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\begin{titlepage}
\hfill Preprint-KUL-TF-94/6 \newline
\vspace{2cm}
\begin{center}
{ \bf {\large ANOMALOUS ANGULAR MOMENTUM FLUCTUATIONS FOR AN IDEAL BOSE GAS IN
A ROTATING BUCKET }}\\[3mm]
P. Tuyls\footnote[1]{I.I.K.W Onderzoeker Belgium}, M. Van
Canneyt\footnotemark[1], A. Verbeure\\[5mm]
K.U. Leuven, Instituut voor Theoretische Fysica, B 3001 Leuven,
Belgium\\[20mm]
{\bf Abstract}
\end{center}
An equilibrium superfluid is usually modelled by a Bose gas below its
transition temperature. Here we compute rigorously the deviation from
normality of the angular momentum fluctuations, and prove that the extremal
phases, or the extremal equilibrium state components, show normal behaviour,
except for special boundary conditions.
\vspace{1 cm}
\underline{ Pacs nrs. :}
05.30, 05.40, 05.70 .
\end{titlepage}
Blatt and Butler [1] showed that an ideal Bose - Einstein gas below its
transition point is an equilibrium superfluid in the sense that it is a
substance whose moment of inertia in thermal equilibrium is less than its
classical value ( see also [2] ).
Blatt, Butler and Schafroth [3] considered the rotating bucket model and
discussed the question whether the effective moment of inertia is the same
for classical as well as for quantum statistical mechanics. By heuristic
computations they showed that this holds above the transition point, but not
below. This result got a rigorous derivation in [4] for quantum statistical
interacting systems with an equilibrium state satisfying suitable
clusterproperties, i.e. at temperatures high enough.
Now the moment of inertia is directly related to the fluctuation of the
angular momentum (see e.g. [3,4]).
In this letter we report on rigorous results about the moment of inertia,
its deviations from its classical behaviour below the transition point in
the model of the rotating bucket. We compute rigorously the angular
momentum fluctuations. We start from the rigorous study of the
average angular momentum in the equilibrium state of this model given by
Lewis and Pul\'e [5]. They study in much detail the thermodynamic limit
of the equilibrium state of the free Boson gas in a rotating bucket given
by the following Hamiltonian. Let \( \Lambda^1\) be the cylindrical region
of unit volume in \( \Rbar ^3\):
\begin{equation}
\Lambda^1 =\{ x\in \Rbar^3 : x_1^2 + x_2^2 < a^2;\mid x_3 \mid < ( 2\pi
a^2)^{-1}\}
\end{equation}
and for each \( L > 0 \), let \(\Lambda^L\) be the region :
\begin{equation}
\Lambda^L = \{x \in \Rbar^3 : L^{-1}x \in \Lambda^1\}
\end{equation}
then :
\begin{equation}
H_L - \Omega_L J_L - \mu_L N_L = \int \limits_{\Lambda^L} dx \frac{1}{2}
\nabla a^+(x)\nabla a(x) - \Omega_L J_L - \mu_L N_L
\end{equation}
where \(J_L\) is the third component of the angular momentum, \( \Omega_L\)
the angular velocity of the system, \( \mu_L \) the chemical potential and
\( N_L \) the number operator, of course \( a^{\pm} (x) \) are boson creation
and annihilation operators.
The main contribution of [5] consists in proving rigorously that the
thermodynamic limit \( L \rightarrow \infty \) makes sense requiring the
constraints of fixed particle density \( \overline{ \rho}\) and average
angular momentum \( \overline{ \lambda}\) such that for all volumes \(
\Lambda^L\) :
\begin{equation}
\langle N_L \rangle_L = \overline{\rho}L^3,\hspace{3mm} \langle J_L \rangle_L =
\overline{\lambda}L^3
\end{equation}
where :
\begin{equation}
\langle A \rangle_L = \frac{Tr\left( e^{-\beta( H_L-\Omega_L J_L-\mu_L
N_L)}A\right) }{Tr \left( e^{-\beta(H_L-\Omega_L J_L- \mu_L N_L)}\right)}
\end{equation}
These constraints determine the L-dependence of the chemical potential
\(\mu_L\) and the angular velocity \( \Omega_L\). The constraints (4) are
written as follows in the limit \( L \rightarrow \infty\) :
\begin{equation}
\overline{\rho} = \lim \limits_{L\rightarrow \infty} \frac{\langle N_L
\rangle_L}{L^3};\hspace{3mm}
\overline{\lambda} = \lim \limits_{L\rightarrow \infty} \frac{\langle J_L
\rangle_L}{L^3}
\end{equation}
In [5], using analogous techniques as in [6], it is proved that these
limits exist and they analyse the condensation problem. Let
\begin{equation}
z \rightarrow g_{\alpha}(z) = \sum \limits_{n=1}^{\infty} n^{-\alpha} z^n;
\hspace{3mm}\alpha > 1, \hspace{3mm}z \in [0,1]
\end{equation}
The function \( z \rightarrow (2 \pi \beta)^{-\frac{3}{2}}g_{\frac{3}{2}}(z)\)
is continuous on \([0,1]\) and increases monotonically to a maximum
\(\rho_c\) at \( z=1\) so that for \( \overline{\rho} \leq \rho_c\) the
equation :
\begin{equation}
\overline{\rho} = (2 \pi \beta)^{-\frac{3}{2}}g_{\frac{3}{2}}(z)
\end{equation}
has a unique root \( z(\overline {\rho})\). If \( \overline {\rho} >
\rho_c\) then :
\begin{equation}
\overline{\rho} = (\overline{\rho} - \rho_c) + \rho_c
\end{equation}
The quantity \( \overline{\rho} - \rho_c \) represents the condensate
density.
It is also proved [5], that for the rotating bucket the condensation
takes place in the two lowest energy levels which we denote simply by
\(\epsilon_1\) and \(\epsilon_2\), such that :
\begin{equation}
\overline{\rho} - \rho_c = \rho_{0,1} + \rho_{0,2}
\end{equation}
where
\begin{equation}
\rho_{0,i}= \lim \limits_{L \rightarrow \infty} \frac{\langle n_i
\rangle}{L^3};\hspace{3mm} i=1,2 \hspace{3mm}
n_i = \mbox{ the number operator for the level}\hspace{1mm} \epsilon_i
\end{equation}
Consequently the formula (6) for the angular momentum can be written as
follows :
\begin{equation}
\overline{\lambda} - \lambda_c = \lambda_{0,1} + \lambda_{0,2}
\end{equation}
where \( \lambda_{0,i}\) represents the contribution to the angular momentum
from the condensate in the level i; depending on the values of \(
\overline{\lambda}\) and \( \overline{\rho} \), one or both of the
\(\lambda_{0,i}\) are different from zero.
In fact \( \overline \lambda\) and \( \overline{\rho} \) are coupled
through the formulae (6).
Now we turn to the fluctuation of the angular momentum \( I_{\delta} \). Define
the
fluctuation by:
\begin{equation}
I_\delta = \lim \limits_{L \rightarrow \infty} \frac{1}{L^{6\delta}}
\langle \frac{ ( J_L - \overline{\lambda}L^3)^2}{L^3} \rangle_L
\end{equation}
Now we solve the following problem :
Find the value of \( \delta \in [0,\frac{1}{2}]\) such that \(
I_{\delta}\) exists and is not trivial, i.e. such that \( 0 < I_{\delta} <
\infty \). The parameter $\delta$ indicates at which level the
fluctuations appear.
If \( \overline{\rho} < \rho_c \), i.e. in the absence of condensate or
under good clusterconditions, or short range correlations it is given in
the thermodynamic limit [3,4] by
\begin{equation}
I_{\frac{1}{3}} = \lim \limits_{L \rightarrow \infty } \frac {1}{L^2} \langle
\frac{(J_L
- \overline{\lambda}L^3)^2}{L^3} \rangle_L
\end{equation}
i.e. \( \delta = \frac{1}{3}\), this value of \( \delta \) refers to the
normal situation coinciding with the classical case. Any other value of \(
\delta \)
, this is \( \delta \neq \frac{1}{3} \), which can be found in particular if
\( \overline{\rho} > \rho_c \), refers to an abnormality of angular
momentum fluctuations. Let us now examine the case \( \overline{\rho} > \rho_c
\),
i.e. in the presence of condensate, in the so-called superfluid
region.
If one computes for \( \overline{\rho} > \rho_c \) straightforwardly the
angular momentum fluctuation in the equilibrium state \( \langle . \rangle_L
\), then one finds that \(
\delta = \frac{1}{2}\) is the solution of the problem, posed in (13). We get
a discontinuous jump in \( \delta \) from \(\delta = \frac{1}{3}\) for \(
\overline{\rho} < \rho_c \)\hspace{1mm} to \(\delta = \frac{1}{2}\) for
\(\overline{\rho} > \rho_c \). This is due to the abrupt appearance of long
range order. Also, \(\delta = \frac{1}{2}\) is the rigorous expression for the
heuristic discussions given in [3] and [4].
On the other hand, the abnormality \(\delta = \frac{1}{2}\), is a typical
value corresponding to the computation of a mean square deviation in a
mixed state i.e. a convex combination of extremal equilibrium states. In
this free Boson gas model, of course, we are in this situation. If there is
condensation it is well known that there is spontaneous breaking of the
gauge symmetry, and the equilibrium state is the integral over the equilibrium
states
with fixed gauge. The intrinsic properties of the equilibrium
states are situated in these gauge breaking extremal states. Therefore also
the study of the deviation from normality of the angular momentum
fluctuation should refer to these gauge breaking states. That is exactly
what we report on in the following.
In particular we use the standard technique to recover the gauge breaking
extremal equilibrium states. One considers the Hamiltonian :
\begin{equation}
H_L - \Omega_L J_L - \mu_L N_L+ \overline{h} \sqrt{L^3} a(f_L) + h \sqrt{L^3}
a^+(f_L)
\end{equation}
where \(H_L\) is given by (3),
\[ a^{\pm} (f_L) = \int \limits_{\Lambda} dx f_L (x) a^{\pm} (x) \]
and \( f_L\) is any element of \(L^2(\Lambda )\) which has nonvanishing
components for the condensate levels (10).
The presence of the factor \( \sqrt{L^3}\) is to make the
perturbation extensive and the complex number \(h \) is an external field
which tends to zero after having taken the thermodynamic limit. Here we
differ from this procedure in the sense that we let \( h\) tend to zero
following a power law in the volume, we take \( h = h_L\)
\begin{equation}
h_L = \frac{h_1}{L^{3\alpha}}
\hspace{3mm}\mbox{with}\hspace{3mm} \alpha > 0,
\hspace{3mm} h_1 \mbox{ fixed.}
\end{equation}
This way of proceeding is treating the perturbation as a boundary condition.
Now we compute
the parameter \(\delta\) using the state :
\begin{equation}
\langle A \rangle_L^{h_L} = \frac{Tr( e^{-\beta(H_L- \Omega_L J_L - \mu_L N_L +
(\overline h
\sqrt{L^3} a(f_L) + h \sqrt{L^3} a^+(f_L))}A)}{Tr \left( e^{-\beta(H_L -
\Omega_L J_L - \mu_L N_L+ \overline h
\sqrt{L^3} a(f_L) + h \sqrt{L^3} a^+(f_L))} \right)}
\end{equation}
instead of (5). One finds the following remarkable
results which hold true in both situations of condensation in one or two
levels :
\[
\left\{ \begin{array}{ll}
\mbox{if}\hspace{3mm} 0 < \alpha < 1 &\mbox{then} \hspace{3mm}\delta =
\max\{\frac{1}{3},\frac{\alpha}{2}\}\\
\mbox{if} \hspace{3mm}\alpha > 1 &\mbox{then}\hspace{3mm} \delta = \frac{1}{2}
\end{array}
\right.
\].
There are a number of remarks to be made, explaining this result.
In general the parameter \( \delta \) depends on the boundary conditions
i.e. on the parameter \( \alpha \); this phenomenon is found also in other
models [7,8,9], as well in quantal as classical ones; this effect is seen
only in the case when the external field drops off quickly enough i.e. if \( 1 >
\alpha \geq \frac{1}{3}\). However if the field tends to zero too fast with
the volume i.e. if \( \alpha \geq 1\), the effect of the boundary condition
is nonexistent and the limit state
\begin{equation}
\lim \limits_{L \rightarrow \infty} \langle . \rangle_L^{h_L}
\end{equation}
behaves as the mixed, nonextremal, equilibrium state, yielding the
parameter value \( \delta = \frac{1}{2}\).
On the other hand, if \( \alpha \) is small, i.e. \( \alpha <
\frac{2}{3}\), or the external field vanishes very slowly, then the field forces
the system into an extremal phase; from the thermodynamic point of view the
state of the system is in a minimum of an harmonic potential, the
distribution of the angular momentum fluctuation is Gaussian and normal (
for details about the distribution see [10]), yielding \( \delta =
\frac{1}{3} \), the same value as in the case of low density \(
\overline{\rho} < \rho_c\), or as in the classical systems.
So far the discussions of the results. It is clear that {\sl non -
classical }
behaviour appears only in the case, \( \frac{2}{3} < \alpha < 1 \), and that
it is solely due to the boundary conditions. Disregarding the effect of
this external field, the angular momentum fluctuations do not show any
abnormal behaviour passing the critical line \( \overline{\rho}=\rho_c\).
These rigorous results cast serious doubts on the ideal Bose - Einstein gas
being a good model for superfluidity under {\sl all circumstances}. This
confirms
other reservations in this direction which were made in the literature (
e.g. already in [1]). The detailed proofs of the above results will appear
in a more elaborate paper [10].
\newpage
{\large References }\\
\noindent
[1] J.M. Blatt, S.T. Butler: Phys. Rev., {\bf 100}, 476 (1955)
\noindent
[2] S.J. Puttermann, M. Kac, G.E. Uhlenbeck: Phys. Rev. Lett.,{\bf 29},546
(1972)
\noindent
[3] J.M. Blatt, S.T. Butler, M.R. Schafroth: Phys. Rev.,{\bf 100}, 481
(1955)
\noindent
[4] B. Nachtergaele: J. Math. Phys., {\bf 26}, 2317 (1985)
\noindent
[5] J.T. Lewis, J.V. Pul\'e: Commun. Math. Phys., {\bf 45}, 115 (1975)
\noindent
[6] J.T. Lewis, J.V. Pul\'e: Commun. Math. Phys., {\bf 36}, 1 (1974)
\noindent
[7] A. Verbeure, V.A. Zagrebnov: J. Stat. Phys.,{\bf 69}, 329 (1992)
\noindent
[8] A. Verbeure, V.A. Zagrebnov: Gaussian, Non-Gaussian Critical
Fluctuations in the Curie-Weiss model; K.U.L.-T.F.-93/14, to appear in
J. Stat. Phys.
\noindent
[9] M. Broidioi: Critical exponents in the Imperfect and the Free Bose Gas;
K.U.L.-T.F.-93/7, to appear in J.M.P.
\noindent
[10] P. Tuyls, M. Van Canneyt, A. Verbeure: Angular Momentum Fluctuations
of the Bose Gas in a Rotating Bucket; in preparation.
\end{document}
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