spots.

We adopt the rotation period for W1738 to be the longer of the two periods observed, $6.0 \pm 0.1$ hours, and interpret the double-sinusoid variability as evidence of there being one or more large features present in its atmosphere. The features evolve over a period of months, such that in June 2013 the variability is dominated by a single spot or storm, and by October 2013 smaller features have developed giving rise to the shorter period variability observed. This configuration of a dominant large system which is long-lived, and multiple smaller surface features,
is similar to what was observed for Neptune by Simon et al. (2016, \S 4.1). We note that  for W1738 the amplitude of the variability, and the suggested wavelength dependence of the amplitude, is well reproduced by models with patchy thin clouds (\S 4.2). The models have one hemisphere slightly more than half-covered and the other slightly less than half-covered with clouds. A hemisphere half-covered in clouds could also be described as a hemisphere with a large spot, depending on the wavelength involved, and the height of the clouds.

\section{Conclusion}

We obtained 12 hours of continuous {\em Spitzer} data on the Y0 brown dwarf W1738 at [3.6], followed by another 12 hours at [4.5]. Two sets of data were obtained
four months apart, on June 30 and October 30 2013. We also obtained interspersed Gemini $Y$ and $J$ data on W1738, with about 1.4 hours on-source at $Y$ and 2.3
hours on-source at $J$, on two occasions separated by about one month, July 17 and August 23 2013.

Fourier and Lomb-Scargle analyses of the mid-infrared [4.5] data suggested the presence of three and six hour periods in the data.  We use a probabilistic method to fit double sinusoids to the [4.5] data (the near-infrared data covers a short time span and the [3.6] data are
too noisy). We constrain the second sinusoid to have half the period of the first, but allow amplitude and phase to vary. We find sinusoids with periods of 5.8
$\pm$ 0.1 hours and 6.13 $\pm$ 0.08 hours, and half those values, reproduce the observed [4.5] light curves on the two epochs well. The amplitudes range from 0.3\%
to 1.1\%, leading to peak-to-peak variability of 3\%. The shorter time-span near-infrared data was inspected visually only. The data suggest that W1738 is also variable in the near-infrared, although only at the $\lesssim 2 \sigma$ confidence level. If real, the implication is that this Y0 dwarf varied by 10 -- 30\% in $Y$ and 5 -- 15 \% in $J$, peak-to-peak, with a period of about three hours, at two epochs during 2013.

The observations are consistent with W1738, a 5 Jupiter-mass $T_{\rm eff} \approx 400$ K Y-type brown dwarf, being seen nearly-equator on, having a rotation
period of $6.0 \pm 0.1$ hours, and having one or more large surface features which give rise to the variability. The features evolve over timescales of months. The
observed variability at $\lambda \sim 4\,\mu$m is likely due to thermal variations caused by atmospheric circulation, while the larger variations at $\lambda
\sim 1\,\mu$m, if real, may be due to the presence of patchy clouds of KCl and Na$_2$S in the lower regions of the atmosphere.

\acknowledgments

Based in part on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for
Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia),
    Minist\'{e}rio da Ci\^{e}ncia, Tecnologia e Inova\c{c}\~{a}o (Brazil)
    and Ministerio de Ciencia, Tecnolog\'{i}a e Innovaci\'{o}n Productiva
    (Argentina).
Based in part on observations obtained with the {\em Spitzer Space Telescope}, which is operated by  the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.
S. L.'s research is supported by Gemini Observatory.  D.S.' work was supported in part by
NASA grant
      NNH12AT89I from Astrophysics Theory.

We thank the referee for comments which greatly improved the paper.


\section{Introduction}

Fundamental laws of physics are usually governed by at most second order differential equations, the reason for that being quite clear. Ostrogradsky's approach \cite{Ostrogradsky1850} to higher-derivative theories produces terms in the Hamiltonian, which are linear in particles' momenta. This imminently leads to an unbounded from below Hamiltonian at classical level, hence instabilities and ghost problem after quantisation. One should not, however, rashly label a higher-derivative theory unphysical having in mind the following argument. Consider a system of two second-order differential equations which gives stable dynamics. Generally, this system is equivalent to one differential equation of fourth order. It may happen that the Ostrogradsky's approach gives boundless negative energy spectrum, which does not necessarily mean unstable system. The explanation is that the Ostrogradsky's energy may not coincide with the energy given by the system of two second-order differential equations.

The Pais-Uhlenbeck oscillator (PUO) \cite{Pais:1950za} has been proposed as a prototype of higher-derivative theories of gravity, which are renormalisable \cite{PhysRevD.16.953}. This simple mechanical model has been subject to renewed interest in recent years due to many works inferring that the PUO can be regarded as a viable physical system. Owing to the presence of higher-order time derivatives, the PUO cannot be quantised using the canonical approach. Instead one uses the Dirac constraints method to construct the correct Hamiltonian, which is diagonal with positive and negative states \cite{Mannheim:2004qz}. Taking a limit corresponding to equal frequencies, in which the second-order derivative vanishes leaving pure fourth-order PUO, the negative norm states go off-shell thus purging the theory.

The first step towards quantisation of the PUO is a suitable Hamiltonian definition, which improves the Ostrogradsky's one. The search for proper Hamiltonian is based on the observation that there always exists at least two quadratic in variables integrals of motion \cite{2005AcPPB..36.2115B}. Since the Hamiltonian is also an integral of motion, one can make a general ansatz for it. Consequent requirement of equivalence between the corresponding equations of motion and the initial Lagrange-Euler equations leads to a derivation of Poisson-like structure. Thus the efforts to find the right Hamiltonian resulted in a bunch of alternative Hamiltonian formulations of the PUO \cite{Mostafazadeh:2010yw,Masterov:2015ija,Masterov:2016jft}. Furthermore, the stability of this system has been studied using both analytical and numerical methods \cite{2013MPLA...2850038P,Pavsic:2013noa,Pavsic:2013mja}.

Another interesting problem is the question of entangled states. Ever since 1935, entanglement \cite{PhysRev.47.777,Schrodinger1935} has been challenging our understanding of quantum phenomena. Taking advantage of the presence of well-defined Hamiltonian formulation of the PUO giving viable physics, in this paper we are interested in studying the quantum entanglement of systems of such oscillators. One convenient way to deal with this problem is to use the framework of thermo-field dynamics (TFD) of a double Hilbert space \cite{2013PhyA..392.3518H}. In this context, it has been shown recently that the entanglement entropy of coupled harmonic oscillators depends on the temperature and coupling parameters \cite{Nakagawa:2016ady}. Alternatively, the entanglement entropy can be derived from holography \cite{Ryu:2006bv,Ryu:2006ef} (for a review see \cite{Nishioka:2009un}). The idea is analogous to the Bekenstein-Hawking entropy of black holes. Specifically, the entanglement entropy of $(d+1)$-dimensional conformal field theory is related by the AdS/CFT correspondence to the area of a static $d$-dimensional minimal surface in $\text{AdS}_{d+2}$. Hence, one can study certain aspects of string theory on microscopic level by making use of thermodynamic quantities such as quantum entanglement entropy.

The rest of this paper is structured as follows. In section \ref{sec:oscHolo} we focus on the appearance of higher-derivative oscillators in the context of the AdS/CFT correspondence. We emphasise the importance of the knowledge one can gain within this approach for understanding of particular corners of information space in this context. In section \ref{sec:two1D4PUOs} we consider two minimally coupled PUOs and compute their entanglement entropy. In section \ref{sec:N1D4PUOs} we refer to the more general case of a ring of PUOs and compute the entanglement entropy of one of the oscillators (first subsystem) and the others (second subsystem). We finalise our discussion with some concluding remarks in section \ref{sec:conclusion}. In appendix \ref{sec:BCM} we set aside some technical details concerning the Hamiltonian diagonalisation.

\section{Appearance of higher-derivative oscillators in holographic models}
\label{sec:oscHolo}

\subsection{Reduction of gauge theories with CS terms to PUOs}
\label{subsec:oscCS}

Higher-derivative theories naturally arise in string theory in the context of holography. One beautiful example of a holographic correspondence is the equivalence between 3D topological Chern-Simons gauge theory and the chiral half of a rational conformal field theory \cite{Witten1:1989, Axelrod:1991}. It appears as a special case of the celebrated AdS/CFT correspondence, namely the case of superstring theories on space-times of the form ${\rm{AdS}}_3\times K^7$, where $K^7$ is some compact space. It is shown that the low-energy supergravity theory on ${\rm{AdS}}_3$ typically contains gauge fields with Chern-Simons terms. The relevance of topological field theories to the gauge/gravity correspondence was first discussed in \cite{Witten2:1998}. Besides in string theory, the importance of massive Chern-Simons theories have also been recognized in condensed matter physics with a view towards quantum computations \cite{Freedman:2004}.

In this subsection we would like to show that under some assumptions the massive Chern-Simons theories naturally reduce to higher-derivative oscillators, namely Pais-Uhlenbeck oscillators. To demonstrate this point we will elaborate on the simplest example of a topologically massive electrodynamics described by the Lagrangian \cite{Schonfeld:1981, Deser:1982, Deser2:1999}:
\begin{equation}\label{topmassive ED}
{\cal L} =  - \frac{1}{4}\,{F_{\mu \nu }}\,{F^{\mu \nu }} - {\varepsilon ^{\mu \nu \lambda }}\,\left( {\frac{m}{4}\,{A_\mu }\,{F_{\nu \lambda }} + \frac{k}{m}\,{F_{\mu \sigma }}\,{\partial ^\sigma }{F_{\nu \lambda }}} \right)\,,\quad {F_{\mu \nu }} = {\partial _\mu }{A_\nu } - {\partial _\nu }{A_\mu }\,,
\end{equation}
where $k$ is a dimensionless constant. The  given (2+1)-dimensional system (\ref{topmassive ED}) can be reduced to a mechanical one via the substitution $A^\mu(x)\to\sqrt{m}\,r^\mu(t)$ and $B^2=m^2\,e^{-1}$. The resulting  Lagrangian is
\begin{equation}\label{mechlag}
L = \frac{1}{2}\,m\,\dot x_i^2 + \frac{{e\,B}}{2}\,{\varepsilon _{ij}}\,{x_i}\,{{\dot x}_j} + k\,{\varepsilon _{ij}}\,{{\dot x}_i}\,{{\ddot x}_j}\,.
\end{equation}
In (\ref{topmassive ED}) we impose the Weyl gauge, $A^0=0$, which  effectively leads to the elimination of the variable $x^0$. Thus it is evident that the Lagrangian (\ref{mechlag}) describes a charged higher-derivative oscillator coupled to an external homogeneous magnetic field $B$. The equations of motion of the effective theory are given by
\begin{align}
&x_1^{(3)}(t) - \frac{{B\,e}}{{2\,k}}\,{{\dot x}_1}(t) - \frac{m}{{2\,k}}\,{{\ddot x}_2}(t) = 0\,,\\
&x_2^{(3)}(t) - \frac{{B\,e}}{{2\,k}}\,{{\dot x}_2}(t) + \frac{m}{{2\,k}}\,{{\ddot x}_1}(t) = 0\,.
\end{align}
Making the substitution $y_i(t)=\dot x_i(t)$ this system transforms to a fourth-order PU oscillator:
\begin{equation}\label{4thorderPUOeq}
y_i^{(4)}(t) + \,\left( {\omega _1^2 + \omega _2^2} \right)\,{{\ddot y}_i}(t) + \omega _1^2\,\omega _2^2\,{y_i}(t) = 0\,.
\end{equation}
The oscillator frequencies are related to the parameters of the mechanical system by the following expressions:
\begin{equation}
\omega _1^2 + \omega _2^2 = \frac{{{m^2}}}{{4\,{k^2}}} - \frac{{B\,e}}{k},\,\quad \omega _1^2\,\omega _2^2 = \frac{{{B^2}\,{e^2}}}{{4\,{k^2}}}\,.
\end{equation}
The system (\ref{mechlag}), however, as  most higher-derivative models, has a spectrum unbounded from below. This can be cured by supplying the coupled system with an appropriate constraints \cite{Horvathy:2005}. Classically this is equivalent to replacing the higher-derivative Lagrangian
(\ref{mechlag}) with the first-order exotic Duval-Horvathy Lagrangian \cite{Duval:2000,Duval:2005},
\begin{equation}\label{HD exotica}
L = {P_i}\,\dot x_i^{} - \frac{1}{{2\,m}}\,P_i^2 + \frac{\theta }{2}\,{\varepsilon _{ij}}\,{P_i}\,{{\dot P}_j} + \frac{{e\,B}}{2}\,{\varepsilon _{ij}}\,{x_i}\,{{\dot x}_j}\,,
\end{equation}
which generates equations of motion with the effective mass $\mu  = m\,(1 - e\,B\,\theta )$, ${P_i} = \mu \,{{\dot x}_i}$, ${{\dot P}_i} = e\,B\,{\varepsilon _{ij}}\,{{\dot x}_j}$. The exotic particle system (\ref{HD exotica}) can also be obtained by a reduction of another (2+1)-dimensional Abelian gauge field theory given by the Lagrangian with several Chern-Simons terms \cite{Hagen:1987, Deser3:1987},
\begin{equation}
{\cal L} = {\varepsilon ^{\mu \nu \lambda }}\,{\Phi _\mu }\,{\partial _\nu }{A_\lambda } - \frac{\lambda }{2}{\Phi _\mu }\,{\Phi ^\mu } - \frac{k}{{2\,m}}\,{\varepsilon ^{\mu \nu \lambda }}\,{\Phi _\mu }\,{\partial _\nu }{\Phi _\lambda } - \frac{{\beta \,m}}{2}\,{\varepsilon ^{\mu \nu \lambda }}\,{A_\mu }\,{\partial _\nu }{A_\lambda }\,.
\end{equation}

In the next subsection we turn our attention to the appearance of higher-derivative oscillators in another interesting holographic model, namely the Pilch-Warner supergravity solution.

\subsection{Higher-derivative oscillators and the Pilch-Warner solution}
\label{subsec:oscPWsol}

The Pilch-Warner (PW) flow geometry \cite{Pilch:2000ej,Pilch:2000fu} is a solution of five-dimensional $\mathcal{N}=8$ supergravity lifted to ten dimensions. It interpolates between two critical points of the scalar potential. In the ultraviolet (UV) critical point the solution reproduces the maximally supersymmetric $\text{AdS}_5\times S^5$ type IIB background. In the infrared (IR) critical point it gives another type IIB supergravity solution, which represents a warped $\text{AdS}_5$ times a squashed five-sphere geometry. The solution preserves $1/8$ of the supersymmetry all over the flow, except for the IR point where the supersymmetry is enhanced to $1/4$. In this section we will focus on the IR point, in which the gravitational theory corresponds by holography to a large $N$ limit of the $\mathcal{N}=1$ superconformal theory of Leigh-Strassler \cite{Leigh:1995ep}.

We will now show that higher-derivative oscillators naturally appear in the Penrose limit of the Pilch-Warner solution in the presence of $B$-field. The pp-wave limit of the PW solution in its IR point is investigated in \cite{Brecher:2002ar} for a particular choice of null geodesic, which corresponds to the moduli space of a D3-brane probe. The relevant fields that will contribute the bosonic part of the world-sheet action are the metric and the NS-NS antisymmetric two-form field $B_2$. For a gauge choice, in which the $B$-field permeates in the $X^1$, $X^2$, $X^3$, $X^4$, and $U$ directions, the action takes the form
\begin{align}
	S_B=-\frac{1}{4\pi\alpha'}\int\mathop{}\!\mathrm{d}\tau\mathop{}\!\mathrm{d}\sigma&\left[\sqrt{-g}g^{\alpha\beta}
	\left(2\partial_\alpha U\partial_\beta V+A_{ij}X^iX^j\partial_\alpha U\partial_\beta U
	+\partial_\alpha X^i\partial_\beta X^i\right)\right.\nonumber\\
	&\left.-2\sqrt{3}E\epsilon^{\alpha\beta}\left(X^1\partial_\alpha U\partial_\beta X^3
	-X^2\partial_\alpha U\partial_\beta X^4\right)\right],
\end{align}
where $\epsilon^{01}=1$ and the pp-wave spectrum matrix is $A_{ij}=\text{diag}[1,1,4]$. The quantity $E$ is the conserved energy corresponding to the Killing vector $\partial/\partial\tau$. A convenient world-sheet gauge is $g_{\alpha\beta}=\eta_{\alpha\beta}$ and the world-sheet coordinates are $\sigma^0=\tau$, $\sigma^1=\sigma$. The equation of motion for $U$ is $\nabla^2U=0$. In light-cone gauge $U=\alpha'p^+\tau+\text{const}$, the equations of motion for the world-sheet scalars are:
\begin{align}
\label{eq:EoMsys}
	\nabla^2X^1-4M^2X^1+\sqrt{3}M\partial_\sigma X^3&=0,\nonumber\\
	\nabla^2X^2-4M^2X^2-\sqrt{3}M\partial_\sigma X^4&=0,\nonumber\\
	\nabla^2X^3-M^2X^3-\sqrt{3}M\partial_\sigma X^1&=0,\nonumber\\
	\nabla^2X^4-M^2X^4+\sqrt{3}M\partial_\sigma X^2&=0,\nonumber\\
	\nabla^2X^p-M^2X^p&=0,
\end{align}
where $p=5,6,7,8$ are the directions unaffected by the $B$-field and $M=E\alpha'p^+$. By making use the ansatz $X^i(\tau,\sigma)=e^{i\sigma}x_i(\tau),~i=1,\dotsc,8$, the system of second-order partial differential equations \eqref{eq:EoMsys} boils down to the following equivalent system:
\begin{align}
\label{eq:EoMPUO}
	x_q^{(4)}+\left(5M^2+2\right)x_q^{(2)}+\left(4M^4+2M^2+1\right)x_q&=0,\nonumber\\
	x_p^{(2)}+\left(M^2+1\right)x_p&=0,
\end{align}
where $q=1,2$ and the derivative is taken with respect to $\tau$. Let us stress that the crucial point here is the inclusion of the Kalb-Ramond two-form $B_2$. Its role is evident from the form of the system of partial differential equations \eqref{eq:EoMsys}---the $B$-field ties up the equations for $X^1,~X^3$ and $X^2,~X^4$ thus making the system equivalent to one, in which some of the differential equations are of higher order. Therefore we conclude that the presence of $B$-field results in dynamics, which can be modeled by higher-derivative Pais-Uhlenbeck oscillators.

On the other hand one can consider quadratic fluctuations around classical solutions. The case of rotating strings in PW background is considered in \cite{Dimov:2003bh}. Since we expect that the Kalb-Ramond field will play again significant role, we will consider only the $S^5$ part of the geometry where the $B$-field is turned on. The Lagrangian, describing the quadratic fluctuations, is
\begin{align}
	\mathcal{L}_{S^5}^\text{tot}=\partial_\alpha\tilde{\zeta}^A\partial^\alpha\tilde{\zeta}^A
	+\tilde{M}^2\left(\tilde{\zeta}_{\tilde{2}}^2+\tilde{\zeta}_{\tilde{4}}^2\right)
	&+\left(4\tilde{M}^2+\frac{3}{2}\bar{\rho'}^2\right)
	\left(\tilde{\zeta}_{\tilde{3}}^2+\tilde{\zeta}_{\tilde{5}}^2\right)\nonumber\\
	&+2\sqrt{3}\tilde{M}\left(\tilde{\zeta}_{\tilde{4}}\partial_1\tilde{\zeta}_{\tilde{5}}
	-\tilde{\zeta}_{\tilde{2}}\partial_1\tilde{\zeta}_{\tilde{3}}\right),
\end{align}
where $\tilde{M}^2=\frac{4}{9}(c_\beta+c_\gamma+c_\phi)^2$ is a constant depending on the angular velocities $c_\beta$, $c_\gamma$, and $c_\phi$ of the string rotating in the directions $\beta$, $\gamma$, and $\phi$, correspondingly\footnote{For further details, please see \cite{Dimov:2003bh}.}. The corresponding equations of motion are:
\begin{align}
	\nabla^2\tilde{\zeta}_{\tilde{1}}&=0,\nonumber\\
	\nabla^2\tilde{\zeta}_{\tilde{2}}-\tilde{M}^2\tilde{\zeta}_{\tilde{2}}+\sqrt{3}\tilde{M}
	\partial_1\tilde{\zeta}_{\tilde{3}}&=0,\nonumber\\
	\nabla^2\tilde{\zeta}_{\tilde{3}}-\left(4\tilde{M}^2+\frac{3}{2}\bar{\rho'}^2\right)
	\tilde{\zeta}_{\tilde{3}}-\sqrt{3}\tilde{M}\partial_1\tilde{\zeta}_{\tilde{2}}&=0,\nonumber\\
	\nabla^2\tilde{\zeta}_{\tilde{4}}-\tilde{M}^2\tilde{\zeta}_{\tilde{4}}-\sqrt{3}\tilde{M}
	\partial_1\tilde{\zeta}_{\tilde{5}}&=0,\nonumber\\
	\nabla^2\tilde{\zeta}_{\tilde{5}}-\left(4\tilde{M}^2+\frac{3}{2}\bar{\rho'}^2\right)
	\tilde{\zeta}_{\tilde{5}}+\sqrt{3}\tilde{M}\partial_1\tilde{\zeta}_{\tilde{4}}&=0,
\end{align}
where $\bar{\rho'}^2$ is $\sigma$-dependent. For the case of point-like string the above equations are equivalent to the pp-wave limit equations \eqref{eq:EoMsys}. Using the ansatz $\tilde{\zeta}_{\tilde{i}}=e^{i\sigma}y_i(\tau)$ leads to the following system of differential equations:
\begin{align}
\label{eq:EoMPUO2}
	y_p^{(4)}+\left[5\tilde{M}^2+2+\frac{3}{2}\bar{\rho}'^2\right]y_p^{(2)}
	+\left[4\tilde{M}^4+2\tilde{M}^2+1+\left(\tilde{M}^2+1\right)
	\frac{3}{2}\bar{\rho'}^2\right]y_p&=0,\nonumber\\
	y_1^{(2)}+y_1&=0,
\end{align}
where $p=2,4$. The first equation in \eqref{eq:EoMPUO2} is slightly different than the EoM of fourth-order PUO since $\bar{\rho'}^2$ is a function of $\sigma$. However, one can consider two limiting cases, i.e. short and long strings. In short-string approximation $\bar{\rho'}^2\approx1/\!\left(2\coth^2\rho_0-2\right)$, while in long-string approximation $\bar{\rho'}^2\approx\frac{1}{\pi^2}\log^2\left(\coth^2\rho_0-1\right)$, $\rho_0$ being the maximal value of $\rho(\sigma)$ \cite{Dimov:2003bh}. Hence, the coefficient in front of $y_p$ is constant in these two approximations and we again arrive at the fourth-order Pais-Uhlenbeck oscillator.
\\
\indent In order to elaborate on the consequences of these results in the following sections we will proceed with a detailed analysis of the fourth-order PUOs given in Eq. (\ref{4thorderPUOeq}).

\section{Two minimally coupled 1D fourth-order PU oscillators}
\label{sec:two1D4PUOs}

As a first step towards description of higher-derivative oscillators in the context of information geometry, we will consider two minimally coupled one-dimensional fourth-order Pais-Uhlenbeck oscillators. Following the Ostrogradsky's approach, the dynamics of such system is governed by an alternative Hamiltonian of decoupled harmonic oscillators \cite{Masterov:2015ija}\footnote{For an alternative Hamiltonian formulation of the odd-order PUO, please see \cite{Masterov:2016jft}.} supplemented with interaction term:
\begin{equation}
\label{eq:H2}
	H_2=\frac{1}{2}\sum_{\mu=1}^2\sum_{k=0}^1\mathrm{sgn}(\alpha_{\mu,k})
	\left(p_\mu^kp_\mu^k+\omega_{\mu,k}^2x_\mu^kx_\mu^k\right)
	+\frac{1}{2}\sum_{\mu\neq\nu=1}^2c_{\mu\nu}x_\mu x_\nu,
\end{equation}
where the index $\mu$ labels the different oscillators of fourth-order, any of them with distinct frequencies $\omega_{\mu,k}$. Since the oscillators are one-dimensional, we do not include spatial index in the canonical variables $\left(x_\mu^k,p_\mu^k\right)$. The canonical coordinates $\left(x_\mu^k,p_\mu^k\right)$ corresponding to the Poisson structure are defined through \cite{Pais:1950za}
\begin{align}
\label{eq:cancoord}
	x_\mu^0&=\sqrt{\frac{|\alpha_{\mu,0}|}{\omega_{\mu,1}^2-\omega_{\mu,0}^2}}
	\left(x_\mu^{(2)}+\omega_{\mu,1}^2x_\mu\right),&
	p_\mu^0&=\mathrm{sgn}(\alpha_{\mu,0})\frac{\mathop{}\!\mathrm{d} x_\mu^0}{\mathop{}\!\mathrm{d} t},\nonumber\\
	x_\mu^1&=\sqrt{\frac{|\alpha_{\mu,1}|}{\omega_{\mu,1}^2-\omega_{\mu,0}^2}}
	\left(x_\mu^{(2)}+\omega_{\mu,0}^2x_\mu\right),&
	p_\mu^1&=\mathrm{sgn}(\alpha_{\mu,1})\frac{\mathop{}\!\mathrm{d} x_\mu^1}{\mathop{}\!\mathrm{d} t}.
\end{align}
Consequently, the coordinate $x_\mu$ is expressed in terms of the canonical coordinates as
\begin{equation}
	x_\mu=\frac{\sqrt{|\alpha_{\mu,1}|}x_\mu^0-\sqrt{|\alpha_{\mu,0}|}x_\mu^1}
	{\sqrt{|\alpha_{\mu,1}||\alpha_{\mu,0}|\left(\omega_{\mu,1}^2-\omega_{\mu,0}^2\right)}}.
\end{equation}
Let us consider the physically interesting case, in which all constants $\alpha_{\mu,k}$ are positive, i.e. $\mathrm{sgn}(\alpha_{\mu,k})=1$. This choice will keep us away from the ghost problem after quantisation. Thus the Hamiltonian \eqref{eq:H2} can be represented as a sum of free and interaction parts, $H_2=H_2^\mathrm{F}+H_2^\mathrm{I}$, each of them given by:
\begin{equation}
	H_2^\mathrm{F}=\frac{1}{2}\sum_{\mu=1}^2\sum_{k=0}^1p_\mu^kp_\mu^k,\quad
	H_2^\mathrm{I}=\frac{1}{2}\sum_{\mu=1}^2\sum_{k=0}^1\omega_{\mu,k}^2x_\mu^kx_\mu^k
	+\sum_{k,l=0}^1\tilde{c}^{kl}x_1^kx_2^l,
\end{equation}
where the new coupling constant is
\begin{equation}
	\tilde{c}^{kl}=(-1)^{k+l}c_{12}\left[\alpha_{1,k}\,\alpha_{2,l}
	\left(\omega_{1,1}^2-\omega_{1,0}^2\right)\left(\omega_{2,1}^2-\omega_{2,0}^2\right)\right]^{-1/2}.
\end{equation}
The next step is to simultaneously diagonalise the kinetic and potential parts of the Hamiltonian. This will be done by introducing uniform coordinates in the phase space $\eta=\left(x_\mu^k,p_\mu^k\right)^\mathrm{T}$ and writing the Hamiltonian in matrix form
\begin{equation}
	H_2=\frac{1}{2}\eta^\mathrm{T}
	\begin{pmatrix}
		\Omega & 0\\
		0 & \mathds{1}_4
	\end{pmatrix}
	\eta,
\end{equation}
where the matrix $\Omega$ is given by
\begin{equation}
	\Omega=
	\begin{pmatrix}
		\omega_{1,0} & 0 & \tilde{c}^{00} & \tilde{c}^{01}\\
		0 & \omega_{1,1} & \tilde{c}^{10} & \tilde{c}^{11}\\
		\tilde{c}^{00} & \tilde{c}^{10} & \omega_{2,0} & 0\\
		\tilde{c}^{01} & \tilde{c}^{11} & 0 & \omega_{2,1}
	\end{pmatrix}.
\end{equation}
In the case of identical PUOs the frequencies are equal in pairs, i.e. $\omega_{1,0}=\omega_{2,0}=\omega_0$ and $\omega_{1,1}=\omega_{2,1}=\omega_1$, and the coefficients $\tilde{c}^{kl}$ become symmetric in their two indices. In order to simplify the notations we rename the coefficients as follows: $\tilde{c}^{00}=c_0$, $\tilde{c}^{11}=c_1$, $\tilde{c}^{01}=\tilde{c}^{10}=c_2$. Thus the matrix $\Omega$ turns into symmetric block circulant matrix with symmetric blocks:
\begin{equation}
\label{eq:OmegaWC}
	\Omega=
	\begin{pmatrix}
		W & C\\
		C & W
	\end{pmatrix},
	\quad W=
	\begin{pmatrix}
		\omega_0 & 0\\
		0 & \omega_1\\
	\end{pmatrix},
	\quad C=
	\begin{pmatrix}
		c_0 & c_2\\
		c_2 & c_1
	\end{pmatrix}.
\end{equation}
Now we can apply the procedure described in appendix \ref{sec:BCM}. The number of blocks that define the symmetric block circulant matrix $\Omega$ are $n=2$, hence $j=0,1$ and we obtain $\mathbf{H_0}=W+C$, $\mathbf{H_1}=W-C$. The corresponding eigenvalues $\lambda_{0\pm}$, $\lambda_{1\pm}$ and eigenvectors $\mathbf{v}_0$, $\mathbf{v}_1$ are given by the expressions:
\begin{align}
\label{eq:freq}
	\lambda_{0\pm}&=a_+\pm c_2h_+\rightarrow\mathbf{v}_{0\pm}=(b_+\pm h_+,1)^\mathrm{T},\nonumber\\
	\lambda_{1\pm}&=a_-\pm c_2h_-\rightarrow\mathbf{v}_{1\pm}=(b_-\pm h_-,1)^\mathrm{T},
\end{align}
where
\begin{align}
	a_\pm&=\frac{1}{2}\left[\omega_0+\omega_1\pm(c_0+c_1)\right],\nonumber\\
	b_\pm&=\frac{1}{2c_2}\left[c_0-c_1\pm(\omega_0-\omega_1)\right],\nonumber\\
	h_\pm&=\frac{1}{2c_2}\sqrt{2c_2^2+\left[c_0-c_1\pm(\omega_0-\omega_1)\right]^2}.
\end{align}
One should keep in mind that the parameters $\omega_0$, $\omega_1$, $c_0$, and $c_1$ are not independent because the eigenvalues $\lambda_{0\pm}$ and $\lambda_{1\pm}$ must be real.

The matrix $P$, which diagonalises the matrix $\Omega$, is constructed from the normalised $\mathbf{w}_j/||\mathbf{w}_j||$ eigenvectors \eqref{eq:eigenvecw} and takes the form
\begin{equation}
	P=\frac{1}{\sqrt{2}}
	\begin{pmatrix}
		\frac{b_++h_+}{\sqrt{1+(b_++h_+)^2}} & \frac{b_+-h_+}{\sqrt{1+(b_+-h_+)^2}} &
		\frac{b_-+h_-}{\sqrt{1+(b_-+h_-)^2}} & \frac{b_--h_-}{\sqrt{1+(b_--h_-)^2}}\\
		\frac{1}{\sqrt{1+(b_++h_+)^2}} & \frac{1}{\sqrt{1+(b_+-h_+)^2}} &
		\frac{1}{\sqrt{1+(b_-+h_-)^2}} & \frac{1}{\sqrt{1+(b_--h_-)^2}}\\
		\frac{b_++h_+}{\sqrt{1+(b_++h_+)^2}} & \frac{b_+-h_+}{\sqrt{1+(b_+-h_+)^2}} &
		\frac{-b_--h_-}{\sqrt{1+(b_-+h_-)^2}} & \frac{-b_-+h_-}{\sqrt{1+(b_--h_-)^2}}\\
		\frac{1}{\sqrt{1+(b_++h_+)^2}} & \frac{1}{\sqrt{1+(b_+-h_+)^2}} &
		\frac{-1}{\sqrt{1+(b_-+h_-)^2}} & \frac{-1}{\sqrt{1+(b_--h_-)^2}}
	\end{pmatrix}.
\end{equation}
Finally, taking into account that the matrix $P$ is orthogonal ($P^{-1}=P^\mathrm{T}$), we obtain the diagonalised Hamiltonian
\begin{equation}
\label{eq:H2diag}
	H_2=\frac{1}{2}\eta^\mathrm{T}
	\begin{pmatrix}
		PDP^\mathrm{T} & 0\\
		0 & \mathds{1}_4
	\end{pmatrix}
	\eta=\frac{1}{2}\widehat{\eta}^\mathrm{T}
	\begin{pmatrix}
		D & 0\\
		0 & \mathds{1}_4
	\end{pmatrix}
	\widehat{\eta},
\end{equation}
where the coordinate transformations and the diagonal matrix $D$ are given by:
\begin{equation}
	\widehat{\eta}=
	\begin{pmatrix}
		P^\mathrm{T} & 0\\
		0 & \mathds{1}_4
	\end{pmatrix}
	\eta,
	\quad D=\text{diag}[\lambda_{0+},\lambda_{0-},\lambda_{1+},\lambda_{1-}].
\end{equation}
Furthermore, we define creation and annihilation operators $\mathfrak{a}_j^\dagger$ and $\mathfrak{a}_j$, $j=1,\ldots,4$, as
\begin{align}
	\mathfrak{a}_j&=\frac{1}{\sqrt{2\hbar}}\left(\sqrt{\lambda_j}\widehat{x}_j
	+\frac{i}{\sqrt{\lambda_j}}\widehat{p}_j\right),\nonumber\\
	\mathfrak{a}_j^\dagger&=\frac{1}{\sqrt{2\hbar}}\left(\sqrt{\lambda_j}\widehat{x}_j
	-\frac{i}{\sqrt{\lambda_j}}\widehat{p}_j\right),
\end{align}
where $\widehat{x}_j$ and $\widehat{p}_j$ are the coordinate and momentum part of the transformed vector $\widehat{\eta}$, and the frequencies $\lambda_j$ are connected to the diagonal elements of the matrix $D$ by $[\lambda_1,\lambda_2,\lambda_3,\lambda_4]=[\lambda_{0+},\lambda_{0-},\lambda_{1+},\lambda_{1-}]^{1/2}$. The creation and annihilation operators satisfy the commutation relations
\begin{equation}
	[\mathfrak{a}_j,\mathfrak{a}_k]=[\mathfrak{a}_j^\dagger,\mathfrak{a}_k^\dagger]=0,\quad
	[\mathfrak{a}_j,\mathfrak{a}_k^\dagger]=\delta_{jk}.
\end{equation}
Thus the Hamiltonian \eqref{eq:H2diag} takes the standard form
\begin{equation}
\label{eq:H2quant}
	H_2=\sum_{j=1}^4\hbar\lambda_j\left(\mathfrak{a}_j^\dagger\mathfrak{a}_j+\frac{1}{2}\right).
\end{equation}
The Fock space is built up from the vacuum in the usual way,
\begin{equation}
	\ket*{\{n_j\}}=\prod_{j=1}^4\frac{(\mathfrak{a}_j^\dagger)^{n_j}}{\sqrt{n_j!}}\ket*{\{0\}},
\end{equation}
where $\ket*{\{n_j\}}=\ket*{n_1}\otimes\cdots\otimes\ket*{n_4}$. The energy spectrum is derived from the eigenvalue problem $H_2\ket*{\{n_j\}}=\mathcal{E}_{\{n_j\}}\ket*{\{n_j\}}$,
\begin{equation}
	\mathcal{E}_{\{n_j\}}=\sum_{j=1}^4\hbar\lambda_j\left(n_j+\frac{1}{2}\right).
\end{equation}
In terms of the excited states, which are orthonormal $\braket{\{m_j\}}{\{n_j\}}=\delta_{\{m_j\},\{n_j\}}$, the diagonal Hamiltonian \eqref{eq:H2quant} can be written in matrix form
\begin{equation}
	H_2=\sum_{\{n_j\}=0}^\infty\sum_{j=1}^4\hbar\lambda_j\left(n_j+\frac{1}{2}\right)\ket*{\{n_j\}}\bra*{\{n_j\}}.
\end{equation}
The partition function of the system is defined as
\begin{equation}
	Z(K_j):=\mathrm{Tr}_{\{j\}}e^{-\beta H_2}=
	\prod_{j=1}^4\frac{\exp(-K_j/2)}{1-\exp(-K_j)},
\end{equation}
where $\beta$ is the inverse temperature and $K_j=\hbar\lambda_j\beta$. Then the standard density matrix in equilibrium, $\rho_\text{eq}(K_j)=e^{-\beta H_2}/Z(K_j)$, takes the form
\begin{equation}
	\rho_\text{eq}(K_j)=\frac{1}{Z(K_j)}\sum_{\{n_j\}=0}^\infty\exp\left[-\sum_{j=1}^4K_j
	\left(n_j+\frac{1}{2}\right)\right]\ket*{\{n_j\}}\bra*{\{n_j\}}.
\end{equation}
Thermo-field dynamics explores a double Hilbert space defined as a direct product of two isomorphic Hilbert spaces---the original one with set of base vectors $\{\ket*{n}\}$ and its copy called the tilde space with bases $\{\ket*{\tilde{n}}\}$---namely $\{\ket*{n}\otimes\ket*{\tilde{m}}\}\equiv\{\ket*{n}\ket*{\tilde{m}}\}\equiv\{\ket*{n,\tilde{m}}\}$. Therefore one can define a statistical state by $\ket*{\Psi}=\sum_n\sqrt{\rho_\text{eq}}\ket*{n}\ket*{\tilde{n}}$, which can be easily computed for our system,
\begin{equation}
	\ket*{\Psi(K_j)}=\frac{1}{\sqrt{Z(K_j)}}\sum_{\{n_j\}=0}^\infty\exp\left[-\sum_{j=1}^4\frac{K_j}{2}
	\left(n_j+\frac{1}{2}\right)\right]\ket*{\{n_j\}}\ket*{\{\tilde{n}_j\}}.
\end{equation}
Using defined this way statistical state one can define extended density matrix by $\widehat{\rho}(K_j)=\ket*{\Psi(K_j)}\bra*{\Psi(K_j)}$:
\begin{equation}
	\widehat{\rho}(K_j)=\frac{1}{Z(K_j)}\sum_{\{n_j\}=0}^\infty\sum_{\{m_j\}=0}^\infty
	\exp\left[-\sum_{j=1}^4\frac{K_j}{2}\left(n_j+m_j+1\right)\right]
	\ket*{\{m_j\}}\bra*{\{n_j\}}\ket*{\{\tilde{m}_j\}}\bra*{\{\tilde{n}_j\}}
\end{equation}
Consequently, the renormalised extended density matrix $\widehat{\rho}_{1,2}(K_j)=\mathrm{Tr}_{3,4}\widehat{\rho}(K_j)$ of the first PUO (equivalently harmonic oscillators 1 and 2) is obtained by tracing out the second PUO (equivalently harmonic oscillators 3 and 4),
\begin{align}
	\widehat{\rho}_{1,2}(K_j)&=\frac{1}{Z(K_j)}
	\prod_{j=3}^4\frac{\exp(-K_j/2)}{1-\exp(-K_j)}\sum_{\substack{n_1,m_1\\n_2,m_2}}^\infty
	\exp\left[-\frac{K_1}{2}(n_1+m_1+1)-\frac{K_2}{2}(n_2+m_2+1)\right]\nonumber\\
	&\times\ket*{m_1,m_2}\bra*{n_1,n_2}\ket*{\tilde{m}_1,\tilde{m}_2}\bra*{\tilde{n}_1,\tilde{n}_2}.
\end{align}
Finally, one derives the extended entanglement entropy of the first PUO from the expression $\widehat{S}_{1,2}=-k_\text{B}\mathrm{Tr}_{1,2}\left[\widehat{\rho}_{1,2}\log\widehat{\rho}_{1,2}\right]$,
\begin{align}
\label{eq:entropy2}
	\widehat{S}_{1,2}(K_1,K_2)=\frac{k_\text{B}}{2}\coth\frac{K_1}{4}\coth\frac{K_2}{4}
	&\left[K_1\left(1+\coth\frac{K_1}{4}\right)+K_2\left(1+\coth\frac{K_2}{4}\right)\right.\nonumber\\
	&\left.-2\log\left[\left(e^{K_1}-1\right)\left(e^{K_2}-1\right)\right]\vphantom{\coth\frac{K_1}{4}}\right].
\end{align}
\begin{figure}[h]
	\centering
	\includegraphics[width=0.75\textwidth]{entropy.pdf}
	\caption{The entanglement entropy as function of $K_1$ and $K_2$ in units $k_\text{B}=1$.}
	\label{fig:entropy}
\end{figure}
The extended entanglement entropy $\widehat{S}_{1,2}$ as function of the inverse scaled temperatures $K_1$ and $K_2$ is visualised in figure \ref{fig:entropy} in units $k_\text{B}=1$. As can be seen from the figure, the entanglement entropy goes to infinity for large temperature (equivalent to small values of $K_1$ and $K_2$). Conversely, when the temperature approaches zero (corresponding to approaching infinity arguments of the function $\widehat{S}_{1,2}$), the entanglement entropy approaches zero, which is exactly the statement of Nernst's theorem.

\section{$N$ minimally coupled 1D fourth-order PU oscillators}
\label{sec:N1D4PUOs}

In this section we will slightly depart from the case considered in section \ref{sec:two1D4PUOs} by generalising the Hamiltonian \eqref{eq:H2} for description of $N$ coupled 1D fourth-order PUOs. Moreover, we will restrict ourselves to interaction only between the nearest neighbours over the chain. The generalisation to such Hamiltonian is obvious, namely
\begin{equation}
\label{eq:HN}
	H_N=\frac{1}{2}\sum_{\mu=1}^N\sum_{k=0}^1\mathrm{sgn}(\alpha_{\mu,k})
	\left(p_\mu^kp_\mu^k+\omega_{\mu,k}^2x_\mu^kx_\mu^k\right)
	+\frac{1}{2}\sum_{\langle\mu,\nu\rangle=1}^Nc_{\mu\nu}x_\mu x_\nu.
\end{equation}
Considering the case of physical significance, i.e. $\alpha_{\mu,k}>0$, one can split again the Hamiltonian to kinetic and interaction parts, $H_N=H_N^\mathrm{F}+H_N^\mathrm{I}$,
\begin{equation}
	H_N^\mathrm{F}=\frac{1}{2}\sum_{\mu=1}^N\sum_{k=0}^1p_\mu^kp_\mu^k,\quad
	H_N^\mathrm{I}=\frac{1}{2}\sum_{\mu=1}^N\sum_{k=0}^1\omega_{\mu,k}^2x_\mu^kx_\mu^k
	+\frac{1}{2}\sum_{\langle\mu,\nu\rangle=1}^N\sum_{k,l=0}^1\tilde{c}_{\mu\nu}^{kl}x_\mu^kx_\nu^l,
\end{equation}
where the constant $\tilde{c}_{\mu\nu}^{kl}$ is defined as
\begin{equation}
	\tilde{c}_{\mu\nu}^{kl}=(-1)^{k+l}c_{\mu\nu}\left[\alpha_{\mu,k}\,\alpha_{\nu,k}
	\left(\omega_{\mu,1}^2-\omega_{\mu,0}^2\right)\left(\omega_{\nu,1}^2-\omega_{\nu,0}^2\right)\right]^{-1/2}.
\end{equation}
Thus defined, the constants $\tilde{c}_{\mu\nu}^{kl}$ are symmetric under exchange of pair upper and lower indices, i.e. $\tilde{c}_{\mu\nu}^{kl}=\tilde{c}_{\nu\mu}^{lk}$. This symmetry has been lost in the definition of the corresponding constant in section \ref{sec:two1D4PUOs} because we explicitly did the sum over the two lower indices. We will focus our attention on a chain of identical PUOs, therefore we have to close the chain by imposing $\tilde{c}_{N\,N+1}^{kl}=\tilde{c}_{N1}^{kl}$. Introducing again the vector $\eta=(x_\mu^k,p_\mu^k)^\mathrm{T}$, the Hamiltonian can be written in matrix form as
\begin{equation}
	H_N=\frac{1}{2}\eta^\mathrm{T}
	\begin{pmatrix}
		\Omega & 0\\
		0 & \mathds{1}_{2N}
	\end{pmatrix}
	\eta,
\end{equation}
where the matrix $\Omega$ has a block tridiagonal structure
\begin{equation}
	\Omega=
	\begin{pmatrix}
		W_1 & C_1 & 0 & \cdots & C_N^\mathrm{T}\\
		C_1^\mathrm{T} & W_2 & C_2 & \cdots & 0\\
		0 & C_2^\mathrm{T} & \ddots & \ddots & \vdots\\
		\vdots & \vdots & \ddots & \ddots & C_{N-1}\\
		C_N & 0 & \cdots & C_{N-1}^\mathrm{T} & W_N
	\end{pmatrix}
\end{equation}
and $W_j$ and $C_j$ are square $2\times2$ matrices given by:
\begin{align}
	W_j&=
	\begin{pmatrix}
		\omega_{j,0} & 0\\
		0 & \omega_{j,1}
	\end{pmatrix},
	~j=1,\ldots,N;\\
	C_j&=
	\begin{pmatrix}
		\tilde{c}_{j\,j+1}^{0\,0} & \tilde{c}_{j\,j+1}^{0\,1}\\
		\tilde{c}_{j\,j+1}^{1\,0} & \tilde{c}_{j\,j+1}^{1\,1}
	\end{pmatrix},
	~j=1,\ldots,N.
\end{align}
Let us consider again the case of identical PUOs, namely $\omega_{1,k}=\omega_{2,k}=\ldots=\omega_{N,k}$ and $\alpha_{1,k}=\alpha_{2,k}=\ldots=\alpha_{N,k}$. As a consequence the coefficients $\tilde{c}_{\mu\nu}^{kl}$ become symmetric under exchange of any two upper or down indices, $\tilde{c}_{\mu\nu}^{kl}=\tilde{c}_{\mu\nu}^{lk}$. Hence the matrices $W_i$ are all equal to each other and the matrices $C_j=C_j^\mathrm{T}$ become symmetric and equal as well. Therefore the problem boils down to diagonalisation of the following $2N\times2N$ symmetric block circulant matrix with symmetric blocks
\begin{equation}
	\Omega=
	\begin{pmatrix}
		W & C & 0 & \cdots & C\\
		C & W & C & \cdots & 0\\
		0 & C & \ddots & \ddots & \vdots\\
		\vdots & \vdots & \ddots & \ddots & C\\
		C & 0 & \cdots & C & W
	\end{pmatrix},
\end{equation}
where $W$ and $C$ are defined as in \eqref{eq:OmegaWC}, and $\Omega\in\mathcal{BC}_{N,2}$. These type of matrices are diagonalised in terms of Discrete Fourier Transform (DFT). As a first step we will block diagonalise the matrix $\Omega$ by making use of an unitary matrix $U$,
\begin{equation}
	\widehat{\Omega}=U^{-1}\Omega U=\text{diag}[D_1,D_2,\ldots,D_N].
\end{equation}
Defining the matrix $U$ as build of the following diagonal $2\times2$ blocks
\begin{equation}
	U_{kl}=\frac{1}{\sqrt{N}}e^{2\pi ikl/N}\mathds{1}_2,\quad k,l=0,1,\ldots,N-1,
\end{equation}
we can treat the matrix $\Omega$ as standard circulant matrix, which is block diagonalised by the matrix $U$, and the diagonal elements of the block-diagonal matrix $\widehat{\Omega}$ consist of the eigenvalues of $\Omega$, namely
\begin{align}
	D_{k+1}&=W+C\rho_k+C\rho_k^{N-1}=W+\left(e^{2\pi ik/N}+e^{-2\pi ik/N}\right)C\nonumber\\
	&=W+2\cos\left(2\pi k/N\right)C.
\end{align}
Substituting with the expressions for $W$ and $C$ we obtain
\begin{equation}
	D_{k+1}=
	\begin{pmatrix}
		\omega_0+2\cos(k\theta)c_0 & 2\cos(k\theta)c_2\\
		2\cos(k\theta)c_2 & \omega_1+2\cos(k\theta)c_1
	\end{pmatrix},
\end{equation}
where $\theta=2\pi/N$ and $k=0,1,\ldots,N-1$. The only thing we need to do is to find the eigenvalues and eigenvectors of the matrices $D_{k+1}$. A straightforward calculation gives
\begin{equation}
	\lambda_{k+1\pm}=a_k\pm2c_2h_k\rightarrow\mathbf{v}_{k+1\pm}=(b_k\pm h_k,1)^\mathrm{T},
\end{equation}
where
\begin{align}
	a_k&=\frac{1}{2}\left[\omega_0+\omega_1+2{c_0+c_1}\cos(k\theta)\right],\nonumber\\
	b_k&=\frac{1}{4c_2}\left[2(c_0-c_1)+(\omega_0-\omega_1)\sec(k\theta)\right],\nonumber\\
	h_k&=\frac{1}{4c_2}\left[2(c_0-c_1)^2+8c_2^2+(\omega_0-\omega_1)^2
	+4(c_0-c_1)(\omega_0-\omega_1)\cos(k\theta)\right.\nonumber\\
	&\left.\hphantom{{}={}\frac{1}{4c_2}}+2\left((c_0-c_1)^2+4c_2^2\right)
	\cos(2k\theta)\right]^{1/2}\sec(k\theta).
\end{align}
The Hamiltonian \eqref{eq:HN} thus becomes (taking into account that the matrix $U$ is unitary, i.e. $U^{-1}=U^\dagger$, and $\eta^\mathrm{T}=\eta^\dagger$)
\begin{equation}
\label{eq:HNdiag}
	H_N=\frac{1}{2}\eta^\dagger\,\text{diag}\left[U\text{diag}[D_1,\ldots,D_N]
	U^\dagger,\mathds{1}_{2N}\right]\eta=\frac{1}{2}\widehat{\eta}^\dagger\,\text{diag}
	\left[\widehat{D}_1,\ldots,\widehat{D}_N,\mathds{1}_{2N}\right]\widehat{\eta},
\end{equation}
where the matrices $D_{k+1}$ are diagonalised by orthogonal matrices $P_{k+1}$ composed of the orthonormal vectors $\mathbf{v}_{k+1\pm}/||\mathbf{v}_{k+1\pm}||$, namely $D_{k+1}=P\widehat{D}_{k+1}P^{-1}$. The explicit form of the matrix $P_{k+1}$ is
\begin{equation}
	P_{k+1}=
	\begin{pmatrix}
		\frac{b_k+h_k}{\sqrt{1+(b_k+h_k)^2}} & \frac{b_k-h_k}{\sqrt{1+(b_k-h_k)^2}}\\
		\frac{1}{\sqrt{1+(b_k+h_k)^2}} & \frac{1}{\sqrt{1+(b_k-h_k)^2}}
	\end{pmatrix}.
\end{equation}
Finally, the diagonal matrices $\widehat{D}_{k+1}$ and the transformation between $\eta$ and $\widehat{\eta}$ are given by the following expressions:
\begin{align}
	\widehat{\eta}&=\text{diag}\left[\text{diag}[P_1^\dagger,\ldots,P_N^\dagger]
	U^\dagger,\mathds{1}_{2N}\right]\eta,\nonumber\\
	\widehat{D}_{k+1}&=\text{diag}[\lambda_{k+1+},\lambda_{k+1-}],\quad k=0,\ldots,N-1.
\end{align}
Furthermore, we define creation and annihilation operators $\mathfrak{a}_j^\dagger$ and $\mathfrak{a}_j$, $j=1,\ldots,2N$, as
\begin{align}
	\mathfrak{a}_j&=\frac{1}{\sqrt{2\hbar}}\left(\sqrt{\lambda_j}\widehat{x}_j
	+\frac{i}{\sqrt{\lambda_j}}\widehat{p}_j\right),\nonumber\\
	\mathfrak{a}_j^\dagger&=\frac{1}{\sqrt{2\hbar}}\left(\sqrt{\lambda_j}\widehat{x}_j
	-\frac{i}{\sqrt{\lambda_j}}\widehat{p}_j\right),
\end{align}
where $\widehat{x}_j$ and $\widehat{p}_j$ are the coordinate and momentum part of the transformed vector $\widehat{\eta}$, and the frequencies $\lambda_j$ are connected to the diagonal elements of the matrix $\widehat{D}_{k+1}$ by $\lambda_{2j-1}=\sqrt{\lambda_{j-1+}}$ and $\lambda_{2j}=\sqrt{\lambda_{j-1-}}$. The creation and annihilation operators satisfy the commutation relations
\begin{equation}
	[\mathfrak{a}_j,\mathfrak{a}_k]=[\mathfrak{a}_j^\dagger,\mathfrak{a}_k^\dagger]=0,\quad
	[\mathfrak{a}_j,\mathfrak{a}_k^\dagger]=\delta_{jk}.
\end{equation}
Thus the Hamiltonian \eqref{eq:HNdiag} takes the standard form
\begin{equation}
\label{eq:HNquant}
	H_N=\sum_{j=1}^{2N}\hbar\lambda_j\left(\mathfrak{a}_j^\dagger\mathfrak{a}_j+\frac{1}{2}\right).
\end{equation}
The Fock space is built up from the vacuum in the usual way,
\begin{equation}
	\ket*{\{n_j\}}=\prod_{j=1}^{2N}\frac{(\mathfrak{a}_j^\dagger)^{n_j}}{\sqrt{n_j!}}\ket*{\{0\}},
\end{equation}
where $\ket*{\{n_j\}}=\ket*{n_1}\otimes\cdots\otimes\ket*{n_{2N}}$. The energy spectrum is derived from the eigenvalue problem $H_N\ket*{\{n_j\}}=\mathcal{E}_{\{n_j\}}\ket*{\{n_j\}}$,
\begin{equation}
	\mathcal{E}_{\{n_j\}}=\sum_{j=1}^{2N}\hbar\lambda_j\left(n_j+\frac{1}{2}\right).
\end{equation}
In terms of the excited states, which are orthonormal $\braket{\{m_j\}}{\{n_j\}}=\delta_{\{m_j\},\{n_j\}}$, the diagonal Hamiltonian \eqref{eq:HNquant} can be written in matrix form
\begin{equation}
	H_N=\sum_{\{n_j\}=0}^\infty\sum_{j=1}^{2N}\hbar\lambda_j\left(n_j+\frac{1}{2}\right)\ket*{\{n_j\}}\bra*{\{n_j\}}.
\end{equation}
The partition function of the system is defined as
\begin{equation}
	Z(K_j):=\mathrm{Tr}_{\{j\}}e^{-\beta H_N}=
	\prod_{j=1}^{2N}\frac{\exp(-K_j/2)}{1-\exp(-K_j)},
\end{equation}
where $\beta$ is the inverse temperature and $K_j=\hbar\lambda_j\beta$. Following the same line of calculations as in section \ref{sec:two1D4PUOs}, we first compute the ordinary density matrix,
\begin{equation}
	\rho_\text{eq}=\frac{1}{Z(K_j)}\sum_{\{n_j\}=0}^\infty\exp\left[-\sum_{j=1}^{2N}K_j
	\left(n_j+\frac{1}{2}\right)\right]\ket*{\{n_j\}}\bra*{\{n_j\}}.
\end{equation}
Taking into account the different range of the sum, the statistical state is
\begin{equation}
	\ket*{\Psi(K_j)}=\frac{1}{\sqrt{Z(K_j)}}\sum_{\{n_j\}=0}^\infty\exp\left[-\sum_{j=1}^{2N}\frac{K_j}{2}
	\left(n_j+\frac{1}{2}\right)\right]\ket*{\{n_j\}}\ket*{\{\tilde{n}_j\}},
\end{equation}
and the obtained from it extended density matrix reads
\begin{equation}
	\widehat{\rho}(K_j)=\frac{1}{Z(K_j)}\sum_{\{n_j\}=0}^\infty\sum_{\{m_j\}=0}^\infty
	\exp\left[-\sum_{j=1}^{2N}\frac{K_j}{2}\left(n_j+m_j+1\right)\right]
	\ket*{\{m_j\}}\bra*{\{n_j\}}\ket*{\{\tilde{m}_j\}}\bra*{\{\tilde{n}_j\}}.
\end{equation}
The next step is to factorise the whole system into two subsystems. Since all PUOs over the ring are absolutely equivalent, we pick up one of them (for definiteness, we choose the PUO described by harmonic oscillators 1 and 2) to be the first subsystem thus all the rest being the second subsystem. Therefore the renormalised extended density matrix with respect to the first subsystem is calculated by taking trace over the second subsystem of harmonic oscillators $3,4,\ldots,2N$:
\begin{align}
	\widehat{\rho}_{1,2}(K_j)&=\frac{1}{Z(K_j)}
	\prod_{j=3}^{2N}\frac{\exp(-K_j/2)}{1-\exp(-K_j)}\sum_{\substack{n_1,m_1\\n_2,m_2}}^\infty
	\exp\left[-\frac{K_1}{2}(n_1+m_1+1)-\frac{K_2}{2}(n_2+m_2+1)\right]\nonumber\\
	&\times\ket*{m_1,m_2}\bra*{n_1,n_2}\ket*{\tilde{m}_1,\tilde{m}_2}\bra*{\tilde{n}_1,\tilde{n}_2}.
\end{align}
Finally, we end up with the extended entanglement entropy of a Pais-Uhlenbeck oscillator inside a ring of $N-1$ other oscillators,
\begin{align}
\label{eq:entropyN}
	\widehat{S}_{1,2}(K_1,K_2)=\frac{k_\text{B}}{2}\coth\frac{K_1}{4}\coth\frac{K_2}{4}
	&\left[K_1\left(1+\coth\frac{K_1}{4}\right)+K_2\left(1+\coth\frac{K_2}{4}\right)\right.\nonumber\\
	&\left.-2\log\left[\left(e^{K_1}-1\right)\left(e^{K_2}-1\right)\right]\vphantom{\coth\frac{K_1}{4}}\right].
\end{align}
At first look expressions \eqref{eq:entropy2} and \eqref{eq:entropyN} seem to be the same since they completely match. However, this is not the case because the frequencies in the definitions of the inverse scaled temperatures $K_1$ and $K_2$ are given by different expressions for the formulae \eqref{eq:entropy2} and \eqref{eq:entropyN}. Of course the behaviour of the entanglement entropy \eqref{eq:entropyN} as function of the inverse scaled temperature mimics that in figure \ref{fig:entropy} and the Nernst theorem is applicable again.

\section{Concluding remarks}
\label{sec:conclusion}

We have considered the quantum entanglement of coupled fourth-order Pais-Uhlenbeck oscillators in the context of thermo-field dynamics. Being defined by Lagrangian which contains higher-than-first time derivatives in variables, such theory is characterised by Ostrogradsky instabilities after passing to Hamiltonian formalism. However, there exist many alternative Hamiltonian formulations of the PUO, which prove that it can be considered as a reliable physical system. One of these formulations states that every PUO of order $k$ is equivalent to a system of $k$ harmonic oscillators with alternating signs or even more general---with the signs of arbitrary non-zero constants coming from the ansatz of the Hamiltonian as a sum of Noether currents. By choosing all constants to be positive, we already have positive-definite Hamiltonian without negative-norm states. It turns out that such Hamiltonian is very convenient for studying quantum entanglement between PUOs.

Possessing well-defined Hamiltonian system, we further introduced simple linear interaction between the closest-neighbour PUOs. This led to the necessity of diagonalisation of symmetric block circulant matrices with symmetric blocks, the matrix form of the Hamiltonian interaction part being such type of matrix. With diagonalised Hamiltonian in hands, we were able to apply the TFD framework after canonical quantisation and building the Fock space of the quantum system. The double Hilbert space has proven to be extremely useful in defining and exploring various statistical quantities as extended density matrix and extended statistical state, because the isomorphic copy of the original Hilbert space somehow played the role of a tracer. Consequently, we computed the extended entanglement entropy of one PUO with the rest of the system under consideration and showed that the obtained physical quantity obeys the Nernst heat theorem.

Let us now comment on the consequences of the results concerning the holographic models. In section \ref{sec:two1D4PUOs} we obtained the diagonalised Hamiltonian \eqref{eq:H2diag} with the new frequencies \eqref{eq:freq} written in terms of the parameters of the two fourth-order PUOs:
\begin{align}
	\lambda_{0\pm}&=\frac{1}{2}\left[\omega_0+\omega_1+(c_0+c_1)\right]\pm
	\frac{1}{2}\sqrt{2c_2^2+\left[c_0-c_1+(\omega_0-\omega_1)\right]^2},\nonumber\\
	\lambda_{1\pm}&=\frac{1}{2}\left[\omega_0+\omega_1-(c_0+c_1)\right]\pm
	\frac{1}{2}\sqrt{2c_2^2+\left[c_0-c_1-(\omega_0-\omega_1)\right]^2}.
\end{align}
Written this way, the Hamiltonian \eqref{eq:H2diag} represents an alternative Hamiltonian formulation of two free Pais-Uhlenbeck oscillators each of fourth order with frequencies $\lambda_{0\pm}$ and $\lambda_{1\pm}$ and corresponding equations of motion\footnote{For more information, please see \cite{Masterov:2015ija}.}:
\begin{align}
\label{eq:PUOeqs}
	x_1^{(4)}+\left(\lambda_{0+}^2+\lambda_{0-}^2\right)x_1^{(2)}
	+\lambda_{0+}^2\lambda_{0-}^2x_1&=0,\nonumber\\
	x_2^{(4)}+\left(\lambda_{1+}^2+\lambda_{1-}^2\right)x_2^{(2)}
	+\lambda_{1+}^2\lambda_{1-}^2x_2&=0,
\end{align}
where $x_1$ and $x_2$ are determined from \eqref{eq:cancoord}. It is noteworthy to mention that, although the diagonalisation procedure leads to an effective system of two free PUOs, the original system is formed of two interacting PUOs. Moreover, every higher-derivative theory is intrinsically interacting. Now we notice that the system of differential equations \eqref{eq:PUOeqs} is exactly equivalent to the systems \eqref{eq:EoMPUO} and \eqref{eq:EoMPUO2} and we can map both systems. Therefore the occurrence of critical point of the $B$-field can be translated into critical values of the parameters of the system of PUOs. Namely, if
\begin{equation}
	2c_2^2\leq\left[c_0-c_1\pm(\omega_0-\omega_1)\right]^2,
\end{equation}
the frequencies become imaginary, which means instabilities. The above inequality gives a certain range of values for the parameter $c_2$, which is proportional to the strength of the coupling between the PUOs. Similar behaviour is observed in the pp-wave limit of the PW geometry for certain finite set of oscillator modes, where the unstable modes appear for large enough values of the $B$-field \cite{Brecher:2002ar}. In other words, the string can only see the $B$-field if it is excited. In studying supergravity backgrounds one normally requires small curvature of the spacetime at string scale. If the curvature is no longer negligible, then the string dynamics goes beyond the supergravity approximation, which forces one to consider the full string theory.

Consequently, holography teaches us that a change in the background/gravity (due to instabilities) changes the dual gauge theory. From field theory point of view such changes could be interpreted as a phase transition.

In the end of the day we would like to look ahead by slightly touching the issue of information geometry. The entanglement entropy naturally generates an emergent parameter space equipped with Riemannian metric. This metric represents the celebrated Fisher information metric, which can be expressed as second derivative of the entanglement entropy \cite{Matsueda:2014lda,Matsueda:2014yza}, $g_{\mu\nu}(K_1,K_2)=\partial_\mu\partial_\nu S(K_1,K_2)$, where $\partial_\mu=\frac{\partial}{\partial K_\mu}$ with $\mu=1,2$. The absence of minus sign in the definition of the Fisher metric accounts for the Riemannian nature of the metric, the reason being the specific type of variables we have used, i.e, the inverse scaled temperatures $K_1$ and $K_2$. For the case under consideration in this paper, the Fisher information metric for the fourth-order PUO takes the form:
\begin{align}
	g_{11}&=\frac{1}{64}k_\text{B}\coth\frac{K_2}{4}\csch^2\frac{K_1}{4}\left[K_1\left(3+5\coth^2\frac{K_1}{4}
	+7\csch^2\frac{K_1}{4}\right)+4\tanh\frac{K_1}{4}\right.\nonumber\\
	&\phantom{{}={}}\left.+4\coth\frac{K_1}{4}\left(K_1+K_2-5+K_2\coth\frac{K_2}{4}
	-2\log\left[\left(e^{K_1}-1\right)\left(e^{K_2}-1\right)\right]\right)\right],\\
	g_{12}&=\frac{1}{32}k_\text{B}\csch^2\frac{K_1}{4}\csch^2\frac{K_2}{4}
	\left[K_1\left(1+2\coth\frac{K_1}{4}\right)+K_2\left(1+2\coth\frac{K_2}{4}\right)-4\right.\nonumber\\
	&\phantom{{}={}}\left.-2\log\left[\left(e^{K_1}-1\right)\left(e^{K_2}-1\right)\right]\vphantom{\coth\frac{K_2}{4}}\right],
\end{align}
and the other two metric components are obtained by symmetry.

Previously we related the instabilities caused by the presence of critical point of the $B$-field to instabilities in the system of PUOs, and argued that the latter may be interpreted as phase transitions. As a consequence of the instabilities, $K_1$ and $K_2$ in the Fisher metric become imaginary, which transforms the hyperbolic functions into trigonometric ones. Roughly speaking, the hyperbolas ``transform into'' circles, which brings the notion of topological order. From holography point of view, the Pilch-Warner solution is one of the most complicated supergravity solutions, thus being very hard instabilities to be investigated directly using entanglement entropy and alike tools. The holography on the other hand, and the AdS/CFT correspondence as a particular example, is a very powerful tool allowing such instabilities to be examined in the field theory, where the entanglement entropy is much better understood and more easily treatable.

In conclusion, the framework of thermo-field dynamics has proved to be very useful for studying the quantum entanglement of higher-derivative theories, the Pais-Uhlenbeck oscillator being an example. As a possible continuation of this work, it would be interesting to use the quantum PUO for analysing different models in string theory by making use of holography. Such microscopic description could contribute significantly to our understanding of the nature of different high-energy phenomena, which are of great interest to modern theoretical and experimental physics.

\section*{Acknowledgements}

We are grateful to Ivan Masterov for valuable comments on the manuscript. This work was supported by the Bulgarian NSF grant T02/6. SM and RR were also supported by Sofia University Research Fund grant \textnumero~85/2016.