The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 105
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 31182)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 105
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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\section{Introduction}
Virtual knot theory, discovered by Kauffman~\cite{Ka99}, is a nontrivial extension of
classical knot theory. Indeed,
Goussarov, Polyak, and Viro proved that any two
classical knots are equivalent as virtual knots if and only if
they are equivalent as classical knots~\cite[Theorem 1.B]{GPV00}.
Their result served to motivate many subsequent developments, because it predicted that many
classical knot and link invariants can be extended to virtual knots and links.
This result from~\cite{GPV00} was originally deduced from the classical Waldhausen's
theorem~\cite[Corollary 6.5]{Wa68},
but it can also be derived from Kuperberg's theorem~\cite{Ku03}.
In the latter formulation, one represents virtual knots
geometrically as knots in thickened surfaces up to stable equivalence, and
Kuperberg's theorem tells us that the minimal genus representative is unique up to diffeomorphism.
Concordance of virtual knots has recently become an area of active interest, and many basic questions are still open.
One important question, which was raised both by Turaev~\cite[Section 2.2]{Tu08} and
by Kauffman~\cite[p.~336]{Ka15}, is the following:
\emph{If two classical knots are concordant as virtual ones, are they concordant in the usual sense?}
Our main result gives an affirmative answer to this question.
\begin{thmintro}
If two classical knots are concordant as virtual knots, then they are also concordant
as classical knots.
\end{thmintro}
This result can be viewed as the analogue in concordance of the earlier
result of Goussarov, Polyak, and Viro~\cite{GPV00}, and consequently we hope
that it will stimulate further research on the problem of extending concordance
invariants from the classical to the virtual setting.
In fact, there are already exciting new developments along these lines, for
instance the extension of the Rasmussen $s$-invariant to virtual knots given by
Dye, Kaestner, and Kauffman~\cite{DKK14}.
We give a brief overview of the rest of the paper.
In Section~\ref{section2}, we introduce virtual knots as knots in thickened
surfaces up to stable equivalence. We recall Turaev's definition of virtual
knot concordance in Subsection~\ref{subsection2-2}, and we state and prove our
main result in Subsection~\ref{subsection2-3}. In Section \ref{section3}, we
introduce long virtual knots and construct the \emph{virtual knot concordance
group} $\mathscr{V}\!\mathscr{C}.$ We show that a long virtual knot $K$ is virtually slice if and
only if its closure $\overline{K}$ is, and we use it to deduce injectivity of
the natural homomorphism $\psi \colon \mathscr{C} \to \mathscr{V}\!\mathscr{C}$ from the classical
concordance group to the virtual concordance group.
\begin{conv}
All manifolds are assumed smooth and all knots are assumed oriented. Throughout the paper, we work with smooth concordance.
\end{conv}
\section{Virtual knots and concordance} \label{section2}
In this section, we introduce stable equivalence of knots in thickened surfaces
and use them to define virtual knots. This gives rise to a natural notion of
concordance for virtual knots, which allows for a bordism between the two
surfaces whose thickenings contain representatives of the two virtual knots,
and requires also an embedded annulus cobounding the two knots.
\subsection{Diagrams and stable equivalence}
It will be convenient for us to regard virtual knots geometrically as knots in
thickened surfaces, and we take a moment to explain this point of view.
\begin{definition}
A \emph{thickened surface}~$\Sigma \times I$ is a product of a closed, connected, oriented surface~$\Sigma$ with
the interval~$I=[-1,1]$.
A knot~$K$ in a thickened surface~$\Sigma \times I$ is a $1$-dimensional submanifold~$K$
in the interior of $\Sigma \times I$ which is diffeomorphic to a circle.
\end{definition}
Just as classical knots in $S^3$ are considered up to ambient isotopy, we consider knots in thickened surfaces up
to stable equivalence~\cite{CKS02}. We take a moment to recall this carefully.
\begin{definition}\label{def:StableEquiv}
\emph{Stable equivalence} on knots in thickened surfaces is generated by the
following operations, which transform a given knot $K$ in a thickened surface
$\Sigma \times I$ into a new knot $K'$ in a possibly different thickened surface
$\Sigma' \times I$.
\begin{enumerate}
\item\label{DiffEquiv}
Let $f\colon \Sigma \times I \to \Sigma' \times I$ be an orientation-preserving diffeomorphism
sending the orientation class of $\Sigma$
to that of $\Sigma'$. (Notice that this implies that $f(\Sigma \times \{1\}) = \Sigma' \times \{1\}$ and $f(\Sigma \times \{-1\}) = \Sigma' \times \{-1\}$.) The knot $K'=f(K)$ in~$\Sigma' \times I$ is said to be obtained from $K$ in $\Sigma \times I$ by a \emph{diffeomorphism}.
\item Let $h \colon S^0 \times D^2 \to \Sigma$ be the attaching region for a
$1$-handle that is disjoint from the image of $K$ under projection $\Sigma \times
I \to \Sigma$, then $0$-surgery on $\Sigma$ along $h$ is the surface
\[ \Sigma' := \Sigma \smallsetminus h(S^0 \times D^2) \cup_{S^0 \times S^1} D^1 \times S^1. \]
The knot~$K'$ is the image of the knot $K$ in $\Sigma' \times I$, and we say that
it is the knot obtained from $K$ by \emph{stabilisation}.
\item \emph{Destabilisation} is the inverse operation, and it involves cutting
$\Sigma \times I$ along a \emph{vertical annulus}~$A$ and attaching two copies of $D^2 \times I$ along the two annuli.
If the resulting thickened surface is disconnected, then we keep only the component containing $K$.
\end{enumerate}
Note that in (3), an annulus $A$ in $\Sigma \times I$ is called \emph{vertical} if there is an embedded circle~$\gamma \subset \Sigma$ such that $A=\gamma \times I \subset \Sigma \times I$.
An equivalence class under the equivalence relation generated by (1), (2), and (3) above is called a \emph{virtual knot}.
\end{definition}
Virtual links admit a similar description as links in $\Sigma \times I$, though $\Sigma$ need not be connected. We abuse notation slightly and use $K$ for the virtual knot, so $K$ refers to an equivalence class of knots in thickened surfaces.
Given a virtual knot $K$, then any knot in its equivalence class will be
called a \emph{representative} for $K$. A representative is therefore a knot in
a thickened surface $\Sigma \times I$.
\begin{definition}
The \emph{virtual genus} of a virtual knot~$K$ is the minimum
\[ vg(K) := \min \{g(\Sigma)\mid \text{$\Sigma \times I$ contains a representative for $K$}\}, \]
where $g(\Sigma)$ denotes the genus of the surface $\Sigma$.
\end{definition}
A classical knot~$K \subset S^3$ can be isotoped to be disjoint from the two points~$\{0, \infty\}$.
Thus, we can view it as a knot in the thickened surface~$S^2 \times I$. The associated virtual knot
is independent of the choice of isotopy, and we call such a knot \emph{classical}.
Therefore a virtual knot is classical if and only if its virtual genus is zero.
Kuperberg~\cite[Theorem 1]{Ku03} proved a strong uniqueness result for
minimal genus representatives. Namely, he showed that if $(K,\Sigma \times I)$ and
$(K',\Sigma' \times I)$ are two minimal genus representatives for the same virtual knot,
then $K'=f(K)$ for some diffeomorphism $f\colon \Sigma \times I \to \Sigma' \times I$ as in (\ref{DiffEquiv}) of Definition~\ref{def:StableEquiv} above.
For the sake of completeness, we relate the geometric definition of virtual knots to the usual
diagrammatic definition.
A \emph{virtual knot diagram} is a regular immersion of the circle $S^1$ in the plane
${\mathbb R}^2$ with double points that are either classical or virtual. Real
crossings are drawn with one arc over the other whereas virtual crossings are
drawn with circles around them.
Two virtual knot diagrams are equivalent if they are related by planar
isotopies and \emph{generalised Reidemeister moves}. These consist of the three
usual Reidemeister moves together with three additional moves just like the
usual Reidemeister moves but with only virtual crossings, and one more move
called the mixed move which is depicted in Figure~\ref{MM}. A virtual knot is
defined to be an equivalence class of virtual knot diagrams, and virtual links
are defined similarly as equivalence classes of virtual link diagrams.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.65]{mixed-crossing.pdf}
\caption{The mixed move.}
\label{MM}
\end{figure}
Given a virtual knot diagram~$D$ of a virtual knot~$K$, there is a canonical surface~$\Sigma$ called the
Carter surface constructed from $D$ as follows~\cite{KK00}: attach two
intersecting bands at every classical crossing and two non-intersecting bands
at every virtual crossing as in Figure~\ref{band-surface}. Attaching
non-intersecting and non-twisted bands along the remaining arcs of $D$, and
filling in all boundary components with 2-disks, we obtain a closed oriented
surface~$\Sigma$ whose thickening~$\Sigma \times I$ contains a representative of $K$.
Conversely, let $K$ be a knot in a thickened surface $\Sigma \times I$ and $U \subset \Sigma$
a neighbourhood of the image of $K$ under projection $\Sigma \times I \to \Sigma$.
If $f\colon U \to {\mathbb R}^2$ is an orientation preserving immersion,
then the image of $K$ under $f$ is a virtual knot diagram $D$ whose classical
crossings correspond to those of $K$ and whose virtual crossings, which are the
rest of them, are the result of the immersion $f$. This virtual knot diagram
$D$ depends on the choice of immersion $f$, but any two such diagrams are
equivalent via detour moves~\cite{Ka15}.
This establishes a one-to-one correspondence between virtual knot
diagrams modulo the generalised Reidemeister moves and knots in thickened
surfaces up to stable equivalence~\cite{CKS02}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.65]{band-crossing.pdf}
\caption{The bands for classical and virtual crossings.}
\label{band-surface}
\end{figure}
\subsection{Virtual concordance} \label{subsection2-2}
In this section, we define concordance and sliceness for virtual knots in terms
of their representative knots in thickened surfaces.
We follow Turaev~\cite[Section 2.1]{Tu08} in defining virtual knot concordance.
If $K$ is an oriented knot in a thickened surface $\Sigma \times I$, its reverse is the
knot $K^r$ obtained by changing the orientation of $K$, and its mirror image is the knot $K^m$ obtained by changing the orientation of $\Sigma \times I$.
These operations commute with one another, and we use
$-K=K^{rm}$ to denote the knot obtained by taking the
mirror image of the reverse knot.
\begin{definition}\label{defn:VirtualSlice}
\begin{enumerate}
\item Two given knots $K_0 \subset \Sigma_0 \times I$ and $K_1 \subset \Sigma_1 \times I$
in thickened surfaces are \emph{virtually concordant} if there exists a connected
oriented $3$-manifold~$W$ with
$\partial W \cong -\Sigma_0 \sqcup \Sigma_1$ and an annulus~$A \subset W \times I$
cobounding $-K_0$ and $K_1$.
\item A knot $K \subset \Sigma \times I$ is called \emph{virtually slice} if it is
concordant to the unknot. Equivalently, the knot $K$ is virtually slice if
there exists a connected $3$-manifold~$W$ with
$\partial W \cong \Sigma$ and a 2-disk~$\Delta \subset W \times I$ cobounding $K$. We call $\Delta$
a \emph{slice disk} for $K$.
\end{enumerate}
\end{definition}
Clearly, virtual concordance is an equivalence relation on knots in thickened
surfaces. For instance, transitivity follows by stacking the two concordances
in the usual way. The next result shows that two stably equivalent knots are
virtually concordant to one another. Thus, it follows that virtual
concordance defines an equivalence relation on virtual knots.
\begin{lemma}\label{lem:ConcordanceWellDefined}
Suppose $K_0 \subset \Sigma_0\times I$ and $K_1 \subset \Sigma_1\times I$ represent the same virtual knot.
Then $K_0$ and $K_1$ are virtually concordant.
\end{lemma}
\begin{proof}
It is enough to find a 3-manifold $W$ and an annulus $A \subset W \times I$ realising a concordance between
knots in surfaces transformed into each other by one of the operations generating
stable equivalence, see Definition~\ref{def:StableEquiv}.
One can verify that this is possible in each case.
\end{proof}
Kauffman~\cite{Ka15} re-expressed concordance purely in terms of virtual knot diagrams. A
\emph{concordance} between two virtual knot diagrams $K_0$ and $K_1$ consists of
a series of generalised Reidemeister moves together with a collection of saddle
moves, births, and deaths that transform $K_0$ into $K_1$.
As usual, one requires
that the total number of births and deaths equals the number of saddle moves, see \cite[Section 3]{Ka15}.
This condition is equivalent to the requirement that the knots cobound an annulus.
\begin{example}
To illustrate this, we recall from \cite{Ka15} the argument that the Kishino knot $K$ is virtually slice. To see this, perform a saddle move along the dotted line on the left of Figure~\ref{Kishino-slice}. The result is a virtual link diagram on the right,
which is easily seen to be equivalent to the unlink with two components.
Filling them in with 2-disks gives a slice disk for $K$, showing that the Kishino knot is virtually slice.
\begin{figure}[ht]
\centering
\includegraphics[scale=1.60]{Kishino.pdf} \qquad \qquad \qquad \includegraphics[scale=1.60]{Kishino2.pdf}
\caption{Slicing the Kishino knot.}
\label{Kishino-slice}
\end{figure}
\end{example}
Although it is not immediately obvious, Kauffman's diagramatic notion of
virtual concordance is equivalent to Definition~\ref{defn:VirtualSlice}.
Indeed, the equivalence of the two definitions of virtual knot concordance had
been established previously by Carter, Kamada, and Saito \cite[Lemma 12]{CKS02}.
We also refer to~\cite[Section 1.2]{CK15} for further discussion
on this point, and we thank Micah Chrisman for sharing this observation.
\subsection{Virtual concordance of classical knots} \label{subsection2-3}
Suppose $K \subset S^3$ is a classical knot and suppose that $K$ is slice. As
explained earlier, we can view $K$ as a virtual knot by arranging that $K$ lies
in a neighbourhood of the standard sphere $S^2 \times I \subset S^3$. If $\Delta \subset B^4$ is a slice disk for $K$,
then we can further arrange that $\Delta$ lies in $(S^2 \times I) \times I \subset
B^4$. Thus, $K$ is seen to be virtually slice in the sense of
Definition~\ref{defn:VirtualSlice}.
The next theorem is our main result, and it gives a characterisation of virtual sliceness for classical knots.
\begin{theorem}\label{thm:ClassicVSlice}
A classical knot is virtually slice if and only if it is slice.
\end{theorem}
An immediate consequence of this theorem is that there are infinitely many
distinct virtual concordance classes of virtual knots. This fact had been
noted by Turaev using his polynomial invariants $u_\pm(K)$
\cite[Theorems 1.6.1 and 2.3.1]{Tu08}, but Theorem~\ref{thm:ClassicVSlice} gives infinitely
many distinct virtual concordance classes for which $u_\pm(K)$ all vanish.
\begin{corollary} \label{cor:ClassicVSlice}
Two classical knots are virtually concordant if and only if they are concordant as classical knots.
\end{corollary}
\begin{proof}
Given two classical knots $K_0$ and $K_1$, apply the Theorem~\ref{thm:ClassicVSlice}
to the connected sum~$K_0 \# -K_1$, where
$-K_1$ denotes the mirror image of $K_1$ with its orientation reversed.
\end{proof}
Suppose the classical knot~$K \subset S^2 \times I$ is virtually slice, then we can
find a $3$-manifold~$W$ which is a filling of $S^2$ and a
slice disk~$\Delta \subset W \times I$ cobounding the knot~$K$. To transfer the slice disk from
$W\times I$ into $D^4$, we construct an embedding of the universal cover~$\widetilde
W$ into $D^3$. The universal cover~$\widetilde W$ will have boundary~$\partial \widetilde W$ consisting of
many copies of $S^2$. A \emph{compression} of $\widetilde W$ is
a smooth embedding~$\phi \colon \widetilde W \to D^3$
which restricts to an orientation-preserving diffeomorphism $S \to S^2$ on one of
the boundary components~$S$ of $\partial W$.
We will construct compressions of $W$ from compressions of the prime parts of $W$.
\begin{lemma}\label{lem:PrimeNeat}
Let $W$ be a connected, compact, oriented and prime $3$-manifold with boundary~$\partial W \cong S^2$. Then
its universal cover~$\widetilde W$ admits a compression.
\end{lemma}
\begin{proof}
We can fill the boundary component of $W$ with a $3$-ball~$B$ and
obtain a closed $3$-manifold~$W \cup B$.
The universal cover of $W \cup B$ is diffeomorphic to one of the
manifolds $S^3$, $S^2 \times {\mathbb R}$ or ${\mathbb R}^3$. If $W \cup B$ is a geometric piece
in the sense of Thurston, this
can be deduced from geometrisation and checking each geometry~\cite[Section 5]{Sc83}.
If $W$ contains an incompressible torus, its universal cover is
diffeomorphic to the space~${\mathbb R}^3$~\cite[Theorem 1]{HRS89}.
As $S^2 \times {\mathbb R}$ and ${\mathbb R}^3$ embed into $S^3$, we may assume that we have an embedding of
$\widetilde {W \cup B}$ into $S^3$. By post-composing with a diffeomorphism,
we may assume that a lift $B'$ of $B$ is mapped to the standard $3$-ball $D^3 \subset S^3$.
Denote the boundary of $B'$ by $S$. We have the following chain of embeddings
\[ \widetilde W \subset \widetilde {W \cup B} \smallsetminus B' \subset S^3 \smallsetminus B' = D^3, \]
which gives a compression of $\widetilde W \subset D^3$.
\end{proof}
\begin{lemma}\label{lem:NeatExists}
Let $W$ be a connected, oriented, compact $3$-manifold with boundary~$\partial W \cong S^2$. Then
its universal cover $\widetilde W$ admits a compression.
\end{lemma}
\begin{proof}
We fix a prime decomposition~$S:=\{S_i\}$ of the $3$-manifold~$W$. This
is a finite collection~$S$ of disjointly embedded separating $2$-spheres~$S_i$ such that $2$-surgery on these
spheres gives a $3$-manifold whose components $W_1, \ldots, W_k$ are all prime $3$-manifolds.
After relabeling, we may assume that $W_1$ has boundary~$\partial W_1 \cong S^2$.
Take $\pi\colon \widetilde W \to W$ to be a universal cover.
The components of the preimages~$\pi^{-1}(S_i)$ are again $2$-spheres, which form the
collection $\widetilde S$. The spheres~$C \in \widetilde S$ are again separating: each sphere~$C$
cuts~$\widetilde W$ into two half-spaces.
Given an orientation~$\sigma$ on the sphere~$C$, there is a unique half-space
$C^\sigma$ whose boundary orientation on $C$ is $\sigma$. To any subset
\[ I \subseteq \left\{ (C, \sigma) \mid C \in \widetilde S \text{ and } \sigma \text{ an orientation on } C \right\} \]
we can associate the submanifold
${\bigcap_{(C, \sigma) \in I}} C^\sigma$, which is
an intersection of half-spaces of $\widetilde W$.
We call a submanifold $B \subseteq \widetilde W$ \emph{chunked} if $B ={\bigcap_{(C, \sigma) \in I}} C^\sigma$ for a subset $I$.
If $B$ is chunked, then its boundary components are contained in $\widetilde S$ or in the boundary $\partial \widetilde W$ of $\widetilde W$ itself.
Note that if $I$ is empty, then $\bigcap_{I} C^\sigma=\widetilde W$, thus $\widetilde W$ is chunked.
Given a chunked submanifold~$B$ and a boundary sphere~$C \in \widetilde S$ of $B$, there
is a unique smallest chunked submanifold $B' \supset B$ such that $C$ is in the interior of $B'$.
It is of the form $B' = B \cup_C P$ for a universal cover $P$ of a prime $3$-manifold.
We call $B'$ an \emph{elementary extension} of $B$ along $C$.
Fix a boundary component~$T \subset \partial \widetilde W$. Consider the following set
\[ Z:= \left\{ \phi \colon B \to D^3 \mid T \subset B, \phi(T) = S^2, B \text{ is chunked, and $\phi$ is a compression} \right\}.\]
We give $Z$ the partial order of the poset of maps, i.e. for $\phi \colon B \to D^3$
and $\phi' \colon B' \to D^3$, we declare $\phi' \geq \phi$ if and only if
$B \subset B'$ and $\phi'$ restricts to $\phi$.
By Lemma \ref{lem:PrimeNeat}, the set $Z$ is non-empty. Also totally ordered chains have
a maximal element, so $Z$ has a maximal element. Let $\phi \colon B \to D^3$ be maximal.
We claim $B = \widetilde W$, which proves the lemma.
Pick a boundary sphere~$C \in \widetilde S$ of $B$ and denote by $B' = B \cup_C P$ the elementary extension of $B$ along $C$.
We construct a compression of $B'$ restricting to $\phi$. Consider $\phi(C) \subset D^3$. It is
a smoothly embedded $2$-sphere in $D^3$.
It separates the $3$-ball~$D^3$ into two components: an annulus and another $3$-ball~$D'$. Consequently,
the interior of the $3$-ball~$D'$ is disjoint from the image of $\phi(B)$. By
Lemma~\ref{lem:PrimeNeat}, we can embed $\phi_P \colon P \to D'$. As $\operatorname{Diff}^+(S^2)$
is path-connected, we can make $\phi_P$ agree with $\phi$ on $C$ and thus we obtain a compression
\[ \phi \cup_C \phi_P \colon B' \to D^3 \]
extending $\phi$.
\end{proof}
Using the compression of Lemma~\ref{lem:NeatExists},
we show how to transfer a slice disk for a virtually slice classical knot to the 4-ball.
\begin{proof}[Proof of Theorem~\ref{thm:ClassicVSlice}]
Let $K$ be a classical knot which is virtually slice.
By definition, the knot~$K$ is embedded in a thickened $2$-sphere and
there is a filling~$W$ of $S^2$ together with a slice disk~$\Delta \subset W \times I$
cobounding the knot $K$ in the boundary~$\partial W \times I$.
Let $\widetilde W \to W$ be a universal cover
and $\phi \colon \widetilde W \to D^3$ be a compression which exists by Lemma~\ref{lem:NeatExists}.
Let $T \subset \partial \widetilde W$ be a boundary sphere which is mapped via $\phi$ to the boundary of $D^3$.
The product map $\widetilde W \times I \to D^3 \times I$ is also covering map.
As the slice disk $\Delta$ is contractible, it lifts to a disk $\widetilde \Delta \subset \widetilde W \times I$
with boundary $\partial \widetilde \Delta \subset T \times I$. Note that $\partial \widetilde \Delta$ is
still the knot~$K$.
Now $\phi( \Delta ) \subset D^3 \times I \cong D^4$ is a slice disk for $K$.
\end{proof}
\section{The virtual knot concordance group}\label{section3}
In this section, we introduce concordance of long virtual knots and the virtual
knot concordance group~$\mathscr{V}\!\mathscr{C}$.
We then apply Theorem~\ref{thm:ClassicVSlice} to deduce injectivity of the natural
homomorphism~$\mathscr{C} \to \mathscr{V}\!\mathscr{C}$, where $\mathscr{C}$ is the classical concordance group.
\subsection{Long virtual knots}
The group operation in $\mathscr{C}$ and $\mathscr{V}\!\mathscr{C}$ is given by connected sum. For round virtual knots, this operation is not well-defined because it depends on the choice of diagram and on where the diagrams are connected.
These ambiguities disappear if one instead works with long virtual knots.
Recall that a long virtual knot diagram is a regular immersion of ${\mathbb R}$ in the plane
${\mathbb R}^2$ which is identical with the $x$-axis outside a compact set, which we will principally take to be the closed ball $B_0(R)$ of radius $R$ centered at the origin.
Double points of the immersion can occur only inside $B_0(R),$ and each one is
labelled either classical or virtual, indicated as before with an over- or
undercrossing if classical or by encircling the crossing if virtual.
Two such diagrams are equivalent if one can be related to the other by
a compactly supported planar isotopy and a finite sequence of generalised
Reidemeister moves. A \emph{long virtual knot} $K$ is defined to be an equivalence class of long
virtual knot diagrams.
We call the long knot given by the $x$-axis the \emph{long unknot}.
Note that by convention, all long virtual knots are oriented from left to right.
The connected sum of two long virtual knots $K_0$ and $K_1$, denoted $K_0 \# K_1$, is defined by concatenation with $K_0$ on the left and $K_1$ on the right.
It is easy to verify that long virtual knots form a monoid under connected sum with identity given by the long unknot.
\begin{remark}
The connected sum on long virtual knots is not a commutative operation~\cite[Theorem 9]{Ma08}.
\end{remark}
\subsection{The virtual knot concordance group}
We now extend the notion of virtual concordance to long virtual knots, following Kauffman \cite{Ka15}.
\begin{definition}
\begin{enumerate}
\item Two long knots~$K_0$ and $K_1$ are \emph{virtually concordant} if one can
be obtained from the other by generalised Reidemeister moves and a finite
sequence of saddle moves, births, and deaths such that the number of saddle
moves equals the sum of births and deaths.
\item A long virtual knot is \emph{virtually slice} if it is virtually concordant to the long unknot.
\end{enumerate}
\end{definition}
We will use $[K]$ to denote the concordance class of a long virtual knot $K$ and
\[ \mathscr{V}\!\mathscr{C} = \{[K] \mid \text{$K$ is a long virtual knot} \}\]
for the set of concordance classes of long virtual knots. It is immediate from the definition
that the concordance class of the connected sum $K_0 \#K_1$ depends only on the
concordance classes of $K_0$ and $K_1$. This shows that the operation of
connected sum descends to a well-defined operation on $\mathscr{V}\!\mathscr{C}$.
Thus $(\mathscr{V}\!\mathscr{C},\#)$ is a monoid.
Turaev observes that $(\mathscr{V}\!\mathscr{C},\#)$ is actually a group~\cite[Section 5.2]{Tu08}.
Just as with classical knots, the inverse of $[K]$ is obtained by
taking the mirror image and reversing the orientation. Specifically, given a
long virtual knot $K$, let $K^m$ be the long virtual knot obtained by
reflecting $K$ through the vertical line $x=R$, and let $-K$ be the
result of reversing the orientation of $K^m$.
Chrisman~\cite[Theorem 1]{Ch16} proves that $K \# -K$ is virtually
slice, and thus it follows that $[-K]$ is the inverse of $[K]$ in
$(\mathscr{V}\!\mathscr{C},\#)$.
Given a long virtual knot $K$, let $\overline{K}$ denote its closure. Thus, $\overline{K}$ is the round virtual knot obtained by discarding the parts of $K$ outside the closed ball $B_0(R)$ and joining the points $(R,0)$ to $(-R,0)$ on $K$ with the semicircle $(R \cos(\th), -R \sin(\th)) \subset {\mathbb R}^2$ for $0 \leq \th \leq \pi$.
\begin{lemma} \label{lem:virtual-slice-equivalence}
A long virtual knot $K$ is virtually slice if and only if its closure $\overline{K}$ is virtually slice.
\end{lemma}
\begin{proof}
Suppose $K$ is virtually slice. Then there is a finite sequence of births, deaths, and saddles and planar isotopies taking $K$ to the trivial long knot. We can choose $R$ sufficiently large so that all births, deaths, and saddles take place in the ball $B_0(R)$. Since planar isotopies are compactly supported, we can assume that $K$ is unchanged outside of $B_0(R)$.
Thus, the same set of births, deaths, and saddle moves and planar isotopies show that $\overline{K}$ is virtually concordant to the round unknot.
To see the other direction, represent $K$ as a long virtual knot diagram which
coincides with the $x$-axis outside the open ball $B_0(R)$. Construct a new
diagram for $K$ by translating the original diagram vertically and connecting the points $(-R,2R)$ and $(R,2R)$
on the new diagram to the $x$-axis using vertical lines. Now perform a saddle move
and replace the vertical line segments from
$(-R,0)$ to $(-R,R)$ and from $(R,R)$ to $(R,0)$ with the horizontal line
segments from $(-R,0)$ to $(R,0)$ and from $(R,R)$ to $(-R,R)$. This saddle
move transforms $K$ into a 2-component link with one component the trivial
long knot and the other component the round virtual knot $\overline{K}$, which by hypothesis
bounds a slice disk $\Delta$. Capping $\overline{K}$ off with $\Delta$ gives a
virtual concordance from $K$ to the trivial long knot. It follows that $K$ is
virtually slice.
\end{proof}
Recall that for classical knots, the map $K \mapsto \overline{K}$ gives a one-to-one correspondence between long knots and round knots.
From the definition of virtual concordance, one deduces that
the natural inclusion map from classical knots to virtual knots induces
a well-defined homomorphism~$\psi \colon \mathscr{C} \to \mathscr{V}\!\mathscr{C}$.
The next result is then an immediate consequence of Theorem~\ref{thm:ClassicVSlice}
and Lemma~\ref{lem:virtual-slice-equivalence}.
\begin{corollary}
The homomorphism $\psi\colon \mathscr{C} \to \mathscr{V}\!\mathscr{C}$ is injective.
\end{corollary}
It is an open question whether the concordance group $\mathscr{V}\!\mathscr{C}$ of long virtual
knots is abelian, see \cite[Section 6.5]{Tu08}. Another interesting open
problem is to determine the structure of $\mathscr{V}\!\mathscr{C}$, for instance can one
describe the cokernel of the map $\psi$? Does it contain torsion elements?
Turaev introduces many useful invariants of virtual knot concordance
in~\cite{Tu08}. These include the polynomials $u_\pm(K)$ and the graded genus
$\sigma(T)$ of the graded matrix $T=(G,s,b)$ associated to a virtual knot $K$. Any
virtual knot $K$ with $u_+(K) \neq 0$ or $u_-(K)\neq 0$ will have infinite
order in $\mathscr{V}\!\mathscr{C}$ \cite[Proposition 2]{Ch16}. However, if
$K$ is classical, then these invariants vanish, and we view it as an
interesting challenge to derive new invariants of virtual knot concordance to shed light on these questions.
\begin{ackn}
We thank Stefan Friedl for bringing us together and for many fruitful discussions.
The authors would also like to thank Micah Chrisman, Isabel Gaudreau, and Mark Powell for their input and feedback.
H. Boden is grateful to the University of Regensburg for its hospitality. He was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
M. Nagel thanks McMaster University for its hospitality.
He was supported by a CIRGET postdoctoral fellowship
and by SFB 1085 at the University of Regensburg funded by the DFG.
\end{ackn}
\bibliographystyle{alpha}
|
{
"timestamp": "2016-06-22T02:05:43",
"yymm": "1606",
"arxiv_id": "1606.06404",
"language": "en",
"url": "https://arxiv.org/abs/1606.06404"
}
|
\section{Introduction}
Max Born and Pascual Jordan proposed a f\/irst model of quantization in~\cite{BJ}, restricted to one-dimensional systems, a model generalized to $n$-dimensional systems, together with Werner Heisenberg, in~\cite{BHJ}. Hermann Weyl introduced in~\cite{W,W1} a general quantization scheme based on the Fourier transform formula. The Born--Jordan and Weyl quantizations produce in general dif\/ferent operators from the same classical function of momenta and coordinates. The two quantization methods, however, coincide on natural Hamiltonians in $n$-dimensional Euclidean spaces. This property and many others of the two quantizations are studied in the recent book~\cite{dG}, that is our main source for the following discussion.
We consider here the Born--Jordan and Weyl quantizations applied to the algebra of constants of motion of the 2D anisotropic harmonic oscillator. This system admits the maximal number of three functionally independent constants of motion (and it is said to be superintegrable) whenever the ratio of the parameters is a rational number. Otherwise, the independent constants of motion are just two.
Our aim here is to check the behaviour of the Born--Jordan and Weyl quantizations of the anisotropic harmonic oscillator with respect to the integrability and superintegrability of the resulting quantum system for some particular choice of the ratio of the parameters. This aspect of the two quantization procedures is not considered in~\cite{dG}.
We use here the same expression for the classical 2D anisotropic harmonic oscillator and its independent constants of motion that is employed in \cite{CDRraz}. The nD anisotropic harmonic oscillator is there considered as an ``extended system'', a particular structure of some natural Hamiltonians that allows the existence of polynomial constants of motion of higher degree, described in~\cite{CDRraz, CDRTTW}. The interest in using such a construction here comes from other our studies in progress on the quantization of extended systems (see for example~\cite{CDRgen}).
We consider brief\/ly also the factorization in annihilation-creation operators for the 2D aniso\-tro\-pic harmonic oscillator as given in~\cite{JH}, to check how the corresponding classical integrals are quantized by applying again Born--Jordan and Weyl procedures. The examples are confronted with those arising from the extension procedure.
We can apply here the simpler formulas for Born--Jordan and Weyl quantizations for monomials in coordinates and momenta discussed in~\cite{dG}.
The quantization procedures of classical quantities are not exhausted by Born--Jordan's and Weyl's approaches. Several techniques have been developed since the beginning of the quantum era, many of them specif\/ic for the particular system considered.
Indeed, there is no unique way to assign quantum operators to classical quantities in a~mea\-ning\-ful way. Hermiticity of the operator is, usually, a~necessary requirement and it is obtained by some symmetrization procedure, however not uniquely determined, see for example~\cite{PW}.
The problem of preserving the algebra of the constants of motion of a Hamiltonian system after quantization is object of many recent studies, see for example~\cite{Herranz, KN, DV,MPW} and references therein, for solutions in f\/lat and non-f\/lat manifolds.
For superintegrability and quantization in classical and quantum systems see also~\cite{MPW}, in particular for the def\/inition of quantum superintegrable systems, and~\cite{KN}.
\section{Born--Jordan and Weyl quantizations of monomials}
In \cite{BJ,BHJ, dG} the Born--Jordan quantization of monomials in coordinates $(x_i,p_i)$ is determined by the following general rules,
\begin{gather*}
[\hat x_i,\hat p_j]=i\hbar \delta_{ij},\qquad [\hat x_i, \hat x_j]=0,\qquad [\hat p_i, \hat p_j]=0,
\end{gather*}
where $\delta_{ij}=1$ for $i=j$ and zero otherwise, for any quantization of the coordinates $x_i\rightarrow \hat x_i$ and of the momenta $p_i\rightarrow \hat p_i$, and by
\begin{gather}\label{BJ}
x_i^rp_i^s\rightarrow \frac 1{s+1}\sum_{k=0}^s \hat p_i^{s-k}\hat x_i^r\hat p_i^k,
\end{gather}
for the monomials with same indices. When the indices are dif\/ferent, the operators commute by the general quantization rules of above and their quantization is therefore straightforward.
For the Weyl quantization \cite{dG, W}, the general rules are the same of above and, for the monomials, we have instead
\begin{gather}\label{W}
x_i^rp_i^s\rightarrow \frac 1{2^s}\sum_{k=0}^s \binom{s}{k} \hat p_i^{s-k}\hat x_i^r\hat p_i^k.
\end{gather}
The standard realization of the operators $\hat x_i$ and $\hat p_i$, at least for Cartesian coordinates, are
\begin{gather*}
\hat x_i \phi= x_i \phi, \qquad \hat p_i \phi =-i\hbar \frac {\partial}{\partial x_i} \phi,
\end{gather*}
for any function $\phi(x_j)$. We employ in the following this standard quantization of the canonical coordinates.
It is easy to check that for $r=s=1$ the Born--Jordan and Weyl quantizations of monomials coincide, but dif\/fer for $r,s \geq 2$.
Many properties of the Born--Jordan quantization, generalized to any function of coordinates and momenta, and of the Weyl quantization are considered into details in~\cite{dG} and the dif\/ferent characteristics are discussed.
We focus here our analysis on the ef\/fect of the two quantizations on the f\/irst integrals of a~particular superintegrable system. We check in some examples if the quantized f\/irst integrals commute with the Hamiltonian operator, that is, if the algebraic structure of the constants of motion is preserved by the dif\/ferent formulas of quantization, an issue not considered in~\cite{dG}.
\section{The superintegrable 2D anisotropic harmonic oscillator}
In order to express the f\/irst integrals of the system, we can write the Hamiltonian of the superintegrable 2D anisotropic harmonic oscillator in the form of an extended Hamiltonian \cite{CDRraz} as follows
\begin{gather}\label{Hu}
H_{m,n}=\frac 12\left( p_u^2+\left(\frac mn\right)^2p_x^2\right)+\omega^2\left(\frac mn \right)^2\left(x^2+u^2\right),
\end{gather}
where $(x,u)$ are coordinates in the Euclidean plane, $\omega\in \mathbb R$ and $m,n \in \mathbb N\setminus \{0\}$.
Two independent f\/irst integrals of $H_{m,n}$ are $H_{m,n}$ itself and
\begin{gather*}
L=\frac 12 p_x^2+\omega^2x^2,
\end{gather*}
that is associated with the separability of the Hamilton--Jacobi equation of $H_{m,n}$ in coordina\-tes~$(u,x)$.
If we put
\begin{gather}\label{G}
G_n=\sum_{k=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2k+1}\big({-}2\omega^2\big)^k x^{2k+1}p_x^{n-2k-1},
\end{gather}
where $[a]$ denotes the integer part of $a$, and $X_L$ is the Hamiltonian vector f\/ield of the function~$L$, then, adapting to our case the more general theorem proved in~\cite{CDRraz}, we have
\begin{prop}\label{comp}
For any couple of positive integers $(m,n)$, the function $K_{m,n}$ is a first integral of $H_{m,n}$, where
\begin{gather*
K_{m,n}=P_{m,n}G_n+D_{m,n}X_{L}(G_n),
\end{gather*}
with
\begin{gather*}
P_{m,n}=\sum_{k=0}^{[m/2]}\binom{m}{2k} \left(-\frac mn u \right)^{2k}p_u^{m-2k}\big({-}2\omega^2\big)^k,\\
D_{m,n}=\frac 1{n}\sum_{k=0}^{[(m-1)/2]}\binom{m}{2k+1}\, \left(-\frac mn u \right)^{2k+1}p_u^{m-2k-1}\big({-}2\omega^2\big)^k, \qquad m>1,
\end{gather*}
and $D_{1,n}=-\frac m{n^2}u$.
\end{prop}
We can introduce the usual Cartesian coordinates $(x,y)$ by leaving $x$, $p_x$ unchanged and putting
\begin{gather}\label{uy}
u=\frac nm y, \qquad p_u=\frac mn p_y,
\end{gather}
so that
\begin{gather*}
H_{m,n}=\left(\frac mn \right)^2\left(\frac 12\left( p_x^2+p_y^2 \right)+\omega^2\left(x^2+\left(\frac nm \right)^2y^2 \right) \right).
\end{gather*}
In the following, we consider the $H_{m,n}$ of above by dropping the negligible overall factor $\left(\frac mn \right)^2$.
The classical and quantum superintegrability of the $n$D anisotropic harmonic oscillator has been studied in~\cite{JH}. The quantization of the classical system is there obtained by intro\-du\-cing creation and annihilation operators. This technique is widely in use today (see for example~\cite{KN} and~\cite{Ba2}, where the anisotropic harmonic oscillator is generalized to 2D constant-curvature manifolds obtaining new classical and quantum superintegrable systems) and can have some application towards the quantization of extended systems.
The Jauch--Hill Hamiltonian of the anisotropic harmonic oscillator is
\begin{gather*}
H_{JH}=\frac 12\left(\frac{p_1^2}{M_1}+\frac{p_2^2}{M_2}+M_1\omega_1^2q_1^2+M_2\omega_2^2q_2^2 \right),
\end{gather*}
and coincide with $H_{m,n}$ if we put
\begin{gather*}
x=\sqrt{M_1}q_1, \qquad p_1=\sqrt{M_1}p_x,\qquad y=\sqrt{M_2}q_2, \qquad p_2=\sqrt{M_2}p_y,
\end{gather*}
with
\begin{gather*}
m\omega_2=n\omega_1, \qquad \omega_1=\sqrt{2}\omega.
\end{gather*}
The classical f\/irst integrals given in \cite{JH} become
\begin{gather}\label{F}
F_1(m,n)=\frac 12 \big(b_1^nb_2^{*m}+b_1^{*n}b_2^m \big),\qquad F_2(m,n)=-\frac i2 \big(b_1^nb_2^{*m}-b_1^{*n}b_2^m \big),
\end{gather}
where
\begin{gather*}
b_{1}=\frac 1{\sqrt{2\omega_1}}\left( p_x-i\omega_1 x \right),\qquad b_{1}^*=\frac 1{\sqrt{2\omega_1}}\left( p_x+i\omega_1 x \right),
\end{gather*}
and similarly $b_2$, $b_2^*$ in function of $y$, $p_y$, $\omega_2$. The corresponding quantum operators are obtained simply by substituting $p_x$, $p_y$ with $\hat p_x$, $\hat p_y$ in the expressions of above.
The f\/irst integrals obtained with the two methods of above are dif\/ferent, having for example, up to constant factors,
\begin{gather*}
K_{1,1}=xp_y-yp_x, \qquad F_1(1,1)=p_xp_y+\omega^2xy.
\end{gather*}
\section[The two quantizations of the constants of motion of the oscillator]{The two quantizations of the constants of motion\\ of the oscillator}
The Born--Jordan and Weyl quantizations of both $H_{m,n}$ and $L$ clearly coincide, being in both cases
\begin{gather*}
H_{m,n} \rightarrow \hat H_{m,n}= -\frac {\hbar^2}2\left(\frac {\partial^2}{\partial x^2}+\frac {\partial^2}{\partial y^2} \right)+\omega^2\left(x^2+\left(\frac nm \right)^2y^2 \right),\\\
L \rightarrow \hat L=-\frac {\hbar^2}2 \frac {\partial^2}{\partial x^2}+\omega^2 x^2.
\end{gather*}
The operators are clearly independent and commuting, so that the integrability of the system is preserved by both the quantizations.
The Born--Jordan and Weyl quantizations of $K_{1,1}$, $K_{2,1}$ and $K_{3,1}$ coincide and commute with the corresponding Hamiltonian operators $\hat H_{m,n}$. Things become dif\/ferent for $(m,n)=(4,1)$. Indeed
\begin{prop} The first integral $K_{4,1}$ of $H_{4,1}$ is
\begin{gather}\label{Kmn}
K_{4,1}=256\left(xp_y^4-yp_xp_y^3-\frac 34\omega^2 xy^2p_y^2+\frac{\omega^2}8y^3p_xp_y+\frac {\omega^4}{64}xy^4\right).
\end{gather}
By applying to $K_{4,1}$ the Weyl quantization formula \eqref{W}, we have the Weyl operator
\begin{gather}
\hat K_{4,1}^{\rm W}=256\left(\hbar^4 \left( x \frac{\partial^4}{\partial y^4}- y \frac{\partial^4}{\partial x \partial y^3} \right)-\frac {3\hbar^4}2 \frac{\partial^3}{\partial x \partial y^2}+\frac {\hbar^2 \omega^2}8\left( 6xy^2 \frac{\partial^2}{\partial y^2} -y^3 \frac{\partial^2}{\partial x\partial y} \right)\right. \nonumber\\
\left.\hphantom{\hat K_{4,1}^{\rm W}=}{}+ \hbar^2\omega^2y \left( \frac 32 x \frac{\partial}{\partial y} -\frac 3{16}y\frac{\partial}{\partial x} \right)+\frac{\omega^4}{64}xy^4+\frac {3\hbar^2 \omega^2}8 x\right),\label{A}
\end{gather}
and, by applying to $K_{4,1}$ the formula \eqref{BJ}, we get the Born--Jordan operator
\begin{gather}\label{B}
\hat K_{4,1}^{\rm BJ}=\hat K_{4,1}^{\rm W}+32\hbar^2\omega^2x.
\end{gather}
The computation of the commutators gives
\begin{gather}\label{comm}
\big[\hat H_{4,1},\hat K^{\rm W}_{4,1}\big]=0, \qquad \big[\hat H_{4,1},\hat K^{\rm BJ}_{4,1}\big]=-32\hbar^4\omega^2\frac{\partial}{\partial x}.
\end{gather}
\end{prop}
\begin{proof}We have from (\ref{G}) $G_1=x$, therefore $X_LG_1=p_x$. The application of Proposition~\ref{comp} to~(\ref{Hu}) with $(m,n)=(4,1)$ followed by the transformation of coordinates (\ref{uy}) gives
\begin{gather*
P_{4,1}=256p_y^4-192\omega^2y^2p_y^2+4\omega^2y,\qquad D_{4,1}=-256yp_y^3+32\omega^2y^3p_y,
\end{gather*}
and we get (\ref{Kmn}). We observe now that, both for Born--Jordan and Weyl quantizations, we have
\begin{gather*}
\hat K_{4,1}= \hat P_{4,1}\hat x+\hat D_{4,1} \hat p_x,
\end{gather*}
where the order of the operators is immaterial, because $P_{4,1}$ and $D_{4,1}$ depend uniquely on~$(p_y,y)$. Therefore, we need to apply the two quantizations to $P_{4,1}$ and $D_{4,1}$ only, since the quantizations coincide on $x$ and $p_x$. By applying (\ref{BJ}) and (\ref{W}) to the monomials $y^2p_y^2$, $yp_y^3$ and $y^3p_y$, we have
\begin{gather*}
\hat P_{4,1}=256\hat p_y^4-192\omega^2\hat Q_1+4\omega^2\hat y,\qquad \hat D_{4,1}=-256\hat Q_2+32\omega^2\hat Q_3,
\end{gather*}
where
\begin{gather*}
\hat Q_2=i\frac {\hbar^3}2 \left( 2y\frac{\partial ^3}{\partial y^3}+3 \frac{\partial ^2}{\partial y^2} \right),\qquad
\hat Q_3=-i\frac \hbar 2 y^2 \left( 2 y \frac {\partial}{\partial y} +3\right),
\end{gather*}
coincide for both quantizations, and, for the Weyl case,
\begin{gather*}
\hat Q_1=-\frac {\hbar^2}2 \left(2y^2\frac{\partial ^2}{\partial y^2}+4y \frac {\partial}{\partial y} +1 \right),
\end{gather*}
while, for the Born--Jordan case,
\begin{gather*}
\hat Q_1=-\frac {\hbar^2}3 \left(3y^2\frac{\partial ^2}{\partial y^2}+6y \frac {\partial}{\partial y} +2 \right).
\end{gather*}
We obtain in this way (\ref{A}) and (\ref{B}).
We can f\/inally compute the commutators of $\hat H_{4,1}$ with $\hat K_{4,1}^{\rm W}$ and $\hat K_{4,1}^{\rm BJ}$ with the help of the formula
\begin{gather*}
\big[\hat H_{4,1},\hat K_{4,1}\big]=\hat p_x\big({-}i\hbar \hat P_{4,1}+\big[\hat H_{4,1}, \hat D_{4,1}\big]\big) +\hat x \big( 2\omega^2 i \hbar \hat D_{4,1}+\big[\hat H_{4,1},\hat P_{4,1}\big]\big),
\end{gather*}
after making the suitable substitutions, obtaining the~(\ref{comm}).
\end{proof}
We computed the quantizations for several other values of $(m,n)$, for example $(5,1)$, $(6,1)$, $(1,4)$, $(3,4)$, such that $\hat K_{m,n}^{\rm BJ}\neq \hat K_{m,n}^{\rm W} $, obtaining always commutation with $\hat H_{m,n}$ for $\hat K^{\rm W}_{m,n}$ and no commutation with $\hat H_{m,n}$ for $\hat K^{\rm BJ}_{m,n}$. In these last cases, it can be observed that both $\omega$ and $\hbar$ always appear as factors in the commutator, meaning that for $\omega=0$ and for $\hbar\rightarrow 0$ the operators commute.
Analogous results are obtained from the quantizations of the f\/irst integrals $F_1(m,n)$ or $F_2(m,n)$ given in (\ref{F}) for several values of $(m,n)$: the Weyl formula produces symmetry ope\-rators of the Hamiltonian, the Born--Jordan one, when giving dif\/ferent operators, does not.
Actually, one can conjecture that the Weyl quantizations of $K_{m,n}$ and $F_i(m,n)$ always commute with the Hamiltonian operator $\hat H_{m,n}$.
It can be observed, as one of the referees of this article pointed out, that ``for the Jauch--Hill approach the quantization using creation and annihilation operators shows very easily that the quantum extensions of $F_1$ and $F_2$ are still integrals. It
is less obvious to prove that this quantization is nothing but Weyl's one''.
\begin{rmk} The failure of the Born--Jordan quantization formula in reproducing the algebra of constants of motion at the quantum level is, actually, restricted to the particular set of generators of the algebra that we choose. We do not know in general if another choice of independent f\/irst integrals can lead to dif\/ferent results.
\end{rmk}
\begin{rmk} The failure in reproducing the algebra of the constants of motion after quantization appears also in the case of natural Hamiltonian systems on curved manifolds. In these cases, quantum corrections of the Hamiltonian operator are necessary in order to preserve the integrable or superintegrable algebraic structure \cite{Herranz, DV}.
\end{rmk}
\section{Conclusions}
From the examples computed, it appears that the Born--Jordan quantization formula fails in preserving the high-degree constants of motion of the 2D anisotropic harmonic oscillator, and therefore its superintegrability, to the quantum level, dif\/ferently from the Weyl formula. A~study of this problem in full generality is then desirable.
\subsection*{Acknowledgements}
I am grateful to the referees of this article for their comments and suggestions.
\pdfbookmark[1]{References}{ref}
|
{
"timestamp": "2016-08-18T02:04:52",
"yymm": "1606",
"arxiv_id": "1606.06474",
"language": "en",
"url": "https://arxiv.org/abs/1606.06474"
}
|
\section{Introduction}
\label{sec:intro}
Observations of the highly-redshifted 21-cm emission are considered
the most powerful probe of the birth of the first luminous sources and
the consequent epoch of reionization \citep[for recent reviews
see][]{mcquinn15,furlanetto15}. The 21-cm emission precisely traces and times the evolution of the average Hydrogen neutral fraction and the growth of the HII regions around ionizing sources throughout reionization \citep[e.g.,][]{ciardi03,mellema06,mcquinn07,lidz08}.
Prior to reionization, during the so-called Cosmic Dawn, the 21-cm signal marks the Ly$\alpha$ coupling and the X-ray heating eras respectively. The Ly$\alpha$ coupling occurs with the birth of the first luminous sources that are expected to be highly effective in coupling the spin temperature to the InterGalactic Medium (IGM) temperature, generating 21-cm emission via the Wouthuysen--Field effect \citep[]{wouthuysen52,field59}. Most models anticipate this transition to occur around $z \simeq 25-30$, when the IGM is colder than the Cosmic Microwave Background (CMB), causing a 21-cm signal in absorption against the CMB. The 21-cm emission is here sensitive to the nature of the first luminous sources as well as the details of the formation of the first galaxies in the first minihalos \citep[e.g.,][]{ciardi03b,furlanetto06,fialkov13}. In particular, the redshift and amplitude of the peak in the 21-cm emission strongly depends on whether the dominant contribution to the Ly$\alpha$ coupling comes from atomically-cooled galaxies or minihalos, and how much the Lyman--Werner background suppresses star formation \citep[e.g.,][]{haiman00,ricotti01,fialkov13}.
As star formation progresses, X-rays are generated in the first galaxies by either early black holes or the diffuse, hot interstellar medium. Although other sources of energy injection due to dark matter annihilation \citep[]{valdes07,valdes13} and shocks from fluid motions may be present in the early Universe \citep{mcquinn12}, X-ray emission is commonly believed to be the most significant source of IGM heating that would, eventually, drive its temperature above the CMB temperature \citep[]{pritchard07,mesinger13,pacucci14,tanaka16}. The relative timing of this process is, however, very uncertain due to the essentially unknown properties of the first galaxies \citep{mesinger16}. In particular, most models assume that the IGM is heated well above the CMB temperature by the onset of reionization. However, if the first galaxies show a hard X-ray spectrum or their X-ray efficiency (commonly parameterized as the number of X-ray photons produced per stellar baryon) is low, then heating becomes inefficient and reionization begins when the IGM is still colder than the CMB \citep[the `cold reionization' scenario,][]{fialkov14,mesinger14}. Such a scenario would also impact the subsequent morphology of reionization \citep{iliev12,ewall-wice16b}.
This theoretical landscape is still completely unconstrained by observations, but first upper limits to the 21-cm fluctuations in the $12 < z < 18$ range are starting to appear at approximately three orders of magnitude higher than the expected signal \citep{ewall-wice16a}. There is also initial evidence of heating prior to reionization provided by recent 21-cm power spectrum upper limits at $z = 8.4$ that constrain the IGM to be warmer than 8~K \citep[]{ali15,pober15,greig16}.
While current experiments targeting 21-cm fluctuations are well placed to constrain the reionization process statistically, only the upcoming interferometric arrays like the Hydrogen Epoch of Reionization Array \citep[]{pober14,deboer16} and the Square Kilometre Array \citep{koopmans15} will have sufficient sensitivity and frequency coverage to probe the Ly$\alpha$ and X-ray heating epochs \citep{mesinger15,ewall-wice16b}. Therefore increased attention has recently been devoted to observations targeting the global (sky-averaged) 21-cm emission \citep[e.g.,][]{pritchard10,morandi12,liu16}, including novel ways to use interferometric arrays to probe the global 21-cm signal \citep[][]{mckinley13,presley15,sing15,vedantham15}. Albeit challenged by the same requirements of accurate subtraction of bright foreground emission and control over systematic effects that affect its sibling 21-cm fluctuation observations, the global 21-cm signal may represent an alternative, relatively inexpensive way to achieve the milliKelvin sensitivity needed to access the pre-reionization epoch. Instruments like the Experiment to Detect the Global
Epoch-of-Reionization Signature \citep[EDGES,][]{bowman08,bowman10}, the Large aperture Experiment to detect the
Dark Ages \citep[LEDA,][]{greenhill12,bernardi15,kocz15,price2016}, SCI--HI \citep{voytek14} and the Dark Age Radio Explorer \citep[DARE,][]{mirocha15,harker16} are (or will be) targeting such an epoch.
In this paper we present a Bayesian foreground separation method and show that it can extract the 21-cm signal from the Cosmic Dawn even in the presence of non--spectrally-smooth foreground emission parameterized through high-order polynomials in frequency. We apply the algorithm to early LEDA data to derive upper limits on the global 21-cm signal in the $13.2 < z < 27.4$ range.
The paper is organized as follows. In Section~\ref{sec:bayes} we describe the Bayesian method and its application to simulated data. In Section~\ref{sec:observations}, we apply it to LEDA data; our results are discussed in Section~\ref{sec:conclusions}.
\section{Bayesian framework and simulations}
\label{sec:bayes}
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{figs/model_comparison.pdf}
\caption{Comparison between a 21-cm empirical Gaussian model (dashed line) used in our current analysis and a physical model derived from ARES (solid line). The physical model is taken from \citet{mirocha15} and is defined by four parameters: the minimum virial temperature for star-forming halos $T_{\rm min}$, the efficiencies of Ly$\alpha$ and X-ray photon production, $\xi_{\rm LW}$ and $\xi_{\rm X}$ respectively, and the IGM ionization efficiency $\xi_{\rm ion}$. We set $T_{\rm min} = 10^4$~K, $\xi_{\rm LW} = 969$, $\xi_{\rm X} = 0.02$ and $\xi_{\rm ion} = 40$ respectively \citep[see][for details]{mirocha15} and this can be considered as a reference model. The Gaussian model parameters are $A_{\rm HI} = -125$~mK, $\nu_{\rm HI} = 71$~MHz and $\sigma_{\rm HI} = 8$~MHz, similar to the model chosen for our simulations. The agreement between the two profiles is at the 10--20-per-cent level across most of the LEDA band.}\label{fig:model_comparison}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{figs/triangle-publn-simulated_160618.pdf}
\caption{Posterior probability distribution, marginalized into one and
two dimensions, for the $N=7$ foreground and the 21-cm models fitted
to the simulated data. The dark and light shaded regions indicate
the 68- and 95-per-cent confidence regions. The simulated parameter values are indicated in red. The marginalized probability distributions are plotted in the $[0,1]$ range.}\label{fig:triangle_sim_plot}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{figs/triangle-publn-simulated-zoomin_160618.pdf}
\caption{Zoom-in on the 21-cm parameters from
Figure~\ref{fig:triangle_sim_plot}. The dark and light shaded regions indicate
the 68- and 95-per-cent confidence regions. The simulated parameter values
are indicated in red. The marginalized probability distributions are plotted in the $[0,1]$ range.}
\label{fig:triangle_sim_zoomin_plot}
\end{figure*}
Bayesian Monte Carlo sampling (for example using Markov Chains; MCMC) has become a
standard method for exploring a likelihood surface and reconstructing the
posterior distribution in order to extract cosmological parameters
from CMB observations and, recently, also in the 21-cm field
\citep[]{harker12,greig15}.
Bayes' theorem indeed relates the posterior probability distribution
$\mathcal{P}({\bf \Theta} | {\bf D}, \mathcal{H})$ of a set of
parameters ${\bf \Theta}$ given the data ${\bf D}$ and a model
${\mathcal H}$, that includes the hypothesis and any related
assumptions, to the likelihood
$\mathcal{L}({\bf D} | {\bf \Theta}, \mathcal{H})$ as:
\begin{equation}
\label{eqn:bayes}
\mathcal{P}\left(\mathbf{\Theta}|\mathrm{\mathbf{D}},\mathcal{H}\right)
= \frac{
\mathcal{L}\left(\mathrm{\mathbf{D}}|\mathbf{\Theta},\mathcal{H}\right)
\mathit{\Pi}\left(\mathbf{\Theta}| \mathcal{H}\right)}
{\mathcal{Z}\left(\mathrm{\mathbf{D}}| \mathcal{H}\right)},
\end{equation}
\noindent
where the priors
${\mathit \Pi}\left(\mathbf{\Theta}|\mathcal{H}\right)$ encode
existing knowledge of parameter values and the evidence
${\mathcal Z}\left(\mathrm{\mathbf{D}}| \mathcal{H}\right)$ is the
integral of the likelihood
${\mathcal L}({\bf D} | {\bf \Theta}, {\mathcal H})$ over the prior
space, allowing not only normalization of the posterior but also model
selection via its inherent ability to quantify Occam's razor \citep[e.g.][]{mackay03,liddle06,trotta08,parkinson13}.
We implemented an algorithm for extracting the global 21-cm signal
following \cite{harker12}, who assume Gaussian measurement noise and
hence write the likelihood $\mathcal{L}_j$ of measuring the observed
sky temperature $T_{\rm ant}(\nu_j)$ at a single frequency $\nu_j$ as:
\begin{equation}
{\mathcal L}_j\left(T_{\rm ant} (\nu_j) | {\bf \Theta}\right) = \frac{1} {\sqrt{2 \pi \sigma^2(\nu_j)}} \mathrm{e}^{-\frac{[T_{\rm ant}(\nu_j) - T_m(\nu_j,\mathbf{\Theta})]^2}{2 \sigma^2(\nu_j)}},
\end{equation}
\noindent
where $T_m(\nu_j,\mathbf{\Theta})$ is the model spectrum and
$\sigma(\nu_j)$ is the standard deviation of the frequency-dependent
instrumental noise,
\begin{eqnarray}
\sigma(\nu_j) = \frac{T_{\rm ant} (\nu_j)}{\sqrt{\Delta \nu \Delta t}}, \nonumber
\end{eqnarray}
where $\Delta \nu$ is the channel width and $\Delta t$ is the total integration time.
Assuming that $T_{\rm ant}(\nu)$ is measured at $M$ discrete frequency
channels and that the noise is uncorrelated between frequency
channels, the (log-)likelihood for the full frequency spectrum
becomes:
\begin{equation}
{\ln\mathcal L}\left({\bf T_{\rm ant}} | {\bf \Theta}\right) = \sum_{j=1}^M {\ln\mathcal L}_j\left(T_{\rm ant}(\nu_j) | {\bf \Theta}\right).
\end{equation}
\noindent
The sky model at each frequency channel $\nu_j$ is the sum of the foreground $T_f$ and the 21-cm signal $T_{\rm HI}$:
\begin{eqnarray}
T_m (\nu_j) = T_f (\nu_j) + T_{\rm HI} (\nu_j).
\end{eqnarray}
Single-dipole observations measure the integrated Galactic foreground spectrum averaged over the whole sky, losing information about its spatial structure and how to separate foregrounds from the 21-cm global signal is still a very active debate in the community. \cite{liu13} and \cite{switzer14} suggest taking advantage of the spatial structure of the Galactic foreground in order to improve its separation from the spatially-constant 21-cm global signal. The most commonly adopted approach is to simply leverage the different spectral behaviour of foregrounds and the 21-cm signal, parameterizing the foreground spectrum through a principal component analysis \citep[e.g.][]{vedantham14} or a log-polynomial \citep[e.g.][]{pritchard10,bowman10,harker12,voytek14,bernardi15,presley15}. In this paper we therefore model the foreground emission as a $N^{\rm th}$ order log-polynomial:
\begin{equation}
\log_{10} T_f (\nu_j) = \sum_{n=0}^N p_n \left[ \log_{10} \left( \frac{\nu_j}{\nu_0} \right) \right]^n,
\end{equation}
where we have adopted the convention $\nu_0 = 60$\,MHz.
The choice of the polynomial order is critical in order to correctly model the foreground spectrum. Although earlier works showed that the foreground spectrum can be well described by very few components in frequency \citep[e.g.][]{de-oliveira-costa08,pritchard10}, more recent simulations suggest that most of the frequency structure present in the observed sky arises from the coupling between the sky and the antenna beam pattern \citep[]{bernardi15,mozdzen15}.
Our implementation is focused on the pre-reionization, Cosmic Dawn signal at
$15 \lesssim z \lesssim 30$, where the IGM is expected to be colder then the CMB. The 21-cm signal can be modelled as a Gaussian absorption profile
\citep[]{bernardi15,presley15}:
\begin{eqnarray}
T_{\rm HI}(\nu_j) = A_{\rm HI} \, \mathrm{e}^{-\frac{(\nu_j - \nu_{\rm HI})^2}{2 \sigma^2_{\rm HI}}},
\label{eq:hi_profile}
\end{eqnarray}
\noindent
where $A_{\rm HI}$, $\nu_{\rm HI}$ and $\sigma_{\rm HI}$ are the
amplitude, peak position and standard deviation of the 21-cm spectrum.
We investigated how well this empirical model reproduces a physical 21-cm spectrum by using the publicly-available code ARES\footnote{https://bitbucket.org/mirochaj/ares} \citep[][]{mirocha12,mirocha14,mirocha15}. Figure~\ref{fig:model_comparison} shows that a Gaussian profile closely resembles the physical reference model defined in \cite{mirocha15} across most of the considered observing band. Deviations between the two models start to become noticeable at high redshift when collisional coupling drives the 21-cm signal negative with respect to the Gaussian model. Although we plan to incorporate physical models in future analyses, the Gaussian profile is sufficiently accurate for the purpose of testing our signal-extraction method and applying it to establish first-order upper limits on the 21-cm signal (Section~\ref{sec:observations}).
In order to efficiently explore the posterior probability distribution, we use the
sampler \textsc{MultiNEST} \citep[]{feroz08,feroz09}; crucially, it is
an efficient calculator of the Bayesian evidence (with the posterior
samples coming as a by-product) in relatively-low-dimensionality
parameter spaces such as ours, and it robustly uncovers any
degeneracies, skirts, wings or multimodalities in the posterior. We
use an MPI-enabled python wrapper for \textsc{MultiNEST}
\citep{buchner14} that allows a full model fit to be evaluated in just
a few minutes on a typical desktop machine. We have released a python
implementation of our software,
\textsc{hibayes}\footnote{\url{http://github.com/ska-sa/hibayes}
\citep{ascl_hibayes}.}, that incorporates the models described here,
although the inclusion of different models is straightforward and will
be the goal of future work.
We tested the signal extraction on a simulated case where we considered
a $N = 7$ polynomial foreground model, representing the level of
corruption of the intrinsic sky spectrum due to the primary beam for a
simulated LEDA case \citep{bernardi15}. Such an assumption may be representative of other experiments --- or considered a somewhat pessimistic case.
We adopted the 21-cm model
labelled as `A' in \cite{bernardi15}, which has an amplitude
$A_{\rm HI} = -100$\,mK, a peak frequency $\nu_{\rm HI} = 67$\,MHz, a
width $\sigma_{\rm HI} = 5$\,MHz and similar to the fiducial
model of \cite{pritchard10} and \cite{mirocha15} plotted in Figure~\ref{fig:model_comparison}. We considered a 400-hour integration time
with a 1-MHz channel width and a dual-polarization dipole. We also
assumed the total bandwidth to span the 40--89\,MHz range. These
assumptions, although tuned to the LEDA case, can generally represent
the observing specifications of any ground-based 21-cm global
experiment that targets the pre-reionization era, with the 89-MHz
cutoff being due to the radio frequency interference (RFI) caused by
the radio FM band.
We assumed uniform priors on all the parameters and, in order (solely) to reduce the computing load, we set conservative priors on the 21-cm signal to be $-400 < A_{\rm HI} < 0$~mK, $40 < \nu_{\rm HI} < 89$~MHz and $0 < \sigma_{\rm HI} < 35$~MHz. Whereas the priors on the peak position and width are essentially due to the observational constraints, the amplitude prior can be theoretically motivated by assuming an extreme (and somewhat unlikely) model with no gas heating occurring in the redshift range of interest.
The peak amplitude of the 21-cm signal may be estimated analytically from the
expression for the 21-cm brightness temperature
\citep[e.g.][]{mesinger15}:
\begin{eqnarray}
A_{\rm HI} & \approx & 27 \, x_{\rm HI} \left(1 - \frac{T_\gamma}{T_s} \right) (1 + \delta) \left( \frac{H}{\mathrm{d}v_r/\mathrm{d}r + H} \right) \nonumber \\
& & \sqrt{\frac{1 + z}{10} \frac{0.15}{\Omega_{\rm M} h^2} } \left( \frac{\Omega_{\rm b} h^2}{0.023} \right) \left( \frac{1-Y_p} {0.75} \right) \, {\rm mK}.
\label{eq:delta_Tb}
\end{eqnarray}
For the global-signal case, we can ignore density fluctuations
(i.e.~set $\delta = 0$), peculiar velocities
(i.e.~set $\mathrm{d}v_r/\mathrm{d}r = 0$) and safely assume the Helium
fraction $Y_p = 0.25$. Assuming the IGM is fully neutral during this epoch (i.e.~$x_{\rm HI} = 1$), Equation~\ref{eq:delta_Tb} becomes
\begin{eqnarray}
A_{\rm HI} \approx 27 \left(1 - \frac{T_\gamma}{T_s} \right) \sqrt{\frac{1 + z}{10} \frac{0.15}{\Omega_{\rm M}} } \, \frac{\Omega_{\rm b} h}{0.023} \, {\rm mK},
\label{eq:delta_Tb_1}
\end{eqnarray}
where $T_\gamma$ is the CMB temperature, $T_s$ is the spin
temperature, $\Omega_{\rm M} = 0.315$ is the matter density, $\Omega_{\rm b} = 0.049$ is the baryon density and $h \equiv H/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}}) = 0.673$ is the normalized Hubble parameter \citep{planck15}.
Assuming no gas heating, the gas temperature $T_K$ can be calculated from thermal decoupling (where $T_\gamma = T_K$) following the $(1+z)^2$ adiabatic cooling. Also assuming that the Ly$\alpha$
emission from the first luminous sources is very effective in completely coupling the spin
temperature $T_s$ to the gas temperature, we can write
\begin{eqnarray}
T_s = T_K = T_{\gamma,0} (1 + z_d) \left[ \frac{1 + z}{1 + z_d} \right]^2,
\end{eqnarray}
with $T_{\gamma,0} = 2.73$\,K the CMB temperature at the present time
and $z_d \approx 200$ is the redshift of the thermal decoupling
between the IGM and the CMB. Substituting everything into
Equation~\ref{eq:delta_Tb_1} we obtain
\begin{eqnarray}
A_{\rm HI} \approx 27 \left(1 - \frac{1 + z_d}{1 + z} \right) \sqrt{ \frac{(1 + z)}{10} \frac{0.15}{\Omega_{\rm M}}} \, \frac{\Omega_{\rm b} h}{0.023} \, {\rm mK},
\label{eq:delta_Tb_2}
\end{eqnarray}
which gives $A_{\rm HI} \approx -380$~mK at $z = 15.7$, corresponding to the lowest redshift of the considered observing band.
Results of the \textsc{hibayes} fit to the simulated data are shown in
Figures~\ref{fig:triangle_sim_plot}
and~\ref{fig:triangle_sim_zoomin_plot}. Most of the parameters are
well recovered to within the 68-per-cent contours, although some of
the best-fitting values are marginally offset from their true values.
Correlations between some parameters are apparent, although the
one-dimensional marginalized distributions are fairly smooth for all
the parameters, with no evidence for multimodality. Most of the
foreground parameters are very tightly constrained, as are the 21-cm
peak frequency and width. The 21-cm amplitude shows the largest
relative errors --- at the 12-per-cent level --- and noticeable
anti-correlation with the foreground amplitude and slope. Such
anti-correlation can be explained by the degeneracy between these
parameters at the peak frequency $\nu_{\rm HI}$: if the foreground
amplitude is overestimated (underestimated), the 21-cm amplitude will
be underestimated (overestimated) or the foreground slope will be
steeper (flatter), in order to preserve the same observed spectrum
value. We also note that there are correlations between the
higher-order polynomial coefficients, although they can be
disentangled given the high level of sensitivity simulated here. The
results presented here are in agreement with the Fisher matrix
estimates from \cite{bernardi15} and show that, even in the presence
of spectrally-unsmooth foreground emission requiring high-order
polynomials to be modelled, our method is able to extract the
21-cm signal provided sufficient signal-to-noise ratio. Our results
are also broadly consistent with the MCMC analysis presented by
\cite{harker12} and \cite{harker15}, although they used a fairly distinct
frequency band and 21-cm model to ours.
\section{Analysis of LEDA data}
\label{sec:observations}
We next applied our method to preliminary LEDA data. The LEDA instrument is described in detail in upcoming papers
\citep[][]{schinzel16,price2016}; here, we briefly describe the system along with the
observations and data-reduction approach.
LEDA is a sub-instrument of the Long Wavelength Array at the Owens Valley Radio Observatory \citep[LWA-OVRO,][]{hallinan2016}. LWA-OVRO is primarily an all-sky imaging radio interferometer, designed to operate in the frequency range 10--88~MHz. It consists of a `core' of 251 dual-polarization dipole-type antennas within a 200-m diameter, plus an additional 5~`outrigger' antennas, located a few hundred metres from the core, customized for LEDA. In addition, 32~`expansion' antennas are quasi-randomly distributed up to 1500~m from the core.
Each LEDA outrigger antenna is equipped with a receiver board designed for precision radiometry.
Here we present the total-power data taken from a single outrigger antenna, whereas future LEDA analyses will make use of data from all five outrigger antennas, supported by the analysis of
cross-correlations with the core antennas in order to improve the instrument calibration by measuring the antenna primary beam \citep[see the discussion in][]{bernardi15} and ionospheric distortions.
Observations were made during the nights of 2016 February 11 and 12 over a 2-hour period centred at ${\rm LST} = 10^{\rm h} 30^{\rm m}$ when the Galactic centre, Cassiopeia~A and Cygnus~A were near or below the horizon. The antenna total-power data were digitized at a rate of 196.608\,MHz, giving a 0--98.304\,MHz bandwidth, covered by 4096 channels, each of them 24-kHz wide. Data were integrated over 1~second.
We calibrated spectra using a multi-stage approach, as follows. The first calibration stage was a modified version of the three-state switching calibration technique employed by EDGES \citep{rogers12}. The three-state switching removes the effect of time variations in the system gain $G(\nu, t)$ and receiver temperature $T_{\rm{rx}}(\nu,t)$, and imposes an absolute temperature scale on the data. The LEDA outrigger antennas switch between the sky, and two calibration references --- referred to as `hot' and `cold' --- with different noise-equivalent temperatures $T_{\rm{hot}}(\nu,t)$ and $T_{\rm{cold}}(\nu,t)$ respectively. The power measured in each state is then given by:
\begin{align}
P_{\rm{ant}} (\nu,t) & =G(\nu, t) \, \Delta\nu \, k_{\rm{B}} (T'_{\rm{ant}}(\nu, t) + T_{\rm{rx}}(\nu, t)) \nonumber \\
P_{\rm{hot}} (\nu,t) & =G(\nu, t) \, \Delta\nu \, k_{\rm{B}} (T_{\rm{hot}}(\nu, t) + T_{\rm{rx}}(\nu, t)) \nonumber \\
P_{\rm{cold}} (\nu,t) & =G(\nu, t) \, \Delta\nu \, k_{\rm{B}} (T_{\rm{cold}}(\nu, t) + T_{\rm{rx}}(\nu, t)),
\label{eq:3ss_EDGES}
\end{align}
where $P_{\rm{ant}}$, $P_{\rm{hot}}$ and $P_{\rm{cold}}$ are the powers for the antenna, hot calibration reference and cold calibration reference states respectively; $T'_{\rm{ant}}$ is the antenna noise-equivalent temperature; $\Delta \nu = 24$~kHz is the channel width and $k_{\rm{B}}$ is the Boltzmann constant. The first-stage calibrated antenna temperature $T'_{\rm{ant}}$ is recovered via
\begin{equation}
T'_{\rm{ant}} (\nu, t) = (T_{\rm{hot}} - T_{\rm{cold}})\frac{P_{\rm{ant}}-P_{\rm{cold}} }{P_{\rm{hot}}-P_{\rm{cold}}}+T_{\rm{cold}}.\label{eq:3ss}
\end{equation}
where $T_{\rm{hot}}$ and $T_{\rm{cold}}$ are measured before the observation and where we drop the explicit dependence on time and frequency on the r.h.s.~of the equation for simplicity.
As the receiver switched between the three states every 5~seconds, with the first second of data in each state blanked, the total on-sky time was eventually 1152~seconds ($\approx 19$~minutes).
The second-stage calibrated antenna temperature $T''_{\rm{ant}}$ is obtained by correcting $T'_{\rm{ant}}$ for the reflection coefficient $\Gamma$:
\begin{equation}
T''_{\rm{ant}} (\nu,t) = T'_{\rm ant} (\nu,t) [1-|\Gamma|^{2} (\nu) ],
\label{eq:vna}
\end{equation}
where $\Gamma (\nu)$ measures the impedance mismatch between the receiver and the antenna and was determined using a vector network analyzer \citep{price2016}.
At this point spectra were flagged for RFI using the \textsc{SumThreshold} algorithm \citep{offringa2010}, then averaged in frequency to achieve a final resolution of 768~kHz. The $40$--$85$~MHz band of interest was subsequently extracted, with a few MHz lost at the upper end of the bandwidth due to filter roll-off.
The final calibration stage was performed using a sky spectrum model $\hat{T}_{\rm{ant}}(\nu,t)$:
\begin{equation}
\hat{T}_{\rm{ant}}(\nu,t)=\frac{\int d\Omega B(\theta,\phi,\nu)T_{\rm{sky}}(\theta,\phi,\nu, t)}{\int d\Omega B(\theta,\phi,\nu)},
\end{equation}
where $B(\theta,\phi,\nu)$ is the antenna beam pattern and $T_{\rm{sky}}(\theta, \phi, \nu, t)$ is the model sky brightness distribution evaluated using the \cite{de-oliveira-costa08} global sky model\footnote{Our python-based implementation that includes observer-centred sky models is available at \url{https://github.com/telegraphic/pygsm} \citep{ascl_pygsm}.}. We used the dipole beam model from \cite{dowell2012}. \cite{dowell16} have recently completed a sky survey covering the frequency range $35-80$~MHz using the LWA and these data may be used to improve the calibration in the future.
The calibrated antenna temperature $T_{\rm ant} (\nu,t)$ measured at time $t$ is then obtained as:
\begin{align}
T_{\rm ant} (\nu,t) & = \frac{\int_T \hat{T}_{\rm ant}(\nu,t') dt'} {\int_T T''_{\rm ant}(\nu,t') dt'} T''_{\rm ant}(\nu,t)\nonumber \\
& = \alpha(\nu) T''_{\rm ant}(\nu,t) \label{eqn:calvec},
\end{align}
where the average occurs over the full $T = 2$~hours of observation. The calibration $\alpha(\nu)$ is calculated from observations on 2016 February 11, and then applied to the observations on February 12.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{figs/calibrated_spectra_mine_with_z.pdf}
\caption{Measured sky spectrum for a 2-hour observation ($\approx 19$~minutes effective integration time, 2016 February 12 at $9.5^{\rm h} < {\rm LST} < 11.5^{\rm h}$). Note that the error bars have been inflated by a factor of 1000 in order to make them visible.}\label{fig:observed_spectrum}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{figs/diode-load-uncert.pdf}
\caption{Comparison of the estimated thermal noise (dashed line) and the noise measured as the standard deviation of the observed data (solid line).}
\label{fig:rms-comparison}
\end{figure}
The final calibrated sky spectrum $T_{\rm ant} (\nu)$ after averaging
in time is shown in Figure~\ref{fig:observed_spectrum} and becomes the input for \textsc{hibayes}. The thermal noise can be estimated by propagating the uncertainties of Equation~\ref{eq:3ss_EDGES}:
\begin{eqnarray}
\sigma^{2} (\nu) & = & \left(\frac{\partial T'_{\rm ant}}{\partial P_{\rm ant}}\right)^{2} (\Delta P_{\rm ant})^2 + \left(\frac{\partial T'_{\rm ant}}{\partial P_{\rm cold}}\right)^{2} (\Delta P_{\rm cold})^2 \nonumber \\
& + & \left(\frac{\partial T'_{\rm ant}}{\partial P_{\rm hot}}\right)^{2} (\Delta P_{\rm hot})^2.
\label{eq:rms-unc}
\end{eqnarray}
Figure~\ref{fig:rms-comparison} compares the estimated thermal noise with that derived as the standard deviation of the calibrated antenna temperature as a function of time for each frequency channel. The measured and the expected thermal noise levels are consistent above 55\,MHz, whereas the measured noise is higher than expected at lower frequencies. The large spikes below 50\,MHz correlate with known RFI sources, where a larger fraction of data are flagged, causing an effective decrease in integration time.
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{figs/evidence_plot_dec2015.pdf}
\caption{Bayesian evidence for foreground models fitted to the LEDA data, relative to the $N=7^{\rm th}$ order polynomial foreground model, as a function of polynomial order $N$. The uncertainties on the evidence are of the same magnitude as the filled circle size.}\label{fig:evidence_plot}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{figs/triangle-publn_160618.pdf}
\caption{Posterior probability distribution, marginalized into one and
two dimensions, for the $N=7^{\rm th}$ order polynomial foreground
and 21-cm models, fitted to the LEDA data. The dark and light shaded
regions indicate the 68- and 95-per-cent confidence
regions. The marginalized probability distributions are plotted in the $[0,1]$ range.}\label{fig:triangle_plot}
\end{figure*}
In the \textsc{hibayes} analysis, we first looked to confirm the foreground parametrization used in our simulations (section~\ref{sec:bayes}). Following \cite{harker15}, we sought to establish the foreground model by fitting the data with increasing polynomial order assuming that the 21-cm signal is fairly described by Equation~\ref{eq:hi_profile}. We found that the evidence (Figure~\ref{fig:evidence_plot}) increases sharply as a function of polynomial order until $N=6$, after which it starts to flatten.
According to the scale of \cite{jeffreys39}, the $N=7$ model is still decisively (odds $> 100:1$) preferred over the $N=6$ model, whereas the $N=8$ model is disfavoured (negative odds) over the $N=7$ model. In practice the evidence remains essentially flat as the polynomial increases beyond the $N=7^{\rm th}$ order and small (positive or negative) variations are likely due to sampling accuracy. We therefore fitted the data using an evidence-motivated model that includes the 21-cm signal and a $N=7^{\rm th}$ order polynomial foreground model and we used the measured noise as a function of frequency. We emphasize here that the chosen $N=7^{\rm th}$ order polynomial is not intended to represent the spectral structure of the intrinsic sky emission but rather the `observed foregrounds', i.e.~the convolution of the intrinsic sky emission with the instrumental response. It is also worth noticing that the evidence here favours a model that is in fair agreement with earlier LEDA simulations presented in \cite{bernardi15}. In a future work we will investigate the possibility of parameterizing the instrument and the sky emission
separately and of using the evidence to indicate the best choice of sub-models, rather than assuming a combined parameterization as here.\\
After having established the foreground model, we set the priors on the width of the 21-cm signal to be the same used for simulations as they encompass the full breadth of theoretical predictions. We set uniform priors on the 21-cm peak position to be $50 < \nu_{\rm HI} < 100$~MHz as such a range brackets both models with the most extreme star-formation efficiency --- which would shift the peak at low frequencies --- and with the most extreme X-ray efficiency --- which would shift the peak at high frequencies \citep[][]{pritchard10,mirocha15}. We relaxed the constraints on the depth of the 21-cm peak amplitude that we used for the simulated case because we seek to derive data-driven upper limits even in the case of models that are disfavoured by theory. We still assumed that the 21-cm signal cannot be positive, i.e.~that $T_s = T_k < T_\gamma$, which is accepted in any model for the redshift range considered here.
We ran different chains with decreasing lower bounds of the $A_{\rm HI}$ prior. In the initial case we set $-380 < A_{\rm HI} < 0$~mK and found that the whole prior range is within the 95-per-cent confidence region. We found that the two-dimensional posterior distributions for both the 21-cm amplitude and the width showed monotonically-decreasing profiles with increasing prior range until an area of the prior range is clearly excluded at a confidence level greater than 95-per-cent for $-1000 < A_{\rm HI} < 0$~mK.
The posterior probability distribution for this final run is displayed in Figure~\ref{fig:triangle_plot}.
The foreground parameters are very well constrained and all their marginalized, one-dimensional distributions are Gaussian-like, similar to the simulated case, with the exception of the foreground amplitude $p_0$, whose marginalized posterior is slightly asymmetric.
The best-fit foreground parameters are fairly different from the simulated case apart from the first two coefficients that indicate the foreground amplitude at 60\,MHz and its power-law slope. This is not unexpected as higher-order polynomials are most likely compensating for limitations in the instrumental calibration that were not included in the simulations (e.g.~errors on the reflection coefficient or calibration load). It is interesting, however, that the best-fit foreground coefficients show correlations between foreground parameters similar, albeit at a qualitative level, to the simulated case, for example between the $p_3$ and $p_5$, and $p_5$ and $p_7$, coefficients.
The 68- and 95-per-cent confidence contours of the 21-cm parameters are fairly different and, essentially, identify upper limit regions, as expected given the noise levels. Whereas no constraints can be placed on the peak position $\nu_{\rm HI}$ within the prior range, bright, narrow 21-cm Gaussian profiles are disfavoured by the data. Quantitatively, we constrain $A_{\rm HI} > -890$~mK and $\sigma_{\rm HI} > 6.5$~MHz at the 95-per-cent level in the $13.2 < z < 27.4$ range: this amplitude limit is only a factor of $\approx 2.5$ away from constraining the extreme model with no heating described in Section~\ref{sec:bayes}.
In the absence of a detection, the root-mean-square of the residuals is a metric often used in observations --- although less statistically rigorous than limits derived directly from the posterior probability distribution. Figure~\ref{fig:residuals} shows the residual spectrum after subtraction of the best-fit, maximum \textit{a posteriori} foreground model. We find a 470~mK rms residual over the whole redshift range, that, if considered a proxy for the 68-per-cent confidence level, is approximately consistent with the \textsc{hibayes} constraints on the 21-cm peak amplitude.
\begin{figure}
\includegraphics[width=1.\columnwidth]{figs/residual_spectrum_160530.pdf}
\caption{Residual spectrum after subtraction of the best-fit, maximum \textit{a posteriori} foreground model. Error bars are plotted at the $2$-$\sigma$ confidence level and include both the measured and the best-fit parameter uncertainties. All the data points but one at 57.7~MHz are compatible with zero.}
\label{fig:residuals}
\end{figure}
\section{Summary and Conclusions}
\label{sec:conclusions}
We have presented a fully-Bayesian algorithm for simultaneously fitting the global 21-cm signal in the presence of sky foregrounds. Our algorithm capitalizes on the Bayesian evidence's Occam's razor effect for model selection, with posterior probability distributions coming as a by-product.
We tested the method on simulated data and showed that, assuming a 7$^{\rm th}$-order polynomial foreground spectrum, the 21-cm global signal --- parameterized as a Gaussian absorption profile --- can be strongly constrained with a 400-hour integration time for a LEDA-like observing setup. This result more quantitatively confirms the Fisher matrix analysis previously carried out by \cite{bernardi15}. Although here we presented a specific application for the 21-cm signal from the Cosmic Dawn, the code can easily be extended to the full redshift range of interest for global-signal measurements. The code \citep{ascl_hibayes} is publicly available at \url{http://github.com/ska-sa/hibayes}.
We applied the method to observations in order to derive upper limits on the 21-cm signal from the Cosmic Dawn. We showed that the Bayesian evidence can guide the choice of foreground model \citep{harker15}, with a maximum for a $7^{\rm th}$-order polynomial in the present case. We emphasize that such a model reflects the combination of the intrinsic foregrounds and the spectral structure introduced by the instrument. In this respect, the evidence does not yet constrain the intrinsic foreground spectrum as suggested by \cite{harker15} and future work will be dedicated to incorporating both the intrinsic sky and instrument models in the analysis and placing constraints on the intrinsic foreground spectrum.
The best-fit foreground parameters are very well constrained; in particular we derive a spectral index for the diffuse Galactic emission $\beta (\equiv p_1) = 2.27 \pm 0.04$. This value is consistent with early measurements of the Galactic radio background at 81.5\,MHz by \cite{bridle67}, but is noticeably flatter than what was measured at 150~MHz by \cite{rogers08}. This possible flattening of the spectral index may be good news for foreground subtraction for future 21-cm (global and interferometric) observations targeting the pre-reionization epoch.
\cite{voytek14} report the only other broadband measurements at these frequencies. A direct comparison with their results is not straightforward as they do not report either r.m.s.~residuals or direct upper limits on 21-cm parameters. We note, however, that our best-fit spectral index is consistent with theirs, although their spectrum normalization is about 40-per-cent greater than what we report here. Our measurements, however, are within 10-per-cent of the carefully absolutely-calibrated Galactic spectrum measured by EDGES at 150~MHz \citep{rogers08} once it is scaled down to 60~MHz using the EDGES spectral index. We therefore believe our absolute flux density scale to be appropriate and its uncertainty to be negligible at the present level of sensitivity.
Our analysis constrains the 21-cm signal amplitude and width to be $-890 < A_{\rm HI} < 0$~mK and $\sigma_{\rm HI} > 6.5$~MHz respectively at the 95-per-cent confidence level in the $13.2 < z < 27.4$ ($100 > \nu > 50$~MHz) range. Note that the constraint on $\sigma_{\rm HI}$ corresponds to a redshift width $\Delta z\approx 1.9$ at redshift $z\simeq 20$.
Our results are the tightest upper limits on the 21-cm signal from the Cosmic Dawn to date and are encouraging in terms of achieving a factor of a few improvement in the sensitivity necessary to start placing significant constraints on structure prior reionization and on the thermal history of the IGM and the related sources of heating.
\section*{Acknowledgments}
\label{sec:acknowledgments}
We thank an anonymous referee for helpful comments that considerably improved the manuscript. GB thanks Judd Bowman, Andrea Ferrara, Adrian Liu and Aaron Ewall-Wice for useful inputs and comments on this work and Jordan Mirocha for help with ARES. The LEDA experiment is supported by NSF grants AST/1106059 and PHY/0835713. JZ gratefully acknowledges a South Africa National Research Foundation Square Kilometre Array Research Fellowship. This research was supported by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence `Origin and Structure of the Universe'. With the support of the Ministry of Foreign Affairs and International Cooperation, Directorate General for the Country Promotion (Bilateral Grant Agreement ZA14GR02 - Mapping the Universe on the Pathway to SKA).
|
{
"timestamp": "2016-06-21T02:13:36",
"yymm": "1606",
"arxiv_id": "1606.06006",
"language": "en",
"url": "https://arxiv.org/abs/1606.06006"
}
|
\section{Introduction}
Masses of free fermions arise from local fermion bilinear terms in the action. If symmetries of the theory prevent such terms, fermions remain massless perturbatively. However, these symmetries can break spontaneously and generate non-zero fermion bilinear condensates that can make fermions massive. This traditional mechanism of fermion mass generation is well known and is used in the standard model of particle physics to give quarks and leptons their masses. In QCD, along with confinement, this mechanism also helps explain the existence of light pions while making nucleons heavy. In this work we explore a different mechanism of fermion mass generation where fermions acquire their mass through four-fermion condensates, while fermion bilinear condensates vanish. This alternate mechanism has been the focus of many recent studies in 3D lattice models \cite{Slagle:2014vma,He:2015bda,Ayyar:2014eua,Catterall:2015zua,Ayyar:2015lrd,He:2016sbs}. Here we explore if these results extend to 4D. The 3D studies also show that no spontaneous symmetry breaking of any lattice symmetries is necessary for fermions to become massive. The presence of a new second order critical point makes the mechanism interesting even in the continuum.\footnote{The fermion mass generation mechanism we explore in this work is different from the one proposed in \cite{Stern:1998dy,PhysRevD.59.016001,Kanazawa:2015kca} where chiral symmetry is spontaneously broken due to four-fermion condensates instead of fermion bilinear condensates and massless bosons are present. In the massive fermion phase we explore here all particles, including bosons are massive.}
We believe that the alternate mechanism of mass generation can be understood qualitatively if we view the four-fermion condensate as a fermion bilinear condensate between a composite fermion (consisting of three fundamental fermions) and a fundamental fermion. When three fundamental fermions bind to form a composite fermion state, the four-fermion condensate can begin to act like a conventional mass term. However, since such composite states can only form at sufficiently strong couplings a non-perturbative approach is required to uncover it. At weak couplings, when composite states do not form, four-fermion condensates cannot act like mass terms although they are still non-zero. Since there are no local order parameters that signal the formation of the composite fermion bound states, the massive phase does not require spontaneous symmetry breaking. All these arguments are consistent with the results in 3D lattice models mentioned above.
Generating fermion masses through interactions but without spontaneous symmetry breaking is a subtle problem from the perspective of 4D continuum quantum field theories due to anomaly matching arguments \cite{tHooft:1979bh,PhysRevLett.45.100,Banks:1991sh,Banks:1992af}, but 4D lattice models that display such a mechanism of mass generation are well known and have been studied extensively in the context of lattice Yukawa models with both staggered fermions \cite{Hasenfratz:1988vc,Hasenfratz:1989jr,Lee:1989mi} and Wilson fermions \cite{Bock:1990tv,Bock:1990cx,Golterman:1990yb,Gerhold:2007gx,Bulava:2011jp}. These models contain a massless fermion phase at weak couplings (referred to as the paramagnetic weak or PMW phase), and a non-traditional symmetric massive fermion phase at strong couplings (referred to as the paramagnetic strong or PMS phase). The fermion mass in the PMS phase can be argued as being generated due to four fermion condensates since fermion bilinear condensates vanish in that phase. A review of the early work can be found in \cite{Shigemitsu:1991tc}.
In order for the PMS phase found in previous lattice calculations to become interesting from the point of view of continuum quantum field theory, it must be possible to tune the fermion mass to zero in lattice units. In units where the fermion mass remains fixed, this would imply that the lattice spacing vanishes. This can be accomplished in the presence of a direct second order transition between the PMW and the PMS phase. Such a transition was proposed as an important ingredient for realizing chiral fermions on the lattice \cite{Eichten:1985ft,Golterman:1992yha}. Unfortunately, all previous studies found that there was always an intermediate phase (referred to as the ferromagnetic of FM phase) where the symmetry that protected the fermions from becoming massive at weak couplings, was broken spontaneously. In the presence of the FM phase, fermions in the PMS phase cannot be made arbitrarily light in lattice units. We believe this was the reason the PMS phase was abandoned as merely a lattice artifact. In fact earlier studies in 3D also found an intermediate FM phase \cite{Alonso:1999hh}. Hence the recent discovery of a possible direct second order PMW-PMS phase transition in 3D is exciting, and raises the possibility that such transitions may exist even in 4D.
A second order PMW-PMS transition does not fall under the usual Landau-Ginzburg paradigm, since both the phases have the same global symmetries and there is no local order parameter that distinguishes them. For this reason it must be different from the usual Gross Neveu universality class. Such transitions are known in condensed matter literature and usually driven due to a change in the topological properties of the ground state \cite{Senthil:2014ooa}. The PMW-PMS transition could occur due to a similar reason although it does not seem to involve any topological order \cite{He:2016sbs}. From a condensed matter perspective, the PMS phase can be viewed as a trivial insulator where the ground state does not break any lattice symmetries since it is formed by local singlets. In contrast, the traditional massive fermion phase with fermion bilinear condensates is like a gapped semi-metal or a topological insulator. Topological insulators can have chiral zero modes attached to domain walls where the sign of the condensate changes \cite{Callan:1984sa}. Such zero modes are extensively used today in lattice QCD studies through the domain wall formulation introduced by Kaplan \cite{Kaplan:1992bt}. Many interesting properties of such topological insulators in background fields have also been studied by particle physicists many years ago \cite{Golterman:1992ub,Kaplan:1999jn}. Recently the focus has shifted to the classification of the topological insulators in the presence of fermion self interactions \cite{Fidkowski:2009dba,PhysRevB.83.075103,1367-2630-15-6-065002,Fidkowski:2013jua,PhysRevB.92.125104,PhysRevB.89.195124}. These studies suggest that when the fermion content of the theory is chosen appropriately, such interactions can smoothly deform a topological insulator to a trivial insulator. During such a change massless chiral fermions on the edges acquire non-traditional masses due to the formation of four-fermion condensates since fermion bilinear condensates are forbidden. The associated phase transition on the edge need not involve any spontaneous symmetry breaking \cite{Slagle:2014vma}. This has prompted many applications of the alternate mass generation mechanism to particle physics \cite{Wen:2013ppa,You:2014vea,You:2014oaa,BenTov:2015gra}.
A direct second order PMW-PMS phase transition has remained elusive in 4D so far. Given the discovery of such a transition in 3D \cite{Ayyar:2015lrd}, we believe it is worth searching for it even in 4D. The first step obviously would be to explore if the same lattice model that showed its presence in 3D, also contains it in 4D. Interestingly, this model contains sixteen Weyl fermions, which has been argued to be the right number necessary for the possible existence of the transition in 4D \cite{You:2014vea}. However, this model was already studied long ago in the context of Higgs-Yukawa models and a {\em wide} intermediate FM phase was found \cite{Lee:1989mi}, implying that one needs to explore extensions to it. It may be possible to add new couplings to the model that have the effect of narrowing the width of the FM phase. Unfortunately, the conclusions of the earlier work were mostly drawn from mean field theory and crude Monte Carlo calculations. Hence, in this work we focus on accurately determining the phase boundaries of the model so as to get a sense of how far away is the possible critical point in the extended parameter space. By working in the limit where the Higgs field can be integrated out explicitly, we can accurately study the model in the chiral limit with Monte Carlo methods on lattices up to $12^4$ using the fermion bag approach \cite{PhysRevD.82.025007,Chandraepja13}. In contrast to the earlier work, our results shows a surprisingly {\em narrow} intermediate FM phase, assuming it exists.
Our paper is organized as follows. In section \ref{sec2} we provide a new view point for our lattice model and discuss its symmetries. We also discuss observables that shed light on the phase structure of the model. In section \ref{sec3} we present the fermion bag approach and show that fermion bags have interesting topological properties. In particular we discuss an index theorem very similar to the one in non-Abelian gauge theories with massless fermions. In section \ref{sec4} we explain how the fermion bag approach provides a new theoretical perspective on the physics of the PMS phase and the alternate mass generation mechanism. In particular we explain how all fermion bilinear mass order parameters in the model must vanish at sufficiently strong couplings, although fermions are massive. In section \ref{sec5} we present our Monte Carlo results and in section \ref{sec6} we present our conclusions.
\section{The Model}
\label{sec2}
Our model was originally studied within the context of lattice Higgs-Yukawa models \cite{Lee:1989mi}. However, it can also be obtained directly by discretizing naively the continuum four-fermion action containing a single Dirac fermion field $\psi^a(x)$ and ${\overline{\psi}}^a(x)$ where $a$ labels the four spinor indices. We believe this alternate view point sheds more light on the mechanism of mass generation with four-fermion condensates (or equivalently the PMS phase) at strong couplings. Consider the continuum Euclidean action given by
\begin{equation}
S_{\rm cont} = \int d^4x \ \Big\{\ {\overline{\psi}}(x)\gamma_\alpha\partial_\alpha \psi(x) \ -\ U\Big(\psi^4(x) \psi^3(x) \psi^2(x) \psi^1(x) + {\overline{\psi}}^4(x) {\overline{\psi}}^3(x) {\overline{\psi}}^2(x) {\overline{\psi}}^1(x)
\Big) \Big\}.
\label{contact}
\end{equation}
where $\gamma_\alpha$ are the usual $4 \times 4$ Hermitian Dirac matrices. Note that the continuum model breaks the $U(1)$ fermion number symmetry
\begin{equation}
\psi(x) \rightarrow \exp(i\theta)\psi(x), \ \ {\overline{\psi}}(x) \rightarrow \exp(-i\theta){\overline{\psi}}(x)
\label{u1fs}
\end{equation}
explicitly, but it is invariant under Euclidean rotations and the $U(1)$ chiral symmetry
\begin{equation}
\psi(x) \rightarrow \exp(i\theta \gamma_5)\psi(x), \ \ {\overline{\psi}}(x) \rightarrow
{\overline{\psi}}(x)\exp(i\theta \gamma_5).
\label{u1cs}
\end{equation}
Perturbatively, no fermion bilinear mass term can be generated through radiative corrections since all such terms break either the $U(1)$ chiral symmetry or the rotational symmetry. Thus, the model must contain a massless fermion phase (or the PMW phase) at weak couplings. At strong couplings, assuming we can perform a perturbative (strong coupling) expansion in the kinetic term (similar to the hopping parameter expansion on the lattice), we can see the presence of a symmetric massive fermion phase (or the PMS phase).
In particular the leading order theory is trivial since all fermion fields are bound into local space-time singlets under the symmetries of the action. Introduction of the kinetic term can create excitations that transform non-trivially under both chiral and rotational symmetries, but all of these must be massive since energetically favored singlets need to be broken to create them. But, can the strong coupling expansion as described above be justified after the subtleties of UV divergences are taken into account? Although we cannot answer this question for a single Dirac field, ignoring the fermion doubling problem, we can easily discretize the continuum action (\ref{contact}) naively on the lattice and ask the same question in a controlled setting in the lattice theory. In particular we can even explore if the fermion mass of the lattice theory at strong couplings (i.e., in the PMS phase) can be made light as compared to the cutoff.
Discretizing (\ref{contact}) naively on a space-time lattice we obtain
\begin{equation}
S_{\rm naive} = \sum_{x,y} \ \Big\{\ {\overline{\psi}}_x \gamma_\alpha \frac{1}{2}(\delta_{x+\hat{\alpha},y} - \delta_{x-\hat{\alpha},y})\psi_y\ -\ U \Big(\psi^4_x \psi^3_x \psi^2_x \psi^1_x + {\overline{\psi}}^4_x {\overline{\psi}}^3_x {\overline{\psi}}^2_x {\overline{\psi}}^1_x\Big) \Big\},
\end{equation}
where we use the notation $\psi^a_x$ to denote the lattice Grassmann fields. Using the well known spin diagonalization transformation
\begin{equation}
\psi_x \rightarrow (\gamma_1)^{x_1}(\gamma_2)^{x_2}(\gamma_3)^{x_3}(\gamma_4)^{x_4} \psi_x,\ \
{\overline{\psi}}_x \rightarrow {\overline{\psi}}_x (\gamma_4)^{x_4}(\gamma_3)^{x_3}(\gamma_2)^{x_2}(\gamma_1)^{x_1} ,\ \
\end{equation}
used to define staggered fermions \cite{Sharatchandra:1981si}, we obtain the lattice action,
\begin{equation}
S_{\rm naive} = \sum_{x,y} \ \Big\{\ {\overline{\psi}}_x M_{x,y}\psi_y\ -\ U\Big(\psi^4_x \psi^3_x \psi^2_x \psi^1_x + {\overline{\psi}}^4_x {\overline{\psi}}^3_x {\overline{\psi}}^2_x {\overline{\psi}}^1_x\Big) \Big\}.
\end{equation}
where $M_{x,y}$ is the free staggered fermion matrix
\begin{equation}
M_{x,y} \ =\ \frac{1}{2}\ \sum_{\alpha} \ \eta_{\alpha,x}\ \big(\delta_{x+\hat{\alpha},y} - \delta_{x-\hat{\alpha},y}\big).
\end{equation}
The phase factors $\eta_{1,x}=1$, $\eta_{2,x}=(-1)^{x_1}$, $\eta_{3,x}=(-1)^{x_1+x_2}$, $\eta_{4,x}=(-1)^{x_1+x_2+x_3}$ are well known. Since $\psi^a_x$ on even sites only connect with ${\overline{\psi}}^a_x$ on odd sites and vice versa, we can eliminate half the degrees of freedom by defining $\psi^a_x$ only on even sites and ${\overline{\psi}}^a_x$ only odd sites. We can go a step further and stop distinguishing between $\psi^a_x$ and ${\overline{\psi}}^a_x$ since every site has an identical single four component Grassmann variable. This finally leads to the Euclidean action,
\begin{equation}
S \ =\ \frac{1}{2}\sum_{x,y,a}\ \psi^a_x \ M_{x,y} \ \psi^a_y \ - \ U\ \sum_x \ \psi^4_x \psi^3_x \psi^2_x \psi^1_x.
\label{act}
\end{equation}
We can also view the above action as being constructed directly with four reduced flavors of staggered fermions with an onsite four-fermion interaction. In this interpretation, the spinor indices $a=1,2,3,4$ are viewed as labels of the four reduced staggered flavors. When $U=0$ the above model describes eight flavors of Dirac fermions (or equivalently sixteen flavors of Weyl fermions) in the continuum. This matches the required number of fermions that allows for a non-traditional massive phase according to recent insights \cite{You:2014vea}.
Lattice symmetries of staggered fermions are well known \cite{Golterman:1984cy}. These include: \\
\noindent (1) {\em Shift Symmetry:}
\begin{equation}
\psi^a_x \rightarrow \xi_{\rho,x} \psi^a_{x+\rho},
\end{equation}
where $\xi_{1,x}=(-1)^{x_2+x_3+x_4}$, $\xi_{2,x}=(-1)^{x_3+x_4}$, $\xi_{3,x}=(-1)^{x_4}$, $\xi_{4,x}=1$. This symmetry is based on the identity $\xi_{\rho,x} \eta_{\alpha,x}\xi_{\rho,x+\hat{\alpha}} = \xi_{\rho,x} \eta_{\alpha,x}\xi_{\rho,x-\hat{\alpha}} = \eta_{\alpha,x+\hat{\rho}}$.\\
\noindent (2) {\em Rotational Symmetry:}
\begin{equation}
\psi^a_x \rightarrow S_R(R^{-1}\ x)\psi^a_{R^{-1}\ x},
\end{equation}
where $R \equiv R^{(\rho\sigma)}$ is the rotation $x_\rho \rightarrow x_\sigma$, $x_\sigma = -x_\rho$, $x_\tau \rightarrow x_\tau$ for $\tau \neq \rho,\sigma$ and
\begin{equation}
S_R(x) = \frac{1}{2}[1\pm \eta_{\rho,x} \eta_{\sigma,x} \mp \xi_{\rho,x}\xi_{\sigma,x} +
\eta_{\rho,x} \eta_{\sigma,x}\xi_{\rho,x}\xi_{\sigma,x}].
\end{equation}
This symmetry follows from the relation
$S_R(R^{-1}\ x) \eta_{\alpha,x} S_R(R^{-1}\ x + R^{-1} \hat{\alpha}) = R_{\mu\nu} \eta_{\nu,R^{-1}\ x}$. \\
\noindent (3) {\em Axis Reversal Symmetry:}
\begin{equation}
\psi^a_x \rightarrow (-1)^{x_\rho}\psi^a_{I\ x},
\end{equation}
where $I \equiv I^{(\rho)}$ is the axis reversal $x_\rho \rightarrow -x_\rho$, $x_\tau = x_\tau$, $\tau \neq \rho$.\\
\noindent (4) {\em Global Chiral Symmetry:}
\begin{equation}
\psi^a_x \rightarrow (V)^{ab}\psi^b_x, \ x\ \in \ \mbox{even},\ \
\psi^a_x \rightarrow (V^*)^{ab}\psi^b_x,\ x\ \in \mbox{odd}.\ \
\end{equation}
where $V$ is an $SU(4)$ matrix in the fundamental representation. Note that the fields at even and odd sites transform differently.
As in the continuum, the above symmetries forbid fermion bilinear mass terms to be generated through radiative corrections. The corresponding mass order parameters were constructed long ago \cite{vandenDoel:1983mf,Golterman:1984cy} and were studied recently in \cite{Catterall:2015zua}. They are given by
\begin{subequations}
\begin{eqnarray}
O^0_{ab}(x) \ &=&\ \psi^a_x\psi^b_x \\
O^1_{\mu,a}(x) \ &=&\ \epsilon_x \xi_{\mu,x}\psi^a_x\ S_\mu\psi^a_{x} \\
O^{2A}_{\mu\nu,a}(x) \ &=&\ \xi_{\mu,x} \xi_{\nu,x+\hat{\mu}} \psi^a_x S_\mu S_\nu\psi^a_{x} \\
O^{2B}_{\mu\nu,a}(x) \ &=&\ \ \epsilon_x\xi_{\mu,x} \xi_{\nu,x+\hat{\mu}} \psi^a_x S_\mu S_\nu\psi^a_{x} \\
O^{3}_{\mu\nu\lambda,a}(x) \ &=&\ \xi_{\mu,x} \xi_{\nu,x+\hat{\mu}} \xi_{\nu,x+\hat{\mu}+\hat{\nu}} \psi^a_x S_\mu S_\nu S_\lambda\psi^a_{x},
\end{eqnarray}
\label{orderparam}
\end{subequations}
where $x$ is a lattice site, $\epsilon_x = (-1)^{x_1+x_2+x_3+x_4}$ and $S_\mu \psi^a_x = \psi^a_{x+\hat{\mu}} + \psi^a_{x-\hat{\mu}}$. Further we assume $\mu\neq\nu\neq \lambda$ in the above expressions. These order parameters naturally vanish in the PMW phase, when fermions are massless. However, we will argue in section \ref{sec4} that even the PMS phase they vanish where fermions become massive.
\section{Fermion Bags, Topology and an Index Theorem}
\label{sec3}
The partition function of our lattice model whose action is given in (\ref{act}), can be written in the fermion bag approach \cite{PhysRevD.82.025007,Chandraepja13}. In addition to providing an alternate Monte Carlo method to solve lattice fermion field theories, this alternative approach also gives new theoretical insight into the fermion mass generation mechanism involving four-fermion condensates \cite{Chandrasekharan:2014fea}. While the details of this approach was already discussed in \cite{Ayyar:2014eua}, here we repeat the steps for reduced staggered fermions instead of regular staggered fermions. Although the final expression is identical, here it is written in terms of Pfaffians instead of determinants. We first write
\begin{equation}
Z \ =\ \sum_{[n]}\ \int \prod_x \ [d\psi_x^1 d\psi_x^2 d\psi_x^3 d\psi_x^4]\
\ \prod_a \exp\Big(-\frac{1}{2}\ \sum_{x,y} \psi_x^a \ M_{x,y}\ \psi_x^a\Big)\
\prod_x\ \Big(U \psi^4_x \psi^3_x \psi^2_x \psi^1_x \Big)^{n_x}.
\label{pf}
\end{equation}
where $[n]$ is a configuration of monomers defined by the binary field $n_x=0,1$ that represents the absence ($n_x=0$) or presence ($n_x=1$) of a monomer (see Fig.~\ref{fig:1} for an illustration). We can perform the Grassmann integral at the sites that contain the monomer first to obtain
\begin{equation}
Z \ =\ \sum_{[n]}\ U^{N_m}\ \prod_a \int [d\psi_{x_1}^a d\psi_{x_2}^a...]
\ \exp\Big(-\frac{1}{2}\ \sum_{x,y} \psi_x^a \ W_{x,y}\ \psi_y^a\Big),
\end{equation}
where $N_m$ is the total number of monomers in the configuration $[n]$, and sum in the exponent is only over free sites (i.e., sites without monomers) ordered in a convenient way say $x_1,x_2,...$ and $W_{x,y}$ is the reduced staggered Dirac matrix $M_{x,y}$ connecting only the free sites. Performing the remaining Grassmann integration over the free sites we obtain
\begin{equation}
Z \ =\ \sum_{[n]} U^{N_m} \big(\mathrm{Pf}(W)\big)^4
\label{fbpf}
\end{equation}
where $\mathrm{Pf}(W)$ refers to the Pfaffian of the matrix $W$. Note that the sign of $\mathrm{Pf}(W)$ is ambiguous and depends on the order in which the free sites are chosen in the definition of the matrix $W$. However, this ambiguity cancels in the full partition function and all physical correlation functions. Since $W$ is an anti-symmetric matrix connecting only even and odd sites, the matrix $W$ can be expressed as a block matrix by separating even and odd sites into separate blocks with non-zero entries only in the off-diagonal block. Hence $\mathrm{Pf}(W)$ is the determinant of this off-diagonal block. When the number of odd and even sites are different then $\mathrm{Pf}(W) = 0$.
Free fermion bags refer to the connected set of free sites which do not belong to a monomer. A monomer configuration can in principle have different disconnected fermion bags, which implies that
\begin{equation}
\mathrm{Pf}(W)\ =\ \prod_{\cal B}\ \mathrm{Pf}(W_{\cal B})
\end{equation}
where the matrix $W_{\cal B}$ refers to the reduced staggered Dirac matrix $W$ connecting only the sites within the bag ${\cal B}$ and the product is over all bags. We will refer to $W_{\cal B}$ as the {\em fermion bag matrix}. If $S_{\cal B}$ is the total number of sites of the fermion bag, then $W_{\cal B}$ is also an $S_{\cal B} \times S_{\cal B}$ anti-symmetric matrix with non-zero entries only between even and odd sites. It is easy to see that $\mathrm{Pf}(W_{\cal B})=0$ for a bag with an unequal number of even and odd sites.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1.pdf}
\end{center}
\caption{\label{fig:1} An illustration of a fermion bag configuration. The sites with monomers are marked with filled circles. Connected sites without monomers form fermion bags.}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.5\textwidth]{fig2.pdf}
\end{center}
\caption{\label{fig:2} An illustration of a $\nu=1$ topological fermion bag configuration. The Dirac matrix $W_{\cal B}$ associated with this bag will contain at least one zero mode. This connection between topology and zero modes is analogous to the index theorem of the massless Dirac operator in non-Abelian gauge theories.}
\end{figure}
Let us now discuss a curious connection between the zero modes of the fermion bag matrix $W_{\cal B}$ and the topology of the fermion bag ${\cal B}$. This connection is analogous to the well known index theorem of the massless Dirac operator in non-Abelian gauge theories \cite{Atiyah:1971rm,Smit:1986fn,Adams:2009eb}. Note that when a bag does not contain an equal number of even and odd sites $W_{\cal B}$ is a matrix with zero modes. If we introduce the concept of a topological charge for the fermion bag through the integer $\nu = n_e - n_o$ (where $n_e$ ($n_o)$ refer the number of even (odd) sites of the bag), then it is easy to argue that the fermion bag matrix $W_{\cal B}$, will have at least $|\nu|$ zero modes, similar to the index of the massless Dirac operator in non-Abelian gauge theories. An example of a $\nu = 1$ topological fermion bag is shown in Fig.~\ref{fig:2}. The analogy with non-Abelian gauge theories extends even further. For example, in certain massless four-fermion models where a chiral symmetry forbids the presence of the chiral condensate, they can still acquire non-zero expectation values due to the presence of fermion bags with topological charge $\nu = \pm 1$ \cite{Chandrasekharan:2014fea}. This is similar to the fact that chiral condensates obtain a non-zero contribution in QCD with a single massless quark flavor due to the presence of gauge field configurations with topological charge $\nu = \pm 1$ \cite{Leutwyler:1992yt}.
\section{Absence of SSB at Strong Couplings}
\label{sec4}
The conventional wisdom is that when fermions become massive, one or more of the fermion bilinear mass order parameters given in (\ref{orderparam}) acquire a non-zero expectation value due to spontaneous breaking of some of the lattice symmetries. However, it was discovered long ago that the usual single site order parameter vanishes at sufficiently strong couplings even though fermions are massive \cite{Hasenfratz:1988vc,Hasenfratz:1989jr,Lee:1989mi,Bock:1990tv,Bock:1990cx}. More recently the vanishing of all bilinear mass order parameters at strong couplings was studied in \cite{Catterall:2015zua}. In this section we give analytic arguments for this result within the fermion bag approach. Our aim is to illustrate the importance of topology and zero modes of the fermion bag matrix in some of these arguments. A simple extension of these arguments allow us to also conclude the absence of any spontaneous symmetry breaking.
All bilinear mass order parameters given in (\ref{orderparam}) can be written compactly in the form
\begin{equation}
O^\alpha(x) = \sum_y f^\alpha_{a,b}(x,y)\psi^a_x\psi^b_y,
\end{equation}
where $\alpha = 0,1,2A,2B,3$ and $f^\alpha_{a,b}(x,y)$ is appropriately defined with non-zero values only when $x$ and $y$ lie within a hypercube. On a finite lattice we expect $\langle O^\alpha(x)\rangle = 0$ purely from symmetry transformations on Grassmann fields, assuming boundary conditions do not break the symmetries \footnote{our choice of anti-periodic boundary conditions fall in this class}. In the fermion bag approach this vanishing of the symmetry order parameter can be understood through the following three facts:
\begin{enumerate}
\item $\langle O^\alpha(x)\rangle$ can get non-zero contributions from a fermion bag configuration only when both the Grassmann fields in $O^\alpha(x)$ are present within the same fermion bag. To show this let us prove that the weight of a fermion bag vanishes due to the insertion of a single $\psi^a_x$. First note that inserting $\psi^a_x$ in the path integral means that $x$ must be a free site within a fermion bag which we refer to as ${\cal B}_x$. Inserting $\psi^a_x$ and performing the Grassmann integration within the bag gives
\begin{equation}
\mathrm{Pf}(W_{{\cal B}_x}([x]))\ =\ \int\ \prod_{x'\in {\cal B}_x}\ [d\psi^a_{x'}]\ \psi^a_x\ \exp\Big(-\frac{1}{2}\sum_{x,y\in {\cal B'}_x} \psi^a_x W_{x,y} \psi^a_y\Big).
\end{equation}
Note that this is equivalent to removing the site $x$ from the bag and the matrix $W_{{\cal B}_x}([x])$ refers to the fermion bag matrix without the site $x$. Note this matrix has one row and one column less than $W_{{\cal B}_x}$ which contains the site $x$. Without $\psi^a_x$, the above Grassmann integral would give $\mathrm{Pf}(W_{{\cal B}_x})$. Since $x$ will either be an even or an odd site, removing it changes the topology of the fermion bag as defined in the previous section. Thus, if $\mathrm{Pf}(W_{{\cal B}_x}) \neq 0$, then $\mathrm{Pf}(W_{{\cal B}_x}([x])) = 0$ and vice versa. Since the weight of the fermion bag involves a product of four Pfaffians, one for each flavor, the fermion bag weight always vanishes in the presence of a single $\psi^a_x$ source term inside it.
\item When $\alpha=0,2A,2B$, contribution to $\langle O^\alpha(x)\rangle$ from every single fermion bag configuration vanishes because the fermion bag weight that contains the fermion source terms vanishes. In the fermion bag approach these expectation values are given by
\begin{eqnarray}
\langle O^0(x)\rangle &=& \frac{1}{Z}
\Big\{\sum_{[n]} U^{N_m} \Big(\mathrm{Pf}(W_{{\cal B}_{x}}[x])\Big)^2\Big(\mathrm{Pf}(W_{{\cal B}_{x}})\Big)^2
\prod_{{\cal B} \neq {\cal B}_{x}} \Big(\mathrm{Pf}(W_{\cal B})\Big)^4\Big\},
\\
\langle O^{2A,2B}(x)\rangle &=& \frac{1}{Z}
\Big\{\sum_{y} f^{2A,2B}_{a,a}\sum_{[n]} U^{N_m}
\Big(\mathrm{Pf}(W_{{\cal B}_{x,y}}[x,y])\Big)\Big(\mathrm{Pf}(W_{{\cal B}_{x,y}})\Big)^3
\prod_{{\cal B} \neq {\cal B}_{x}} \Big(\mathrm{Pf}(W_{\cal B})\Big)^4\Big\},
\nonumber \\
\end{eqnarray}
where ${\cal B}_{x}$, $\mathrm{Pf}(W_{{\cal B}_x})$ and $\mathrm{Pf}(W_{{\cal B}_x}([x]))$ were already defined above. We now define ${\cal B}_{x,y}$ as the free fermion bag containing both the sites $x,y$, $\mathrm{Pf}(W_{{\cal B}_{x,y}})$ is the Pfaffian of that fermion bag matrix, and $\mathrm{Pf}(W_{{\cal B}_{x,y}}([x,y]))$ is the Pfaffian of the fermion bag matrix where $x$ and $y$ are also dropped from the bag ${\cal B}_{x,y}$.
Mathematically,
\begin{equation}
\mathrm{Pf}(W_{{\cal B}_{x,y}}([x,y]))\ =\ \int\ \prod_{x'\in {\cal B}_x}\ [d\psi^a_{x'}]\ \psi^a_x \ \psi^a_y\ \exp\Big(-\frac{1}{2}\sum_{x,y \in {\cal B}_{x,y}} \psi^a_x W_{x,y} \psi^a_y\Big),
\end{equation}
and
\begin{equation}
\mathrm{Pf}(W_{{\cal B}_{x,y}})\ =\ \int\ \prod_{x'\in {\cal B}_x}\ [d\psi^a_{x'}]\ \exp\Big(-\frac{1}{2}\sum_{x,y \in {\cal B}_{x,y}} \psi^a_x W_{x,y} \psi^a_y\Big).
\end{equation}
Since $O^0(x)$ contains $\psi^a_x\psi^b_x$ where $a\neq b$, out of the four Pfaffians contributing to the weight of the fermion bag ${\cal B}_x$, two involve matrices that contain the site $x$ and two involve matrices that do not contain it. Their product vanishes for topological reasons like before, since one of the two matrices will have an extra even or odd site and its pfaffian will vanish. In the case of $O^{2A,2B}(x)$ the weight function $f^{2A,2B}_{a,b}(x,y) \neq 0$ when $a=b$, but when both $x,y$ belong to either even sites or odd sites. Thus, either $W_{{\cal B}_{x,y}}[x,y]$ or $W_{{\cal B}_{x,y}}$ has two extra even or odd sites. Again for topological reasons like before, the pfaffian of one of these two matrices will vanish.
\item The contribution to $\langle O^\alpha(x)\rangle$ for $\alpha = 1,3$ from a given fermion bag configuration may be non-zero, since in these two cases the fermion bag weight that contains the fermion source terms can have a non-zero weight. For these bilinears, since $a=b$ the expression for the expectation value is given by
\begin{equation}
\langle O^{1,3}(x)\rangle = \frac{1}{Z} \sum_y \ f^\alpha_{a,a}(x,y)
\Big\{\sum_{[n]} U^{N_m}
\Big(\mathrm{Pf}(W_{{\cal B}_{x,y}}[x,y])\Big)\Big(\mathrm{Pf}(W_{{\cal B}_{x,y}})\Big)^3
\prod_{{\cal B} \neq {\cal B}_{x,y}} \Big(\mathrm{Pf}(W_{\cal B})\Big)^4\Big\}
\end{equation}
where ${\cal B}_{x,y}$, $\mathrm{Pf}(W_{{\cal B}_{x,y}})$ and $\mathrm{Pf}(W_{{\cal B}_{x,y}}([x,y]))$ were defined above. Note that now both $\mathrm{Pf}(W_{{\cal B}_{x,y}})$ and $\mathrm{Pf}(W_{{\cal B}_{x,y}}([x,y]))$ can be non-zero.
\end{enumerate}
Using the above three facts we can understand why $\langle O^\alpha(x)\rangle = 0$ for all values of $\alpha$. When $\alpha = 0,2A,2B$, contribution from each fermion bag configuration vanishes. However, when $\alpha =1,3$ a further sum over contributions from all symmetry transformations of the fermion bag configuration is necessary to show that the expectation value vanishes. Under these transformations fermion bags transform as a classical extended objects in space-time. For example under a shift in some direction, all monomers and fermion bags get shifted by one lattice unit in that direction. Similarly under rotation by $90^o$ about some axis, the full fermion bag configuration rotates by the same amount. All such configurations obtained by symmetry transformations will have the same weight in the absence of source insertions. This means $\mathrm{Pf}(W_{\cal B})$ remains the same for all fermion bags ${\cal B} \neq {\cal B}_{x,y}$. On the other hand $\mathrm{Pf}(W_{{\cal B}_{x,y}}([x,y]))$ transforms like $\psi^a_x\psi^a_y$ because of the source insertions and hence will cancel due to the sum over symmetry transformations.
Interestingly, if fermion bags are sufficiently far apart symmetry operations can be performed on a single fermion bag without affecting other bags. Such symmetry fluctuations of the single fermion bag that contains the fermion source terms then naturally lead to $\langle O^\alpha(x)\rangle = 0$. Let us illustrate this by considering the calculation of $\langle O^1_{\mu,a}(x)\rangle$ which is given by
\begin{equation}
\langle O^1_{\mu,a}(x)\rangle \ =\ \epsilon(x)\xi_\mu(x) \ \Big(
\langle \psi^a_x \psi^a_{x+\hat{\mu}}\rangle - \langle \psi^a_{x-\hat{\mu}} \psi^a_x\rangle \Big).
\end{equation}
Note that under the shift symmetry we expect
\begin{equation}
\langle \psi^a_x \psi^a_{x+\hat{\mu}}\rangle = \langle \psi^a_{x-\hat{\mu}} \psi^a_x\rangle.
\end{equation}
which is the reason for $\langle O^1_{\mu,a}(x) \rangle$ to vanish. In the fermion bag approach we have
\begin{equation}
\langle \psi^a_x\psi^a_{x+\hat{\mu}}\rangle = \frac{1}{Z}
\Big\{\sum_{[n]} U^{N_m} \
\Big[\mathrm{Pf}(W_{{\cal B}_{x,x+\hat{\mu}}}[x,x+\hat{\mu}]) \Big(\mathrm{Pf}(W_{{\cal B}_{x,x+\hat{\mu}}})\Big)^3\Big] \ \prod_{{\cal B} \neq {\cal B}_{x,x+\hat{\mu}}} \Big(\mathrm{Pf}(W_{\cal B})\Big)^4
\Big\}.
\end{equation}
If the fermion bags are sufficiently far apart, then we can map the bag ${\cal B}_{x-\hat{\mu},x}$ that contributes to $\langle\psi^a_{x-\hat{\mu}}\psi^a_x\rangle$ to another unique bag ${\cal B}^\mu_{x,x+\hat{\mu}}$ obtained by translating ${\cal B}_{x-\hat{\mu},x}$ by one lattice spacing in the $\hat{\mu}$ direction without disturbing any of the other bags. An illustration of such a translation is shown in Fig.~\ref{fig:fluct}. Due to translational invariance we must have
\begin{equation}
\mathrm{Pf}(W_{{\cal B}_{x-\hat{\mu},x}}([x-\hat{\mu},x]))\ =\
\mathrm{Pf}(W_{{\cal B}^\mu_{x,x+\hat{\mu}}}([x,x+\hat{\mu}])),
\ \
\mathrm{Pf}(W_{{\cal B}_{x-\hat{\mu},x}})\ =\
\mathrm{Pf}(W_{{\cal B}^\mu_{x,x+\hat{\mu}}}).
\end{equation}
Since none of the other bags are disturbed, we see that $\langle O^1_\mu(x)\rangle = 0$ simply due the sum over all symmetry fluctuations of the single fermion bag containing the fermion source terms.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=.4\textwidth]{fig1.pdf}
\hspace{0.5in}
\includegraphics[width=.4\textwidth]{fig3.pdf}
\end{center}
\caption{\label{fig:fluct} An illustration of a symmetry fluctuation of a fermion bag when other fermion bags are sufficiently far apart. The fermion bag in the center of the figure on the left has been translated by one unit to the right and shown in the figure on the right. Such a change in a fermion bag is referred to as a symmetry fluctuation and the sites affected during the fluctuation are shown with a different color in the right figure.}
\end{figure*}
The above discussion sheds light on why $\langle O^\alpha(x)\rangle=0$ in a finite system due to symmetry transformations. However, the more important question is to address whether some of these symmetries can break spontaneously. For this one must compute the two point correlation function of local order parameters,
\begin{equation}
C^{\alpha}(x,x') \ =\ \langle O^\alpha(x) O^\alpha(x')\rangle.
\end{equation}
If $C^{\alpha}(x,x') \neq 0$ in the limit when $|x-x'| \rightarrow\infty$, then we say the order parameter is non-zero and the corresponding symmetry is spontaneously broken. Let us now argue that $C^{\alpha}(x,x') = 0$ for all fermion bilinear mass order parameters when $|x-x'| \rightarrow\infty$ at sufficiently strong couplings. We will assume that fermion bags of large size are exponentially suppressed and that fermion bags are well separated from each other so that when fermion bags fluctuate due to symmetry transformations they rarely touch each other. Emperical evidence shows that this assumption is quite reasonable. Hence, contribution from configurations where $x$ and $x'$ lie within the same fermion bag should also be exponentially suppressed in the limit where $|x-x'|$ is large. Thus, a non-zero order parameter requires a non-vanishing contribution from configurations where $x$ and $x'$ are in two different fermion bags. However, we have already argued that $\langle O^\alpha(x)\rangle=0$ within each fermion bag once fluctuations of these two bags are taken into account. Thus, all fermion bilinear mass order parameters must vanish at sufficiently strong coupling, (in the PMS phase) even though fermions are massive. The fact that fermion masses and fermion bilinear condensates need not be related to each other was first presented in \cite{Chandrasekharan:2014fea}.
A straightforward generalization of the above arguments show that any symmetry local order parameter that vanishes within a fermion bag (after taking into account symmetry fluctuations of the fermion bag), cannot develop long range order, as long as fermion bags are well separated from each other and large fermion bags are exponentially suppressed. Since distinct fermion bags are always separated by local singlets (monomers), correlations between them are screened. The only way for long range correlations to arise is due to topology that requires the presence of another fermion bag far away, whose weight vanishes due to zero modes that arise through an index theorem. In such cases while the order parameter vanishes on a finite lattice, two point correlations can develop long range correlations. Examples of lattice models that contain such topological correlations are easy to construct \cite{Chandrasekharan:2014fea}. However, as we have discussed above, our model is different and such topological correlations in symmetry order parameters are absent. Hence there can be no spontaneous symmetry breaking of any lattice symmetries at sufficiently large couplings.
\section{Width of the Intermediate Phase}
\label{sec5}
The arguments of the previous section no longer apply when free fermion bags become large and are not well separated from each other. This occurs in the intermediate coupling region where fermion bilinear condensates can in principle form and lattice symmetries can break spontaneously. As explained in the introduction, it would be exciting to find a 4D lattice model without such an intermediate FM phase, but with a direct PMW-PMS second order phase transition. Unfortunately, earlier studies suggest that our lattice model (\ref{act}) has a {\em wide} intermediate FM phase \cite{Lee:1989mi}, although the phase boundaries were not accurately determined. Using the fermion bag approach, in this section we determine them. Since our algorithms scale badly with system size (especially in 4D), we have been able to perform Monte Carlo calculations only up to $L=12$ (we have one result at $L=14$ at $U=1.75$). Assuming the presence of an intermediate phase and using finite size scaling, we are still able to determine the phase boundaries accurately. In contrast to earlier work, our results point to a surprisingly {\em narrow} intermediate FM phase.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{fig4.pdf}
\end{center}
\caption{\label{fig:rmno} The monomer density (left) and the condensate susceptibility $\chi_1$ (right) plotted as a function of $U$ in the intermediate coupling region for various lattice sizes. There is no sign of a first order transition, but the rapid growth of the susceptibility suggests an intermediate phase with spontaneous breaking of the $SU(4)$ symmetry.}
\end{figure}
We first show results for the four-fermion condensate defined through the monomer density $\rho_m$ in the fermion bag approach using the relation
\begin{equation}
\rho_m = \frac{U}{L^4}\ \sum_x \ \langle \ \psi^4_x\psi^3_x\psi^2_x\psi^1_x \ \rangle.
\end{equation}
Note that with our normalization $\rho_m=0$ at $U=0$ and $\rho_m=1$ at $U=\infty$. In Fig.~\ref{fig:rmno} (on the left side) we plot the behavior of $\rho_m$ as a function of $U$ for various lattice sizes. The condensate increases rapidly but smoothly between $U=1.5$ and $1.9$ suggesting the absence of any large first order transitions. However, with this data alone it is unclear if there is a single transition due to the absence of an intermediate phase, or two transitions due its presence. For this purpose we compute the two independent susceptibilities
\begin{equation}
\chi_1 \ =\ \frac{1}{2}\ \sum_{x} \ \langle \psi^1_0\psi^2_0\ \psi^1_x\psi^2_x \rangle,\ \
\chi_2 \ =\ \frac{1}{2}\ \sum_{x} \ \langle \psi^1_0\psi^2_0\ \psi^3_x\psi^4_x \rangle,
\end{equation}
that can help in determining if bilinear condensate $\Phi = \langle O^0_{ab}(x) \rangle \neq 0$. In general, $\chi_1 \neq \chi_2$, as can be easily verified for small values of $U$, but for large values of $U$ they become almost similar. Assuming a fermion bilinear condensate forms, the leading behavior at large volumes is expected to scale as $\chi_1 \sim \chi_2 \sim \Phi^2 L^4/4$, since only half the lattice volume contributes in the sum. In other words, a clear signature for the formation of the condensate is the volume scaling of the susceptibilities and that for large $L$ the two susceptibilities become identical.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{fig5.pdf}
\end{center}
\caption{\label{fig:susvsL} The plots on the left show $2\chi_1/L^4$ and $2\chi_2/L^4$ as a function of $L$ at $U=1.67$ (squares) and $1.75$ (circles). Also $\chi_1$ is higher than $\chi_2$. The plot on the right shows the condensate $\Phi = \langle O^0_{ab}(x) \rangle$ as a function of $U$. We see the intermediate FM phase extends roughly from $1.60 \leq U \leq 1.81$.}
\end{figure}
\begin{figure}[t]
\begin{center}
\vbox{
\includegraphics[width=\textwidth]{fig6.pdf}
\includegraphics[width=\textwidth]{fig7.pdf}
}
\end{center}
\caption{\label{fig:suscrit} Plots of $\chi_1/L^2$ (top row) and $\chi_2/L^2$ (bottom row) as a function of $U$ for various lattice sizes near the two transitions. The value of $U$ where the curves cross is shown as the dotted line and indicates the rough location of the critical point.}
\end{figure}
In Fig. (\ref{fig:rmno}) (on the right side) we show the behavior of $\chi_1$ as a function of $U$ for various lattice sizes. For these couplings we find $\chi_2$ to be qualitatively similar. In Fig.\ref{fig:susvsL} (on the left side) we plot both $2\chi_1/L^4$ and $2\chi_2/L^4$ as a function of $L$ at $U=1.67$ and $1.75$. We take the fact that the data seems to be saturating as a sign that a condensate is forming. Further, we observe that $\chi_1 \sim \chi_2$ for the highest two lattices, which provides further evidence for this view point. In contrast, in three dimensions we never found evidence that $\chi_i/L^3$ saturates \cite{Ayyar:2015lrd}. Assuming that the bilinear condensate does form, we fit our data to the form
\begin{equation}
\chi = \frac{1}{4} \Phi^2 L^4 + b L^2,
\end{equation}
which we found empirically to be a good form for the behavior of the susceptibilities in the intermediate region, to extract the condensate $\Phi = \langle O^0_{ab}(x) \rangle$. This plotted in Fig. \ref{fig:susvsL} (on the right side).
The fact that $\Phi = \langle O^0_{ab}(x) \rangle \neq 0$ implies that the $SU(4)$ symmetry is broken in the range $1.60 \leq U \leq 1.81$. However, note that this region is much narrower than what was computed in the earlier work. It also means we should have two transitions in our model in quick succession (the PMW-FM transition and the FM-PMS transition). Here we assume that both transitions are second order since we have not seen any reason to believe one of them is first order, but with our small lattice results we cannot rule out the possibility of weak first order transitions. This is especially true for the FM-PMS transition, where the condensate seems to rapidly reducing. Assuming they are second order the PMW-FM transition would follow the Gross Neveu universality while the FM-PMS transition could follow the $SU(4) \sim SO(6)$ spin model universality, both of which would show mean field exponents up to logarithmic corrections. This means
\begin{equation}
\chi_i /L^{2-\eta} \sim f_i((U-U_c) L^{1/\nu})
\end{equation}
where $\eta = 0$ and $\nu = 1/2$ (up to log corrections). In Fig. \ref{fig:suscrit} we plot $\chi_i /L^2 $ versus $U$ for different $L$ values. As the figure shows, all these curves (for large $L)$ appear to intersect through $U_c$ as expected. We see that $U_c$ for the PMW-FM transition is at roughly $1.60$, and for FM-PMS phase is at around $1.81$, in agreement with our previous conclusion based on computing $\Phi$.
\section{Conclusions}
\label{sec6}
In this work we have studied a lattice field theory model where fermions are massless at weak couplings, but become massive at sufficiently strong couplings even though all fermion bilinear condensates vanish. Fermions seem to acquire their mass through four-fermion condensates. The presence of an intermediate FM phase does not rule out the possibility that this alternate mechanism of mass generation is only a lattice artifact in 4D. On the other hand since the intermediate phase is quite narrow in bare coupling constant space, extending only from $1.60 \leq U \leq 1.81$, it is likely that an extension of the model may reveal the absence of the intermediate phase and may even show the presence of a direct second order PMW-PMS phase transition like in 3D. Such a transition would make the mass generation mechanism through four-fermion condensates interesting even in the continuum.
We can use the continuum model (\ref{contact}) to understand this alternate mechanism of mass generation better. We view four-fermion condensates as a fermion bilinear condensate between a fundamental fermion field and a composite fermion fermion field. For example if $\psi_a(x), a=1,2,3,4$ represents the four components of a Dirac field in four dimensions, then we can view the composite field
\begin{equation}
\bar{\chi}_a(x) = \varepsilon_{abcd} \psi^b(x) \psi^c(x) \psi^d(x),
\end{equation}
as an independent Dirac field such that $\bar{\chi}\psi$ acts as the chirally invariant mass term for a theory that contains both $\psi(x)$ and $\bar{\chi}(x)$. Note that the $U(1)$ fermion number symmetry (\ref{u1fs}) acts as the chiral symmetry for this fermion mass term, while the $U(1)$ chiral symmetry (\ref{u1cs}) acts as the fermion number symmetry of this mass term. Since the continuum model (\ref{contact}) breaks the $U(1)$ fermion number symmetry explicitly, the new type of mass term is always allowed by interactions. However, at weak couplings composite states do not form and the mass term continues to behave as an irrelevant four-fermion coupling. At sufficiently strong couplings, when the composite states form the four-fermion coupling begins to behave like a mass term and becomes relevant.
This fermion mass generation mechanism where fundamental fermions pair with composite fermions is an old idea \cite{Eichten:1985ft,Golterman:1992yha}. The fact that such a mass generation mechanism can occur without any spontaneous symmetry breaking within a phase (PMS phase) of a regulated microscopic field theory that also contains a phase (PMW phase) with massless fermions was also known before but not emphasized. We find the existence of both these phases within the same regulated microscopic theory exciting, since it means that fermion mass generation can be a dynamical phenomenon purely related to renormalization group arguments rather than symmetry breaking. For all this to be of interest in continuum quantum field theory, there must be a direct second order PMW-PMS transition in the regulated theory. Search for it in 4D would be an interesting research direction for the future.
\acknowledgments
We would like to thank M.~Golterman for pointing us to the lattice literature on the subject. We also thank S.~Catterall, U.-J.~Wiese and C.~Xu for helpful discussions at various stages of this work. SC would like to thank the Center for High Energy Physics at the Indian Institute of Science for hospitality, where part of this work was done. The material presented here is based upon work supported by the U.S. Department of Energy, Office of Science, Nuclear Physics program under Award Number DE-FG02-05ER41368. An important part of the computations performed in this research was done using resources provided by the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy's Office of Science \cite{Pordes2008,Sfiligoi2009}.
|
{
"timestamp": "2016-06-22T02:00:46",
"yymm": "1606",
"arxiv_id": "1606.06312",
"language": "en",
"url": "https://arxiv.org/abs/1606.06312"
}
|
"\\section{Introduction}\n\nBiochemical gradients are ubiquitous in biological systems. \nGradients (...TRUNCATED)
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"\\section{Introduction}\n\\label{sec:intro}\nMalware detection has evolved as one of the challengin(...TRUNCATED)
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"\\section{Introduction}\n\\label{sec:intro}\nThe long-awaited first direct detection of gravitation(...TRUNCATED)
| {"timestamp":"2016-06-22T02:10:57","yymm":"1606","arxiv_id":"1606.06526","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction} \n\nThe recent detection of gravitational waves (GW) \\citep{2016PhRvL.116(...TRUNCATED)
| {"timestamp":"2016-11-29T02:06:31","yymm":"1606","arxiv_id":"1606.06124","language":"en","url":"http(...TRUNCATED)
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"\n\\section{Introduction}\n\\label{secintro}\n\nThe aim of this paper is to draw connections betwee(...TRUNCATED)
| {"timestamp":"2016-06-22T02:04:53","yymm":"1606","arxiv_id":"1606.06382","language":"en","url":"http(...TRUNCATED)
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