\section{Introduction}

Entanglement, as measured by, {\em e.g.}, bipartite block entanglement entropy, is
playing an increasingly important role in the study of
condensed-matter or quantum many-body physics, both conceptually and
quantitatively. It has been  used as a very useful and in
some cases indispensable way to characterize phases and phase
transitions, especially for phases and quantum phase transitions in
strongly correlated fermionic or spin systems (for a review, see Ref. [\onlinecite{amico:517}]).
For bosonic systems, studies of entanglement entropy have mostly
focused on {\em relativistic} free bosonic field theories \cite{wilczek94,calabrese-2004-0406,casini-2005-0512},
which are equivalent to coupled harmonic oscillator systems (for reviews,
see Refs. [\onlinecite{adesso-2007-40}] and [\onlinecite{RevModPhys.77.513}]).

In this paper we study the entanglement properties of free {\em
  non-relativistic} Bose gases. In addition to interest in its own
right, our motivation also comes in part from the
following consideration. In recent studies it has been shown that
entanglement is enhanced at quantum critical points \cite{QPT_Sachdev} and
strongly correlated phases with topological order \cite{xgwen1990}, in the form of
either violation of area law \cite{PhysRevD.34.373, area-law,
  eisert-2008, wolf:010404, gioev:100503,  Barthel:2006, Li:2006, wilczek94,
  PhysRevLett.90.227902, calabrese-2004-0406, PhysRevLett.93.260602,
  santachiara-2006-L06002, feiguin:160409, bonesteel:140405},
or subleading corrections to the area law that diverges with block
size \cite{levin:110405, kitaev:110404, fradkin:050404} (usually in a
logarithmic fashion). On the other hand there have been relatively
few studies of the behavior of entanglement entropy in states with
traditional long-range order \cite{vidal05, vidal06, vidal07}. In a
recent work \cite{ding:052109}, we calculated the block entanglement
entropy of some exactly soluble spin models that exhibit
ferromagnetic or antiferromagnetic long-range order in the ground
state, and found that such conventional orders also lead to
logarithmically divergent contribution to the entropy. Bose-Einstein
condensation (BEC) is perhaps the simplest example of conventional
ordering. It is thus natural to study its entanglement properties.
As we are going to show, a Bose-Einstein condensate (referred to as
a condensate from now on) indeed makes a logarithmically divergent
contribution to the entropy as well.

Besides the entanglement entropy of the ground state, the entanglement
properties of system at finite temperature are also of great interest. However, the entanglement
entropy is only well-defined for a pure state. For a system that is
described by a mixed density matrix, the von Neumann entropy of the
reduced density matrix becomes different for the two parts of the
bipartite systems. In such cases, there is a natural extension of the
entanglement entropy that one can work with - the mutual information
\cite{dreissig,Wolf2008}. We will show that a condensate, when
present, makes a logarithmically divergent contribution to the mutual
information.

This paper is organized as follows. In Sec. II we study the ground state
entanglement entropy of a generic free boson model that is
translationally invariant \cite{klich06}. In Sec. III we introduce an infinite-range
hopping model for bosons which is exactly solvable, and calculate the
mutual information analytically. In Sec. IV we introduce a long-range
hopping model for bosons in one-dimension (1D) which exhibits a finite temperature BEC
for a certain parameter range, then we present a numerical
study of the mutual information for this model. In the end, we
summarize and discuss the results of this paper in Sec. V.

\section{Zero Temperature: Entanglement Entropy of Free Bosons}
Consider a general Hamiltonian of free bosons hopping on a lattice
of size $L$: \begin{equation}\label{Eq:GeneralH} H = - \sum_{ij} t_{ij}
\hat{a}^\dagger_i \hat{a}_j, \end{equation} where $t_{ij} > 0$, $\hat{a}_i(\hat{a}^\dagger_i)$'s
are the bosonic annihilation (creation) operators. If the system is
translationally invariant, $t_{ij} = t_{i-j}$, then the Hamiltonian
can be diagonalized by Fourier transformation:
\begin{equation} H = \sum_{k}
\varepsilon(k) \hat{b}^\dagger_k \hat{b}_k, \end{equation}
where $\hat{b}_k = \frac{1}{\sqrt{L}}\sum_j e^{-ijk} \hat{a}_j$ is the annihilation operator
in $k$ space. In most generic cases, the ground state is the $k=0$
state. At zero temperature, all particles fall into the ground
state. For a system containing $N$ particles, the ground state is
given by:
\begin{equation} \ket{\Psi_0} = \frac{1}{\sqrt{N!}} (\hat{b}^\dagger_0)^N
\ket{0} = \frac{1}{\sqrt{N!}} (\frac{1}{\sqrt{L}} \sum_j
\hat{a}^\dagger_j)^N \ket{0}. \end{equation}

To consider its bipartite block entanglement entropy, we divide the
system of size $L$ in two parts, and label them $A$ and $B$
respectively. Let the
sizes of each part be $L_A$ and $L_B$, $L_A + L_B = L$, and define
\begin{equation}
\hat{a}^\dagger_A = \frac{1}{\sqrt{L_A}} \sum_{j \in A}
\hat{a}^\dagger_j,\ \hat{a}^\dagger_B = \frac{1}{\sqrt{L_B}} \sum_{j \in B}
\hat{a}^\dagger_j.
\end{equation}
Then we can write $\ket{\Psi_0}$ as:
\begin{equation}
\begin{split}
\ket{\Psi_0} & = \frac{L^{-N/2}}{\sqrt{N!}} (\sqrt{L_A} \hat{a}_A +
\sqrt{L_B} \hat{a}_B )^N \ket{0} = \frac{L^{-N/2}}{\sqrt{N!}}\sum_{l=0}^{N}
\frac{N!}{(N-l)!l!} (\sqrt{L_A} \hat{a}^\dagger_A)^{l} (\sqrt{L_B}
\hat{a}^\dagger_B)^{N-l} \\
& = L^{-N/2} \sum_{l} \sqrt{\frac{N!}{(N-l)!l!}} L_A^{l/2}
L_B^{(N-l)/2}  \left[ \frac{1}{\sqrt{l!}} \hat{a}^{\dagger l}_A \frac{1}{\sqrt{(N-l)!}}
\hat{a}^{\dagger N-l}_B \ket{0} \right] \\
& = \sum_l \sqrt{\lambda_l} \ket{l}_A \otimes \ket{N-l}_B,
\end{split}
\end{equation}
where $\lambda_l = L^{-N} \frac{N!}{(N-l)!l!} L_A^{l} L_B^{N-l}$,
$\ket{l}_A = \frac{1}{\sqrt{l!}} \hat{a}^{\dagger l}_A \ket{0}_A$,
$\ket{N-l}_B = \frac{1}{\sqrt{(N-l)!}} \hat{a}_B^{^\dagger N-l}\ket{0}_B$,
and $\ket{0} = \ket{0}_A \otimes \ket{0}_B$.

This is an explicit Schmidt decomposition, and therefore the
entanglement entropy is readily given by:

\begin{equation}
E = - \sum_l \lambda_l \ln \lambda_l.
\end{equation}

We are interested in the asymptotic behavior in two limiting cases: (1) the
equal partition case; (2) size of $B$ is substantially larger than
$A$, i.e., $L_B \gg L_A$.

(i) Equal partition, $L_A = L_B = \frac{L}{2}$:
\begin{equation}
\lambda_l = \frac{N!}{l!(N-l)!} \frac{L_A^l L_B^{N-l}}{L^N} =
\frac{N!}{l!(N-l)! 2^N} = \frac{N!}{(\frac{N}{2})!}
\frac{(\frac{N}{2})! 2^N}{(\frac{N}{2} - (\frac{N}{2} - l))! (\frac{N}{2}
  + (\frac{N}{2} - l))!}.
\end{equation}
Let $x = \frac{N}{2} - l$, then $x \in [-\frac{N}{2},
  \frac{N}{2}]$, and we can denote $\lambda_l$ as $\lambda_x =
\frac{2^N N!}{(\frac{N}{2})!} \frac{(\frac{N}{2})!}{(\frac{N}{2} -
  x)!(\frac{N}{2} + x)!}$ which can be approximated by a Gaussian
distribution factor
$\lambda_x \sim e^{\frac{- 2x^2}{N}}$ when $N$ is large.
In the limit $N \rightarrow \infty$, the summation
over $n$ (or $x$) can be approximated by an integral. Also in this
limit, the Gaussian factor is sharply peaked around $x = 0$, the
integral region can be extended to from minus infinity to
infinity. Using the fact that $\sum_x \lambda_x\simeq
\int^{\infty}_{-\infty}\lambda(x) dx = 1$, we arrive at
\begin{equation}
\lambda(x) \simeq \sqrt{\frac{2}{N \pi}} e^{-\frac{2 x^2}{N}}.
\end{equation}

The entanglement entropy is then
\begin{equation} \label{eqn:EEEP}
E \simeq - \int^{\infty}_{-\infty}\lambda(x)\ln \lambda(x) dx =
\frac{1}{2} \left(1 + \ln(\frac{N \pi}{2}) \right) = \frac{1}{2} \ln N
+ \mathcal{O}(1).
\end{equation}

(ii) Unequal partition, $L_B \gg L_A$:

If $L_B \gg L_A$, $L \rightarrow \infty$, but keep $\frac{N}{L}
\rightarrow \ev{n}$ (fixed), the distribution of $\lambda_l$
approaches a Poisson distribution:
\begin{equation}
\lambda_l = \frac{N!}{l!(N-l)!} \frac{L_A^l L_B^{N-l}}{L^N}
\xrightarrow{^{N \rightarrow \infty}} \frac{(L_A \ev{n} )^l e^{-L_A \langle n \rangle }}{l!}.
\end{equation}
The entropy of the Poisson distribution, which in this case is our
entanglement entropy, is known to be:
\begin{equation}\label{Eq:EE}
\begin{split}
E &= \frac{1}{2} [1 + \ln (2\pi L_A \ev{n})] - \frac{1}{12 L_A \ev{n}} +
O(\frac{1}{(L_A \ev{n})^2})\\
&= \frac{1}{2}\left[1 + \ln (2 \pi N_A )\right] -\frac{1}{12 N_A} +
  O(\frac{1}{(N_A)^2}) = \frac{1}{2}\ln N_A + \mathcal{O}(1),
\end{split}
\end{equation}
where $N_A = L_A \ev{n}$ is the average particle number in subsystem $A$.

Therefore, we find, in both cases, that the leading term of the
mutual information goes as $\frac{1}{2} \ln N_A$ for $L_A \le L_B$.

\section{Mutual Information: Analytic Study of an Infinite-Range Hopping Model}
In this section, we will study the natural generalization of
entanglement entropy at finite temperature: the mutual
information, which is defined as
\begin{equation} E_M = \frac{1}{2} (E_A + E_B - S), \end{equation}
where $E_A$ and $E_B$ are the von Neumann entropy of the reduced density
matrices of subsystems A and B, respectively, and $S$ is the entropy of
the whole system. Note that at finite temperature $E_A$ and $E_B$ are no
longer the same due to the fact that the system is described by a mixed
density matrix. We must emphasize here that our definition of
mutual information differs from its usual definition \cite{minote} by a factor of 2
so that it will converge to the entanglement entropy when the system
approaches a pure state.

\subsection{Model, spectrum and thermodynamic properties}
In order to facilitate an exact solution, we consider the following
infinite-range hopping model which is obtained by setting $t_{ij}$
in Eq. (\ref{Eq:GeneralH}) to a constant properly scaled by the
system size $t_{ij} = t / L$ so that the thermodynamic limit is
well-defined. The Hamiltonian is then

\begin{equation}
H = -\frac{t}{L} \sum_{i,j}\hat{a}_i^\dagger \hat{a}_j = - \frac{t}{L} (\sum_i
\hat{a}_i^\dagger) (\sum_j \hat{a}_j).
\end{equation}
By substituting the Fourier transform of $\hat{a}_j$'s defined in Sec. II,
$\hat{b}_k = \frac{1}{\sqrt{L}}\sum_j e^{-ijk} \hat{a}_j$, one obtains:
\begin{equation}
H = -t \hat{b}^\dagger_0 \hat{b}_0.
\end{equation}
This model has a very simple spectrum with a ground state with energy
$-t$, and all the other excited states are degenerate with zero
energy. This particularly simplified spectrum makes an exact solution possible.

To study the finite temperature properties of this model, we will work
with the grand canonical ensemble (GCE),
in which the chemical potential
$\mu$ is introduced to control the average
density of the system. This model exhibits BEC
at finite temperature $T_C$. To determine $T_C$, we start by
considering a system of finite size $L$, and its occupation numbers are:
\begin{equation}
\ev{N_{k=0}} = \ev{N_0} = \frac{1}{e^{\beta (-t -
    \mu)} - 1},\text{ } \ev{N_{k}} = \frac{1}{e^{\beta(-\mu)} -
  1}\text{ for $k \ne 0$}.
\end{equation}
Here $\ev{N_{0}}$ and $\ev{N_{k}}$ denote average occupation numbers for
the corresponding states in $k$-space; $\beta = \frac{1}{T}$ is the
inverse temperature. From this point on, when we write
$\ev{N_k}$, it immediately indicates $k \ne 0$. The average total particle number of the
system will be denoted as $\ev{N}$. To identify $T_C$, we know in the
thermodynamic limit, when $T \rightarrow T_C + 0^+$, $\mu \rightarrow
E_{k=0} = -t$, and $\frac{\langle N_0 \rangle}{N} \rightarrow 0$. Therefore,
$\ev{N_{k}} = \frac{1}{e^{\beta_c t} - 1} \simeq \frac{ \langle N \rangle }{L}
= \ev{n}$ where $\ev{n}$ is the average particle density. So we
obtain:
\begin{equation}
T_C = \frac{t}{\ln (1 + 1/\ev{n})}.
\label{Eq:inf_Tc}
\end{equation}

Above $T_C$, $\ev{n} = \frac{L - 1}{L} \frac{1}{e^{-\beta \mu} - 1} +
\frac{1}{L} \frac{1}{e^{\beta (-t - \mu)} - 1}$, from which in the
large $L$ limit we can derive that
\begin{equation}
\mu = -T \ln\left(1+\frac{1}{\ev{n}}\right).
\end{equation}
$\mu$ has a finite size correction which is negligible above $T_C$,
but will become important below $T_C$.

We also know that the partition function of the system in GCE. bears
the following form:
\begin{equation}
Z = (\frac{1}{1 - e^{\beta \mu}})^{L-1} \frac{1}{1 - e^{\beta(t+\mu)}},
\end{equation}
from which it is easy to show that the entropy in GCE. takes the following form:
\begin{equation}\label{Eq:thermal entropy}
\begin{split}
S &= - \frac{\partial \Omega}{\partial T} = \ln Z - \frac{1}{T}
\frac{\partial}{\partial \beta} \ln Z\\
&= (1 + \ev{N_0}) \ln(1 + \ev{N_0}) - \ev{N_0} \ln N_0 + (L - 1)
\left[(1 + \ev{N_k}) \ln (1 + \ev{N_k}) - \ev{N_k} \ln \ev{N_k} \right].
\end{split}
\end{equation}

Anticipating later relevance, we are particularly interested in the behavior of
finite size systems near $T_C$. For a finite system, the chemical
potential $\mu$ is no longer strictly equal the ground state energy
below $T_C$, but picks up a finite size correction $\delta \mu$
determined by the following condition:
\begin{equation}
\frac{1}{e^{-\beta \delta \mu} - 1} = \ev{N_0},
\end{equation}
from which we can easily derive that
\begin{equation}
\delta \mu = - T \ln\left(1 + \frac{1}{\ev{N_0}}\right).
\end{equation}
Consider $T = T_C = \frac{t}{\ln(1 + 1/\langle n \rangle)}$, and make use of the
following fact
\begin{equation}
\ev{N_0}= \ev{N} - \sum_{k \ne 0} \ev{N_k} = \ev{N} - \frac{L -
  1}{e^{\beta (t - \delta \mu)} - 1},
\end{equation}
we obtain
\begin{equation}
\ev{N_0} = \ev{N} - \frac{L - 1}{e^{\beta_c (t - \delta \mu)}
  - 1} = \ev{N} - \frac{L - 1}{(1 + L / \ev{N}) (1 + 1 / \ev{N_0}) - 1}.
\end{equation}
This equation can be solved to give $\ev{N_0}$ as a function of system
size $L$ at a given density $\ev{n} = \ev{N} / L$, at $T=T_C$:
\begin{equation}\label{Eq:N_0@Tc}
\ev{N_0} = \sqrt{L} \sqrt{\left(\frac{\ev{N}}{L}\right)^2 +
  \frac{\ev{N}}{L} + \frac{1}{4L}} \simeq \sqrt{L} \sqrt{ \langle n
  \rangle ^2 + \langle n \rangle }.
\end{equation}
Even though this divergent $N_0$ does not affect the thermodynamic behavior
of the system, as we will see later it makes a (leading) divergent contribution to the mutual information
at $T=T_C$ depending on how the system is partitioned, or specifically
how large is the subsystem size $L_A$ compared with this $\sqrt{L}$ divergence.

\subsection{Formalism and issues}
In the following part, we will use Peschel's result \cite{Peschel2003}
on the reduced density matrix of a Gaussian state:
\begin{equation}\label{Eq:RDM}
\rho_A = \mathcal{K} e^{\{\ln((1+G)G^{-1})\}^T_{ij}\hat{a}^\dagger_i \hat{a}_j},
\end{equation}
where $G_{ij} = \ev{\hat{a}^\dagger_i \hat{a}_j}$ is the two point correlation
function matrix truncated within the subsystem, and $\mathcal{K}$ is the
normalization factor. The entropy is given as
\begin{equation}\label{Eq:entropy}
E_A = \sum_l \left[(1 + g_l)\ln (1 + g_l) - g_l \ln g_l\right],
\end{equation}
where $g_l$'s are the complete set of eigenvalues of $G$'s (after
truncation). Actually this formula also applies to the original
system.

We must note that, this formula does not lead to the correct
zero temperature limit for the entropy. At zero temperature,
$G_{ij} = \ev{a_i^\dagger a_j} = \ev{n}$. Its eigenvalues are all zero
except for one: $g_0 = \ev{n} L = \ev{N}$, which gives us a non-zero entropy
$S_{T=0} =(\ev{N} + 1) \ln (\ev{N} + 1) - \ev{N} \ln \ev{N} = \ln \ev{N} + \ev{N} \ln (1 +
\frac{1}{\ev{N}}) \sim \ln \ev{N}$ at $T = 0$. This reflects
the fact that we are working with GCE where the particle-number
fluctuation is still permissible at $T = 0$ and the fluctuation
amplitude $\delta N \sim \ev{N}$. However, as we show below, the mutual information still converges to
the correct zero temperature limit, the entanglement entropy, at least
to the leading order.

The von Neumann entropy for a subsystem $A$ is given by
\begin{equation}
E_{A}^{(\text{GCE})} = (N_A + 1) \ln (N_A + 1) - N_A \ln N_A = \ln N_A + N_A \ln (1 +
\frac{1}{N_A}),
\end{equation}
where $N_A = \ev{n} L_A$ is the average total particle number in the subsystem
$A$. In the large $N$ limit, the second term converges to $1$. So the
mutual information is given by:
\begin{equation}
E_{M} \equiv \frac{1}{2} (E_A + E_B - S_{GCE\ T=0}) = \frac{1}{2} \left(\ln
\frac{N_A N_B}{N} + 1\right).
\end{equation}
For $N_A \le N_B$, we have
\begin{equation}
E_M \simeq \frac{1}{2} \ln N_A + \mathcal{O}(1).
\end{equation}
This agrees with Eq. (\ref{Eq:EE}) at the leading order.

\subsection{Mutual information}

According to our Eq. (\ref{Eq:RDM}) and Eq. (\ref{Eq:entropy}), to
obtain the von Neumann entropy of the reduced density matrix, all what we
have to do is to diagonalize the truncated two-point correlation
function matrix. Fortunately, within this infinite-range hopping
model, this is rather simple. For a finite system, we can obtain a
general result for all temperatures:
\begin{equation}
\begin{split}
G_{ij} &= \ev{\hat{a}^\dagger_i \hat{a}_j} = \ev{\frac{1}{L} \sum_k e^{-i k (i-j)}
  \hat{b}^\dagger_k \hat{b}_k} \\
& = \frac{1}{L} \ev{\hat{b}^\dagger_0 \hat{b}_0} + \frac{1}{L} \sum_{k \neq 0}
e^{-ik(i-j)}\ev{\hat{b}^\dagger_k \hat{b}_k} = \frac{\ev{N_{0}}}{L} +
\frac{\ev{N_{k}}}{L} \sum_{k \neq 0} e^{-ik(i-j)}\\
& = \frac{\ev{N_0}}{L} + (\delta_{ij} - \frac{1}{L}) \ev{N_k}.
\end{split}
\end{equation}
In the above calculation, we have made use of the fact that $\ev{N_k}$
is $k$-{\em independent}. This matrix is easily diagonalized. For a
system of size $L$, and a $G$ truncated to a size of $L_A \times L_A$
denoted by $G_{A}$, the eigenvalues are
\begin{equation}
g_1 = \frac{L_A \ev{N_0}}{L} + \frac{L - L_A}{L} \ev{N_k},\ g_l =\ev{N_k} \text{
  for $l = 2, \dots , L_A$}.
\end{equation}
Now the von Neumann entropy of subsystem $A$ can be calculated
directly from above result:
\begin{equation}
\begin{split}
E_A &= \sum_{l = 1}^{L_A} \left((1 + g_l) \ln (1 + g_l) - g_l \ln g_l
\right)\\
&= \left( 1 + \frac{L_A \ev{N_0}}{L} + \frac{L - L_A}{L} \ev{N_k} \right) \ln
\left(1 + \frac{L_A \ev{N_0}}{L} + \frac{L - L_A}{L} \ev{N_k} \right) \\
&- \left(\frac{L_A \ev{N_0}}{L} + \frac{L - L_A}{L} \ev{N_k} \right)
\ln \left(\frac{L_A \ev{N_0}}{L} + \frac{L - L_A}{L} \ev{N_k} \right)\\
& + (L_A - 1)\left[(1 + \ev{N_k}) \ln (1 + \ev{N_k}) - \ev{N_k} \ln
(\ev{N_k}) \right].\\
\end{split}
\end{equation}
Combining the above with Eq. (\ref{Eq:thermal entropy}), we can obtain the mutual
information for a general bipartite system:
\begin{equation}\label{Eq:MI}
\begin{split}
&E_M = \frac{1}{2} (E_A + E_B - S) \\
&= \frac{1}{2} \biggl[ \left(1 + \frac{L_A
  \ev{N_0}}{L} + \frac{L_B}{L} \ev{N_k}\right) \ln\left(1 + \frac{L_A
  \ev{N_0}}{L} + \frac{L_B}{L} \ev{N_k}\right)\\
& - \left(\frac{L_A \ev{N_0}}{L} + \frac{L_B}{L} \ev{N_k}\right) \ln\left(\frac{L_A
  \ev{N_0}}{L} + \frac{L_B}{L} \ev{N_k}\right) \\
&+ \left(1 + \frac{L_B \ev{N_0}}{L} + \frac{L_A}{L} \ev{N_k}\right) \ln\left(1 + \frac{L_B
  \ev{N_0}}{L} + \frac{L_A}{L} \ev{N_k}\right)\\
&-\left(\frac{L_B \ev{N_0}}{L} + \frac{L_A}{L} \ev{N_k}\right) \ln\left(\frac{L_B
  \ev{N_0}}{L} + \frac{L_A}{L} \ev{N_k}\right) \\
&  -(1 + \ev{N_k}) \ln (1 + \ev{N_k}) + \ev{N_k} \ln \ev{N_k} - (1 +
\ev{N_0}) \ln (1 + \ev{N_0}) + \ev{N_0} \ln \ev{N_0} \biggl]. \\
\end{split}
\end{equation}

Next, we shall discuss the asymptotic behavior of $E_M$ in different
temperature regions and with different partitions.

\begin{enumerate}
\item[(1)] $L_A \ll L$, $T > T_C$:

in this case, $\ev{N_0}$ and $\ev{N_k} \simeq \ev{n}$ are both of
order one , so $\frac{L_A \langle N_0 \rangle }{L} \rightarrow 0$, $\frac{L_A
  \langle N_k \rangle }{L} \rightarrow 0$, $L_B \simeq L$. Thus
\begin{equation}
E_M \simeq \mathcal{O}\left(\frac{L_A}{L}\right).
\end{equation}

\item[(2)] $L_A = L_B = \frac{L}{2}$, $T > T_C$:
in this case, $E_M$ is reduced to
\begin{equation}
\begin{split}
E_M &= \frac{1}{2} \biggl(\left(2 + \ev{N_0} + \ev{N_k}\right) \ln \left(1 +
\frac{1}{2}\left(\ev{N_0} + \ev{N_k}\right)\right) \\
&- \left(\ev{N_0} + \ev{N_k}\right) \ln \left(\ev{N_0}/2 + \ev{N_k}/2\right) \\
& - (1 + \ev{N_k}) \ln\left(1 + \ev{N_k}\right) +\ev{N_k} \ln \ev{N_k} \\
&- \left(1 + \ev{N_0}\right) \ln (1 + \ev{N_0}) + \ev{N_0} \ln \ev{N_0} \biggl)
\end{split}
\end{equation}
which can also be written as an explicit function of $L$, $\ev{n}$ and $T$ using previous results.
\item[(3)] $L_A \ll L$, $T < T_C$:
in this case, $\ev{N_0}$ becomes a macroscopic number, while $\ev{N_k}$ remains
 to be of order $1$. Therefore we isolate the contribution from \ev{N_0},
and other terms are of order $\mathcal{O}(1)$:
\begin{equation}\label{Eq:E_m3}
E_M = \frac{1}{2} \ln \left(1 + \frac{L_A \langle N_0 \rangle }{L}
\right) + \mathcal{O}(1).
\end{equation}
If we define $\ev{N_{A0}} = \frac{L_A \langle N_0 \rangle }{L}$ to be the
average particle number in the condensate of the subsystem, then
\begin{equation}
E_M = \frac{1}{2} \ln \ev{N_{A0}} + \mathcal{O}(1).
\end{equation}
\item[(4)] $L_A = L_B = \frac{L}{2}$, $T < T_C$:
the leading contribution is again obtained by keeping $\ev{N_0}$'s contribution only:
\begin{equation}\label{Eq:E_m4}
E_M = \frac{1}{2} \ln \left(\frac{1}{4}\ev{N_0} + 1 \right) + \mathcal{O}(1)
= \frac{1}{2} \ln \ev{N_{A0}} + \mathcal{O}(1).
\end{equation}

\item[(5)] $L_A \ll L$, $T = T_C$:
as we calculated before, \ev{N_0} diverges as $\sqrt{L}$. When $L_A
\ll L$, according to Eqs. (\ref{Eq:E_m3}) and (\ref{Eq:N_0@Tc}),
we have
\begin{equation}
E_M = \frac{1}{2} \ln \left(1 + \frac{L_A \sqrt{(\langle n
\rangle + 1) \langle n \rangle}}{\sqrt{L}} \right) + \mathcal{O}(1) = \frac{1}{2}
\ln\left(\langle n \rangle \frac{L_A}{\sqrt{L}}\right) + \mathcal{O}(1).
\end{equation}
For such partition, the scaling behavior of mutual information depends
on the ratio $\frac{L_A}{\sqrt{L}}$. If we consider $L_A$ is a small
but still finite fraction of $L$, the scaling behavior of the mutual
information still persists: $E_M = \frac{1}{4} \ln (\langle n \rangle L_A) + \mathcal{O}(1)$.

\item[(6)] $L_A = L/2$, $T = T_C$:
by referring to Eqs. (\ref{Eq:E_m4}) and (\ref{Eq:N_0@Tc}), we
have \begin{equation} E_M = \frac{1}{2} \ln \left(\frac{1}{4} \sqrt{(\langle n \rangle +
1)\langle
  n \rangle L} \right) + \mathcal{O}(1) = \frac{1}{4} \ln (\langle n \rangle L_A) + \mathcal{O}(1).
\end{equation}
\end{enumerate}

To sum up, from this study, we find that the extensive part of the
thermal entropy of the whole system is canceled out in the mutual
information. Below $T_C$ the mutual information is dominated by
contribution from the condensate. Even above $T_C$, contribution
from the condensate is of the same order as that from the excited
states. The mutual information really characterizes the quantum
feature of the system.

To visualize the behavior of mutual information, we present a numerical calculation for the
mutual information of this model in Fig. (\ref{Fig:mi_inf}). This is
done by numerically diagonalizing the truncated two-point correlation function
matrix then computing the von Neumann entropy of the reduced density
matrix from those eigenvalues. The system is equally partitioned,
$\ev{n} = 1$, so $T_C = \frac{1}{\ln 2}$. We see exactly what our analytic
results tell us: above $T_C$ the mutual information saturates; below
$T_C$, $E_M \simeq \frac{1}{2} L_A$; at $T_C$, $E_M \simeq \frac{1}{4}
L_A$. Note that in the plot, the analytic results (dash lines) deviate
from the numerical calculation because we only keep terms to the
subleading order; terms that goes to zero [i.e. of order
$\mathcal{O}(\frac{1}{L_A})$] in the thermodynamic limit are
neglected.

\begin{figure}

  \includegraphics[width=8cm]{inf_hopping.eps}
  \caption{(Color online) Numerical calculation of mutual information
    for the infinite-range hopping model for equally partitioned
    systems with average density $\ev{n} = 1$. The scatters are
    numerical data, while the dash lines are obtained from our analytic
    results corresponding to the particular temperature. We see
    exactly what our analytic results tell us: above $T_C =
    \frac{1}{\ln 2}$ the mutual information saturates; below
    $T_C$, $E_M \simeq \frac{1}{2} L_A$; at $T_C$, $E_M \simeq
    \frac{1}{4} L_A$. Note that the analytic results (dash lines) deviate
    from the numerical calculation because we only keep terms to the
    subleading order; terms that goes to zero [i.e., of order
    $\mathcal{O}(\frac{1}{L_A})$] in the thermodynamic limit are
    neglected.}
  \label{Fig:mi_inf}
\end{figure}

\section{Numerical Study of Mutual Information in One-dimension with Long-range Hopping}

In this section, we shall present our results of numerical study of
the mutual information of free bosons living on a one dimensional
lattice. In this numerical study, we adopt our previous method, and
calculate the von Neumann entropy of the reduced density matrix from
the eigenvalues of the truncated two-point correlation function
matrix. Throughout this calculation, we hold the average density
fixed at $\ev{n} = 1$ (which means we keep adjusting the chemical potential at
different temperatures) and consider equal partition only.

It is well known that for nearest-neighbor (NN) (or other short-range) hopping models whose
dispersion relation at long-wave length takes the form $\epsilon (k) \sim k^2$, a finite
temperature BEC can only exist in three dimensions (3D). However, 3D is in general very
challenging for a numerical study that requires large system sizes.
Moreover, in 3D the mutual information is dominated by area
law \cite{area-law}, which renders the logarithmic divergence
suggested by our study in Sec. III sub-leading and thus difficult to
isolate. For both of these reasons, it is
desirable to study a model in 1D with BEC at finite $T$. In 1D, the
short-range hopping model does {\em not} support BEC at finite
temperature. To stabilize a condensate in 1D, we introduce power-law
long-range hopping in our free boson model to modify its long-wave
length dispersion. This is similar to what was done in
Ref. [\onlinecite{yusuf}], in which the authors introduced long-range
interaction between spins to stabilize magnetic order in 1D. The
Hamiltonian with long-range hopping is obtained by setting $t_{ij}$ in
Eq. (\ref{Eq:GeneralH}) to the following form, tuned
by a parameter $\gamma$:
\begin{equation}
H = -\sum_{ij} \frac{t}{\abs{i-j}^\gamma} a^\dagger_i a_j = -2t \sum_k
(\sum_{n=1}^{L-1} \frac{\cos (nk)}{n^\gamma}) b^\dagger_k b_k = \sum_k
\varepsilon_\gamma(k) b^\dagger_k b_k.
\end{equation}
We will show that the long wave-length dispersion is modified to be
$\varepsilon_\gamma(k) \sim k^{\gamma - 1}$  for $\gamma < 3$, as a
result if which a finite temperature BEC exists for $\gamma < 2$.

Consider the eigenenergy function $\varepsilon_\gamma(k) =
- 2 t \sum_{n=1}^{L-1} \frac{\cos (nk)}{n^\gamma}$ in the thermodynamic
limit:
\begin{equation}
\begin{split}
\varepsilon_\gamma(k) = -2t\ \sum_{n=1}^{\infty} \frac{\cos (nk)}{n^\gamma}
= -2t\ \text{Re}\left[ \sum_n \frac{e^{i n k}}{n^\gamma} \right]
= -2t\ \text{Re}\left[ F(\gamma, i k) \right],
\end{split}\label{Eq:ek}
\end{equation}
where $F(\gamma, v)$ is the Bose-Einstein integral function \cite{PhysRev.83.678} defined as:
\begin{equation}
F(\gamma, v) = \frac{1}{\Gamma(\gamma)} \int dx \frac{x^{\gamma -
    1}}{e^{x + v} - 1} = \sum_{n=1}^{\infty} \frac{e^{- n v}}{n^\gamma}.
\end{equation}
The analytic properties of $F(\gamma, v)$ near $v = 0$ are known \cite{PhysRev.83.678}:
\begin{equation}\label{Eq:vare_expansion}
F(\gamma, v) =
\begin{cases}[1.5]
  \displaystyle{\Gamma(1 - \gamma) v^{\gamma - 1} + \sum_{n=0}^{\infty}
  \frac{\zeta(\gamma - n)}{n!} (-v)^n}, &
  \gamma \notin \mathbb{Z}, \\
  \displaystyle{\frac{(-v)^{\gamma - 1}}{(\gamma - 1)!} \left[\sum_{r = 1}^{\gamma -
      1} \frac{1}{r} - \ln(v) \right] + \sum_{n \neq \gamma -
    1}^{\infty} \frac{\zeta(\gamma - n)}{n!} (-v)^n}, &
  \gamma \in \mathbb{Z},
\end{cases}
\end{equation}
where $\zeta(x)$ is the Riemann zeta function. Thus we find that
$\varepsilon_\gamma(k) \rightarrow k^{\gamma-1}$ for small $k$ when $1
< \gamma <3$. When $\gamma > 3$, the low energy dispersion is dominated by
the $k^2$ term. When $\gamma \le 1$, $\varepsilon_\gamma(k)$ is not
well-defined in the thermodynamic limit; in order to have a well-defined
thermodynamic limit, the hopping energy $t$ must be properly scaled by
the system size in this case.

Next we will consider the thermodynamics of this model with different
$\gamma$ and demonstrate that for $\gamma < 2$, we indeed have a
finite temperature BEC. At low temperature, only the small
$k$ part of the spectrum is important. For $1 < \gamma < 3$ we consider free bosons
with a dispersion $\sigma k^{\gamma-1}$. Here $\sigma = -2 t \Gamma(1
- \gamma)$ is given in Eq. (\ref{Eq:vare_expansion}). The average
density of such system in the thermodynamic limit is given by:
\begin{equation}
\frac{ \langle N \rangle}{L} = \frac{1}{2 \pi}\int dk \ev{N_k} =
\frac{1}{2\pi} \frac{\sigma^{\frac{1}{1 - \gamma}} }{(\gamma - 1)}
\int_0^\infty d\varepsilon  \frac{\varepsilon^{\frac{1}{\gamma -1}
    -1}}{z^{-1}e^{\beta \varepsilon} - 1} = \frac{(\beta
  \sigma)^{\frac{1}{1-\gamma}}}{2 \pi (\gamma -1)}
\Gamma(\frac{1}{\gamma -1}) g_{\frac{1}{\gamma - 1}}(z),
\end{equation}
where $z = e^{\beta \mu}$, $\beta = \frac{1}{T}$ is the inverse
temperature, and $g_v(z) = \frac{1}{\Gamma(v)} \int_0^\infty dx
\frac{x^{v-1}}{z^{-1} e^x - 1}$ is the Bose-Einstein integral
function. To have a finite temperature BEC,
$\ev{N} / L = \ev{n}$ must remain finite when $z \rightarrow 1$, which
indicates $\frac{1}{\gamma - 1} > 1$, because for $v \le 1$,
$g_{v}(1)$ diverges.

To have a better understanding of the thermodynamics of this
model, in Fig. (\ref{Fig:Tc}) we present a numerical calculation of
$T_C$. The exact average density of the system is:
\begin{equation}
\frac{\langle N \rangle}{L} = \frac{1}{2\pi} \int_{-\pi}^{\pi} dk \frac{1}{e^{\beta(\varepsilon_\gamma(k) - \mu)} - 1},
\end{equation}
where $\varepsilon_\gamma(k)$ is the eigenenergy function given in
Eq. (\ref{Eq:ek}). $T_C$ is computed by setting $\mu =
\varepsilon_\gamma(0)$ and then solving this equation
numerically. According to Fig. \ref{Fig:Tc}, $T_C$ grows
monotonically from $0$ to $\infty$ as $\gamma$ goes from $2$ to
$1$. The divergent behavior of $T_C$ as $\gamma \rightarrow 1$ is a
consequence of the divergent bandwidth in that limit.

\begin{figure}

  \includegraphics[width=8cm]{Tc.eps}
  \caption{(Color online) Numerical calculation of $T_C$ (line + symbols) for the
    long-range hopping model in the thermodynamic limit. $T_C$ is
    measured in unit of the hopping energy $t$ which is set to 1. As one can
    see, $T_C$ grows monotonically from $0$ to $\infty$ as $\gamma$
    goes from $2$ to $1$. The divergent behavior of $T_C$ as $\gamma
    \rightarrow 1$ is a consequence of the divergent bandwidth.}
  \label{Fig:Tc}
\end{figure}

According to our study of the infinite range hopping model, above
$T_c$ the mutual information should saturate as the system size
grows. Below $T_C$, the mutual information has a scaling
behavior $E_M \simeq \frac{1}{2} \ln L_A$; for $T = T_C$ $E_M
\simeq \frac{1}{4} \ln L_A$ for equally partitioned system in that model. We expect
the $\ln L_A$ scaling behavior both below $T_C$ and at $T_C$ to
persist in the long-range hopping model. As we shall see later, this
is indeed the case. However, the details of the scaling behavior
(i.e. the prefactor) can be different for different $\gamma$.
To study this scaling behavior, we fix the temperature and examine the
mutual information as a function of system size. This is desirable
because, if our conjecture according to the study of infinite-range
hopping model is correct, the mutual information will be
proportional to $\ln L_A$ when $T \le T_C$.

Before we demonstrate our results in the long-range hopping model,
first let us verify our analysis in the NN hopping
model in which no BEC would occur. This case actually corresponds to
$\gamma \rightarrow \infty$. The Hamiltonian is given by
restricting the hopping in Eq. (\ref{Eq:GeneralH}) to the nearest
neighbors only:
\begin{equation}
H_{n.n} = - t \sum_{\langle ij \rangle} \hat{c}_i^\dagger \hat{c}_j.
\end{equation}
Figure \ref{Fig:fixT_k2} is a linear-log plot of mutual information
against subsystem size at different temperatures. Throughout our study we shall consider equal
partition only, i.e., $L_A = L / 2$. The average density is also set to
$\ev{n} = 1$ here, so are the other results we will show
later. Clearly, at a fixed temperature, the mutual information
saturates as the system size grows. At low temperatures, small systems can be considered in the zero
temperature limit. This leads to the mutual information growing as
$\sim \frac{1}{2}\ln L_A$, until saturation kicks in.

\begin{figure}

  \includegraphics[width=8cm]{mi_n1_k2.eps}
  \caption{(Color online) Mutual information of the nearest neighbor hopping model
    plotted against subsystem size on a logarithmic scale. Average
    density is set to $\ev{n} = 1$, and the system is equally
    partitioned, $L = 2 L_A$. The black dash line is $E_M \sim
    \frac{1}{2}\ln L_A$. This line will be in other graphs for comparison as
    well. Clearly, the mutual information saturates when the system
    size grows large enough.}
  \label{Fig:fixT_k2}
\end{figure}

Next we consider $ 1 < \gamma < 2$. Now the finite temperature transition
emerges, and the signature for the transition in mutual information -
the logarithmic scaling with (sub-)system size also emerges. In
Fig. \ref{Fig:fixT_g1.7}, we plot the mutual information (scatters) for $\gamma
= 1.7$ at different temperatures. The best fit (cyan dash
line corresponding to the diamond data points) for $\beta = 0.5 > \beta_C$
gives $E_M = 0.2405 \ln L_A + 0.214$. At $\beta_C = 0.297$, the
scaling behavior is fit (magenta dash line) as $E_M = 0.1226 \ln
L_A + 0.1688$. Both behaviors agrees qualitatively with what has been
suggested by our analytic study of the infinite-range hopping
model. When the temperature is well below $T_C$, we have the
logarithmic scaling behavior: a set of parallel linear lines on this
logarithm scale plot for different temperatures. However, the
prefactor is significantly different from that of the infinite range
hopping model which is $\frac{1}{2}$. In fact, by calculating mutual
information of different $\gamma$'s, we find that the prefactor varies as $\gamma$ changes.
For very low temperature, small systems again effectively fall into the zero
temperature region, and the mutual information restores to the $\sim
\frac{1}{2} \ln L_A$ behavior as in zero temperature. But when the
system size becomes large it crosses back to the finite temperature
scaling behavior again. This is evident for $\beta > 1$ in the
figure. For temperature close to but still below $T_C$, small systems
behave differently: the mutual information scales more like the line
for $\beta = \beta_C$, and it bends up as the system size increases and finally
crosses back to its genuine behavior below $T_C$. For temperature
close to $T_C$ but now above, small systems behave the other way: the
mutual information bends downwards and saturates at large system
size. $\beta_c$ serves as a very distinctive boundary between the
above two different bending behaviors. The latter two bending features
are also present in Fig. \ref{Fig:mi_inf}, our numerical verification of the infinite-range
hopping model. But the first feature at very low temperature is
missing in Fig. \ref{Fig:mi_inf} since in that case the mutual
information scales the same way as the entanglement entropy.

\begin{figure}

  \includegraphics[width=8cm]{gamma_1.7.eps}
  \caption{(Color online) Mutual information of the long-range hopping
    model with the parameter $\gamma = 1.7$ as a function of
    subsystem size on a logarithmic scale, at various (inverse)
    temperatures. The average boson density \ev{n} is set to 1, and
    the system is equally partitioned, $L_A = L /2$. The scaling behavior
    for inverse temperature $\beta = 0.5$ goes as $E_M = 0.2405 \ln L_A + 0.214$ (cyan
    dash line corresponding to the diamond data points). At the
    transition point, $\beta=\beta_C = 0.297$, the scaling law is fit
    to be $E_M = 0.1226 \ln L_A + 0.1688$ (red dash line corresponding
    to the right triangle data points).}
  \label{Fig:fixT_g1.7}
\end{figure}

Very similar behaviors were observed for the entire range $1 < \gamma <
2$. Representative results are presented in Figs. \ref{Fig:fixT_g1.5}
and \ref{Fig:fixT_g1.3} for $\gamma=1.5$ and $1.3$ respectively. We
thus conclude that for the entire range $1 < \gamma < 2$, mutual
information saturates for $T > T_C$, while it diverges logarithmically
with increasing subsystems size, for both $T < T_C$ and $T=T_C$. The
coefficients in front of the logarithms are $\gamma$-dependent.

\begin{figure}

  \includegraphics[width=8cm]{mi_gamma1.5_n1.eps}
  \caption{(Color online) Mutual information of the long-range hopping
    model with the parameter $\gamma = 1.5$ as a function of
    subsystem size on a logarithmic scale, at various (inverse)
    temperatures. The average boson density \ev{n} is set to 1, and
    the system is equally partitioned, $L_A = L /2$. The mutual
    information for $\beta = 1$ is fit to scale as $E_M \simeq 0.324
    \ln L + 0.445$ (orange dash line). At $\beta_C = 0.16843$, we observer a
    weaker scaling behavior which is fit to be $E_M
    = 0.064 \ln L_A + 0.084$ (red dash line).}
  \label{Fig:fixT_g1.5}
\end{figure}

\begin{figure}

  \includegraphics[width=8cm]{gamma_1.3.eps}
  \caption{(Color online) Mutual information of the long-range hopping
    model with the parameter $\gamma = 1.3$ as a function of
    subsystem size on a logarithmic scale, at various (inverse)
    temperatures. The average boson density \ev{n} is set to 1, and
    the system is equally partitioned, $L_A = L /2$. The best
    fitting for $\beta = 0.5 > \beta_C$ (cyan dash line) gives $E_M = 0.378 \ln
    L_A + 0.2$. At $\beta_C = 0.0954$, the scaling behavior is fit
    (red dash line) as $E_M = 0.023 \ln L_A + 0.03535$.}
  \label{Fig:fixT_g1.3}
\end{figure}

\section{Summary and Concluding Remarks}

In this paper we have studied entanglement properties of free {\it
  non-relativistic} Bose gases. At zero temperature, all
particles fall into the ground state, and we find the entanglement entropy
diverges as the logarithm of the particle number in the
subsystem. At finite temperatures, we studied the
natural generalization of entanglement entropy - the mutual
information. We find the mutual information
has a similar divergence in the presence of a Bose-Einstein
condensate. When the system is above $T_C$ or does not have a
condensate, the mutual information saturates for large subsystem
size. It should be noted that for the special models we studied in
this paper there is no area-law contribution to the mutual
information, thus the contribution from the condensate, when present,
dominates the mutual information. In more generic models in two- or
three-dimensions where an area-law contribution is present, we expect
such logarithmic divergent contribution from the condensate to be
present as a sub-leading term in the subsystem-size dependence of the
entanglement entropy and mutual information.

Physically it is easy to understand why the condensate makes such an
important contribution to entanglement. First of all, BEC is
intrinsically a quantum process, just like entanglement reflects the
intrinsically quantum nature of the system. More specifically, when a
(macroscopically) large number of particles occupy the same state (at
$k=0$), they are necessarily delocalized throughout the sample, giving
rise to entanglement between blocks.

Just like in our previous work on a very different
system \cite{ding:052109}, our results here suggest that {\em conventional}
ordering, like BEC, makes a logarithmic
contribution to entanglement. One thus needs to take caution when using
entanglement as a diagnostic for exotic phases (such as topological
phases) or quantum criticality.

\section*{ACKNOWLEDGMENTS}

We thank Dr. Libby Heaney for a useful correspondence. This work was
supported by NSF Grant No. DMR-0704133. The authors thank the Kavli Institute for
Theoretical Physics (KITP) for the warm hospitality during the
completion of this work. The work at KITP was supported in part by
National Science Foundation Grant No. PHY-0551164.


\section{ Introduction}

Scalar fields play essential role in many branches of physics, from
cosmology to condensed matter physics to particle physics --  there
is an unremitting interest in models of self-interacting scalar
fields.  The rich variety of such models includes some that have
been studied only recently, e.g., the so called K-fields with a
nonstandard kinetic part \cite{1}, or models with a non-smooth
V-shaped self-interaction  \cite{2}. The signum-Gordon model
considered in the present paper is probably the simplest example
from the latter class. The pertinent field potential has the form
$U(\varphi)=  g |\varphi|,$ where $g
>0$ is a coupling constant and $|\varphi|$ is the modulus of the real
scalar field $\varphi$. Such potential is V-shaped with the sharp
minimum at the vacuum field $\varphi=0$. Models of this kind were
discovered while playing with the well-known classical systems of
harmonically coupled pendulums in order to illustrate the phenomenon
of spontaneous symmetry breaking and topological defects \cite{3}.
Subsequent investigations have revealed that the V-shaped form of
the potential has very interesting consequences for the dynamics of
the scalar field. One of them is the existence of strictly periodic
oscillons \cite{4}. The motivation, various results and further
references for the V-shaped self-interaction can be found in
\cite{2,3,4}.

The present note is a follow-up to the paper \cite{4}. The oscillons
described in that paper did not move in space (apart from the
trivial uniform motion obtained by applying Lorentz boosts). Rather
unexpectedly, we have found that there exist also oscillons that
periodically move to and fro in the space with arbitrary constant
velocity $\pm v$, where $0 < |v| \leq 1$. For the oscillons
presented in \cite{4}  $v=0$. The  new oscillons appear naturally
when the particular solution reported in \cite{4} is put in the
framework of polynomial solutions of the signum-Gordon equation.

Comparing with other oscillons discussed in literature \cite{5},
several differences should be pointed out. First, our oscillons are
strictly periodic in time, in particular they do not emit any
radiation. Second, they have  strictly finite size because the field
assumes the vacuum value  at a finite distance exactly. Third, they
have relatively simple, explicitly given form composed of several
linear and quadratic functions of the time $t$ and the spatial
coordinate $x$.

The swaying oscillon reminds  the wobbling kink in the $\varphi^4$
model \cite{6}.  However, one should note that the wobbling kink is
an excitation of static kink, while all the swaying oscillons are
degenerate in energy, and moreover there is no static oscillon --
even for the presented in \cite{4} non-swaying one  the field
oscillates in time.

The plan of our paper is as follows. Section 2 is devoted to a
preliminary discussion of the signum-Gordon equation and of its
solutions. The swaying oscillons are presented in Section 3. Section
4 contains the conclusion.

\section{Preliminaries }

The Lagrangian of the signum-Gordon model (s-G) has the form
\begin{equation} L = \frac{1}{2} (\partial_t \varphi\partial_t
\varphi  -   \partial_x \varphi\partial_x \varphi)
 - g\:|\varphi|, \end{equation} where $\varphi$ is a real  scalar field,
$t, x$ are  time and position coordinates in the two-dimensional
Minkowski space-time $M$. For convenience, $t, x, \varphi, g$ are
dimensionless --  this can  be achieved by redefinitions of the
physical position, time, field and the coupling constant
(multiplication by constants of appropriate dimensions).  The
signum-Gordon equation
\begin{equation}
\partial_t^2 \varphi     -  \partial_x^2 \varphi + \mbox{sign}(\varphi(x,
t))=0
\end{equation}
is the Euler-Lagrange equation corresponding to Lagrangian (1) (from
now on we put $g=1$). The sign function $\mbox{sign}(\varphi)$ has
the values $\pm 1$ for $\varphi \neq 0$ and $\mbox{sign}(0)=0$. The
simplest way to obtain Eq.\ (2) from Lagrangian (1) is first to
regularize the field potential $U(\varphi)= |\varphi|$, e.g.,
$U(\varphi) =  \sqrt{\epsilon^2 + \varphi^2}$ or $U(\varphi) =
\epsilon\: \ln(\cosh(\varphi/\epsilon))$, and to take the limit
$\epsilon \rightarrow 0_+ $ in the Euler-Lagrange equation obtained
from the regularized Lagrangian. Direct computation of the variation
of the action $S = \int dt dx L$ is more subtle because of the
$|\varphi|$ term, but  it  gives the signum-Gordon equation (2) too.

The l.h.s.  of  Eq.\ (2) is not continuous with respect to
$\varphi$. Because such equations are not very common in field
theory, let us briefly comment  on the related mathematical aspects.
First, it is clear that in general one should expect non smooth
solutions: the value of at least one of the second derivatives
$\partial_t^2\varphi, \partial_x^2\varphi$ has to jump when the
function $\mbox{sign}(\varphi)$  changes its value. Second, the use
of the stationary action principle implies that in general we
consider so called weak solutions of the Euler-Lagrange equation,
\cite{7}. For the weak ones it is sufficient that
\[ \delta S = \int_M dt dx\; \left(\frac{\partial L}{\partial
\phi}\:
\delta\phi(x,t)
+ \frac{\partial L}{\partial (\partial_{\mu}\phi)}  \;
\partial_{\mu} \delta\phi(x,t)\right) =0 \] for all test functions $\delta\phi(x,t)$ from a
certain class (typically one uses the  $D(M)$ class of smooth
functions on $M$ with compact support).  This condition is
equivalent to $\int_M dt dx\:  {\cal E\!\!L}\: \delta \varphi =0$,
where $ {\cal E\!\!L} =
\partial L/\partial \phi - \partial_{\mu}(\partial L/
\partial(\partial_{\mu}\varphi))$, only if the derivative $\partial_{\mu}(\partial L/
\partial(\partial_{\mu}\varphi))$ exists for a given probed function
$\varphi(x,t)$. Then  the Euler--Lagrange equation ${\cal
E\!\!L}=0$, in our case the signum-Gordon Eq.\ (2), has to be
satisfied at almost all points $(x,t)$ in the two-dimensional
space-time  $M$, but not necessarily at all points as it would be
the case with strong solutions.  Of course, the set of weak
solutions contains  the strong ones as a subset.

In the case of signum-Gordon equation the weak solutions that are
not strong are ubiquitous.  For instance, $\varphi_0 = x^2/2$ is a
smooth static solution of Eq.\ (2) in the weak sense, but not in the
strong sense. The point is that $\partial_t^2 \varphi_0 -
\partial_x^2 \varphi_0 + \mbox{sign}(\varphi_0)=0$ everywhere in
$M$ except the line $x=0$ in $M$. On this line $\partial_t^2
\varphi_0 -
\partial_x^2 \varphi_0 + \mbox{sign}(\varphi_0)= -1 $
because $\partial_x^2 \varphi_0 =1$, $ \mbox{sign}(0) =0$.
Nevertheless,
\[ \int_{M} dt dx \;[\partial_t^2 \varphi_0 -
\partial_x^2 \varphi_0 + \mbox{sign}(\varphi_0)] \; \delta\phi(x,t)
=0 \] for arbitrary test function $\delta\phi$.

In general, physically relevant are the weak solutions.  To see
this, consider the following simple example from classical mechanics
of a point particle on a plane with Cartesian coordinates  $(x,y)$.
The particle is free except when it crosses the $y$-axis, where  it
is subjected to a finite constant force $\vec{F}_0$ parallel to the
$y$-axis. Thus, the force $\vec{F}=0$ at all points $(x, y)$ with
$x\neq0$,  and $\vec{F}= \vec{F}_0$ when  $x =0$. It is clear that
integrating the Newton's equation $d\vec{p}/dt = \vec{F}$ we obtain
$\vec{p} = const$ even if the trajectory crosses the $y$-axis.  The
physical reason is that the finite force $\vec{F}_0$ acts on the
particle only during infinitesimally short time when the particle is
exactly on the $y$-axis, hence it is not able to perturb the free
motion. Such trajectories are the weak solutions of the Newton's
equation (now the test functions are denoted as $\delta\vec{r}(t)$
and we integrate over $t$).  On the other hand, the trajectories
which do not intersect  the $y$-axis are solutions in the strong
sense. Notice that such Newton's equation is not equivalent to the
free equation, in which $\vec{F}=0$ everywhere,  because our
particle is accelerated if it moves along the $y$-axis.

Coming back to the signum-Gordon model, in the case  the field
$\varphi$ is constant in the space  Eq. (2) acquires the form of one
dimensional Newton's equation $ \ddot{\varphi}(t) = -
\mbox{sign}(\varphi)$ that describes nonlinear oscillations around
$\varphi=0$. Notice that there is no linear regime even for
arbitrarily small values of $\varphi$. Newton's equation of this
kind appears in the elementary problem of a ball vertically bouncing
from a floor in a constant gravitational field (the elevation above
the floor is given by $|\varphi|$) \cite{2}.   Many examples of
oscillatory systems from classical mechanics that do not have the
linear small amplitude regime can be found in \cite{8}.

Because the function $\mbox{sign}(\varphi)$ is piece-wise constant,
it is natural first to solve Eq.\ (2) in the regions in which
$\varphi$ has a constant sign. For instance, if $\varphi <0$, Eq.\
(2) acquires the form
\begin{equation}
\partial_t^2 \varphi     -  \partial_x^2 \varphi - 1 =0.
\end{equation}
The oscillon solutions are constructed from second order polynomials
in $x, t$.  The most general second order polynomial  that obeys Eq.
(3) has the form
\begin{equation}
\varphi_2(x, t) = a_0 x^2 + a_1 t x + (a_0+\frac{1}{2}) t^2 + b_0 x
+ b_1 t + c_0,
\end{equation}
where $a_0, a_1, b_0, b_1, c_0$ are constants  (beware that they are
not completely arbitrary because of the condition $\varphi_2 <0$).
It is rather exceptional feature of the signum-Gordon equation that
non-trivial and interesting solutions can be constructed from such
simple elementary functions.  Note that the class of functions of
the form (4) is invariant with respect to Lorentz boosts, space-time
translations, and the reflections  $x \rightarrow -x, \; t
\rightarrow -t$. It contains the static solutions of the form
\begin{equation}
\varphi_s = - \frac{1}{2} (x-b_0)^2 + c_0 + \frac{1}{2} b_0^2,
\end{equation}
where $c_0 + b_0^2/2 <0$ in order to keep $\varphi_s <0$.

The oscillons are constructed by patching together several such
polynomial solutions. The patching conditions have the standard
form: the field $\varphi$ is continuous all over  $M$, also the
derivatives $\partial_t\varphi, \partial_x\varphi$ are continuous
function of $x, t$ except perhaps at the border line between two
patches. If the border line  is  a (segment of)   characteristic
line ($x = \pm t + \mbox{const}$) for the signum-Gordon equation,
the  derivative in the direction perpendicular to that line does not
have to be continuous --  a finite jump is allowed.

\section{The swaying oscillons }

Hint that new oscillons  may exist comes from the following
procedure for constructing periodic solutions of the signum-Gordon
equation. Let $\varphi_-(x,t) $ be a  solution of Eq.\ (3) negative
for all $t$ from an open interval  $(0, T),  \; T>0,$ and such that
\begin{equation}
\varphi_-(x, 0) = \: 0 \: = \varphi_-(x, T).
\end{equation}
It is clear that the function $\varphi_+$ defined by
\begin{equation}
\varphi_+(x, t) = -  \varphi_-(x, -t)
\end{equation}
is a positive solution of the equation $\partial_t^2 \varphi     -
\partial_x^2 \varphi +1 =0   $ for all $t \in (-T,0)$.
The functions $\varphi_-, \varphi_+$ as well as their time
derivatives match each other at the time $t=0$:
\[
\varphi_+(x, 0) = \: 0 \: = \varphi_-(x, 0), \]
\[
\lim_{t \rightarrow 0_-} \partial_t\varphi_+(x, t ) = - \lim_{t
\rightarrow 0_-} \partial_t\varphi_-(x, -t )  = \lim_{s \rightarrow
0_+} \partial_s\varphi_-(x, s ),
\]
where $s$ stands for $-t$, and $t\in (-T,0)$.  The crucial
observation is that also $\varphi_+(x, -T), \:  \varphi_-(x, T)$
match each other:
\[
\varphi_+(x, -T) = \: 0 \: = \varphi_-(x, T), \]
\[
\lim_{t \rightarrow -T_+} \partial_t\varphi_+(x, t ) =  \lim_{s
\rightarrow T_-} \partial_s\varphi_-(x, s ).
\]
Therefore, we may extend our partial solutions $\varphi_{\pm}$ to
all times $t \geq T$ and $ t \leq -T$ just by applying time
translations (by multiples of $\pm T$) to $\varphi_{\pm}$. In this
way we obtain periodic solutions of the signum-Gordon equation  (2)
with the period equal to $2T$, provided that there exists
$\varphi_-(x,t)$ with the properties specified above.  It turns out
that a class of the solutions  $\varphi_-(x,t)$ with the desired
properties can be constructed by patching together several solutions
of the form (4). Also the trivial solution $\varphi=0$ is involved.
The schematic picture of such `patchwork' for the swaying oscillon
is presented in Fig.\ 1.

Note that so far we have not made any assumption about the behavior
of $\varphi_-$ at large $|x|$. In \cite{4} certain restrictive
boundary conditions were imposed right at the start of calculations
and they forced the oscillons to stay still.  The method adopted in
the present note is radically different from the one used in
\cite{4} -- in that paper the main tool was d'Alembert formula for
solutions with given initial data.

In order to ensure finiteness of the total energy we assume that
$\varphi_-(x,t) =0$ outside a certain compact region. This is
related to the general observation that in the case of models with
the V-shaped potential there are no exponential or long range tails.
The field reaches its vacuum value rather abruptly, the tails have a
parabolic shape and a strictly finite length \cite{2}. Thus, our
first task is to find the polynomials of the form (4) which match
the trivial solution $\varphi =0$. The matching conditions imposed
on a line $x(t)$ in $M$ can be written in the form
\[
\varphi_2(x(t), t) =0, \;\;\; \left. \partial_x \varphi_2(x,
t)\right|_{x = x(t)} =0. \]  They give the following two equations
\[
a_0 x^2(t) + a_1\: t\: x(t)  + (a_0+\frac{1}{2}) t^2 + b_0 x(t) +
b_1 t + c_0 =0, \;\;\; 2 a_0 x(t) + a_1 t + b_0 =0.  \] Simple
calculations show that the solution $x(t)$ exists only if $a_0
\neq0$, and then
\begin{equation}
x(t) = v t + x_0, \;\; \varphi_2(x,t) = -\frac{(x - x(t))^2}{2
(1-v^2)},
\end{equation} where  $ v= - a_1/(2 a_0), \; x_0 = - b_0/(2 a_0)$,
and $v^2 <1$  in order to satisfy the condition $\varphi_2 < 0$.
Here we have assumed that $x(t)$ does not coincide with a
characteristic line.

Thus we have found that the boundary of our oscillon has to move
with the constant velocity $v$, and close to the boundary the field
has the parabolic shape (as expected). Note that $\varphi$ given by
formula (8) coincides with the Lorentz boosted and translated in the
space the static solution
\[
\varphi_s = \left\{ \begin{array}{cc} 0  &  x \leq 0, \\
- \frac{x^2}{2} &  x>0. \end{array}\right.
\]
(but the swaying oscillon is not the Lorentz boosted oscillon of
\cite{4}).

The structure of the solution $\varphi_-$ is shown in Fig.\ 1, in
which the support of $\varphi_-$ for the swaying oscillon of unit
length and vanishing total momentum is depicted. The period of this
oscillon is equal to its spatial size, i.e., to 1. As discussed in
\cite{4}, we may use the symmetries of the signum-Gordon equation,
such as Poincar\'e or scaling transformations in order to obtain
more general oscillons. The interior of the parallelogram is divided
into seven sectors $\emph{a}\div \emph{f}\:$ by the four
characteristic lines drawn from its corners. Each sector has
different causal neighborhood. For instance, the field in the
triangular sector \emph{c} is completely determined by Cauchy data
on the segment $[1/2 + v/2, 1]$ of the $x$ axis; the sector \emph{e}
is controlled by Cauchy data in the future, i.e., on the segment
$[1/2, 1+v/2]$ of the $t=1/2$ line which lies in the future of the
sector \emph{e}; the sectors \emph{a } and \emph{d} are controlled
by the boundaries of the oscillon, etc. The parallelogram shown in
Fig.\ 1 has the height equal to one half of its length. In this case
the characteristic lines drawn from the lower (upper) corners meet
at a point lying on the upper (lower) edge.

In the case of the non-swaying oscillon presented in \cite{4} we
have $v=0$ and a rectangle in Fig.\ 1. The parallelogram is the
simplest deformation of that rectangle  consistent with the
conditions $ \varphi_-(x,0) = 0 = \varphi_-(x, \frac{1}{2})$,  and
with the fact that the both sides of the oscillon have to move with
a constant velocity, as has been shown above. Such generalization --
the parallelogram instead of the rectangle -- is very suggestive in
the `patchwork approach' adopted in the present paper, but it is not
obvious at all in the based on the d'Alembert formula approach used
in \cite{4}. Let us also note that a Lorentz boost of the oscillon
considered in \cite{4} gives a uniform rectilinear motion, and not
the swaying one.  Moreover, it deforms the rectangle into a
hyperbolically rotated parallelogram with the upper and bottom sides
not parallel to the $x$-axis.

The building blocks of $\varphi_-$ are denoted as $\varphi_a, \ldots
, \varphi_g$ after the sectors of the parallelogram. The fields
$\varphi_a, \varphi_d$ in the sectors $a$ and $ d$  have the form
(8) with $x(t) = v t$ or $x(t)= v t +1$, respectively, i.e.,
\begin{equation}
\varphi_a = -\frac{(x-vt)^2}{2(1-v^2)}, \;\;\; \varphi_d = -
\frac{(x-vt-1)^2}{2(1-v^2)}.
\end{equation}
In the regions $x \leq vt$ and $ x \geq v t +1$, i.e. on both sides
of the parallelogram, the field has the vacuum value  $\varphi=0$.
\begin{center}
\begin{figure}[tph!]
 \hspace*{1.7cm}
\includegraphics[height=5.5cm, width=8cm]{fig1.eps}
\caption{\small The support of the  solution $\varphi_-(x,t)$. The
field  $\varphi_-(x,t)$ vanishes on the continuous lines that form
the boundary of the parallelogram. In each sector $\emph{a}\div
\emph{f}$ the function $\varphi_-$ is given by a different formula.
The matching conditions that relate the functions in neighboring
sectors are imposed along the dotted lines. These four lines  are
characteristic lines of the signum-Gordon equation. They have the
slopes $\pm 1$. }
\end{figure}
\end{center}
The fields $\varphi_b, \varphi_c,  \varphi_e,  \varphi_f$ are
determined by imposing on the  solution (4) the condition $\varphi_2
=0$ on the lines $t=0$ or $t=1/2$, and  the  conditions of matching
with $\varphi_a$ or $\varphi_d$ on the characteristic lines. As the
example let us determine $\varphi_b$. The condition $\varphi_2(x,0)
=0$ gives $a_0=b_0=c_0=0$. Next, $\varphi_2 = t^2/2 + a_1 t x +b_1
t$ is compared to $\varphi_a$ on the part of the characteristic line
$x=t$ with $t \in (0, 1/4 + v/4)$:
\[ \frac{t^2}{2} + a_1 t^2 +b_1 t = - \frac{1-v}{2(1+v)} t^2. \] Therefore,
$b_1=0$, $a_1 = -1/(1+v)$, and  $\varphi_b = t^2/2 - tx/(1+v)$.
Similar calculations give $\varphi_c, \varphi_e, \varphi_f$.
Finally, we compute $\varphi_g$ by comparing $\varphi_2$ to
$\varphi_b, \varphi_c,  \varphi_e, \varphi_f$  along the four
characteristic lines that form the boundary of the sector $g$. The
results have the following form:
\begin{equation}
\varphi_b = \frac{t^2}{2} - \frac{t x }{1+v}, \;\;\; \varphi_c =
\frac{t^2}{2} + \frac{t (x-1)}{1-v},
\end{equation}
\begin{equation}
\varphi_e = \frac{1}{2} (t-\frac{1}{2}) \: \left( \frac{1 }{2 } + t
+ \frac{1-2x }{1+v}\right), \end{equation}
\begin{equation}
\varphi_f = \frac{1}{2} (t-\frac{1}{2}) \: \left( \frac{1 }{2 } + t
+ \frac{2x-1}{1-v}\right),
\end{equation}
\begin{equation}
\varphi_g =  \frac{(v x +t)^2 }{2(1-v^2)} + \frac{x^2 + t^2}{2 } +
\frac{1+v}{8(1-v)}  - \frac{x+t }{2(1-v)}.
\end{equation}
All these functions are negative inside their domains.  The shape of
the swaying oscillon is depicted in Fig.\ 2.
\begin{center}
\begin{figure}[tph!]
\hspace*{1.8cm}
\includegraphics[height=5.5cm, width=7cm]{fig2.eps}
\caption{\small The shape of the swaying oscillon at the times
$t=1/8$ (the dashed line) and $t=3/8$ (the continuous line). The
velocity of the swaying motion  $v = 1/2$. }
\end{figure}
\end{center}
The evolution of our oscillon is described by the function
$\varphi_-(x, t)$ in the time interval $[0, 1/2]$, and by
$\varphi_+(x,t)$,  formula (7),  for $t\in[-1/2,0]$. In particular,
the field $\varphi_+$ at the boundaries of the oscillon has the form
\[
\varphi_{+,a}(x,t) = \frac{(x+v t)^2}{2(1-v^2)}, \;\;\;
\varphi_{+,d}(x,t) = \frac{(x+v t-1)^2}{2(1-v^2)}.  \] We see that
now the boundaries of the oscillon move with  the velocity $-v$. The
world-sheet of the oscillon is depicted in Fig.\ 3. Note that at the
times $t=k/2$, $k$ - integer, when the sharp turns take place, the
field $\varphi$ vanishes everywhere.
\begin{center}
\begin{figure}[tph!]
 \hspace*{1.8cm}
\includegraphics[height=5.5cm, width=7cm]{fig3.eps}
\caption{\small The world-sheet of the swaying oscillon.  In the
interior of the parallelograms $\varphi_+ >0$ and $\varphi_- <0$,
whereas on their boundaries (the thick continuous lines)
$\varphi_{\pm}=0$.}
\end{figure}
\end{center}

In the case  $x(t)$ is a  characteristic line we have $x(t) = v t +
x_0$, where  $|v| =1$. There is just one matching condition
$\varphi_2(x(t),t) =0$. Solving it we obtain relations between the
constant coefficients present in $\varphi_2$, and finally
\[
\varphi_2(x,t) = (x-x(t))\; \left(a_0 \:(x - x(t)) - \frac{1}{2}
x(t) + (\frac{1}{2} + 2 a_0) x_0 + b_0\right) \] (in the region
where $\varphi_2(x,t) <0$).  Next steps are similar to those
described above, but the situation is much simpler. In particular,
when $v=1$, the left and right hand sides of the parallelogram in
Fig.\ 1 coincide with characteristic lines. Therefore, the sectors
$a, f, c, d, g$ are absent. The remaining sectors $ b, e$ meet at
the line $x=1-t$. The corresponding functions $\varphi_b, \varphi_e$
are given by formulas (10), (11) with $v=1$, and they correctly
match each other on that line.

The total energy $E$ and momentum $P$ of the oscillon can easily be
calculated from formulas
\[
E= \frac{1}{2}  \int^{\infty}_{-\infty}dx\; [(\partial_t\varphi)^2 +
(\partial_x\varphi)^2]+  \int^{\infty}_{-\infty}dx\; |\varphi|,
\;\;\; P = -  \int^{\infty}_{-\infty}dx\;\partial_t\varphi
\partial_x\varphi,  \]
considered at the time $t=0$ when $\varphi=0= \partial_x\varphi$. We
see that $P=0$, in spite of the swaying motion of the oscillon. This
can be understood if we regard the swaying oscillon as a nonlinear
bound state of the basic oscillon, that is the one with  $v=0$, with
a wave packet traveling along the basic oscillon. If the swaying
oscillon has  $P=0$, the nonzero momentum of the wave packet is
compensated by the momentum related with the motion of the basic
oscillon. The wave packet bounces from the boundaries of the basic
oscillon and does not leave its interior. Then the basic oscillon
has to move accordingly in order to keep $P=0$.

In order to compute the total energy we need
$\partial_t\varphi|_{t=0}$. Formulas (9), (10)  give
$\partial_t\varphi_b|_{t=0}= -x/(1+v)$ for $ x\in [0,(1+v)/2]$, and
$\partial_t\varphi_c|_{t=0}= (x-1)/(1-v)$ for $ x\in [(1+v)/2, 1]$.
In the case $v=\pm1$ the part with $\varphi_c$ is absent.  Simple
integration gives $E=1/24$. Thus the total energy does not depend on
$v$ --  all our swaying oscillons have the same energy. We have not
found any explanation for such a degeneracy. One may suspect that
there exists a hidden symmetry. Note that it would be sufficient if
it works only in the subspace of the polynomial solutions
$\varphi_2$, not necessarily on the level of Lagrangian or action.
The bound state interpretation offers the following picture. The
basic oscillon set in motion would have an energy larger that its
rest energy equal to 1/24. Apparently, the binding energy
compensates the kinetic energy of the basic oscillon as well as the
energy of the bouncing wave packet, so that the total energy remains
equal to 1/24.

\section{Conclusion}

 We have shown that oscillons in   the (1+1)-dimensional
signum-Gordon model can periodically move to and fro in the space
(the x-line) with a constant speed $v$ from the interval $[0,1]$.
The amplitude of such swaying motion is equal to $v l/2$, where $l$
is the length of the oscillon. The pertinent analytic solutions of
the field equation have been constructed from the second order
polynomials in $t$ and $x$.

The present paper is a follow-up to \cite{4}, and the remarks and
comments given there apply also to the swaying oscillons. Our new
findings contribute to the already substantial evidence that the
models  of the signum-Gordon type have rather amazing properties. In
particular, it is quite surprising that one can find simple,
explicit solutions that describe very nontrivial objects like the
oscillons, or $Q$-balls \cite{9}, and this happens in spite of the
unpleasant $\mbox{sign}(\varphi)$ form of the nonlinear term in the
field equation.